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arXiv:1411.5327v3 [] 11 Jan 2017 Almost algebraic actions of algebraic groups and applications to algebraic representations Uri Bader∗1 , Bruno Duchesne†2 , and Jean Lécureux‡3 1 Uri Bader, Weizmann Insitute of Science, Rehovot, Israel. Université de Lorraine, Institut Élie Cartan, B.P. 70239, 54506 Vandoeuvre-lès-Nancy Cedex, France. 3 Département de Mathématiques, Bâtiment 425, Faculté des Sciences d’Orsay, Université Paris-Sud 11, 91405 Orsay, France. 2 April 4, 2018 Abstract Let G be an algebraic group over a complete separable valued field k. We discuss the dynamics of the G-action on spaces of probability measures on algebraic G-varieties. We show that the stabilizers of measures are almost algebraic and the orbits are separated by open invariant sets. We discuss various applications, including existence results for algebraic representations of amenable ergodic actions. The latter provides an essential technical step in the recent generalization of Margulis-Zimmer super-rigidity phenomenon [BF13]. 1 Introduction This work concerns mainly the dynamics of an algebraic group acting on the space of probability measures on an algebraic variety. Most (but not all) of our results are known for local fields (most times, under a characteristic zero assumption). Our main contribution is giving an approach which is applicable also to a more general class of fields: complete valued fields. On our source of motivation, which stems from ergodic theory, we will elaborate in §1.2, and in particular Theorem 1.16. First we describe our objects of consideration and our main results, put in some historical context. Setup 1.1. For the entire paper (k, | · |) will be a valued field, which is assumed to be complete and separable as a metric space, and b k will be the completion of its algebraic closure, endowed with the extended absolute value. ∗ [email protected] † [email protected] ‡ [email protected] 1 Note that b k is separable and complete as well (see the proof of Proposition 2.2). The most familiar examples of separable complete valued fields are of course R and C, but one may also consider the p-adic fields Qp , as well as their finite extensions. Considering k = Cp = “ Qp one may work over a field which is simultaneously complete, separable and algebraically closed. Another example of a complete valued field is given by fields of Laurent series K((t)), where K is any field (this field is local if and only if K is finite, and separable if and only if K is countable), or more generally the field of Hahn series K((tΓ )), where Γ is a subgroup of R (see for example [Poo93]). This field is separable if and only if K is countable and Γ is discrete (see [hmb]). Convention 1.2. Algebraic varieties over k will be identified with their b k-points and will be denoted by boldface letters. Their k-points will be denoted by corresponding Roman letters. In particular we use the following. Setup 1.3. We fix a k-algebraic group G and we denote G = G(k). We are interested in algebraic dynamical systems, which we now briefly describe. For a formal, pedantic description see §2.1 and in particular Proposition 2.2. By an algebraic dynamical system we mean the action of G on V , where V is the space of k-points of a k-algebraic variety V on which G acts k-morphically. Such a dynamical system is Polish: G is a Polish group, V a Polish space and the action map G × V → V is continuous (see §2.1 for proper definitions). The point stabilizers of such an action are algebraic subgroups, and by a result of BernsteinZelevinski [BZ76], the orbits of such an action are locally closed (see Proposition 2.2). Following previous works of Furstenberg and Moore, Zimmer found a surprising result: for the action of an algebraic group G on an algebraic variety V , all defined over R, consider now the action of G on the space Prob(V ) of probability measures on V . Then the point stabilizers are again algebraic subgroups and the orbits are locally closed. However, this result does not extend trivially to other fields. For example, with k = C, consider the Haar measure on the circle S 1 < C∗ . For the action of C∗ on itself, the stabilizer of that measure is S 1 , which is not a C-algebraic subgroup. Similarly, for k = Qp , consider the Haar measure on the p-adic integers Zp < Qp . For the action of Qp on itself, the stabilizer of that measure is Zp , which is not a Qp -algebraic subgroup. Definition 1.4. A closed subgroup L < G is called almost algebraic if there exists a k-algebraic subgroup H < G such that L contains H = H(k) as a normal cocompact subgroup. A continuous action of G on a Polish space V is called almost algebraic if the point stabilizers are almost algebraic subgroups of G and the collection of G-invariant open sets separates the G-orbits, i.e the quotient topology on G\V is T0 . Remark 1.5. If k is a local field then G is locally compact and by [Eff65, Theorem 2.6] the condition G\V is T0 is equivalent to the (a priori stronger) condition that every G-orbit is locally closed in V . 2 Remark 1.6. If k = R then every compact subgroup of G is the real points of a real algebraic subgroup of G (see e.g. [VGO90, Chapter 4, Theorem 2.1]). It follows that every almost algebraic subgroup is the real points of a real algebraic subgroup of G. We get that a continuous action of G on a Polish space V is almost algebraic if and only if the stabilizers are real algebraic and the orbits are locally closed. Two obvious classes of examples of almost algebraic actions are algebraic actions (by the previously mentioned result of Bernstein-Zelevinski) and proper actions (as the stabilizers are compact and the space of orbits is T2 , that is, Hausdorff). The notion of almost algebraic action is a natural common generalization. It is an easy corollary of Prokhorov’s theorem (see Theorem 2.3 below) that if the action of G on V is proper then so is its action on Prob(V ), see Lemma 2.7. The main theorem of this paper is the following analogue. Theorem 1.7. If the action of G on a Polish space V is almost algebraic then the action of G on Prob(V ) is almost algebraic as well. The following corollary was obtained by Zimmer, under the assumptions that k is a local field of characteristic 0 and V is homogeneous, see [Zim84, Chapter 3]. Corollary 1.8. Assume G has a k-action on a k-variety V. Then the induced action of G = G(k) on Prob(V(k)) is almost algebraic. In the course of the proof of Theorem 1.7 we obtain in fact a more precise information. A k-G-variety is a k-variety with a k-action of G. Proposition 1.9. Fix a closed subgroup L < G. Then there exists a k-subgroup H0 < G which is normalized by L such that L has a precompact image in the Polish group (NG (H0 )/H0 )(k) and such that for every k-G-variety V, any Linvariant finite measure on V(k) is supported on the subvariety of H0 -fixed points, VH0 ∩ V(k). This proposition is a generalization of one of the main results of Shalom [Sha99], who proves it under the assumptions that k is local and L = G. For the case L = G the following striking corollary is obtained. Corollary 1.10. If for every strict k-algebraic normal subgroup H/G, G(k)/H(k) is non-compact, then every G-invariant measure on any k-G-algebraic variety V(k) is supported on the G-fixed points. In particular we can deduce easily the Borel density theorem. Corollary 1.11. Let G be a k-algebraic group and Γ < G = G(k) be a closed subgroup such that G/Γ has a G-invariant probability measure. If for every proper k-algebraic normal subgroup H / G, G(k)/H(k) is non-compact, then Γ is Zariski dense in G. To deduce the last corollary from the previous one, consider the map Z G/Γ → (G/Γ )(k), 3 Z where Γ denotes the Zariski closure of Γ, and push forward the invariant measure Z from G/Γ to obtain a G-invariant measure on (G/Γ )(k). The homogeneous space Z Z G/Γ must contain a G-fixed point, hence must be trivial. That is Γ = G. 1.1 Applications: ergodic measures on algebraic varieties A classical theme in ergodic theory is the attempt to classify all ergodic measures classes, given a continuous action of a topological group on a Polish space. In this regard, the axiom that the space of orbits is T0 has strong applications. Recall that, given a group L acting by homeomorphisms on a Polish space V , a measure on V is L-quasi-invariant if its class is L-invariant. The following proposition is well known. Proposition 1.12. Let V be a Polish G-space and assume that the quotient topology on G\V is T0 . Let L < G be a subgroup and µ an L-quasi-invariant ergodic probability (or σ-finite) measure. Then there exists v ∈ V such that µ(V −Gv) = 0. Indeed, G\V is second countable, as V is, and for a countable basis Bi , denoting the push forward of µ to G\V by µ̄, the set \ \ {Bi | µ̄(Bi ) = 1} ∩ {Bic | µ̄(Bi ) = 0} is clearly a singleton, whose preimage in V is an orbit of full measure. In particular, we get that for a subgroup L < G and an algebraic dynamical system of G, every L-invariant measure is supported on a single G-orbit. Another striking result is that an algebraic variety cannot support a weakly mixing probability measure. Recall that an L-invariant probability measure µ is weakly mixing if and only if µ × µ is L-ergodic. Corollary 1.13. Assume G has a k-action on the k-variety V. Fix a closed subgroup L < G and let µ be an L-invariant weakly mixing probability measure on V = V(k). Then there exists a point x ∈ V L such that µ = δx . This corollary follows at once from Proposition 1.9, as the action of L on VH0 ∩ V(k) is via a compact group. We end this subsection with the following useful application, obtained by composing Proposition 1.12 with Theorem 1.7. This corollary is in fact our main motivation for developing the material in this paper. It deals with measure on spaces of measures, and is the main tool in deriving Theorem 1.16 below. Corollary 1.14. Assume G has a k-action on the k-variety V. Denote V = V(k). Let L < G be a subgroup and ν be an L-ergodic quasi-invariant measure on Prob(V ). Then there exists µ ∈ Prob(V ) such that ν(Prob(V ) − Gµ) = 0. 1.2 Applications to algebraic representations of ergodic actions. A main motivation for us to extend the foundation outside the traditional local field zone is the recent developments in the theory of algebraic representations of 4 ergodic actions, and in particular its applications to rigidity theory. In [BF13] the following theorem, as well as various generalizations, are proven. Theorem 1.15 ([BF13, Theorem 1.1], Margulis super-rigidity for arbitrary fields). Let l be a local field. Let T to be the l-points of a connected almost-simple algebraic group defined over l. Assume that the l-rank of T is at least two. Let Γ < T be a lattice. Let k be a valued field. Assume that as a metric space k is complete. Let G be the k-points of an adjoint simple algebraic group defined over k. Let δ : Γ → G be a homomorphism. Assume δ(Γ) is Zariski dense in G and unbounded. Then there exists a continuous homomorphism d : T → G such that δ = d|Γ . The proofs in [BF13] are based on the following, slightly technical, theorem which will be proven here. Theorem 1.16. Let R be a locally compact group and Y be an ergodic, amenable Lebesgue R-space. Let (k, | · |) be a valued field. Assume that as a metric space k is complete and separable. Let G be a simple k-algebraic group. Let f : R×Y → G(k) be a measurable cocycle. Then either there exists a k-algebraic subgroup H G and an f -equivariant measurable map φ : Y → G/H(k), or there exists a complete and separable metric space V on which G acts by isometries with bounded stabilizers and an f equivariant measurable map φ0 : Y → V . A more friendly, cocycle free, version is the following. Corollary 1.17. Let R be a locally compact, second countable group. Let Y be an ergodic, amenable R-space. Suppose that G is an adjoint simple k-algebraic group, and there is a morphism R → G = G(k). Then : • Either there exists a complete and separable metric space V , on which G acts by isometries with bounded stabilizers, and an R-equivariant measurable map Y →V; • or there exists a strict k-algebraic subgroup H and an R-equivariant measurable map Y → G/H(k). Taking Y to be a point in the above corollary, we obtain the following. Corollary 1.18. Suppose R < GLn (k) is a closed amenable subgroup. Then the Z image of R in R modulo its solvable radical is bounded. Z Indeed, upon moding out the solvable radical of R , the latter is a product of simple adjoint factors, and by the previous corollary the image of R in each factor is bounded. Note that over various fields, such as Cp and F̄p ((t)), every bounded group is amenable, being the closure of an ascending union of compact groups, while for other fields there exist bounded groups which are not amenable. For example SL2 (Q[[t]]), which is bounded in SL2 (Q((t))), factors over the discrete group SL2 (Q) which contains a free group. 5 1.3 The structure of the paper. The paper has two halves: the first half consisting of §2,§3 is devoted to the proof of Theorem 1.7 and the second half is devoted to the proof of Theorem 1.16. In §2 we collect various needed preliminaries, in particular we discuss the Polish structure on algebraic varieties, and on spaces of measures. The most important results in this section are Proposition 2.2 that discusses algebraic varieties and and Corollary 2.14 that uses disintegration as a replacement for a classical ergodic decomposition argument (which is not applicable in our context, due to the lack of compactness). The heart of the paper is §3, where the concept of almost algebraic action is discussed. Theorem 1.7 is proven at §3.4. In §4, we give a thorough discussion of bounded subgroups of algebraic groups, and in §5, we discuss a suitable replacement of a compactification of coset spaces. In §6, we prove Theorem 1.16. Acknowledgements U.B was supported in part by the ERC grant 306706. B.D. was supported in part by Lorraine Region and Lorraine University. B.D & J.L. were supported in part by ANR grant ANR-14-CE25-0004 GAMME. 2 Preliminaries 2.1 Algebraic varieties as Polish spaces Recall that a topological space is called Polish if it is separable and completely metrizable. For a good survey on the subject we recommend [Kec95]. We mention that the class of Polish spaces is closed under countable disjoint unions and countable products. A Gδ -subset of a Polish space is Polish so, in particular, a locally closed subset of a Polish space is Polish. A Hausdorff space which admits a finite open covering by Polish open sets is itself Polish. Indeed, such a space is clearly metrizable (e.g. by Urysohn metrization theorem [Kec95, Theorem 1.1]) so it is Polish by Sierpinski theorem [Kec95, Theorem 8.19] which states that the image of a continuous open map from a Polish space to a separable metrizable space is Polish. A topological group which underlying topological space is Polish is called a Polish group. Sierpinski theorem also implies that for a Polish group K and a closed subgroup L, the quotient topology on K/L is Polish. Effros Lemma [Eff65, Lemma 2.5] says that the quotient topology on K/L is the unique K-invariant Polish topology on this space. Another important result of Effros concerning Polish actions (that are continuous actions of Polish groups on Polish spaces) is the following. Theorem 2.1 (Effros theorem [Eff65, Theorem 2.1]). For a continuous action of a Polish group G on a Polish space V the following are equivalent. 1. The quotient topology on G\V is T0 . 6 2. For every v ∈ V , the orbit map G/ StabG (v) → Gv is a homeomorphism. Our basic class of Polish actions will be given by actions of algebraic groups on algebraic varieties. As mentioned in Setups 1.1 & 1.3, we fixed a complete and separable valued field (k, | · |), that is a field k with an absolute value | · | which is complete and separable (in the sense of having a countable dense subset). See [EP05, BGR84]1 for a general discussion on these fields. It is a standard fact that a complete absolute value on a field F has a unique extension to its algebraic closure “ F [BGR84, §3.2.4, Theorem 2] and Hensel lemma implies that the completion F of this algebraic closure is still algebraically closed [BGR84, §3.4.1, Proposition 3]. Recall that we identify each k-variety V with its set of b k-points. In particular, this identification yields a topology on V. Identifying the affine space An (b k) with b k n , any affine k-variety can be seen as a closed subset of An (b k). More generally, a k-variety has a unique topology making its affine charts homeomorphisms. Observe that with this topology, the set of k-points V of V is closed. Topological notions, unless otherwise said, will always refer to this topology. In particular, for the k-algebraic group G we fixed, G and G = G(k) are topological groups. We note that V actually carries a structure of a k-analytic manifold, G is a k-analytic group and the action of G on V is k-analytic. We will not make an explicit use of the analytic structure here. The interested reader is referred to the excellent text [Ser06], in which the theory of analytic manifolds and Lie groups over complete valued fields is developed (see in particular [Ser06, Part II, Chapter I]). We will discuss the category of k-G-varieties. A k-G-variety is a k-variety endowed with an algebraic action of G which is defined over k. A morphism of such varieties is a k-morphism which commutes with the G-action. Proposition 2.2. A k-variety V and its set of k-points V are Polish spaces. In particular, G and G are Polish groups. If V is a k-G-variety then the G-orbits in V are locally closed and the quotient topology on G\V is T0 . For v ∈ V , the orbit Gv is a k-subvariety of V. There exists a k-subgroup H < G contained in the stabilizer of v such that the orbit map G/H → Gv is defined over k and the induced map G/H → Gv is a homeomorphism, where H = H(k), G/H is endowed with the quotient space topology and Gv is endowed with the subspace topology. Proof. Let us first explain how the extended absolute value makes b k Polish. In our situation k has a countable dense subfield k0 . The algebraic closure k0 of k0 is still countable and thus its completion k“0 is separable and algebraically closed. By the universal property of the algebraic closure, k embeds in k“0 and by uniqueness of the extension of the absolute value, this embedding is an isometry. Thus b k is algebraically closed, complete and separable. Since b k is Polish, so is the affine space An (b k) ' b k n . It follows that V (respectively V ) is a Polish space, as this space is a Hausdorff space which admits a finite open covering by Polish open sets — the domains of its k-affine charts (respectively their k-points). 1 In the second reference, the word valuation is used for what we call an absolute value. 7 The fact that the G-orbits in V are locally closed is proven in the appendix of [BZ76]. Note that in [BZ76] the statement is claimed only for non-Archimedean local fields, but the proof is actually correct for any field with complete non-trivial absolute value, which is the setting of [Ser06, Part II, Chapter III] on which [BZ76] relies. Another proof can be found in [GGMB13, §0.5]. It is then immediate that the quotient topology on G\V is T0 . For v ∈ V the orbit Gv is a k-subvariety of V by [Bor91, Proposition 6.7]. Z We set H = StabG (v) (note that if char(k) = 0 then H = StabG (v)). By [Bor91, AG, Theorem 14.4], H is defined over k, and it is straightforward that H = H(k) = StabG (v). By [Bor91, Theorem 6.8] the orbit map G/H → Gv is defined over k, thus it restricts to a continuous map from G/H onto Gv. The fact that the latter map is a homeomorphism follows from Effros theorem (Theorem 2.1) since G\V is T0 . We emphasize that, as a special case of Proposition 2.2, we get that for every kalgebraic subgroup H of G, the embedding G/H → G/H(k) is a homeomorphism on its image. We will use this fact freely in the sequel. 2.2 Spaces of measures as Polish spaces In this subsection V denotes a Polish space. We let Prob(V ) be the set of Borel probability measures on V , endowed with the weak*-topology (also called the topology of weak convergence). This topology comes from the embedding of Prob(V ) in the dual of the Banach space of bounded continuous functions on V . If d is a complete metric on V which is compatible with the topology (the metric topology coincides with the original topology on V ), the corresponding Prokhorov metric d on Prob(V ) is defined as follows: for µ, ν ∈ Prob(V ), d(µ, ν) is the infimum of ε > 0 such that for all Borel subset A ⊆ V , µ(A) ≤ ν(Aε ) + ε and symmetrically ν(A) ≤ µ(Aε ) + ε, where Aε is the ε-neighborhood (for d) around A. The following theorem summarizes some standard results, see Chapter 6 and Appendix III of [Bil99]. Theorem 2.3 (Prokhorov). The metric space (Prob(V ), d) is complete and separable and the topology induced by d on Prob(V ) is the weak*-topology. In particular the space Prob(V ) endowed with the weak*-topology is Polish. A subset C in Prob(V ) is precompact if and only if it is tight: for every  > 0 there exists compact K ⊂ V such that for every µ ∈ C, µ(K) > 1−. In particular Prob(V ) is compact if V is. Remark 2.4. Replacing if necessary d by a bounded metric, we note that there is another metric on Prob(V ) with the same properties (metrizing the weak*topology and being invariant under isometries): the Wasserstein metric [Vil09, Corollary 6.13]. We endow Homeo(V ) with the pointwise convergence topology. The following is a standard application of the Baire category theorem, see [Kec95, Theorem 9.14]. 8 Theorem 2.5. Assume G is acting by homeomorphisms on V . Then the action map G × V → V is continuous if and only if the homomorphism G → Homeo(V ) is continuous. Lemma 2.6. If G acts continuously on V then it also acts continuously on Prob(V ) and if the action G y (V, d) is by isometries, the action G y (Prob(V ), d) is also by isometries. Proof. The fact that G acts by isometries on Prob(V ) when G acts by isometries on V is straightforward from the definition of the Prokhorov metric. In order to prove that G acts continuously on Prob(V ) when it acts continuously on V it is enough, by Theorem 2.5, to show that for every µ ∈ Prob(V ) and every sequence gn in G, gn → e in G implies gn µ → µ in Prob(V ). Fix µ ∈ Prob(V ) and assume gn → e in G. For every bounded continuous function f on V , we have by Lebesgue bounded convergence theorem Z Z Z f (x) d(gn µ)(x) = f (gn x) dµ(x) → f (x) dµ(x) as for every x ∈ V , gn x → x. Thus, by the definition of the weak*-topology gn µ → µ. We observe that Lemma 2.6 and Proposition 2.2 show that if V is a k-G-variety then G acts continuously on V = V(k) and on Prob(V ). The following is a nice application of Prokhorov theorem (Theorem 2.3). Lemma 2.7. If the action of G on V is proper then the action of G on Prob(V ) is proper as well. Proof. For a compact C ⊂ Prob(V ) we can find a compact K ⊂ V with µ(K) > 1/2 for every µ ∈ C by Theorem 2.3. Then for g ∈ G and µ ∈ C such that gµ ∈ C we get that both µ(K) > 1/2 and µ(gK) = gµ(K) > 1/2, thus gK ∩ K 6= ∅. We conclude that {g ∈ G | gC ∩ C 6= ∅} is precompact, as it is a subset of the precompact set {g ∈ G | gK ∩ K 6= ∅}. 2.3 Polish extensions and disintegration Definition 2.8. A Polish fibration is a continuous map p : V → U where U is a T0 -space and V a Polish space. An action of G on such a Polish fibration is a pair of continuous actions on V and U such that p is equivariant. Let p : V → U be a Polish fibration. Let ProbU (V ) be the set of probability measures on V which are supported on one fiber. We denote p• : ProbU (V ) → U the natural map. Lemma 2.9. The map p• is a Polish fibration. If the group G acts on the Polish fibration V → U , then it also acts on p• . 9 Proof. Since U is T0 , fibers of p are separated by a countable family (Cn ) of closed saturated subsets of V . A probability measure µ is supported on one fiber if and only if for all n, µ(Cn )µ(V \ Cn ) = 0. The set {µ ∈ Prob(V ), µ(Cn ) = 1} is closed and {µ ∈ Prob(V ), µ(V \ Cn ) = 1} is Gδ since for all 0 < r < 1, {µ ∈ Prob(V ), µ(V \ Cn ) > r} is open. So ProbU (V ) is a Gδ -subset of Prob(V ) and thus Polish. Let us show that p• is continuous. Assume µn → µ in ProbU (V ). Let u = p• (µ) and un = p• (µn ). Let O ⊆ U be an open set containing u. For n large enough, µn (p−1 (O)) > 1/2 and thus un ∈ O. If G acts on V → U , it is clear that G acts on ProbU (V ). The continuity of the action on ProbU (V ) follows from Lemma 2.6. Let (U, ν) be a probability space and X be a Polish space, we denote by L0 (U, X) the space of classes of measurable maps from U to X, under the equivalence relation of equality ν-almost everywhere. Note that the dependence on ν is implicit in our notation. We endow that space with the topology of convergence in probability. Fixing a compatible metric d on X, this topology is metrized as follows: for φ, φ0 ∈ L0 (U, X), the distance between φ and φ0 is Z 0 δ(φ, φ ) = min(d(φ(v), φ0 (v)), 1) dν(v). X This topology can be also defined using sequences: φn → φ if for any ε > 0, there is A ⊆ U such that ν(A) > 1 − ε and for all n sufficiently large and all v ∈ A, d(φ(v), φn (v)) < ε. We note that this topology on L0 (U, X) does not depend on the choice of an equivalent metric on V . This turns L0 (U, X) into a Polish space. Lemma 2.10. Assume (αn ) is a sequence converging to α in probability in L0 (U, X). Then there exists a subsequence αnk which convergence ν-a.e. to α, that is for νalmost every u ∈ U , αnk (u) converges to α(u) in X. The proof of the lemma is standard, but in most textbooks it appears only for the cases X = R or X = C, see for example [Fol99, Theorem 2.30]. Even though the standard proof works mutatis-mutandis, we give below a short argument, reducing the general case to the case X = R. Proof. Observe that the sequence d(αn , α) (which denotes the map u 7→ d(αn (u), α(u))) converges in probability to 0 in L0 (U, R). Thus there exists a subsequence d(αnk , α) converging to 0 a.e, and we get that αnk converges to α a.e. If p : V → U is a Polish fibration, and ν is a measure on U , we denote L0p (U, V ) the space of measurable (identified if agree almost everywhere) sections of p, i.e. maps which associates to u ∈ U a point in p−1 (U ), endowed with the induced topology from L0 (U, V ). If G acts on the Polish fibration p, it also acts on L0p (U, V ) via the formula (gf )(u) = gf (g −1 u) where u ∈ U and f ∈ L0p (U, V ). The following theorem is a variation of the classical theorem of disintegration of measures. It is essentially proven in [Sim12]. 10 Theorem 2.11. Let p : V → U be a Polish fibration and ν be a probability measure on U . Let P = {µ ∈ Prob(V ) | p∗ µ = ν}. For every α ∈ L0p• (U, ProbU (V )) the R formula U α(u) dν defines an element of P . The map thus obtained L0p• (U, ProbU (V )) → P is a homeomorphism onto. Definition 2.12. For µ ∈ P , the element of L0p• (U, ProbU (V )) obtained by apR plying to µ the inverse map of α 7→ U α(u) dν is denoted u 7→ µu . It is called the disintegration of µ with respect to p : V → U . R Proof. We first claim that the map α 7→ U α(u) dν is continuous, and then we argue to show that it is invertible, and its inverse is continuous as well. For the continuity, given a converging sequence αn → α in L0p• (U, ProbU (V )) R R with µn = U αn (u) dν, µ = U α(u) dν, it is enough to show that every subsequence of µn has a subsequence that converges to µ. Since every sequence that converges in measure has a subsequence that converges almost everywhere, abusing our notation and denoting again αn and µn for the resulting sub-subsequences, we may assume that αn converges to α ν-almost everywhere. Picking bounded function f on V , we obtain that for ν-a.e u ∈ U , Ran arbitrary continuous R dα (u)f → dα(u)f . Thus by Lebesgue bounded convergence theorem we n V V get Z Z Z Z Z Z dµn f = dν dαn (u)f → dν dα(u)f = dµf. V U V U V V This shows that indeed µn → µ. R We now argue that the map α 7→ U α(u) dν is invertible and its inverse is continuous. Without loss of generality, we can assume that p is onto. Hence U is second countable. Since it is also T0 , it follows that U is countably separated. By [Zim84, Proposition A.1], there exists a Borel embedding φ : U → [0, 1]. We consider [0, 1] with the measure φ∗ ν. Precomposition by φ gives a homeomorphism
L0(φ◦p)∗ [0, 1], Prob[0,1] (V ) → L0p• (U, ProbU (V )). Thus, in what follows we may and do assume that U ⊂ [0, 1].2 Under this assumption [Sim12, Theorem 2.1] guarantees that the map L0p• (U, ProbU (V )) → P is invertible. We denote the preimage of µ ∈ P by u 7→ µu . We are left to show that this association is continuous. To this end we embed V in a compact metric space V 0 and extend p by setting p(v 0 ) = 1 for v 0 ∈ V 0 − V . Then [Sim12, Theorem 2.2] proves that for almost every u ∈ U , µu is obtained as the weak*-limit of the normalized restrictions, denoted by µu,η , of µ on p−1 (u − η, u + η) as η → 0. Assume that µn → µ is a converging sequence in P . We know that for ν-a.e. u, d(µu,η , µu ) → 0 when η → 0 and similarly for all n ∈ N, d(µnu,η , µnu ) → 0 when η → 0. Fix ε > 0. For n ∈ N, we set An = {u ∈ U ∃η0 > 0 ∀k ≥ n ∀η ∈ (0, η0 ); d(µku,η , µu ) ≤ ε }. Then ν(∪An ) = 1 and An ⊆ An+1 . Thus there is n such that ν(An ) ≥ 1 − ε and for u ∈ An , d(µu , µku ) ≤ ε for all k ≥ n. This shows that the image sequence of (µn ) in L0p• (U, ProbU (V )) indeed converges to the image of µ. 2 Since the embedding U → [0, 1] is only Borel, when we assume U ⊂ [0, 1], the fibration V → U cannot be assumed to be Polish anymore. Since our argument does not depend on the topology of U , this does not matter here. 11 We note that if G acts on the fibration V → U (that is, G acts on U and V and p is equivariant) then the disintegration homeomorphism is also equivariant with respect to the natural action of G on L0p (U, V ) given by (gf )(u) = g(f (g −1 u)). Lemma 2.13. Let p : V → U be a Polish fibration with an action of G such that the G-action on U is trivial. Let ν be a probability measure on U , and let f ∈ L0p (U, V ). Then there exists U1 ⊂ U of full measure such that \ Stab(f ) = Stab(f (u)). u∈U1 Proof. Let L be the stabilizer of f in G. If L0 is a countable dense subgroup of L, T then there is a full measure subset U1 ⊂ U such that L0 ⊂ u∈U1 Stab(f (u)) (for any g ∈ L0 , there is such a subspace Ug . Choose U1 to be the intersection over L0 ). T Since all these stabilizers are closed, and L0 is dense in L, we actually have L ⊂ u∈U1 Stab(f (u)). Since the reverse inclusion is clear, we conclude that L= \ Stab(f (u)). u∈U1 Corollary 2.14. Assume G acts continuously on the Polish space V and the quotient topology on G\V is T0 . Let L < G be a closed subgroup and µ be an L-invariant probability measure on V . Then there exist a point v ∈ V and an L-invariant probability measure on G · v ' G/ Stab(v). Proof. Let ν be the pushforward measure of µ on U . By Theorem 2.11, we may consider the disintegration of µ as an element (µu ) ∈ L0p• (U, ProbU (V )) and this element is clearly L-invariant. By Lemma 2.13, the stabilizer of (µu ) is an intersection of stabilizers of the measures µu , for u in a subset of U . In particular L stabilizes some µu , which is a measure supported on an orbit G · v. The latter is equivariantly homeomorphic to G/ Stab(v) thanks to Effros theorem (Theorem 2.1). 3 Almost algebraic groups and actions The goal of this section is the proof of Theorem 1.7. Starting with an almost algebraic action of G on a Polish V , we aim to prove that the action G y Prob(V ) is algebraic as well. So we have to prove that stabilizers of probability measures on V are almost algebraic and the quotient G\ Prob(V ) is T0 . Going toward wider and wider generality, we prove the first point in §3.2 and the second one in §3.3 3.1 Almost algebraic groups Recall that by our setup 1.1, (k, | · |) is a fixed complete and separable valued field and G is a fixed k-algebraic group. By Proposition 2.2, G = G(k) has the structure 12 of a Polish group. Recall that a closed subgroup L < G is called almost algebraic if there exists a k-algebraic subgroup H < G such that L contains H = H(k) as a normal cocompact subgroup (Definition 1.4). Lemma 3.1. An arbitrary intersection of almost algebraic subgroups is again almost algebraic. More precisely, let (Li )i∈I be a collection of almost algebraic subgroups and Hi algebraic subgroups such that Hi = Hi (k) is normal and cocompact in Li . Then one can find a finite subset I0 such that, defining H = ∩i∈I0 Hi , we have that H = ∩i∈I Hi and H(k) is normal and cocompact in ∩i∈I Li . Proof. Let L = ∩Li and H = ∩Hi which coincides with (∩i∈I Hi )(k). Then it is straighforward to check that H C L. Thanks to the Noetherian property of G, there exists a finite subset I0 ⊂ I such that ∩i Hi coincides with ∩i∈I0 Hi . Let L be theQZariski closure of L and Li the one of Li . The diagonal image of L(k) in i∈I0 Li (k)/Hi is locally closed by Proposition 2.2 and it is a group. Thus it is actually closed. Moreover it is homeomorphic to L(k)/H. To conclude, it suffices to
observe that L/H is closed in L(k)/H and lies in T Q (L(k)/H) i∈I0 Li /Hi which is compact. Remark 3.2. Actually the proof of this lemma shows that any almost algebraic subgroup L has a minimal subgroup among all cocompact normal subgroups N which can be written N = N(k) for some algebraic subgroup N ≤ G. This group is actually the intersection of all such subgroups and it is invariant under the normalizer NG (L) of L in G. Lemma 3.3. Let H, L be closed subgroups of G such that H is almost algebraic, H C L and L/H is compact. Then L is almost algebraic. Proof. There is a algebraic subgroup N of G such that N = N(k) is normal and cocompact in H. Moreover thanks to Remark 3.2, N may be choosen to be invariant under NG (H) and thus N is cocompact and normal in L. 3.2 Almost algebraicity of stabilizers of probability measures Let V be a Polish space endowed with a continuous G-action. Recall that the action G y V is called almost algebraic if the stabilizers are almost algebraic subgroups of G and the quotient topology on G\V is T0 (Definition 1.4). Remark 3.4. For a continuous action of G on a Polish space V , the action is almost algebraic if and only if the stabilizers are almost algebraic and for every v ∈ V and any sequence gn ∈ G, gn v → v implies gn → e in G/ StabG (v). This equivalent definition is much easier to check, and we will allow ourselves to use it freely in the sequel. The two definitions are indeed equivalent by Effros’ Theorem 2.1. Example 3.5. Let I be a k-algebraic group and φ : G → I a k-morphism. Let L be an almost algebraic group in I = I(k). Then the action of G on I/L is almost algebraic. This fact is proved after Lemma 3.7. 13 Lemma 3.6. Let K be a compact group acting continuously on a T0 -space X. Then the orbit space K\X is T0 as well. Proof. Continuity of the action means that the action map K ×X → K ×X which associates (k, kx) to (k, x) is a homeomorphism. Compactness of K implies that the projection (k, x) 7→ x from K × X to X is closed. Composing the two yields closedness of the map (k, x) 7→ kx. This implies that if F ⊂ X is closed, then KF is again closed. Let x, y ∈ X in different K-orbits. Let us consider Y = Kx ∪ Ky with the induced topology. This is a compact T0 -space. Now, consider the set of closed non-empty subspaces of Y with the order given by inclusion. By compactness any decreasing chain has a non-empty intersection and thus Zorn’s Lemma implies there are minimal elements, that are points since Y is T0 . Thus Y has at least a closed point. Without loss of generality we may and shall assume that {x} is closed in Y . This means that there exists a closed subset F of X such that F ∩ Y = {x}. In particular F ∩ Ky = ∅, and therefore Ky ∩ KF = ∅. Finally, KF is a closed K-invariant set separating Kx from Ky. Lemma 3.7. Let J be a topological group acting continuously on a topological space X. If N is a closed normal subgroup of J, the induced action of J/N on N \X is continuous and the orbits spaces J\X and (J/N )\(N \X) are homeomorphic. Proof. The map (g, x) 7→ N gx from J ×X to N \X is continuous and goes through the quotient space J/N ×N \X which is the orbit space of N ×N acting diagonally on J × X. Thus, (gN, N x) 7→ N gx is continuous, that is the action of J/N on N \X is continuous. By the universal property of the topological quotient, the continuous map x 7→ (J/N )N x from X to (J/N )\(N \X) induces a continuous map J\X → (J/N )\(N \X). Conversely, the continuous map N \X → J\X induces also a continuous map (J/N )\(N \X) → J\X which is the inverse of the previous one. Proof of Example 3.5. Since φ−1 (L) and its conjugates are almost algebraic in G, it is clear that the stabilizers are almost algebraic. So we are left to prove that the topology on G\I/L is T0 . Let H be a cocompact normal subgroup in L with H = H(k) for some k-algebraic subgroup H of I. By Lemma 3.7 the orbit space G\I/L is homeomorphic to the space of orbits of the action of G × (L/H) on I/H. Note that the action of G on I/H ⊂ I/H(k) has locally closed orbits (and therefore G\I/H is T0 ) by Proposition 2.2, as the action of G on I/H is k-algebraic. Now the T0 property of G\I/L follows from Lemma 3.6 for the compact group L/H acting continuously on the T0 -space G\I/H. Lemma 3.8. Let J be a countable set, (Li )i∈J a family of almost algebraic subQ groups of G. Then the diagonal action of G on i∈J G/Li is almost algebraic. Q Proof. Stabilizers of points in i∈J G/Li are intersections of almost algebraic subgroups of G. Hence by Lemma 3.1 they are almost algebraic. So we just have Q to prove that G\ ( i∈J G/Li ) is T0 . 14 For i ∈ J, let Hi be an algebraic subgroup of GQsuch that Hi = Hi (k) is a cocompact normal subgroup of Li . Consider V = i∈J G/Hi . We first prove that the topology on G\V is T0 , by proving that orbit maps are homeomorphisms (Theorem 2.1). Let (hi Hi )i∈J be an element of V and (gn ) be a sequence of elements of G such that gn · (hi Hi ) converges to (hi Hi ) in V . T Let H = i∈J hi Hi h−1 = Stab((hi Hi )i∈J ). We have to prove that gn converges i to e in G/HT(see Remark 3.4). By Noetherianity, there exists a finite J0 ⊂ J such Q that H = i∈J0 hi Hi h−1 . Set V = G/H . We see that, in V0 , we have 0 i i∈J0 i that gn .(hi Hi )i∈J0 converges to (hi Hi )i∈J0 . By Proposition 2.2, it follows that gn converges to the identity in G/H. Q Now let K be the compact group i∈J Li /Hi . The group K acts also continuously on V via the formula (li Hi ) · (gi Hi ) = (gi li−1 Hi ) and this action commutes with the action of G. Thus we can apply Lemma 3.6 to KQacting on G\V and get that the space of orbits for the G-action on V /K ' i∈J Gi /Li is T0 , as desired. Our main goal in this subsection is proving the following theorem, which is an essential part of our main theorem, Theorem 1.7. Theorem 3.9. Let V be a Polish space with an almost algebraic action of G. Then stabilizers of probability measures on V are almost algebraic subgroups of G. We first restate and prove Proposition 1.9, discussed in the introduction. Proposition 3.10. Fix a closed subgroup L < G. Then there exists a k-subgroup H0 < G which is normalized by L such that L has a precompact image in the Polish group (NG (H0 )/H0 )(k) and such that for every k-G-variety V, any L-invariant finite measure on V(k) is supported on the subvariety of H0 -fixed points. Proof. Replacing G by the Zariski closure of L, we assume that L is Zariski-dense in G and consider the collection {H < G | H is a k-algebraic subgroup, Prob(G/H(k))L 6= ∅}. By the Noetherian property of G there exists a minimal element H0 in this collection. We let µ0 be a corresponding L-invariant measure on G/H0 (k). We first claim that H0 is normal in G. Assuming not, we let N G be the normalizer of H0 and consider the set U = {(xH0 , yH0 ) | y −1 x ∈ / N} ⊂ G/H0 × G/H0 . This set is a non-empty Zariski-open set which is invariant under the diagonal Gaction, as its complement is the preimage of the diagonal under the natural map G/H0 × G/H0 → G/N × G/N. Since the support of µ0 × µ0 in G/H0 × G/H0 is invariant under L×L which is Zariski-dense in G×G we get that (µ0 ×µ0 )(U(k)) 6= 0. It follows from Corollary 2.14 that there exist u ∈ U(k) and an L-invariant finite measure on G/ StabG (u) ⊂ (G/ StabG (u))(k). By the definition of U we get a contradiction to the minimality of H0 , as point stabilizers in U are properly contained in conjugates of H0 . This proves that H0 is normal in G. 15 Next we let V be a k-G-variety and µ be an L-invariant measure on V(k). We argue to show that µ is supported on VH0 ∩ V(k). Indeed, assume not. Let V0 be the Zariski-closure of V(k) ∩ VH0 , and V00 = V − V0 . Then we see that V0 is defined over k [Bor91, AG, 14.4]. Furthermore, H0 acts on V0 trivially, so that we have V0 (k) = V(k) ∩ VH0 . Hence by assumption we get that µ(V00 (k)) > 0. Replacing V by V00 and restricting and normalizing the measure, we may and shall assume that VH0 ∩ V(k) = ∅. We consider the variety G/H0 × V as a k-G-variety. The measure µ0 × µ is an L-invariant measure on (G/H0 × V)(k). It follows from Corollary 2.14 that there exists u ∈ (G/H0 × V)(k) and an L-invariant measure on G/ StabG (u). By Proposition 2.2 there exist a k-algebraic subgroup H < G with H = H(k) = StabG (u) and an orbit map G/H → Gu inducing a homeormorphism G/H → G/ StabG (u). Thus we obtain an L-invariant probability measure on G/H(k). Now, H is contained in some conjugate gH0 g −1 , for some g ∈ G. Hence we get that g −1 Hg < H0 is such that G/g −1 Hg has an L-invariant probability measure. By minimality, this implies that g −1 Hg = H0 , hence by normality of H0 , H = H0 . Therefore u belongs to V(k) ∩ VH0 , which was assumed to be empty. Hence we get a contradiction. This proves that µ is supported on VH0 . We set S = (G/H0 )(k) and let T be the closure of the image of L in S. We are left to show that T is compact. S is a Polish group and T is a closed subgroup. The quotient topology on T \S is Hausdorff, and in particular T0 . The measure µ0 is an L-invariant finite measure on S, hence it is also T -invariant. Substituting S = V and T = G = L in Corollary 2.14 we find a finite measure µ1 on S which is supported on a unique T -coset, T s. The measure (Rs )∗ µ1 , given by pushing µ1 by the right translation by s−1 is then a T -invariant probability measure on T . It is well-known result due to A. Weil (see [Oxt46] where the result is attributed to Ulam) that a Polish group that admits an invariant measure class is locally compact, and a locally compact group that admits an invariant probability measure is compact. Thus T is indeed compact. Corollary 3.11. Fix a k-G-algebraic variety V, and set V = V(k). Let µ ∈ Prob(V ). Then Stab(µ) is almost algebraic. Proof. Let L = Stab(µ). We may and shall assume L to be Zariski-dense in G, and we can find H0 as in Proposition 1.9. We know that µ is supported on the set of VH0 thus H0 = H0 (k) < L. Since G/H0 is acting on VH0 ∩ V(k) and the stabilizer of µ is closed in G/H0 , we conclude that L has a closed image. We know that the image of L is precompact, thus it is actually compact, and we conclude that L is almost algebraic. Lemma 3.12. Let L < G be an almost algebraic group, with H = H(k) a normal cocompact algebraic subgroup of L. Then there is a G-equivariant continuous map φ : Prob(G/L) → Prob(G/H). Furthermore, we have, for every µ ∈ Prob(G/L), Stab(µ) = Stab(φ(µ)). Proof. Let λ be a Haar probability measure on L/H. For a continuous bounded function Rf on G/H let f be the continuous bounded function on G/L defined by f (gL) = L/H f (gh) dλ(h) and finally φ(µ)(f ) = µ(f ). 16 Then it is clear that φ is equivariant, and we deduce that Stab(µ) ⊂ Stab(φ(µ)). In the other direction, we note that if π : G/H → G/L is the projection, we have π∗ (φ(µ)) = µ. Hence the other inclusion is also clear. To check the continuity, let µn → µ ∈ Prob(G/L), and take f a continuous bounded function on G/H. Then φ(µn )(f ) = µn (f ) → µ(f ) = φ(µ)(f ). Hence φ(µn ) converges to φ(µ). Proof of Theorem 3.9. Choose µ ∈ Prob(V ) and denote L = StabG (µ), H = FixG (supp(µ)). Set U = G\V , and let ν = p• µ, where p : V → U is the projection. Note that p is a Polish fibration. By Theorem 2.11, L is equal to the stabilizer of an element f ∈ L0p• (U, ProbU (V )). By Lemma 2.13 there exists a ν-full measure T set U1 ⊂ U such that L = u∈U1 Stab(f (u)). For a fixed u ∈ U1 , f (u) is a measure on a G-orbit in V which we identify with G/L0 for some almost algebraic subgroup L0 < G. Let H0 < G be a k-algebraic subgroup such that H 0 = H0 (k) is a cocompact normal subgroup of L0 . By Lemma 3.12, Stab(f (u)) is also the stabilizer of a probability measure on G/H 0 ⊂ G/H0 (k). By Corollary 3.11, it follows that Stab(f (u)) is almost algebraic. We conclude that L is almost algebraic by Lemma 3.1. 3.3 Separating orbits in the space of probability measures In this subsection, we prove the following theorem. Theorem 3.13. Let L < G be an almost algebraic subgroup. Then the action of G on Prob(G/L) is almost algebraic. The proof of Theorem 3.13 consists in several steps, proving particular cases of the theorem, each of them using the previous one. First we start with the case when L = G (Lemma 3.14). Then we treat the case when L is a normal algebraic subgroup of G (Lemma 3.15). The main step is then to deduce the theorem when L is any algebraic subgroup of G (Proposition 3.20), before concluding with the general case. Lemma 3.14. The G-action on Prob(G) is almost algebraic. Proof. The regular action of G on itself is proper, so by Lemma 2.7 it follows that the action of G on Prob(G) is proper. Any proper action is almost algebraic. Lemma 3.15. Let H < G be a normal k-algebraic subgroup. Then the G-action on Prob((G/H)(k)) is almost algebraic. Proof. Denoting I = G/H and I = I(k), we know that the I-action on Prob(I) is almost algebraic (Lemma 3.14). Since G/H is a subgroup of I, G stabilizes each I-orbit. It is thus enough to show that G acts almost algebraically on each I-orbit. We know that such an orbit is of the form I/L where L is almost algebraic (Theorem 3.9), so this follows from Example 3.5. An essential technical tool for proving Theorem 3.13 and Theorem 1.7 is given by the following proposition. 17 Proposition 3.16. Let V be a Polish space, with a continuous action of G. Assume that • The quotient topology on G\V is T0 , and • For any v ∈ V , the action of G on Prob(G.v) is almost algebraic. Then the quotient topology on G\ Prob(V ) is T0 . The proposition will directly follow from the following lemma. Lemma 3.17. Let p : V → U be a Polish fibration with an action of G, and let ν be a probability measure on U . Assume that the action of G on U is trivial and that the action of G on Prob(p−1 (u)) is almost algebraic for almost every u ∈ U . Let P = {µ ∈ Prob(V ) | p∗ µ = ν}. Then the topology on G\P is T0 . This proof is similar to the proof presented in [Zim84, Proof of Proposition 3.3.1]; see also [AB94, Lemma 6.7]. Proof. The set P is Polish, as a closed subset of Prob(V ). By Theorem 2.1 we need to show that the orbit maps are homeomorphisms. By Theorem 2.11, P is equivariantly homeomorphic to L0p• (U, ProbU (V )). Fixing f ∈ L0p• (U, ProbU (V )) and letting gn ∈ G be such that gn f → f , we will show that gn converges to the identity in G/ Stab(f ) by proving that every subsequence of (gn ) has a sub-subsequence which converges to the identity in G/ Stab(f ). Doing so, we are free to replace (gn ) by any subsequence. Relying on Lemma 2.10, we replace (gn ) by a subsequence such that gn f (u) → f (u) for every u in some ν-full subset U0 ⊂ U . Let U1 ⊂ U0 be a full measure subset such that the action of G on p−1 (u) is almost algebraic for every u ∈ U1 . Let u ∈ U1 . By definition, we know that f (u) ∈ Prob(p−1 (u)) and that the action of G on Prob(p−1 (u)) is almost algebraic. By Proposition 2.2, the orbit map G/ Stab(f (u)) → Gf (u) is a homeomorphism thus gn f (u) → f (u) implies that gn converges to the identity in G/ Stab(f (u)). By Lemma 2.13, there is also a full measure subset U2 , that we may and do assume to be contained in U1 , such that \ Stab(f ) = Stab(f (u)) u∈U2 and since G is second countable, one can find U3 countable in U2 such that \ Stab(f ) = Stab(f (u)). u∈U3 By assumption, for every u ∈ U3 , theQgroup Stab(f (u)) is almost algebraic. Hence by Lemma 3.8, the action of G on u∈U3 G/ Stab(f (u)) is almost algebraic. In particular, we see that gn converges to e in G/ Stab(f ). Proof of Proposition 3.16. Let U = G\V and p : V → U be the projection. Consider the G-invariant continuous map p∗ : Prob(V ) → Prob(U ). Clearly the fibers of p∗ are closed and G-invariant, so it is enough to prove that for a given ν ∈ Prob(U ), the quotient space G\p−1 ∗ ({ν}) has a T0 -topology. This is precisely Lemma 3.17. 18 Let π : V → V 0 be a continuous G-map between Polish spaces, µ ∈ Prob(V ) and ν = π∗ µ. Then ν has a unique decomposition ν = νc + νd where P νc and Pνd are the continuous and discrete parts of ν. Moreover νd can be written λ∈Λ λ f ∈Fλ δf , where Λ = {λ ∈ R+ | ∃u ∈ V 0 , π∗ µ({u}) = λ} and Fλ = {u ∈ P V 0 | ν({u}) = λ}. −1 Defining µλ to be the restriction of µ to π (Fλ ) and µc = µ − λ∈Λ µλ , we have P a unique decomposition µ = µc + λ∈Λ µλ , where π∗ (µc ) is non-atomic and each P π∗ (µλ ) is a finitely supported, uniform measure of the form λ f ∈Fλ δf . Lemma 3.18. Let π : V → V 0 be a continuous G-map between Polish spaces and T µ ∈ Prob(V ). Using the above decomposition, we have Stab(µ) = Stab(µc ) ∩ ( λ Stab(µλ )). If gn µ → µ then gn µc → µc and for each λ ∈ Λ, gn µλ → µλ . Proof. The statement about Stab(µ) is straightforward from the uniqueness of the decomposition of µ. Let (gn ) be a sequence such that gn µ → µ. Once again, we use a sub-subsequence argument: we prove that any subsequence of (gn ) contains a sub-subsequence such that gn µλ → µλ for every λ. Hence we start by replacing (gn ) by an arbitrary subsequence. Observe that gn µ → µ implies gn ν → ν because π∗ : Prob(V ) → Prob(V 0 ) is continuous. Let K 0 be a compact metrizable space in which V 0 is continuously embedded as a Gδ -subset (see [Kec95, Theorem 4.14]). Then Prob(V 0 ) embeds as a Gδ -subset in Prob(K 0 ) as well [Kec95, Proof of Theorem 17.23]. We begin with the following observation. Assume νn is a sequence of probability measures converging to ν ∈ Prob(V 0 ) and νn decomposes as δun + νn0 with un ∈ V 0 and νn0 ∈ Prob(V 0 ). Up to extraction un converges to some k ∈ K 0 and thus ν({k}) > 0 which implies that k ∈ V 0 . Let λ1 be the maximum of Λ. The above observation implies that up to extraction we may assume that for any f ∈ Fλ1 , gn f converges to some l(f ) ∈ V 0 . Since gn ν → ν, we have that l(f ) ∈ Fλ1 thus gn νλ1 converges to νλ1 , where νλ1 = π∗ (µλ1 ). An induction on Λ (countable and well ordered with the reverse order of R) shows that (after extraction) gn νλ → νλ for any λ ∈ Λ. Once again, we embed V in some compact metrizable space K. Fix λ ∈ Λ and let µ0 be an adherent point of (gn µλ ) in Prob(K). As π is G-equivariant, we have that π∗ gn µλ = gn π∗ µλ = gn νλ which converges to νλ . Hence π∗ µ0 = νλ . Furthermore, we also see that µ0 is supported on π −1 (Fλ ), hence µ0 ∈ Prob(V ). The same argument proves that µ−µ0 , which is an adherent point of gn (µ−µλ ), is supported on V \ π −1 (Fλ ). As µ can be written uniquely as a sum of a measure supported on π −1 (Fλ ) and a measure supported on V \π −1 (Fλ ), we see, writing µ = (µ−µ0 )+µ0 = P(µ−µλ )+µλ , that necessarily µ0 = µλ . This concludes the proof since µc = µ − λ∈Λ µλ . Lemma 3.19. Let H < G be a k-algebraic subgroup. Set N = NG (H), H = H(k) and N = N(k). Let V = G/H, V0 = G/N, V = V(k) and V 0 = V0 (k). P Consider the map π : V → V 0 . Let F ⊂ V 0 be a finite set, ν = 1/|F | f ∈F δf and µ ∈ Prob(V ) be a measure with π∗ µ = ν. Let (gi ) be a sequence with gi µ → µ. Then gi → e ∈ G/ Stab(µ). Proof. Denote m = |F |. We know that (V0 )m / Sym(m) is an algebraic variety, hence by Proposition 2.2, every G-orbit in (V 0 )m / Sym(m) is locally closed. It 19 follows in particular that gi → e in G/ Stab(F ). Again, it is enough to show that every subsequence of (gi ) contains a subsequence which tends to e modulo Stab(µ). We start by extracting an arbitrary subsequence of (gi ). Let us number f1 , f2 , . . . , fm the elements of F and denote F 0 = (f1 , . . . , fm ) ∈ 0 m (V ) . Since gi converges to e in G/ Stab(F ), it follows that, passing to a subsequence, there exists σ ∈ Sym(m) such that gi F 0 tends to σ(F 0 ) = (fσ(1) , . . . , fσ(m) ). This means GF 0 ⊃ Gσ(F 0 ) and thus GF 0 ⊃ Gσ(F 0 ) ⊃ · · · ⊃ Gσ n (F 0 ) = GF 0 for some n ∈ N. In particular GF 0 = Gσ(F 0 ) and since orbits are locally closed we have that GF 0 = Gσ(F 0 ). This shows that there exists g ∈ Stab(F ) such that gF 0 = σ(F 0 ). Hence we have gi F 0 → gF 0 , and by almost algebraicity of the action on (V 0 )m it follows that T gi tends to g modulo Stab(F 0 ) = f ∈F Stab(f ). Let us fix some notations. For f ∈ F we denote by µf the restriction of µ to π −1 ({f }) and fix f ∈ G such that f N = f and denote by Hf ≤ G the −1 conjugate of H by f . Observe that f Nf = StabG (f ), Hf C StabG (f ) and −1 π ({f }) ' StabG (f )/Hf where π : G/H → G/N is the projection and StabG (f ) is the stabilizer of f under the action of G on G/N. We also denote µ0f = g −1 µσ(f ) T T and gi0 = g −1 gi . Since gi0 → e ∈ G/ f ∈F Stab(f ) there exist ni ∈ f ∈F Stab(f ) such that gi0 n−1 converges to e (in G). We observe that ni µf = ni (gi0 )−1 gi0 µf . As i 0 0 gi µf tends to µf and ni (gi0 )−1 tends to e, we have that ni µf converges to µ0f . Those measures are supported on π −1 ({f }) ' (StabG (f )/Hf ) (k). By Lemma 3.15, Stab(f ) acts almost algebraically on Prob ((StabG (f )/Hf ) (k)). So we have that ni tends to some n in Stab(f )/ Stab(µf ). We conclude that gi0 = gi0 n−1 in G/ Stab(µf ). Arguing similarly for i ni tends to nT every f , it follows that gi T tends to gn in G/ f ∈F Stab(µf ). Hence (gi ) converges also in G/ Stab(µ), since f ∈F Stab(µf ) ≤ Stab(µ). Let h be the limit point of (gi ) modulo Stab(µ). Then we have that gi µ converges to hµ by continuity of the action. Hence h ∈ Stab(µ), meaning that h = e modulo Stab(µ). In other words, gi converges to e in G/ Stab(µ). Proposition 3.20. Let H < G be a k-algebraic subgroup and set H = H(k). Then the action of G on Prob(G/H) is almost algebraic. Proof. Assume the proposition fails for an algebraic subgroup H. We also assume, as we may, that H is minimal in the collection of k-subgroup of G with the property that the G-action on Prob(G/H) is not almost algebraic. By Theorem 3.9, G acts on Prob(G/H) with almost algebraic stabilizers. Hence we have to show that for every measure µ ∈ Prob(G/H) and sequence gn with gn µ → µ then gn tends to e in G/ Stab(µ) (Remark 3.4). We fix such a measure µ and a sequence gn . We will achieve a contradiction by showing that gn does tend to e in G/ Stab(µ). We set N = NG (H), N = N(k), V = G/H, V0 = G/N, V = V(k) and V 0 = 0 V (k). We consider the natural inclusion G/H ⊂ V and view µ as a measure on V . We consider the projection map π : V → V 0 and set ν = π∗ µ. We use the notation introduced T in the discussion before Lemma 3.18. The lemma gives: Stab(µ) = Stab(µc )∩( λ∈Λ Stab(µλ )) where Λ is countable subset of [0, 1], gn µc → µc and for 20 each λ ∈ Λ, gn µλ → µλ . By Lemma 3.19, for each λ ∈ Λ, gi → e ∈ G/ Stab(µλ ). Assume given also that gn → e ∈ G/ Stab(µc ). Since by Theorem 3.9 the groups Stab(µλ ) and Stab(µc ) are almost Q algebraic, we will get by Lemma 3.8 that the action of G on G/ Stab(µc ) × λ G/ Stab(µλ ) is almost algebraic. Hence, , !! \ Stab(µλ ) gn → e ∈ G Stab(µc ) ∩ = G/ Stab(µ), λ achieving our desired contradiction. We are thus left to show that indeed gn → e ∈ G/ Stab(µc ). For the rest of the proof we will assume as we may µ = µc , that is ν ∈ Prob(V 0 ) is atom-free. We consider the measure µ × µ ∈ Prob(V × V ) and the subset U = {(xH, yH) | y −1 x ∈ / N} ⊂ G/H × G/H = V × V defined and discussed in the proof of Proposition 1.9. We set U = U(k). Note that the diagonal in V 0 × V 0 is ν × ν-null as ν is atom-free, thus U is µ × µ-full. We view as we may µ × µ as a probability measure on U . We now consider the G-action on U and claim that the G-orbits are locally closed and for every u ∈ U , G acts almost algebraically on Prob(Gu). The fact that the G-orbits are locally closed follows from Proposition 2.2, as U is a k-subvariety of V. Fix now a point u = (xH, yH) ∈ U for some x, y ∈ G, and consider −1 the G-action on Prob(Gu). By the definition of U, H ∩ Hy x H, thus by the −1 minimality of H the G-action on Prob(G/H∩H y x ) ' Prob(G/H x ∩H y ) is almost algebraic. Since by Proposition 2.2 G/H x ∩ H y is equivariantly homeomorphic to Gu we conclude that indeed, G acts almost algebraically on Prob(Gu), and the claim is proved. By Proposition 3.16, we conclude that G acts on Prob(U ) almost algebraically. Hence Effros’ Theorem 2.1 implies that gn → e in G/ Stab(µ × µ) as gn (µ × µ) → µ × µ. Observing that Stab(µ × µ) = Stab(µ), the proof is complete. Proof of Theorem 3.13. By Theorem 3.9 we know that the point stabilizers in Prob(G/L) are almost algebraic. We are left to show that for every µ ∈ Prob(G/L), for every sequence gn ∈ G satisfying gn µ → µ we have gn → e modulo Stab(µ) (see Remark 3.4). Fix µ ∈ Prob(G/L) and a sequence gn ∈ G satisfying gn µ → µ. Let H < G be a k-algebraic subgroup with H = H(k) normal and cocompact in L, and recall that by Lemma 3.12 we can find a G-equivariant continous map φ : Prob(G/L) → Prob(G/H) such that Stab(µ) = Stab(φ(µ)). We get that gn φ(µ) → φ(µ). By Proposition 3.20, the G-action on Prob(G/H) is almost algebraic, thus gn → e modulo Stab(φ(µ)). This finishes the proof, as Stab(µ) = Stab(φ(µ)). 3.4 Proof of Theorem 1.7 For the convenience of the reader we restate Theorem 1.7. Theorem 3.21. If the action of G on V is almost algebraic then the action of G on Prob(V ) is almost algebraic as well. 21 Proof. By Theorem 3.9, we know that the G-stabilizers in Prob(V ) are almost algebraic. We need to show that the quotient topology on G\ Prob(V ) is T0 . By Proposition 3.16, it is enough to check that the quotient topology on G\V is T0 , which is guaranteed by the assumption that the action of G on V is almost algebraic, and, as we will see, that for any v ∈ V , the action of G on Prob(Gv) is almost algebraic. We note that by Effros theorem (Theorem 2.1), the orbit Gv is equivariantly homeomorphic to the coset space G/ StabG (v), and thus Prob(Gv) ' Prob(G/ StabG (v)). Since StabG (v) is an almost algebraic subgroup of G, the fact that the G-action on Prob(Gv) is almost algebraic now follows from Theorem 3.13. 4 On bounded subgroups In this section, we essentially retain the setup 1.1 & 1.3: we fix a complete (k, | · |) valued field and a k-algebraic group G. Nevertheless there is no need for us to assume that (k, | · |) is separable, so we will refrain from doing so. Definition 4.1. A subset of k is called bounded if its image under | · | is bounded in R. For a k-variety V, a subset of V(k) is called bounded if its image by any regular function is bounded in k. Remark 4.2. Note that the collection of bounded sets on a k-variety forms a bornology. Remark 4.3. For a k-variety V it is clear that a subset of V(k) is bounded if and only if its intersection with every k-affine open set is bounded, so in what follows we will lose nothing by considering exclusively k-affine varieties. We will do so. Remark 4.4. Note that if (k 0 , | · |0 ) is a field extension of k endowed with an absolute value extension of | · | and V is a k-variety, we may regard V(k) as a subset V(k 0 ) and, as one easily checks, a subset of V(k) is k-bounded if and only if it is k 0 -bounded. Thus it causes no loss of generality assuming k is algebraically closed since b k is so. Nevertheless, we will not assume that. It is clear that every k-regular morphism of k-varieties is a bounded map in the sense that the image of a bounded set is bounded. For a k-closed immersion of k-varieties f : U → V also the converse is true: a subset of U(k) is bounded if and only if its image is bounded, as f ∗ : k[V] → k[U] is surjective. This is a special case of the following lemma. Lemma 4.5. For a finite k-morphism f : U → V a subset of U(k) is bounded if and only if its image is bounded. Proof. Assume there exists an unbounded set L in U(k) with f (L) being bounded in V(k). Then we could find p ∈ k[U] and a sequence un ∈ L with |p(un )| → ∞. The function p is integral over f ∗ k[V] so there exist q1 , . . . qm ∈ f ∗ k[V] with P m−i pm + m = 0. Thus, i=1 qi p 1= m m X X |qi (un )| qi (un ) ≤ → 0, i p (un ) |pi (un )| i=1 i=1 22 as the sequences qi (un ) are uniformly bounded. This is a contradiction. Recall that a seminorm on a k-vector space E is a function k · k : E → [0, ∞) satisfying 1. kαvk = |α|kvk, for α ∈ k, v ∈ E. 2. ku + vk ≤ kuk + kvk, for u, v ∈ E. A seminorm on E is a norm if furthermore we have 3. kvk = 0 ⇔ v = 0, for v ∈ E. Two norms on a vector space, k · k, k · k0 , are called equivalent if there exists some C ≥ 1 such that C −1 k · k ≤ k · k0 ≤ Ck · k. It is a general fact that any linear map between two Hausdorff topological (k, | · |)-vector spaces of finite dimensions is continuous [Bou87, I, §2,3 Corollary 2] and thus we get easily the following. Theorem 4.6. All the norms on a finite dimensional k-vector space are equivalent. Proof. It suffices to use that the identity map (E, || · ||) → (E, || · ||0 ) is continuous and observe that every continuous linear map is bounded. The latter is an easy exercise in case | · | is trivial, and standard if it is not. Recall that, if (e1P , . . . , en ) is a basis, then the norm || · ||∞ (relative to this basis) is defined as || xi ei ||∞ = max{|xi |}. Corollary 4.7. For a subset B ⊂ E = k n the following properties are equivalent. 1. B is a bounded set of An . 2. All elements of E ∗ are bounded on B. 3. All the coordinates of the elements of B are uniformly bounded. 4. The norm || · ||∞ is bounded on B. 5. Every norm on E is bounded on B. 6. Some norm on E is bounded on B. Theorem 4.8. For a subgroup L of GLn (k) the following are equivalent: 1. L is bounded in GLn (k). 2. L is bounded as a subset of Mn (k). 3. L preserves a norm on k n . 4. L preserves a spanning bounded set in k n . For a subgroup L of G = G(k) the following are equivalent: 23 1. L is bounded. 2. L preserves a norm in all k-linear representations of G. 3. L preserves a norm in some injective k-linear representation of G. 4. L preserves a spanning bounded set in some injective k-linear representation of G. Proof. Note that the second part of the theorem follows from the first once we recall that any injective homomorphism of algebraic groups is a closed immersion. We prove the equivalence of the first four conditions. (1) ⇔ (2) : Clearly, if L is bounded in GLn (k) then it is bounded in Mn (k). Assume L is bounded in Mn (k). Then it has a bounded image under both morphisms ι det GLn → − GLn ,→ Mn , GLn ,→ Mn −−→ A1 , ι where GLn → − GLn is the group inversion. We conclude that L has a bounded image under the product morphism GLn → Mn ×A1 . But the latter morphism id ⊕ det−1 is the composition of the isomorphism ι and the closed immersion GLn −−−−−−→ Mn ⊕A1 . Thus L is bounded in GLn (k). (2) ⇔ (3) : If L is bounded in Mn (k) then, by Corollary 4.7(3) all its matrix elements are uniformly bounded, hence for all v ∈ k n , supg∈L kgvk∞ is finite. This expression forms an L-invariant norm on k n . On the other hand, if L preserves a norm on k n , by the equivalence of this norm with k · k∞ , all matrix elements of L are uniformly bounded, thus it is bounded in Mn (k). (3) ⇔ (4) : If L preserves a norm then it preserves its unit ball which is a bounded spanning set. If L preserves a bounded spanning set B than it also preserves its symmetric convex hull: ( n ) n X X αi vi vi ∈ B, αi ∈ k, |αi | ≤ 1 . i=1 i=1 The latter is easily seen to be the unit ball of an L-invariant norm. Note that if L is a compact subgroup of G then L is bounded, as the k-regular functions of G are continuous on G. Corollary 4.9. Every bounded subgroup of G admits a bi-invariant metric. Proof. Let L be a bounded subgroup of G. Fix an injective k-linear representation G → GL(V ) and consider L as a subset of Endk (V ). Endk (V ) is a representation of G × G, hence admits a norm which is invariant under the bounded group L × L by Theorem 4.8. This norm gives an L × L-invariant metric on Endk (V ) and on its subset L. Proposition 4.10. Assume V is an affine k-variety with a k-affine action of G. Z Let B ⊂ V(k) be a bounded set and denote by B its Zariski closure. Then the Z Z image of StabG (B) is bounded in the k-algebraic group StabG (B )/ FixG (B ). 24 Z Proof. Without loss of generality we may replace G by StabG (B ) and then asZ Z Z sume V = B . We then may further assume G = StabG (B )/ FixG (B ). We do so. By [Bor91, Proposition 1.12] there exists a k-embedding of V into some vector space, which we may assume having a spanning image, equivariant with respect to some k-representation G → GLn , which we thus may assume injective. The proof then follows from the implication (4) =⇒ (1) in the second of equivalence of Theorem 4.8. Corollary 4.11. Let L < G be a bounded subgroup. Then NG (L)/ZG (L) is bounded. 5 The space of norms and seminorms In this section we study a compact space on which an algebraic group over a complete valued field acts by homeomorphisms, the space of seminorms. This space was already considered in the case when k is local, in [Wer04]. We fix a finite dimensional vector space E over k. Given two norms n, n0 on E we denote ™ ß n(y)n0 (x) x, y ∈ E \ {0} . d(n, n0 ) = log sup n0 (y)n(x) This number is finite by the fact that n and n0 are equivalent norms. Recall that two seminorms on E are called homothetic if they differ by a multiplicative positive constant. The relation of being homothetic is an equivalence relation. We denote the set consisting of all homothety classes of norms on E by I(E). Observe that d(n, n0 ) only depends on the homothety classes of n and n0 and thus define a function on I(E). Lemma 5.1. The function d : I(E) × I(E) → [0, ∞) defines a metric on I(E). The group PGL(E) acts continuously and isometrically on I(E) and the stabilizers in PGL(E) of bounded subsets in I(E) are bounded as well. Proof. The fact that d is a metric and PGL(E) acts by isometries on I(E) is a straightforward verification. To prove the continuity part, it suffices to show that the orbit map g 7→ gn is continuous for all n ∈ I(E). Fix a norm n on E. Let (gi ) be a sequence converging to e in PGL(E). By an abuse of notation we identify gi with an element of GL(E) such that gi → e ∈ GL(E), and also still is n. Using that d(gi n, n) = n denote n a norm whose homothety class o log sup n(gi−1 y)n(x) n(gi−1 x)n(y) | x, y ∈ E \ {0}, n(x), n(y) < 1 and that gi−1 converges uni- formly to e on the unit ball of E with respect to n, we see that indeed d(gi n, n) → 0. Let L be the stabilizer of some bounded set N ⊆ I(E). Fix v 6= 0 and identify N with a set N 0 of norms on E satisfying n(v) = 1 for every n ∈ N 0 . The set B = {x ∈ E | ∀n ∈ N 0 , n(x) ≤ 1} is clearly bounded in E. By Theorem 4.8, its stabilizer L0 ∈ GL(E) is bounded, hence also its image in PGL(E), namely L. Remark 5.2. The space I(E) actually contains the affine Bruhat-Tits building I(E) associated to PGL(E) [Par00] and there is a metric d0 on I(E) such that 25 (I(E), d0 ) is CAT(0) —not necessarily complete. The metric d is similar to the one considered by Goldman and Iwahori in [GI63]. The two metrics d and d0 are Lipschitz-equivalent. This can be checked first on an apartment and extended to the whole building using that any two points actually lie in some apartment. Thus, Lemma 5.1 and Theorem 4.8 are a reminiscence of the Bruhat-Tits fixed point theorem. Let S 0 (E) be the space of non-zero seminorms on E, and S(E) be its quotient by homotheties. We endow S 0 (E) with the topology of pointwise convergence and S(E) with the quotient topology. Proposition 5.3. The space S(E) is compact and metrizable. The action of PGL(E) on S(E) is continuous. Proof. Let m be the dimension of E. Fix a basis (e1 , . . . , em ) of E. Let S1 (E) be the set of all s ∈ S 0 (E) such that s(ei ) ≤ 1 for every 1 ≤ i ≤ d, and s(ej ) = 1 for some j. We first claim that the quotient map S 0 (E) → S(E) restricts to a surjection S1 (E) → S(E). This follows from the fact that if a seminorm is zero on all the vectors ei , then it is zero everywhere, by triangle inequality. Furthermore, the map S1 (E) → S(E) is actually an injection. Indeed if s ∈ S1 (E) and λs ∈ S1 (E) it is easy to conclude that λ = 1. We now claim Indeed, let k · k1 be the norm P that the Pspace S1 (E) is compact. P defined as k xi ei k1 = i |xi |. Let v = xi ei . Then we see that s(v) ≤ kvk1 for every v ∈ E. So we get that S1 (E) is homeomorphic to a closed subset of Q v∈E [0, kvk1 ]. This proves the compactness of S1 (E) and therefore of S(E). Note that it also proves that every element of S1 (E) is 1-Lipschitz with respect to k · k1 . It follows that S1 (E) is homeomorphic to S(E). The metrizability of S1 (E) comes from the fact that S1 (E) is a closed subset of the space of continuous functions on E, which is metrizable because E is separable. Now, let (gn , sn ) be a sequence converging to (e, s) ∈ GL(E)×S1 (E) then gn sn tends to s. Indeed, for every v ∈ E, |sn (gn v)−s(v)| ≤ |sn (gn v)−sn (v)|+|sn (v)−s(v)| ≤ kgn v−vk1 +|sn (v)−s(v)| → 0. Each non-zero seminorm s has a kernel ker(s) = {v ∈ E | s(v) = 0}, which is a proper linear subspace of E depending only of the homothety class of s. The map S(E) → N, s 7→ dim(ker(s)) is obviously PGL(E)-invariant. Denote by Sm (E) the space of homothety classes of seminorms s such that dim(ker(s)) = m. Note that S0 (E) = I(E). We denote by Grm (E) the Grassmannian of m-dimensional linear subspaces of E. The map Sm (E) → Grm (E), s 7→ ker(s) is clearly PGL(E)equivariant. Grm (E) is the k-points of a k-algebraic variety, thus carries a Polish topology by Proposition 2.2. Proposition 5.4. The maps S(E) → N, s 7→ dim(ker(s)) and Sm (E) → Grm (E), s 7→ ker(s) are measurable. 26 Proof. We first note that the space S(E) is covered by (countably many) open sets which are homeomorphic images of sets of the form {s ∈ S 0 (E) | s(v) = 1}, for v ∈ E, under the quotient map S 0 (E) → S(E). It is therefore enough to establish 0 that the corresponding maps S 0 (E) → N, Sm (E) → Grm (E) are measurable 0 (where Sm (E) denotes the preimage of Sm (E)). Fix a basis for E and a countable dense subfield k0 < k. Let E0 = E(k0 ) be the k0 -span of the fixed basis of E. A subspace of E is said to be defined over k0 if it has a basis in E0 . E0 is a k0 -vector space and it is a countable dense subset of E. Note that for every d, Grm (E0 ) is countable. Observe that for s ∈ S 0 (E), dim(ker(s)) ≤ m if and only if we can find a codimension m subspace F < E which is defined over k0 , such that s restricts to a norm on F . The latter condition is equivalent by Theorem 4.6 to the condition that there exists n ∈ N such that for every v ∈ F , s(v) ≥ |v|/n for some fixed norm | · |. Note that it is enough to check this for every v ∈ F0 = F (k0 ), thus we obtain [ [ \ {s ∈ S 0 (E) | dim(ker(s)) ≤ m} = {s ∈ S 0 (E) | s(v) ≥ |v|/n}. F0 ∈Grdim(E)−m (E0 ) n v∈F0 This shows that the map s 7→ dim(ker(s)) is measurable. 0 (E) → Grm (E) is measurable, we make two In order to prove that the map Sm observations. We first observe that the topologies of pointwise convergence and uniform convergence give the same Borel structure on S 0 (E). In fact, for every separable topological space X, the pointwise and uniform convergence topologies on Cb (X) give the same Borel structure (as uniform balls are easily seen to be Borel for the pointwise convergence topology), and S 0 (E) could be identified with a closed (for both topologies) subspace of bounded continuous functions on the unit ball of E. Our second observation is that we may identify Grm (E) with a subset of the space of closed subsets of the unit ball of E. Endowing it with the Hausdorff metric topology, we get a PGL(E)-invariant Polish topology on Grm (E). Since the Polish group PGL(E) acts transitively on Grm (E), by Effros Lemma [Eff65, Lemma 2.5] the quotient topology is the unique PGL(E)-invariant Polish topology on this space, thus the topology on Grm (E) given by the Hausdorff metric coincides with the one discussed in Proposition 2.2. The proof is now complete, observing further that with respect to the uniform 0 convergence topology on Sm (E) and the Hausdorff metric topology on Grm (E), the map s 7→ ker(s) is in fact continuous (moreover, it is C-Lipschitz on {s ∈ 0 Sm (E) | s is C-Lipschitz}). 6 Existence of algebraic representations This section is devoted to the proof of Theorem 1.16, which we restate below. The reader who is unfamiliar with the notion of measurable cocycles and amenable actions might consult with profit Zimmer’s book [Zim84, Chapter 4]. The following theorem provides a so-called algebraic representation of the space R, thus allowing to start the machinery developed in [BF13] and prove cocycle super-rigidity for the group G. 27 Theorem 6.1. Let R be a locally compact group and Y an ergodic, amenable Lebesgue R-space. Let (k, | · |) be a valued field. Assume that as a metric space k is complete and separable. Let G be a simple k-algebraic group. Let f : R×Y → G(k) be a measurable cocycle. Then either there exists a k-algebraic subgroup H G and an f -equivariant measurable map φ : Y → G/H(k), or there exists a complete and separable metric space V on which G acts by isometries with bounded stabilizers and an f equivariant measurable map φ0 : Y → V . Furthermore, in case k is a local field the G-action on V is proper and in case k = R and G is non-compact the first alternative always occurs. Proof. We first note that the isogeny G → G, where G is the adjoint group associated to G, is a finite morphism. Thus, by Lemma 4.5 we may assume that G is an adjoint group. We do so. By [Bor91, Proposition 1.10] we can find a k-closed immersion from G into some GLn . By the fact that G is simple, we may assume that this representation is irreducible. By the fact that G is adjoint, the associated morphism G → PGLn is a closed immersion as well. We will denote for convenience E = k n . Via this representation, G acts continuously and faithfully on the metric space of homothety classes of norms, I(E), and on the compact space of homothety classes of seminorms, S(E), introduced in §5. By the amenability of the action of R on Y there exists a f -map, that is a f -equivariant map, φ : Y → Prob(S(E)), which we now fix. By Proposition 5.4, there is a measurable partition S(E) = ∪n−1 d=0 Sd (E), given by the dimension of the kernels of the seminorms. For a given d, the function Y → [0, 1] given by y 7→ φ(y)(Sd (E)) is R-invariant, hence almost everywherePequal to some constant, by ergodicity. We denote this constant by αd . Note that n−1 d=0 αd = 1. We choose d such that αd > 0 and define ψ : Y → Prob(Sd (E)), ψ(y) = 1 φ(y)|Sd (E) . αd Note that ψ is a f -map. We will consider two cases: either d > 0 or d = 0. This is a first bifurcation leading to the two alternatives in the statement of the theorem. We first consider the case d > 0. We use the map Sd (E) → Grd (E) discussed in Proposition 5.4 to obtain the push forward map Prob(Sd (E)) → Prob(Grd (E)). By post-composition, we obtain a f -map Ψ : Y → Prob(Grd (E)). By Theorem 1.7 the action of G on Prob(Grd (E)) is almost algebraic (as the action of G on Grd (E) is almost algebraic by Proposition 2.2), and the quotient topology on G\ Prob(Grd (E)) is T0 . We claim that there exists µ ∈ Prob(Grd (E)) such that the set Ψ−1 (Gµ) has full measure in Y . The standard argument is similar to the prof of Proposition 1.12: for a countable basis Bi for the topology of G\ Prob(Grd (E)), the set \ \ {Bi | Ψ−1 (Bi ) is full in Y } ∩ {Bic | Ψ−1 (Bi ) is null in Y } is clearly a singleton, whose preimage is of full measure in Y . Let µ be a preimage of this singleton in Prob(Grd (E)). 28 By the fact that G acts almost algebraically on Prob(Grd (E)), we may identify Gµ with a coset space G/L, for some almost algebraic subgroup L = StabG (µ) < G, and view Ψ as an f -map from Y to G/L. By Proposition 1.9, there exists a k-subgroup H0 < G which is normalized by L such that L has a precompact image in the Polish group (NG (H0 )/H0 )(k) and such that µ is supported on the subvariety of H0 fixed points in Grd (E). Note that by the irreducibility of the representation G → GLn we have no G-fixed points in Grd (E), thus H0 G. Assume moreover that H0 6= {e} and let H be the Zariski-closure of L. By [Bor91, Theorem AG14.4], H is a k-subgroup of G. By the simplicity of G, H G, as H normalizes H0 . Post-composing the f -map Ψ with the map G/L → G/H(k) we obtain a k-algebraic subgroup H G and an f -equivariant measurable map φ : Y → G/H(k), as desired. Assume now H0 = {e}. In that case L is compact, and in particular bounded in G. It follows by Theorem 4.8 that L fixes a norm on E. Thus we may map the coset space G/L G-equivariantly into S0 (E) = I(E). Using the δ-measure embedding I(E) ,→ Prob(I(E)) and obtain a new f -map Y → Prob(I(E)). We are then reduced to the case d = 0, to be discussed below. We consider now the case d = 0, that is we assume having an f -map Y → Prob(I(E)). We set V = Prob(I(E)). By Lemma 5.1, G acts isometrically and with bounded stabilizers on I(E). By Lemma 2.6, G acts isometrically on V . Let us check that stabilizers are bounded. Fix µ ∈ Prob(I(E)), and let L be its stabilizer in G. Since I(E) is Polish there is a ball B of I(E) such that µ(B) > 1/2. It follows that for any g ∈ L, gB intersects B. Thus the set LB is bounded in I(E), and by Lemma 5.1 its stabilizer is bounded in G. It follows that L is bounded. Thus we have found an f -map from Y to a complete and separable metric space V on which G acts by isometries with bounded stabilizers as desired. Acknowledgement U.B was supported in part by the ERC grant 306706. B.D. is supported in part by Lorraine Region and Lorraine University. B.D & J.L. are supported in part by ANR grant ANR-14-CE25-0004 GAMME. References [AB94] N. A’Campo & M. Burger – “Réseaux arithmétiques et commensurateur d’après G. A. Margulis”, Invent. Math. 116 (1994), no. 1-3, p. 1–25. [BF13] U. Bader & A. Furman – “Algebraic representations of ergodic actions and super-rigidity”, (2013). [BGR84] S. Bosch, U. Güntzer & R. Remmert – Non-Archimedean analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 261, Springer-Verlag, Berlin, 1984, A systematic approach to rigid analytic geometry. 29 [Bil99] P. Billingsley – Convergence of probability measures, second éd., Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, Inc., New York, 1999, A Wiley-Interscience Publication. [Bor91] A. Borel – Linear algebraic groups, second éd., Graduate Texts in Mathematics, vol. 126, Springer-Verlag, New York, 1991. [Bou87] N. Bourbaki – Topological vector spaces. Chapters 1–5, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1987, Translated from the French by H. G. Eggleston and S. Madan. [BZ76] I. N. Bernšteı̆n & A. V. Zelevinskiı̆ – “Representations of the group GL(n, F ), where F is a local non-Archimedean field”, Uspehi Mat. Nauk 31 (1976), no. 3(189), p. 5–70. [Eff65] E. G. Effros – “Transformation groups and C ∗ -algebras”, Ann. of Math. (2) 81 (1965), p. 38–55. [EP05] A. J. Engler & A. Prestel – Valued fields, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005. [Fol99] G. B. Folland – Real analysis, second éd., Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1999, Modern techniques and their applications, A Wiley-Interscience Publication. [GGMB13] O. Gabber, P. Gille & L. Moret-Bailly – “Fibrés principaux pour les corps valués henséliens”, (2013). [GI63] O. Goldman & N. Iwahori – “The space of p-adic norms”, Acta Math. 109 (1963), p. 137–177. [hmb] L. M.-B. (http://mathoverflow.net/users/7666/laurentmoret bailly) – “When is a valued field second-countable?”, MathOverflow, URL:http://mathoverflow.net/q/96999 (version: 2012-0515). [Kec95] A. S. Kechris – Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. [Oxt46] J. C. Oxtoby – “Invariant measures in groups which are not locally compact”, Trans. Amer. Math. Soc. 60 (1946), p. 215–237. [Par00] A. Parreau – “Immeubles affines: construction par les normes et étude des isométries”, in Crystallographic groups and their generalizations (Kortrijk, 1999), Contemp. Math., vol. 262, Amer. Math. Soc., Providence, RI, 2000, p. 263–302. [Poo93] B. Poonen – “Maximally complete fields”, Enseign. Math. (2) 39 (1993), no. 1-2, p. 87–106. 30 [Ser06] J.-P. Serre – Lie algebras and Lie groups, Lecture Notes in Mathematics, vol. 1500, Springer-Verlag, Berlin, 2006, 1964 lectures given at Harvard University, Corrected fifth printing of the second (1992) edition. [Sha99] Y. Shalom – “Invariant measures for algebraic actions, Zariski dense subgroups and Kazhdan’s property (T)”, Trans. Amer. Math. Soc. 351 (1999), no. 8, p. 3387–3412. [Sim12] D. Simmons – “Conditional measures and conditional expectation; Rohlin’s disintegration theorem”, Discrete Contin. Dyn. Syst. 32 (2012), no. 7, p. 2565–2582. [VGO90] È. B. Vinberg, V. V. Gorbatsevich & A. L. 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arXiv:1707.09854v1 [math.KT] 27 Jul 2017 The IA-congruence kernel of high rank free Metabelian groups David El-Chai Ben-Ezra August 1, 2017 Abstract The congruence subgroup problem for a finitely generated group Γ and G ≤ Aut(Γ) asks whether the map Ĝ → Aut(Γ̂) is injective, or more generally, what is its kernel C (G, Γ)? Here X̂ denotes the profinite completion of X. In this paper we investigate C (IA(Φn ), Φn ), where Φn is a free metabelian group on n ≥ 4 generators, and IA(Φn ) = ker(Aut(Φn ) → GLn (Z)). We introduce surjective representations of IA(Φn ) onto the group x7→1 ker(GLn−1 (Z[x±1 ]) −→ GLn−1 (Z)) which come via the classical Magnus representation of IA(Φn ). Using this representations combined with some methods and results from Algebraic K-theory, we prove that for every n ≥ 4, C (IA(Φn ), Φn ) contains a product of n copies of the congruence \ \ ±1 ])) which is central in IA(Φ \n ). kernel ker(SLn−1 (Z[x±1 ]) → SLn−1 (Z[x It enables us to show that contrary to free nilpotent cases, C (IA(Φn ), Φn ) is not trivial and not even finitely generated. We note that using some results of this paper we show in an upcoming paper that actually, all the elements of C (IA(Φn ), Φn ) lie in the center \n ). of IA(Φ Mathematics Subject Classification (2010): Primary: 19B37, 20H05, Secondary: 20E36, 20E18. Key words and phrases: congruence subgroup problem, automorphism groups, profinite groups, free metabelian groups. Contents 1 Introduction 2 2 Some background in algebraic K-theory 6 3 IA (Φn ) and its subgroups 8 4 The subgroups Ci 13 1 5 The centrality of Ci 20 m 6 Some elementary elements of hIA (Φn ) i 7 A main lemma 7.1 Decomposing the proof . . 7.2 Some needed computations 7.3 Elements of form 2 . . . . . 7.4 Elements of form 3 . . . . . 7.5 Elements of form 4 . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 26 30 34 41 47 Introduction The classical congruence subgroup problem (CSP) asks for, say, G = SLn (Z) or G = GLn (Z), whether every finite index subgroup of G contains a principal congruence subgroup, i.e. a subgroup of the form G (m) = ker (G → GLn (Z/mZ)) for some 0 6= m ∈ Z. It is a classical 19th century result that the answer is negative for n = 2. On the other hand, quite surprisingly, it was proved in the sixties by Mennicke [Men] and by Bass-Lazard-Serre [BaLS] that for n ≥ 3 the answer to the CSP is affirmative. A rich theory of the CSP for more general arithmetic groups has been developed since then. By the observation GLn (Z) ∼ = Aut (Zn ), the CSP can be generalized to automorphism groups as follows: Let Γ be a group and G ≤ Aut (Γ). For a finite index characteristic subgroup M ≤ Γ denote: G (M ) = ker (G → Aut (Γ/M )) . Such a G (M ) will be called a “principal congruence subgroup” of G and a finite index subgroup of G which contains G (M ) for some M will be called a “congruence subgroup”. The CSP for the pair (G, Γ) asks whether every finite index subgroup of G is a congruence subgroup. One can easily see that the CSP is equivalent to the question: Is the congruence map Ĝ = limG/U → limG/G (M ) injective? Here, U ranges over all ←− ←− finite index normal subgroups of G, and M ranges over all finite index characteristic subgroups of Γ. When Γ is finitely generated, it has only finitely many subgroups of given index m, and thus, the charateristic subgroups: Mm = ∩ {∆ ≤ Γ | [Γ : ∆] = m} are of finite index in Γ. Hence, one can write Γ̂ = 1 lim ←−m∈N Γ/Mm and have : limG/G (M ) = ←− ≤ lim G/G (Mm ) ≤ limm∈N Aut(Γ/Mm ) ←−m∈N ←− Aut(lim (Γ/M )) = Aut(Γ̂). m∈N m ←− 1 By the celebrated theorem of Nikolov and Segal which asserts that every finite index subgroup of a finitely generated profinite group is open [NS], the second inequality is actually an equality. However, we do not need it. 2 Therefore, when Γ is finitely generated, the CSP is equivalent to the question: Is the congruence map: Ĝ → Aut(Γ̂) injective? More generally, the CSP asks what is the kernel C (G, Γ) of this map. For G = Aut (Γ) we will also use the simpler notation C (Γ) = C (G, Γ). The classical congruence subgroup result n mentioned above can therefore
be reformulated as C (Z ) = {e} for n ≥ 3, and 2 it is also known that C Z = F̂ω , where F̂ω is the free profinite group on a countable number of generators (cf. [Mel], [L]). Very few results are known when Γ is non-abelian. Most of the results are related to Γ = π(Sg,n ), the fundamental group of the closed surface of genus g with n punctures (see [DDH], [Mc], [A], [Bo1], [Bo2]). As observed in [BER], the result of Asada in [A] actually gives an affirmative solution to the case Γ = F2 , G = Aut (F2 ) (see also [BL]). Note that for every n > 0, π(Sg,n ) ∼ = F2g+n−1 = the free group on 2g + n − 1 generators. Hence, the above mentioned results relate to various subgroups of the automorphism group of finitely generated free groups. However, the CSP for the full Aut (Fd ) when d ≥ 3 is still unsettled. Denote now the free metabelian group on n generators by Φn = Fn /Fn′′ . Considering the metabelian case, it was shown in [BL] (see also [Be1]) that C (Φ2 ) = F̂ω . In addition, it was proven there that C (Φ3 ) ⊇ F̂ω . The basic motivation which led to this paper was to complete the picture in the free metabelian case and investigate C (Φn ) for n ≥ 4. Now, denote IA(Φn ) = ker(Aut(Φn ) → GLn (Z)). Then, the commutative exact diagram: 1 → IA (Φn ) → Aut (Φn ) ց ↓ Aut(Φ̂n ) → GLn (Z) → 1 ↓ → GLn (Ẑ) gives rise to the commutative exact diagram (see Lemma 2.1 in [BER]): \ \ \ IA (Φn ) → Aut (Φn ) → GL n (Z) ց ↓ ↓ Aut(Φ̂n ) → GLn (Ẑ) → 1 . \ Hence, by using the fact that GL n (Z) → GLn (Ẑ) is injective for n ≥ 3, one can obtain that C (Φn ) is an image of C (IA (Φn ) , Φn ). Thus, for investigating C (Φn ) it seems to be worthwhile to investigate C (IA (Φn ) , Φn ). The main result of the paper is the following: Theorem 1.1. For every n ≥ 4, C (IA (Φn ) , Φn ) contains a subgroup C which satisfies the following properties: Qn 1. C is isomorphic to a product C = i=1 Ci of n copies of: \ \ ±1 ])). (Z[x±1 ]) → SLn−1 (Z[x Ci ∼ = ker(SLn−1 \ 2. C is contained in the center of IA (Φn ). 3. C is a direct factor of C (IA (Φn ) , Φn ), i.e. there is a normal subgroup N ⊳ C (IA (Φn ) , Φn ) such that C (IA (Φn ) , Φn ) = N × C. 3 We note that as for every n ≥ 4, IA (Φn ) is finitely generated [BM], the \ action of Aut (Φn ) on IA (Φn ) by conjugation induce an action of Aut (Φn ) \ \ on IA (Φn ), such that the closer IA (Φn ) of IA (Φn ) in Aut (Φn ) acts trivially \ \ \ on Z(IA (Φn )), the center of IA (Φn ). Thus, as we have Aut (Φn )/IA (Φn ) = \ \ \ GLn (Z) we obtain a natural action of GLn (Z) on Z(IA (Φn )). It will be clear from the description in the paper that the permutation matrices permute the copies Ci through this natural action. As by using the techniques of Kassabov and Nikolov in [KN] one can show that Ci are not finitely generated, this gives the imediate following theorem as a corollary: Theorem 1.2. For every n ≥ 4, C (IA (Φn ) , Φn ) is not finitely generated. It will be shown in an upcoming paper that when Γ is a finitely generated free nilpotent group of class c, then C (IA (Γ) , Γ) = {e} is always trivial. So the free metabelian cases behave completely different from free nilpotent cases. The following problem is still open: Qn Problem 1.3. Is C (IA (Φn ) , Φn ) = i=1 Ci or it contains more elements? We get close to solve Problem 1.3 in an upcoming result, which asserts [Be2]: \ Theorem 1.4. For every n ≥ 4, C (IA (Φn ) , Φn ) is central in IA (Φn ). We note that considering our basic motivation, as C (Φn ) is an image of C (IA (Φn ) , Φn ) we actually obtain that when n ≥ 4, the situation is dramatically different from the cases of n = 2, 3 described above, and: Theorem 1.5. For every n ≥ 4, C (Φn ) is abelian. We remark that despite the result of the latter theorem, we do not know whether C (Φn ) is also not finitely generated. In fact we can not even prove at this point that it is not trivial. The main line of the proof of Theorem 1.1, is organized along the paper as follows. For a ring R, ideal J ⊳ R and d ∈ N denote: GLd (R, J) = ker(GLd (R) → GLd (R/J)). ±1 n For n ∈ N denote also the ring Rn = Z[x±1 1 , . . . , xn ] = Z[Z ]. Using the Magnus embedding of IA (Φn ), in which IA (Φn ) can be viewed as:      x1 − 1  x1 − 1        .. .. IA (Φn ) = A ∈ GLn (Rn ) | A  =    . .     xn − 1 xn − 1 u we obtain in 3, for every 1 ≤ i ≤ n, a natural embedding of: GLn−1 (Rn , (xi − 1)Rn ) ֒→ IA (Φn ) 4 and a surjective natural homomorphism: ρi IA (Φn ) ։ GLn−1 (Z[xi±1 ], (xi − 1)Z[x±1 i ]) in which the obvious copy of the subgroup GLn−1 (Z[xi±1 ], (xi − 1)Z[x±1 i ]) in GLn−1 (Rn , (xi − 1)Rn ) is mapped onto itself via the composition map. This description, u after quoting some u classical notions and results from Algebraic Ktheory in 2, enables us in 4 to show that for every n ≥ 4 and 1 ≤ i ≤ n, C (IA (Φn ) , Φn ) contains a copy of: Ci ∼ = ±1 \ ker(GLn−1 (Z[xi±1\ ], (xi − 1)Z[x±1 i ]) → GLn−1 (Z[xi ])) ∼ = ±1 \ \ ker(SLn−1 (Z[x±1 i ]) → SLn−1 (Z[xi ])) such that C (IA (Φn ) , Φn ) is maped onto Ci through the map ρ̂i which induced by ρi . The second isomorphism in the obove equation is obtained by using a main lemma which is left to the end of the paper and some classical results from Algebraic K-theory. In particular, we get that for every 1 ≤ i ≤ n: u C (IA (Φn ) , Φn ) = (C (IA (Φn ) , Φn ) ∩ ker ρ̂i ) ⋊ Ci . In 4 we aslo show that the copies Ci intersect each other trivially. Then, following the techniques of Kassabov and Nikolov in [KN] we show that Ci is not finitely generated, and therefore deduce that C (IA (Φn ) , Φn ) is not finitely u generated either. Then, in 5 we show that the copies Ci lie in the center of \ IA (Φn ), using classical results from Algebraic K-theory and the main reminded lemma. In particular we obtain that: C (IA (Φn ) , Φn ) = (C (IA (Φn ) , Φn ) ∩ni=1 ker ρ̂i ) × n Y Ci i=1 This complete the proof of Theorem 1.1. u m In 6 we compute some elements u in hIA (Φn ) i which help us later to prove the main reminded lemma. In 7, using classical results from algebraic Ktheory, we end the paper by proving a main lemma which asserts that for every 1 ≤ i ≤ n, we have:
m GLn−1 (Rn , (xi − 1)Rn ) ∩ E n−1 Rn , Hn,m2 ⊆ hIA (Φn ) i when: • GLn−1 (Rn , (xi − 1)Rn ) denotes its appropriate copy in IA (Φn ) described above.
• E n−1 Rn , Hn,m2 is the subgroup of En−1 (Rn ) = hIn−1 + rEi,j | r ∈ Rn i which is generated as a normal subgroup by the elementary
matrices of the form In−1 + hEi,j for h ∈ Hn,m2 = ker Rn → Zm2 [Znm2 ] , 1 ≤ i 6= j ≤ n. Here, In−1 is the (n − 1) × (n − 1) unit matrix and Ei,j is the matrix which has 1 in the (i, j)-th entry and 0 elsewhere. 5 • The intersection is obtained by viewing the copy of GLn−1 (Rn , (xi −1)Rn ) in IA(Φn ) as a subgroup of GLn−1 (Rn ). Acknowledgements: I wish to offer my deepest thanks to my great supervisor Prof. Alexander Lubotzky for his sensitive and devoted guidance, and to the Rudin foundation trustees for their generous support during the period of the research. 2 Some background in algebraic K-theory In this section we fix some notations and recall some definitions and background in algebraic K-theory which will be used through the paper. One can find more general information in the references ([Ros], [Mi], [Bas]). In this section R will always denote a commutative ring with identity. We start with recalling the following notations: Let R be a commutative ring, H ⊳ R an ideal, and d ∈ N. Then: • GLd (R) = {A ∈ Mn (R) | det (A) ∈ R∗ }. • SLd (R) = {A ∈ GLd (R) | det (A) = 1}. • Ed (R) = hId + rEi,j | r ∈ R, 1 ≤ i 6= j ≤ di. • GLd (R, H) = ker (GLd (R) → GLd (R/H)). • SLd (R, H) = ker (SLd (R) → SLd (R/H)). • Ed (R, H) = the normal subgroup of Ed (R), which is generated as a normal subgroup by the elementary matrices of the form Id + hEi,j for h ∈ H. As for every d ≥ 3, Ed (R, H) is normal in GLd (R) (see Corollary 1.4 in [Su]), one can have the following definitions: K1 (R; d) = GLd (R) /Ed (R) SK1 (R; d) = SLd (R) /Ed (R) K1 (R, H; d) = GLd (R, H) /Ed (R, H) SK1 (R, H; d) = SLd (R, H) /Ed (R, H) . We now go ahead with the following definition: Definition 2.1. Let R be a commutative ring, and 3 ≤ d ∈ N. We define the “Steinberg group” Std (R) to be the group which generated by the elements xi,j (r) for r ∈ R and 1 ≤ i 6= j ≤ d, under the relations: • xi,j (r1 ) · xi,j (r2 ) = xi,j (r1 + r2 ). • [xi,j (r1 ) , xj,k (r2 )] = xi,k (r1 · r2 ). • [xi,j (r1 ) , xk,l (r2 )] = 1. 6 for every different 1 ≤ i, j, k, l ≤ d and every r1 , r2 ∈ R. As the elementary matrices Id + rEi,j satisfy the relations which define Std (R), the map xi,j (r) 7→ Id + rEi,j defines natural homomorphism φd : Std (R) → E d (R). The kernel of this map is denoted by K2 (R; d) = ker (φd ). Now, for two invertible elements u, v ∈ R∗ and 1 ≤ i 6= j ≤ d define the “Steinberg symbol” by: {u, v}i,j = hi,j (uv) hi,j (u)−1 hi,j (v)−1 ∈ Std (R) when hi,j (u) = wi,j (u) wi,j (−1) and
wi,j (u) = xi,j (u) xj,i −u−1 xi,j (u) . One can show that {u, v}i,j ∈ K2 (R; d) and lie in the center of Std (R). In addition, for every 3 ≤ d ∈ N, the Steinberg symbols {u, v}i,j do not depend on the indices i, j, so they can be denoted simply by {u, v} (see [DS]). The Steinberg symbols satisfy many identities. For example: {uv, w} = {u, w} {v, w} , {u, vw} = {u, v} {u, w} . (2.1) In the semi-local case we have the following ([SD], Theorem 2.7): Theorem 2.2. Let R be a semi-local commutative ring and d ≥ 3. Then, K2 (R; d) is generated by the Steinberg symbols {u, v} for u, v ∈ R∗ . In particular, K2 (R; d) is central in Std (R). Let now R be a commutative ring, H ⊳ R ideal
and d ≥ 3. Denote R̄ = R/H. Clearly, there is a natural map E d (R) → E d R̄ . It is easy to see that Ed (R, H) is in the kernel of the latter map, so we have a map:
πd : Ed (R) /Ed (R, H) → Ed R̄ . In addition, it is easy to see that we have a surjective map:
ψd : Std R̄ ։ Ed (R) /Ed (R, H)

defined by: xi,j (r̄) → Id + rEi,j , such that φd : Std R̄ → Ed R̄ satisfies: φd = πd ◦ ψd . Therefore, we obtain the surjective map:
ψd K2 R̄; d = ker (φd ) ։ ker (πd ) = ≤ (Ed (R) ∩ SLd (R, H)) /Ed (R, H) SK1 (R, H; d) . In particular, it implies that if Ed (R) = SLd (R), then we have a natural surjective map: K2 (R/H; d) ։ SK1 (R, H; d) . From here one can easily deduce the following corollary, which will be needed later in the paper. 7 Corollary 2.3. Let R be a commutative ring, H ⊳ R ideal of finite index and d ≥ 3. Assume also that Ed (R) = SLd (R). Then: 1. SK1 (R, H; d) is a finite group. 2. SK1 (R, H; d) is central in GLd (R) /Ed (R, H). 3. Every element  of SK1 (R, H; d) has a representative in SLd (R, H) of the A 0 such that A ∈ SL2 (R, H). form 0 Id−2 Proof. The ring R̄ = R/H is finite. In particular, R̄ is Artinian and hence semi
local. Thus, by Theorem 2.2, K2 R̄; d is an abelian group which generated by the Steinberg symbols {u, v} for u, v ∈ R̄∗ . As R̄ is finite, so is the number of the Steinberg symbols. By equation
2.1 we obtain that the order of any abelian group Steinberg symbol is finite. So K2 R̄; d is a finitely generated
d is finite. Moreover, as which its generators are of finite order. Thus, K2 R̄;

R̄ is semi-local, Theorem 2.2 implies that K2 R̄; d is central Std R̄
. Now, as we assume Ed (R) = SLd (R), SK1 (R, H; d) is the image of K2 R̄; d under the surjective map
Std R̄ ։ Ed (R) /Ed (R, H) = SLd (R) /Ed (R, H) . This implies part 1 and that SK1 (R, H; d) is central in SLd (R) /Ed (R, H). Moreover, as d ≥ 3, we have {u, v} = {u, v}1,2 for every u, v ∈ R̄∗ . Now, it is easy to check from the definition that the image of {u, v}1,2 under the  
A 0 · Ed (R, H) for map Std R̄ ։ SLd (R) /Ed (R, H) is of the form 0 Id−2 some A ∈ SL2 (R, H). So as SK1 (R, H; d) is generated by the images of the Steinberg symbols, the same holds for every element in SK1 (R, H; d). So we obtain part 3. Now, as d ≥ 3 we can write: GLd (R) = SLd (R) · {Id + (r − 1) E3,3 | r ∈ R∗ } ∗ and as all the elements of theform Id + (r  − 1) E3,3 for r ∈ R commute with A 0 for A ∈ SL2 (R, H), the centrality all the elements of the form 0 Id−2 of SK1 (R, H; d) in SLd (R) /Ed (R, H) shows that actually SK1 (R, H; d) is central in GLd (R) /Ed (R, H), as required in part 2. 3 IA (Φn ) and its subgroups We start to deal with the IA-automorphism group of the free metabelian group, G = IA (Φn ) = ker (Aut (Φn ) → Aut (Φn /Φ′n ) = GLn (Z)), by presenting some of its properties and subgroups. We start with the following notations: • Φn = Fn /Fn′′ = the free metabelian group on n elements. Here Fn′′ denotes the second derivative of Fn , the free group on n elements.. ′ m ′ m • Φn,m = Φn /Mn,m , where Mn,m = (Φ′n Φm n ) (Φn Φn ) . 8 • IGn,m = G(Mn,m ) = ker (IA (Φn ) → Aut (Φn,m )) . • IAn,m = ∩ {N ⊳ IA (Φn ) | [IA (Φn ) : N ] | m} ±1 n • Rn = Z[Zn ] = Z[x±1 1 , . . . , xn ] where x1 , . . . , xn are the generators of Z . • Zm = Z/mZ. • σi = xi − 1 for 1 ≤ i ≤ n. We also denote by ~σ the column vector which has σi in its i-th entry. Pn • An = i=1 σi Rn = the augmentation ideal of Rn . Pn • Hn,m = ker (Rn → Zm [Znm ]) = i=1 (xm i − 1) Rn + mRn . By the well known Magnus embedding (see [Bi], [RS], [Ma]), one can identify Φn with the matrix group: ) (  n X g a 1 t1 + . . . + a n tn ai (xi − 1) | g ∈ Zn , ai ∈ Rn , g − 1 = Φn = 0 1 i=1 where ti is a free basis for Rn -module, under the identification of the generators of Φn with the matrices   xi ti 1 ≤ i ≤ n. 0 1 Moreover, for every α ∈ IA (Φn ), one can describe α by its action on the generators of Φn , by:     xi ti xi ai,1 t1 + . . . + ai,n tn α: 7→ 0 1 0 1 and this description gives an injective homomorphism (see [Bac], [Bi]): IA (Φn ) ֒→ defined by α 7→ GLn (Rn )  a1,1 · · ·  ..  . an,1  a1,n ..  .  · · · an,n which gives an identification of IA (Φn ) with the group: IA (Φn ) = = {A ∈ GLn (Rn ) | A~σ = ~σ } n o In + A ∈ GLn (Rn ) | A~σ = ~0 . Proposition 3.1. Let In + A ∈ IA (Φn ) and denote Pnthe entries of A by ak,l for 1 ≤ k, l ≤ n. Then, for every 1 ≤ k, l ≤ n, ak,l ∈ l6=i=1 σi Rn ⊆ An . 9 Proof. For a given 1 ≤ k ≤ n, the condition A~σ = ~0 gives the equality: 0 = ak,1 σ1 + ak,2 σ2 + . . . + ak,n σn . Thus, for a given 1 ≤ l ≤ n, the map Rn → Z[x±1 l ] which defined by xi 7→ 1 for ±1 every i 6= l, maps 0 = ak,1 σ1 + ak,2 σ2 + . . . + ak,n σn 7→ āk,l σP l ∈ Z[xl ]. Hence, n ±1 ±1 as Z[xl ] is a domain, āk,l = 0 ∈ Z[xl ]. Therefore: ak,l ∈ l6=i=1 σi Rn ⊆ An , as required. Proposition 3.2. Qn Let In + A ∈ IA (Φn ). Then det (In + A) is of the form: det (In + A) = r=1 xsrr for some sr ∈ Z. Qn Proof. The invertible elements in Rn are the elements of the form ± i=1 xsrr (see Qn[CF], chapter 8). Thus, as In + A ∈ GLn (Rn ) we have: det (In + A) = ± i=1 xsrr . However, according to Proposition 3.1, for every entry ak,l of A we have: ak,l ∈ An . Hence, under the xi 7→ 1 for every 1 ≤ i ≤ n, Q projection sr one has In + A 7→ In and thus, ± ni=1 x = det (In + A) 7→ det (In ) = 1. Qn r Therefore, the option det (In + A) = − i=1 xsrr is impossible, as required. Consider now the map:    Pn g a 1 t1 + . . . + a n tn n Φn = | g ∈ Z , ai ∈ Rn , g − 1 = i=1 ai (xi − 1) 0 1 ↓    Pn g a 1 t1 + . . . + a n tn n n | g ∈ Zm , ai ∈ Zm [Zm ], g − 1 = i=1 ai (xi − 1) 0 1 which induced by the projections Zn → Znm , Rn = Z[Zn ] → Zm [Znm ]. Using result of Romanovskiı̆ [Rom], it is shown in [Be1] that this map is surjective and that Φn,m is canonically isomorphic to its image. Therefore, we can identify the principal congruence subgroup of IA (Φn ), IGn,m , with: IGn,m = = {A ∈ ker (GLn (Rn ) → GLn (Zm [Znm ])) | A~σ = ~σ } n o In + A ∈ GLn (Rn , Hn,m ) | A~σ = ~0 . Let us step forward with the following definitions: Definition 3.3. Let A ∈ GLn (Rn ), and for 1 ≤ i ≤ n denote by Ai,i the minor which obtained from A by erasing its i-th row and i-th column. Now, for every 1 ≤ i ≤ n, define the subgroup IGLn−1,i ≤ IA (Φn ), by:   The i-th row of A is 0, IGLn−1,i = In + A ∈ IA (Φn ) | . In−1 + Ai,i ∈ GLn−1 (Rn , σi Rn ) Proposition 3.4. For every 1 ≤ i ≤ n we have: IGLn−1,i ∼ = GLn−1 (Rn , σi Rn ). Proof. The definition of IGLn−1,i gives us a natural projection from IGLn−1,i → GLn−1 (Rn , σi Rn ) which maps an element In + A ∈ IGLn−1,i to In−1 + Ai,i ∈ 10 GLn−1 (Rn , σi Rn ). Thus, all we need is to explain why this map is injective and surjective. Injectivity: Here, It is enough to show that given an element In + A ∈ IA (Φn ), every entry in the i-th column is determined uniquely by the other entries in its row. Indeed, as A satisfies the condition A~σ = ~0, P for every 1 ≤ − n i6=l=1 ak,l σl , i.e. k ≤ n we have: ak,1 σ1 + ak,2 σ2 + . . . + ak,n σn = 0 ⇒ ak,i = σi we have a formula for ak,i by the other entries in its row. Surjectivity: Without loss of generality we assume i = n. Let In−1 + σn C ∈ GLn−1 (Rn , σn Rn ), and denote by ~cl the column vectors of C. Define:   Pn−1 In−1 + σn C − l=1 σl~cl ∈ IGLn−1,n 0 1 and this is clearly a preimage of In−1 + σn C. Under the above identification of IGLn−1,i with GLn−1 (Rn , σi Rn ), we will use through the paper the following notations: Definition 3.5. Let H ⊳ Rn . We define: ISLn−1,i (H) = IGLn−1,i ∩ SLn−1 (Rn , H) IEn−1,i (H) = IGLn−1,i ∩ E n−1 (Rn , H) ≤ ISLn−1,i (H) .
Observe now that as for every 1 ≤ i ≤ n, GLn−1 Z[xi±1 ], σi Z[x±1 i ] ≤ GLn−1 (Rn , σi Rn ), the latter isomorphism gives also a natural embedding of
±1 ], σ Z[x ] as a subgroup of IA (Φn ). Actually: GLn−1 Z[x±1 i i i Proposition 3.6. For every 1 ≤ i ≤ n, there is a canonical surjective homomorphism:

ρi ±1 ±1 ±1 GLn−1 Z[x±1 i ], σi Z[xi ] ֒→ IA (Φn ) ։ GLn−1 Z[xi ], σi Z[xi ] such that the composition map is the identity. Hence:
±1 IA (Φn ) = ker ρi ⋊ GLn−1 Z[x±1 i ], σi Z[xi ] . Proof. Without loss of generality we
assume i = n. First, consider the homomorphism IA (Φn ) → GLn Z[x±1 n ] , which induced by the projection Rn → Z[x±1 ] which defined by x → 7 1 for every j 6= n. By Proposition 3.1, given j n Pn−1 In + A ∈ IA (Φn ), all the entries of the n-th column of A are in j=1 σj Rn .
Hence, the above map IA (Φn ) → GLn Z[x±1 n ] is actually a map: o n
IA (Φn ) → In + Ā ∈ GLn Z[xn±1 ] | the n-th column of Ā is ~0 . Observe now,
that the right side of the above map is mapped naturally to GLn−1 Z[x±1 n ] by erasing the n-th column and the n-th row from every element. Hence we obtain a map:
IA (Φn ) → GLn−1 Z[x±1 n ] . 11 Now, by Proposition 3.1, every entry of A such that In + A ∈ IA (Φn
), is in An . Thus, the entries of every Ā such that In−1 + Ā ∈ GLn−1 Z[x±1 n ] is an image of In + A ∈ IA (Φn ), are all in σn Z[x±1 ]. Hence, we actually obtain a n homomorphism:
±1 ρn : IA (Φn ) → GLn−1 Z[x±1 n ], σn Z[xn ] .
Observing that the copy of GLn−1 Z[xn±1 ], σn Z[x±1 n ] in IGLn−1,n is mapped isomorphically to itself by ρn , finishes the proof. Let have now the following propositions: ∼ Proposition 3.7. For 1 ≤ i ≤ n denote R̄ = Z[x±1 i ]. Then, viewing Im(ρi ) =
GLn−1 R̄, σi R̄ , for every m ∈ N one has:
Im(ρi ) ∩ IGn,m = GLn−1 R̄, σi ((xm i − 1)R̄ + mR̄) Proof. It is easy to see that the elements of IGL
n−1,i which correspond to the elements of GLn−1 R̄, σi ((xm i − 1)R̄ + mR̄) are clearly in Imρi ∩ IGn,m . On the other hand, without loss of generality we assume that i = n and let In + A ∈ Imρn ∩ IGn,m . Then In + A has the form:   Pn−1 In−1 + σn C − l=1 σl~cl ∈ IGLn−1,n 0 1 Pn−1 when the entries of C admit ck,l ∈ R̄ and j=1 σl ck,l ∈ Hn,m . So ck,l ∈ m Hn,m ∩ R̄ = (xn − 1)R̄ + mR̄, and the claim follows. Proposition 3.8. For 1 ≤ i ≤ n denote R̄ = Z[xi±1 ]. Then, for every m ∈ N one has:
ρi IGn,m2 ⊆ Im(ρi ) ∩ IGn,m ⊆ ρi (IGn,m ) Proof. As every element in Imρi ∩ IGn,m is mapped to itself via ρi we clearly have Imρi ∩ IGn,m ⊆ ρi (IGn,m
). On the other hand, if In + A ∈ IGn,m2 then viewing Imρi ∼ = GLn−1 R̄, σi R̄ , the entries of ρi (In + A) = In−1 +B belong to Pm−1 2 2 ⊆ (xm (xm −1) R̄+m σi R̄. Observe now that we have: r=0 xmr i i −1)R̄+mR̄, i so: 2 xm i − 1 = σi 2 m −1 X r=1 xri = σi m−1 X r=0 xri m−1 X r=0
xmr ∈ σi (xm i − 1)R̄ + mR̄ . i So by proposition 3.7 ρi (In + A) ∈ Imρi ∩ IGn,m as required. Proposition 3.9. For every m ∈ N and 1 ≤ i ≤ n one has: ρi (IAn,m ) = Im(ρi ) ∩ IAn,m . 12 Proof. As every element in the intersection Imρi ∩ IAn,m is mapped to itself via ρi we clearly have Imρi ∩IAn,m ⊆ ρi (IAn,m ). For the opposite, assume that α ∈ IAn,m , and denote ρi (α) = β ∈ Imρi . We want to show that β ∈ IAn,m . So let N ⊳ IA(Φn ) such that [IA(Φn ) : N ]|m. Then obviously [Imρi : (N ∩ Imρi )]|m. −1 Thus, as ρi is surjective [IA(Φn ) : ρ−1 i (N ∩ Imρi )]|m so α ∈ ρi (N ∩ Imρi ) and hence β = ρi (α) ∈ N ∩ Imρi ≤ N . As this is valid for every such N , we have: β ∈ IAn,m as required. We close this section with the following definition: Definition 3.10. For every 1 ≤ i ≤ n, denote: IGL′ n−1,i = {In + A ∈ IA (Φn ) | The i-th row of A is 0} . Obviously, IGLn−1,i ≤ IGL′ n−1,i , and by the same injectivity argument as in the proof of Proposition 3.4, one can deduce that: Proposition 3.11. The subgroup IGL′ n−1,i ≤ IA (Φn ) is canonically embedded in GLn−1 (R), by the map: In + A 7→ In−1 + Ai,i . Remark 3.12. Note that in general IGLn−1,i  IGL′ n−1,i . For example, I4 + σ3 E1,2 − σ2 E1,3 ∈ IGL′ 3,4 \ IGL3,4 . 4 The subgroups Ci In this section we define the subgroups Ci ≤ C (IA (Φn ) , Φn ), and we show that C (IA (Φn ) , Φn ) can be viewed as a semi-direct product of each one of them. We also show that when n ≥ 4: ±1 ])) \ \ (Z[x±1 ]) → SLn−1 (Z[x Ci ∼ = ker(SLn−1 and use it to show that C (IA (Φn ) , Φn ) is not finitely generated. Along the section we will simplify the notations and write IAm = IAn,m and IGm = IGn,m . It is proven in [Be1] that Φ̂n = lim ←−Φn,m . So, as for every m ∈ N ker(Φn → Φn,m ) is characteristic in Φn , we can write explicitly: \ = ker(IA (Φn ) → Aut(Φ̂n )) \ = ker(IA (Φn ) → lim ←−Aut (Φn,m )) \ = ker(IA (Φn ) → lim ←− (IA (Φn ) /IGm )) \ = ker(IA (Φn ) → lim (IA (Φn ) /IGm2 )). ←− Now, as for every n ≥ 4 we know that IA (Φn ) is finitely generated (see [BM]), \ we have IA (Φn ) = lim ←− (IA (Φn ) /IAm ). Hence: C(IA (Φn ) , Φn ) = ker(lim ←− (IA (Φn ) /IAm ) → lim ←− (IA (Φn ) /IGm )) C (IA (Φn ) , Φn ) = lim (IAm · IGm /IAm ) ←− 2 = lim ←− (IAm · IGm /IAm ) . 13 Observe now that for every 1 ≤ i ≤ n the composition map:

ρi ±1 ±1 ±1 GLn−1 Z[x±1 i ], σi Z[xi ] ֒→ IA (Φn ) ։ GLn−1 Z[xi ], σi Z[xi ] induce a similar composition map for the profinite completions:

±1 ±1 \ ±1 ±1 \ \ ρ̂i GLn−1 Z[x i ], σi Z[xi ] ֒→ IA (Φn ) ։ GLn−1 Z[xi ], σi Z[xi ] . \ This enables us to write: IA (Φn ) = ker ρi ⋊ Imρi and IA (Φn ) = ker ρ̂i ⋊ Imρ̂i . Define now: Definition 4.1. Ci = C (IA (Φn ) , Φn ) ∩ Imρ̂i = ker(Imρ̂i → Aut(Φ̂n )). Proposition 4.2. If 1 ≤ i 6= j ≤ n, then Ci ∩ Cj = {e}. Proof. By the above description for C(IA (Φn ) , Φn ), one can write: Ci = ker(Imρ̂i → Aut(Φ̂n )) = 2 ker(lim ←− (IAm · Imρi /IAm ) → lim ←− (IA (Φn ) /IGm )) 2 lim ← − ((IAm · Imρi ) ∩ (IAm · IGm )) /IAm . = We claim now that: (IAm · Imρi ) ∩ (IAm · IGm2 ) ⊆ IAm · (Imρi ∩ IGm ) . So we have to show that if ar = bs such that a, b ∈ IAm , r ∈ Imρi and s ∈ IGm2 , then there exist c ∈ IAm and t ∈ Imρi ∩ IGm such that
ar = bs = ct. Indeed, write: Imρi ∋ r = a−1 bs. Then: r = ρi (r) = ρi a−1 b ρi (s), and:
ρi a−1 b ∈ ρi (IAm ) = Imρi ∩ IAm ρi (s) ∈ ρi (IGm2 ) ⊆ Imρi ∩ IGm .
Therefore, by defining c = a · ρi a−1 b , and t = ρi (s) we get the required inclusion. Thus, we have: Ci ∩ Cj ⊆ lim ←− ((IAm · (Imρi ∩ IGm ) ∩ IAm · (Imρj ∩ IGm )) /IAm ) . We claim now that: IAm · (Imρi ∩ IGm ) ∩ IAm · (Imρj ∩ IGm ) = IAm . Indeed, if ar = bs such that a, b ∈ IAm , r ∈ Imρi ∩ IGm and s ∈ Imρj ∩ IGm , then:
ar = aρi (r) = aρi a−1 b ρi (s)
and we have aρi a−1 b ∈ IAm . I addition, it is not difficult to observe that: ρi (s) ∈ ρi (Imρj ∩ IGm ) ⊆ hIn + m(σi Ek,j − σj Ek,i ) | k 6= i, ji m = hIn + σi Ek,j − σj Ek,i | k 6= i, ji Hence, ar ∈ IAm , and Ci ∩ Cj = {e} as required. 14 ⊆ IAm . The proof of the above proposition shows that for every 1 ≤ i ≤ n and for every m ∈ N we have: (IAm · Imρi )∩(IAm · IGm2 ) ⊆ IAm ·(Imρi ∩ IGm ) ⊆ (IAm · Imρi )∩(IAm · IGm ) and by proposition 3.8 we have also ρi (IGm2 ) ⊆ Im(ρi ) ∩ IGn,m ⊆ ρi (IGm ). It ρ̂i follows that Ci ֒→ C(IA (Φn ) , Φn ) ։ Ci such that the composition map is the identity. Indeed: Ci = = = ֒→ ρ̂i ։ = = lim ← − ((IAm · Imρi ) ∩ (IAm · IGm )) /IAm lim ← −IAm · (Imρi ∩ IGm ) /IAm limIA · (Imρi ∩ IGm2 ) /IAm ←− m 2 lim ←−IAm · IGm /IAm = C(IA (Φn ) , Φn ) 2 lim ←−ρi (IAm ) · ρi (IGm ) /ρi (IAm ) lim ← − (Imρi ∩ IAm ) · (Imρi ∩ IGm ) / (Imρi ∩ IAm ) limIA · (Imρi ∩ IGm ) /IAm = Ci . ←− m Hence, one gets the following corollary: Corollary 4.3. For every 1 ≤ i ≤ n we have: C (IA (Φn ) , Φn ) = (ker ρ̂i ∩ C (IA (Φn ) , Φn )) ⋊ Ci . We would like now to compute Ci and to show that it is canonically isomorphic to: \ \ ±1 ])). ker(SLn−1 (Z[x±1 ]) → SLn−1 (Z[x So set n ≥ 4, 1 ≤ i0 ≤ n, denote: x = xi0 , σ = σi0 = xi0 − 1, R = Z[x±1 ] = m Z[x±1 i0 ] and for m ∈ N denote: Hm = (x − 1) R + mR. Denote also: ρ = ρ̂ = ρi0 : IA (Φn ) ։ GLn−1 (R, σR) \ \ (Φn ) ։ GLn−1 (R, σR). ρ̂i : IA 0 Now, write (the last equality is by proposition 3.7): Ci0 = = ker(Imρ̂ → Aut(Φ̂n )) \ ker(GLn−1 (R, σR) → Aut(Φ̂n )) = \ ker(GLn−1 (R, σR) → lim ←− (IA (Φn ) /IGm )) = \ ker(GLn−1 (R, σR) → lim ←− (GLn−1 (R, σR) · IGm /IGm )) = \ ker(GLn−1 (R, σR) → limGLn−1 (R, σR) / (GLn−1 (R, σR) ∩ IGm )) ←− \ ker(GLn−1 (R, σR) → lim ←−GLn−1 (R, σR) /GLn−1 (R, σHm )). = 15 Now, by the same computation as in Proposition 3.8 one can show that for every m ∈ N one has (Hm2 ∩ σR) ⊆ σHm ⊆ (Hm ∩ σR), so the latter is equals to: \ ker(GLn−1 (R, σR) → lim ←−GLn−1 (R, σR) / (GLn−1 (R, σR) ∩ GLn−1 (R, Hm ))) \ = ker(GLn−1 (R, σR) → lim ←−GLn−1 (R) /GLn−1 (R, Hm ) \ = ker(GLn−1 (R, σR) → limGLn−1 (R/Hm )). ←− Now, if R̄ is a finite quotient of R, then as x is invertible in R, its image x̄ ∈ R̄ is invertible in R̄. Thus, there exists r ∈ N such that x̄r = 1R̄ . In addition, there exists s ∈ N such that 1R̄ + . . . + 1R̄ = 0R̄ . Therefore, for m = r · s the map | {z } s R → R̄ factorizes through Zm [Zm ] ∼ lim (R/Hm ), = R/Hm . Thus, we have R̂ = ← − which implies that: GLn−1 (R̂) = ← lim GL (R/H ). Therefore: n−1 m − \ (R, σR) → GLn−1 (R̂)). Ci0 = ker(GLn−1 Now, the short exact sequence: 1 → GLn−1 (R, σR) → GLn−1 (R) → GLn−1 (Z) → 1 gives rise to the exact sequence (see [BER], Lemma 2.1): \ \ GLn−1 (R, σR) → GL\ n−1 (R) → GLn−1 (Z) → 1 which gives rise to the commutative diagram: \ \ GLn−1 (R, σR) → GL\ n−1 (R) → GLn−1 (Z) ց ↓ ↓ GLn−1 (R̂) → GLn−1 (Ẑ) → 1 → 1 . Assuming n ≥ 4 and using the affirmative answer to the classical congruence subgroup problem ([Men], [BaLS]), the map: GL\ n−1 (Z) → GLn−1 (Ẑ) is injec\ tive. Thus, by diagram chasing we obtain that the kernel ker(GLn−1 (R, σR) → \ GLn−1 (R̂)) is mapped onto ker(GLn−1 (R) → GLn−1 (R̂)). Let have now the following lemma: 
Lemma 4.4. Let d ≥ 3 and denote: Dm = Id + xk·m − 1 E1,1 | k ∈ Z for m ∈ N. Then: \ GL d (R) = lim (GLd (R) / (Dm Ed (R, Hm ))) ←− \ SL d (R) = lim ← − (SLd (R) /Ed (R, Hm )) . Proof. We will prove the first part and the second is similar but easier. We first claim that Dm Ed (R, Hm ) is a finite index normal subgroup of GLd (R): 16 By well-known result of Suslin [Su], SLd (R) = Ed (R). Thus, by Corollary 2.3, SK1 (R, Hm ; d) = SLd (R, Hm ) /Ed (R, Hm ) is finite. As the subgroup SLd (R, Hm ) is of finite index in SLd (R), so is Ed (R, Hm ). Now, it is not difficult elements of R is equals to R∗ =  k to see that the group of invertible  k·m ±x | k ∈ Z (see [CF], chapter 8). So as x | k ∈ Z is of finite index in R∗ , Dm SLd (R) is of finite index in GLd (R). We deduce that also Dm Ed (R, Hm ) is of finite index in GLd (R). It remains to showuthat Dm Ed (R, Hm ) is normal in GLd (R). We already stated previously (see 2) that Ed (R, Hm ) is normal in GLd (R). Thus, it is easy to see that it is enough to show that the commutators of the elements of Dm with any set of generators of GLd (R), are in Ed (R, Hm ). By the above mentioned result of Suslin and as R∗ = {±xr | r ∈ Z}, GLd (R) is generated by the elements of the forms: 1. Id + (±x − 1) E1,1 2. 3. Id + rEi,j Id + rE1,j r ∈ R, 2 ≤ i 6= j ≤ d r ∈ R, 2 ≤ j ≤ d 4. Id + rEi,1 r ∈ R, 2 ≤ i ≤ d. Now, obviously, the elements of Dm commute with the elements of the forms 1 and 2. In addition, for the elements of the forms 3 and 4, one can easily compute that: 

= Id + r xk·m − 1 E1,j ∈ Ed (R, Hm ) Id + xk·m − 1 E1,1 , Id + rE1,j 

Id + xk·m − 1 E1,1 , Id + rEi,1 = Id + r x−k·m − 1 Ei,1 ∈ Ed (R, Hm ) for every 2 ≤ i, j ≤ d, as required. Now, every finite index normal subgroup of GLd (R) contains Dm for some m ∈ N. In addition, It is not hard to show that when d ≥ 3, every finite index normal subgroup N ⊳ GLd (R) contains Ed (R, H) for some finite index ideal H ⊳ R (see [KN], Section 1). Thus, as every finite index ideal H ⊳ Rn contains \ Hm for some m, GL d (R) = lim (GLd (R) / (Dm Ed (R, Hm ))), as required. ←− Let have now the following proposition: \ Proposition 4.5. Let n ≥ 4. Then, the map GLn−1 (R, σR) → GL\ n−1 (R) is injective. Hence, the surjective map: \ (R, σR) → GLn−1 (R̂)) ։ ker(GL\ Ci0 = ker(GLn−1 n−1 (R) → GLn−1 (R̂)) is an isomorphism. Proof. We showed in the previous lemma that GL\ n−1 (R) = lim ←−GLn−1 (R) / (Dm En−1 (R, Hm )) 
where: Dm = In−1 + xk·m − 1 E1,1 | k ∈ Z and Hm = (xm − 1) R + mR. \ Hence, the image of GLn−1 (R, σR) in GL\ n−1 (R) is: lim ←−GLn−1 (R, σR) / (GLn−1 (R, σR) ∩ Dm En−1 (R, Hm )) 17 which by using that Dm ⊆ GLn−1 (R, σR), one can see that it is equals to: lim ←−GLn−1 (R, σR) / (Dm (GLn−1 (R, σR) ∩ En−1 (R, Hm ))) . Now, by the surjective map ρ : IA (Φn ) ։ GLn−1 (R, σR) we have: m hIA (Φn ) i IEn−1,i0 (Hn,m ) ρ m ։ hGLn−1 (R, σR) i ρ ։ GLn−1 (R, σR) ∩ En−1 (R, Hm ) .
m So as by the main Lemma (Lemma 7.1) we have IEn−1,i0 Hn,m2 ⊆ hIA (Φn ) i, we have also: m GLn−1 (R, σR) ∩ En−1 (R, Hm2 ) ⊆ hGLn−1 (R, σR) i . m As obviously Dm2 ⊆ hGLn−1 (R, σR) i, we deduce the following natural surjective maps: 2 2 lim ←−GLn−1 (R, σR) / (Dm (GLn−1 (R, σR) ∩ En−1 (R, Hm ))) m ։ lim ←−GLn−1 (R, σR) / hGLn−1 (R, σR) i \ ։ GLn−1 (R, σR) ։ limGLn−1 (R, σR) / (Dm (GLn−1 (R, σR) ∩ En−1 (R, Hm ))) ←− 2 2 = lim ←−GLn−1 (R, σR) / (Dm (GLn−1 (R, σR) ∩ En−1 (R, Hm ))) such that the composition gives the identity map. Hence, these maps are also injective, and in particular, the map: \ GLn−1 (R, σR) → lim ←−GLn−1 (R, σR) / (Dm (GLn−1 (R, σR) ∩ En−1 (R, Hm ))) is injective, as required. Proposition 4.6. Let d ≥ 3. Then, the natural embedding SLd (R) ≤ GLd (R) induce a natural isomorphism: \ \ ∼ ker(SL d (R) → SLd (R̂)) = ker(GLd (R) → GLd (R̂)). Proof. By Lemma 4.4 we have: \ ker(GL d (R) → GLd (R̂) = lim ←−GLd (R/Hm )) = ker(limGLd (R) /Dm Ed (R, Hm ) → limGLd (R) /GLd (R, Hm )) ←− ←− =← lim GL (R, H ) /D E (R, H ) . d m m d m − Notice now that under the map R → Zm [Zm ], every A ∈ GLd (R, Hm ) is mapped to Id and hence det (A) 7→ 1, which implies that det (A) = xk·m for some k ∈ Z (provided m > 2. When m = 2, det (A) = ±x2k , but this exception does 18 not influence the inverse limit below). Thus: GLd (R, Hm ) = Dm SLd (R, Hm ) (for every m > 2). Therefore, since Dm ∩ SLd (R, Hm ) = {Id }, we deduce that: \ ker(GL d (R) → GLd (R̂)) = = lim ← −Dm SLd (R, Hm ) /Dm Ed (R, Hm ) lim ←−SLd (R, Hm ) /Ed (R, Hm ) = \ lim ←− ker(SLd (R) → SLd (R̂)). The immediate corollary from Propositions 4.5 and 4.6 is: Corollary 4.7. For every n ≥ 4, Ci0 ∼ = ker(SL\ n−1 (R) → SLn−1 (R̂)). We close the section by showing that ker(SL\ n−1 (R) → SLn−1 (R̂)) is not finitely generated, using the techniques in [KN]. It is known that the group ring Z[x±1 ] = Z [Z] is Noetherian (see [I], [BrLS]). In addition, it is known that dim (Z) = 1 and thus dim (Z [Z]) = 2 (see [Sm]). Therefore, by Proposition 1.6 in [Su], as n − 1 ≥ 3, for every H ⊳ R, the canonical map: SK1 (R, H; n − 1) → SK1 (R, H) := lim −→ SK1 (R, H; d) d∈N is surjective. Hence, the canonical map (when H ⊳ R ranges over all finite index ideals of R): ker(SL\ n−1 (R) → SLn−1 (R̂)) = = lim ←− (SLn−1 (R, H) /En−1 (R, H)) lim ←−SK1 (R, H; n − 1) → lim ←−SK1 (R, H) is surjective, so it is enough to show that lim ←−SK1 (R, H) is not finitely generated. By a result of Bass (see [Bas], chapter 5, Corollary 9.3), for every H ⊳ K ⊳ R of finite index in R, the map: SK1 (R, H) → SK1 (R, K) is surjective. Hence, it is enough to show that for every l ∈ N there exists a finite index ideal H ⊳ R such that SK1 (R, H) is generated by at least l elements. Now, as SK1 (R) = 1 [Su], we obtain the following exact sequence for every H ⊳ R (see Theorem 6.2 in [Mi]): K2 (R) → K2 (R/H) → SK1 (R, H) → SK1 (R) = 1. In addition, by classical result of Quillen ([Q], [Ros] Theorem 5.3.30), we have:
K2 (R) = K2 Z[x±1 ] = K2 (Z) ⊕ K1 (Z) so by the classical facts K2 (Z) = K1 (Z) = {±1} (see [Mi] chapters 3 and 10) we deduce that K2 (R) is of order 4. Hence, it is enough to prove that for every l ∈ N there exists a finite index ideal H ⊳ R such that K2 (R/H) is generated by at least l elements. Following [KN], we state the following proposition (by the proof of Theorem 2.8 in [SD]): 19 Proposition 4.8. Let p be a prime, l ∈ N and denote by P ⊳ Z [y] the ideal
l which generated by p2 and y p . Then, for R̄ = Z [y] /P , K2 R̄ is an elementary abelian p-group of rank ≥ l. Observe now that for every l ≥ 0: pl+1 (y + 1) l = (y p + 1 + p · a (y))p = 1 mod P so y + 1 is invertible in R̄. Therefore we have a well defined sujective homomorphism R
→ R̄ which defined by sending x 7→ y + 1. In particular, H = ker R → R̄ is a finite index ideal of R which satisfies the above requirements. This shows that indeed ker(SL\ n−1 (R) → SLn−1 (R̂)) is not finitely generated, and C (IA (Φn ) , Φn ) is not finitely generated either. 5 The centrality of Ci In this section we will prove that for every n ≥ 4, the copies Ci lie in the \ center of IA (Φn ). Along the section we will assume that n ≥ 4 is constant, and will show it for i = n. Symmetrically, it will be valid for every i. So for ±1 simplifying we will denote: R = Rn = Z[x±1 1 , . . . , xn ] and Hm = Pnnotations m Hn,m = i=1 (xi − 1) Rn + mRn (note that in the previous section we used these notations for different objects, i.e. for Z[x±1 ] and (xm −1)Z[x±1 ]+mZ[x±1 ] respectively). Also here we will simplify the notations and write IAm = IAn,m and IGm = IGn,m . We saw in Section 4 that we can write: C(IA (Φn ) , Φn ) = ker(lim (IA (Φn ) /IAm ) → lim (IA (Φn ) /IGm2 )) ←− ←− 2 /IAm ) . = lim (IA · IG m m ←− Similarly, we can write: Cn = ≤ ≤ 2 lim ← − (IAm · (Imρn ∩ IGm ) /IAm ) 2 lim ←− (IAm · IGm /IAm ) \ lim ←− (IA (Φn ) /IAm ) = IA (Φn ). \ Hence, if we want to show that Cn lies in the center of IA (Φn ), it suffices to show that for every m ∈ N, the group IAm · (Imρn ∩ IGm2 ) /IAm lies in the center of IA (Φn ) /IAm .
Denote R̄ = Z[x±1 n ], and recall Imρn ∩ IGn,m2 ≃ GLn−1 R̄, σn Hm2 ∩ R̄) (see Proposition 3.7). We claim that by this isomorphism one has:

IAm · Imρn ∩ IGn,m2 /IAm = IAm · SLn−1 R̄, σn Hm2 ∩ R̄) /IAm . Indeed, if α ∈ Imρn ∩ IGm2 then det(α) ∈ 1 + σn Hm2 ∩ R̄. In particular, it is 2 s in 1 + Hm2 ∩ R̄ so it must be of the form xm for some s ∈ Z (see proposition n 3.1). Consider now the element
2 (In + σn E1,1 − σ1 E1,n )m s ∈ IAm ∩ GLn−1 R̄, σn Hm2 ∩ R̄) . 20 2 Then by replacing α with (In + σn E1,1 − σ
1 E1,n )−m s · α we deduce that Imρn ∩ IGn,m2 ⊆ IAm · SLn−1 R̄, σn Hm2 ∩ R̄) and we get the above equality. It \ follows that if we want to show that Cn lies in the center of IA (Φn ) it suffices
\ (Φn ). to show that IAm · SLn−1 R̄, σn Hm2 ∩ R̄) /IAm lies in the center of IA However, we are going to show even more. We will show that IAm · ISLn−1,n (σn Hm2 ) /IAm lies in the center of IA (Φn ) /IAm . Let Fn be the free group on f1 , . . . , fn . It is a classical result by Magnus ([MKS], Chapter 3, Theorem N4) that IA (Fn ) is generated by the automorphisms of the form: ( fr 7→ [ft , fs ] fr αr,s,t = fu 7→ fu u 6= r when 1 ≤ r, s 6= t ≤ n (notice that we may have r = s). In their paper [BM], Bachmuth and Mochizuky show that for every n ≥ 4, IA (Φn ) is generated by the images of these generators under the natural map Aut (Fn ) → Aut (Φn ). I.e. IA (Φn ) is generated by the elements of the form: Er,s,t = In + σt Er,s − σs Er,t 1 ≤ r, s 6= t ≤ n. Therefore, for showing the centrality of Cn , it is enough to show that given: • an element: λ̄ ∈ IAm · ISLn−1,n (σn Hm2 ) /IAm , • and one of generators: Er,s,t = In + σt Er,s − σs Er,t for 1 ≤ r, s 6= t ≤ n, there exists a representative of λ̄, λ ∈ ISLn−1,n (σn Hm2 ), such that [Er,s,t , λ] ∈ IAm . So assume that we have an element: λ̄ ∈ IAm · ISLn−1,n (σn Hm2 ) /IAm . Then, a representative for λ̄, λ ∈ ISLn−1,n (σn Hm2 ), has the form:   Pn−1 In−1 + σn B − i=1 σi~bi λ= 0 1 for some (n − 1) × (n − 1) matrix B which its entries bi,j admit bi,j ∈ Hm2 and its column vectors denoted by ~bi . Let have now the following proposition: Proposition 5.1. Let λ̄ ∈ IAm · ISLn−1,n (σn Hm2 ) /IAm . Then, for every 1 ≤ l < k ≤ n − 1, λ̄ has a representative in ISLn−1,n (σn Hm2 ), of the following form (the following notation means that the matrix is similar to the identity matrix, except the entries in the l-th and k-th rows):   Il−1 0 0 0 0 0  0 1 + σn a 0 σn b 0 −σl a − σk b    ← l-th row  0  0 I 0 0 0 k−l−1    0 σn c 0 1 + σn d 0 −σl c − σk d    ← k-th row  0  0 0 0 In−k−1 0 0 0 0 0 0 1 (5.1) 21 for some a, b, c, d ∈ Hm2 . Proof. We will demonstrate the proof in the case l = 1, k = 2, and symmetrically, the arguments hold for arbitrary 1 ≤ l < k ≤ n − 1. Consider an arbitrary representative of λ̄:  Pn−1 ~  In−1 + σn B − i=1 σi bi λ= ∈ ISLn−1,n (σn Hm2 ) . 0 1 Then In−1 + σn B ∈ SLn−1 (R, σn Hm2 ). Consider now the ideal: ′ R ⊲ Hm 2 = n−1 X 2 2 m 2 (xm r − 1)R + σn (xn − 1)R + m R. r=1 ′ ′ Observe that σn Hm2 ⊳ Hm 2 ⊳ H m2 ⊳ R and that Hm2 ∩ σn R = σn Hm2 . In addition, by similar computations as in the proof of proposition 3.8, for 4 2 every x ∈ R we have xm − 1 ∈ (x − 1) (xm − 1)R + (x − 1) m2 R, and thus ′ ′ H m 4 ⊆ Hm 2 , so Hm2 is of finite index in R.
′ Now, In−1 + σn B ∈ SLn−1 (R, σn Hm2 ) ⊆ SLn−1 R, Hm 2 . Thus, by the ′ third part of Corollary 2.3, as R ⊲ Hm 2 is an ideal of finite index, n − 1 ≥ 3 and En−1 (R) = SLn−1 (R) [Su], one can write the matrix In−1 + σn B as:   ′ A 0 In−1 + σn B = AD when A = 0 In−3

′ ′ Now, consider the and D ∈ E n−1 R, Hm for some A′ ∈ SL2 R, Hm 2 . 2 images of D and A under the projection σn → 0, which we denote by D̄ and Ā,
′ respectively. Observe that obviously, D̄ ∈ E n−1 R, Hm 2 . In addition, observe that: AD ∈ GLn−1 (R, σn R) ⇒ ĀD̄ = In−1 . Thus, we have In−1 + σn B = AĀ−1 D̄−1 D. Therefore, by replacing D by D̄−1 D and A by AĀ−1 we can assume that:   ′ A 0 In−1 + σn B = AD for A = 0 In−3
′ where: A′ ∈ SL2 R, Hm ∩ GL2 (R, σn R) = SL2 (R, σn Hm2 ), and: 2 D ∈ ⊆ ′ E n−1 (R, Hm 2 ) ∩ GLn−1 (R, σn R) E n−1 (R, H m2 ) ∩ GLn−1 (R, σn R) = IEn−1,n (Hm2 ) . Now, as we prove in the main lemma (Lemma 7.1) that IEn−1,n (Hm2 ) ⊆ m hIA (Φn ) i ⊆ IAm , this argument shows that λ can be replaced by a representative of the form 5.1. We now return to our initial mission. Let λ̄ ∈ IAm ·ISLn−1,n (σn Hm2 ) /IAm , and let Er,s,t = In + σt Er,s − σs Er,t for 1 ≤ r, s 6= t ≤ n be one of the above 22 generators for IA (Φn ). We want to show that there exists a representative of λ̄, λ ∈ ISLn−1,n (σn Hm2 ), such that [Er,s,t , λ] ∈ IAm . We separate the treatment to two cases: The first case is: 1 ≤ r ≤ n − 1. In this case one can take an arbitrary representative λ ∈ ISLn−1,n (σn Hm2 ) ∼ = SLn−1 (R, σn Hm2 ). Considering the embedding of IGL′n−1,n in GLn−1 (R), we have: Er,s,t ∈ IGL′n−1,n ⊆ GLn−1 (R). Thus, as by Corollary 2.3 SK1 (R, Hm2 ; n − 1) = SLn−1 (R, Hm2 ) /En−1 (R, Hm2 ) is central in GLn−1 (R) /En−1 (R, Hm2 ), we have: [Er,s,t , λ] ∈ [GLn−1 (R) , SLn−1 (R, σn Hm2 )] ⊆ En−1 (R, Hm2 ) . In addition, as SLn−1 (R, σn Hm2 ) ≤ GLn−1 (R, σn R) and GLn−1 (R, σn R) is normal in GLn−1 (R), we have: [Er,s,t , λ] ∈ [GLn−1 (R) , GLn−1 (R, σn R)] ⊆ GLn−1 (R, σn R) . Thus, we obtain from Lemma 7.1 that: [Er,s,t , λ] ∈ En−1 (R, Hm2 ) ∩ GLn−1 (R, σn R) = IEn−1,n (Hm2 ) ⊆ hIA (Φn )m i ⊆ IAm . The second case is: r = n. This case is a bit more complicated than the previous one, as Er,s,t is not in IGL′n−1,n . Here, by Proposition 5.1 one can choose λ ∈ ISLn−1,n (σn Hm2 ) which its t-th row is equals to the standard vector ~et . As t 6= r = n, we obtain thus that both λ and Er,s,t are in the canonical embedding of IGL′n−1,t in GLn−1 (R). Moreover, considering the above mentioned embedding of IGL′n−1,t in GLn−1 (R), we have: Er,s,t ∈ GLn−1 (R, σt R) , λ ∈ SLn−1 (R, Hm2 ) . Note that when considering λ ∈ IGL′n−1,n ֒→ GLn−1 (R), i.e. when considering λ ∈ GLn−1 (R) through the embedding of IGL′n−1,n in GLn−1 (R), we have λ ∈ GLn−1 (R, σn Hm2 ) ≤ GLn−1 (R). However, when we consider λ ∈ IGL′n−1,t ֒→ GLn−1 (R) we do not necessarily have: λ ∈ GLn−1 (R, σn Hm2 ), but we still have λ ∈ GLn−1 (R, Hm2 ). Thus, by similar arguments as in the first case: [Er,s,t , λ] ∈ [GLn−1 (R, σt R) , SLn−1 (R, Hm2 )] ⊆ En−1 (R, Hm2 ) ∩ GLn−1 (R, σt R) = IEn−1,t (Hm2 ) ⊆ hIA (Φn )m i ⊆ IAm . \ This finishes the argument which shows that Ci are central in IA (Φn ). 23 6 Some elementary elements of hIA (Φn )m i m In this section we compute some elements in hIA (Φn ) i, which needed through the proof of the main lemma. Additionally to the previous notations, on the section, and also later on, we will use the notation: σr,m = xm r − 1 for 1 ≤ r ≤ n. Proposition 6.1. Let n ≥ 4, 1 ≤ u ≤ n and m ∈ N. Denote by ~ei the i-th row standard vector. Then, the elements of IA (Φn ) of the form (the following notation means that the matrix is similar to the identity matrix, except the entries in the u-th row):   Iu−1 0 0  au,1 · · · au,u−1 1 au,u+1 · · · au,n  ← u-th row 0 0 In−u when (au,1 , . . . , au,u−1 , 0, au,u+1 , . . . , au,n ) is a linear combination of the vectors: 1. {m (σi~ej − σj ~ei ) | i, j 6= u, i 6= j} 2. {σ k,m (σi~ej − σj ~ei ) | i, j, k 6= u, i 6= j} 3. {σu σu,m (σi~ej − σj ~ei ) | i, j 6= u, i 6= j} m with coefficients in Rn , belong to hIA (Φn ) i. Proof. Without loss of generality, we assume that u = 1. Observe now that for every ai , bi ∈ Rn for 2 ≤ i ≤ n one has:      1 a2 · · · an 1 b2 · · · bn 1 a2 + b 2 · · · an + b n = . 0 In−1 0 In−1 0 In−1 Hence, it is enough to prove that the elements of the following forms belong to m hIA (Φn ) i (when we write a~ei we mean that the entry of the i-th column in the first row is a):   1 mf (σi~ej − σj ~ei ) i, j 6= 1, i 6= j, f ∈ Rn 1. 0 In−1   1 σ k,m f (σi~ej − σj ~ei ) i, j, k 6= 1, i 6= j, f ∈ Rn 2. 0 In−1   1 σ1 σ1,m f (σi~ej − σj ~ei ) i, j 6= 1, i 6= j, f ∈ Rn 3. 0 In−1 We start with the elements of form 1. Here we have: m    1 f (σi~ej − σj ~ei ) 1 mf (σi~ej − σj ~ei ) m ∈ hIA (Φn ) i . = 0 In−1 0 In−1 24 We pass to the elements of form 2. In this case we have: " −1  m # 1 f (σi~ej − σj ~ei ) xk −σ1~ek m hIA (Φn ) i ∋ , 0 In−1 0 In−1   1 σ k,m f (σi~ej − σj ~ei ) . = 0 In−1 We finish with the elements of form 3. The computation here is more complicated than in the previous cases, so we will demonstrate it for the private case: n = 4, i = 2, j = 3. It is clear that symmetrically, with similar argument, the same holds in general when n ≥ 4 for every i, j 6= 1, i 6= j. By similar arguments as in the previous case we get:   1 0 0 0  0 1 0 0  m . hIA (Φ4 ) i ∋   0 0 1 0  0 σ3 σ1,m f −σ2 σ1,m f 1 Therefore, we also have:  x4 0 0 −σ1  0 1 0 0 m  hIA (Φ4 ) i ∋   0 0 1 0 0 0 0 1  1 −σ3 σ1 σ1,m f  0 1 =   0 0 0 0 7   1   0 ,   0 0 0 1 0 σ3 σ1,m f  σ2 σ1 σ1,m f 0 0 0  . 1 0  0 1 0 0 1 −σ2 σ1,m f  0  0   0  1 A main lemma In this section we prove the main lemma which asserts that: Lemma 7.1. For every n ≥ 4, m ∈ N and 1 ≤ i ≤ n we have:
m IEn−1,i Hn,m2 ⊆ hIA (Φn ) i . For simplifying the proof and the notations, we will prove the lemma for the private case i = n, and symmetrically, all the arguments are valid for every 1 ≤ i ≤ n. Also in this section n will be constant, so we Pncan use the notations: m IAm = hIA (Φn ) i, R = Rn and Hm = Hn,m = r=1 Ur,m + Om where: Ur,m = σr,m R and Om = mR. In addition, using the isomorphism IGLn−1,n ∼ = GLn−1 (R, σn R) (Proposition 3.4), we will identify IGLn−1,n with GLn−1 (R, σn R), and IEn−1,n (Hm ) 25 with GLn−1 (R, σn R)∩E n−1 (R, Hm ). So the goal of this section is proving that GLn−1 (R, σn R) ∩ En−1 (R, Hm2 ) ⊆ IAm . Through the proof we use also elements of IGL′ n−1,n , so as GLn−1 (R, σn R)∩ En−1 (R, Hm2 ) ≤ GLn−1 (R, σn R) ≤ IGL′ n−1,n ≤ GLn−1 (R) (Proposition 3.11), in this section, we will make all the computations in GLn−1 (R), and omit the n-th row and column from each matrix. 7.1 Decomposing the proof In this subsection we will start the proof of Lemma 7.1. In the end of the subsection, there will be remained a few tasks, which will be accomplished in the forthcoming subsections. We start with the following definition: Definition 7.2. For every m ∈ N, define the following ideal of R: Tm = n X σr2 Ur,m + r=1 n X 2 σr Om + Om . r=1 By the observation that for every x ∈ R we have one can have the following computation: m2 x −1 = (x − 1) 2 m −1 X xj = (x − 1) m−1 X j=0 j=0 ∈ ⊆ Pm−1 j=0 xj xj ∈ (x − 1)R + mR, m−1 X xjm j=0 (x − 1) ((x − 1)R + mR) ((xm − 1)R + mR) (x − 1)2 (xm − 1)R + (x − 1)2 mR + (x − 1)m2 R which shows that Hm2 ⊆ Tm . Hence, it is enough to prove that: GLn−1 (R, σn R) ∩ En−1 (R, Tm ) ⊆ IAm . Equivalently, it is enough to prove that the group: (GLn−1 (R, σn R) ∩ En−1 (R, Tm )) · IAm /IAm is trivial. We continue with the following lemma, which is actually a lemma of Suslin (Corollary 1.4 in [Su]) with some elaborations of Bachmuth and Mochizuki [BM]: Lemma 7.3. Let R be a commutative ring, d ≥ 3, and H ⊳ R ideal. Then, Ed (R, H) is generated by the matrices of the form: (Id − f Ei,j ) (Id + hEj,i ) (Id + f Ei,j ) (7.1) for h ∈ H, f ∈ R and 1 ≤ i 6= j ≤ d. Proof. By Suslin’s proof to Corollary 1.4 in [Su], and the remark which follows Proposition 3.5 in [BM], Ed (R, H) is generated by the matrices of the form: t Id + h (f1~ei + f2~ej ) (f2~ei − f1~ej ) 26 for f1 , f2 ∈ R, h ∈ H and 1 ≤ i 6= j ≤ d. So it is enough to show that these matrices are generated by the matrices of the form 7.1. We will show it for the case i, j, d = 1, 2, 3 and it will be clear that the general argument is similar. So we have the matrix:   1 + hf1 f2 −hf12 0 Id + h (f1~e1 + f2~e2 )t (f2~e1 − f1~e2 ) =  hf22 1 − hf1 f2 0  0 0 1 for some f1 , f2 ∈ R and h ∈ H,  1 0 1 = 0 f2 −f1   1 0 0 1 0 1 0  0 1 = 0 f2 −f1 1 0 0 As the matrix  1  0 0 which is equals to:   −hf1 1 0 hf1 −hf2   0 1 hf2  1 −f2 f1 1   −hf1 1 0 0 1 0 −hf2   0 1 0  0 1 1 −f2 f1 1 0 0   0 hf1 1 0 1 hf2  =  0 1 0 1 0 0 is generated by the matrices of the   1 0 0 1  0 1 0  0 f2 −f1 1 0 =  hf1 hf2  . 1  0 hf2  1  hf1 1 0 0  0 1 0 0 1 form 7.1, it remains to show that:   0 −hf1 1 0 0 1 −hf2   0 1 0  0 1 −f2 f1 1     1 0 0 1 0 −hf1 1 0 0  0 1 0  0 1 0  0 1 0  f2 −f1 1 0 0 1 −f2 f1 1     1 0 0 1 0 0 1 0 0 1 0   0 1 −hf2   0 1 0  · 0 0 0 1 f2 −f1 1 −f2 f1 1 is generated by the matrices of the   1 0 0 1  0 1 0  0 f2 −f1 1 0  1 + hf1 f2 0 = hf1 f22 form 7.1. Now:  0 −hf1 1 1 0  0 0 1 −f2 −hf12 1 −hf12 f2 27 0 1 f1  −hf1  0 1 − hf1 f2  0 0  1  1 = 0 0  −hf12 1 −hf12 f2 1 0 1 =  0 0 −hf12 f2  1 0 0 ·  0 1 0 f2 0 1  0 1 + hf1 f2 0 0  hf1 f22 1  0 0  1  1  0 0 is generated by the matrices of the   1 0 0 1  0 1 0  0 f2 −f1 1 0 1 −hf12 0 1 0 0 0 1 0 0 1 0  −hf1  0 1 − hf1 f2  0 0  1  1 −hf1 0  0 1 −f2  0 1 0  0 1 form 7.1, and by similar computation:   0 0 1 0 0 1 −hf2   0 1 0  0 1 −f2 f1 1 is generated by these matrices as well. We proceed with the followimg lemma: Lemma 7.4. Let n ≥ 4. Then, every element of GLn−1 (R, σn R)∩En−1 (R, Tm ) can be decomposed as a product of elements of the following four forms: 1. A−1 (In−1 + hEi,j ) A h ∈ σn Om 2. A−1 (In−1 + hEi,j ) A h ∈ σn2 Un,m , σn σr2 Ur,m for 1 ≤ r ≤ n − 1 3. A−1 [(In−1 + hEi,j ) , (In−1 + f Ej,i )] A 2 h ∈ Ōm , f ∈ σn R 4. A−1 [(In−1 + hEi,j ) , (In−1 + f Ej,i )] A h ∈ σr2 Ūr,m , σr Ōm for 1 ≤ r ≤ n − 1, f ∈ σn R when A ∈ GLn−1 (R), i 6= j, and Ūr,m , Ōm are the projections of Ur,m , Om ±1 to Rn−1 = Z[x±1 1 , . . . , xn−1 ], induced by the projection σn 7→ 0. Remark 7.5. Notice that as GLn−1 (R, σn R) is normal in GLn−1 (R), every element of the above forms is an element of GLn−1 (R, σn R) ∼ = IGLn−1,n ≤ IA (Φn ). Proof. (of Lemma 7.4) Let B ∈ GLn−1 (R, σn R) ∩ En−1 (R, Tm ). We first claim that for proving the lemma, it is enough to show that B can be decomposed as a product of the elements in the lemma (Lemma 7.4), and arbitrary elements in GLn−1 (Rn−1 ). Indeed, assume that we can write B = C1 D1 · · · Cn Dn for some Di of the forms in the lemma and Ci ∈ GLn−1 (Rn−1 ) (notice that C1 or Dn might be equal to In−1 ). Observe now that we can therefore write: B = C1 D1 C1−1 · . . . · (C1 · . . . · Cn ) Dn (C1 · . . . · Cn )−1 (C1 · . . . · Cn ) 28 and by definition, the conjugations of the Di -s can also be considered as of the forms in the lemma. On the other hand, we have: −1 (C1 · . . . · Cn ) Dn−1 (C1 · . . . · Cn ) · . . . · C1 D1−1 C1−1 B = C1 · . . . · Cn and as the matrices of the forms in the lemma are all in GLn−1 (R, σn R) (by Remark 7.5) we deduce that: C1 · . . . · Cn ∈ GLn−1 (R, σn R) ∩ GLn−1 (Rn−1 ) = {In−1 } i.e. C1 · . . . · Cn = In−1 . Hence: −1 B = C1 D1 C1−1 · . . . · (C1 · . . . · Cn ) Dn (C1 · . . . · Cn ) i.e. B is a product of matrices of the forms in the lemma, as required. So let B ∈ GLn−1 (R, σn R) ∩ En−1 (R, Tm ). According to Lemma 7.3, as B ∈ En−1 (R, Tm ) and n − 1 ≥ 3, we can write B as a product of elements of the form: (In−1 − f Ei,j ) (In−1 + hEj,i ) (In−1 + f Ei,j ) Pn Pn 2 2 for some f ∈ R, h ∈ Tm = r=1 σr Ur,m + r=1 σr Om + Om and 1 ≤ i 6= j ≤ n − 1. We will show now that every element of the above form can be writen as a product of the elements of the forms in the lemma and elements of GLn−1 (Rn−1 ). Observe first that as: Tm = n X σr2 Ur,m + r=1 ⊆ σn n X 2 σr Om + Om r=1 n−1 X σr2 Ur,m + σn Un,m + Om r=1 ! + n−1 X σr2 Ūr,m + r=1 n−1 X 2 σr Ōm + Ōm r=1 Pn−1 we can decompose h = σn h1 + h2 for some: h1 ∈ r=1 σr2 Ur,m + σn Un,m + Om Pn−1 Pn−1 2 . Therefore, we can write: and h2 ∈ r=1 σr2 Ūr,m + r=1 σr Ōm + Ōm (In−1 − f Ei,j ) (In−1 + hEj,i ) (In−1 + f Ei,j ) = (In−1 − f Ei,j ) (In−1 + σn h1 Ej,i ) (In−1 + f Ei,j ) · (In−1 − f Ei,j ) (In−1 + h2 Ej,i ) (In−1 + f Ei,j ) . Thus, as the matrix (In−1 − f Ei,j ) (In−1 + σn h1 Ej,i ) (In−1 + f Ei,j ) is clearly a product of elements of forms 1 and 2 in the lemma, it is enough to deal with the matrix: (In−1 − f Ei,j ) (In−1 + h2 Ej,i ) (In−1 + f Ei,j ) Pn−1 Pn−1 2 2 . Let us now write: f = σn f1 + f2 σr Ōm + Ōm when h2 ∈ r=1 σr Ūr,m + r=1 for some f1 ∈ R and f2 ∈ Rn−1 , and write: (In−1 − f Ei,j ) (In−1 + h2 Ej,i ) (In−1 + f Ei,j ) = (In−1 − f2 Ei,j ) (In−1 − σn f1 Ei,j ) · (In−1 + h2 Ej,i ) (In−1 + σn f1 Ei,j ) (In−1 + f2 Ei,j ) . 29 Now, as (In−1 ± f2 Ei,j ) ∈ GLn−1 (Rn−1 ), it is enough to deal with the element: (In−1 − σn f1 Ei,j ) (In−1 + h2 Ej,i ) (In−1 + σn f1 Ei,j ) which can be writen as a product of elements of the form: (In−1 − σn f1 Ei,j ) (In−1 + kEj,i ) (In−1 + σn f1 Ei,j ) 2 k ∈ Ōm , σr2 Ūr,m , σr Ōm , for 1 ≤ r ≤ n − 1. Finally, as for every such k one can write: (In−1 − σn f1 Ei,j ) (In−1 + kEj,i ) (In−1 + σn f1 Ei,j ) = (In−1 + kEj,i ) [(In−1 − kEj,i ) , (In−1 − σn f1 Ei,j )] and (In−1 + kEj,i ) ∈ GLn−1 (Rn−1 ), we actually finished. Corollary 7.6. For proving Lemma 7.1, it is enough to show that every element in the forms of Lemma 7.4 is in IAm . We start to deal here with the elements of form 1: Proposition 7.7. The elements of the following form are in IAm : A−1 (In−1 + hEi,j ) A, for A ∈ GLn−1 (R) , h ∈ σn Om and i 6= j. Proof. In this case we can write h = σn mh′ for some h′ ∈ R. So as: A−1 (In−1 + σn h′ Ei,j ) A ∈ GLn−1 (R, σn R) ≤ IA (Φn ) we obtain that: A−1 (In−1 + hEi,j ) A = = as required. A−1 (In−1 + σn mh′ Ei,j ) A =
m A−1 (In−1 + σn h′ Ei,j ) A ∈ IAm We will devote the remaining sections to deal with the elements of the three other forms. In these cases the proof will be much more difficult, and we will need the help of the following computations. 7.2 Some needed computations Proposition 7.8. For every f, g ∈ R we have  1 − f g −f g  fg 1 + fg 0 0 30 the following equalities:  0 0  1  1 0 0 =  fg 1 0 f g2 0 1  1 −f g 1 · 0 0 −f g 2   1 0 0   0 1 + fg −f  2 0 fg 1 − fg   0 1 − fg 0 f 1 0  0 1 0  0 1 0 1 −f g 2 0 1 + f g 0 0  1 0 0  fg 1 0  −f g 2 0 1   1 −f g 0 1 0  · 0 0 f g2 1 =  1  0 f  1 · 0 0 = =  1 0 0 0 1 + fg −f g 2 1 − fg 0 f g2  0 1 − fg 0 0  0 1 1 −f 0  0 0 1 + fg f g2   −f 1 − fg 0 1 f   1 0 0 1 − fg  0 1 0  0 −f −f 1 f   1 0 0 ·  0 1 + f g −f g 2   0 f 1 − fg 0 1 0  0  f 1 − fg  −f 1  0 0 1 + fg 0  f g2 1 0   fg 1 0 1 + fg 0 0  2 1 −f g −f g 0 1 0  0 0 1  0 −f g 2 1   fg 1 0 0 1 + fg 0  2 1 −f g f g 0 1 0  0 0 1 Proof. Here is the computation for Equation 7.2:     1 − fg −f g 0 1 0 f  fg 1 + f g 0  =  0 1 −f   0 0 1 g g 1    1 0 0 1 0 f 1 0 0 1 0 =  0 1 0   0 1 −f   0 g g 1 0 0 1 −g −g 1   −f f  1 (7.3)  0 f 1 −f  0 1  0 −f g 2  1 (7.4)  0 0 1 f g2  0 1 (7.5)  1 0 −f 0 1 f  −g −g 1   1 0 −f  0 1 f  0 0 1    1 0 0 1 0 0 1 0 0 1 0  =  0 1 0   0 1 −f   0 g g 1 0 0 1 −g −g 1     1 0 0 1 0 f 1 0 0 1 0 1 0  0 1 · 0 1 0  0 1 0  0 g g 1 0 0 1 −g −g 1 0 0 31 (7.2)  −f f  1  1 =  fg f g2 =  0 1 − fg −f   0 1 − fg −f g 2 0 1 + fg f g2  1 0 0  fg 1 0 f g2 0 1  1 −f g 1 · 0 0 −f g 2 −f g 1 −f g 2  f 1  0 0 1 + fg 0  0 −f 1 f  0 1   1 0 0   0 1 + fg −f  2 0 fg 1 − fg   0 1 − fg 0 f 1  0 0  0 1 0 1 −f g 2 0 1 + f g 0  0 −f 1 f . 0 1 Equation 7.3 is obtained Similarly by changing the signs of f and g simultaneously. Here is the computation for Equation 7.4:      1 − fg −f g 0 1 0 g 1 0 −g  fg 1 + f g 0  =  0 1 −g   0 1 g  0 0 1 f f 1 −f −f 1      1 0 0 1 0 g 1 0 0 1 0 −g 1 0  0 1 g  =  0 1 0   0 1 −g   0 f f 1 0 0 1 −f −f 1 0 0 1  =  1  0 f  1 · 0 0 1 = 0 f =   0 1 0 0  0 1 1 0 0  g 1 −g   0 1 0  1 − fg   fg −f 0 1 f 0 1 f 1  0 f  1 · 0 0 0 1 0 0 0 1  g 1 0 −g   0 1 1 −f 0  0 0 1 0 1 0  0 1 −f 1 0 0   0 f g2 1 1 −f g 2   0 0 1 + fg 0  0 1 − fg 0 0  0 1 1 −f 0  0 0 1 + fg f g2   −f 1 − fg 0 1 f  0 1 0  0 1 0  −g g  1 −f g 1 + fg −f  f g2 1 0   fg 1 0 1 + fg 0 0  2 1 −f g −f g 0 1 0  0 0 1 0 1 0  −g g  1  −f g 2 f g2  1 − fg  0 −f g 2  1 and Equation 7.5 is obtained similarly by changing the signs of f and g simultaneously. In the corollary, a 3 × 3 matrix B ∈ GL3 (R) denotes the blocks  following  B 0 ∈ GLn−1 (R). matrix 0 In−4 32 Corollary 7.9. Let n ≥ 4, f ∈ σn P n−1 r=1 σr Ur,m + Un,m + Om  and g ∈ R. Then, mod IAm we have the following equalities (the indices are intended to help us later to recognize forms of matrices: form 7, form 12 etc.):   1 − f g −f g 0  fg 1 + fg 0  (7.6) 0 0 1 13     1 0 0 1 − fg 0 f  −f   0 1 0 ≡  0 1 + fg 2 2 0 fg 1 − fg 1 −f g 0 1 + fg 2     1 0 0 1 − fg 0 −f    f 0 1 0 ≡  0 1 + fg 0 −f g 2 1 − f g 3 f g2 0 1 + fg 4     1 − fg 0 f g2 1 0 0   0 1 + fg 0 1 0 f g2  ≡ −f 0 1 + fg 5 0 −f 1 − fg 6     2 1 − f g 0 −f g 1 0 0   0 1 + f g −f g 2  0 1 0 ≡ f 0 1 + fg 7 0 f 1 − fg 8 and:   1 − fg 0 ≡ f  fg 0 1 + fg 0  0 1 14   2 −f g 1 0   0 1 + fg 0 1 + fg 7 0 −f 1 − fg  −f g 0 0 1 0 (7.7)  0 f g2  . 1 − fg 6 Moreover (the inverse of a matrix is denoted by the same index - one can observe that the inverse of each matrix in these equations is obtained by changing the sign of f ):   1 − f g 0 −f g   0 1 0 (7.8) fg 0 1 + f g 15     1 0 0 1 − fg f 0 f g 2   −f g 2 1 + f g 1  ≡  0 1 − fg 0 −f 1 + fg 8 0 0 0 9     1 0 0 1 − fg −f 0 1 + fg 0  ≡  0 1 − f g −f g 2   f g 2 0 f 1 + fg 6 0 0 1 10     1 − fg f g2 0 1 0 0 1 + fg 0   0 1 − fg −f  ≡  −f 2 0 0 1 11 0 fg 1 + fg 3 33  and: 1 − fg f ≡ 0  and: 1 − fg f ≡ 0   0 1 0   0 1 12 0   0 fg  1 0 0 1 + f g 16   0 1 0 0   0 1 − fg 1 12 0 f g2 0 1 − fg −f g 2 1 − fg  0 −f g −f g 2 1 + fg 0  0  f 1 + fg 1 (7.9)  0 −f  1 + fg 3   and: −f g 2 1 + fg 0 1 − fg 0 ≡ −f  1 0 0  0 1 − fg −f g  0 fg 1 + f g 17   0 f g2 1 + f g −f g 2   1 0 f 1 − fg 0 1 + fg 5 0 0 (7.10)  0 1  0 11   1 + fg ≡  −f g 2 0  1 0 0  0 1 − fg fg  0 −f g 1 + f g 18   f 0 1 − fg 0 1 − fg 0   0 1 0 1 10 f g2 0 (7.11)  −f  0 1 + fg 4 Remark 7.10. We remark that as f ∈ σn R, then every matrix which takes part in the above epualities is indeed in GLn−1 (R, σn R) ∼ = IGLn−1,n ≤ IA (Φn ).  P n−1 Proof. As f ∈ σn r=1 σr Ur,m + Un,m + Om , Equation 7.6 is obtained by applying Proposition 7.8 combined with Proposition 6.1. Equation 7.7 is obtained similarly by transposing all the computations which led to the first part of Equation 7.6. Similarly, by switching the roles of the second row and column with the third row and column, one obtains Equations 7.8 and 7.9. By switching one more time the roles of the first row and column with the second row and column, we obtain Equations 7.10 and 7.11 as well. 7.3 Elements of form 2 Proposition 7.11. The elements of the following form, belong to IAm : A−1 (In−1 + hEi,j ) A where A ∈ GLn−1 (R), h ∈ σn σr2 Ur,m , σn2 Un,m for 1 ≤ r ≤ n − 1 and i 6= j. 34 

Notice that for every n ≥ 4, the groups En−1 σn2 Un,m and En−1 σn σr2 Ur,m for 1 ≤ r ≤ n − 1 are normal in GLn−1 (R), and thus, all the above ele

ments are in En−1 σn2 Un,m and En−1 σn σr2 Ur,m . Hence, for proving Proposition 7.11, it
is enough to show
that for every 1 ≤ r ≤ n − 1 we have, En−1 σn2 Un,m , En−1 σn σr2 Ur,m ⊆ IAm . Therefore, by Lemma 7.3, for proving Proposition 7.11, it is enough to show that the elements of the following form are in IAm : (In−1 − f Ej,i ) (In−1 + hEi,j ) (In−1 + f Ej,i ) when h ∈ σn σr2 Ur,m , σn2 Un,m for 1 ≤ r ≤ n − 1, f ∈ R and i 6= j. We will prove it in a few stages, starting with the following lemma. Lemma 7.12. Let h ∈ σn σr Ur,m , σn Un,m for 1 ≤ r ≤ n − 1 and f1 , f2 ∈ R. Assume that the elements of the forms: (In−1 ± f1 Ej,i ) (In−1 + hEi,j ) (In−1 ∓ f1 Ej,i ) (In−1 ± f2 Ej,i ) (In−1 + hEi,j ) (In−1 ∓ f2 Ej,i ) for every 1 ≤ i 6= j ≤ n − 1, belong to IAm . Then, the elements of the form: (In−1 ± (f1 + f2 ) Ej,i ) (In−1 + hEi,j ) (In−1 ∓ (f1 + f2 ) Ej,i ) for 1 ≤ i 6= j ≤ n − 1, also belong to IAm . Proof. Observe first that by Proposition 6.1, all the matrices of the form In−1 + hEi,j for h ∈ σn σr Ur,m , σn Un,m belong to IAm . We will use it in the following computations. Without loss of generality, we will show that for i, j = 2, 1 we have: (In−1 − (f1 + f2 ) E1,2 ) (In−1 + hE2,1 ) (In−1 + (f1 + f2 ) E1,2 ) ∈ IAm and the general argument is similar. In the following computation we use the following notations:   B 0 ∈ GLn−1 (R) is denoted by B ∈ GL3 (R). - A matrix 0 In−4 - “=” denotes an equality between matrices in GLn−1 (R). - “≡” denotes an equality in IA (Φn ) /IAm . (In−1 − (f1 + f2 ) E1,2 ) (In−1 + hE2,1 ) (In−1 + (f1 + f2 ) E1,2 )   2 1 − h (f1 + f2 ) −h (f1 + f2 ) 0 = h 1 + h (f1 + f2 ) 0  0 0 1    1 0 − (f1 + f2 ) 1 0 (f1 + f2 )  0  1 1 1 −1 = 0 −h −h (f1 + f2 ) 1 h h (f1 + f2 ) 1 35  1 =  0 −h  1 ·  0 0  0 0  −h (f1 + f2 ) 1  0 − (f1 + f2 ) 1  0 1 1 0 1 h 0 1 0 1 h (f1 + f2 )  0 1 0  0 1 0  0 (f1 + f2 )  . 1 −1 0 1 As the product of the elements in the square brackets is a conjugation of an element of GLn−1 (R, σn R) it is also an element of GLn−1 (R, σn R) ∼ = IGLn−1,n ≤ IA (Φn ). In addition, by the remark in the beginning of the proof, the remained element is in IAm . So mod IAm , the above expresion is equals to:     1 0 − (f1 + f2 ) 1 0 0 1 0 f1 + f2  0 1  0 1 −1  1 0  0 1 0 0 1 0 0 1 h h (f1 + f2 ) 1 = =     1 0 0 1 0 −f1 1 0 −f2  0 1 1  0 1 0  0 1 0  0 0 1 0 0 1 0 hf2 1     1 0 0 1 0 f1 1 0 f2 · 0 1 0   0 1 −1   0 1 0  h hf1 1 0 0 1 0 0 1   1 0 −f2 1  0 1 0  0 0 0 1 0   1 1 0 f1 ·  0 1 −1   0 h 0 0 1   1 0 −f1 1 1  0 · 0 1 0 0 1 h  −f1 1 0 0 1  0 1 0 1 0 hf2 1  0 0 1 0 1 0  0 1 hf1 1 −h −hf1  0 0 1 0 f1 1 0   0 1 −1 hf1 1 0 0 1 0 1 0   1 1 0 −f1 1 0 −f2 1  0 0  0 1 =  0 1 0 0 1 0 0 1 0   2 1 0 0 1 − hf1 −hf1 1 0  h 1 + hf1 · 0 h hf1 1 0 0  36    0 0  1  1  0 0  0 0 1 0  hf2 1  0 1 0 0  0 1 0 0 1  0 f2 1 0  0 1  1 0 f1 0 1 −1  0 0 1  f2 0  1  1 1 0 −f2 0  0 =  0 1 0 0 0 1   1 0 0 1 1 0  0 · 0 0 h hf1 1    1 0 −f2 1 =  0 1 0  0 0 0 1 0   1 0 0 1 1 0  0 · 0 h hf1 1 0    1 0 0 0 −f1 1 0 f1 1 1  0 1 0   0 1 −1  0 1 0 0 1 0 hf2 1   2 1 − hf1 −hf1 −hf1 f2 0 f2 h 1 + hf1 hf2  1 0  0 1 0 0 1  0 −f1 1 1 1  0 0 1 0  0 f2 1 1 0  0 0 1 0  0 1 0  0 1 0  −hf1 f2 hf2   1 0 1 hf2 0 1 0  f1 −1  1 0 1 0 1 − hf1 h 0 −hf12 1 + hf1 0 Notice now that by assumption, the right matrix in the above exspression belongs to IAm , and the nearer matrix also belong to IAm by the remark in the beginning of the proof. So, mod IAm , the latter expression is equals to:     1 0 −f2 1 0 −f1 1 0 0  0 1 0  0 1 1  0 1 0  0 0 1 0 0 1 0 hf2 1     1 0 f1 1 0 0 1 0 f2 1 0  0 1 0  ·  0 1 −1   0 0 0 1 h hf1 1 0 0 1     1 0 −f2 1 0 −f1 1 0 0 1 0  0 1 1  0 1 0  0 =  0 1 0 0 1 0 0 1 0 hf2 1 0     1 0 f2 1 0 −f2 1 0 0 1 0  0 1 0  0 · 0 1 0  0 1 0 0 1 0 0 1 h hf1 1 0 = =   1 0 − (f2 + f1 ) 1 0  0 1  0 1 1 0 0 1 0 hf2   1 − hf2 −hf1 f2 −hf22  0 1 0 · h hf1 1 + hf2   0 1 0  0 1 0 0 1 0 0 1 0  f1 −1  1  f2 0  1  0 f2 + f1 1 −1  0 1   1 0 0 1 −hf2 (f1 + f2 ) hf2 (f1 + f2 )  0 1 + hf2  −hf2   0 1 0 0 0 1 0 hf2 1 − hf2    1 −hf1 f2 0 1 − hf2 0 −hf22  1 0  0 1 0 · 0 0 hf1 1 h 0 1 + hf2 37  0 0 . 1  1 0 ≡  0 1 + hf2 0 hf2  0 −hf2  . 1 − hf2 Consider now the last part of Equation 7.6 in Corollary 7.9, and switch the roles of f, g by h, f2 respectively. Then, by switching the roles of the first row and column with the third row and column in the equation, and transposing the expresion, one deduces that:   1 0 0  0 1 + hf2 −hf2  0 hf2 1 − hf2    1 − hf2 −hf22 0 1 + hf2 0 −hf22  ≡ In−1 h 1 + hf2 0   0 1 0 ≡ 0 0 1 h 0 1 − hf2 and this finishes the proof of the lemma. We pass to the next stage: Proposition 7.13. The elements of the following form belong to IAm : (In−1 − f Ej,i ) (In−1 + hEi,j ) (In−1 + f Ej,i ) where h ∈ σn σr2 Ur,m , σn2 Un,m for 1 ≤ r ≤ n − 1, f ∈ Z and i 6= j. Proof. According to Lemma 7.12, it is enough to prove the proposition for f = ±1. Without loss of generality, we will prove the proposition for r = 1, i.e. h ∈ σn σ12 U1,m , and symmetrically, the same is valid for every 1 ≤ r ≤ n − 1. The case h ∈ σn2 Un,m will be considered separately. So let h ∈ σn σ12 U1,m and write: h = σ1 u for some u ∈ σn σ1 U1,m . We will prove the proposition for i 6= j ∈ {1, 2, 3} - as one can see below, we will do it simultaneously for all the options for i 6= j ∈ {1, 2, 3}. The treatment in the other cases in which i 6= j ∈ {1, k, l} such that 1 < k 6= l ≤ n − 1 is obtained symmetrically, so we get that the proposition is validfor every 1 ≤  i 6= j ≤ n−1. B 0 ∈ GLn−1 (R) As before, we denote a given matrix of the form 0 In−4 by B ∈ GL3 (R). Let have now the following computations (the indices of the matrices below intended to recognize the appropriate form of matrix in Corollary 7.9, as will be explained below. We remind that the inverse of a matrix is denoted by the same index, and one can observe that the inverse of each indexed matrix is obtained by changing the sign of u). We remind that u ∈ σn σ1 U1,m ⊆ σn R. Thus, by Proposition 6.1 we have:      1 − σ1 u −σ12 u 0 x2 −σ1 0 1 0 0  u 1 + σ1 u 0  1 0   ux2 1 0  =  0 0 0 1 12 0 0 1 0 0 1  −1  −1 x2 x2 σ1 0 · 0 1 0  ∈ IAm 0 0 1 38  1 − σ1 u  0 u  0 −σ12 u  1 0 0 1 + σ1 u 7   0 0  1 + σ1 u u 2 −σ1 u 1 − σ1 u 3   0 −σ12 u  1 + σ1 u 6 1  0 0 1 0  0 1 − σ1 u 0 u   1 0 0 x3 0 −σ1 0  0 1 0  =  0 1 0 0 1 ux3 0 1  −1  −1 x3 0 x3 σ1  ∈ IAm · 0 1 0 0 0 1    0 0 1 0  0 1  1  0 −σ2 1 0 σ2   0 0 1 0  0 1  1   −σ3 0 1 σ3 0 1 =  uσ2 −uσ1 σ2  1 0 0 · 0 1 u 0 0 1 = 1  −uσ1 σ3 uσ3  1 0 0 · 0 1 0 0 u 1  0 0 1 0  −σ1 1  0 0 1 0  ∈ IAm σ1 1 0 1 0  0 −σ1  1  0 0 1 σ1  ∈ IAm . 0 1 By switching the signs of σ1 , σ2 and σ3 in the two latter computations we obtain also that:     1 0 0 1 0 0  0 1 − σ1 u  ,  0 1 + σ1 u −σ12 u  ∈ IAm . u 2 0 −σ1 u 1 + σ1 u 1 0 u 1 − σ1 u 8 Consider now the identities which we got in Corollary 7.9, and switch the roles of f, g in the corollary by u,σ1 , respectively. Remember now that u ∈ σn σ1 U1,m . Hence, as by the computations above the matrices which correspond to forms 7 and 8 belong to IAm , we obtain from the last part of Equation 7.6 that also form 13 belong to IAm . Thus, as we showed that forms 1, 3, 6 also belong to IAm , Equation 7.6 shows that also forms 2, 4, 5 belong to IAm . Similar arguments show that Equations 7.6-7.11 give that all the 18 forms belong to IAm . In particular, the matrices which correspond to forms 13 − 18 belong to IAm , and these matrices (and their inverses) are precisely the matrices of the form (we remind that h = σ1 u): (In−1 ± Ej,i ) (In−1 + hEi,j ) (In−1 ∓ Ej,i ) , i 6= j ∈ {1, 2, 3} . Clearly, by similar arguments, the proposition holds for every 1 ≤ i 6= j ≤ n − 1 and every h ∈ σn σr2 Ur,m for 1 ≤ r ≤ n − 1. The case h ∈ σn2 Un,m is a bit different, but easer. In this case one can consider the same computations we built for r = 1, with the following fittings: Firstly, write h ∈ σn2 Un,m as h = σn u 39 for some u ∈ σn Un,m . Secondly, change σ1 to σn , change σ2 , σ3 to 0 and change x2 , x3 to 1 in the right side of the above equations. It is easy to see that in this situation we obtain in the left side of the equations the same matrices, just that instead of σ1 we will have σn . From here we continue exactly the same. Proposition 7.14. The elements of the following form belong to IAm : (In−1 − f Ej,i ) (In−1 + hEi,j ) (In−1 + f Ej,i ) where h ∈ i 6= j. σn2 Un,m , σn σr2 Ur,m for 1 ≤ r ≤ n − 1, f ∈ σs R for 1 ≤ s ≤ n and Proof. We will use again the result of Corollary 7.9, when we switch the roles of f, g in the corollary by h, σs u respectively for some u ∈ R and 1 ≤ s ≤ n. Similarly to proposition 7.13, we  will prove it for s = 1, i 6= j ∈ {1, 2, 3}, and B 0 ∈ GLn−1 (R) by B ∈ GL3 (R). denote a given matrix of the form 0 In−4 As h ∈ σn σr2 Ur,m , σn2 Un,m , we have also σ1 uh ∈ σn σr2 Ur,m , σn2 Un,m . Hence, we obtain from the previous proposition, that the matrices of the forms 13 − 18 belong to IAm . In addition:      1 0 0 1 0 0 1 0 0  0 1 − uσ1 h  =  −huσ2 h 1 0  0 1 0  0 −u2 σ12 h 1 + uσ1 h 1 −hu2 σ1 σ2 0 1 −uσ2 uσ1 1    1 0 0 1 0 0 1 0  ∈ IAm · 0 1 h  0 0 0 1 uσ2 −uσ1 1   1 0 0  0 1 − uσ1 h −u2 σ12 h  0 h 1 + uσ1 h 6 =  1 0  −hu2 σ1 σ3 1 huσ3 0   1 0 0 · 0 1 0  0 h 1  0 1 0   uσ3 1 0 1 −uσ3 0 0 1 0  0 0 1 −uσ1  0 1  0 uσ1  ∈ IAm 1 and by switching the signs of u and h simultaneously, we get also forms 3 and 8. So we easily conclude from Corollary 7.9 (Equations 7.6 and 7.8) that also the matrices of the other eight forms are in IAm . In particular, the matrices of the form: (In−1 − σ1 uEj,i ) (In−1 + hEi,j ) (In−1 + σ1 uEj,i ) , i 6= j ∈ {1, 2, 3} belong to IAm . The treatment for every i 6= j and 1 ≤ s ≤ n − 1 is similar, and the treatment in the case s = n is obtained by replacing σ1 by σn and σ2 , σ3 by 0 in the above equations. Pn Corollary 7.15. As every f ∈ R can be decomposed as f = s=1 σs fs + f0 for some f0 ∈ Z and fi ∈ R, we obtain from Lemma 7.12 and from the above two propositions that we actually finised the proof of Proposition 7.11. 40 7.4 Elements of form 3 Proposition 7.16. The elements of the following form, belong to IAm : A−1 [(In−1 + hEi,j ) , (In−1 + f Ej,i )] A 2 where A ∈ GLn−1 (R), f ∈ σn R, h ∈ Ōm and i 6= j, when Ōm is the projection ±1 ±1 of Om to Rn−1 = Z[x1 , . . . , xn−1 ] which induced by the projection σn 7→ 0. We will prove the proposition in the case i, j = 2, 1, and the same arguments are valid for arbitrary i 6= j. In this case one can write: h = m2 h′ for some h′ ∈ Rn−1 , and thus, our element is of the form:    1 − f m2 h ′ f 0 1 −f 0
2 0 A A−1  −f m2 h′ 1 + f m2 h ′ 0  0 1 0 0 In−3 0 0 In−3 for some A ∈ GLn−1 (R), f ∈ σn R and h′ ∈ Rn−1 . The proposition will follow easily from the following lemma: Lemma 7.17.  Let h1 , h2 ∈ R, f ∈ σn R and denote a given matrix  B 0 ∈ GLn−1 (R) by B ∈ GL3 (R). Then: 0 In−4   1 − f m (h1 + h2 ) f 0 1 −f 2 A−1  −f (m (h1 + h2 )) 1 + f m (h1 + h2 ) 0   0 1 0 0 0 0 1   1 − f mh1 f 0 ≡ A−1  −f (mh1 )2 1 + f mh1 0   0 0 1   1 − f mh2 f 0 1 ·  −f (mh2 )2 1 + f mh2 0   0 0 0 0 1 1 −f 0 1 0 0 −f 1 0  0 0  1  of the form  0 0 A 1 0 0  A mod IAm 1 Now, if one proves the lemma, he can deduce that:    1 − f m2 h ′ f 0 1 −f 0
2 0 A A−1  −f m2 h′ 1 + f m2 h ′ 0  0 1 0 0 In−3 0 0 In−3      m 1 − f mh′ f 0 1 −f 0 ≡ A−1  −f (mh′ )2 1 + f mh′ 0  A mod IAm 0  0 1 0 0 In−3 0 0 In−3 and as the latter element is obviously belong to IAm we obtain that:    1 −f 0 1 − fh f 0 0  A ∈ IAm 0  0 1 A−1  −f h2 1 + f h 0 0 In−3 0 0 In−3 as required. So it is enough to prove Lemma 7.17. 41 Proof. (of Lemma 7.17) Notice through the computation the reminder (Remark 7.5) that as GLn−1 (R, σn R) is normal in GLn−1 (R) every conjugation of an element of GLn−1 (R, σn R) ≤ IA (Φn ) by an element of GLn−1 (R), belongs to GLn−1 (R, σn R) ≤ IA (Φn ). We remark also that through the computation, we will use freely the result of Proposition 7.7 and the three notations which we used in the proof  of Lemma7.12: B 0 ∈ GLn−1 (R) is denoted by B ∈ GL3 (R). - A matrix 0 In−4 - “=” denotes an equality between matrices in GLn−1 (R). - “≡” denotes an equality in IA (Φn ) /IAm .    1 − f m (h1 + h2 ) f 0 1 −f 0 2 A−1  −f (m (h1 + h2 )) 1 + f m (h1 + h2 ) 0   0 1 0  A 0 0 1 0 0 1 =  1 0 −f 0 1 −f m (h1 + h2 ) A−1  −m (h1 + h2 ) 1 1   1 0 f 0 1 f m (h1 + h2 )   · m (h1 + h2 ) −1 1   1 0 0 0 1 0  = A−1  −m (h1 + h2 ) 1 1   1 0 0 1 · 0 1 0  0 m (h1 + h2 ) −1 1 0 1 0 0 0 1 0   1 −f 0 1 0 0 −f −f m (h1 + h2 ) 1  0 f 1 f m (h1 + h2 )   0 1  0 0 A 1   1 −f 0 1 0 0  0 0 A 1     1 0 0 1 0 0 1 0 −f 0 1 0  0 1 0   0 1 −f mh1  = A−1  −mh2 0 1 0 0 1 −mh1 1 1     1 0 0 1 0 f 1 0 −f 1 0 1 0   0 1 f mh1   0 1 −f mh1   0 1 · 0 mh1 −1 1 0 0 1 0 0 1 mh2 0    1 0 0 1 0 0 1 0 0 1 0   0 1 −f mh2   0 1 ·A−1  −m (h1 + h2 ) 1 1 0 0 1 m (h1 + h2 ) −1    m   1 0 0 1 −f f · A−1  0 1 f (h1 + h2 )  A A−1  0 1 0  A 0 0 1 0 0 1 42  0 0 A 1  0 0 A 1    1 − f mh1 f 0 1 0 0 0 1 0   −f (mh1 )2 1 + f mh1 0  ≡ A−1  −mh2 0 1 0 0 1    1 0 −f 1 0 0 1 0 A ·  0 1 −f mh1   0 0 0 1 mh2 0 1      m 1 0 0 1 0 0 1 0 0 · A−1  0 1 0   0 1 −f h2   0 1 0  A −m (h1 + h2 ) 1 1 0 0 1 m (h1 + h2 ) −1 1   1 −f f ·A−1  0 1 0  A 0 0 1  1 0 ≡ A−1  −mh2  1 0 0 1 0 · 0 mh2 0 1 =   1 − f mh1 0 0 1 0   −f (mh1 )2 0 1 0   1 −f f  0 1 0 A 0 0 1 1 0 A−1  −mh2  1 0 0 1 0 · 0 mh2 0 1  1 0 0 1 0 · 0 mh2 0 1  1 0 0 1 0 · 0 mh2 0 1 f 1 + f mh1 0  0 1 0   0 1 0 0 0 1   1 − f mh1 f 0 0 0 1 0   −f (mh1 )2 1 + f mh1 0  0 1 0 0 1    1 0 0 1 0 −f  0 1 0  0 1 0  −mh2 0 1 0 0 1    1 0 0 1 0 0  0 1 0   0 1 −f mh1  −mh2 0 1 0 0 1   1 −f f  0 1 0 A 0 0 1 43  −f −f mh1  1     m  1 − f mh1 f 0 1 0 0 0 1 0  A = A−1  −f (mh1 )2 1 + f mh1 0  A · A−1  f mh1 h2 −f h2 1 0 0 1     1 0 0 1 0 −f 1 0 0 0 1 0  0 1 0  0 1 0 A ·A−1  −mh2 0 1 0 0 1 mh2 0 1      m 1 0 0 1 0 0 1 0 0 · A−1  0 1 0   0 1 −f h1   0 1 0  A −mh2 0 1 0 0 1 mh2 0 1   1 −f f ·A−1  0 1 0  A 0 0 1 ≡  A−1   1 · 0 0   1 − f mh1 f 0 1 0 0 0 1 0  −f (mh1 )2 1 + f mh1 0   −mh 0 0 1 2 0 1    0 −f 1 0 0 1 −f f 1 0  0 1 0  0 1 0 A 0 1 mh2 0 1 0 0 1  1 − f mh1 f = A−1  −f (mh1 )2 1 + f mh1 0 0   1 f 0 1 0 ·A−1  0 1 0   0 0 1 −mh2   1 0 0 1 0 1 0  0 ·A−1  −mh2 0 1 0  1 − f mh1 = A−1  −f (mh1 )2 0   1 0 · A−1  0 1 0 f h2  1 − f mh2 −1  0 ·A 2 f (mh2 )  0 1 −f 0 0  0 1 0 0 0 1 1  0 0 1 −f 1 0  0 1 0 1 0 0  f −f 1 1 0  0 0 1 mh2 f 1 + f mh1 0  m 0 0  A 1 f 1 −f mh2 44  A  0 1 0  0 1 mh2  0 0 1 1 0  0 0 1 0  0 1 −f 0  0 1 0 0 1  −f 1  0 0 0 1 + f mh2  0 0 1 0 A 0 1  −f f 1 0 A 0 1  0 0 A 1 −f 1 0  f 0 A 1    1 − f mh1 f 0 1 −f 0 A−1  −f (mh1 )2 1 + f mh1 0   0 1 0  A 0 0 1 0 0 1      m 1 f 0 1 0 0 1 0  A ·A−1  0 1 0  A · A−1  0 0 0 1 0 −f h2 1    1 − f mh2 0 −f 1 −f f  0 1 0 A 0 1 0 ·A−1  2 0 0 1 f (mh2 ) 0 1 + f mh2 ≡     1 − f mh1 f 0 1 −f 0 1 f 0 ≡ A−1  −f (mh1 )2 1 + f mh1 0   0 1 0   0 1 0  0 0 1 0 0 1 0 0 1     1 − f mh2 0 −f 1 −f 0 1 0 f  0 1 0  0 1 0 A 0 1 0 · 2 0 0 1 0 0 1 f (mh2 ) 0 1 + f mh2 = ≡   1 − f mh1 f 0 A−1  −f (mh1 )2 1 + f mh1 0   0 0 1   2 1 f mh2 0 1 0 A ·A−1  0 2 2 0 −f (mh2 ) 1   1 − f mh2 0 −f  0 1 0 ·A−1  2 f (mh2 ) 0 1 + f mh2 1 0 0 −f 1 0 1 0 0 1 0 0  0 0 A 1  f 0 A 1    1 − f mh1 f 0 1 −f 0 A−1  −f (mh1 )2 1 + f mh1 0   0 1 0  0 0 1 0 0 1    1 − f mh2 0 −f 1 0 f   0 1 0  A. 0 1 0 · 2 0 0 1 f (mh2 ) 0 1 + f mh2 Now, by similar computation as for Equation 7.5, we have (switch the roles of f, g in the equation by f, mh2 respectively, and then switch the roles of the first row and column with the third row and column):     1 0 0 1 −f −f  0 1 + f mh2 f mh2  =  0 1 0  0 −f mh2 1 − f mh2 0 0 1 45   f 1   f (mh2 )2 0 0 1 − f mh2  f 0 1 0 1 + f mh2 0   2 0 1 f (mh2 ) 1 + f mh2 0 · −f (mh2 )2  1 − f mh2 ·  −f (mh2 )2 0 0 1 0 Therefore, using Proposition 7.7, we have:    1 1 0 0 f mh2  A ≡ A−1  0 A−1  0 1 + f mh2 0 0 −f mh2 1 − f mh2  1 + f mh2 0 · 2 −f (mh2 ) Now, observe that:  1 − f mh2 0 f   −f (mh2 )2 1 0 0 0 1 − f mh2   1 A−1  0 0 0 1 + f mh2 −f mh2  1 0 = A−1  0 1 + f h2 0 −f h2  0 0 1 f mh2  0 1  0 0 1 0 . −f mh2 1 −f 1 0 f 1 + f mh2 0  −f 0  1  0 0  A. 1  0 f mh2  A 1 − f mh2  m 0 f h2  A ∈ IAm . 1 − f h2 Hence we obtain that mod IAm we have:    1 + f mh2 0 f 1 −f −f  0 1 0 0  A−1  0 1 2 0 0 1 −f (mh2 ) 0 1 − f mh2   1 − f mh2 f 0 ·  −f (mh2 )2 1 + f mh2 0  A ≡ In−1 . 0 0 1 But as the inverse of every matrix in the above equation is obtained by replacing f by −f , we have:    1 −f 0 1 0 −f A−1  0 1 0   0 1 0  A 0 0 1 0 0 1  1 + f mh2 2 −1  ≡A f (mh2 ) 0 −f 1 − f mh2 0  0 1 − f mh2   0 0 1 f (mh2 )2 46  0 −f A 1 0 0 1 + f mh2 and thus:  1 − f mh2 0 −f 1 0 f  0 1 0 0 1 0 A−1  2 0 0 1 f (mh2 ) 0 1 + f mh2   1 − f mh2 f 0 1 −f ≡ A−1  −f (mh2 )2 1 + f mh2 0   0 1 0 0 0 0 1   A  0 0  A. 1 This finishes the proof of the lemma, and hence, also the proof of Proposition 7.16. 7.5 Elements of form 4 Proposition 7.18. The elements of the following form, belong to IAm : A−1 [(In−1 + hEi,j ) , (In−1 + f Ej,i )] A where A ∈ GLn−1 (R), f ∈ σn R, h ∈ σr2 Ūr,m , σr Ōm for 1 ≤ r ≤ n− 1 and i 6= j, ±1 when Ūr,m , Ōm are the projections of Ur,m , Om to Rn−1 = Z[x±1 1 , . . . , xn−1 ] which induced by the projection σn 7→ 0.  through the subsection we denote a given matrix of the form  As before, B 0 ∈ GLn−1 (R) by B ∈ GL3 (R). We start the proof of this propo0 In−4 sition with the following computation. Lemma 7.19. Let f, g ∈ R and A ∈ GLn−1 (R). Then:   1 − fg −f g 0 1 + fg 0  A A−1  f g 0 0 1     1 0 0 1 0 0 −f   = A−1  f g 1 0  AA−1  0 1 + f g 2 2 fg 0 1 0 fg 1 − fg      1 0 0 1 −f g 0 1 0  AA−1  ·A−1  0 1 −f  AA−1  0 2 0 0 1 0 −f g 1    1 − fg 0 f 1 0 0  AA−1  f 2 g 2 1 −f 2 g 0 1 0 ·A−1  −f g 2 0 1 + f g 0 0 1  0 f A 1  1 0 0 0 1 f A 0 0 1    1 0 −f  AA−1  0 1 0  A. 0 0 1 1 0 0 1 0 0 Proof. The lemma follows from the following computations, which based on Proposition 7.8, Equation 7.2:   1 − f g −f g 0  fg 1 + fg 0  0 0 1 47  1 =  fg f g2  1 · 0 0  1 · 0 0  0 0 1 0 1 0   0 1 + fg 0 1 0 f g2  0 0 1 −f g 1 −f   0 1 0 1 0 −f g 2  0 0 1 − fg 0 1 −f   0 1 0 1 −f g 2 0  1 1 0 0 =  fg 1 0   0 0 f g2 0 1   1 0 0 1 ·  0 1 −f   0 0 0 1 0  1 − fg 0 f 0 1 0 · −f g 2 0 1 + f g  0 −f 1 − fg  0 0  1  1  0 0  0 0 1 f  0 1  1 0 0 0 1 f  0 0 1   f 1 0 0 1 0  0 1 f  0 1 0 1 + fg 0 0 1 0 0 0 1 + fg f g2 −f g 1 −f g 2  0 −f 1 − fg  0 0  1 1   f 2g2 0 0 1 0  1  0 0 1 0 0 1 0 0  −f 0  1  0 0 1 f  0 1  0 f  1  0 1 0 −f 2 g   0 1 1 0 0  −f 0 . 1 Observe now that if we have f ∈ σn R and instead of g we have h ∈ σr2 Ūr,m , σr Ōm for 1 ≤ r ≤ n − 1, then by Propositions 7.7 and 7.11, we have:   1 − f h −f h 0 1 + fh 0  A A−1  f h 0 0 1     1 0 0 1 −f h 0 1 0 0 1 0   1 1 0  A ∈ IAm . = A−1  −1 1 0   0 0 0 1 0 0 1 0 0 1 Therfore, by Propositions 7.7, 7.11 and the previous GLn−1 (R) we have the following equation mod IAm :   1 0 0 1 0 −f   0 1 A−1  0 1 + f h 0 f h2 1 − fh 0 0  1 − fh 0 ≡ A−1  −f h2  0 f 1 0  0 1 1 0 0 1 + fh 0 0 lemma, for every A ∈  0 f A 1 −1 −f 0  A. 1 I.e. we have the following corollary (notice the switching of the sign of f ): 48 Corollary 7.20. For every h ∈ σr2 Ūr,m , σr Ōm for 1 ≤ r ≤ n − 1, f ∈ σn R and A ∈ GLn−1 (R) , the following is equals mod IAm : A−1 [(In−1 + hE3,2 ) , (In−1 + f E2,3 )] A ≡ A−1 [(In−1 − f E1,3 ) , (In−1 + hE3,1 )] A. Let have now the following proposition: Proposition 7.21. Let h ∈ σ12 Ū1,m , σ1 Ōm and f ∈ σn R. Then: [(In−1 + hE3,2 ) , (In−1 + f E2,3 )] ∈ IAm . Proof. Denote h = σ1 u  1 0 1 IAm ∋  0 −σ2 u σ1 u = for some u ∈ σ1 Ū1,m , Ōm . Then:    0 1 0 0 1 0 0 1 0 0  0 1 f  0 1 0  0 1 1 0 0 1 σ2 u −σ1 u 1 0 0  0 −f  1   1 0 0  f σ2 u 1 0  2 σ1 σ2 u f 0 1   1 0 0 1 1 0  0 · 0 0 σ1 u 1 0 and thus:  0 0 1 1 f  0 0 1 0  1 [(In−1 + hE3,2 ) , (In−1 + f E2,3 )] =  0 0 as required. 0 1 h  0 0 1 1 0  0 −σ1 u 1 0   0 1 0 , 0 1 0  0 0 1 −f  0 1  0 0 1 f  ∈ IAm 0 1 We can now pass to the following proposition. Proposition 7.22. Let h ∈ σ12 Ū1,m , σ1 Ōm , f ∈ σn R and A ∈ GLn−1 (R). Then: A−1 [(In−1 + hE3,2 ) , (In−1 + f E2,3 )] A ∈ IAm . Proof. We will prove the proposition by induction. By result of Suslin [Su], as n − 1 ≥ 3, SLn−1 (R) is generated by the elementary matrices of the form: In−1 + rEl,k for r ∈ R, and 1 ≤ l 6= k ≤ n − 1. Qn So as the invertible elements of R are the elements of the form ± i=1 xsi i for si ∈ Z (see [CF], chapter 8), GLn−1 (R) is generated by the elementary matrices and the matrices of the form: In−1 + (±xi − 1) E1,1 for 1 ≤ i ≤ n. Therefore, it is enough to show that if: A−1 [(In−1 + hE3,2 ) , (In−1 + f E2,3 )] A ∈ IAm 49 and E is one of the above generators, then mod IAm we have: A−1 E −1 [(In−1 + hE3,2 ) , (In−1 + f E2,3 )] EA −1 ≡A (7.12) [(In−1 + hE3,2 ) , (In−1 + f E2,3 )] A. So if E is of the form In−1 + (±xi − 1) E1,1 , we obviously have property 7.12. If E is an elementary matrix of the form In−1 + rEl,k such that l, k ∈ / {2, 3} then we also have property 7.12 in an obvious way. Consider now the case l, k = 2, 3. In this case, by Corollary 7.20 we have:     1 0 0 1 0 0 A−1 E −1  0 1 0  ,  0 1 f  EA 0 h 1 0 0 1         1 0 0 1 0 −f 1 0 0 1 0 0 ≡ A−1  0 1 −r   0 1 0  ,  0 1 0   0 1 r  A 0 0 1 0 0 1 h 0 1 0 0 1     2 2 2 1 0 0 1 − hf + h f 0 −hf 1 0 0  0 1 r A 0 1 0 = A−1  0 1 −r   2 0 0 1 −h f 0 1 + hf 0 0 1     2 2 2 1 − hf + h f 0 −hf 1 0 0  AA−1  rh2 f 1 −rhf  A 0 1 0 = A−1  2 −h f 0 1 + hf 0 0 1       1 0 −f 1 0 0 1 0 0 = A−1  0 1 0  ,  0 1 0  AA−1  rh2 f 1 −rhf  A. 0 0 1 h 0 1 0 0 1 So by applying Propositions 7.7, 7.11 and Corollary 7.20 once again on the opposite way, we obtain that this element belongs to IAm by the induction hypothesis. The other cases for l, k are treated by similar arguments: if l, k = 3, 2 we do exactly the same, and if l or k are different from 2 and 3, then the situation is easier - we use similar arguments, but without passing to [(In−1 − f E1,3 ) , (In−1 + hE3,1 )] through Corollary 7.20. Corollary 7.23. Let h ∈ σ12 Ū1,m , σ1 Ōm , f ∈ σn R and A ∈ GLn−1 (R). Then, for every i 6= j we have: A−1 [(In−1 + hEi,j ) , (In−1 + f Ej,i )] A ∈ IAm . Proof. Denote a permutation matrix, which its action on GLn−1 (R) by conjugation, moves 2 7→ j and 3 7→ i, by P . Then: A−1 [(In−1 + hEi,j ) , (In−1 + f Ej,i )] A = A−1 P −1 [(In−1 + hE3,2 ) , (In−1 + f E2,3 )] P A ∈ IAm . Now, as one can see that symmetrically, the above corollary is valid for every h ∈ σr2 Ūr,m , σr Ōm for 1 ≤ r ≤ n − 1, we actually finished the proof of Proposition 7.18. 50 References [A] M. Asada, The faithfulness of the monodromy representations associated with certain families of algebraic curves, J. Pure Appl. Algebra 159 (2001), 123–147. [Bac] S. Bachmuth, Automorphisms of free metabelian groups, Trans. Amer. Math. Soc. 118 (1965) 93-104. [Bas] H. Bass, Algebraic K-theory, W. A. Benjamin, Inc., New YorkAmsterdam, 1968. [Be1] D. E-C. 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On Existence of Solutions to Structured Lyapunov Inequalities Aivar Sootla and James Anderson arXiv:1603.07686v1 [math.OC] 24 Mar 2016 A condensed version of this paper will appear in the Proceedings of the 2016 American Control Conference Abstract— In this paper, we derive sufficient conditions on drift matrices under which block-diagonal solutions to Lyapunov inequalities exist. The motivation for the problem comes from a recently proposed basis pursuit algorithm. In particular, this algorithm can provide approximate solutions to optimisation programmes with constraints involving Lyapunov inequalities using linear or second order cone programming. This algorithm requires an initial feasible point, which we aim to provide in this paper. Our existence conditions are based on the so-called H matrices. We also establish a link between H matrices and an application of a small gain theorem to the drift matrix. We finally show how to construct these solutions in some cases without solving the full Lyapunov inequality. I. I NTRODUCTION Lyapunov equations and matrix inequalities play a central role in control theory, since they are used for, e.g., verifying stability of a dynamical systems, optimal control, and model order reduction (cf. [2]). Lyapunov matrix inequalities with sparsity constraints on the decision variables are used in in the context of distributed control [3], structured model reduction [4] etc. In such applications, a typical constraint on the decision variables is block-diagonality of a matrix. The major bottleneck in solving optimisation programmes with a Lyapunov inequality constraint is scalability, since it is a semidefinite programme (SDP). There exist a number of methods addressing scalability of SDPs (cf. [5], [6], [7]), and in one of them, it was proposed to replace the constraints in the cone of positive semidefinite matrices with conic innerapproximations [8], [9]. There are two main conic approximations: one which results in a linear programme (LP), and another which results in a second order cone programme (SOCP). Since we are dealing with inner approximations of the cone of positive semidefinite matrices, even if the LP or SOCP solution can be computed, this solution is usually conservative with respect to the optimal SDP solution. This limitation was partially addressed using the basis pursuit algorithm [1], which is iterates over LPs or SOCPs and provides a guarantee of improvement with each iteration. This algorithm requires an initial feasible point in order J. Anderson is with St John’s College, Oxford and the Department of Engineering Science, University of Oxford, Parks Road, Oxford, OX1 3PJ, U.K. e-mail: [email protected] A. Sootla is with the Montefiore Institute, University of Liège, B28, Liège Belgium, B4000 e-mail: [email protected]. A. Sootla holds an F.R.S–FNRS fellowship. This paper is partially funded through the Belgian Network DYSCO, and the Interuniversity Attraction Poles Programme initiated by the Belgian Science Policy Office. The authors would like to thank Prof. Amir Ali Ahmadi for valuable discussions, and specifically for pointing out the reference [1]. to start the iterations. Hence, major questions still remain concerning existence theorems and scalable computation of block-diagonal solutions to Lyapunov inequalities. Necessary and sufficient conditions for block-diagonal stability were described more than 20 years ago in [10]. However, these results do not provide a constructive way to build block-diagonal Lyapunov functions. This perhaps explains why these results are relatively unused in the control theory literature. Besides some simple cases, such as, the drift matrix being block-triangular matrix (cf. [11]), it is known that the closed loop interconnection of strictly passive systems has a drift matrix which admits a block-diagonal solution to Lyapunov inequalities [12]. It is also well-known that stable Metzler matrices admit diagonal solutions to the Lyapunov inequality [13]. Additional special cases are covered in [14], [15] and revisited in what follows. In this paper, we aim at identifying additional cases, when a block-diagonal solution to a Lyapunov inequality can be found using algebraic methods or LPs. We start by studying a generalisation of Metzler matrices known as H matrices. Stable H matrices possess many properties of stable Metzler matrices, for example they also admit diagonal solutions to Lyapunov inequalities [16]. We provide another such property, namely we show that for H matrices, diagonal solutions to Lyapunov inequalities can be computed using algebraic methods and/or LPs. We then investigate conditions on specific blocks in block-partitioned matrices. We establish a link between the H matrix conditions and a version of the small gain theorem before extending this intuition to blockpartitioned case. In the 2 by 2 block partitioned case, we provide an explicit way to construct block-diagonal solutions to the Lyapunov inequalities without the need to solve the full inequality. An extension to n by n block partitioned case is one of the future work directions. The rest of the paper is organised as follows. In Section II, we cover some preliminaries and motivate our problem formulation in Section III. We show how to construct diagonal solutions to Lyapunov inequalities for H drift matrices in Section IV. We provide stability results for block partitioned matrices and link the condition for H matrices with the small gain theorem in Section V. In Section VI we provide a largescale numerical example and we conclude in Section VII, where we discuss linear programming solutions to Lyapunov inequalities. Notation: Our notation is mostly standard: ρ(A) stands for the spectral radius of a matrix A, A ≥ 0 (respectively, A  0) means that all entries aij of A are nonnegative (respectively, positive), A  0 (respectively, A  0) means that A is positive semidefinite (respectively, positive definite). n Let S n denote the set of symmetric n by n matrices, S+ denotes the cone of positive semidefinite n by n matrices. Let kBk2 be the matrix induced norm, that is kBk2 is equal to the maximum singular value of B, and let σ(B) denote the minimum singular value of B. The H∞ norm of a transfer matrix G is defined as kGkH∞ = max kG(s)k2 and s∈C:Re(s)≥0 kGkH∞ = max kG(ω)k2 for stable G. For a space X, its ω∈R dual is denoted as X ∗ . Finally, I is the identity matrix of an appropriate dimension. II. P RELIMINARIES Consider the linear time invariant dynamical system ẋ(t) = Ax(t), x(0) = x0 (1) where x(t) ∈ Rn . An important concept associated with the system (1) is stability, which is typically verified by solving a linear matrix inequality (LMI). Proposition 1: System (1) is stable if and only if there exists an X  0 that satisfies the LMI AX + XAT ≺ 0. (2) A matrix X which satisfies (2) defines a Lyapunov function of the form V (x) = x(t)T X −1 x(t) for system (1). In this paper, we aim at describing some sufficient conditions of solvability of the LMI (2) when the decision variable X satisfies additional sparsity constraints. In order to simplify the presentation we say that a matrix A ∈ RN ×N has α = {k1 , . . . , kn }-partitioning with N = n P ki , if the matrix A is written as follows i=1  A11  A21  A= .  .. A12 A22 .. . ... ... .. .  A1n A2n   ..  .  An1 An2 ... Ann where Aij ∈ Rki ×kj . We say that A is α-diagonal if it is α-partitioned and Aij = 0 if i 6= j, and α-lower triangular if Aij = 0 if i < j. We aim at characterising α-diagonally stable matrices A ∈ RN ×N , which are such that there exists an α-diagonal positive definite X ∈ RN ×N satisfying (2). If α = {1, . . . , 1}, we say that an α-diagonal (respectively, αlower triangular, α-diagonally stable) matrix A is diagonal (respectively, lower-triangular, diagonally stable). We will make use of so-called scaled diagonally dominant matrices. Definition 1: A matrix A ∈ Rn×n is called strictly row scaled diagonally dominant if there exist positive scalars d1 , . . . , dn such that X di |aii | > dj |aij | j6=i for all i = 1, . . . , n. The matrix A is strictly row diagonally dominant if di = 1 for all i. A related class to scaled diagonally dominant matrices is the class of H matrices. In order to define this class we require the following definitions: Definition 2 ([17]): Given an α-partitioned matrix A with nonsingular Aii for all i, we define the α-comparison matrix Mα (A) as  −1 kA−1 if i = j, α ii k2 Mij (A) = (3) −kAij k2 otherwise, When α = {1, . . . , 1}, we will simply write M(A). −1 Note that kA−1 = σ(Aii ). Hence using a continuity ii k2 −1 argument we can assume that kA−1 = 0 for a singuii k2 lar Aii , and Definition 2 is well-posed. The α-partitioned matrices allow a version of Gershgorin circle theorem: Proposition 2 ([18]): For an α-partitioned matrix A ∈ n P RN ×N , where α = {k1 , . . . , kn } and N = ki , every i=1 eigenvalue of A satisfies k(λI − Aii )−1 k−1 2 ≤ n X kAij k2 j=1,j6=i for at least one i where i = 1, . . . , n. Definition 3: A matrix A ∈ Rn×n is said to be Metzler if all the off-diagonal elements are positive. Definition 4: A matrix A ∈ Rn×n is said to be an H matrix, if the minimal real part of the eigenvalues of M(A) is greater than or equal to zero. It is clear that stable Metzler matrices are also H matrices. It is also straightforward to show that A is strictly row and column scaled diagonally dominant if and only if M(A) has eigenvalues with positive real part [19]. We also refer the reader to [20], [16] for additional information on H matrices. Let DD+ denote the cone of matrices A such that A and AT are strictly diagonally dominant and the elements on the diagonal of A are positive (that is, Aii > 0). Similarly, let H+ denote H matrices A with positive elements on the diagonal of A. If A is a symmetric DD+ matrix, then by Proposition 2 with α = {1, . . . , 1}, it is easy to show that A  0. Moreover, the constraint that A = AT ∈ DD+ can be written as a set of linear constraints n X aii > cij ∀i, (4) j6=i − cij ≤ aij ≤ cij , and cij = cji ∀i 6= j. Hence, a constraint A  0 can be replaced by a more restrictive but scalable linear constraints. This approach was proposed in [8], [9] to restrict some sum-of-squares optimisation problems which are naturally SDPs to LPs. A symmetric H+ matrix is also positive semidefinite, and this constraint can be imposed by a number of second order cone constraints [21]. That is A = AT ∈ H+ if and only if A= N X 2 EiT Xi Ei , with X ∈ S+ , Ei ∈ T2 i=1 where E ∈ T2 , if E ∈ Rn×2 and every column of E has only one non-zero element equal to one, and N = |T2 |. To summarise the subsection, we will mention this strong result on diagonal stability of H+ matrices, which we will revisit in the sequel. Proposition 3 ([16]): Let −A be an H+ matrix. Then A is diagonally stable if and only if A is nonsingular. III. M OTIVATION AND P ROBLEM F ORMULATION A. Conic Programming Fig. 1. Conic optimisation problems take the generic form of optimising a linear functional over the intersection of an affine subspace and a proper cone. Typically conic programmes have the following primal and dual formulations: minimise cT x s.t. Ax = b x∈K maximise bT y s.t. c − AT y = s Lk+1 = decomp(Xk ) where K is a proper cone (i.e. closed, non-empty, pointed, convex) and K∗ is the dual cone of K defined as K∗ := {y | hy, xi ≥ 0, ∀x ∈ K} . It is well known, that the cone of positive semidefinite n n ∗ n . The ) = S+ is self dual meaning that (S+ matrices S+ cone of symmetric DD+ matrices KLP = {X ∈ S n ∩ DD+ } , n . however, is not self-dual and it is larger than the cone S+ More specifically:  ∗ KLP = X ∈ S n viT Xvi ≥ 0, ∀vi ∈ T1 , where T1 is the set of all vectors in Rn with a maximum of two non-zero elements, each of which is ±1. The cone of symmetric H+ matrices defined as X=X has the dual  ∗ KSOCP = X ∈ S n EiT XEi  0, ∀Ei ∈ T2 . We will make use of these cones and their duals in the remainder of the paper. B. Structured Gramians via Basis Pursuit The standard from primal SDP [22] is written as s.t X ∈ Sn+ , (5) hAi , Xi = bi , where decomp(Xk ) is a Cholesky decomposition of Xk , the optimal solution decision variable from iteration k. In some cases, Xk can have singular values close to zero, thus creating numerical problems in the iterative procedure. In order to avoid such cases, we can remove a k-th column of L with the k-th entry close to zero (we assume that L is lower triangular). One can also use LDL decomposition to avoid dealing with negative eigenvalues of Xk . This approach also slightly improves numerical complexity of the conic programme by lowering the number of decision variables and constraints. We finally note that the method relies on the fact that at the first iteration a feasible solution X0 exists. In many applications, it is desirable to solve the following problem using an LP or SOCP rather than the more natural SDP: min trace(X) such that: AX + XAT ≺ 0 KSOCP = {X ∈ S n ∩ H+ } , X Note that Z ∈ K(L) ⇒ Z  0. The algorithm in [1] solves a sequence of optimization problems of the form (5) but with the conic constraint replaced by X ∈ K(Lk ) where the sequence {Lk } is given by L0 = I (y, s) ∈ (Rm , K∗ ) min hC, Xi Block-diagram of the boiler-header system. i = 1, . . . , m where Sn+ is the cone of n×n positive semidefinite matrices. The basis pursuit algorithm proceeds as follows: At each iteration the algorithm re-parameterizes the simpler cone that approximates Sn+ and then solves an optimization problem over this cone, the solution of which is then used to update the cone for the next iteration. In particular, the algorithm specifies for a fixed matrix L, the cone  K(L) = X | X = LT QL, Q = QT ∈ DD+ . T (6) is α-diagonal, where A is Hurwitz, and α is a given partitioning. Note that since A is Hurwitz, then the condition X  0 is implied by the solvability of (6). The basis pursuit can be applied given an X satisfying the constraints of the programme (6). Therefore, we set up our problem: find X satisfying the constraints of (6) with algebraic or linear programming methods. There are many practical applications, where the problem of the form (6) appears and one of them is structured model reduction. For example, consider the boiler-header system described in [12] and schematically depicted in Figure 1. The state space can be partitioned according to dimensions of the subsystems, which are {3, 3, 1}. The system always admits diagonal generalised Gramians, since the drift matrix of the closed loop system is a stable H+ matrix. In order to perform structured model reduction [12] the authors computed a {3, 3, 1}-diagonal generalised controllability Gramian P = diag{P1 , P2 , P3 } such that P1 , P2 ∈ R3×3 and P3 ∈ R. The optimal trace of such a Gramian computed using semidefinite programming is equal to 1.2817 · 104 . Using linear programming, we found the minimum trace of 2.4132 · 104 signifying a loss of quality of almost 100%. After solving one iteration of the basis pursuit algorithm we obtain objective equal to 1.5893 · 104 , an additional iteration of the algorithm gives 1.3172 · 104 , and one more provides a value equal to 1.3093 · 104 , which comes really close to the optimal value. Naturally, on this example we do not need basis pursuit or linear programming to obtain an optimal solution due to the low complexity of the problem. However, this example indicates that the basis pursuit algorithm can be beneficial to obtain an approximate solution of large scale Lyapunov inequalities using linear programmes. IV. H M ATRICES AND D IAGONAL S TABILITY The main result of this section concerns diagonal stability of H matrices, where we sharpen the results from [16] by providing an explicit diagonal Lyapunov function for a class of H+ matrices. Theorem 1: Let −A be an H+ matrix with a nonsingular M(A). Then the following conditions hold
T 1) There exist positive vectors v = v1 . . . vn ,
T w = w1 . . . wn such that M(A)v, wT M(A) are also positive. 2) There exists a diagonal X such that −(AX + XAT ) is an H+ matrix. Moreover, we can choose it as X = Pv Pw−1 , where Pv = diag{v1 , . . . , vn }, Pw = diag{w1 , . . . , wn }, and v, w satisfy point 1). 3) There exists a diagonal positive definite matrix Y such that −Pw APw−1 Y − Y Pw−1 AT Pw ∈ DD+ (7) Proof: 1) By definition −M(A) is a Metzler matrix with all eigenvalues λi (M(A)) ≤ 0, since M(A) is nonsingular by the premise, −M(A) is a Hurwitz Metzler matrix. Hence the claim follows by applying the results from [23]. 2) Let X = Pv Pw−1 , then (M(A)X+XM(AT ))w = (M(A)v+XM(A)T w)  0, where the inequality follows since M(A)v and M(A)T w are positive and X is nonnegative. Hence S = −M(A)X − XM(AT ) is a Metzler matrix and there exists a positive vector such that Sw is negative. This implies that S is a symmetric Hurwitz and Metzler matrix, which means that M(A)X + XM(AT ) is positive definite. Note that aii < 0 for all i, let (−AX − XAT )ij = −aij xj − aji xi ( −aij xj − aji xi (M(A)X + XM(AT ))ij = −|aij |xj − |aji |xi or equal to the minimal eigenvalue of M(−AX − XAT ). This implies that M(−AX − XAT ) has eigenvalues with positive real part, hence −AX − XAT is an H+ matrix. 3) Consider the matrix R = Pw M(A)Pv , and e the vector of ones. Now it is easy to see that Re  0: Pw M(A)Pv e = Pw M(A)v  0. This implies that the matrix Pw M(A)Pv is row strictly diagonally dominant. Similarly, we can show that Pw M(A)Pv is column strictly diagonally dominant. This by definition implies that the matrix −Pw APv is a row and column diagonally dominant matrix with positive elements on the diagonal or a DD+ matrix. Hence the matrix −Pw APv − Pv AT Pw is positive definite. Since we can set Y = Pv Pw the result follows. We showed that there exists a diagonal X matrix such that the matrix Z = −Pw APw−1 X − XPw−1 AT Pw is a DD+ matrix and hence positive definite. Note that the constraint Z = Z T ∈ DD+ is linear and if needed we can relax the sparsity constraints on X. This implies that given an H drift matrix, we can compute an α-diagonal Lyapunov function with an arbitrary α using linear programming. If the entries of the A matrix are poorly scaled then solving a linear programme can be numerically challenging. Using our methods, this can be avoided if we compute an initial point using the right and left eigenvectors of M(A), instead of the positive vectors v and w satisfying point 1). Having an initial point re-scales the optimisation programme and can provide feasible points as shown on a specific example in [24]. Theorem 1 is a direct generalisation of the similar result for Metzer matrices (cf. [23]), but our result can be applied to a broader class of matrices including lower-triangular matrices. Using Theorem 1 other results for Metlzer matrices can be extended to problems such as construction of sumand max-separable Lyapunov functions (cf. [23]). The state-space transformation Pw is essential in order to guarantee the diagonal dominance of the inequality. Consider an asymptotically stable matrix   −1 −2 A= . 2 −5 and a positive definite X = diag{x1 , x2 }. The matrix −A is an H+ matrix and it is stable. The diagonal dominance of AX +XAT requires the following inequalities to be fulfilled 2 · x1 > 2x2 + 2x1 , i=j i 6= j It is straightforward to show that M(A)X + XM(AT ) ≤ M(−AX − XAT ), moreover the elements on the diagonal are equal. This means that we can write M(A)X + XM(AT ) = sI − R1 , M(−AX − XAT ) = sI − R2 , where the matrices R1 and R2 satisfy R1 ≥ R2 ≥ 0. According to Weilandt’s theorem ρ(R1 ) ≥ ρ(R2 ) (cf. [16]). Therefore the minimal eigenvalue of M(A)X + XM(AT ) is smaller 2 · 5x2 > 2x2 + 2x1 for some positive x1 , x2 . The first inequality is equivalent to 0 > 2x2 , which is impossible to fulfil. V. α-D IAGONAL S TABILITY AND H+ M ATRICES A. A Motivating Example In this section, we cover two main classes of results for diagonal stability and compare them to a classical example from [14] for cyclic systems. These classes stem from two arguments based on the passivity and the small gain theorem. In this section, we will argue that the H+ matrix condition Fig. 2. Feedback interconnection of two stable systems G1 , G2 . is an implicit constraint in these stability proofs. In order to explain our motivation consider an example studied in [14] and let:   01×n−1 −β1 A0n = − diag{α1 , . . . , αn } diag{β2 , . . . , βn } 0n−1×1 where αi , βi are positive scalars. This matrix represents the dynamics of a negative feedback of a cascade of transfer βi . First, let us consider the 2 by 2 functions Gi (s) = s+α i case, which gives   −α1 −β1 A02 = β2 −α2 β2 β1 and G2 = (s+α . and two transfer functions G1 = (s+α 1) 2) According to the small gain theorem, the system is stable if β1 s + α1 H∞ β2 s + α2 = H∞ β1 β2 < 1. α1 α2 This argument can be extended to an arbitrary size matrix resulting in the condition β1 · · · βn < 1. (8) α1 · · · αn Surprisingly, it is straightforward to verify by definition that A0n is an H+ matrix if and only if (8) holds. Hence on this loop the H+ matrix condition is a small gain condition. Alternatively, using passivity arguments it was shown in [14], that A0n is asymptotically stable if and only if β1 · · · βn < (sec(π/n))n . (9) α1 · · · αn This in particular means that for n = 2 all matrices in the form A02 are not only stable, but also diagonally stable, however, they may not be H matrices. This analysis is based on passivity arguments and has been extended to less restrictive classes of systems in [15]. It is easy to verify that with n → ∞ the limit (sec(π/n))n converges to one. Hence, it appears (for this class of system) that for large dimensions, H matrices constitute a large subset of diagonally stable matrices. We will pursue the relation between H+ matrices and small gain argument in the αdiagonal case in the remainder of the paper. B. Passivity and Small Gain Conditions for α-Diagonal Stability Let Gc be the closed loop transfer function depicted in Figure 2, which is an interconnection of two Linear Time Invariant (LTI) subsystems   Ai B i , Gi = Ci Di where Ai ∈ Rki ×ki , Bi ∈ Rki ×mi , Ci ∈ Rli ×ki , Di ∈ Rli ×mi and m1 = l2 , m2 = l1 . The closed loop transfer function from [u1 , u2 ] to [y1 , y2 ] has the following statespace realisation   c c c A11 Ac12 B11 B12 c c   Ac21 Ac22 B21 B22  Gc =  c c c c ,  C11 C12 D11 D12 c c c c C21 C22 D21 D22 where Ac11 = A1 − B1 R21 D2 C1 , Ac12 = −B1 R21 C2 Ac21 Ac22 = B2 R12 C1 , (10) = A2 − B2 R12 D1 C2 and R12 = (I + D1 D2 )−1 , R21 = (I + D2 D1 )−1 and the rest of the matrices are computed accordingly. For the sake of simplicity we assume that this realisation is minimal. Passivity and small gain arguments can both be used to determine if the closed loop system is stable but we will focus on the small gain condition. Passivity results in this direction will be addressed in future work, similar ideas were pursued in [25], [26]. It is straightforward to verify that stability of the system with inputs u1 , u2 and outputs y1 , y2 depends on stability of the transfer function L = (I − G2 G1 )−1 . Proposition 4 (Small Gain Theorem): Suppose B is a Banach-algebra and Q ∈ B. If kQk < 1, then (I − Q)−1 exists and ∞ X (I − Q)−1 = Qk . k=0 Applying Proposition 4 we can verify that if Q := kG2 G1 kH∞ ≤ kG2 kH∞ kG1 kH∞ < 1 then the function L and hence the closed loop are stable (cf. [27]). We can apply the small gain condition to the closed transfer function, which would result in a condition on α-diagonal stability of the matrix Ac . However, given only a partitioning α = {k1 , k2 } and a realisation of the closed loop transfer function Gc , these conditions again will be hard to verify. We can apply a small gain theorem in another way, namely apply it to the matrix Ac directly. In this case, we do not need to know the realisation of transfer functions G1 , and G2 , all we need to know is the matrix Ac and the partitioning α. The conditions on α-diagonal stability of Ac are established in the following proposition. Proposition 5: Let Ac be α partitioned with α = {k1 , k2 }  c  A11 Ac12 c A = . Ac21 Ac22 Let K1 (s) = −(sI −Ac11 )−1 Ac12 , K2 (s) = (sI −Ac22 )−1 Ac21 with Hurwitz Ac11 , Ac22 . If there exists a γ > 0 such that kK1 kH∞ < 1/γ and kK2 kH∞ < γ, then the matrix Ac is α-diagonally stable. Proof: We need to show that there exists an α-diagonal Lyapunov function for the system ẋc = Ac xc . For the sake of clarity we drop the superscript c from Acij and simply write Aij . The inequality kK1 kH∞ < 1/γ and the Bounded Real Lemma imply that X1  0 solves the Riccati equation 0 = A11 X1 + X1 AT11 + γ 2 X1 X1 + A12 AT12 = (11)  2  
X γ I 0 1 k1 A11 X1 + X1 AT11 + X1 A12 , 0 Ik2 AT12 which has always has a solution since (I, A11 ) is a controllable pair (cf. [2]), since the control matrix is equal to I and we can control every state independently. Again, due to the Bounded Real Lemma the inequality kK2 kH∞ < µ is equivalent to A22 Y2 + Y2 AT22 + µ−2 Y2 Y2 + A21 AT21 = 0, (12) where Y2  0 since (I, A22 ) is a controllable pair (cf. [2]). Let µ = γ − ε for some ε > 0 such that µ > kK2 kH∞ , which implies that µ−2 Y2 Y2  γ −2 Y2 Y2 and consequently: A22 Y2 + Y2 AT22 + Y2 γ −2 Y2 + A21 AT21 ≺0 By multiplying the equation by γ −2 setting X2 = Y2 γ −2 A22 X2 + X2 AT22 + X2 X2  + +γ  T −2 A21 AT21  ≺0⇔ γ 2 Ik1 0 A21 (AT22 X2 + X2 A22 )−1 A21 X2 0 Ik2 
X2  0 (13) Combining the inequalities (11) and (13) yields  
AT21 T X A 0  A11 X1 + X1 A11 − 1 12 X2  
X1 · (A22 X2 + X2 AT22 )−1 A21 X2 A12 = A11 X1 + X1 AT11 − (X1 AT21 + A12 X2 ) · (X2 A22 + AT22 X2 )−1 (A21 X1 + X2 AT12 ). (14) Applying the Schur complement properties to (14) yields   A11 X1 + X1 AT11 A12 X2 + X1 AT21 ≺ 0, A21 X1 + X2 AT12 A22 X2 + X2 AT22 thus the blocks on the diagonal are negative definite which completes the proof. Our proof is constructive, and shows how to build an αdiagonal Lyapunov function by solving two Riccati equations (11) and (12) instead of solving an LMI. Next we link a simplified version of these conditions with α-partitioned and H+ matrices. C. Conditions for α-Diagonal Stability via H+ Matrices The authors in [18] showed that A is Hurwitz if it is an α-partitioned matrix such that Mα (A) ∈ DD+ , and the matrices Aii are Hurwitz and Metzler for all i. In particular, this result shows that stability of A is implied by stability of all the blocks Aii . We provide a generalisation of this result. Lemma 1: Let A be α-partitioned matrix and Mα (A) be an H+ matrix. Let also Aii be Hurwitz matrices, and the Hamiltonian matrices   Aii γi−2 I Hi = (15) −I −ATii have no purely imaginary eigenvalues with γi = kA−1 ii k2 + ε for all ε > 0. Then A is a Hurwitz matrix. Proof: We prove the result by contradiction. Let A have eigenvalues with a positive real part. Since Mα (A) is an H+ matrix, there exists positive scalars di such that for every i X dj −1 (16) kA−1 kAij k2 . ii k2 > di i6=j The matrix A is unstable if and only if D−1 AD is unstable with D = diag{d1 Ik1 , . . . , dn Ikn }. Let λ be the eigenvalue of D−1 AD with a positive real part. By Proposition 2 there exists an index i such that X X dj dj Aij = kAij k2 . k(λI − Aii )−1 k−1 2 ≤ di 2 di i6=j i6=j (17) Now since the Hamiltonian matrix Hi has no purely imaginary eigenvalues for all ε > 0 and Aii is Hurwitz, this implies that k(sI − Aii )−1 kH∞ = kA−1 ii k2 . Therefore the maximum of k(zI − Aii )−1 k2 over z with Re(z) ≥ 0 is −1 equal to kA−1 k2 ≤ kA−1 ii k2 , and k(λI − Aii ) ii k2 . Hence due to (16) X dj −1 −1 k(λI − Aii )−1 k−1 kAij k2 . 2 ≥ kAii k2 > di i6=j We arrive at the contradiction with (17), which completes the proof. Lemma 1 allows us to determine stability of A by verifying stability of the blocks Aii subject to the condition (15) and Mα (A) being an H+ matrix. This, however, does not directly imply that there exists an α-diagonal Lyapunov function. In what follows, we only present the result for α = {k1 , k2 } partitioning. Theorem 2: Let A be α partitioned with α = {k1 , k2 }, then under the premise of Lemma 1 the matrix A is αdiagonally stable. Proof: The proof is using the small gain argument for the systems G1 (s) = (sI − A11 )−1 A12 , G2 (s) = (sI − A22 )−1 A21 . We have that kG1 kH∞ kG2 kH∞ ≤ ∆ where ∆ := kA21 k2 k(sI − A11 )−1 kH∞ kA12 k2 k(sI − A22 )−1 kH∞ . Under the premise of Lemma 1 we have that γkA12 k2 < −1 −1 kA−1 and γ −1 kA21 k2 < kA−1 11 k2 22 k2 . Hence kG1 kH∞ < −1 γ , while kG2 kH∞ < γ. Proposition 5 proves the claim. Note that if A is such that Mα (A), Mα (AT ) ∈ DD+ , it is not generally true that Mα (A + AT ) ∈ DD+ . This property holds for α = {1, . . . , 1} and was used in the proof of Theorem 1. Hence the absence of this property for a general α is the major obstacle for extending Theorem 1 to the α-diagonal case. VI. N UMERICAL E XAMPLE Consider the one-dimensional heat equation in the form ∂ 2 T (t, x) ∂T (t, x) =α + u(x, t) ∂t ∂x2 T (0, t) = T (1, t) = 0, t≥0 T (x, 0) = 0 x ∈ [0, 1] x ∈ (0, 1), t > 0 TABLE I T IME TO COMPUTE THE GENERALISED CONTROLLABILITY G RAMMIAN Size of the system SDP solution SOCP relaxation LP relaxation LP relaxation w scaling 50 0.94 0.74 0.01 0.01 100 22.7 4.11 0.02 0.03 150 310.7 11.9 0.05 0.05 200 NA 31.2 0.10 0.10 with α = −0.01, where T (t, x) denotes the temperature at time t at x. Assume, we want to heat (i.e. apply an input) at a point of the rod located at 1/3 of its length across, and observe the temperature at a point on the rod located at 2/3 of its length. Then as in [28], we can obtain the following spatially discretised model: Ẋ(t) = AX(t) + Bu(t), X(0) = 0, Y (t) = CX(t), where X(t) ∈ Rn is the temperature n spatial discretisation points, and  2 −1 −1 2 −1   .. .. A = α(n + 1)2  . .   −1 at time t at each of the  .. . 2 −1     ∈ Rn×n ,  −1 2 (18) The matrices B ∈ Rn×1 , C ∈ R1×n are equal to zero except for the entries dn/3e and d2n/3e, respectively, which are equal to one. Our goal is to compute the diagonal controllability Gramians P for various n with a minimal trace, which we will do in the dual form: max trace(BB T Y ), Y s.t. diag{Y A + AT Y + I} = 0 Y ≺ 0. In the dual form, we have an LP relaxation where −Y belongs to the dual to the cone of symmetric DD+ matrices, and an SOCP relaxation, −Y belongs to the dual to the cone of symmetric H+ matrices. We solve only the dual SDP formulation and the corresponding relaxation. Due to the structure of the system, the trace of its Gramians does not change much with dimensions and we always get the optimal values in the range between 6.5 to 6.6 for the SDP programme. Remarkably the results for the SOCP relaxation are only slightly higher, but in the same range of values. This however, is due to structure of the system, where the drift is Metzler and the matrix BB T has only one non-zero entry on the diagonal. The optimal solutions for the LP relaxation are in the range between 10.7 − 10.9, hence there is a drop in quality when using this relaxation. In Table I, we provide the computational times for various systems sizes n. The entry “NA” means that the programme terminated due to running out of memory. Note however, that we do not take into account the time for parsing the constraints (that is, we plot only the variable “solvertime” in Yalmip [29]). Since A is a Metlzer matrix it is straightforward to find a transformation T such that T AT −1 becomes a diagonally dominant matrix (see Theorem 1). We have implemented the LP relaxation while transforming the A, B matrices with such a transformation T . The optimal solutions for the trace vary between 7.1 and 7.3, thus drastically improving the quality of the relaxation with a mild loss (if any) in computational time. VII. D ISCUSSION AND C ONCLUSION We have provided some sufficient conditions on A, which guarantee the existence of feasible points in (6) and interpreted these results as small gain like conditions. Moreover, our sufficient conditions also provide computationally cheap solutions, for example Proposition 5 replaces an LMI constraint with two Riccati Equation solutions. If we drop the “X is α-diagonal” constraint and set Q = AX + XAT , then the LMI (2) has a solution for any Q ≺ 0 if and only if λi (A) + λ̄j (A) 6= 0. Since Q is arbitrarily negative definite, we can replace the constraint AX + XAT ≺ 0 with −AX − XAT ∈ DD+ . Thus our solvability LMI becomes a linear program. Finally we showed how our constructive proofs can be used to initiate a recently developed basis pursuit algorithm for solving large scale optimization problems. R EFERENCES [1] A. A. Ahmadi and G. Hall, “Sum of squares basis pursuit with linear and second order cone programming,” 2015, to appear in Contemporary Mathematics. [2] K. Zhou, J. C. Doyle, and K. Glover, Robust and optimal control. Prentice Hall New Jersey, 1996, vol. 40. [3] F. Lin, M. Fardad, and M. R. 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Prioritized Sweeping Neural DynaQ with Multiple Predecessors, and Hippocampal Replays arXiv:1802.05594v1 [] 15 Feb 2018 Lise Aubin, Mehdi Khamassi, and Benoı̂t Girard Sorbonne Université, CNRS, Institut des Systèmes Intelligents et de Robotique (ISIR), F-75005 Paris, France [email protected] Abstract. During sleep and awake rest, the hippocampus replays sequences of place cells that have been activated during prior experiences. These have been interpreted as a memory consolidation process, but recent results suggest a possible interpretation in terms of reinforcement learning. The Dyna reinforcement learning algorithms use off-line replays to improve learning. Under limited replay budget, a prioritized sweeping approach, which requires a model of the transitions to the predecessors, can be used to improve performance. We investigate whether such algorithms can explain the experimentally observed replays. We propose a neural network version of prioritized sweeping Q-learning, for which we developed a growing multiple expert algorithm, able to cope with multiple predecessors. The resulting architecture is able to improve the learning of simulated agents confronted to a navigation task. We predict that, in animals, learning the world model should occur during rest periods, and that the corresponding replays should be shuffled. Keywords: Reinforcement Learning, Replays, DynaQ, Prioritized Sweeping, Neural Networks, Hippocampus, Navigation 1 Introduction The hippocampus hosts a population of cells responsive for the current position of the animal within the environment, the place cells (PCs), a key component of the brain navigation system [1]. Since the seminal work of [2], it has been shown that PCs are reactivated during sleep – obviously without any locomotion – and that these reactivations are functionally linked with the improvement of the learning performance of a navigation task [3]. Similar reactivations have been observed in the awake state [4], while the animal is immobile, either consuming food at a reward site, waiting at the departure site for the beginning of the next trial or stopped at a decision point. These reactivations contain sequences of PCs’ activations experienced in the awake state (forward reactivations) [5], sequences played in the reverse order (backward reactivations) [4], and sometimes never experienced sequences (resulting from the concatenation of experienced sequences) [6]. These reactivations have been interpreted in the light of the act 1 .6 .2 dist 0 T2 R1 T1 R2 . . . 0 0.2 0.6 1 φ= 0.6 0.2 0 . . 1 0.5 Location Memory Fig. 1. Model of the rat experiment used in [6]. The maze is discretized into 32 positions (squares). The agent can use 4 discrete actions (N,E,S,W). The input state φ is the concatenation of 32 location components and two reward memory components. The location part of φ represents the activation of 32 place cells co-located with the maze discrete positions, their activity act depends on the Manhattan distance of the agent to the cell. All figures by Aubin & Girard, 2018; available at https://doi.org/10.6084/m9.figshare.5822112.v2 under a CC-BY4.0 license. memory consolidation theory [7]: they would have the role of copying volatile hippocampal memories into the cortex [8] for reorganization and longer-term storage [9]. Some recent results have however shown that these reactivations also have a causal effect on reinforcement learning processes [3,10]. A number of reinforcement learning (RL) algorithms make use of reactivations of their inputs, reminiscent of hippocampal reactivations, that are thus candidates to explain this phenomenon [11]. Among them, the Dyna family of algorithms [12] is of special interest because it was specifically designed to make the best possible use of alternation between on-line and off-line learning phases (i.e. phases during which the agent acts in the real world or in simulation). We concentrate here on the Q-learning version of Dyna (Dyna-Q). When operating on-line, Dyna-Q is indistinguishable from the original model-free Q-learning algorithm: it computes reward prediction error signals, and uses them to update the estimated values of the (state, action) couples, Q(s, a). In its original version [12], when off-line, the Dyna algorithm reactivates randomly chosen quadruplets composed of an initial state, a chosen action, and the predicted resulting state and reward (produced by a learned world-model, this phase being thus modelbased ), in order to refine the on-line estimated values. However, when the number of reactivations is under a strict budget constraint, it is more efficient to select those that will provide more information: those that effectively generated a large reward prediction error in the last on-line phase, and those that are predicted to do so by the world model, a principle called prioritized sweeping [13,14]. We are here interested in mimicking the process by which the basal ganglia, which is central for RL processes [15], can use the state representations of the world that are provided by the hippocampus. The manipulated state descriptor will thus be a population activity vector, and we will represent the Q-values and the world model with neural network approximators [16]. In the following, we describe the rat experimental setup proposed in [6], and how we simulated it. In this task, a state can have multiple predecessor states resulting from the execution of a single action, we thus present a modified Dyna-Q learning algorithm, with a special stress on the neural-network algorithm we designed to learn to approximate binary relations (not restricted to functions) with a growing approach: GALMO for Growing Algorithm to Learn Multiple Outputs. Our results successively illustrate three main points. First, because of interferences between consecutively observed states during maze experience, themselves due to the use of a neural-network function approximator, the world model had to be learned with shuffled states during off-line replay. Second, GALMO allows to efficiently solve the multiple predecessor problem. Third, the resulting system, when faced with a training schedule similar to [6], generates a lot of disordered state replays, but also a non-negligible set of varied backward and forward replay sequences, without explicitly storing and replaying sequences. 2 2.1 Methods Experimental task We aim at modeling the navigation task used in [6]: two successive T-mazes (T1 and T2 on Fig. 1), with lateral return corridors. The left and right rewarding sites deliver food pellets with different flavors. The training involves daily changing contingencies, forcing rats to adapt their choice to turn either left or right at the final choice (T2) based on the recent history of reward. These contingencies are: 1) always turn right, while the left side of the maze is blocked; 2) always turn left, while the right side of the maze is blocked; 3) always turn right; 4) always turn left; 5) alternate between left and right on a lap-by-lap basis. Rats attempting to run backward on the maze were physically prevented to do so by the experimenter. They had forty trials the first day to learn task 1, and forty trials the second day to learn task 2. Then, depending on their individual learning speed, rats had between seventeen and twenty days to learn task 3, 4 and 5 (a single condition being presented each day). Once they reached at least 80% success rate on all tasks, rats were implanted with electrodes; after recovery, recording sessions during task performance lasted for six days. During the six recording sessions, the reward contingency was changed approximately midway through the session and hippocampal replays were analyzed when rats paused at reward locations. Original analyses of replayed sequences [6] revealed that: during same-side replays (i.e., replays representing sequences of previously visited locations on the same arm of the maze as the current rat position) forward and backward replays started from the current position; during opposite-side replays (i.e., representing locations on the opposite arm of the maze) forward replays occurred mainly on the segment leading up to reward sites, and backward replays covered trajectories ending near reward sites. In general, the replay content did not seem to only reflect recently experienced trajectories, since trajectories experienced 10 to 15 minutes before were replayed as well. Indeed, there were more opposite-side replays during task 3 and 4 than during the alternation task. Finally, among all replays, a few were shortcuts never experienced before which crossed a straight path on the top or bottom of the maze between the reward sites. 2.2 Simulation We have reproduced the T-maze configuration with a discrete environment composed of 32 squares (Fig. 1, left), each of them represents a 10 × 10 cm area. States are represented by a vector φ, concatenating place cells activity and a memory of past rewards (Fig. 1, right). The modeled place cells are centered on the discrete positions, their activity (color-coded on Fig. 1) decreases with the Manhattan distance between the simulated rat position to the position they encode (top of Fig. 1). When a path is blocked (contingencies 1 and 2), the activity field does not expand beyond walls and will thus shrink, as is the case of real place cells [17]. To represent the temporal dimension, which is essential during the alternation task, we have added two more components in the state’s vector representation (Fig. 1, right): the left side reward memory (L) and the right side reward memory (R). They take a value of 1 if the last reward was obtained on that side, 0.5 if the penultimate reward was on that side, and 0 if that side has not been rewarded during the last two reward events. Therefore, after two successful laps, the task at hand can be identified by the agent based on the value of this memory (Tab. 1). This ability to remember the side of the rewards is supposed to be anchored both on the different position and flavor cues that characterize each side. Since it has been shown that, beyond purely spatial information, the hippocampus contains contextual information important for the task at hand [18], we hypothesize that this memory is also encoded within the hippocampus, along with the estimation of the agent’s current position. The agent can choose between four actions: North, South, East and West. As in the real experiment, the agent cannot run backward. Table 1. State of the L and R memory components of φ and corresponding meaning in terms of task at hand, after two successful laps. L 1 0 0.5 1 R 0 1 1 0.5 Task identification (after 2 laps) Always turn right (Tasks 1 & 3) Always turn left (Tasks 2 & 4) Alternation (Task 5), go left next time Alternation (Task 5), go right next time 2.3 Neural DynaQ with a prioritized sweeping algorithm Our algorithm is based on a Dyna architecture [12] which means that, as in model-based architectures, we need to learn a world model composed of a reward and a transition model [19]. In order to implement prioritized sweeping [13,14], the transition model must be designed so as to allow the prediction of the predecessors of a state s given an action a, because it will be needed to backpropagate the reward prediction computed in state s to its predecessors. Hence, our architecture is composed of two distinct parts: one dedicated to learning the world model, and the other one to learning the Q-values. Algorithm 1 LearnWM: learn the world model collect S // a set of (φt , φt−1 , a, r) quadruplets for k ∈ {N, S, E, W } do SPk ← {(φt , φt−1 ) : (φt , φt−1 , a, r) ∈ S and a = k} k ← {(φt , r) : (φt , φt−1 , a, r) ∈ S and a = k} SR for f ∈ {P, R} do // P,R: Predecessor and Reward types of networks Nfk ← null // list of networks (outputs) Gfk ← null // list of networks (gates) k k create Nnew ; append Nnew to Nfk k k create Gnew ; append Gnew to Gfk GALMO(Sfk , Nfk , Gfk ) // refer to Algo 2 for this specific training procedure end for end for Learning the world model. Two sets of neural networks compose the world model. Four reward networks NRa , one for each action a, learn the association between (state, action) couples and rewards (NRa : s → r(s, a)). Four other networks NPa learn the states for which a transition to a given state s is produced after execution of action a, i.e., the predecessors of s (NPa : s → {s0 }). Owing to the nature of the task (navigation constrained by corridors) and the states’ representation, the data that must be learned are not independent. Indeed, successive state vectors are very similar due to the overlap between place-fields, and are always encountered in the same order during tasks execution (because the agent always performs the same stereotyped trajectories along the different corridors). However, it is well known that the training of a neural network is guaranteed to converge only if there is no correlation in the sequence of samples submitted during learning, a condition that is often not respected when performing on-line reinforcement learning [20]. We indeed observed that in the task at hand, despite its simplicity, it was necessary to store the successive observations and to train the world model off-line with a shuffled presentation of the training samples (for the general scheme of the off-line training, see Algo. 1). For that reason, we created a dataset S compiling all transitions, i.e (φt , φt−1 , a, r) quadruplets from all tasks. When there is no predecessor of φt by action a (as can be the case when this action would require to come through a wall), the transition is represented as (φt , 0, a, r): those ”null” transitions allow NPa networks to represent the fact that the transition does not exist. Correct alternation from left to right T2 T2 T1 Memory part of Φ: Followed by a wrong choice at T2 R R T2 T1 L=1 R=0.5 L=0.5 R=1 L=0.5 R=1 L=1 R=0.5 T1 same action different predecessors L=1 R=0.5 L=1 R=0.5 R Fig. 2. Example of multiple predecessors in the alternation task. The agent first correctly goes to the right (left). It then goes to the left (middle) where, at the reward site, its predecessor state has a (L = 0.5, R = 1) memory component. It then makes a wrong decision and goes to the left again (right), but is not rewarded: at this last position, the location component of the predecessor state (white mouse) is identical but the memory component is different (L = 1, R = 0.5) from the previous lap. Violet gradient: past trajectory; white mouse: previous position; gray mouse: current position; white R: agent rewarded; black R: current position of the reward. Despite its simplicity, the navigation task modeled here has some specificities: during task 5 (alternation), some states have more than one predecessor for a given action (see an example on Fig. 2), the algorithm must thus be capable of producing more than one output for the same input. To do that, we have created a growing type of algorithm inspired by mixture of expert algorithms [21] (which we call here the GALMO algorithm, see Algo. 2), based on the following principles: – The algorithm should allow the creation of multiple Ni networks (if needed) so that a single input can generate multiple outputs. Each of these network is coupled with a gating network Gi , used after training to know if the output of Ni has to be taken into account when a given sample is presented. – When a sample is presented, the algorithm should only train the Ni network that generates the minimal error (to enforce network specialization), and remember this training event by training Gi to produce 1 and the other Gk6=i to produce 0. – The algorithm should track the statistics of the minimal training errors of each sample during an epoch, so as to detect outliers (samples whose error is much higher than the others’). GALMO assumes that these outliers are caused by inputs who should predict multiple outputs and are stuck in predicting the barycenter of the expected outputs. A sample is considered an outlier when its error is larger than a threshold θ, equal to the median of the current error distribution, plus w times the amplitude of the third quartile (Q3 − median). When such a detection occurs, a new network is created on the fly, based on a copy of the network that produced the minimal error for the sample. The new network is then trained once on the sample at hand. Algorithm 2 GALMO: Growing algorithm to learn multiple outputs INPUT: S, N , G OUTPUT: N , G // S = h(in0 , out0 ), ..., (inn , outn )i : list of samples // N = hN0 i : lists of neural networks (outputs) // G = hG0 i : lists of neural networks (gates) θ ← +∞ for nbepoch ∈ {1, maxepoch} do M ← null // M is a list of the minimal error per sample for each (in,out)∈ S do E ← null // E is a list of errors for a sample for each N ∈ N do append kN (in) − outkL1 to E end for if min(E) < θ then backprop(Nargmin(E) , in, out) backprop(Gargmin(E) , in, 1) for each G ∈ G with G 6= Gargmin(E) do backprop(G, in, 0) end for else create Nnew ; append Nnew to N Nnew ← copy(Nargmin(E) ) backprop(Nnew , input =in, target =out) create Gnew ; append Gnew to G backprop(Gnew , in, 1) end if end for θ ← median(M) + w ∗ (Q3(M) − median(M)) end for In principle, the algorithm could be modified to limit the maximal number of created networks, or to remove the networks that are not used anymore, but these additions were not necessary here. Neural Dyna-Q. The second part of the algorithm works as a classical neural network-based Dyna-Q [16] with prioritized sweeping [13,14]. As in [16], a the Q-values are represented by four 2-layer feedforward neural networks NQ (one per action). During on-line phases, the agent makes decisions that drive its movements within the maze, and stores the samples in a priority queue, their priority is the absolute value of the reward prediction error, i.e., |δ|. Every time the agent receives a reward, similarly to rats, it stops and replays are simulated with a budget B (Algo. 3): the samples with the highest priority are replayed Algorithm 3 Neural Dyna-Q with prioritized sweeping & multiple predecessors INPUT: φt=0 , NP , GP , NR , GR a∈{N,S,E,W } OUTPUT: NQ PQueue ← {} // PQueue: empty priority queue nbTrials ← 0 repeat a ← softmax(NQ (φt )) take action a, receive r, φt+1 a backprop(NQ , input = φt , target = r + γmaxa (NQ (φt+1 )) a Put φt in PQueue with priority |NQ (φt ) − (r + γmaxa (NQ (φt+1 )))| if r> 0 then nbReplays ← 0 Pr = hi // empty list of predecessors repeat φ ← pop(PQueue) for each GP ∈ GP do if GP (φ) > 0 then k ← index(GP ) append NPk (φ) to Pr end if end for for each p ∈ Pr s.t norm(p) >  do for each a ∈ {N, S, E, W } do a a a (φ))) (p) + γmaxa (NQ , input = p, target = NR backprop(NQ a a a (φ)))| (p) + γmaxa (NQ (p) − (NR Put p in PQueue with priority |NQ nbReplays ← nbReplays + 1 end for end for until PQueue empty OR nbReplays ≥ B end if φt ← φt+1 nbTrials ← nbTrials +1 until nbTrials = maxNbTrials first, their potential predecessors are then estimated and placed in the queue with their respective priorities, and so on until the replay budget is exhausted. The various parameters used in the simulations are summarized in Tab. 2. 3 3.1 Results Learning the world model Because of correlations in sample sequences, the world model is learned off-line: the samples are presented in random order, so as to break temporal correlations. We illustrate this necessity with the learning of the reward networks NR : when trained on-line (Fig. 3, left), the reward networks make a lot of erroneous pre- Table 2. Parameter values. value parameter 4000 maxepch: number of epoch replays to train the world model 3 w: gain of the outlier detector threshold in GALMO 20 B: replay budget per stop at reward sites 2 number of layers in NP , NR and NQ 10, 16, 26 size of the hidden layers in NQ , NR and NP (respectively) ±0.05, ±0.0045, ±0.1 weight initialization bound in NQ , NR and NP (resp.) 0.5, 0.1, 0.1 learning rate in NQ , NR and NP (resp.) 0.9, 1, 1 sigmoid slope in NP , NR and NQ (resp.) (hidden layer) 0.5, 0.4, 0.4 sigmoid slope in NP , NR and NQ (resp.) (output layer) Fig. 3. Reward predictions are inaccurate when the model is trained on-line (Left panel) and accurate when it is trained off-line (Right panel). L, R: memory configuration. Note the use of a logarithmic scale, so as to make visible errors of small amplitude. dictions for each possible task, while when trained off-line with samples presented in randomized order, the predictions are correct (Fig. 3, right). With a single set of NP networks, the error of the whole set of states decreases steadily with learning, except for four states which have multiple predecessors (Fig. 4, top left). With the GALMO algorithm, when the error of these states reaches the threshold θ (in red on Fig. 4, top right), networks are duplicated and specialized for each of the possible predecessors. We repeated the experiment 10 times. It always converged, with a number of final networks comprised between 2 and 5 (3 times 2 networks, 1 time 3, 5 times 4, and 1 time 5). 3.2 Reinforcement learning with multiple predecessors We compare the efficiency of the Dyna-Q model we developed with the corresponding Q-learning (i.e. the same architecture without replays with the world model). As expected, Q-learning is able to learn the task, measured here with the proportion of erroneous choices at the decision point T2 (Fig. 4, bottom left). On average it does not fully converge before 1000 epochs of training, the Dyna-Q learns much faster, thanks to the replay mechanism (Fig. 4, bottom right), converging on average after 200 trials. Error evolution at state T2 Error evolution at state T2 1 0.9 0.8 0.8 0.7 0.7 0.6 0.6 Errors Errors 1 0.9 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0 0 0.1 200 400 600 800 1000 1200 1400 1600 1800 2000 0 0 200 400 600 800 #episodes 1000 1200 1400 1600 1800 2000 #episodes Fig. 4. Top: Learning error dynamics without (left) and with (right) GALMO. Errors of all samples (gray) during epochs of training. GALMO allows for the creation of multiple prediction networks to handle the states where multiple outputs have to be generated. Bottom: Learning without (left) and with (right) replays. Evolution of the proportion of decision errors at point T2 during the alternation task. Blue: 10 run average, light blue: standard deviation. 3.3 Preliminary analysis of generated replays Always Right B RND Same Opposite Always Left B Alternate B RND Same Opposite Same Central F RND Opposite Central O Fig. 5. Type of replays. B: backward, F: forward, RND: random. We analyze a posteriori the state reactivations caused by the prioritized sweeping DynaQ algorithm in always turn right, always turn left and alternate tasks. Prioritized sweeping does not rely on explicit replay of sequences, but it however favors them. We considered sequences or replays implying three or more consecutive steps; with 128 possible states, a 3-state sequence has a chance level of 0.01% of being produced by a uniform random selection process. We observed in all cases (Fig. 5) that a bit more than 80% of the state reactivations did not correspond to actual sequences. Most of the sequences are backward, except for the alternate task, which also generated 4.5% of forward ones. As in [6], we classified these sequences as being on the same side as the current agent location, on the opposite side, or in the central part of the maze. There is no clear pattern here, except that central reactivations were observed in the alternate task only. 4 Discussion We proposed a new neural network architecture (GALMO) designed to associate multiple outputs to a single input, based on the multiple expert principle [21]. We implemented a neural version of the DynaQ algorithm [16], using the prioritized sweeping principle [13,14], using GALMO to learn the world model. This was necessary because the evaluation task, adapted from [6], contained some states that have multiple predecessor. We showed that this system is able to learn the multiple predecessors cases, and to solve the task faster than the corresponding Q-learning system (i.e., without replays). This required learning the world-model off-line, with data presented in shuffled order, so as to break the sequential correlations between them, which prevented the convergence of the learning process. A neuroscience prediction derives from this result (independently from the use of GALMO): if the learning principles of the rat brain are similar to those of the gradient descent for artificial neural network, then the world model has to be learned off-line, which would be compatible with non-sequential hippocampal replays. Besides, the part of the DynaQ algorithm that uses the world model to update the Q-values predicts a majority of non-sequential replays, but also 15 to 20% of sequential reactivations, both backward and forward. Concerning GALMO, it has been tested with a quite limited set of data, and should thus be evaluated against larger sets in future work. In our specific case, the reward networks NRa did not require the use of GALMO; a single network could learn the full (s, a) → r mapping as rewards were deterministic. But should they be stochastic, GALMO could be used to learn the multiple possible outcomes. Note that, while it has been developed in order to learn a predecessor model in a DynaQ architecture, GALMO is much more general, and would in principle be able to learn any one-to-many mapping. Finally, in the model-based and dyna reinforcement learning contexts, having multiple predecessors or successors is not an exceptional situation, especially in a robotic paradigm. The proposed approach is thus of interest beyond the task used here. Acknowledgements The authors would like to thank Olivier Sigaud for fruitful discussions. This work has received funding from the European Unions Horizon 2020 research and innovation programme under grant agreement No 640891 (DREAM Project). This work was performed within the Labex SMART (ANR-11-LABX-65) supported by French state funds managed by the ANR within the Investissements d’Avenir programme under reference ANR-11-IDEX-0004-02. References 1. O’Keefe, J., Dostrovsky, J.: The hippocampus as a spatial map. preliminary evidence from unit activity in the freely-moving rat. Brain research 34(1) (1971) 171–175 2. Wilson, M.A., McNaughton, B.L., et al.: Reactivation of hippocampal ensemble memories during sleep. Science 265(5172) (1994) 676–679 3. Girardeau, G., Benchenane, K., Wiener, S.I., Buzsáki, G., Zugaro, M.B.: Selective suppression of hippocampal ripples impairs spatial memory. 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Towards Data Quality Assessment in Online Advertising Sahin Cem Geyik 1,+,∗ , Jianqiang Shen 2,∗ , Shahriar Shariat Ali Dasdan 4,+ , Santanu Kolay 2 arXiv:1711.11175v1 [] 30 Nov 2017 2 3,+ , 1 LinkedIn Corporation, [email protected] Turn Inc., {jianqiang.shen, santanu.kolay}@turn.com 3 Uber, [email protected] 4 Vida Health, [email protected] ABSTRACT 1. INTRODUCTION In online advertising , our aim is to match the advertisers with the most relevant users to optimize the campaign performance. In the pursuit of achieving this goal, multiple data sources provided by the advertisers or third-party data providers are utilized to choose the set of users according to the advertisers’ targeting criteria. In this paper, we present a framework that can be applied to assess the quality of such data sources in large scale. This framework efficiently evaluates the similarity of a specific data source categorization to that of the ground truth, especially for those cases when the ground truth is accessible only in aggregate, and the user-level information is anonymized or unavailable due to privacy reasons. We propose multiple methodologies within this framework, present some preliminary assessment results, and evaluate how the methodologies compare to each other. We also present two use cases where we can utilize the data quality assessment results: the first use case is targeting specific user categories, and the second one is forecasting the desirable audiences we can reach for an online advertising campaign with pre-set targeting criteria. Online advertising strives to serve the most beneficial advertisement (ad) to the most relevant online users in the appropriate context (a specific website, mobile application, etc.). This typically results in attaining higher return-oninvestment (ROI) for the advertisers [10], where the value is generated either from a direct response such as a click or conversion (e.g. the purchase of a product, subscription to a newsletter, etc.), or through delivering a branding message. For this purpose, advertisers receive help from multiple entities in the domain. Supply-side platforms (SSP) provide ad-space (inventory) on websites or mobile apps, to serve ad impressions to users. Ad-exchanges run auctions on available inventory from SSPs. Demand-side platforms (DSP) act on behalf of the advertisers and aim to bid on the most valuable inventory. Advertisers often get performance reports from an independent evaluation agency # . For privacy reasons, these reports, in most cases, only contain aggregate metrics (e.g. click-through rate, percentage of female audiences). In order to reach the right audience usually defined by the advertiser, which in general would improve direct response and branding metrics, the advertisers need to utilize various data sources to label the users in the most accurate way possible. Data management platforms (DMP) have been emerging as a central hub to seamlessly collect, integrate and manage large volumes of user data [6]. Such user data could be first-party (i.e. historical user data collected by advertisers in their private customer relationship management systems), or third-party (i.e. data provided by third-party data partners, typically each specializing in a specific domain, e.g., demographics, credit scores, buying intentions). While first-party data is proprietary to the advertiser and free to utilize, third-party data often carries a pre-negotiated cost per impression (ad served to a user in a website or application). In both cases, it is important for the advertiser to know how accurate a data source is. That is, if a data source has tagged a user to be in category ci (user property, e.g. gender, age, income), how likely it is for the user to actually be in that category. In this paper, we are investigating the above problem which we call data quality assessment in online advertising. The main issue in evaluating the accuracy of a data source Categories and Subject Descriptors H.3.5 [Information Storage and Retrieval]: Online Information Services; I.2.1 [Artificial Intelligence]: Applications and Expert Systems General Terms Algorithms, Application Keywords Online Advertising; Data Quality; Targeting; Forecasting ∗ The authors contributed to this work equally. + This work was completed when the authors were at Turn. # This work has been presented in the KDD 2016 Workshop on Enterprise Intelligence. There are certain independent evaluation agencies in online advertising domain, whose names we cannot list here to comply with the company policy. Advertisers trust these organizations to collect the ground truth. Table 1: Confusion matrix of data source s for tagging users with category c. Ground Truth Predicted by data source s c not c Unknown c n+,+ n+,− n+, not c n−,+ n−,− n−, n n n Unknown ,+ ,− , is the lack of ground truth in the user-level granularity. For example, the advertisers, in reality, never have access to the confusion matrix (Table 1) of a data source in either first or third-party cases. Therefore, the only way for an advertiser to evaluate the quality of a data source is to run an advertising campaign on a set of users and then evaluate the performance in hindsight. Even in those cases, the post-campaign data is often constrained (mostly due to privacy concerns) and in aggregate, that is, only the total number of users in different categories is provided, and not a granular user-tocategory assignment. If it were possible to have the granular data, it would then be trivial to just use the ground truth data source to come up with the accuracy metrics, e.g. filling in the entries of the confusion matrix. Therefore, utilizing the aggregate performance statistics makes the data quality evaluation task quite challenging, and somewhat similar to aggregate learning tasks in machine learning [5], few of which are also directly applicable to this problem. The main contributions of this work are as follows: • formal definition of data quality assessment problem, and the challenges of solving it in online advertising domain, • multiple approaches for evaluating the quality of a data source, which also take into account the efficiency requirements due to the large number of possible data sources∗ to be evaluated, • several use cases where data quality assessment comes in handy for online advertising, and, • initial evaluation of our methodology utilizing simulated data and real-world advertising campaigns. Rest of the paper is organized as follows. In Section 2, we give a more formal definition of the data quality assessment problem. Next, we discuss the literature that deals with either data quality assessment, or aggregate learning (which, as aforementioned, is relevant to our problem) in Section 3. We present our two proposed assessment methodologies in Section 4 and Section 5, and later give some use cases on how we can utilize our data quality assessment output in Section 6. Finally, we present some initial results in Section 7 and conclude the paper along with some potential future work in Section 8. 2. RESEARCH PROBLEMS As we have explained in the previous section, we seek to evaluate first or third-party data sources available for online advertising using multiple accuracy metrics. * While we cannot list the exact number due to the company policy, there are currently over 200k active data sources in Turn’s system. Definition 1. A sound data source tags each virtual user (cookie ID that might be specific to a browser and device) with one and only one of the 3 labels – {Positive, Negative, Unknown}. The tagging process could be explicit or deductive, but cannot be self-contradictory. For example, a user can have a positive tag – Age25, or a negative tag – NotAge25, but cannot be tagged as both Age25 and NotAge25. The data source can also simply indicate that it has no knowledge on a user by tagging it as Unknown. In real-time bidding, the positive tags are the most important, as advertisers usually utilize them to target the desirable audiences. The problem of data quality assessment is defined as the following: Definition 2. Given a sound data source S, its data quality assessment is defined as a measurement mS that has error no more than ǫ over user examples drawn from user set U , with probability of at least δ,   P ( Eui ∈U Ω(ui , Si ) − mS ≤ ǫ) ≥ δ, where Ω(ui , Si ) is a metric to measure the granular targeting performance when this data source tag user ui with Si . As an example, suppose we have a data source which we utilize to tag a user as Male (positive example) or Not Male (negative example). Consider two evaluation metrics, which are Accuracy (percentage of correct taggings by our data source), and True Positive Rate (percentage of positive examples, i.e. Males, that our data source also tags as males). For the accuracy metric, we have the following Ω: Ω(ui , S) = I(GT (ui ) == Si ), which does an exact comparison of the ground truth tagging of user ui (GT (ui )) against the tagging by data source S (Si ). On the other hand, if we were to calculate true positive rate, then we would have the following Ω: Ω(ui , Si ) = I(GT (ui ) == Si == Male), which counts only those cases where both ground truth and the data source tag the user as Male. Note that the above problem definition is a very general formulation, which is typically used in evaluating Machine Learning models [7, 8, 12]. As long as both the ground truth category of a user and that of the data source are available, one can  come up with a perfect data quality assessment, i.e. Eui ∈U Ω(ui , Si ) ≈ mS from Def. 2. The problem occurs when we don’t have direct access to the ground truth category of every single user. Typically, Ω(ui , Si ) is unknown, but rather the category distribution of groups of users is provided. The main reason is to protect the privacy of users [1]. In these kind of situations, especially in online advertising, we may utilize a specific data source to make smart advertising decisions to choose the most appropriate set of users, and in the end, we can receive an aggregated report from a third-party evaluator, which is considered as the ground truth and provides a non-granular distribution of the audience we have reached over many categories of interest. As an example, the report may provide that over all users we have 20% Male, and 80% Not Male. When this occurs, we can no longer do a one-to-one comparison between ground truth and data source in the user granularity, but rather need to come up with alternative methods that can deal with aggregated data, which is our main focus in this paper. In many cases, we need to select the best data source from a large set of candidates with the same semantic goal and adopt it for targeting. For example, given a set of data sources that tag users as male, female, or unknown, we may care more about their relative performance and less about their absolute measurements. The data quality assessment can then be simplified as a ranking problem: Definition 3. Given two sound data sources S 1 and S 2 , and an accuracy metric Ω, a data quality ranking system outputs a rank measurement r 1 for S 1 and r 2 for S 2 such that     r 1 > r 2 ⇐⇒ Eui ∈U Ω(ui , Si1 ) > Eui ∈U Ω(ui , Si2 ) . Once we have the rank measurements for each sound data source, we can order them and select the best one. 3. PREVIOUS WORK As we have explained in the problem definition, evaluation of a data source can be taken as any other machine learning model evaluation task, provided that we have the ground truth information in the user granularity. A detailed evaluation of 18 performance metrics for classification problems is given in [7]. These 18 metrics can be listed as accuracy, kappa statistic, mean F-measure, macro average arithmetic, macro average geometric, AUC of each class against the rest (two variants), AUC of each class couples (two variants), scored AUC, probabilistic AUC, macro average mean probability rate, mean probability rate, mean absolute error, mean squared error, LogLoss, calibration loss, and calibration by bins. The paper provides a detailed correlation analysis and noise sensitivity analysis . Also, the survey by Gunawardana et al. [8] discusses both the evaluation settings and proper evaluation metrics for different classes of recommendation problems, of which online advertising is a sub-problem. When we only have access to aggregated ground truth data, evaluation of a data source is much harder. There has been significant work in aggregate learning tasks which utilize aggregate assignments of classes to groups of samples to train a model. Our aim in this paper is significantly different from such works, since we already have a model (i.e. data source), and we are trying to evaluate its performance utilizing many campaigns and multiple aggregates of ground truth data. Cheplygina et al. [5] provides an overview of aggregate learning methodologies, which may utilize granular response variables/feature vectors (single instance) or aggregate response variables/feature vectors for groups (multiple instance) to train their models, and later, testing them. Musicant et al. [11] utilizes aggregate outputs for the response variables to specialize the training process of k-nearest neighbors, decision trees, and support vector machines. In [3], the authors utilize aggregate views of data, which consist of a choice of different combinations of features, response variables, and combining machine learning models learned from these views. Another interesting work is presented in [16], which gives error bounds on how a model learned from aggregate data can perform. They assert that a machine learning model should minimize empirical proportion risk, and prove that under certain assumptions for the class distributions, learning in the aggregate setting can actually improve individual classification performance. Finally, specific to the online advertising domain, we can list [14] as being a relevant work to ours. In this paper, similar to aggregate learning techniques, the authors aim to learn a predictive model to decide whether a user is in a specific ground truth category using the aggregate data over many campaigns, by assigning the most likely label to all users in the aggregate, or assigning a probabilistic single label. They utilize logistic regression with L2 -norm regularization, where the response variables are the artificially generated labels. 4. BRUTE FORCE EVALUATION In this section we will present our first proposal for data quality assessment, which includes setting up specialized campaigns for a data source and utilizing the targeting results directly for evaluation. Note that we typically rely on the independent survey agencies to collect the ground truth analysis data on our audience population. Such agencies use offline data (such as credit card information) and online data (such as information filled in social networking websites) to profile an Internet user. Reports from these survey agencies are generally considered as the ground truth by advertisers. Such reports are aggregated statistics and contain no user-level information due to privacy reasons. 4.1 Performance Campaign for Data Source An intuitive and straightforward way to evaluate a data source is to set up a campaign that only targets certain users which are tagged by data source S to be in category c. This way we can calculate the quality of the data source N(c ) as p(cg |cs ) = N(cgs ) , where N (cg ) is the number of users in category c reached by this campaign as reported by the ground truth and N (cs ) is the total number of users reached by the campaign via at least one impression. Note that we can put a limit on the number of impressions to be served to a user so that we can increase the unique user reach and have more reliable results. We applied this methodology to evaluate age and gender categorizations of some well-established data providers in online advertising. Table 2 demonstrates some results for one of the better performing such data providers in age categorization. We anonymize the name of the data provider and exact age ranges listed in the table to comply with the company’s regulations. In Table 2, R1,· · · · R10,· represent the age ranges such that they are mutually exclusive and sorted in ascending order, e.g. R5 is the range that is the immediate higher range after R4 (i.e. minimum age in range R5 is one larger than maximum age in range R4 ), and the immediate lower range before R6 . Ri,p represents the predicted results by the data source for age range i, while Ri,g stands for ground truth data for the same age range. For example, we can observe from the table that p(R4,g |R4,p ) = 0.345, that is, when our data source classifies the user to be in age range R4 , 34.5% of the time it is correct, or in other words, 34.5% of the users reached by campaign that targets category R4 , as predicted by the data source, were actually in category R4 , as provided by ground truth. Note that for this particular data source, the exact match, highlighted in bold, is quite high compared to random. Data providers utilize various models and online/offline information to tag users. Please note that data sources from Table 2: Ground truth distributions of anonymous data provider’s category taggings. Age Ranges R1,p R2,p R3,p R4,p R5,p R6,p R7,p R8,p R9,p R10,p R1,g 0.414 0.058 0.031 0.031 0.037 0.056 0.08 0.065 0.05 0.035 R2,g 0.245 0.297 0.053 0.041 0.057 0.049 0.067 0.074 0.044 0.036 R3,g 0.033 0.246 0.298 0.089 0.056 0.061 0.054 0.082 0.066 0.055 R4,g 0.018 0.037 0.227 0.345 0.063 0.041 0.035 0.043 0.065 0.048 the same data provider may have quite diverse accuracy values. For example, the accuracy results in Table 2 range from 0.29 to 0.49. Thus we cannot evaluate one data source and assume its sibling data sources have similar predictive power. Each data source needs to be evaluated individually. This causes a significant disadvantage with the above methodology, which is the fact that we need to set up a separate campaign for each category so that we can gather the accuracy statistics. To remedy this problem, we propose a second methodology in Section 5, which solves an optimization problem to come up with the best fitting accuracy probabilities based on the aggregate reports. 4.2 Cost Analysis In this subsection, we further analyze the cost of the brute force method discussed in Section 4.1. It must be noted that obtaining the ground truth, that is the aggregated labeled data, is costly on its own right. However, in the following lemma, we focus only on the ad serving costs to underline the utility and benefit of our approach. Lemma 1. Given a data source that can tag the user by one of the possible c categories (e.g. the data source gives a positive/negative output on one age group of possible c age groups), then to observe a significant difference between the calculated accuracy of one category versus others, we need at least ⌈ 4202.969 (1 − 1c )⌉ impressions. c Proof. Assuming a uniform distribution, we assume the average on-target rate is 1c for each data source (although the intention of a data source is to increase this value). Each impression can be considered as a Bernoulli trial with 1c probability of success. The sample variance is, thus, 1c (1 − 1c ). We would like to detect a significant difference between the prediction accuracy of the correct category versus the rest of possible tags. The industry standard accepts a 5% error. Then, for a significance level of α = 5% for a two tailed hypothesis test and to attain at least 90% power, we have √ n √0.05 > (z0.975 + z0.9 ), where n stands for the number 1 1 c (1− c ) of users, z0.975 and z0.9 are the values of the quantile function of the standard normal distribution for 97.5% and 90%, respectively [2]. Therefore, the number of users that receive the ad impressions must be more than ⌈ 4202.969 (1 − 1c )⌉ for c each data source. Based on Lemma 1, for d data sources that provide information on one of the possible c categories, we need at least d⌈ 4202.969 (1 − 1c )⌉ impressions. As we discussed before, it is c necessary to evaluate each data source individually, considering the diversity of their predictive power. Given the very large number of data sources, this causes the brute-force approach to incur a very significant cost. R5,g 0.012 0.021 0.049 0.182 0.337 0.053 0.03 0.03 0.048 0.039 R6,g 0.032 0.043 0.042 0.057 0.212 0.339 0.041 0.04 0.033 0.048 R7,g 0.043 0.051 0.042 0.037 0.047 0.23 0.332 0.048 0.048 0.061 R8,g 0.064 0.089 0.049 0.039 0.038 0.041 0.204 0.339 0.044 0.06 R9,g 0.075 0.091 0.132 0.116 0.082 0.065 0.077 0.203 0.488 0.106 R10,g 0.03 0.046 0.054 0.041 0.05 0.046 0.048 0.052 0.101 0.492 5. ACCURACY INFERENCE High-quality data sources can enable advertisers to reach the right audience at the right moment. Because they have become an important component of online advertising, more and more online/offline data are being ingested into Turn’s data management platform. As we have mentioned previously, there are currently over 200k active data sources in our system. Lemma 1 established that explicitly evaluating each of these data sources by running performance campaigns is overwhelmingly costly: not only a large amount of money is required to run the performance campaigns, but also enormous manual effort to set up and manage those campaigns is essential as well. We need an efficient way to simultaneously infer the accuracy of multiple data sources. As we have presented in Section 2, our focus in this paper is to calculate the accuracy metrics of a data source for single or multiple categories. In essence, we are trying to calculate a set of probabilities, which represent the likelihood of a data source predicting correctly/incorrectly that a user belongs to a category ci . In Figure 1, we have shown the set of probabilities that we aim to predict. For representational purposes, we have shown the accuracy probabilities of a data source which denote its capabilities to tag a user as Male or not, though the same logic follows for any category. The probabilities in the figure can be listed as follows: • α1 : The probability of the user actually being Male when the data source tags it as Male. This value can also be called precision or positive predictive value. • α2 : The probability of the user actually being Not Male when the data source tags it as Male. This value can also be called false discovery rate. • α3 : The probability of the user being Unknown (i.e. ground truth does not exist) when the data source tags it as Male. • β1 : The probability of the user actually being Male when the data source tags it as Not Male. • β2 : The probability of the user actually being Not Male when the data source tags it as Not Male. This value can also be called negative predictive value. • β3 : The probability of the user being Unknown (i.e. ground truth does not exist) when the data source tags it as Not Male. • γ1 : The probability of the user actually being Male when the data source tags it as Unknown. • γ2 : The probability of the user actually being Not Male when the data source tags it as Unknown. • γ3 : The probability of the user being Unknown (i.e. ground truth does not exist) when the data source tags it as Unknown. As it can be seen from the figure, and trivial from the defini- α1 Male D1 by Data Source S α2 Male by Ground Truth N1 Not Male by Ground Truth N2 α3 β1 Not Male D2 by Data Source S β2 β3 γ1 D3 Unknown by Data Source S γ2 γ3 Unknown by Ground Truth N3 Figure 1: Objective probabilities of a data source S for a specific category (is user Male). We log the received report data along with the first-party campaign data into our in-house data warehousing system called Cheetah [4], which is built on top of the Hadoop framework [13]. Cheetah is designed specifically for our online advertising application to allow various simplifications and custom optimizations. Campaign facts are stored within nested relational data tables. Fast MapReduce jobs are designed to extract key features of the performance campaigns, compare them with the ground truth and infer the accuracy of a data source. Utilizing the collected campaign information, we present two approaches in this section to efficiently infer the quality of data sources: one that ranks data sources, and another which directly deduces precision (α1 ) of a data source. 5.2 Ranking Based Assessment tions, we have α1 +α2 +α3 = β1 +β2 +β3 = γ1 +γ2 +γ3 = 1. Among these nine variables, α1 , i.e. precision, is often the most important value for the advertisers, since it denotes the goodness of a data source to be used for their advertising purposes. To calculate this value, we presented the methodology, which is based on creating specific campaigns to evaluate a singular data source for a specific category in Section 4. In this section we are proposing an optimization scheme, which utilizes the aggregated category distributions over multiple campaigns, for both the data source we want to evaluate, and the ground truth. In the following subsections, we call the above nine variables predictive values of a data source. 5.1 Setup for Inference We propose to set up multiple performance campaigns without using any data source for targeting, so the audience will not be explicitly skewed by any data source. We compare the ground truth of each campaign against the hypothesis of each data source, and infer the quality of the data source. We follow a set of rules to set up a performance campaign. First, the targeting criteria should be minimal and cannot be biased by any third-party data. For example, targeting the online users in U.S. is fine, since this is purely based on IP address and not biased; however, using a data source to limit audience to middle-aged men is not acceptable as the quality of this data source is what we want to assess. In general, only geographical location should be used as targeting criteria. Second, the targeted websites must be discriminative, that is, the population of the visiting users should be largely skewed towards one of our possible tags. This way, we will not mistakenly estimate a data source to be accurate, when in fact it is predicting the label in a random manner. For example, a website is beneficial for such an experiment if 70% of visitors are female and 30% of visitors are male, and not necessarily if the distribution is 50% to 50% (since, in such a case, a random prediction of female vs. male will closely fit the overall audience in aggregate). One should note that obtaining such knowledge is not always feasible before running the campaign. Therefore, in our system, we target only the websites that are known, based on our domain experience and verified by independent reports, to be popular among a certain group of audience. After creating such a campaign, we run it for a certain period to collect data. The ground truth is collected through independent agencies as described in Section 4. Ranking Data Sources. In many instances, we only need to choose the best data source from a large set of candidates with similar semantic purposes. If this is the case, then data quality assessment becomes a ranking problem. A data source’s absolute precision α1 is of less importance then, and rather its rank among others is critical. Since the independent evaluation agency sends us the aggregate statistics on a campaign, we can similarly construct such statistics using a data source. Note that this approach will only represent the view of the data source, and not the ground truth, unlike the independent agency case. We can then evaluate this data source based on how close the constructed statistics are to that of the ground truth, and therefore rank data sources based on such closeness measure. This logic is presented in Algorithm 1. Algorithm 1: Measurement calculation for ranking. Input: ψ: aggregation function, f : closeness function Input: S: data source Output: µ̄: score for quality of data source S 1 foreach performance campaign C do 2 U ← Retrieve audience of C; 3 R̂ ← ψ(U, S); 4 R ← Retrieve the ground truth report; 5 µC ← f (R̂, R); 6 7 µ̄ ← Calculate the average value of µC ; return µ̄; There are multiple ways to design the closeness function f to compare two aggregated statistics. Since the positive taggings are the most valuable for online advertising purposes, we propose to compare the positive population distributions between the ground truth and the data source. We define the percentage of population marked as positive by a data source as count+ (S, U) R̂ = , (1) count+ (S, U) + count− (S, U) where count+ counts the number of users in U marked as positive by S, and count− counts negatives. Given that R is the ground truth ratio of positive population, a simple way to calculate the closeness can be defined as: R̂ − R . (2) However, this does not consider the scale of R̂ or R, which are usually quite small for a rare positive group. To make the measurements more comparable, instead we propose to calculate the relative error as the closeness: ˆ + (S, U) count+ (U) − count , RelativeErr = count+ (U) two predictive values. Since the data sources are unbiased, the order of metrics for negatives are also preserved. Averaging is monotonic, therefore we can expand the previous statements to micro and macro averaging cases as well. where count+ (U) is the number of positives reported by the ground truth, count− (U) is the number of ground truth negˆ + (S, U) is the scaled number of positives atives, and count marked by the given data source. We want to scale the number of positives of the data source, because the population recognized by S might be quite different from the one recognized by the ground truth. For example, the independent evaluation agency might have data on 10000 users, while S might only have data on 1000 users. Therefore, we need to extrapolate the population unrecognized by S to scale up the populations: 5.3 Precision Inference Approach RelativeErr = count+ (U) − R̂ · (count+ (U) + count− (U)) count+ (U) = 1− R̂ count+ (U)/(count+ (U) + count− (U)) = 1− R − R̂ R̂ = R R (3) The above value reflects the potential error rate if we scale the data source’s recognizable population to the size of the ground truth population. Per Algorithm 1, we calculate average relative error (RelativeErr) across all performance campaigns for each data source. We can then rank data sources based on their average relative errors. Soundness Analysis. A ranking algorithm needs to be sound to ensure the optimal assessment: Given two data sources S 1 and S 2 , a ranking algorithm is sound if it outputs measurements r 1 for S 1 and r 2 for S 2 , such that r 1 < r 2 if and only if S 1 is more likely to perform better than S 2 , as we defined in Def. 3. We will show that the RelativeErr based ranking algorithm is sound in many cases. First, let us define the notion of unbiasedness for a data source: Definition 4. A data source S is unbiased if and only if its positive predictive value equals to its negative predictive value: α1 ≈ β2 . Our experience suggests that many data sources we utilize are on demographics and can be considered as unbiased. For example, the accuracy that a data source claims someone as male, in general, is close to the accuracy that it claims someone as female (Not Male as in Figure 1). In real-time bidding, better on-target metrics, i.e. improving the ratio of audience that really have the data source’s claimed characteristics, is the endeavor of any data provider. We can show that the RelativeErr based ranking algorithm is sound: 1 k Lemma 2. Given a set of sound data sources hS , .., S i, the RelativeErr based ranking algorithm is sound for precisions and orders the data sources based on their expected performance. In addition, if the data sources are all unbiased, the algorithm is sound for any on-target metrics. Proof. Per definition, R̂ of the better data source is closer to the reality, thus R − R̂ is smaller. Since R is constant, the order of RelativeErr is preserved for precisions. On-target metrics can be the precision, or the negative predictive value, or simply the micro or macro average of the Although the ranking methodology is able to pinpoint the highest performing data sources, the output ranking measurement is only a surrogate of precision. It correlates with the underlying precision, but is inherently different. As we will show later, in online advertising, it is often necessary to forecast the campaign performance as well as evaluate whether a third-party data source is worth the extra amount of money that an advertiser has to pay in order to utilize it. In such cases we need an accurate estimation of a data source’s precision. Direct Inference of Data Source Precision. We propose an efficient way to directly estimate the predictive values of a data source. As shown in Figure 1, a data source’s hypothesis on the audience population can be mapped into the ground truth using its predictive values. Given a performance campaign C i , let the size of Positive, Negative and i i Unknown audiences identified by data source S be D+ , D− and Di correspondingly (these are scaled values to cover the whole population of C i ), and the size of ground truth Positive, Negative and Unknown audiences be Gi+ , Gi− and Gi correspondingly. When the audience population size is large, it is clear that we have i i Gi+ ≈ D+ · α1 + D− · β1 + Di · γ1 i i Gi− ≈ D+ · α2 + D− · β2 + Di · γ2 i i Gi ≈ D+ · α3 + D− · β3 + Di · γ3 Combining with the probability simplex constraint and the unbiasedness constraint, we can estimate a data source’s predictive values by solving a quadratic optimization problem: given n performance campaigns hC 1 , C 2 , .., C n i, we search for predictive values α1 , α2 , α3 , β1 , β2 , β3 , γ1 , γ2 , γ3 so that min α,β,γ n  X i=1 i i (D+ · α1 + D− · β1 + Di · γ1 − Gi+ )2 i i +(D+ · α2 + D− · β2 + Di · γ2 − Gi− )2 i i +(D+ · α3 + D− · β3 + Di · γ3 − Gi )2 s.t. 0 ≤ αj , βj , γj ≤ 1 3 X j=1 αj = 1, 3 X ∀j = 1, 2, 3 βj = 1, j=1 − ξ ≤ α1 − β2 ≤ ξ . 3 X γj = 1  (4) (5) (6) j=1 (7) Here, (4) is our optimization objective which aims to find the best mapping between the data source’s hypothesis and the ground truth. Note that here we assume the size of audiences of campaigns C i to be similar, which can be controlled at campaign set up time. Otherwise we need to normalize by the audience size of each campaign. (5) and (6) enforce the probability simplex. (7) attempts to help us find the unbiased solution, and predefined constant ξ controls our confidence on the unbiasedness of the predictive values. We, therefore, can run a few performance campaigns, extract each data source’s hypothesis on those campaigns, compare with the ground truth and solve the above optimization problem. As we will show, this will efficiently give us the estimated predictive values of data sources in batch (among those, precision is the most valuable for online advertising). Performance Analysis. The proposed inference approach is efficient, in terms of both computation complexity and money. First, it is straightforward to show that the quadratic programming problem has a semi-definite Hessian with a bowl shape. The optimization problem is convex and can be solved efficiently with polynomial time complexity. Additionally, we only need to run a limited number of performance campaigns to simultaneously estimate the predictive values of multiple data sources. In practice, it is possible that a data source’s predictive values are slightly different in different performance campaigns due to variance. Given a campaign C i , it is natural to assume a data source’s predictive values for this specific campaign are αij = αj + ε1j , ∀j = 1, 2, 3, βji = βj + ε2j , ∀j = 1, 2, 3, γji = γj + ε3j , ∀j = 1, 2, 3, where ε is normally distributed with zero mean. In such cases, we can get the unbiased estimate of a data source’s predictive values by running a limited number of performance campaigns: Theorem 3. Given k ≥ 3 performance campaigns, our direct inference method can get the unique and unbiased estimate of a data source’s predictive values. Furthermore, given any predictive value αi and its estimation α̂i , we have P ( αi − α̂i ≤ si · Tδ/2,2k−6 ) ≥ 1 − δ, where 0 < δ < 1 is a constant, si is the standard error of the estimation, and Tδ/2,2k−6 is (1 − δ/2)-th quantile of Student Distribution with 2k − 6 degrees of freedom. Proof. The optimization can be converted into a linear regression problem within a simplex search space. This regression problem contains 6 free regressors and each campaign provides 2 points in the space. When we have k ≥ 3 campaigns, the quadratic matrix is positive-definite and we will have a unique global optimal solution. A Bias-VarianceNoise decomposition shows the solution is unbiased. Since the errors are normally distributed, the sum of the regression residuals is then distributed proportional to Student Distribution with 2k − 6 degrees of freedom: t = (αi − α̂i )/si ∼ T2k−6 We then construct the confidence levels for the estimated regressors. By running more campaigns, we can quickly reduce the estimation errors and get highly reliable predictive value estimations of multiple data sources. Given its computational and economic efficiency, we adopted the direct inference method and utilize it continuously to generate the quality report on data sources. 6. USE CASES In this section, we will discuss some use cases where the quality assessment of first or third-party data sources can be useful. First we will talk about targeting in online advertising, and the amount that an advertiser should be willing to pay for a data source. Then we will give a very general use case in campaign forecasting, i.e. to predict, before an online advertising campaign starts, what category of users will actually be reached by a pre-set targeting criteria. 6.1 Targeting in Online Advertising Advertisers aim to reach the best audiences to promote their products, so that they can increase the likelihood of a click or an action happening. The automated way of grouping users into beneficial and non-beneficial subsets is often called audience segmentation. For an informative work on how this kind of audience segmentation can improve click rates, refer to [15]. In this paper, however, we focus on a different kind of targeting where the advertisers already have a pre-defined set of users they want to target. As an example, suppose that an advertiser wants to reach only female audiences within the age range 21-35. There are multiple data sources this advertiser can utilize to reach this group, but as discussed in this paper, none of these data sources gives a definitive classification. Intuitively, an accurate prediction of the quality of a data source is essential for advertisers to choose it over others. Also, note that here we mostly care about the precision or positive predictive value of a data source (i.e. α1 , when a data source suggests that a user is in category c, the likelihood that this user actually belongs to category c), since this is the signal that the advertiser uses to bid on a user. Here, we would also like to discuss the consideration of data cost. In general, when an advertiser wants to utilize a third-party data source for bidding purposes, it should pay the third-party provider a certain amount of money. This cost is generally per impression served using this data source, hence can have significant effect on the ROI of an advertiser (i.e. advertiser needs to pick up extra clicks/conversions to make up for the money paid to the third-party for the targeting information it provides). An important point for an advertiser to consider is if using a data contract is “worth” its price. We will give a simple calculation here for the case when the advertiser utilizes no data sources to reach a specific audience (i.e. free targeting), and whether adding the data source and paying for it makes sense. Our main argument is that, by paying for the data source to target a specific audience, the reduced cost of the mis-targeted impressions (i.e. those impressions that are served to the audience that are out of our desired audience) should make up for the data cost. In other words, for the same amount of money we should get more of the desired impressions, although our total number of impressions is less due to data cost. Please note that below we assume the effective cost per impression (cpi) to be the same for both free targeting and data source assisted targeting, just that the data source has the additional data cost per impression (cpidata ): totalSpend (1 − errorRate(dataSource)) ≥ cpi + cpidata totalSpend (1−errorRate(freeTargeting)) cpi α1,freeTargeting α1,data ≥ . cpi + cpidata cpi Above, totalSpend is the amount of money that the campaign spends, cpi is effective cost per impression, and cpi data totalSpend is data cost per impression (hence cpi+cpi is the numdata ber of impressions picked up by data source assisted targetis the number of impressions that can be ing, and totalSpend cpi picked via free targeting, i.e. no data cost). errorRate is indeed the inaccuracy of free targeting (percentage of audience that is not desired), and percentage of the cases when the data source predicts a user to be in desired audience, while, in fact, it is not. In the next inequality, we actually translated (1 - errorRate) into α1 from Section 5. After further reorganizing the above inequality, we get the following:   α1,data −1 . (8) cpidata ≤ cpi × α1,freeTargeting This means that for a data source to be beneficial for a campaign, its data cost per  impression should be less than  α1,data cpi × α1,freeTargeting − 1 . Please note that we have the assumption here that effective cost per impression would be the same for free targeting vs. data source which is not always valid, i.e. we may have to pay more to show ads (impression) to those users that the data source tagged to be desirable. Also, it can be seen that the benefit of the data source often depends on how expensive the impressions are for a campaign, hence is campaign specific. Finally, the above calculations do not take into account the cost of data evaluation utilizing our proposed two methodologies, which was also mentioned in Section 4.2. However, this evaluation can be performed once for each data source, and hence is not of significance for each campaign that utilizes it. 6.2 Forecasting Forecasting the performance (return-on-investment), reach (unique users we can show an ad to), and delivery (amount of money we can spend on advertising given the targeting criteria) of a campaign is a significant problem that has to be dealt in online advertising [9]. Here we show that by utilizing the accuracy metrics (i.e. α1→3 , β1→3 , and γ1→3 from Section 5) over multiple data sources that may tag a user, we can actually predict the expected number of users that will fall into a specific audience/category, on top of the total spend/reach as in the traditional forecasting problem, once the advertising campaign goes live. Here is how the forecasting process in online advertising works. Once the advertiser sets some targeting criteria (filtering of users to show ads according to anonymous user properties) and goals (in terms of clicks and conversions) for a campaign, we can utilize our system as explained in [9] to find out which users this campaign is likely to reach. We can already calculate expected number of unique users and delivery for the campaign from this information alone. Furthermore, one problem that we can solve for the advertisers is the prediction of what percentage of these users will fall into a specific user category/class c. Note that the approach mentioned in [14] does work on this problem of predicting likelihood of a user belonging to a specific category via utilizing many features, but their focus is on targeting rather than forecasting. Here, we suggest that rather than training a simple model for predicting the membership in a category, we can utilize multiple data sources and their estimated predictive values to forecast the expected number of users that will fall into a category. This information is not used in bidding time, hence contrary to the targeting use case we explained in Section 6.1, there is no data cost. Figure 2 summarizes the overall idea. In the first step, Targeting Criteria T Users which obey the Targeting Criteria T Figure 2: Forecasting Example. we communicate the targeting criteria set by the advertiser to our set of users. For real-time forecasting, we often need to use a sampled set of users [9]. Once we filter the users that are appropriate for the targeting criteria, we can go through each of these users and see whether these users are tagged by a first or third-party category cd . Once we see these taggings, we can calculate the probability of this user belonging to a desired ground-truth category cg by the probability p(cg |cd ) which is the α1 from Section 5, i.e. precision. If we assume that each user is tagged by one and only one ci , we can forecast the expected number of users that will belong to category cg as: X X X I(u has ci ) p(cg |ci ) . (9) p(cg |u) = E[ |cg | ] = u∈T u∈T ci ∈C In (9), T is the set of users that belong to a certain set of targeting criteria T. cg is the category that the advertiser desires to forecast how many of their targeted users will belong to. ci is a first/third-party category from a list of categories C for which we have the prediction values. I is an indicator to see whether a user has category ci , and above formula is valid only because we assume that each user has only one of possible ci s. If each user can have multiple first/third-party categories (as is the case in real situations), we need to aggregate multiple p(cg |ci )s, where we can utilize combination methods such as getting the maximum, minimum, average or median of the precision values. 7. EXPERIMENTAL RESULTS In this section we will give some preliminary results for our optimization-based evaluation technique. We have already given some preliminary results for our first methodology (Section 5.1) in Table 2. As aforementioned, we do aim to calculate all nine prediction values of a data source for a category, but for purposes of online advertising, the most important one is the precision (α1 ). Only if this value is high we can reliably use this data source to reach a certain category of users. Simulation Results. To evaluate our methodology we ran several simulated campaigns, where for each campaign we create an audience of 100 users and assign them to predicted categories as follows: • Random, disjoint sets of 20 users each are assigned to category c, not c, and unknown, • The rest 40 users are assigned to either category c, not Difference 0.10 0.20 0.30 0.00 High Quality Low Quality 1 2 3 4 5 6 7 8 9 10 Number of Campaigns 0.12 Figure 3: Number of campaigns versus difference between predicted and actual precision values. 0.00 Difference 0.04 0.08 High Quality Low Quality 0.00 0.10 0.20 Noise Level 0.30 Figure 4: Noise versus difference between predicted and actual precision values for six campaigns scenario. c, and unknown in a uniform manner. Then, we generate the ground truth categories for two types of data sources with the following actual probability values: • High Quality: This data set has the following underlying nine probability values: α1→3 = (0.8, 0.15, 0.05), β 0.7, 0.1), γP 1→3 = (0.2,P 1→3 = (0.4, 0.5, 0.1), note that P α = β = i i 1→3 1→3 1→3 γi = 1, • Low Quality: This data set has the following underlying nine probability values: α1→3 = (0.4, 0.5, 0.1), β1→3 = (0.3, 0.6, 0.1), and γ1→3 = (0.5, 0.4, 0.1). Please note that this kind of synthetic data generation is quite counter-intuitive. We first create the predicted values using some pre-set distribution, and then generate these users’ actual categories using the predictive values of the two data sources. For example, if the user in our synthetically generated audience has a predicted category of not c by High Quality data source, then we assign it to ground truth category of c by probability β1 = 0.2, not c by probability β2 = 0.7, and unknown by probability β3 = 0.1. Once we generate the dataset, we actually have aggregated values of category counts for each data source. Using these category counts, we can utilize our data quality assessment method we described in Section 5.3, and look into the difference between our computed predictive values, and the actual predictive values as given above. The results of the above described simulations are given in Figures 3 and 4. In Figure 3, we estimate the nine predictive values for each of the data sources using our methodology, by utilizing multiple campaigns (the results are averaged over 100 trials at each value in x-axis). As we have proven in Section 5.3, we do need at least three campaigns to get a Figure 5: Pearson correlations between the estimated positive population and the ground truth. unique solution for all nine probabilities. We can observe that starting with four campaigns, the difference of the real values versus our predicted values fall to zero. We have plotted the difference between α1 s (precision values, since α1 = p(actually positive | predicted positive)), but the results are similar for other α2→3 , β1→3 , and γ1→3 . Next, we performed another experiment where we introduced a uniform noise between ±ζ (where we changed ζ between 0 and 0.35) to the above nine real predictive values, and then generated the ground truth assignments. We tried to recover the nine predictive values using six campaigns and present the difference (averaged over 100 trials) between real and predicted α1 values in Figure 4. We can see that even under significant noise levels, our methodology can recover the precision values accurately. Because α1 for the high quality data source is higher, we can also observe that the noise effect is slightly less. Real-World Results. Following the methods we discussed in Section 5.1, we ran 156 performance campaigns, each of which targeted a specific website. We used half of the campaigns to calculate RelativeErr and predictive values for around 100 data sources. Then, we tried to estimate the positive population in the rest of campaigns using these 100 data sources. We utilized the average of positives predicted by the sources, and calculated the correlation with the ground truth positive sizes. For the direct inference method, it is clear that for each campaign C i , its estimated positive i i population is Gi+ ≈ D+ · α1 + D− · β1 + Di · γ1 (for a single data source). For the RelativeErr method, by deducing (3), we can roughly estimate Gi+ = M · τ̂ R̂/(1 − µ̄), where M is the population size, τ̂ is the percentage of population recognized by the ground truth in the training set, and µ̄ is the average R − R̂/R (i.e. RelativeErr from Eq. 3) of the training set for a single data source. Each method’s Pearson correlation coefficient is shown in Figure 5. The direct inference method gives a significantly more accurate estimate of the positive population (p < .001) and it correlates well with the ground truth. Next, we utilized two popular data sources S1 and S2 as the targeting criterion and ran test campaigns individually. The reported positive rates from the independent evaluation agency can be treated as the ground truth of their precisions. The ground-truth precisions and our estimated values are listed in Figure 6. The direct inference method yields a much closer estimation to the ground truth (p < .001), while the ranking method preserves the orders but the values are substantially different from the ground truth. Our proposed approach was deployed into Turn’s data management platform and generates weekly reports on the of combining multiple data sources and their quality assessments to come up with a better model. Acknowledgments We thank many talented scientists and engineers at Turn for their help and feedback in this work. 9. REFERENCES Figure 6: Reported measurements on the positive population from different methods. Figure 7: Inferred precision of a data source over a three month period. quality of our top data sources. We have received positive feedback from our campaign optimization managers in the field, commenting that the reported precisions are close to the real campaign results. Interestingly, by evaluating our data sources periodically, we are forming a positive reinforcement loop over their data quality: feeling the pressure, data providers work consistently to improve their data quality. For example, the estimated precision of one data source over a three month period is plotted in Figure 7. It is clear that the data source’s quality has been improved over this time period. 8. CONCLUSIONS AND FUTURE WORK In this paper, we have presented a novel framework to evaluate first or third-party data sources on user properties for online advertising, which is a particularly challenging task when the ground truth is reported in aggregate form. We call this problem data quality assessment, and presented two solutions, one utilizing the data sources directly in a campaign, and another one, which utilizes outputs from multiple online advertising campaigns to optimize a set of probabilities which represent the “goodness” of a data source. 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Artifact reduction for separable non-local means Sanjay Ghosh and Kunal N. Chaudhury1 arXiv:1710.09552v1 [] 26 Oct 2017 Department of Electrical Engineering, Indian Institute of Science, Bangalore. Abstract. It was recently demonstrated [J. Electron. Imaging, 25(2), 2016] that one can perform fast non-local means (NLM) denoising of one-dimensional signals using a method called lifting. The cost of lifting is independent of the patch length, which dramatically reduces the run-time for large patches. Unfortunately, it is difficult to directly extend lifting for non-local means denoising of images. To bypass this, the authors proposed a separable approximation in which the image rows and columns are filtered using lifting. The overall algorithm is significantly faster than NLM, and the results are comparable in terms of PSNR. However, the separable processing often produces vertical and horizontal stripes in the image. This problem was previously addressed by using a bilateral filter-based post-smoothing, which was effective in removing some of the stripes. In this letter, we demonstrate that stripes can be mitigated in the first place simply by involving the neighboring rows (or columns) in the filtering. In other words, we use a two-dimensional search (similar to NLM), while still using one-dimensional patches (as in the previous proposal). The novelty is in the observation that one can use lifting for performing two-dimensional searches. The proposed approach produces artifact-free images, whose quality and PSNR are comparable to NLM, while being significantly faster. Keywords: Denoising, non-local means, fast algorithm, lifting, artifact. 1 Introduction We consider the problem of denoising grayscale images corrupted with additive white Gaussian noise. A popular denoising method is the non-local means (NLM) algorithm,1 where image patches are used to perform pixel aggregation. While NLM is no longer the state-of-the-art, it is still used in the image processing community due to its simplicity, decent denoising performance, and the availability of fast implementations. The NLM of an image f = {f (i) : i ∈ Ω}, where  Ω = i = (i1 , i2 ) : 1 ≤ i1 , i2 ≤ N }, is given by1 P j∈S(i) wij f (j) NLM[f ](i) = P (i ∈ Ω). (1) j∈S(i) wij where S(i) = i + [−S, S]2 is a search window around the pixel of interest. The weights wij are set to be  1 X
2  wij = exp − 2 f (i + k) − f (j + k) , (2) α k∈P where α is a smoothing parameter and P = [−K, K]2 is a two-dimensional patch. A direct implementation of (1) has the per-pixel complexity of O(S 2 K 2 ), where S and K are typically in the range [7, 20] and [1, 3].1 Several computational tricks and approximations have been proposed to speedup the direct implementation.2–8 A particular means to speed up NLM is using a separable approximation, which in fact is a standard trick in the image processing literature.9–12 In separable filtering, the rows are processed first followed by the columns (or in the reverse order). Of course, if the original filter is non-separable, then the output of separable filtering is generally different from that of the original filter, since a natural image typically contains diagonal details.12 This is the case with NLM since expression (2) is not separable. The present focus is on a 1 Email: [email protected], [email protected] recent separable approximation of NLM.13 At the core of this proposal is a method called lifting, which computes the NLM of a one-dimensional signal using O(S) operations per sample. In other words, the complexity of lifting is independent of the patch length K. Extending lifting for NLM denoising of images, however, turns out to be a difficult task. Therefore, we proposed a separable approximation, called separable NLM (SNLM),13 in which the rows and columns of the image are independently filtered using lifting. In particular, we separately computed the “rows-then-columns” and “columns-then-rows” filtering, which were then optimally combined. The per-pixel complexity of SNLM is O(S), which is a dramatic reduction compared to the O(S 2 K 2 ) complexity of NLM. A flip side of SNLM (as is the case with other separable formulations14 ) is that often vertical and horizontal stripes are induced in the processed image. The stripes are more prominent along the last filtered dimension.14 In SNLM, this problem was alleviated using the optimal recombination mentioned above followed by a bilateral filter-based post-smoothing. In this work, we demonstrate that the stripes can be mitigated in the first place simply by involving the neighboring rows (or columns) in the filtering. In other words, we use a two-dimensional search (similar to classical NLM1 ), while still using one-dimensional patches (as done previously13 ). The present novelty is in the observation that one can use lifting for performing a two-dimensional search. In particular, the per-pixel complexity of the proposed approach is O(S 2 ), which is higher than our previous proposal, but still substantially lower than that of classical NLM. Importantly, the proposed approach no longer exhibits the visible artifacts that are otherwise obtained using SNLM. The rest of the paper is organized as follows. We recall the SNLM algorithm in Section 2 and its fast implementation using lifting. We also illustrate the artifact problem with an example. The proposed solution is presented in Section 3, along with some algorithmic details. In Section 4, we report the denoising performance of our approach and compare it with classical NLM and SNLM. We end the paper with some concluding remarks in Section 5. 2 Separable Non-Local Means To set up the context, we briefly recall the SNLM algorithm.13 Suppose we have a one-dimensional signal g = {g(i) : 1 ≤ i ≤ N }, corresponding to a row or column. The one-dimensional analogue of (1) is given by P j∈S(i) wij g(j) NLM1D[g](i) = P (1 ≤ i ≤ N ), (3) j∈S(i) wij and  K
2  1 X g(i + k) − g(j + k) , wij = exp − 2 β k=−K (4) where S(i) = i + [−S, S] and β is a smoothing parameter. In other words, both the search window and patch are one-dimensional in this case. It was observed in our previous work that the weights {wij : 1 ≤ i ≤ N, i − S ≤ j ≤ i + S} can be computed using O(1) operations with respect to K. In particular, consider the N × N matrices: F(i, j) = g(i)g(j) and F(i, j) = K X k=−K (1 ≤ i, j ≤ N ), F(i + k, j + k). (5) (6) 3 (a) Clean. (b) Noisy, 20.19. (e) Proposed (RC), 26.67. (f) Proposed (CR), 26.66. (c) 1D search and 1D patch. (d) 2D search and 1D patch. (g) SNLM,13 26.57. (h) NLM,1 26.02. Fig 1 Denoising of Peppers at noise standard deviation σ = 25. We see stripes in (c) in which both the patch and search window are one-dimensional (both are along rows). As seen in (d), the stripes can however be reduced using a two-dimensional search in place of the one-dimensional counterpart (though we still see some noise). The image obtained by further processing (d) using a two-dimensional search and one-dimensional patches (along columns) is shown in (e). The visual quality and PSNR (mentioned below each image) of (e) is comparable to NLM. In (f), we have reversed the order of processing: we first use one-dimensional patches along columns and then along rows (the search is two-dimensional). Notice that the order (RC/CR) has no visible impact on the final output. Also notice that residual stripes can be seen in SNLM. We see see that that F F is is the the smoothed smoothed version version of of F, F, obtained obtained by We by box box filtering filtering F F along along its its sub-diagonals. sub-diagonals. 13 13 The important observation is that we can write The important observation is that we can write K X K 2 X g(i + k) g(j + k)
2 = F(i, i) + F(j, j) 2F(i, j). g(i + k) − g(j + k) = F(i, i) + F(j, j) − 2F(i, j). k= K k=−K (7) (7) In particular, using this so-called lifting, we can compute the patch distance using just three In particular, this so-called and lifting, can compute the patch distance using just samples of F,using one multiplication, twowe additions. The computational gain comes fromthree the samples of F, one multiplication, and two additions. The computational gain comes the fact that the box filtering in (6) can be computed using O(1) operations with respect to from K using fact that the13box filtering following in (6) can the be computed using operations with using recursions. Moreover, observation that O(1) not all samples of F respect are usedtoinK(3), an 13 recursions. Moreover, following the observation that not all samples of F are used in (3), an 13 efficient mechanism for computing (and storing) just the required samples was proposed. 13 The efficient forcomputing computing storing) just the to required samples was proposed. The per-pixelmechanism complexity of (3)(and using lifting reduces O(S) from the brute-force complexity per-pixel complexity of computing (3) using lifting reduces to O(S) from thepatches brute-force of O(SK). Unfortunately, extending lifting to handle two-dimensional turnscomplexity out to be of O(SK). Unfortunately, extending lifting to handle two-dimensional patches turns out using to be difficult. Instead, we proposed to use separable filtering, where the rows (columns) are filtered difficult. Instead, we columns proposed(row). to use The separable filtering,outputs where are the rows (columns) combined are filteredto using (3) followed by the two distinct then optimally get (3) followed by the columns (row). The two distinct outputs are then optimally combined to the final image. In fact, the reason behind the averaging was to suppress artifacts in the formget of the final image. In fact, the reason behind the averaging was to suppress artifacts in the form of stripes arising from the separable filtering. This is demonstrated with an example in Fig. 1, where we have compared NLM, SNLM, and the proposed approach. We used bilateral filtering to remove the stripes in SNLM, at an additional cost. However, the final image still has some residual artifacts. 3 Proposed Approach We see less stripes in Fig. 1(d) precisely because we use a two-dimensional search. In other words, we use a cross between classical NLM and SNLM in which we use (8) for the aggregation and (4) for the weights. The two-dimensional search results in the averaging of pixels from across rows (and columns). This does not happen in SNLM, which causes the stripes to appear in Fig. 1(c). v u j = (j1, j2) q = (i1, q2 ) i = (i1, i2) Rows (2 K + 1) (2 S + 1) (2 S + 1) Fig 2 Illustration of the idea behind the proposed method (see text for details). The working of our proposal is explained in Fig 2. The pixel of interest in this case is the pixel at position i = (i1 , i2 ) marked with a red dot. The search window of length 2S + 1 is marked with a green bounding box. Two neighboring pixels at locations j = (j1 , j2 ) and q = (i1 , q2 ) are marked with red dots. The former pixel is on a neighboring row, while the latter is on the same row as the pixel of interest. Similar to SNLM,13 we can consider either horizontal or vertical patches. For our example, the patches (of length 2K + 1) are aligned with the image rows; they are marked with light blue rectangles. For our proposal, the denoising at i = (i1 , i2 ) is performed using the formula: P `∈S(i) wi` f (`) P , (8) `∈S(i) wi` and K  1 X
2  wi` = exp − 2 f (`1 , `2 + k) − f (i1 , i2 + k) , β k=−K (9) where S(i) = i + [−S, S]2 and ` = (`1 , `2 ). To compute (8), we group the neighboring patches into two categories: (i) patches with row index i1 , e.g., patch q in Fig. 2, and (ii) patches with a different row index, e.g., patch j in the figure. Let {u(t) : 1 ≤ t ≤ N } and {v(t) : 1 ≤ t ≤ N } be the i1 -th and j1 -th row, where N is the length of a row (see Fig 2). Similar to (5) and (6), we define the N × N matrices: Fuu (p, q) = u(p)u(q), Fvv (p, q) = v(p)v(q), and Fuv (p, q) = u(p)v(q), (10) and the corresponding matrices Fuu , Fvv , and Fuv , where, for example, Fuu (p, q) = K X k=−K u(p + k)u(q + k) (1 ≤ p, q ≤ N ). (11) As in (7), the (squared) distance between patches centered at i = (i1 , i2 ) and q = (i1 , q2 ) is Fuu (i2 , i2 ) + Fuu (q2 , q2 ) − 2Fuu (i2 , q2 ). (12) On the other hand, the distance between patches centered at i = (i1 , i2 ) and j = (j1 , j2 ) is Fuu (i2 , i2 ) + Fvv (j2 , j2 ) − 2Fuv (i2 , j2 ). (13) In other words, we can compute the distance between patches centered at i and q using Fuu . To compute the distance between patches centered at i and j, we require the matrices Fuu , Fvv , and Fuv . Moreover, using these matrices, we can compute patch distances for different i, j, and q, provided the row index of i and q is i1 , and the row index of j is j1 . Thus, an efficient way of computing (8) is to sequentially process the rows. For each row (fixed u), we compute Fuu , Fvv , and Fuv , where v corresponds to neighboring rows that are separated by at most S. We compute 2S + 1 matrices of the form Fvv and another 2S matrices of the form Fuv . As mentioned in Section 2, we can compute each matrix using O(1) operations with respect to K. Moreover, as per the sum in (3), we only require entries within the diagonal band {1 ≤ i, j ≤ N : |i − j| ≤ S} of each matrix. The cost of computing the banded entries is thus O(N S) for each matrix. The overall cost of processing N rows is O(N 2 S 2 ). The per-pixel complexity of computing (8) using the proposed approach is thus O(S 2 ). We can efficiently compute (and store) the banded entries using the method in Section 2.2 of the original paper.13 The main difference with SNLM is that we require a total of 4S + 1 matrices for processing each row; whereas, just one matrix is required in SNLM. As shown in Fig. 1(d), some residual noise can still be seen after the processing mentioned above. We perform a similar processing once more, except this time we use one-dimensional patches along columns. The visual quality and PSNR of the final image (Fig. 1(e)) are comparable to NLM (Fig. 1(h)). Moreover, we see from Figs. 1(e) and 1(f) that if we first use one-dimensional patches along columns and then along rows, then the outputs are similar. We empirically corroborate these observations in the next section. Therefore, we propose to first process the rows using (8) and then process the columns of the intermediate image using (8). A precise description of the proposed approach for processing the (noisy) image along rows using lifting is provided in Algorithm 1. We then perform column processing on the intermediate image to obtain the final output of our algorithm. That is, we simply apply Algorithm 1 on the intermediate image, where we logically switch the rows and columns in the algorithm. Suppose S1 and S2 are the corresponding search windows for the row-aligned and column-aligned processing. Then we set the search parameter in Algorithm 1 as: S = S1 for the row-aligned processing, and S = S2 for the column-aligned processing. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 Data: Image f of size M × N , and parameters K, S, β. Result: Row-processed image f˜ of size M × N given by (8). for i1 = 1, . . . , M do % lifting for i2 = 1, . . . , N do u(i2 ) = f (i1 , i2 ); end Compute matrices Fuu and Fuu using (10) and (11); for j1 ∈ { i1 − S, . . . , i1 + S}\{i1 } do for j2 = 1, . . . , N do v(i2 ) = f (j1 , j2 ); end Compute matrices Fvv , Fuv , Fvv and Fuv using (10) and (11); end % weight computation and pixel aggregation for i2 = 1, . . . , N do Set P = 0 and Q = 0; for j1 = i1 do for k2 ∈ {i2 − S, . . . , i2 + S} do Compute weight wi2 k2 using (12) and (9); P = P + wi2 k2 f (j1 , k2 ) ; Q = Q + wi2 k2 ; end end for j1 ∈ { i1 − S, . . . , i1 + S}\{i1 } do for j2 ∈ {i2 − S, . . . , i2 + S} do Compute weight wi2 j2 using (13) and (9); P = P + wi2 j2 f (j1 , j2 ) ; Q = Q + wi2 j2 ; end end f˜(i1 , i2 ) = P/Q. end end Algorithm 1: Proposed processing along rows using lifting. 4 Experiments The denoising performance of the proposed method is compared with NLM and SNLM in Table 1. We have used standard grayscale images from15, 16 for our experiments. The Matlab implementation used to generate the results in this section is publicly available2 . The search windows for the three methods were set as follows. Suppose S be the search window for NLM (which we take as 2 http://in.mathworks.com/matlabcentral/fileexchange/64856 σ Method 5 10 20 30 50 5 10 House (256 × 256) 20 30 50 Montage (256 × 256) Noisy 34.1/83 28.1/60 22.1/34 18.6/22 14.2/12 34.2/83 28.1/61 22.1/36 18.6/25 14.1/15 NLM1 36.9/90 34.1/87 29.7/82 26.8/77 24.0/69 39.1/97 34.3/94 29.6/89 26.5/85 22.2/76 Darbon et al.4 36.1/90 31.4/75 26.1/51 22.8/36 18.6/21 38.4/89 30.9/76 25.8/52 22.6/38 18.6/24 SNLM13 36.6/89 33.6/86 29.4/81 26.5/76 23.7/69 39.3/97 34.6/94 29.7/89 26.7/84 22.5/77 Proposed 36.6/89 34.1/86 30.4/82 27.3/77 24.1/70 39.4/97 34.8/94 30.2/90 27.3/86 23.4/79 BM3D19 38.6/95 34.7/93 31.3/88 29.3/85 26.6/78 41.1/98 37.3/96 33.5/94 31.2/91 27.4/85 Method Boat (512 × 512) Man (1024 × 1024) Noisy 34.1/97 28.1/90 22.1/73 18.6/59 14.2/41 34.1/99 28.1/97 22.1/91 18.6/85 14.1/70 NLM1 35.1/96 30.8/89 26.7/78 24.7/70 23.0/62 35.3/98 31.1/95 27.5/88 25.8/83 24.2/76 Darbon et al.4 34.4/97 30.3/94 25.4/82 22.4/71 18.4/53 35.1/97 30.4/98 25.6/95 22.5/90 18.5/80 SNLM13 35.0/96 30.7/89 26.6/77 24.5/69 22.7/61 35.3/98 31.0/95 27.2/87 25.4/83 23.8/75 Proposed 34.9/96 30.7/89 26.8/77 24.7/70 22.9/62 35.1/98 31.1/95 27.5/88 25.8/ 84 24.2/78 BM3D19 37.3/98 33.9/96 30.8/92 29.0/88 26.7/81 37.3/99 34.1/98 31.2/96 29.5/94 27.4/89 Table 1 Comparison of the denoising performances on various images15 in terms of PSNR/SSIM at various noise standard deviations σ. The PSNRs are rounded to one decimal place, while the SSIMs (in %) are rounded to integer. S 7 Method NLM1 10 12 7 K=2 44 87 10 12 K=3 124 45 88 126 Darbon et al.4 0.33 0.60 0.84 0.33 0.62 0.85 SNLM13 0.31 0.39 0.45 0.32 0.40 0.46 Proposed 1.20 2.30 3.20 1.30 2.40 3.30 Table 2 Comparison of the run-time (in seconds) of the proposed approach with classical NLM for a 256 × 256 image. The computations were performed using Matlab on a 3.40 GHz Intel quad-core machine with 32 GB memory. reference). Following the original proposal,13 the window for SNLM is also set as S. For a fair comparison with NLM, we ensure that equal number of pixel are averaged in both methods. This is achieved if (2S1 + 1)2 + (2S2 + 1)2 = (2S + 1)2 . Moreover, following,14 we set S2 = S1 /2. These equations uniquely determine S1 and S2 (up to an integer rounding). Moreover, we normalize 8 (a) Clean (512 ⇥ 512). (b) Noisy, 18.6/60. (c) NLM,1 25.3/71. (d) SNLM, 25.3/71. (e) SNLM13 (smoothed), 24.9/69. (f) Proposed, 25.4/70. Fig smoothing (d) (d) using using aa bilateral-filter. bilateral-filter. Fig33Denoising Denoisingof ofMan Man1515 at atσ = = 30. 30. Notice Notice that that stripes stripes can be seen in (e) after smoothing The our proposal proposal (f) (f) isis visually visually ThePSNR/SSIM PSNR/SSIMvalues valueswith withreference reference to to the the clean clean image image are also provided. The result from our similar 335, 1.7, 1.7, and and 9.8 9.8 seconds. seconds. similartotoclassical classicalNLM NLM(c). (c). The The runtime runtime for for NLM, NLM, SNLM, and the proposed method are 335, We proposed approximation approximation(f) (f) Weused usedthe theparameter parametersettings settings mentioned mentioned in in the the main main text. The PSNR/SSIM between the proposed andthe theclassical classicalNLM NLM(c) (c)are are40.58/70.22, 40.58/70.22, whereas whereas the the values are 37.96/69.90 for SNLM (d). and (d). proposed approach gives comparable results inthe terms of PSNR comparison the smoothing parameters in (2) and (9) using relation β 2 =and α2SSIM. /(2K 17 +A 1).visual For the results in of the1,denoising is provided and4,4.and Weαcan clearly seenotice somefrom stripes in the images Table we set Kresults = 3, S = 10, S1 in = Fig. 9, S23 = = 10σ. We Table 1 that the 17 obtained approach using SNLM, with andresults without the boxed In contrast, proposed givesboth comparable inpost-processing terms of PSNR (see and SSIM. A areas). visual comparison there is hardly any artifacts present in the denoised image obtained using our method. A images timing of the denoising results is provided in Fig. 3 and 4. We can clearly see some stripes in the comparison is provided in Table While the proposed method(see is slower than areas). SNLM In (this is the obtained using SNLM, both with2.and without post-processing the boxed contrast, price we pay for removing the stripes), it is nevertheless significantly faster than NLM. there is hardly any artifacts present in the denoised image obtained using our method. A timing 4 We noteisthat thoughin Darbon is generally fastermethod than our proposal, its (this denoising comparison provided Table et 2. al. While the proposed is current slower than SNLM is the performance starts deteriorating with the increase in noise variance. This is evident from Table price we pay for removing the stripes), it is nevertheless significantly faster than NLM. 18 1 and Wethough also note that NLM SNLM fall short of our KSVD and BM3D19itsindenoising terms of WeFig. note4.that Darbon et al.4 and is generally faster than current proposal, denoising performance. Nevertheless, NLM continues to be variance. of interest This due toisits decentfrom denoising performance starts deteriorating with the increase in noise evident Table 20–23 18 19 capability, and importantly, the availability of fast approximations. As reported by other 1 and Fig. 4. We also note that NLM and SNLM fall short of KSVD and BM3D in terms of authors,24performance. NLM is quite Nevertheless, effective in preserving fine details, successfully removing noise. denoising NLM continues to bewhile of interest due to its decent denoising 20–23 capability, and importantly, the availability of fast approximations. As reported by other 24 authors, NLM is quite effective in preserving fine details, while successfully removing noise. 9 (a) Clean (512 ⇥ 512). (b) Noisy, 28.1/83. (c) NLM,1 34.7/94. (d) SNLM,13 34.6/94. (e) BM3D,19 37.2/97. (f) KSVD,18 36.9/97. (g) Darbon et al.,4 30.9/91. (h) Proposed, 34.6/94. kodim2316 Fig classical Fig 44 Denoising Denoising of kodim23 at σ = 10. The result obtained through our proposal (h) is visually similar to classical NLM PSNR/SSIM NLM (c). (c). The runtime for NLM, SNLM, and the proposed method are 335, 1.7, and 9.8 seconds. The PSNR/SSIM between 42.4/99.4 between the proposed approximation (h) and the classical NLM (c) are 43.33/99.7, whereas these values are 42.4/99.4 for some for (d) (d) and and 30.85/89.3 for (g). We have zoomed the region around the beak in (c), (d), (g), and (h). We can see some artifacts in (d) and residual noise in (g); the zooms in (c) and (h) are visually indistinguishable. artifacts 5 Conclusion We proposed a method that uses the idea of lifting from previous work13 to perform fast non-local means denoising of images. The proposed method does not give rise to undesirable artifacts (as was the case with the original proposal), and produces images whose denoising quality and PSNR/SSIM are comparable to non-local means. While this comes at the expense of added computation, the proposed method nevertheless is much faster than non-local means. In fact, the speedup is about 40x for practical parameter settings. 6 Acknowledgements The last author was supported by a Startup Grant from IISc and EMR Grant SB/S3/EECE/281/2016 from DST, Government of India. References 1 A. Buades, B. Coll, and J.-M. Morel, “A non-local algorithm for image denoising,” Proc. IEEE Conference on Computer Vision and Pattern Recognition, 2, pp. 60-65 (2005). 2 M. Mahmoudi and G. Sapiro, “Fast image and video denoising via nonlocal means of similar neighborhoods,” IEEE Signal Processing Letters, 12(12), pp. 839-842 (2005). 3 J. Wang, Y. Guo, Y. Ying, Y. Liu, and Q. Peng, “Fast non-local algorithm for image denoising,” Proc. IEEE International Conference on Image Processing, pp. 1429-1432 (2006). 4 J. Darbon, A. Cunha, T. F. Chan, S. Osher, and G. J. Jensen, “Fast nonlocal filtering applied to electron cryomicroscopy,” Proc. 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Asynchronous Programming in a Prioritized Form Mohamed A. El-Zawawy1,2 of Computer and Information Sciences Al Imam Mohammad Ibn Saud Islamic University Riyadh, Kingdom of Saudi Arabia arXiv:1501.00669v1 [] 4 Jan 2015 1 College 2 Department of Mathematics Faculty of Science Cairo University Giza 12613, Egypt [email protected] Abstract: Asynchronous programming has appeared as a programming style that overcomes undesired properties of concurrent programming. Typically in asynchronous models of programming, methods are posted into a post list for latter execution. The order of method executions is serial, but nondeterministic. This paper presents a new and simple, yet powerful, model for asynchronous programming. The proposed model consists of two components; a context-free grammar and an operational semantics. The model is supported by the ability to express important applications. An advantage of our model over related work is that the model simplifies the way posted methods are assigned priorities. Another advantage is that the operational semantics uses the simple concept of singly linked list to simulate the prioritized process of methods posting and execution. The simplicity and expressiveness make it relatively easy for analysis algorithms to disclose the otherwise un-captured programming bugs in asynchronous programs. Key–Words: Prioritized posting, asynchronous programming, operational semantics, context-free grammar, concurrent programming. 1 Introduction All contemporary appliances (mobile, desktop, or web applications) require high responsiveness which is conveniently provided by asynchronous programming. Hence application program interfaces (APIs) enabling asynchronous, non-blocking tasks, such as web access or file operations) are accommodated in dominant programming languages. APIs provide asynchronous programming but mostly in a hard way. For example consider the following situation. A unique user interface (UI) task thread is typically used to design and implement user interfaces. Hence events on that thread simulate tasks that change the UI state. Therefore when the UI cannot be redrawn or respond, it get freezed. This makes it sensible, in order to keep the application responding continuously to UI tasks, to run blocking I/O commands and long-lasting CPUbound asynchronously. Asynchronous programming has multi-threaded roots. This is so as APIs have been implemented using multi-threaded programs with shared-memory. Software threads execution is not affected by the number of processors in the system. This is justified by the fact that the threads are executed as recursive sequential softwares running concurrently with interleaved write and reads commands. The many possible interleavings in this case cause the complexity of models of concurrent programming [8]. In a complex process, atomic locking commands can be added for prevention and prediction of bad thread interleavings. The non-deterministic style of interleaving occurrence creates rarely appearing programming-errors which are typically very hard to simulate and fix. This difficulty lead researchers to design multi-threaded programs (of APIs) in the framework of asynchronous programming models [6, 7]. The relative simplicity of asynchronous programming makes it a convenient choice to implement APIs or reactive systems. This is proved by recent years intense use of asynchronous programming by servers, desktop applications, and embedded systems. The idea of asynchronous programming is to divide cumulative program executions into tasks that are brieflyrunning. Moreover accessing the shared memory, each task is executed as a recursive sequential software that specifies (posts) new methods to be executed later. Many attempts were made to present formal asynchronous programming models. However few attempts [5] were made to express and formalize the fact that posted tasks in asynchronous programs may well have different execution priorities. A big disadvantage of existing work [5] that considers execution priorities is the complexity of the models hosting such priorities. For example the work in [5] considers the execution priorities using several task-buffers which makes the solution a bit involved. This paper presents a simple, yet powerful, model for asynchronous programming with priorities for task posting. We call the proposed model AsynchP . The paper also presents a novel and robust operational semantics for the constructs of AsynchP . A simple singly linked list of prioritized posted tasks is used to precisely capture the posting process. Our proposed asynchronous model is to simplify analyzing asynchronous programs [4]. Motivating Example A motivating example of designing our model comes form the way hardware interactions take place in operating systems (more specifically in Windows) kernels. Concepts of prioritized interrupt sets are used to simulate these hardware interactions in an asynchronous style. For such applications a simple, yet powerful and mathematically well founded, model for prioritized asynchronous programming is required. Contribution The contributions of this paper are: 2 Prioritized Asynchronous gramming Model Pro- This section presents our model, AsynchP , for prioritized asynchronous programming. In AsynchP , each posted method has an execution priority. An asynchronous program execution is typically divided into quick-running methods (tasks). Tasks of higher priority get executed first and task of equal priorities are executed using the first come first serviced strategy. Asynchronous programming has an important application in reactive systems where a single task must not be allowed to run too long and to prevent executing other (potentially) highly prioritized tasks. Figure 1 presents the simple and powerful model AsynchP for prioritized asynchronous programming. Considering single local and global variables and using free syntax of expressions does not cause any generality lose. However each expression is built using the global variable of the program and the local variable of the active method. A AsynchP program P consists of a single global variable g and a sequence of methods denoted M1 , . . . , Mn . The Provided(e) statement continues executing the program provided that the expression e evaluates to a non-zero value. Each method M is expressed as a structure meth(m, l, S) of a method name, a single local variable l, and a top-level statement S. The sets of all program methods and statements are denoted by Meths and Stmts, respectively. Intuitively, the asynchronous call is modeled by the statement Synch(m(e),p) where: • the called method name is m, • the calling parameter is the expression e, and • the execution priority of this call is p. • A prioritized asynchronous programming model; AsynchP . • A novel operational semantics for AsynchP programs. Organization The rest of the paper is organized as follows. Section 2 presents the proposed prioritized asynchronous programming model – AsynchP . The semantics of prioritized asynchronous programming model; AsynchP is shown in Section 3 which is followed by Section 4 that reviews related work and presents directions for future work. The last section (Section 5) of the paper concludes it. We assume three levels of execution priorities; {high(1), medium(2), low(3)}. 3 Mathematical AsynchP Framework for This section presents a novel operational semantics for asynchronous programs built using AsynchP . Our semantics is based on a singly liked-list (which we call Asynchronous Linked List (ALL)) to host the posted methods. ALL is divided into three regions using pointers. The first, the middle, and last regions of ALL host posted methods that have high, medium, and low execution priorities, respectively. Definition 1 introduces formally the concept of (Asynchronous Node (AN)) to be used to build ALL. g ∈ G = Global variable names. l ∈ L = Method local variable names. p ∈ P = {high(1), medium(2), low(3)} = The set of all synchronization priorities. m ∈ M = The set of all method names. v ∈ V al = the set of possible values of local and global variables. S ∈ Stats ::= S1 ; S2 | g := e | l := e | Provided e | if e then St else Sf | while e do S | run m(e) | return() | Synch(m(e),p). M ∈ Meths ::= meth(m, l, S). P ∈ Programs ::= program(g, M ∗ ). Figure 1: AsynchP : a language model for a simple prioritized asynchronous programming. Definition 1 An asynchronous node (AN), n, is a single linked list node whose data contents are two locations containing: • x1 : a method name, and • x2 : a parameter expression. For a method call m(e) in a AsynchP program, we let Nodem(e) denotes the asynchronous node whose locations x1 and x2 contain m and e, respectively. The set of all asynchronous nodes is denoted by NodesA . Definition 2 introduces formally the concept of Asynchronous Linked List (ALL) that is to be used to accurately capturing the semantics of the constructs of the proposed asynchronous model. Definition 2 An asynchronous linked list (ALL), li =< f, c, eh , em >, (1) is a singly linked list whose nodes are asynchronous nodes (in NodesA ) such that: • f is a pointer to the first node of the list, • c is a pointer to the current node, and • eh , em are pointers to the last node in the list hosting a method of high and medium priorities, respectively. The set of all asynchronous linked lists is denoted by ListsA . Whenever a method gets posted, an asynchronous node is created and inserted into an asynchronous list. If the posted method is of priority h or m, the created node gets inserted after the nodes pointed to by eh or em , respectively. If the posted method is of priority l, the created node gets inserted at the end of the list. Whenever a posted method is to be executed, the method corresponding to the head of an asynchronous node is executed and that head gets removed form the list. These two operations are assumed to be carried out by the functions defined in Definition 3. Definition 3 Let li =< f, c, eh , em > be a asynchronous linked list (in ListsA ). We let • addA : NodesA ×P ×ListsA → ListsA denotes a map that adds a given node n of a given priority p after the node pointed to be li.ep in a given list li1 . • removeA : ListsA → NodesA × ListsA denotes a map that removes the first node of a given list li and return the removed node and the resulting linked list. Definition 4 introduces the states of our proposed operational semantics. Definition 4 Let program(g, M1 , . . . , Mn ), where Mi = meth(mi , li , Si ), be a program in AsynchP . An asynchronous program state (APS) is a triple (s,li,sk), where: • s is a partial map from G ∪ (M × L) to Val. • li is an asynchronous linked list. • sk is stack of method names. We let Mi .l and Mi .l denote li and Si , respectively. Each semantic state is a triple of a partial map captures the contents of global and local variables, an asynchronous linked list, and a stack of method names. 1 list Note that p ∈ {h, m, l}. If p = l, el is the last node in the li′ = addA (Node(m(e)), p, li) Synch(m(e), p)) : (s, li, sk) → (s, li′ , sk) is-empty(sk) = false sk′ = pop(sk) return() : (s, li, sk) → (s, li, sk′ ) (synchs ) (returns ) m′ = peek(sk) v = ke(s(g), s(m′ .l)k s′′ = s[m.l 7→ v] ′′ ′′ ′′ sk = push(sk, m) m.S : (s , li, sk ) → (s′ , li′ , sk′ ) (runs ) run m(e) : (s, li, sk) → (s′ , li′ , sk′ ) m′ = peek(sk) v = ke(s.g, m′ .l)k g := e : (s, li, sk) → (s[g 7→ v], li, sk) m′ = peek(sk) (:=sg ) v = ke(s.g, m′ .l)k l := e : (s, li, sk) → (s[m′ .l 7→ v], li, sk) m′ = peek(sk) ke(s.g, m′ .l)k = 6 0 (:=sl ) (ps ) Provided e : (s, li, sk) → (s, li, sk) m′ = peek(sk) ke(s.g, m′ .l)k = 0 (w1s ) while e do S : (s, li, sk) → (s, li, sk) m′ = peek(sk) ke(s.g, m′ .l)k = 6 0 S : (s, li, sk) → (s′′ , li′′ , sk′′ ) while e do S : (s′′ , li′′ , sk′′ ) → (s′ , li′ , sk′ ) (ws ) 2 while e do S : (s, li, sk) → (s′ , li′ , sk′ ) m′ = peek(sk) ke(s.g, m′ .l)k = 0 Sf : (s, li, sk) → (s′ , li′ , sk′ ) (ifs1 ) if e then St else Sf : (s, li, sk) → (s′ , li′ , sk′ ) m′ = peek(sk) ke(s.g, m′ .l)k = 6 0 ′ St : (s, li, sk) → (s , li′ , sk′ ) (ifs2 ) if e then St else Sf : (s, li, sk) → (s′ , li′ , sk′ ) S1 : (s, li, sk) → (s′′ , li′′ , sk′′ ) S2 : (s′′ , li′′ , sk′′ ) → (s′ , li′ , sk′ ) ′ ′ ′ (;s ) S1 ; S2 : (s, li, sk) → (s , li , sk ) Figure 2: Transition rules for statements. The stack is meant to keep the order in which methods call each other. Figures 2 and 3 present the transition rules of the proposed operational semantics. Some comments on the rules are in order. The rule synchs creates an asynchronous node corresponding to the method m and the parameter e. Using the map addA , the node then is added to the asynchronous list li to get the new list li′ . The rule (returns ), pops an element from the method stack as the return statement means that the top element of the stack is executed. The rule (runs ) first peeks the first element of the stack to get the local variable (m′ .l) of the currently active method. This local variable is then used together with the global variable to evaluate the expression e. The resulting value is used to modify the local variable of the method (m) that is to be executed. Then m is pushed into the stack and the statement of m is executed. The rule (progs ) first runs the statements of all methods of the program being executed then runs all statements of the methods that is posted. The posted statements are executed via the rules (⇒s1 ) and (⇒s2 ). 4 Related and Future Work Parallel, distributed, reactive, and concurrent programming have been attracting much researcher activities. The asynchronous programming methodologies include: • multi-threaded light-weight orchestration programming [19], • thread Join-based allocation, • typed synchronous programming languages [20], • functional sensible programming, (⇒s1 ) (s, empty, empty) ⇒ (s, empty, empty) (n, li′ ) = removeA (li) n.x1 .S : (s, li′ , empty) → (s′′ , li′′ , empty) (s′′ , li′′ , empty) ⇒ (s′ , empty, empty) (⇒s2 ) (s, li, empty) ⇒ (s′ , empty, empty) sk′′ = push(m, sk) S : (s, li, sk′′ ) → (s′ , li′ , sk′ ) meth(m, l, S) : (s, li, sk) → (s′ , li′ , sk′ ) (meths ) ∀1 ≤ i ≤ n. Mi : (si , lii , ski ) → (si+1 , lii+1 , ski+1 ) (sn+1 , lin+1 , empty) ⇒ (s′ , empty, empty) (progs ) program(g, M1 , . . . , Mn ) : (s1 , li1 , sk1 ) → (s′ , empty, empty) Figure 3: Transition rules for methods and programs. • promises, and • co-methods and futures agents [21, 22]. Event-based techniques for programming have been using continuations which are delimited monadic [17, 18]. Fork-join, task, async, and event functions appear not to rely on a specific language design. There is a big research debate about the relationship between threads and events in systems research [16]. In an asynchronous program, executions containing context switches bounded by a user-specified limit are explored by context-bounded verification [14, 15]. This context-bounding idea is not reasonable for programs with big number of events. Several treatments for this problem have been proposed for this problem. Without losing decidability, [13] proposed a context-minimizing technique permitting unbounded context switches. For asynchronous concurrent programs, in [12], the round-robin technique for scheduling is used to enable unbounded context switches. Sequential techniques are also used to analyze asynchronous programs. In [14], a source-to-source technique building sequential programs from multithreaded programs was proposed via under approximating the possible set of executions of the input program. A novel source-to-source transformation providing for any context-bound, a context-bounded under approximation was presented in [11]. A main issue of the work in [11] is that the resulting sequential program may host main states unreachable in the given asynchronous program. Other techniques like [10] treat this problem by repeatedly running the code to the control points where used-defined valued are needed. The work in [9] compares the techniques of asynchronous programs verifications that use verification-condition-checking against that use model-checking. One major results of this work is that eager approaches outperforms lazy ones. The work in [8] uses the construction using a bound on the task number, to reduce asynchronous programs into sequential programs via priority-preemptive schedulers. The work presented in this paper is close to sequentialization [14]; the concept describing compositional reductions to get sequential programs from concurrent ones. Although sequentialization started by checking multi-threaded programs with one contextswitch, it was developed later to treat a user-specified number of context-switches. These switches occur among statically-specified group of threads running using RR order [11]. In [2], a technique for treating context switches among an unspecified number of dynamically-created tasks was presented. This technique (in [2]) hence explicitly treats event-oriented asynchronous programs. For future work, it is interesting to devise static analyses for asynchronous programs using the model AsynchP [1]. Initial experiments show that our proposed model is expected to support devising robust and powerful analysis techniques. An examples of targeted analyses is dead-posting elimination which aims at removing the unnecessary posting statements from asynchronous programs. 5 Conclusion Main reason to use asynchronous programming is to overcome some problems of concurrent programming. The main idea of asynchronous programming is to post methods into a post list for latter execution. The order of executing these methods is nondeterministically serial. A new and simple, yet powerful, model for asynchronous programming was presented in this paper. More precisely, the paper proposed a contextfree grammar and an operational semantics for asynchronous programming. 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On the importance of graph search algorithms for DRGEP-based mechanism reduction methods Kyle E. Niemeyera , Chih-Jen Sungb,∗ arXiv:1606.07802v1 [] 24 Jun 2016 a Department of Mechanical and Aerospace Engineering Case Western Reserve University, Cleveland, OH 44106, USA b Department of Mechanical Engineering University of Connecticut, Storrs, CT 06269, USA Abstract The importance of graph search algorithm choice to the directed relation graph with error propagation (DRGEP) method is studied by comparing basic and modified depth-first search, basic and R-value-based breadth-first search (RBFS), and Dijkstra’s algorithm. By using each algorithm with DRGEP to produce skeletal mechanisms from a detailed mechanism for nheptane with randomly-shuffled species order, it is demonstrated that only Dijkstra’s algorithm and RBFS produce results independent of species order. In addition, each algorithm is used with DRGEP to generate skeletal mechanisms for n-heptane covering a comprehensive range of autoignition conditions for pressure, temperature, and equivalence ratio. Dijkstra’s algorithm combined with a coefficient scaling approach is demonstrated to produce the most compact skeletal mechanism with a similar performance compared to larger skeletal mechanisms resulting from the other algorithms. The computational efficiency of each algorithm is also compared by applying the DRGEP method with each search algorithm on the large detailed mechanism for n-alkanes covering n-octane to n-hexadecane with 2115 species and 8157 reactions. Dijkstra’s algorithm implemented with a binary heap priority queue is demonstrated as the most efficient method, with a CPU cost two orders of magnitude less than the other search algorithms. Keywords: Mechanism reduction, Skeletal mechanism, DRG, DRGEP, Graph search algorithm, Dijkstra’s algorithm ∗ Corresponding author Email address: [email protected] (Chih-Jen Sung) Preprint submitted to Combustion and Flame June 28, 2016 1. Introduction The directed relation graph (DRG) method for the skeletal reduction of large reaction mechanisms, originally developed by Lu and Law [1, 2], has been shown to be applicable to relevant transportation fuel components [3, 4]. Development of this approach has since focused on variants of the method [5–8], but DRG with error propagation (DRGEP) in particular has received much attention [6, 8–13]. The DRGEP differs mainly from DRG in that it takes error propagation down graph pathways into account. Further improvement of the DRGEP method is motivated by the large and continually increasing size of detailed reaction mechanisms for liquid transportation fuels [14]; for instance, see the recent biodiesel surrogate mechanism of Herbinet et al. [15] that contains 3299 species and 10806 reactions. The DRG method maps the coupling of species onto a directed graph and finds unimportant species for removal by eliminating weak graph edges using an error threshold, where the graph nodes represent species and graph edge weights indicate the dependence of one species on another. Following a simple graph search initiated at certain preselected target species (e.g., fuel, oxidizer, important pollutants), species unreached by the search are considered unimportant and removed from the skeletal mechanism. DRGEP modifies this approach slightly by considering the dependence of a target on other species due to a certain path as the product of intermediate edge weights and the overall dependence is the maximum of all path dependencies. One point that has not received much attention is the choice of graph search algorithm used to calculate this overall dependence. While the DRG method only needs to find connected graph nodes and can therefore use any search algorithm, caution must be taken when selecting the method used with DRGEP. The issue of calculating the overall dependence is actually a single-source shortest path problem [16] where the “distance” of the path is the product of intermediate edge weights rather than the sum, and the “shortest” path is that which has the maximum overall dependence rather than the minimum. Search algorithms that in general do not correctly calculate the overall dependence will underestimate the importance of species and cause premature removal from the skeletal mechanism. Further, the resulting skeletal mechanism is independent of the order of the species in the mechanism only if a specific algorithm can correctly capture and calculate 2 the overall dependence of species. Therefore, the reliability of various algorithms needs to be studied by determining whether each is dependent on the order of species in a detailed mechanism. The efficiency of the graph search algorithm is also an important factor due to the recent use of DRGEP in dynamic skeletal reduction approaches [10–12]. In the worst case, DRGEP is applied at every spatial grid location and each time step to generate locally relevant skeletal mechanisms, although as Shi et al. [12] demonstrated this can be eased by combining dynamic DRGEP-based reduction with an adaptive multi-grid chemistry model. Regardless, the computational cost of the search algorithm increases with the number of species and must be considered when comparing algorithms due to the large and ever-increasing size of detailed reaction mechanisms. Most DRGEP studies reported using algorithms based on either depthfirst search (DFS) [8] or breadth-first search (BFS) [10–13], but no comparison has been performed. Here we compare such methods with Dijkstra’s algorithm, the classical solution to the single-source shortest path problem [16, 17], in order to demonstrate the weaknesses of DFS- and BFS-based approaches and the subsequent effectiveness and reliability of Dijkstra’s algorithm. First, by randomly shuffling the order of species in the detailed mechanism for n-heptane of Curran et al. [18], we show that DFS- and BFSbased algorithms can generate results dependent on the order of species in the reaction mechanism while Dijkstra’s algorithm generates consistent results regardless of species order. Second, we demonstrate that, for a given error limit, more compact skeletal mechanisms are possible with DRGEP by using Dijkstra’s algorithm. This is done by comparing skeletal mechanisms generated with different search algorithms for n-heptane covering a comprehensive range of autoignition conditions for pressure, temperature, and equivalence ratio. Third, we compare the computational efficiency of the various search algorithms by calculating the CPU time required to perform the graph search on the large detailed mechanism of Westbrook et al. [19] covering oxidation of n-alkanes from n-octane to n-hexadecane containing 2115 species and 8157 reactions. 2. Methodology 2.1. DRGEP Method In the current work we use the DRGEP method of Pepiot-Desjardins and Pitsch [6], described here in brief. Accurate calculation of the production of a 3 species A that is strongly dependent on another species B requires the presence of species B in the reaction mechanism. This dependence is expressed with the direct interaction coefficient (DIC): PnR i i=1 νA,i ωi δB , (1) rAB = max (PA , CA ) where PA = nR X max (0, νA,i ωi ) , (2) max (0, −νA,i ωi ) , (3) i=1 CA = nR X i=1 ( 1 if reaction i involves species B, i δB = 0 otherwise, (4) A and B represent the species of interest (with dependency in the A → B direction meaning that A depends on B ), i the i th reaction, νA,i the stoichiometric coefficient of species A in the i th reaction, ωi the overall reaction rate of the i th reaction, and nR the total number of reactions. After calculating the DIC for all species pairs, a graph search is performed starting at user-selected target species to find the dependency paths for all species from the targets. A path-dependent interaction coefficient (PIC) represents the error propagation through a certain path and is defined as the product of intermediate DICs between the target T and species B through pathway p: n−1 Y (5) rT B,p = rSj Sj+1 , j=1 where n is the number of species between T and B in pathway p and Sj is a placeholder for the intermediate species j starting at T and ending at B. An overall interaction coefficient (OIC) is then defined as the maximum of all PICs between the target and each species of interest: RT B = max (rT B,p ) . all paths p (6) Pepiot-Desjardins and Pitsch [6] also proposed a coefficient scaling procedure to better relate the OICs from different points in the reaction system 4 evolution that we adopt here. A pseudo-production rate of a chemical element a based on the production rates of species containing a is defined as X Pa = Na,S max (0, PS − CS ) , (7) all species S where Na,S is the number of atoms a in species S and PS and CS are the production and consumption rates of species S as given by Eqs. 2 and 3, respectively. The scaling coefficient for element a and target species T at time t is defined as Na,T |PT − CT | . (8) αa,T (t) = Pa For the set of elements {E}, the global normalized scaling coefficient for target T at time t is   αa,T (t)  αT (t) = max  . (9) a∈{E} max αa,T (t) all time Given a set of kinetics data {D} and target species {T }, the overall importance of species S to the target species set is   RS = max max (αT RT S ) . (10) T ∈{T } k∈{D} all time, k We employ the same sampling method as given by Niemeyer et al. [8], where constant volume autoignition simulations are performed using SENKIN [20] with CHEMKIN-III [21]. Species are considered unimportant and removed if their RS value falls below a cutoff threshold εEP , which is selected using an iterative procedure based on a user-defined error limit for ignition delay prediction [8]. Reactions are eliminated if any participating species are removed. 2.2. Graph search algorithms In this study we compare the results of DRGEP using basic DFS, modified DFS, basic BFS, R-value-based BFS (RBFS), and Dijkstra’s algorithm. Cormen et al. [16] presented detailed discussion of and pseudocode for DFS, BFS, and Dijkstra’s algorithm, while the modified DFS used by Niemeyer et al. [8] and RBFS of Liang et al. [10] differ only slightly from the basic DFS and BFS. 5 2.2.1. DFS-based algorithms The DFS initiates at a root node, in this case a target species, and explores the graph edges connecting to other nodes. The first node found is added to a last in, first out stack, then the search moves to this node and repeats. In this manner, the search continues deeper down the graph pathway until it either reaches a node with no connections or all the connecting nodes have been explored, then backtracks one position up the stack. The search is performed separately using each target species as the root node and the maximum OIC is stored for each species. Lu and Law first introduced the DFS in this context for the DRG method [1]. The modified DFS used by Niemeyer et al. [8] for the DRGEP method (but not described in detail) differs from the basic DFS in that the OIC values for all target species are set to unity before starting the search at the first target. The search then only repeats starting at other target species not discovered in the initial search. The resulting OIC values combine the dependence of all targets on the species and this prevents the use of coefficient scaling based on individual target species activity. 2.2.2. BFS-based algorithms The BFS initiates at the root node and explores all adjacent graph edges, adding connected nodes to a first in, first out queue. After discovering all connected nodes, the search moves to the first node in the queue and restarts the procedure, moving to the next node in the queue after discovering all connected nodes. Previously discovered nodes are not repeated. As with the DFS, the search initializes at each target separately and the maximum OIC for each species is stored. The R-value-based BFS (RBFS), first introduced by Liang et al. [10, 11] and also used by Shi et al. [12, 13], differs from the standard BFS in that PICs smaller than the error threshold εEP are not explored. In other words, the search ignores graph pathways that would lead to an OIC below the cutoff value and only allows discovery of important pathways. This increases efficiency by avoiding exploration of unimportant pathways but causes the RBFS to depend on the value of εEP , unlike the other search methods. As such, the primary application of this algorithm lies in dynamic reduction where the threshold value is known a priori, rather than general comprehensive skeletal reduction where the threshold is to be determined iteratively based on a user-defined error limit [8]. 6 2.2.3. Dijkstra’s algorithm Dijkstra’s algorithm, originally introduced by Dijkstra [17] and discussed in detail by Cormen et al. [16], differs from DFS, BFS, and their variants because it was specifically designed to calculate the shortest paths from a single source node to all other nodes rather than simply search the graph to find connected nodes. As mentioned previously, calculating OIC values is a shortest path problem where the “shortest” path is that with the maximum product of intermediate edge weights. This is the only modification needed to apply Dijkstra’s algorithm to the current problem. In brief, Dijkstra’s algorithm functions similarly to BFS except that it starts with the set of graph nodes stored in a max-priority queue. The algorithm finds and removes the node with the maximum OIC value (initially, the root node) from the queue and explores adjacent nodes to calculate or update OIC values. When all neighboring nodes are explored, the procedure restarts at the node with the next-highest OIC in the queue. The search completes when the queue is empty. As with previously-discussed search algorithms, this is performed separately for each target species as root node. See the Appendix for pseudocode describing the algorithm as applied to the DRGEP method. Transportation and internet routing applications routinely use Dijkstra’s algorithm and so much effort has been focused on optimizing its performance. A naive implementation requires a running time of O(V 2 ) where V is the number of nodes. For sparse graphs, i.e. where the number of edges E follows E ∼ O(V 2 / log V ), the runtime can be improved significantly to O(E log V ) by using an adjacency list to store the immediate neighbors of nodes and a binary max-heap priority queue [16]. Using a Fibonacci heap can reduce the runtime further to O(E + V log V ) [16, 22], although in many cases binary heaps prove more efficient [23]. These will be compared in order to determine the most efficient implementation for DRGEP. Briefly, a binary max-heap is a tree where each node has at most two child nodes and the associated key (in this case, OIC) is always greater than or equal to the keys of the child nodes. By storing the nodes in this manner, a costly search is not required to find the node with the highest key during the operation of Dijkstra’s algorithm. Fibonacci heaps function similarly, but rather than a single tree, the heaps are made up of a collection of trees with more relaxed structures designed to avoid unnecessary operations. Detailed discussion of the heap implementations is beyond the scope of this paper and we refer interested readers to Cormen et al. [16]. 7 3. Results and discussion 3.1. Reliability of graph search algorithms In order to demonstrate the dependence of DFS- and BFS-based methods on the order of species in a reaction mechanism, all five graph search algorithms are used with DRGEP to generate skeletal mechanisms from the detailed mechanism for n-heptane of Curran et al. [18], containing 561 species and 2539 reactions, where the order of species is randomly shuffled. Basic DFS, basic BFS, RBFS, and Dijkstra’s algorithm are used both with and without the target-based coefficient scaling while the modified DFS is not compatible with this and so is used without it. Autoignition chemical kinetics data are sampled from initial conditions at 1000 K, 1 atm, and equivalence ratios of 0.5–1.5. Oxygen, nitrogen, n-heptane, and the hydrogen radical are selected as target species. Figure 1 shows the ignition delay error of skeletal mechanisms at varying levels of detail generated by DRGEP using RBFS and Dijkstra’s algorithm with coefficient scaling and the modified DFS. Basic DFS and BFS demonstrate similar dependence on species order to the modified DFS and thus are omitted from Fig. 1 for clarity, but the skeletal mechanism sizes from each may be different as will be shown. Similarly, the coefficient scaling comparison is also omitted because its application does not affect dependence on species order, although the resulting skeletal mechanism sizes are different. Comparison of results from the original mechanism and five mechanisms with randomly shuffled species illustrates the dependence of the modified DFS on the order of species in the mechanism while RBFS and Dijkstra’s algorithm produce consistent results regardless of species order. This is because DFS and BFS explore graphs based on the order of nodes while Dijkstra’s algorithm follows the strongest path independent of order [16]. In addition, Fig. 1 shows that both Dijkstra’s algorithm and RBFS produce smaller skeletal mechanisms before the error becomes unacceptably large. The dependence of DFS, modified DFS, and BFS on species order stems from the fact that these algorithms do not in general calculate the correct OIC for every species and as such can underestimate the importance of species, causing unwarranted removal. RBFS produces the same results as Dijkstra’s algorithm because it prevents exploration of unimportant paths, i.e. paths with PICs (see Eq. 5) less than the threshold, and therefore only discovers paths that lead to an OIC greater than the threshold value. While this value may be smaller than the correct OIC as calculated by Dijkstra’s algorithm, 8 the species is nonetheless considered important and retained in the skeletal mechanism. This subtle error could be significant, though, if the DRGEP method is followed by a further reduction stage such as sensitivity analysis that depends on the OIC values [8]. 3.2. Effectiveness of graph search algorithms Skeletal mechanisms for n-heptane are generated covering a comprehensive range of conditions using basic DFS, modified DFS, basic BFS, and Dijkstra’s algorithm. Basic DFS, BFS, and Dijkstra’s algorithm are used with and without the coefficient scaling in order to determine its effect as well. Autoignition chemical kinetics data are sampled from initial conditions at 600–1600 K, 1–20 atm, and equivalence ratios of 0.5–1.5. Oxygen, nitrogen, n-heptane, and the hydrogen radical are selected as target species, and the error limit in ignition delay prediction is 30%. RBFS is not compared here because it is not suited for a comprehensive skeletal reduction due to its dependence on threshold value; additionally, based on the results in the previous section we assume the resulting skeletal mechanism would match that from Dijkstra’s algorithm. The results using the original mechanism of Curran et al. [18] are summarized in Table 1. All search methods lead to skeletal mechanisms with similar performance, but Dijkstra’s algorithm with coefficient scaling produces the most compact skeletal mechanism. BFS and modified DFS produce similar results while basic DFS is unable to generate a comparable skeletal mechanism for the given error limit. This weakness is most likely due to the fact that basic DFS finds only the first path from target to species, which typically contains many intermediate species and severely underestimates the OICs for some important species. The modified DFS performs better by inserting the other targets into the pathways and therefore increasing the OIC values for important species, while BFS finds the path with the shortest number of intermediate nodes [16]. Table 1 also demonstrates the need for coefficient scaling. Both basic DFS and Dijkstra’s algorithm generate more compact skeletal mechanisms with the scaling, while BFS actually produces a slightly larger skeletal mechanism. Without scaling, Dijkstra’s algorithm is unable to produce a more compact skeletal mechanism than the other methods, despite its ability to generate results independent of species order in the parent mechanism. 9 3.3. Efficiency of graph search algorithms The computational costs of DFS, modified DFS, basic BFS, RBFS, and Dijkstra’s algorithm are compared by applying the DRGEP method to the detailed mechanism of Westbrook et al. [19] for n-alkanes covering n-octane through n-hexadecane, which contains 2115 species and 8157 reactions. Kinetics data are sampled from constant volume autoignition of n-decane with initial conditions covering 600–1600 K, 1–20 atm, and equivalence ratios of 0.5–1.5. The efficiency improvements available to Dijkstra’s algorithm including use of adjacency list, binary heap, and Fibonacci heap are also compared to the naive implementation. The average computational costs of DFS, modified DFS, BFS, Dijkstra’s algorithm, and its improvements are listed in Table 2. In order to perform a fair comparison, a version of Dijkstra’s algorithm modified to similarly depend on the threshold value is compared with RBFS. Figure 2 shows the average computational cost of RBFS, Dijkstra’s algorithm, and its improvements as a function of threshold value. The most important result shown in Table 2 and Fig. 2 is that Dijkstra’s algorithm implemented with binary heaps is faster than all other search algorithms, and is in fact more efficient than the same algorithm implemented with Fibonacci heaps even though the theoretical time limit with binary heaps is higher. This result is consistent with literature comparisons for sparse graphs [24, 25]. 4. Concluding remarks Graph search algorithms used in the DRGEP method for skeletal mechanism reduction are compared. Basic DFS, basic BFS, modified DFS, RBFS, and Dijkstra’s algorithm are implemented in DRGEP and used to generate (1) skeletal mechanisms covering a limited range of conditions using a randomly shuffled detailed mechanism for n-heptane to determine the dependence of results on species order and (2) skeletal mechanisms covering a comprehensive range of conditions to determine the effectiveness of each algorithm. The RBFS algorithm is not used to generate a comprehensive skeletal mechanism because it is not suitable for use when the cutoff threshold is not known a priori. Both Dijkstra’s algorithm and RBFS are able to generate consistent results independent of species order, and Dijkstra’s algorithm used with target-based coefficient scaling generates a more compact skeletal mechanism than the other methods. Even with the improved search algorithm, however, the size of the resulting skeletal mechanism (131 species and 651 10 reactions) is larger than that of DRGEP with sensitivity analysis (108 species and 406 reactions) [8], suggesting that the post-DRGEP sensitivity analysis is still required. In addition to the reliability and effectiveness of the graph search algorithms, the efficiency is also compared due to its importance to dynamic DRGEP approaches. Efficiency improvements available to Dijkstra’s algorithm are also implemented and compared to the other search algorithms. Dijkstra’s algorithm with a binary heap priority queue runs two orders of magnitude faster than the other search methods and is the most efficient implementation of Dijkstra’s algorithm. As such, this approach is recommended for use in dynamic skeletal reduction using DRGEP. Dynamic approaches that avoid entirely repeating the search by updating graph edge values have also been developed [23, 26] and will be the focus of future work. Acknowledgments This work was supported by the National Science Foundation under Grant No. 0932559 and the Department of Defense through the National Defense Science and Engineering Graduate Fellowship program. Appendix The following pseudocode describes Dijkstra’s algorithm for use in the DRGEP method to calculate the OICs for all species relative to a target species T using the set of DIC values rAB , adapted from Cormen et al. [16]. The adjacency list adj contains the list of adjacent (directly connected) species in the directed graph. Operations are left in general form to be applicable to other implementations of Dijkstra’s algorithm using heaps as well as the basic version. The procedure MAKEQ (Q) generates the maxpriority queue Q, MAXQ (R, Q) returns the node with the highest OIC value, REMQ (u, Q) removes the node u from Q, and UPDATE (R, v, Rtmp ) increases the OIC value for node v to Rtmp . subroutine Dijkstra ( rAB , adj, T ) R ← zeros R(T ) ← 1.0 call MAKEQ (Q) while Q 6= ∅ do u ← MAXQ (R, Q) 11 call REMQ (u, Q) for each node v ∈ adj(u) do Rtmp ← R(u) · ruv if Rtmp > R(v) then call UPDATE (R, v, Rtmp ) end if end for end while return R end subroutine Dijkstra References [1] T. Lu, C. K. Law, Proc. Combust. Inst. 30 (2005) 1333–1341. [2] T. Lu, C. K. Law, Combust. Flame 146 (2006) 472–483. [3] T. Lu, C. K. Law, Combust. Flame 144 (2006) 24–36. [4] T. Lu, C. K. Law, Combust. Flame 154 (2008) 153–163. [5] X. L. Zheng, T. Lu, C. K. Law, Proc. Combust. Inst. 31 (2007) 367–375. [6] P. Pepiot-Desjardins, H. Pitsch, Combust. Flame 154 (2008) 67–81. [7] W. Sun, Z. Chen, X. Gou, Y. Ju, Combust. Flame 157 (2010) 1298–1307. [8] K. E. Niemeyer, C. J. Sung, M. P. Raju, Combust. Flame 157 (2010) 1760–1770. [9] I. G. Zsély, T. Nagy, J. M. Simmie, H. J. Curran, in: 4th European Combustion Meeting, 810045, 2009. [10] L. Liang, J. Stevens, J. T. Farrell, Proc. Combust. Inst. 32 (2009) 527– 534. [11] L. Liang, J. G. Stevens, S. Raman, J. T. Farrell, Combust. Flame 156 (2009) 1493–1502. [12] Y. Shi, L. Liang, H.-W. Ge, R. D. Reitz, Combust. Theor. Model. 14 (2010) 69–89. 12 [13] Y. Shi, H.-W. Ge, J. L. Brakora, R. D. Reitz, Energy Fuels 24 (2010) 1646–1654. [14] T. Lu, C. K. Law, Prog. Energy Comb. Sci. 35 (2009) 192–215. [15] O. Herbinet, W. J. Pitz, C. K. Westbrook, Combust. Flame 157 (2010) 893–908. [16] T. H. Cormen, C. E. Leiserson, R. L. Rivest, C. Stein, Introduction to Algorithms, MIT Press, Cambridge, MA, 2nd edition, 2001. [17] E. W. Dijkstra, Numer. Math. 1 (1959) 269–271. [18] H. Curran, P. Gaffuri, W. Pitz, C. K. Westbrook, Combust. Flame 114 (1998) 149–177. [19] C. K. Westbrook, W. J. Pitz, O. Herbinet, H. J. Curran, E. J. Silke, Combust. Flame 156 (2009) 181–199. [20] A. E. Lutz, R. J. Kee, J. A. Miller, SENKIN: A FORTRAN program for predicting homogeneous gas phase chemical kinetics with sensitivity analysis, Sandia National Laboratories Report No. SAND 87-8248, 1988. [21] R. J. Kee, F. M. Rupley, E. Meeks, J. A. Miller, CHEMKIN-III: A FORTRAN chemical kinetics package for the analysis of gas-phase chemical and plasma kinetics, Sandia National Laboratories Report No. SAND 96-8216, 1996. [22] M. L. Fredman, R. E. Tarjan, J. ACM 34 (1987) 596–615. [23] D. Wagner, T. Willhalm, Lect. Notes Comput. Sci. 4393 (2007) 23–36. [24] B. Cherkassky, A. Goldberg, T. Radzik, Math. Program. 73 (1996) 129– 174. [25] A. V. Goldberg, R. E. Tarjan, Expected performance of Dijkstra’s shortest path algorithm, Technical Report TR-530-96, NEC Research Institute, Princeton University, 1996. [26] L. Roditty, U. Zwick, Lect. Notes Comput. Sci. 3221 (2004) 580–591. 13 Algorithm DFS no scaling scaling mod. DFS # Species # Reactions Max. Error 461 449 2304 2267 19% 19% 173 868 28% BFS no scaling scaling 180 207 891 921 29% 25% Dijkstra no scaling scaling 178 131 986 651 23% 17% Table 1: Comparison of n-heptane skeletal mechanism sizes generated by the DRGEP method using DFS, BFS, and Dijkstra’s algorithm with and without coefficient scaling and modified DFS. The original mechanism contains 561 species and 2539 reactions [18]. 14 Algorithm Mean cost (sec) Cost normalized by Dijkstra DFS 1.48 ×10−1 2.20 mod. DFS 1.46 ×10−1 2.18 BFS 8.30 ×10−2 1.24 Dijkstra (naive) Dijkstra (adj) Dijkstra (binary heap) Dijkstra (Fibonacci heap) 6.72 ×10−2 1.18 ×10−2 1.5 ×10−3 3.4 ×10−3 1.00 0.176 0.028 0.051 Table 2: Comparison of average CPU time costs using DFS, modified DFS, BFS, and Dijkstra’s algorithm, which includes the naive implementation, use of adjacency list, binary heaps, and Fibonacci heaps. The kinetics data used are generated from n-decane autoignition over a range of initial temperatures and pressures, and at varying equivalence ratios, from a detailed mechanism consisting of 2115 species and 8157 reactions [19]. List of Figures 1 2 Error in autoignition delay prediction as a function of number of species for skeletal mechanisms of n-heptane generated by DRGEP using Dijkstra’s algorithm and RBFS with coefficient scaling and modified DFS. The solid lines indicate the results using the original detailed mechanism while the dashed lines indicate the results obtained from five versions with randomlyshuffled species. All 12 results using Dijkstra’s algorithm and RBFS coincide. . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Comparison of average CPU time cost as a function of threshold value using RBFS and Dijkstra’s algorithm, which includes the naive implementation, use of adjacency list, binary heaps, and Fibonacci heaps. The kinetics data used are generated from n-decane autoignition over a range of initial temperatures and pressures, and at varying equivalence ratios, from a detailed mechanism consisting of 2115 species and 8157 reactions [19]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 15 100 Error (%) 80 modified DFS (five random mechanisms) 60 modified DFS (original mechanism) 40 20 0 Dijkstra's algorithm and RBFS (original + five random mechanisms) 50 100 150 200 250 Number of species Figure 1: Error in autoignition delay prediction as a function of number of species for skeletal mechanisms of n-heptane generated by DRGEP using Dijkstra’s algorithm and RBFS with coefficient scaling and modified DFS. The solid lines indicate the results using the original detailed mechanism while the dashed lines indicate the results obtained from five versions with randomly-shuffled species. All 12 results using Dijkstra’s algorithm and RBFS coincide. 16 RBFS Dijkstra Dijkstra (adj) Dijkstra (binary) Dijkstra (Fibonacci) -1 CPU Time (sec) 1x10 1x10-2 1x10-3 1x10-6 1x10-5 1x10-4 1x10-3 1x10-2 Threshold value Figure 2: Comparison of average CPU time cost as a function of threshold value using RBFS and Dijkstra’s algorithm, which includes the naive implementation, use of adjacency list, binary heaps, and Fibonacci heaps. The kinetics data used are generated from ndecane autoignition over a range of initial temperatures and pressures, and at varying equivalence ratios, from a detailed mechanism consisting of 2115 species and 8157 reactions [19]. 17 

arXiv:1212.4590v1 [math.AP] 19 Dec 2012 RELATIVE PARAMETRIZATION OF LINEAR MULTIDIMENSIONAL SYSTEMS J.F. Pommaret CERMICS, Ecole Nationale des Ponts et Chaussées, 6/8 Av. Blaise Pascal, 77455 Marne-la-Vallée Cedex 02, France E-mail: [email protected], [email protected] URL: http://cermics.enpc.fr/∼pommaret/home.html ABSTRACT : In the last chapter of his book ”The Algebraic Theory of Modular Systems ” published in 1916, F. S. Macaulay developped specific techniques for dealing with ”unmixed polynomial ideals ” by introducing what he called ”inverse systems ”. The purpose of this paper is to extend such a point of view to differential modules defined by linear multidimensional systems, that is by linear systems of ordinary differential (OD) or partial differential (PD) equations of any order, with any number of independent variables, any number of unknowns and even with variable coefficients. The first and main idea is to replace unmixed polynomial ideals by pure differential modules. The second idea is to notice that a module is 0-pure if and only if it is torsion-free and thus if and only if it admits an ” absolute parametrization ” by means of arbitrary potential like functions, or, equivalently, if it can be embedded into a free module by means of an ” absolute localization ”. The third idea is to refer to a difficult theorem of algebraic analysis saying that an r-pure module can be embedded into a module of projective dimension r, that is a module admitting a projective resolution with exactly r operators. The fourth and final idea is to establish a link between the use of extension modules for such a purpose and specific formal properties of the underlying multidimensional system through the use of involution and a ”relative localization ” leading to a ” relative parametrization ”. The paper is written in a rather effective self-contained way and we provide many explicit examples that should become test examples for a future use of computer algebra. KEY WORDS : Unmixed ideal, algebraic analysis, homological algebra, extension module, projective dimension, torsion-free module, pure module, characteristic variety, formal integrability, involution, Spencer operator, inverse system. 1) INTRODUCTION : Let D = K[d1 , ..., dn ] = K[d] be the ring of differential operators with coefficients in a differential field K with n commuting derivations ∂1 , ..., ∂n and commutation relations di a = adi +∂i a, ∀a ∈ K. If y 1 , ..., y m are m differential indeterminates, we may identify Dy 1 + ... + Dy m = Dy with Dm and consider the finitely presented left differential module M with presentation Dp → Dm → M → 0 determined by a given linear multidimensional system with n independent variables, m unknowns and p equations. Applying the functor homD (•, D), we get the exact sequence 0 → homD (M, D) → Dm → Dp −→ N −→ 0 of right differential modules that can be transformed by a side-changing functor to an exact sequence of finitely generated left differential modules. This new presentation corresponds to the formal adjoint ad(D) of the linear differential operator D determined by the initial presentation but now with p unknowns and m equations, obtaining therefore a new finitely generated left differential module N and we may consider homD (M, D) as the module of equations of the compatibility conditions (CC) of ad(D), a result not evident at all (Compare to [24]). Using now a maximum free submodule 0 −→ Dl −→ homD (M, D) 1 and repeating this standard procedure while using the well known fact that ad(ad(D)) = D, we obtain therefore an embedding 0 → homD (homD (M, D), D) → Dl of left differential modules for a certain integer 1 ≤ l < m because K is a field and thus D is a noetherian bimodule over itself, a result leading to l = rkD (homD (M, D)) = rkD (M ) < m as in ([19], p 178,201)(See section 3 for the definition of the differential rankrkD ). Now, the kernel of the map ǫ : M → homD (homD (M, D), D) : m → ǫ(m)(f ) = f (m), ∀f ∈ homD (M, D) is the torsion submodule t(M ) ⊆ M and ǫ is injective if and only if M is torsion-free, that is t(M ) = 0. In that case, we obtain by composition an embedding 0 → M → Dl of M into a free module that can also be obtained by localization if we introduce the ring of fractions S −1 D = DS −1 when S = D − {0}. This result is quite important for applications as it provides a (minimum) parametrization of the linear differential operator D and amounts to the controllability of a classical control system when n = 1 ([24], p 258). This parametrization will be called an ”absolute parametrization ” as it only involves arbitrary ”potential-like ” functions (See [1], [9], [18], [19], [20], [24], [25] and [32] for more details and examples, in particular that of Einstein equations). The purpose of this paper is to extend suh a result to a much more general situation, that is when M is not torsion-free, by using unexpected results first found by F.S. Macaulay in 1916 through his study of ”inverse systems ” for ”unmixed polynomial ideals ”. For this we define the purity filtration : 0 = tn (M ) ⊆ tn−1 (M ) ⊆ ... ⊆ t1 (M ) ⊆ t0 (M ) = t(M ) ⊆ M by introducing tr (M ) = {m ∈ M | cd(Dm) > r} where the codimension of Dm is n minus the dimension of the characteristic variety determined by m in the corresponding system for one unknown. The module M is said to be r-pure if tr (M ) = 0, tr−1 (M ) = M or, equivalently, if cd(M ) = cd(N ) = r, ∀N ⊂ M and a torsion-free module is a 0-pure module. Moreover, when K = k = cst(K) is a field of constants and m = 1, a pure module is unmixed in the sense of Macaulay, that is defined by an ideal having an equidimensional primary decomposition. Example 1.1 : As an elementary example with K = k = Q, m = 1, n = 2, p = 2, the differential module defined by d22 y = 0, d12 y = 0 is not pure because z ′ = d2 y satisfies d2 z ′ = 0, d1 z ′ = 0 while z” = d1 y only satisfies d2 z” = 0 and ((χ2 )2 , χ1 χ2 ) = (χ1 ) ∩ (χ1 , χ2 )2 . We obtain therefore the purity filtration 0 = t2 (M ) ⊂ t1 (M ) ⊂ t0 (M ) = t(M ) = M with strict inclusions as 0 6= z ′ ∈ t1 (M ) while z” ∈ t0 (M ) but z” ∈ / t1 (M ). From the few (difficult) references ([1],[9],[15],[18],[27]) dealing with extension modules extr (M ) = and purity in the framework of algebraic analysis, it is known that M is r-pure if and only if there is an embedding 0 → M → extrD (extrD (M, D), D). Indeed, the case r = 0 is exactly the one already considered because ext0D (M, D) = homD (M, D) and the ker/coker exact sequence: extrD (M, D) 0 −→ ext1 (N ) −→ M −→ ext0 (ext0 (M )) −→ ext2 (N ) −→ 0 allows to test the torsion-free property of M in actual practice by using the double-duality formula t(M ) = ext1 (N ) as in ([19]). Also, when r ≥ 1, a similar construction that we shall recall and illustrate in section 4 provides a finitely generated module L with projective dimension pdD (L) = r, that is a minimum resolution of L with only r operators, and an embedding 0 → M → L that allows to exhibit a relative parametrization of D because now the parametrizing potential-like functions are no longer arbitrary but must only depend on arbitrary functions of n − r variables. Example 1.2 : With K = k = Q, m = 2, n = 3, r = 1, the differential module M defined by the involutive system Φ1 ≡ d3 y 1 = 0, Φ2 ≡ d3 y 2 = 0, Φ3 ≡ d2 y 1 − d1 y 2 = 0 is 1-pure and admits the resolution 0 −→ D −→ D3 −→ D2 −→ M −→ 0. The differential module L defined by the system d3 z = 0 is also 1-pure and admits the resolution 0 −→ D −→ D −→ L −→ 0. We finally obtain the relative parametrization y 1 = d1 z, y 2 = d2 z providing the strict inclusion M ⊂ L. In a simple way, this result can be considered as a measure of how far a module is from being projective, recalling that a module P is projective if there exists another (projective) module Q 2 and a free module F such that P ⊕ Q ≃ F . We adapt the ”relative localization” technique used by Macaulay and combine it with the ”involution” technique used in the formal theory of systems of partial differential equations in order to obtain an explicit procedure for determining L when M is given. Many examples will illustrate these new methods that avoid the previous abstract arguments based on ”double duality”. In particular, original non-commutative examples will also be presented. However, we point out the fact that the latter method can be adapted without any change to the case of systems with variables coefficients as it only depends on the use of adjoint operators but the following example will explain by itself the type of difficulty involved. Example 1.3 : Starting now with K = Q(x1 , x2 ), m = 2, n = 3, r = 1, the new differential module M defined by d3 y 1 = 0, d3 y 2 = 0, d2 y 1 − d1 y 2 + x2 y 2 = 0 is also 1-pure and the differential module L is again defined by d3 z = 0 as in the previous example. However we obtain the totally different relative parametrization y 1 = d12 z − x2 d2 z + z, y 2 = d22 z providing the strict inclusion M ⊂ L. More generally, we may consider a constant parameter a ∈ k = Q and consider the new system d3 y 1 = 0, d3 y 2 = 0, d2 y 1 − d1 y 2 + ax2 y 2 = 0 depending on a. For a = 0 we find back the case of the previous example and we let the reader wonder why the situation only changes when a 6= 0. The content of the paper is just following the introduction. In section 2 we recall the definitions and results from the formal theory of systems of OD/PD equations that will be crucially used in the sequel. We pay a particular emphasize to the definition of involution and the way to introduce the Spencer operator in this framework. We also study the possibility and difficulty to use computer algebra in this framework. In section 3 we recall the basic tools needed from module theory and homological algebra in a way adapted to our purpose, in particular the definition of the extension modules, and provide a few of their properties which, though well known by specialists of algebraic analysis, cannot be found easily in the literature. Meanwhile, we provide a few links with the preceding section which are not so well known. Many explicit examples will illustrate the main concepts in the commutative (constant field k) and the non-commutative (differential field K) framework. In section 4 we shall recall the proof of the theorem already quoted showing how to embed an r-pure module M into another module L with projrective dimension equal to r. We shall provide for the first time explicit computations of this result in order to point out the difficulty encountered in such a procedure as a motivation for avoiding it. In section 5 we extend the work of Macaulay, showing why only pure modules can fit with relative localization in a coherent way with what happens for torsion-free modules. Meanwhile, we shall extend for the first time this work to the non-commutative framework, showing in particular that the operator introduced by Macaulay ([11], §60) for studying inverse systems is nothing else than the Spencer operator. Many explicit examples, including highly non-trivial ones provided by Macaulay himself, will be fully treated in such a way that any engineer, even with a poor knowledge of homological algebra, will nevertheless become intuitively able to understand and apply these new techniques without reading the previous sections, just comparing to the way the same examples have been treated in section 4 by means of another approach. 2) TOOLS FROM SYSTEM THEORY : If X is a manifold of dimension n with local coordinates (x) = (x1 , ...xn ), we denote as usual by T = T (X) the tangent bundle of X, by T ∗ = T ∗ (X) the cotangent bundle, by ∧r T ∗ the bundle of r-forms and by Sq T ∗ the bundle of q-symmetric tensors. More generally, let E be a vector bundle over X, that is (roughly) a manifold with local coordinates (xi , y k ) for i = 1, ..., n and k = 1, ..., m simply denoted by (x, y), projection π : E → X : (x, y) → (x) and changes of local coordinates x̄ = ϕ(x), ȳ = A(x)y. If E and F are two vector bundles over X with respective local coordinates (x, y) and (x, z), we denote by E×X F the fibered product of E and F over X as the new vector 3 bundle over X with local coordinates (x, y, z). We denote by f : X → E : (x) → (x, y = f (x)) a global section of E, that is a map such that π ◦ f = idX but local sections over an open set U ⊂ X may also be considered when needed. Under a change of coordinates, a section transforms like f¯(ϕ(x)) = A(x)f (x) and the derivatives transform like: ∂ f¯l (ϕ(x))∂i ϕr (x) = (∂i Alk (x))f k (x) + Alk (x)∂i f k (x) ∂ x̄r We may introduce new coordinates (xi , y k , yik ) transforming like: ȳrl ∂i ϕr (x) = (∂i Alk (x))y k + Alk (x)yik k We shall denote by Jq (E) the q-jet bundle of E with local coordinates (xi , y k , yik , yij , ...) = (x, yq ) k k k called jet coordinates and sections fq : (x) → (x, f (x), fi (x), fij (x), ...) = (x, fq (x)) transforming like the sections jq (f ) : (x) → (x, f k (x), ∂i f k (x), ∂ij f k (x), ...) = (x, jq (f )(x)) where both fq and jq (f ) are over the section f of E. Of course Jq (E) is a vector bundle over X with projection πq while Jq+r (E) is a vector bundle over Jq (E) with projection πqq+r , ∀r ≥ 0. DEFINITION 2.1: A linear system of order q on E is a vector sub-bundle Rq ⊂ Jq (E) and a solution of Rq is a section f of E such that jq (f ) is a section of Rq . Let µ = (µ1 , ..., µn ) be a multi-index with length |µ| = µ1 + ... + µn , class i if µ1 = ... = µi−1 = 0, µi 6= 0 and µ + 1i = (µ1 , ..., µi−1 , µi + 1, µi+1 , ..., µn ). We set yq = {yµk |1 ≤ k ≤ m, 0 ≤ |µ| ≤ q} with yµk = y k when |µ| = 0. If E is a vector bundle over X with local coordinates (xi , y k ) for i = 1, ..., n and k = 1, ..., m, we denote by Jq (E) the q-jet bundle of E with local coordinates simply denoted by (x, yq ) and sections fq : (x) → (x, f k (x), fik (x), fijk (x), ...) transforming like the section jq (f ) : (x) → (x, f k (x), ∂i f k (x), ∂ij f k (x), ...) when f is an arbitrary section of E. Then both fq ∈ Jq (E) and jq (f ) ∈ Jq (E) are over f ∈ E and the Spencer operator just allows to distinguish them by introducing a kind of ”difference” through the operator D : Jq+1 (E) → T ∗ ⊗ Jq (E) : fq+1 → j1 (fq ) − fq+1 with local components (∂i f k (x) − fik (x), ∂i fjk (x) − fijk (x), ...) k (x). In a symbolic way, when changes of and more generally (Dfq+1 )kµ,i (x) = ∂i fµk (x) − fµ+1 i coordinates are not involved, it is sometimes useful to write down the components of D in the form di = ∂i − δi and the restriction of D to the kernel Sq+1 T ∗ ⊗ E of the canonical projection πqq+1 : Jq+1 (E) → Jq (E) is minus the Spencer map δ = dxi ∧ δi : Sq+1 T ∗ ⊗ E → T ∗ ⊗ Sq T ∗ ⊗ E. The kernel of D is made by sections such that fq+1 = j1 (fq ) = j2 (fq−1 ) = ... = jq+1 (f ). Finally, if Rq ⊂ Jq (E) is a system of order q on E locally defined by linear equations Φτ (x, yq ) ≡ aτk µ (x)yµk = 0 and local coordinates (x, z) for the parametric jets up to order q, the r-prolongation Rq+r = ρr (Rq ) = Jr (Rq ) ∩ Jq+r (E) ⊂ Jr (Jq (E)) is locally defined when r = 1 by the link + ∂i aτk µ (x)yµk = 0 and has symbol ear equations Φτ (x, yq ) = 0, di Φτ (x, yq+1 ) ≡ aτk µ (x)yµ+1 i gq+r = Rq+r ∩ Sq+r T ∗ ⊗ E ⊂ Jq+r (E) if one looks at the top order terms. If fq+1 ∈ Rq+1 is over fq ∈ Rq , differentiating the identity aτk µ (x)fµk (x) ≡ 0 with respect to xi and substracting the k k (x)) ≡ 0 (x)+∂i aτk µ (x)fµk (x) ≡ 0, we obtain the identity aτk µ (x)(∂i fµk (x)−fµ+1 identity aτk µ (x)fµ+1 i i ∗ and thus the restriction D : Rq+1 → T ⊗ Rq ([17],[18],[30]). q+r+1 DEFINITION 2.2: Rq is said to be formally integrable when the restriction πq+r : Rq+r+1 → Rq+r is an epimorphism ∀r ≥ 0 or, equivalently, when all the equations of order q + r are obtained by r prolongations only ∀r ≥ 0. In that case, Rq+1 ⊂ J1 (Rq ) is a canonical equivalent formally integrable first order system on Rq with no zero order equations, called the Spencer form. Finding an intrinsic test has been achieved by D.C. Spencer in 1970 ([30]) along coordinate dependent lines sketched by M. Janet in 1920 ([7]) and W. Gröbner in 1940 ([4],[6]). The key ingredient, missing in the old approach, is provided by the following definition. Let T ∗ be the cotangent vector bundle of 1-forms on X and ∧s T ∗ be the vector bundle of s-forms on X with usual bases {dxI = dxi1 ∧ ... ∧ dxis } where we have set I = (i1 < ... < is ). Moreover, introducing the exterior derivative d : ∧s T ∗ −→ ∧s+1 T ∗ : ω = ωI (x)dxI −→ dω = ∂i ωI (x)dxi ∧ dxI , we have d2 = d ◦ d = 0 and may introduce the Poincaré sequence: 4 d d d d ∧0 T ∗ −→ ∧1 T ∗ −→ ∧2 T ∗ −→ ... −→ ∧n T ∗ −→ 0 PROPOSITION 2.3: There exists a map δ : ∧s T ∗ ⊗ Sq+1 T ∗ ⊗ E → ∧s+1 T ∗ ⊗ Sq T ∗ ⊗ E which restricts to δ : ∧s T ∗ ⊗ gq+1 → ∧s+1 T ∗ ⊗ gq and δ 2 = δ ◦ δ = 0. k k . Proof: Let us introduce the family of s-forms ω = {ωµk = vµ,i dxI } and set (δω)kµ = dxi ∧ ωµ+1 i 2 k i j k We obtain at once (δ ω)µ = dx ∧ dx ∧ ωµ+1i +1j = 0. Q.E.D. The kernel of each δ in the first case is equal to the image of the preceding δ but this may no longer be true in the restricted case and we set: s DEFINITION 2.4: We denote by Hq+r (gq ) the cohomology at ∧s T ∗ ⊗ gq+r of the restricted 1 s δ-sequence which only depends on gq . The symbol gq is said to be s-acyclic if Hq+r = ... = Hq+r = 0, ∀r ≥ 0, involutive if it is n-acyclic and finite type if gq+r = 0 becomes trivially involutive for r large enough. DEFINITION 2.5: Rq is said to be involutive when it is formally integrable and its symbol gq δ δ is involutive, that is to say all the sequences ... → ∧s T ∗ ⊗ gq+r → ... are exact ∀0 ≤ s ≤ n, ∀r ≥ 0. Equivalently, the following procedure, where one may have to change linearly the independent variables if necessary, is the heart towards the next effective definition of involution. It is intrinsic even though it must be checked in a particular coordinate system called δ-regular ([17],[18],[29]) and is particularly simple for first order systems without zero order equations. • Equations of class n: Solve the maximum number βqn of equations with respect to the jets of order q and class n. Then call (x1 , ..., xn ) multiplicative variables. − − − − − − − − − − − − − − −− • Equations of class i ≥ 1: Solve the maximum number βqi of remaining equations with respect to the jets of order q and class i. Then call (x1 , ..., xi ) multiplicative variables and (xi+1 , ..., xn ) non-multiplicative variables. −−−−−−−−−−−−−−−−− • Remaining equations equations of order ≤ q − 1: Call (x1 , ..., xn ) non-multiplicative variables. In actual practice, we shall use a multiplicative board where the multiplicative ”variables” are represented by their index in upper left position while the non-multiplicative variables are represented by dots in lower right position. DEFINITION 2.6: A system of PD equations is said to be involutive if its first prolongation can be achieved by prolonging its equations only with respect to the corresponding multiplicative (q+n−i−1)! −βqi for i = 1, ..., n and variables. In that case, we may introduce the characters αiq = m (q−1)!((n−i)! we have dim(gq+1 ) = α1q + ... + αnq . Moreover, one can exhibit the Hilbert polynomial dim(Rq+r ) in r with leading term (α/d!)rd with d ≤ n when α is the smallest non-zero character in the case of an involutive symbol. Such a prolongation allows to compute in a unique way the principal (pri) jets from the parametric (par) other ones. This definition may also be applied to nonlinear systems as well. REMARK 2.7: For an involutive system with β = βqn < m, then (y β+1 , ..., y m ) can be given arbitrarily and may constitute the input variables in control theory, though it is not necessary to make such a choice. In this case, the intrinsic number α = αnq = m − β > 0 is called the n-character and is the system counterpart of the so-called ”differential transcendence degree” in differential algebra. As we shall see in the next section, the smallest non-zero character and the number of zero characters are intrinsic numbers that cannot be known without bringing the system 5 to involution and we have α1q ≥ ... ≥ αnq ≥ 0. EXAMPLE 2.8: ([11], §38, p 40 where one can find the first intuition of formal integrability) The primary ideal q = ((χ1 )2 , χ1 χ3 − χ2 ) provides the system y11 = 0, y13 − y2 = 0 which is neither formally integrable nor involutive. Indeed, we get d3 y11 − d1 (y13 − y2 ) = y12 and d3 y12 − d2 (y13 − y2 ) = y22 , that is to say each first and second prolongation does bring a new second order PD equation. Considering the new system y22 = 0, y12 = 0, y13 −y2 = 0, y11 = 0, the question is to decide whether this system is involutive or not. One could use Janet or Gröbner algorithm but with no insight towards involution. In such a simple situation, as there is no PD equation of class 3, two evident permutations of coordinates (1, 2, 3) → (3, 2, 1) or (1, 2, 3) → (2, 3, 1) both provide one equation of class 3, 2 equations of class 2 and 1 equation of clas 1. It is then easy to check directly that the first permutation brings the involutive system y33 = 0, y23 = 0, y22 = 0, y13 − y2 = 0 that will be used in the sequel and we have α32 = 0, α22 = 0, α12 = 2. EXAMPLE 2.9: With n = 4, m = 1, q = 1, K = Q(x1 , x2 , x3 , x4 ), let us consider the system R1 : {y4 − x3 y2 − y = 0, y3 − x4 y1 = 0 Again, the reader will check easily that the subsystem R1′ ⊂ R1 :   u≡ v≡  w≡ y4 − x3 y1 − y = 0 y3 − x4 y1 = 0 y2 − y1 = 0 1 2 1 2 1 2 3 4 3 • • • (1) namely the projection R1 of R2 to R1 , is formally integrable and even involutive with one equation of class 4, one equation of class 3 and one equation of class 2. In the situation of the last remark, the following theorem will generalizing for PD control systems the well known first order Kalman form of OD control systems where the derivatives of the input do not appear ([27], VI,1.14, p 802). For this, we just need to modify the Spencer form and we provide the procedure that must be followed in the case of a first order involutive system with no zero order equation, for example an involutive Spencer form. • Look at the equations of class n solved with respect to yn1 , ..., ynβ . • Use integrations by part like: yn1 − a(x)ynβ+1 = dn (y 1 − a(x)y β+1 ) + ∂n a(x)y β+1 = ȳn1 + ∂n a(x)y β+1 • Modify y 1 , ..., y β to ȳ 1 , ..., ȳ β in order to ”absorb” the various ynβ+1 , ..., ynm only appearing in the equations of class n. We have the following unexpected result providing what we shall call reduced Spencer form: THEOREM 2.10: The new equations of class n only contain yiβ+1 , ..., yim with 0 ≤ i ≤ n − 1 while the equations of class 1, ..., n − 1 no longer contain y β+1 , ..., y m and their jets. Accordingly, as we shall see in the next section, any torsion element, if it exists, only depends on ȳ 1 , ..., ȳ β . Proof: The first assertion comes from the absorption procedure. Now, if y m or yim should appear in an equation of class ≤ n−1, prolonging this equation with respect to the non-multiplicative m variable xn should bring ynm or yin and (here involution is essential) we should get a linear combination of equations of various classes prolonged with respect to x1 , ..., xn−1 only, but this is impossible and we get the desired reduced form. Q.E.D. jq Φ When Rq is involutive, the linear differential operator D : E → Jq (E) → Jq (E)/Rq = F0 of order q with space of solutions Θ ⊂ E is said to be involutive and one has the canonical linear Janet sequence ([17], p 144): D D D D n 2 1 Fn −→ 0 ... −→ F1 −→ 0 −→ Θ −→ T −→ F0 −→ 6 where each other operator is first order involutive and generates the compatibility conditions (CC) of the preceding one. As the Janet sequence can be cut at any place, the numbering of the Janet bundles has nothing to do with that of the Poincaré sequence for the exterior derivative, contrary to what many physicists believe. Moreover, the dimensions of the Janet bundles can be computed at once inductively from the board of multiplicative and non-multiplicative variables that can be exhibited for D by working out the board for D1 and so on. For this, the number of rows of this new board is the number of dots appearing in the initial board while the number nb(i) of dots in the column i just indicates the number of CC of class i for i = 1, ..., n with nb(i) < nb(j), ∀i < j. It follows that the successive first order operators D1 , ..., Dn are automatically in reduced Spencer form. EXAMPLE 2.11: Coming back to Example 2.9 and changing slightly our usual notations, we get for D1 the following first order involutive system of CC in reduced Spencer form:  3  φ ≡ φ2 ≡  1 φ ≡ d4 v − d3 u + x4 d1 u − x3 d1 v − v = 0 d4 w − d2 u + d1 u − x3 d1 w − w = 0 d3 w − d2 v + d1 v − x4 d1 w = 0 1 2 1 2 1 2 3 4 3 4 3 • as d4 u does not appear in φ2 and φ3 while u does not appear in φ1 . We finally obtain for D2 the only CC: ψ ≡ d4 φ1 − d3 φ2 − d1 φ3 + x4 d1 φ2 − x3 d1 φ1 − φ1 = 0 DEFINITION 2.12: The Janet sequence is said to be locally exact at Fr if any local section of Fr killed by Dr+1 is the image by Dr of a local section of Fr−1 . It is called locally exact if it is locally exact at each Fr for 0 ≤ r ≤ n. The Poincaré sequence is locally exact, that is a closed form is locally an exact form but counterexamples may exist ([18], p 373). j1 Equivalently, we have the involutive first Spencer operator D1 : C0 = Rq → J1 (Rq ) → J1 (Rq )/Rq+1 ≃ T ∗ ⊗ Rq /δ(gq+1 ) = C1 of order one induced by D : Rq+1 → T ∗ ⊗ Rq . Introducing the Spencer bundles Cr = ∧r T ∗ ⊗ Rq /δ(∧r−1 T ∗ ⊗ gq+1 ), the first order involutive (r + 1)-Spencer operator Dr+1 : Cr → Cr+1 is induced by D : ∧r T ∗ ⊗ Rq+1 → ∧r+1 T ∗ ⊗ Rq : α ⊗ ξq+1 → dα ⊗ ξq + (−1)r α ∧ Dξq+1 and we obtain the canonical linear Spencer sequence ([17], p 150): jq D D D D n 3 2 1 Cn −→ 0 ... −→ C2 −→ 0 −→ Θ −→ C0 −→ C1 −→ as the canonical Janet sequence for the first order involutive system Rq+1 ⊂ J1 (Rq ). The canonical Janet sequence and the canonical Spencer sequence can be connected by a commutative diagram where the Spencer sequence is induced by the locally exact central horizontal sequence which is at the same time the Janet sequence for jq and the Spencer sequence for Jq+1 (E) ⊂ J1 (Jq (E)) ([17], p 153) but this result will not be used in this paper (See [5],[20],[22],[23] for more details on Cosserat and Maxwell equations, see ([16]-[21]) and in particular ([22],[23]) for applications to engineering and mathematical physics). REMARK 2.13: We shall revisit Example 2.8 in order to explain the word ”canonical ” that has been used in the previous definitions. For this, starting with the inhomogeneous system y33 = u, y13 − y2 = v, we obtain easily the following inhomogeneous involutive system with its corresponding board of multiplicative and non-multiplicative variables:  4 Φ    3 Φ Φ2    1 Φ ≡ ≡ ≡ ≡ y33 y23 y22 y13 − y2 = = = = u d1 u − d3 v d11 u − d13 v − d2 v v 1 1 1 1 2 2 2 • 3 • • • Using prolongation with respect to the 4 non-multiplicative variables involved should bring 4 first order CC for the right members and we could wait for 4 third order CC involving u and v. 7 Surprisingly, we need the only CC Ψ ≡ d33 v − d13 u + d2 u = 0 and obtain the differential sequence: 0 −→ Θ −→ 1 −→ 2 −→ 1 −→ 0 as a single CC has no CC for itself (See ([18],p365) for the effective general procedure). Such a differential sequence is quite different from the canonical Janet sequence: D D D 1 2 0 −→ Θ −→ 1 −→ 4 −→ 4 −→ 1 −→ 0 which is the only sequence that can provide the Spencer sequence as we already said and could not be obtained by simply using Gröbner bases. This remark will become essential in mathematical physics (foundations of continuum mechanics, gauge theory, general relativity) where only involutive operators must be used ([20],[22],[23]). We also check that the Euler-Poincaré characteristic, namely the alternate sum of the circled dimensions of the vector bundles involved, does not depend on the differential sequence used as we get 1 − 2 + 1 = 1 − 4 + 4 − 1 = 0 (See [18], p 378). In the same spirit, using certain parametric jet variables as new unknowns, we may set z 1 = y, z 2 = y1 , z 3 = y2 , z 4 = y3 in order to obtain the following involutive first order system with no zero order equation:   class 3 class 2  class 1 d3 z 1 − z 4 = 0, d3 z 2 − z 3 = 0, d3 z 3 = 0, d3 z 4 = 0 d2 z 1 − z 3 = 0, d2 z 2 − d1 z 3 = 0, d2 z 3 = 0, d2 z 4 = 0 d1 z 1 − z 2 = 0, d1 z 4 − z 3 = 0 1 2 1 2 1 • 3 • • where we have separated the classes. Contrary to what could be believed, this operator does not describe the Spencer sequence that could be obtained from the previous Janet sequence. Indeed, introducing the trivial vector bundle E with local coordinates (x1 , x2 , x3 , y), it follows that J1 (E) has local coordinates (x1 , x2 , x3 , z 1 , z 2 , z 3 , z 4 ). Now, the involutive system R2 ⊂ J2 (E) ⊂ J1 (J1 (E)) with involutive symbol g2 ⊂ S2 T ∗ ⊗ E ⊂ T ∗ ⊗ T ∗ ⊗ E ⊂ T ∗ ⊗ J1 (E) projects onto J1 (E) but dim(R2 ) = 6 because par(R2 ) = {y, y1 , y2 , y3 , y11 , y12 } while we have only 4 unknowns (z 1 , z 2 , z 3 , z 4 ). Nevertheless, as R2 projects onto J1 (E), we may construct a canonical Janet sequence for this new system where the successive Janet bundles involved will be the Spencer bundles Cr = ∧r T ∗ ⊗ J1 (E)/δ(∧r−1 T ∗ ⊗ g2 ) with a shift by one step in the numbering of the bundles as now C0 = J1 (E) and the successive operators are induced by the composition of the inclusion R2 ⊂ J2 (E) with the Spencer operator D : J2 (E) −→ T ∗ ⊗ J1 (E) as in ([17],p144,150) or ([18],p356). In any case, it is essential to notice that, both in the canonical Spencer sequence and in the canonical Janet sequence, any intermediate operator can be constructed explicitely without knowing the previous ones. EXAMPLE 2.14: With m = 1, n = 4, q = 2, one could treat similarly the involutive system: y44 = 0, y34 = 0, y33 = 0, y24 − y13 = 0 with one equation of class 4, two equations of class 3 and one equation of class 2. EXAMPLE 2.15: Coming back to the involutive system of Example 2.9 with variable coefficients, we let the reader prove that the Janet sequence is: D D D 1 2 0 −→ Θ −→ 1 −→ 3 −→ 3 −→ 1 −→ 0 EXAMPLE 2.16: Let us finally consider the following involutive system of PD equations with two independent variables (x1 , x2 ) and three unknowns (y 1 , y 2 , y 3 ), where again a is an arbitrary constant parameter and we have set for simplicity yik = di y k :  2  y2 + y12 + y23 − y13 − ay 3 y 1 − y12 − y23 − y13 − ay 3  21 y1 − y12 − 2y13 =0 =0 =0 1 2 1 2 1 • Then the corresponding Janet sequence is: D D 1 1 −→ 0 0 −→ Θ −→ 3 −→ 3 −→ 8 Finally, setting ȳ 1 = y 1 − y 3 , ȳ 2 = y 2 + y 3 , we obtain the new first order involutive system:  2  ȳ2 − ȳ12 − ay 3 ȳ 1 − ȳ12 − ay 3  21 ȳ1 − ȳ12 = 0 =0 =0 1 1 1 2 2 • with two equations of class 2 and one equation of class 1 in which y 3 surprisingly no longer appears. If χ1 , ..., χn are n algebraic indeterminates or, in a more intrinsic way, if χ = χi dxi ∈ T ∗ is a covector and D : E −→ F : ξ −→ aτk µ (x)∂µ ξ k (x) is a linear involutive operator of order q, we may introduce the characteristic matrix a(x, χ) = (aτk µ (x)χµ | µ = q) and the resulting map σχ (D) : E −→ F is called the symbol of D at χ. Then there are two possibilities: • If maxχ rk(σχ (D) < m ⇔ αnq > 0: the characteristic matrix fails to be injective for any covector. • If maxχ rk(σχ (D) = m ⇔ αnq = 0: the characteristic matrix fails to be injective if and only if all the determinants of the m × m submatrices vanish. However, one can prove that this algebraic ideal a ∈ K[χ] is not intrinsically defined and must be replaced by its radical rad(a) made by all polynomials having a power in a. This radical ideal is called the characteristic ideal of the operator. DEFINITION 2.17: For each x ∈ X, the algebraic set defined by the characteristic ideal is called the characteristic set of D at x and V = ∪x∈X Vx is called the characteristic set of D. One has the following important theorem ([18],[29]) that will play an important part later on: THEOREM 2.18: (Hilbert-Serre) The dimension d(V ) of the characteristic set, that is the maximum dimension of the irreducible components, is equal to the number of non-zero characters while the codimension cd(V ) = n − d(V ) is equal to the number of zero characters, that is to the number of ”full ” classes in the board of multiplicative variables of an involutive system. EXAMPLE 2.19: Coming back to Remark 2.12, we obtain a = ((χ3 )2 , χ2 χ3 , (χ2 )2 , χ1 χ3 ) =⇒ rad(a) = (χ2 , χ3 ) and thus cd(V ) = 2. However, if we take only into account Example 2.8, we should only get the radical ideal (χ3 ) and the wrong result cd(V ) = 1. The reason for using the radical can be seen from the equivalent first order system that shoul provide b = ((χ3 )4 , ...) with homogeneous polynomials of degree 4 and thus b ⊂ a with a strict inclusion though rad(a) = rad(b). A similar situation can be obtained with Examples 1.1 and 2.9. 3) TOOLS FROM MODULE THEORY : We may roughly say that, if a reader familiar with Gröbner bases ([4],[6]) and computer algebra looks at the previous section, he will feel embarassed because he will believe that ”intrinsicness is always competing with complexity ” as can be seen from Examples 2.8 + 2.12. However, even if he admits that it may be useful to have intrinsic and thus canonical procedures, then looking at the existing literature on differential modules ([1],[9],12]), he will really feel to be on another planet as the main difficulty involved in the theory of differentia modules is to understand why and where formal integrability and involution become essential tools to apply quite before dealing with the homological background of ”algebraic analysis ” involving extension modules. This is the main reason for which the case of variable coefficients is rarely treated ”by itself ” always refering to Weyl algebras for examples and the main difficulty we found when writing ([18], in particular Chapter IV). The central concept, essential for applications but well hidden in the literature dealing with filtred modules ([14],p 383) and totally absent from the use of Gröbner bases because it amounts to formal integrability by duality, is that of a ”strict morphism ”. Accordingly, the purpose of this section will be to explain why such a definition, which seems to be purely technical, will be so important for studying extension modules and purity. If P = aµ dµ ∈ D = K[d], the highest value of |µ| with aµ 6= 0 is called the order of the operator P and the ring D with multiplication (P, Q) −→ P ◦ Q = P Q is filtred by the order q of the operators. We have the filtration 0 ⊂ K = D0 ⊂ D1 ⊂ ... ⊂ Dq ⊂ ... ⊂ D∞ = D. Moreover, it is clear that D, as an algebra, is generated by K = D0 and T = D1 /D0 with D1 = K ⊕ T if we 9 identify an element ξ = ξ i di ∈ T with the vector field ξ = ξ i (x)∂i of differential geometry, but with ξ i ∈ K now. It follows that D = D DD is a bimodule over itself, being at the same time a left D-module by the composition P −→ QP and a right D-module by the composition P −→ P Q. We define the adjoint functor ad : D −→ Dop : P = aµ dµ −→ ad(P ) = (−1)|µ| dµ aµ and we have ad(ad(P )) = P . It is easy to check that ad(P Q) = ad(Q)ad(P ), ∀P, Q ∈ D. Such a definition can also be extended to any matrix of operators by using the transposed matrix of adjoint operators (See [18],[19],[22] for more details and applications to control theory and mathematical physics). Accordingly, if y = (y 1 , ..., y m ) are differential indeterminates, then D acts on y k by setting k and y0k = y k . We may therefore use he jet coordidi y = yik −→ dµ y k = yµk with di yµk = yµ+1 i nates in a formal way as in the previous section. Therefore, if a system of OD/PD equations is written in the form Φτ ≡ aτk µ yµk = 0 with coefficients a ∈ K, we may introduce the free differential module Dy = Dy 1 + ... + Dy m ≃ Dm and consider the differential submodule I = DΦ ⊂ Dy which is usually called the module of equations, both with the differential module M = Dy/DΦ or D-module and we may set M = D M if we want to specify the ring of differential operators. The work of Macaulay only covers the case m = 1 with K replaced by k ⊆ cst(K). Again, we k + ∂i aτk µ yµk may introduce the formal prolongation with respect to di by setting di Φτ ≡ aτk µ yµ+1 i k k in order to induce maps di : M −→ M : ȳµ −→ ȳµ+1i by residue if we use to denote the residue Dy −→ M : y k −→ ȳ k by a bar as in algebraic geometry. However, for simplicity, we shall not write down the bar when the background will indicate clearly if we are in Dy or in M . k As a byproduct, the differential modules we shall consider will always be finitely generated (k = 1, ..., m < ∞) and finitely presented (τ = 1, ..., p < ∞). Equivalently, introducing the matrix of operators D = (aτk µ dµ ) with m columns and p rows, we may introduce the morphism D Dp −→ Dm : (Pτ ) −→ (Pτ Φτ ) : P −→ P Φ = P D over D by acting with D on the left of these row vectors while acting with D on the right of these row vectors and the presentation of M is defined by the exact cokernel sequence Dp −→ Dm −→ M −→ 0. It is essential to notice that the presentation only depends on K, D and Φ or D, that is to say never refers to the concept of (explicit or formal) solutions. It is at this moment that we have to take into account the results of the previous section in order to understant that certain presentations will be much better than others, in particular to establish a link with formal integrability and involution. It follows from its definition that M can be endowed with a quotient filtration obtained from that of Dm which is defined by the order of the jet coordinates yq in Dq y. We have therefore the inductive limit 0 ⊆ M0 ⊆ M1 ⊆ ... ⊆ Mq ⊆ ... ⊆ M∞ = M with di Mq ⊆ Mq+1 and M = DMq for q ≫ 0 with prolongations Dr Mq ⊆ Mq+r , ∀q, r ≥ 0. DEFINITION 3.1: ([14],p 383) If M and N are two differential modules and f : M −→ N is a morphism over D compatible with the filtration, that is if f (Mq ) ⊂ Nq with induced morphism fq : Mq −→ Nq , then f is a strict morphism if fq (Mq ) = f (M ) ∩ Nq , ∀q ≥ 0. Equivalently, chasing in the following diagram: 0 ↓ 0 ↓ fq Mq ↓ −→ Nq ↓ M −→ f N 0 ↓ −→ coker(fq ) ↓ −→ 0 −→ −→ 0 coker(f ) then f is strict if the induced morphism coker(fq ) −→ coker(f ) is a monomorphism ∀q ≥ 0. DEFINITION 3.2: An exact sequence of morphisms finishing at M is said to be a resolution of M . If the differential modules involved apart from M are free, we shall say that we have a free resolution of M . Moreover, a sequence of strict morphisms is called a strict sequence. LEMMA 3.3: If f is a strict morphism as in the last definition, there are exact sequences 10 0 −→ coker(fq ) −→ coker(fq+1 ), ∀q ≥ 0. Proof: As f is compatible with the filtrations and Mq ⊆ Mq+1 , Nq ⊆ Nq+1 , we have an induced morphism coker(fq ) −→ coker(fq+1 ). Now, as f is also strict, we have the following commutative and exact diagram: 0 −→ coker(fq ) −→ coker(f ) ↓ k 0 −→ coker(fq+1 ) −→ coker(f ) The lemma finally follows from an elementary chase. Q.E.D. Having in mind that K is a left D-module with the standard action (D, K) −→ K : (di , a) −→ ∂i a and that D is a bimodule over itself, we have only two possible constructions: DEFINITION 3.4: We define the system R = homK (M, K) = M ∗ and set Rq = homK (Mq , K) = Mq∗ as the system of order q. We have the projective limit R = R∞ −→ ... −→ Rq −→ ... −→ R1 −→ R0 . It follows that fq ∈ Rq : yµk −→ fµk ∈ K with aτk µ fµk = 0 defines a section at order q and we may set f∞ = f ∈ R for a section of R. For an arbitrary differential field K, such a definition has nothing to do with the concept of a formal power series solution (care). DEFINITION 3.5: We may define the right differential module homD (M, D). PROPOSITION 3.6: When M is a left D-module, then R is also a left D-module. Proof: As D is generated by K and T as we already said, let us define: (af )(m) = af (m), ∀a ∈ K, ∀m ∈ M (ξf )(m) = ξf (m) − f (ξm), ∀ξ = ai di ∈ T, ∀m ∈ M In the operator sense, it is easy to check that di a = adi + ∂i a and that ξη − ηξ = [ξ, η] is the k and thus standard bracket of vector fields. We finally get (di f )kµ = (di f )(yµk ) = ∂i fµk − fµ+1 i recover exactly the Spencer operator of the previous section though this is not evident at all. We k k k =⇒ di dj = dj di , ∀i, j = 1, ..., n and thus + fµ+1 − ∂j fµ+1 also get (di dj f )kµ = ∂ij fµk − ∂i fµ+1 i +1j i j di Rq+1 ⊆ Rq =⇒ di R ⊂ R induces a well defined operator R −→ T ∗ ⊗ R : f −→ dxi ⊗ di f . This result has been discovered (up to sign) by Macaulay in 1916 ([11]). For more details on the Spencer operator and its applications, the reader may look at ([22],[23]). Q.E.D. As D is a bimodule over itself, it follows from this proposition that that D∗ = homK (D, K) is a left D-module. Moreover, using Baer’s criterion ([28]), it is known that D∗ is an injective D-module as there is a canonical isomorphism: M ∗ = homK (M, K) ≃ homD (M, D∗ ) where both sides are well defined ([2], Prop 11, p 18;)([28], p 37). DEFINITION 3.7: With any differential module M we shall associate the graded module G = gr(M ) over the polynomial ring gr(D) ≃ K[χ] by setting G = ⊕∞ q=0 Gq with Gq = Mq /Mq+1 and we get gq = G∗q where the symbol gq is defined by the short exact sequences: 0 −→ Mq−1 −→ Mq −→ Gq −→ 0 =⇒ 0 −→ gq −→ Rq −→ Rq−1 −→ 0 We have the short exact sequences 0 −→ Dq−1 −→ Dq −→ Sq T −→ 0 leading to grq (D) ≃ Sq T and we may set as usual T ∗ = homK (T, K) in a coherent way with differential geometry. Moreover any compatible morphism f : M −→ N induces a morphism gr(f ) : gr(M ) −→ gr(N ). 11 EXAMPLE 3.8: If K = Q(x), m = 1, n = 1, let us consider the system yxxx − yx = 0 for which we may exhibit the basis of sections {f = (1, 0, 0, 0, ...), f ′ = (0, 1, 0, 1, ...), f ” = (0, 0, 1, 0, ...)} as in ([11],§59,p 67) or ([21]). We obtain dx f = 0, dx f ′ = −f −f ”, dx f ” = −f ′ and check that all the sections can be generated by a single one, namely f ” which describes the power series of ch(x)−1. With 1 now m = 2, let us consider the module defined by the system yxx = 0, yx2 = 0. Setting y = y 2 −xy 1 , 2 1 1 1 1 we successively get yx = −xyx − y , yxx = −2yx, yxxx = 0 =⇒ y = x2 yxx − yx , y 2 = x2 yxx − xyx + y and a differential isomorphism with the module defined by the new system yxxx = 0. All the sections of the second system are easily seen to be generated by the single section f = (0, 0, 1, ...), a 2 1 = 0, ..., f 2 (x) = x2 , fx2 = 0, ... result leading to the only generating section f 1 (x) = x2 , fx1 = − 21 , fxx of the initial system but these sections do not describe solutions because ∂x f 1 − fx1 = 1 6= 0 and ∂x f 2 − fx2 = x 6= 0. We do not know any reference in computer algebra dealing with sections (See [21] for more details) Coming back to the presentation of M under study, we notice that the morphism D involved is not compatible unless we shift the index of the filtration by ord(D) = q. In that case, we obtain im(D) = I ⊂ Dy and may set Iq+r = I ∩ Dq+r y but we have in general Drp D ⊆ Iq+r only, that is the equations of order q+r may not be produced by r prolongations only. We have thus obtained: PROPOSITION 3.9: The morphism induced by D is strict if and only if D is formally integrable. Accordingly, the module version of both the Janet sequence and the Spencer sequence are strictly exact sequences. Proof: Using q + r instead of q in Lemma 3.3 and applying homK (•, K), we obtain the epimorphisms Rq+r+1 −→ Rq+r −→ 0, ∀r ≥ 0. Q.E.D. The reader will find in ([18], IV,3) more details on the relations existing between G and M which are needed in order to study the non-commutative situation, at least when K is a differential field as such a case is hard enough. We obtain in particular the Hilbert polynomial α d dimK (Mq+r ) = dimK (Rq+r ) = d! r + ... where α is called the multiplicity of M and we use to set cdD (M ) = cd(M ) = n − r, rkD (M ) = rk(M ) = α if cd(M ) = 0 and 0 otherwise. EXAMPLE 3.10: Coming back to the Example 2.8 of Macaulay, we obtain from Remark 2.13 the free resolution 0 −→ D −→ D2 −→ D −→ M −→ 0 but only the morphism on the left is strict as for the morphism on the right we know that its image is indeed I2 = {y33 , y23 , y22 , y13 − y2 } and not just {y33 , y13 −y2 }. However, bringing the system to involution, we get the strict free resolution 0 −→ D −→ D4 −→ D4 −→ D −→ M −→ 0 as the module version of the Janet sequence and we let the reader exhibit the module version of the corresponding Spencer sequence as an exercise. If M is a differential module over the ring D = K[d] of differential operators and m ∈ M , then the differential submodule Dm ⊂ M is defined by a system of OD or PD equations for one unknown and we may look for its codimension cd(Dm). A similar comment can be done for any differential submodule M ′ ⊂ M . Sometimes, a single element m ∈ M , called differentially primitive element, may generate M if Dm = M . EXAMPLE 3.11: With K = Q, let us consider the differential moduleM defined by the system 1 1 yxx − y 1 = 0, yx2 = 0. Then, setting y = y 1 − y 2 , we get yx = yx1 , yxx = yxx = y 1 , yxx − y = y 2 and thus yxxx − yx = 0 provides another way to describe M by means of a single element as in ([18], p435). We have the following commutative and exact diagram: 0 −→ 0 −→ D2 ↓ D D2 ↓ D −→ −→ (P, Q) −→ (P (d2 − 1), Qd) ↓ ↓ (P d + Q) −→ ((P d + Q)(d3 − d)) 12 −→ M k −→ M −→ 0 −→ 0 1 If now we consider the differential module M defined by yxx − ay 1 = 0, yx2 = 0 where a is a constant parameter, we cannot find a differentially primitive element when K = Q if a = 0 but we can when K = Q(x) for any value of a, as in Example 3.8. We may check the following definition in a constructive way ([27]): DEFINITION 3.12: tr (M ) = {m ∈ M | cd(Dm) > r} is the greatest differential submodule of M having codimension > r. PROPOSITION 3.13: cd(M ) = cd(V ) = r ⇐⇒ αqn−r 6= 0, αn−r+1 = ... = αnq = 0 ⇐⇒ tr (M ) 6= q M, tr−1 (M ) = ... = t0 (M ) = t(M ) = M and this intrinsic result can be most easily checked by using the Spencer form of the system defining M . We are now in a good position for defining and studying purity for differential modules. DEFINITION 3.14: M is r-pure ⇐⇒ tr (M ) = 0, tr−1 (M ) = M ⇐⇒ cd(Dm) = r, ∀m ∈ M . In particular, M is 0-pure if t(M ) = 0 and, if cd(M ) = r but M is not r-pure, we may call M/tr (M ) the pure part of M . It follows that tr−1 (M )/tr (M ) is equal to zero or is r-pure (See the picture in [18], p 545). Finally, when tr−1 (M ) = tr (M ), we shall say that there is a gap in the purity filtration: 0 = tn (M ) ⊆ tn−1 (M ) ⊆ ... ⊆ t1 (M ) ⊆ t0 (M ) = t(M ) ⊆ M PROPOSITION 3.15: tr (M ) does not depend on the presentation or on the filtration of M . EXAMPLE 3.16: If K = Q and M is defined by the involutive system y33 = 0, y23 = 0, y13 = 0, then z = y3 satifies d3 z = 0, d2 z = 0, d1 z = 0 and cd(Dz) = 3 while z ′ = y2 only satisfies d3 z ′ = 0 and cd(Dz ′ ) = 1. We have the purity filtration 0 = t3 (M ) ⊂ t2 (M ) = t1 (M ) ⊂ t0 (M ) = t(M ) = M with one gap and two strict inclusions. We now recall the definition of the extension modules extiD (M, D) that we shall simply denote by exti (M ) and the way to use their dimension or codimension. We point out once more that these numbers cannot be obtained without bringing the underlying systems to involution in order to get informations on M from informations on G. We divide the procedure into four steps that can be achieved by means of computer algebra ([27]): • Construct a free resolution of M , say: ... −→ Fi −→ ... −→ F1 −→ F0 −→ M −→ 0 • Suppress M in order to obtain the deleted sequence: ... −→ Fi −→ ... −→ F1 −→ F0 −→ 0 • Apply homD (•, D) in order to obtain the dual sequence heading backwards: ... ←− homD (Fi , D) ←− ... ←− homD (F1 , D) ←− homD (F0 , D) ←− 0 • Define exti (M ) to be the cohomology at homD (Fi , D) in the dual sequence with ext0 (M ) = homD (M, D). The following nested chain of difficult propositions and theorems can be obtained, even in the non-commutative case, by combining the use of extension modules and bidualizing complexes in the framework of algebraic analysis. The main difficulty is to obtain first these results for the graded module G = gr(M ) by using techniques from commutative algebra before extending them to the filtred module M as in ([1],[9],[18],[19]). THEOREM 3.17: The extension modules do not depend on the resolution of M used. 13 PROPOSITION 3.18: Applying homD (•, D) provides right D-modules that can be transformed to left D-modules by means of the side changing functor and vice-versa. Namely, if ND is a right D-module, then D N = ∧n T ⊗K ND is the converted left D-module while, if D N is a left D-module, then ND = ∧n T ∗ ⊗K D N is the converted right D-module. PROPOSITION 3.19: Instead of applying homD (•, D) and the side changing functor in the module framework, we may use ad in the operator framework. Namely, to any operator D : E −→ F we may associate the formal adjoint ad(D) : ∧n T ∗ ⊗ F ∗ −→ ∧n T ∗ ⊗ E ∗ with the useful though striking relation rkD (ad(D)) = rkD (D). PROPOSITION 3.20: exti (M ) is a torsion module ∀1 ≤ i ≤ n but ext0 (M ) = homD (M, D) may not be a torsion module. EXAMPLE 3.21: When M is a torsion module, we have homD (M, D) = 0 (exercise). When n = 3 and the torsion-free module M is defined by the formally surjective div operator, the formal adjoint of div is −grad which defines a torsion module. Also, when n = 1 as in classical control theory, a controllable system allows to define a torsion-free module M which is free in that case and homD (M, D) is thus also a free module. THEOREM 3.22: exti (M ) = 0, ∀i ≥ n + 1. THEOREM 3.23: cd(exti (M )) ≥ i. PROPOSITION 3.24: THEOREM 3.25: exti (M ) = 0, ∀i < cd(M ). cd(M ) ≥ r ⇔ exti (M ) = 0, ∀i < r. PROPOSITION 3.26: cd(M ) = r =⇒ cd(extr (M )) = r and extr (M )is r-pure. PROPOSITION 3.27: extr (extr (M )) is equal to 0 or is r-pure, ∀0 ≤ r ≤ n. PROPOSITION 3.28: If we set t−1 (M ) = M , there are exact sequences ∀0 ≤ r ≤ n: 0 −→ tr (M ) −→ tr−1 (M ) −→ extr (extr (M )) THEOREM 3.29: If cd(M ) = r, then M is r-pure if and only if there is a monomorphism 0 −→ M −→ extr (extr (M )) of left differential modules. THEOREM 3.30: M is pure ⇐⇒ exts (exts (M )) = 0, ∀s 6= cd(M ). The last two theorems are known to characterize purity but it is however evident that they are not very useful in actual practice. THEOREM 3.31: When M is r-pure, the characteristic ideal is thus unmixed, that is a finite intersection of prime ideals having the same codimension r and the characteristic set is equidimensional, that is the union of irreducible algebraic varieties having the same codimension r. REMARK 3.32: For the reader knowing more about commutative algebra, we add a few details about the localization used in the primary decomposition of a module which are not so well known ([3],[18],[21],[31]). For simplicity, setting k = cst(K), we shall denote by A = k[χ] the polynomial ring isomorphic to D = k[d] and consider a module M over A. We denote as usual by spec(A) the set of proper prime ideals in A, by max(A) the subset of maximal ideals in A and by ass(M ) = {p ∈ spec(A)|∃0 6= m ∈ M, p = annA (m)} the (finite) set {p1 , ..., pt } of associated prime ideals, while we denote by {p1 , ...ps } the subset of minimum associated prime ideals. It is well known that M 6= 0 =⇒ ass(M ) 6= ∅. We recall that an ideal q ⊂ A is pprimary if ab ∈ q, b ∈ / q =⇒ a ∈ rad(q) = p ∈ spec(A). We say that a module Q is p-primary if am = 0, 0 6= m ∈ Q =⇒ a ∈ p = rad(q) ∈ spec(A) when q = annA (Q) or, equivalently, 14 ass(Q) = {p}. Similarly, we say that a module P is p-prime if am = 0, 0 6= m ∈ P =⇒ a ∈ p ∈ spec(A) when p = annA (P ). It follows that any p-prime or p-primary module is r-pure with n − r = trd(A/p), a result generalizing ([11],§4, p 43). Accordingly, a module M is r-pure if and only if a = annA (M ) admits a primary decomposition a = q1 ∩...∩qs and rad(a) = p1 ∩...∩ps with cd(A/pi ) = cd(M ) = r, ∀i = 1, ..., s. In that case, the monomorphism 0 −→ M −→ ⊕p∈ass(M) Mp induces a monomorphism 0 −→ M −→ Q1 ⊕ ... ⊕ Qs called primary embedding where the primary modules Qi are the images of the localization morphisms M −→ Mpi = S −1 M with S = A − p inducing epimorphisms M −→ Qi −→ 0 for i = 1, ..., s. Macaulay was only considering the case M = A/a with primary decomposition a = q1 ∩ ... ∩ qs . EXAMPLE 3.33: With k = Q and n = 3, then a = rad(a) = (χ1 , χ2 χ3 ) = (χ1 , χ2 ) ∩ (χ1 , χ3 ) is unmixed and M = A/a is 2-pure while a = rad(a) = (χ1 χ2 , χ1 χ3 ) = (χ1 )∩(χ2 , χ3 ) is mixed, though an intersection of two minimum prime ideals and M = A/a is not 1-pure. On the contrary, if one has the primary decomposition a = ((χ1 )2 , χ1 χ2 , χ1 χ3 , χ2 χ3 ) = (χ1 , χ2 ) ∩ (χ1 , χ3 ) ∩ (χ1 , χ2 , χ3 )2 = p1 ∩ p2 ∩ m2 and M = A/a, then ass(M ) = {p1 , p2 , m} with pi ⊂ m for i = 1, 2, though rad(a) = p1 ∩ p2 as before. In that case, there is an embedding 0 −→ M −→ Q1 ⊕ Q2 ⊕ Q3 where Qi = A/pi is the image of the localization morphism M −→ Mpi for i = 1, 2 because p1 is killed by χ3 ∈ A − p1 , p2 is killed by χ2 ∈ A − p2 and Q3 = A/m2 is m-primary because rad(m2 ) = m ∈ max(A). We have also an embedding 0 −→ M −→ Mp1 ⊕ Mp2 ⊕ Mm but no element of the multiplicative set A − m = {1 + a|a ∈ m} can kill any element of M and the image of M into Mm is thus isomorphic to M which is not a primay module. It is important to notice that the example of Macaulay q = ((χ3 )2 , χ2 χ3 , (χ2 )2 , χ1 χ3 − χ2 ) provides a p-primary module A/q with p = (χ3 , χ2 ) even though the annihilating ideal of G = gr(M ) is the homogeneous ideal a = ((χ3 )2 , χ2 χ3 , (χ2 )2 , χ1 χ3 ) = ((χ2 )2 , χ3 ) ∩ (χ1 , χ2 , χ3 )2 which is a mixed ideal because ass(A/a) = {(χ2 , χ3 ), (χ1 , χ2 , χ3 )}. However, we get rad(a) = (χ2 , χ3 ) in a coherent way. PROBLEM : Is it possible to have a test for checking whether a differential module is pure or not without using the previous results ? 4) MOTIVATION : As we already said in the introduction and in the previous section, a torsion-free module M is 0-pure because in that case t0 (M ) = t(M ) = 0. Accordingly, M can be embedded into a free module F and the inclusion, which may not be strict when n > 1, provides a parametrization by means of a finite number of potential-like arbitrary functions in the classical language of elasticity (Airy function) or electromagnetism (EM 4-potential). As it is clear that such a situation is only a very particular case of purity, it remains to wonder what can happen for an r-pure module whenever r ≥ 1. One has the following result ([9], [18], compare to [1], p494): THEOREM 4.1: If M is an r-pure differential module with r ≥ 1, there exists a differential module L with pd(L) ≤ r and an embedding M ⊆ L. Proof: First of all we notice that we have r > 0 and thus any element m ∈ M is surely a torsion element because cd(Dm) > 0, that is M = t(M ) is a torsion module with ext0 (M ) = homD (M, D) = 0 because D is an integral domain. Let now ... −→ Fr −→ ... −→ F1 −→ F0 −→ N −→ 0 be a free resolution of the right differential module N = extr (M ). According to Proposition 3.26, we have cd(N ) = r > 0 too and N is also a torsion module with ext0 (N ) = homD (N, D) = 0. Applying the functor homD (•, D) to the previous sequence or , equivalently, constructing the adjoint sequence in the operator framework while using the fact that exti (N ) = 0, ∀i < r according to Theorem 3.25, we obtain the finite long exact sequence with exactly r morphisms because N is finitely presented and extr (N ) 6= 0: 0 −→ homD (F0 , D) −→ ... −→ homD (Fr−1 , D) −→ homD (Fr , D) −→ L −→ 0 where the left differential module L is the cokernel of the last morphism on the right. As homD (F, D) is free whenever F is free because of the bimodule structure of D = D DD , the 15 corresponding deleted complex is: 0 −→ homD (F0 , D) −→ ... −→ homD (Fr−1 , D) −→ homD (Fr , D) −→ 0 Applying again homD (•, D) and using the reflexivity of any free module F , that is the isomorphism homD (homD (F, D), D) ≃ F , we obtain the dual sequence: 0 −→ Fr −→ ... −→ F1 −→ F0 −→ 0 and a similar procedure may be followed with operators as we shall see in the next illustrating examples ([18],[27]). This sequence is exact everywhere but at Fr and at F0 where its cohomology is just N by definition, that is to say extr (L) = N = extr (M ). Looking for the cohomology at homD (Fr , D) in the sequence obtained by duality from the resolution of N with coboundry module Br and cocycle module Zr , we obtain the following commutative and exact diagram: 0 0 ↓ ↓ 0 −→ Br = Br ↓ ↓ 0 −→ Zr −→ homD (Fr , D) ↓ ↓ 0 −→ extr (N ) −→ L ↓ ↓ 0 0 −→ 0 Finally, composing the bottom monomorphism with the monomorphism 0 −→ M −→ extr (N ) provided by Theorem 3.29, we get the desired embedding M ⊆ L. It must be noticed that such a procedure can be followed equally well in the commutative and non-commutative framework, that is when K is a field of constants or a true differential field. Q.E.D. EXAMPLE 4.2: With K = Q, m = 1, n = 4, q = 2, let us study the 2-pure differential module M defined by the involutive system:  4 Φ    3 Φ Φ2    1 Φ ≡ ≡ ≡ ≡ y44 y34 y33 y24 − y13 =0 =0 =0 =0 1 1 1 1 2 2 2 2 3 3 3 • 4 • • • From the board of multiplicative variables we may construct at once the Janet sequence: D D D 1 2 4 −→ 1 −→ 0 0 −→ Θ −→ 1 −→ 4 −→ where D1 is defined by the involutive system:  4 Ψ    3 Ψ  Ψ2   1 Ψ ≡ ≡ ≡ ≡ d4 Φ3 − d3 Φ4 d4 Φ2 − d3Φ3 d4 Φ1 − d2 Φ4 + d1 Φ3 d3 Φ1 − d2 Φ3 + d1 Φ2 =0 =0 =0 =0 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 • and D2 by the (trivially) involutive system:  Ω ≡ d4 Ψ1 − d3 Ψ2 + d2 Ψ4 − d1 Ψ3 =0 1 2 We have therefore the resolution: 0 −→ D −→ D4 −→ D4 −→ D −→ M −→ 0 16 3 4 leading to pd(M ) ≤ 3 and the deleted complex is: 0 −→ D −→ D4 −→ D4 −→ D −→ 0 Applying homD (•, D) to this sequence, we get the sequence: 0 ←− D ←− D4 ←− D4 ←− D ←− 0 which can be described by the following adjoint sequence: ad(D) ←− 0 ←− 1 4 ad(D1 ) ←− 4 ad(D2 ) ←− 1 ←− 0 which is not a Janet sequence. As M is a torsion module, using now Theorem 3.25 we get ext0 (M ) = 0, ext1 (M ) = 0 and we check that N = ext2 (M ) 6= 0. For this, dualizing Ψ by λ and Ω by θ, we have to look for the CC of the inhomogeneous system:  1 Ψ −→ −d4 θ = λ1    2 Ψ −→ d3 θ = λ2 3 Ψ −→ d1 θ = λ3    4 Ψ −→ −d2 θ = λ4 which are not already provided by  1 Φ −→    2 Φ −→ Φ3 −→    4 Φ −→ the system: −d4 λ2 − d3 λ1 −d4 λ3 − d1 λ1 −(d4 λ4 − d2 λ1 ) + (d3 λ3 − d1 λ2 ) d3 λ4 + d2 λ2 =0 =0 =0 =0 One can check that the torsion module N can be generated by {u = d2 λ3 + d1 λ4 , v = d3 λ3 − d1 λ2 } satisfying the involutive system:  4 φ    3 φ φ2    1 φ with the two CC:  ψ2 ψ1 ≡ ≡ ≡ ≡ d4 u − d1 v d4 v d3 u − d2 v d3 v =0 =0 =0 =0 ≡ d4 φ2 − d3 φ4 + d2 φ3 − d1 φ1 ≡ d4 φ1 − d3 φ3 1 1 1 1 2 2 2 2 3 3 3 3 =0 =0 4 4 • • 1 1 2 3 2 3 4 4 Accordingly, we have the following strict free resolution of N : 0 −→ D2 −→ D4 −→ D2 −→ N −→ 0 with deleted complex: 0 −→ D2 −→ D4 −→ D2 −→ 0 Applying homD (•, D), we get the desired resolution of L, namely: 0 ←− L ←− D2 ←− D4 ←− D2 ←− 0 Dualizing ψ by z, we finally discover that L is defined by the involutive system:  −φ1    −φ2 −φ3    φ4 −→ −→ −→ −→ d4 z 1 − d1 z 2 d4 z 2 d3 z 1 − d2 z 2 d3 z 2 17 =0 =0 =0 =0 1 1 1 1 2 2 2 2 3 3 3 3 4 4 • • and is therefore 2-pure with pd(L) ≤ 2 and a strict inclusion M ⊂ L defined by y = z 1 . . REMARK 4.3: In this example, we discover that, if L were also r-pure, we should therefore have an embedding 0 −→ L −→ extr (extr (L)) = extr (N ) and thus an isomorphism extr (N ) = L leading to an isomorphism Zr = homD (Fr , D) and to Fr+1 = 0, as can be checked on this example with r = 2. It has been a challenge for the author during many months to find the following counter-example showing that, sometimes L may not even be a torsion module. EXAMPLE 4.4: According to the proof of the theorem, N = extr (M ) does not depend on the resolution of M used while L does indeed depend on the resolution of N used. Coming back to the system studied in Example 2.8 and Remark 2.12 with r = 2, we may use the shortest finite free resolution of M already presented, namely 0 −→ D −→ D2 −→ D −→ M −→ 0. Therefore, taking the adjoint of the only CC found, we may define N by the system:  v −→ d33 θ =0 −u −→ d13 θ + d2 θ = 0 and obtain the corresponding involutive system:  4 φ    3 φ φ2    1 φ ≡ ≡ ≡ ≡ d33 θ d23 θ d22 θ d13 θ + d2 θ =0 =0 =0 =0 1 1 1 1 2 2 2 • 3 • • • 1 1 1 1 2 2 2 2 We obtain the first order involutive system of CC:  4 ψ    3 ψ ψ2    1 ψ ≡ ≡ ≡ ≡ d3 φ3 − d2 φ4 d3 φ2 − d2 φ3 d3 φ1 − d1 φ4 − φ3 d2 φ1 − d1 φ3 − φ2 =0 =0 =0 =0 3 3 3 • with the only CC : ω ≡ d3 ψ 1 − d2 ψ 2 + d1 ψ 4 + ψ 3 = 0 . We may therefore introduce in reverse order the corresponding adjoint volved in the Janet sequence we have just constructed:  4 ψ −→ −d1 λ = ν 4    3 ψ −→ λ = ν3 2 ψ −→ d2 λ = ν 2    1 ψ −→ −d3 λ = ν 1  4 4 4 2   φ3 −→ µ3 ≡ d2 ν +4 d1 ν 1  φ −→ µ ≡ −(d3 ν − d1 ν ) + (d2 ν 3 − ν 2 ) φ2 −→ µ2 ≡ −d3 ν 3 − ν 1    1 φ −→ µ1 ≡ −d3 ν 2 − d2 ν 1 operators of the ones in- =0 =0 =0 =0 This last operator is defining L but is not involutive. We have the two torsion elements: µ5 ≡ d2 ν 3 − ν 2 , µ6 ≡ d1 ν 3 + ν 4 which are generating ext2 (N ) and are easily seen to satisfy the involutive system: d3 µ6 − µ5 = 0, d3 µ5 = 0, d2 µ6 − d1 µ5 = 0, d2 µ5 = 0 because d2 µ5 ≡ d2 µ3 + d3 µ4 + d1 µ1 = 0. Finally, using the first equation, we may eliminate µ5 and identify µ6 with y because we have indeed d33 µ6 = 0, d13 µ6 − d2 µ6 = 0 in order to obtain the strict inclusion M ⊂ L. Equivalently, we may also eliminate ν 1 and ν 2 respectively from µ2 and µ3 in order to obtain: µ4 −→ d13 (d1 ν 3 + ν 4 ) − d2 (d1 ν 3 + ν 4 ) = 0, 18 µ1 −→ d33 (d1 ν 3 + ν 4 ) = 0 but we may notice that L is not 2-pure and thus a torsion module because ν 3 (similarly ν 4 ) is not by itself a torsion element of L. Such a situation is well known in control theory with the SISO (single input u, single output y) system ẏ − u̇ = 0 because u (similarly y) is not by itself a torsion element but z = y − u is a torsion element because ż = 0 (See the pages 9 and 10 of the introduction in [17] for more details on such a comment). PROBLEM : Is it possible to find an analogue of the previous theorem or of the case r = 0, where L should be also r-pure with a free resolution having exactly r morphisms ?. 5) ABSOLUTE AND RELATIVE LOCALIZATIONS : Surprisingly, the positive answer to such a problem has been given by Macaulay in ([11]) for differential modules defined by systems with constant coefficients and only one unknown. Our purpose in this section is to generalize this resul to arbitrary differential modules defined by systems of PD equations with coefficients in a differential field. Now we hope that, after reading the previous section, the reader is convinced that the use of extension modules is a quite important though striking tool for studying linear multidimensional systems. Of course, as for any new language, it is necessary to apply it on many explicit examples before being familiar with it. However, it is evident that it should be even more important to have a direct approach allowing to exhibit the purity filtration and, in particular, to recognize whether a differential module is pure or not. The purpose of this section is to combine the module approach with the system approach, while taking into account the specific properties of the Spencer form in a way rather similar to the use of the Kalman form of a control system when testing controllability, namely to check that an ordinary differential module is 0-pure. For this, we shall divide the procedure into a few successive constructive steps that will be illustrated on explicit examples. • STEP 1: Whenever a system Rq ⊂ Jq (E) is given, there is no way to obtain informations on the corresponding module without bringing this system to an involutive or at least formally integrable system by means of prolongations and projections as in the Example 2.8 of Macaulay where (2) only the projection R2 ⊂ R2 of R4 to R2 is involutive. Of course, an homogeneous system with constant coefficients is automatically formally integrable and one only needs to use a finite number of prolongations in order to obtain an involutive symbol, though it is known that 2-acyclicity is sufficient to obtain first order generating CC ([17]). However, it is essential to notice that it is only with an involutive system that we are sure that the CC system is first order both with the following ones in the Janet sequence. EXAMPLE 5.1: With K = Q, m = 1, n = 3, q = 2, the homogeneous second order systems y33 = 0, y23 − y11 = 0, y22 = 0 or y33 − y11 = 0, y23 = 0, y22 − y11 = 0 both have a 2-acyclic symbol g3 of dimension 1 at order 3 (exercise) and a trivially involutive symbol g4 = 0 at order 4, such a result leading to only one CC of order 2 with cd(M ) = 3 in both cases. We let the reader treat the system y3 = 0, y12 = 0 similarly and conclude (Hint: (χ3 , χ1 χ2 ) = (χ3 , χ1 ) ∩ (χ3 , χ2 ) is unmixed). It is however not evident that the homogeneous system y11 = 0, y12 = 0, y13 = 0, y23 = 0 of Example 3.33 is involutive. Finally, according to section 2 and 3, this first step provides the characters α1q ≥ ... ≥ αnq ≥ 0 and the smallest non-zero character α = αqn−r 6= 0 providing cd(M ) = r, a result leading at once to tr (M ) ⊂ M with a strict inclusion while tr−1 (M ) = ... = t0 (M ) = t(M ) = M . Of course, if α = αnq 6= 0, then M cannot be a torsion module and t(M ) ⊂ M with a strict inclusion. The following example proves nevertheless that it is much more delicate to study systems with variable coefficients. EXAMPLE 5.2: With K = Q(x2 ), n = 3, m = 1, q = 1, let us consider the differential module M defined by the trivially involutive system y3 − x2 y1 = 0. We have cd(M ) = 1 but we can only say that cd(Dz) ≥ 1, ∀z ∈ M . If we set z = y2 , proceeding as in Remark 2.12, we get the involutive system: 19   y3 y2  y1 = = = x2 z3 − (x2 )2 z1 z z3 − x2 z1 1 2 1 2 1 • 3 • • The differential submodule Dz ⊂ M is defined by the second order involutive system:  z33 − 2x2 z13 + (x2 )2 z11 z23 − x2 z12 − 2z1 =0 =0 1 2 1 2 3 • and we get cd(Dz) = 1 exactly. However, even on such a very elementary example, it is not evident that t0 (M ) = t(M ) = M is 1-pure. We also understand that the decoupling system for any autonomous element in engineering sciences, like in magnetohydrodynamics, cannot be studied without these new techniques if we want intrinsic results. Finally, if we denote by I the left ideal of D = Dy generated by y3 −x2 y1 , we notice the relation ann(G) = (χ3 −x2 χ1 ) = gr(I) = rad(gr(I)). However, we have ann(gr(Dz)) = ((χ3 − x2 χ1 )2 , χ2 (χ3 − x2 χ1 )) with radical equal to the prime ideal (χ3 − x2 χ1 ) as before. Hence, in this example, the strict inclusion Dz ⊂ M does not imply gr((Dz) ⊂ gr(M ) = G because otherwise we should get ann(G) ⊆ ann(gr(Dz) and this is the reason for which only the radical must be considered as it does not depend on the filtration. • STEP 2: Once we have obtained cd(M ) = r, in order to check that M is r-pure, it remains to prove that tr (M ) = 0 as we already know that tr−1 (M ) = M . For this, the second step will be to use the specific properties of the Spencer form Rq+1 ⊂ J1 (Rq ). More generally, it is possible to use any equivalent involutive first order system of the form R1 ⊂ J1 (E) with no zero order equations, that is with an induced epimorphism R1 −→ E −→ 0 and such that the corresponding differential module is isomorphic and thus identified to the initial module as in Remark 2.13 . We have now the characters α11 ≥ ... ≥ αn1 ≥ 0 and the smallest non-zero character is is still α = α1n−r 6= 0 providing of course the same codimension cd(M ) = r as in the first step. Accordingly, the number r of non-zero characters and the number r of full classes is the same as in the previous step. However, it must be noticed that the filtration may be different and the following example explains once more why only the radical of the characteristic ideal must be used. EXAMPLE 5.3: K = Q, n = 2, m = 1, q = 2.For the involutive system y22 = 0, y12 = 0, the characteristic ideal is a = ((χ2 )2 , χ1 χ2 ) =⇒ rad(a) = (χ2 ) =⇒ r = 1. Setting z 1 = y, z 2 = y1 , z 3 = y2 , we get the equivalent first order system d2 z 3 = 0, d2 z 2 = 0, d2 z 1 − z 3 = 0, d1 z 1 − z 2 = 0, d1 z 3 = 0 and the polynomial ideal generated by the 3 × 3 minors of the characteristic matrix is a = ((χ2 )3 , (χ2 )2 χ1 , χ2 (χ1 )2 ). Hence the characteristic ideal is rad(a) = (χ2 ) and r = 1 too. EXAMPLE 5.4: For Example 2.8 we may set z 1 = y, z 2 = y1 , z 3 = y2 , z 4 = y3 and obtain the first order involutive system :   d3 z 4 = 0, d3 z 3 = 0, d3 z 2 − z 3 = 0, d3 z 1 − z 4 = 0 d2 z 4 = 0, d2 z 3 = 0, d2 z 2 − d1 z 3 = 0, d2 z 1 − z 3 = 0  d1 z 4 − z 3 = 0, d1 z 1 − z 2 = 0 1 1 1 2 3 2 • • • with no zero-order equation. We have α31 = 4 − 4 = 0, α21 = 4 − 4 = 0, α11 = 4 − 2 = 2 =⇒ r = 2 too. We let the reader treat Example 4.2 similarly and obtain α41 = 0, α31 = 0, α21 = 2 =⇒ r = 2. It is at this moment that we discover that such systems have particular properties not held by other systems, apart from the fact that a canonical sequence may be constructed exactly like the Spencer sequence or the first order part of the Janet sequence. Shrinking the board of multiplicative variables, we obtain from the definition of involutiveness: PROPOSITION 5.5: For an involutive first order system with no zero order equations and solved with respect to the principal (pri) first order jets expressed by means of the parametric 20 (par) other first order jets, the system obtained by looking only at the PD equations of class 1+ ... + class i only contains d1 , ..., di and is still involutive ∀i = 1, ..., n, after adopting the ordering di+1 , ..., dn , d1 , ..., di . EXAMPLE 5.6: Looking at Example 2.9, we notice that the systems:   y3 − x4 y1 y2 − y1 y2 − y1 =0 =0 =0 4 4 3 1 1 2 3 2 • 4 1 2 are both involutive. Also, looking at Example 5.4, we notice that the system:  d2 z 4 = 0, d2 z 3 = 0, d2 z 2 − d1 z 3 = 0, d2 z 1 − z 3 = 0 d1 z 4 − z 3 = 0, d1 z 1 − z 2 = 0 3 3 1 1 2 • is still involutive and we let the reader treat Example 4.2 similarly. We shall denote the corresponding differential module by Mn−i and we notice that M = M0 is defined by more equations than Mn−i . Accordingly, we have an epimorphism (specialization) Mn−i −→ M −→ 0 and similarly epimorphisms Mn−i −→ Mn−i−1 −→ 0. Finally, as cd(M ) = r, we notice that the classes n − r + 1, ..., n are full and we find therefore (χn )m , ..., (χn−r+1 )m among the m × m minors with lower powers of χ1 , ..., χn−r for the other minors because the numbers of equations of the lower classes are decreasing and thus strictly smaller than m. The characteristic ideal is thus (χn , ..., χn−r+1 ) if we set χ1 , ..., χn−r to zero, in a coherent way. Finally, choosing i = n − r, we get an epimorphism Mr −→ M −→ 0. The background will always indicate clearly the meaning of the lower index and cannot be confused with the filtration index of M . • STEP 3: We are now in position to study Mr with more details as a system in n − r variables ([11], §77, p 86). Its defining system has β1n−r = β = m − α equations of strict class n − r, a smaller number of equations of class n − r − 1, ... , and eventually an even smaller number of equations of class 1. Studying this system for itself, we may look for t(Mr ) exactly following the known techniques working for any differential module, in particular double duality as described in section 4. However, if one could find any (relative) torsion element z ∈ Mr , we could project it to an element z ∈ M and we have N = Dz ⊆ M where we do not put a residue bar on the new z for simplicity. Introducing the respective annihilator ideal a and b of M and N , we should have a ⊆ rad(a) ⊆ rad(b) as it is the only result not depending on the filtration of the modules. However, we know that z must be killed by at least one operator involving only d1 , ..., dn−r , in addition to the operators involving separately (dn )m + ..., ..., (dn−r−1 )m + ... and we should obtain tr−1 (M ) = M but tr (M ) 6= 0, that is M should not be pure. Hence M is r-pure if and only if Mr is torsion-free as a differential module over K[d1 , ..., dn−r ]. In such a case, the system defining Mr can be parametrized by α arbitrary unknowns among {y 1 , ..., y m } by using a so-called minimum parametrization in the sense of ([24]). In actual practice, as shown on all the examples, one can use a relative localization with respect to (d1 , ..., dn−r ) only by keeping (dn−r+1 , ..., dn ) untouched and replacing (d1 , ..., dn−r ) by (χ1 , ..., χn−r ) considered as (constant) ”parameters” in the language of Macaulay ([11], §43, p 45 and §77, p 86 with r instead of n − r and a different ordering). Such a method provides therefore a quite useful and simple test for checking purity by linking it to involutivity. An important intermediate result is provided by the next proposition. PROPOSITION 5.7: The partial localization ”kills” the equations of class 1 up to class n− r − 1 (care) and finally only depends on the equations of strict class n − r. Proof: Instead of using the column n − r in the multiplicative board, we provide the proof when r = 0 by using the column n. Working out as usual the first order CC, we only look at the p < m CC of class n for the equations Φ1 , ..., Φp of class 1 up to class n − 1 if we order the Φ starting from the lower class involved. These p CC will be of a very specific form with a square p × p operator matrix for (Φ1 , ..., Φp ) with diagonal operators of the form dn + ... where the dots denote 21 operators involving only d1 , ..., dn−1 in a quasi linear way with coefficients in K, the remaining of the matrix only depending on d1 , ..., dn−1 for the (Φp+1 , ..., Φm ) of strict class n. Therefore, if we have K = k = cst(K), the absolute localization is simply done by setting di −→ χi and the determinant of the square matrix is equal to (χn )p + ... where the dots denote a polynomial of degree < p in χp with coefficients involving only χ1 , ..., χn−1 . It follows that each Φ1 , ..., Φp can be expressed as a linear combination over k(χ1 , ..., χn ) of the Φ of strict class n. The result is similar for the variable coefficient case by using the graded machinery but needs much more work. In any case, setting the Φ of strict class n equal to zero, we should obtain a zero graded module for the Φ1 , ..., Φp which must be eequal to zero too. It must finally be noticed that the first order CC used are in reduced Spencer form as the dn of the Φ of strict class n do not appear in the CC we have used and these Φ do not appear in the other CC too. Q.E.D EXAMPLE 5.8: Coming back to Example 2.11, we notice that the only CC is an identity in (u, v, w) and we may forget about u in order to obtain the new system for (v, w):  3  φ ≡ d4 v − x3 d1 v − v = 0 φ2 ≡ d4 w − x3 d1 w − w = 0  1 φ ≡ d3 w − x4 d1 w − d2 v + d1 v = 0 1 1 1 2 3 2 3 2 3 4 4 • defining a new module M with cd(M ) = 1 as the class 4 is now full. The same CC as before can be written again in the form: (d4 − x3 d1 − 1)φ1 = (d3 − x4 d1 )φ2 − (d2 − d1 )φ3 that we can localize in (d1 , d2 , d3 ) with: φ1 = 0 =⇒ (d3 − x4 d1 )w = (d2 − d1 )v =⇒ v = (d3 − x4 d1 )y, w = (d2 − d1 )y We let the reader check that we have indeed:  2 φ ≡ (d4 − x3 d1 − 1)(d3 − x4 d1 )y φ3 ≡ (d4 − x3 d1 − 1)(d2 − d1 )y = = (d3 − x4 d1 )u = 0 =⇒ u = 0 (d2 − d1 )u = 0 =⇒ u = 0 We finally obtain for M a relative parametrization with the only constraint u ≡ (d4 −x3 d1 −1)y = 0 in a coherent way with Example 2.11. • STEP 4: In this section we come back to the commutative situation with a field k of constants and generalize the results of Macaulay in the following theorem which is recapitulating the results so far obtained. THEOREM 5.9: One has the commutative and exact diagram: 0 −→ t(Mr ) −→ Mr ↓ ↓ 0 −→ tr (M ) −→ M ↓ ↓ 0 0 −→ k(χ1 , ..., χn−r ) ⊗ ↓ −→ k(χ1 , ..., χn−r ) ⊗ ↓ 0 Mr M Proof: For simplifying the notations in this diagram of modules over D, we have identified Mr as a module over k[d1 , ..., dn−r ] with Mr as a module over k[dn−r+1 , ..., dn , d1 , ..., dn−r ] = k[d1 , ..., dn ] = D while the localization of M just tells that the coefficients are now in the field k(χ1 , ..., χn−r ), exactly following Macaulay. Moreover the central column is exact according to the definitionof Mr and the right column is exact because localization preserves the exactness of a sequence. For exampe, with k = Q, n = 2, m = 2, q = 2, r = 1, the differential module M defined by the involutive system y22 = 0, y12 = 0 may also be defined by the first order involutive system z 1 = y, z 2 = y1 , z 3 = y2 =⇒ d2 z 3 = 0, d2 z 2 = 0, d2 z 1 − z 3 = 0, d1 z 3 = 0, d1 z 1 − z 2 = 0. Then M1 is defined by the first order system d1 z 3 = 0, d1 z 1 − z 2 = 0 with torsion module t(M1 ) generated 22 by z 3 and the tensor product of M by k(χ1 ) is defined by d2 z 3 = 0, d2 z 2 = 0, d2 z 1 = 0, d1 z 3 = 0, z 3 = 0, χ1 z 1 − z 2 = 0 after division by χ1 in a coherent way with Example 1.1. Q.E.D. Finally, when Mr is torsion-free as a differential module over k[d1 , ..., dn−r ], then tr (M ) = 0 and we get the following generalization of a result provided by Macalualy ([11], §41, p 43): COROLLARY 5.10: The differential module M is r-pure if and only if cd(M ) = r and there is a monomorphism 0 −→ M −→ k(χ1 , ..., χn−r ) ⊗ M . • STEP 5: The final idea is to embed Mr into a free module over K[d1 , ..., dn−r ] in order to parametrize the corresponding system and substitute into the equations of class n−r+1, ..., n. However, if we look at Example 1.2, we should find after the substitution Φ1 ≡ z13 = 0, Φ2 ≡ z23 = 0 with one CC d2 Φ1 − d1 Φ2 = 0, that is on one side a module L which is not 1-pure and, on the other side a module L having a finite free resolution with 2 operators. However, we forgot that M , being pure, may be identified with its embedding into its localization. Hence, we get in fact χ1 z3 = 0, χ2 z3 = 0 and thus only z3 = 0 is providing a convenient parametrizing module L. Our purpose is to explain and illustrate this procedure for finding such an L in the general situation. Again, the main idea will be provided by this example. Indeed, we obtain the only CC Ψ ≡ d3 Φ3 − d2 Φ1 + d1 Φ2 = 0. Substituting the parametrization, we get of course Φ3 = 0 ⇐⇒ χ1 y 2 = χ2 y 1 , that is, among the two unknowns y 1 , y 2 we are left with only one, say y 1 and, similarly, among the two equations Φ1 , Φ2 we are left with only one, say Φ1 , because χ1 Φ2 = χ2 Φ1 from the CC which is of course compatible with the localization and we choose z3 = 0 as χ1 z3 = 0 =⇒ z3 = 0. The general situation may be treated similarly. Indeed, according to the previous step, we are only concerned with the equations of class n − r + 1, ... , class n while the localization has only to do with the β equations of strict class n − r (care) allowing to express β unknowns as linear combinations of the α remaining unknowns with coefficients in k(χ1 , ..., χn−r ). To each such equation are associated exactly r dots and each dot of index n − r + i provides a reduction of the respective equations of class n − r + i for i = 1, ..., r. It follows that we are left with α equations of each such class. When we ”delocalize”, replacing χi by di , we have to take into account the need to take out the denominators and may find a few ”simplifications” as in the example just considered.. Finally, the maximum number r − 1 (care again) of dots found for one equation is obtained for the equations of strict class n − r + 1 = n − (r − 1) and we have thus exhibited a system defining a module L which is r-pure and admits a free resolution with exactly (r − 1) + 1 = r operators. In any case, the reader must not forget that the localization of a module is useful only if we already know that this module is torsion-free by means of the double-duality formula t(M ) = ext1 (N ) given in the introduction. EXAMPLE 5.11: Let M be defined by the involutive system:  d3 y 4 + d1 y 2 − d1 y 1     d3 y 3 − d2 y 4 + d1 y 2 − d1 y 1    d3 y 2 + d1 y 2 d3 y 1 − d1 y 4 + d1 y 2     d2 y 2 − d1 y 4 + d1 y 1    d2 y 1 − d1 y 3 + d1 y 1 =0 =0 =0 =0 =0 =0 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 • • with characters α31 = 0, α21 = α = 2, α11 = 4. It follows that cd(M ) = 1 as only the class 4 is full and we obtain the following relative localization showing that M is 1-pure: y 1 = χ1 y, y 2 = χ1 z, y 3 = (χ1 + χ2 )y, y 4 = χ1 y + χ2 z Substituting in the four equations of class 3, we only obtain the two equations: 23  d3 z + χ1 z d3 y + (χ1 − χ2 )z − χ1 y =0 =0 after a division by χ1 , χ2 and χ1 + χ2 . The parametrizing module L is thus defined by the two equations:  d3 z + d1 z =0 d3 y + (d1 − d2 )z − d1 y = 0 which are differentially independent and we have the relative parametrization: y 1 = d1 y, y 2 = d1 z, y 3 = (d1 + d2 )y, y 4 = d1 y + d2 z Finally, M ⊂ L =⇒ ass(M ) ⊂ ass(L) =⇒ ass(M ) = {(d3 + d1 ), (d3 − d1 )} =⇒ annD (M ) = (d3 + d1 ) ∩ (d3 − d1 ), a striking result showing that M can be embedded into the direct sum of two primary differential modules according to Remark 3.32 (See [18] for more details). EXAMPLE 5.12: With k = Q, m = 1, n = 3, let us consider the polynomial map χ1 = u5 , χ2 = u3 , χ3 = u4 as in ([11], p 53). We have the exact sequence 0 −→ p −→ k[χ] −→ k[u] ⊂ k(u) showing that p = ((χ2 )2 (χ3 ) − (χ1 )2 , (χ2 )3 − χ1 χ3 , (χ3 )2 − χ1 χ2 ) is a prime ideal ([18], p 126). The corresponding prime differential module M is defined by the involutive system:  y333 − y123     y  233 − y122   y223 − y11 y222 − y13     y  133 − y112   y33 − y12 =0 =0 =0 =0 =0 =0 1 1 1 1 1 • 2 2 2 2 • • 3 • • • • • and is 2-pure. The localized system is finite type over k(χ1 )[d2 , d3 ] with par = {y, y2 , y3 , y22 , y23 }. One can prove that p2 is not a primary ideal even though rad(p2 ) = p. EXAMPLE 5.13: Similarly but now with k = Q, m = 1, n = 4, let us consider the polynomial map χ1 = uv, χ2 = u, χ3 = uv 3 , χ4 = uv 4 as in ([11], p 47). We have the exact sequence 0 −→ p −→ k[χ] −→ k[u, v] ⊂ k(u, v) showing that p = (χ2 χ4 − χ1 χ3 , (χ1 )3 − (χ2 )2 χ3 , (χ3 )3 − χ1 (χ4 )2 , (χ1 )2 χ4 − χ2 (χ3 )2 ) is a prime ideal. It is not evident at all that the corresponding prime differential module M can be defined by the homogeneous involutive system (exercise):  y444 − y224 − y134 − y123     y  344 − y111    y  334 − y114 − y112   y333 − y124 − y122 − y113 y244 + y224 − y123     y  234 − y133 + y111    y  144 + y124 − y113   y44 + y24 − y13 =0 =0 =0 =0 =0 =0 =0 =0 1 1 1 1 1 1 1 • 2 2 2 2 2 2 • • 3 3 3 3 • • • • 4 • • • • • • • and is thus also 2-pure. The localized system is finite type over k(χ1 , χ2 )[d3 , d4 ] with par = {y, y3 , y4 , y33 } and we have for example χ2 y34 − χ1 y33 + (χ1 )3 y = 0 in a coherent way with the comments of Macaulay in ([11], §78, p 88, formula (A) and §88,89, p 98). 6) CONCLUSION : In 1916, F.S. Macaulay discovered a new localization technique for studying unmixed polynomial ideals. We have been able to generalize this procedure for studying pure differential modules, obtaining in particular a kind of relative parametrization generalizing the absolute parametrization already known for torsion-free modules and equivalent to controllability in classical control theory. In the language of multidimensional systems theory, which is more intuitive, instead of 24 using arbitrary potential-like functions for the parametrization, the idea is now to use potentiallike functions which must satisfy a kind of minimum differential constraint limiting, in some sense, the number of independent variables appearing in these functions, in a way similar to the situation met in the Cartan-Khäler theorem of analysis. For such a purpose, we have exhibited new links between purity and involutivity, providing also a new insight into the primary decomposition of modules and ideals by means of tools from the formal theory of linear multidimensional systems. 7) BIBLIOGRAPHY : [1] J.E. BJORK: Analytic D-modules and Applications, Kluwer, 1993. [2] N. BOURBAKI: Algèbre homologique, chap. X, Masson, Paris, 1980. [3] N. BOURBAKI: Algèbre commutative, chap. I-IV, Masson, Paris, 1985. [4] B. BUCHBERGER: Gröbner bases: an Algorithmic Methods in Polynomial Ideal Theory, in: Recent Trends in Multidimensional System Theory, N.K. Bose ED., Reidel, Dordrecht, 1985, 184232. [5] E. COSSERAT, F. COSSERAT: Théorie des Corps Déformables, Hermann, Paris, 1909. [6] W. GRÖBNER: Über die Algebraischen Eigenschaften der Integrale von Linearen Differentialgleichungen mit Konstanten Koeffizienten, Monatsh. der Math., 47, 1939, 247-284. [7] M. JANET: Sur les Systèmes aux dérivées partielles, Journal de Math., 8, 3, 1920, 65-151. [8] E.R. KALMAN, Y.C. YO, K.S. NARENDA: Controllability of Linear Dynamical Systems, Contrib. Diff. Equations, 1, 2, 1963, 189-213. [9] M. KASHIWARA: Algebraic Study of Systems of Partial Differential Equations, Mémoires de la Société Mathématique de France 63, 1995, (Transl. from Japanese of his 1970 Master’s Thesis). [10] E. KUNZ: Introduction to Commutative Algebra and Algebraic Geometry, Birkhäuser, 1985. [11] F.S. MACAULAY: The Algebraic Theory of Modular Systems, Cambridge Tracts, vol. 19, Cambridge University Press, London, 1916. Stechert-Hafner Service Agency, New-York, 1964. [12] P. MAISONOBE, C. SABBAH: D-Modules Cohérents et Holonomes, Travaux en Cours, 45, Hermann, Paris, 1993. [13] D.G. NORTHCOTT: An Introduction to Homological Algebra, Cambridge University Press, Cambridge, 1966. [14] D.G. NORTHCOTT: Lessons on Rings, Modules and Multiplicities, Cambridge University Press, Cambridge, 1968. [15] V.P. PALAMODOV: Linear Differential Operators with Constant Coefficients, Grundlehren der Mathematischen Wissenschaften 168, Springer, 1970. [16] J.-F. POMMARET,:Dualité Différentielle et Applications, C. R. Acad. Sci. Paris, 320, Série I, 1995, 1225-1230. [17] J.-F. POMMARET: Partial Differential Equations and Group Theory: New Perspectives for Applications, Kluwer, 1994. [18] J.-F. POMMARET: Partial Differential Control Theory, Kluwer, 2001. [19] J.-F. POMMARET: Algebraic Analysis of Control Systems Defined by Partial Differential Equations, in Advanced Topics in Control Systems Theory, Lecture Notes in Control and Information Sciences LNCIS 311, Chapter 5, Springer, 2005, 155-223. [20] J.-F. POMMARET: Parametrization of Cosserat Equations, Acta Mechanica, 215, 2010, 4355. [21] J.-F. POMMARET: Macaulay Inverse Systems Revisited, Journal of Symbolic Computation, 46, 2011, 1049-1069. [22] J.-F. POMMARET: Spencer Operator and Applications: From Continuum Mechanics to Mathematical Physics, in ”Continuum Mechanics-Progress in Fundamentals and Engineering Applications”, Dr. Yong Gan (Ed.), ISBN: 978-953-51-0447–6, InTech, 2012, Available from: http://www.intechopen.com/books/continuum-mechanics-progress-in-fundamentals-and-engineerin-applications/spen [23] J.-F. POMMARET: A Pedestrian Approach to Cosserat/Maxwell/Weyl Theory, Preprint 2012 http://hal.archives-ouvertes.fr/hal-00740314 http://fr.arXiv.org/abs/1210.2675 [24] J.-F. POMMARET, A. QUADRAT: Localization and parametrization of linear multidimensional control systems, Systems & Control Letters 37 (1999) 247-260. [25] J.-F. POMMARET, A. QUADRAT: Algebraic Analysis of Linear Multidimensional Control Systems, IMA Journal of Mathematical Control and Informations, 16, 1999, 275-297. 25 [26] A. 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SAMUEL: Commutative Algebra, Van Nostrand, 1958. [32] E. ZERZ: Topics in Multidimensional Linear Systems Theory, Lecture Notes in Control and Information Sciences 256, Springer, 2000. 26 

Data Augmentation of Railway Images for Track Inspection S. Ritika, Dattaraj Rao General Electric ABSTRACT Regular maintenance of all the assets is pivotal for proper functioning of railway. Manual maintenance can be very cumbersome and leave room for errors. Track anomalies like vegetation overgrowth, sun kinks affect the track construct and result in unequal load transfer, imbalanced lateral forces on tracks which causes further deterioration of tracks and can ultimately result in derailment of locomotive. Hence there is a need to continuously monitor rail track health. Track anomalies are rare with the skew as high as one anomaly in millions of good images. We propose a method to build training data that will make our algorithms more robust and help us detect real world track issues. The data augmentation will have a direct effect in making us detect better anomalies and hence improve time for railroads that is spent in manual inspection. This paper talks about a real-world use-case of detecting railway track defects from a camera mounted on a moving locomotive and tracking their locations. The camera is engineered to withstand the environment factors on a moving train and provide a consistent steady image at around 30 frames per second. An image simulation pipeline of track detection, region of interest selection, augmenting image for anomalies is implemented. Training images are simulated for sun kink and vegetation overgrowth. Inception V3 model pretrained on Imagenet dataset is finetuned for a 2 class classification. For the case of vegetation overgrowth, the model generalizes well on actual vegetation images, though it was trained and validated solely on simulated images which might have different distribution than the actual vegetation. Sun kink classifier can classify professionally simulated sun kink videos with a precision of 97.5%. INTRODUCTION Regular maintenance of all the assets is pivotal for proper functioning of railway. Track is a very important asset for railway. Usually the maintenance is done manually and can be very cumbersome [1]. Also, this leaves room for errors and misses which can cause further deterioration of locomotive as well as track and might ultimately lead to derailment. Track is constructed in a particular way by stacking ties on layers of ballast to ensure the load is transferred equally throughout the construct. But if there is growth of vegetation it can reduce the efficiency of ballast in load transferring and can ultimately affect the track [1]. During extreme climatic changes the track can deform. During summer, due to heat the tracks may expand and if proper clearance is not provided it can get bent. This condition is termed sun kink. Also, if locomotive isn’t designed properly, high imbalanced lateral forces on track from locomotive can be the cause of sun kink. This is very detrimental as it causes high vibration in locomotive running on a sun kinked track and can ultimately cause of derailment. This problem has been attempted using conventional image processing technique in the past. [2][3] But these techniques fail to generalize to different environmental and track conditions. PROBLEM STATEMENT Deep neural networks have the constraint that they require a lot of data to be trained without overfitting. Fine tuning a model pre-trained on datasets like Imagenet, COCO, ILSVRC which have millions of labeled images helps a lot to bring down the number of training images required. This is because a lower level features of model which has been trained for classification of a particular type of image can be reused to a completely new image dataset. Even after leveraging pre-trained weights, the minimum number of images required might range from few hundreds to thousands depending on the complexity of the task at hand. Hence the biggest challenge for image classification becomes availability of trainable data. For the task of railway track health monitoring, we have a lot of good track data available. But anomalies like vegetation overgrowth, sun kink are rare and difficult to find. These can be generated manually using tools like paint but it can be a very cumbersome, labor-intensive process. Hence if synthetic data can be generated for the anomalies mentioned above, it can ease the training process and reduce the problem of overfitting. APPROACH The problem can be divided into two parts: finding the track and augmenting it to add anomalies in the region of interest. Detecting Track Track is a very prominent feature in the image since it is a straight converging line. With the placement of camera as seen in figures below, the track position doesn’t vary a lot. Hence, a region of interest (ROI) can be demarcated where there is a high possibility of track to be present. Once the ROI is clipped, the exact position of track needs to be found. Since track are straight lines, edge detection can be used for the same. Canny edge detection [4] is found to give very good results once the thresholds are tuned properly. Image can be filtered before edge detection to remove noise. Edge detection results in a cluster of number of lines. We need to extract the tracks out of it. Hough’s line transform [5] can be used for the same. Thus, the number of lines have reduced but we didn’t get the two lines corresponding to the track yet. The feature that helps at this juncture is that tracks converge. Hence all the lines can be grouped depending on whether their slope is positive or negative. The zero slope lines (mostly due to the track tie) can be ignored in this process. This gives the coordinates of the two lines corresponding to the track. The ROI can be specified utilizing these points. Simulating Vegetation Overgrowth Once the ROI is selected, vegetation needs to be simulated. Vegetation can disrupt the balance of the ballast and can be detrimental to the track. The level of vegetation can be anything from sparse to high. The region can be inside the tracks and over the rails of the track also. Vegetation is usually a cluster of many small grass blobs. Hence the simulation can be generated in the same way. To generate the grass blob greenish pixels can be placed at random places iteratively. Another efficient way for doing this will be using gaussian distribution. Normal, skewed distribution can also be experimented. Random number of such blobs can be placed in the ROI. The number of blobs can control the level of vegetation. Figure 1 Step by step process for simulating vegetation overgrowth Figure 2 Examples of images simulated with vegetation overgrowth Simulating Sun kink Once the track coordinates are detected, the task can be divided into two parts: removing the track and adding a kinked track. To remove the track, a part of ROI can be copied and shifted sideways. This will ensure that the ties look complete. This step can also be used to generate missing rail condition. To generate broken rail condition, two random points on the track can be connected with a dark line. To generate the kinked track, cubic splines can be utilized. Depending on the size of kink, two random points can be chosen on the track and a spline can be drawn utilizing these. Figure 3 Step by step process for simulating sun kink Figure 4 Examples of images simulated for sun kinks EXPERIMENTAL RESULTS The experiments were done on the video feed from a front facing camera mounted on the locomotive. An ROI 600*300 was extracted from the image. The image was filtered using gaussian filter. Canny edge detection was applied followed by Hough’s line transform. The results for these steps are shown in Figure 1 Step by step process for simulating vegetation overgrowthFigure 1 and Figure 3 above. Vegetation Overgrowth A balanced dataset of 500 images were created comprising of images with vegetation generated by the process described above and healthy track images. The dataset was augmented by varying the brightness, flipping the image horizontally. Inception [6] V3 network pretrained on Imagenet dataset was taken as baseline. The network was fit on the training data by finetuning the last classification layer to obtain a two class classifier. The performance of the model on the test data can be seen in Table 1. An important point to note is that the validation set while training the data was solely the simulated image while the model was tested on real vegetation images. The model was seen to give a good performance even though validation and test data might have different distributions. Table 1 Test results on actual vegetation images by Inception V3 model trained solely on images simulated for vegetation overgrowth Actual\Predicted Positive Positive 10 Negative 1 Negative 4 5 Figure 5 Vegetation overgrowth predicted correctly by the model Sun kink A balanced dataset of 500 images was created for a 2-class classification using Inception V3 network similar to the approach taken for vegetation overgrowth. The network was finetuned on the dataset with augmentation of flipping and random brightness adjustments. The trained model was tested on track video professionally simulated for sun kink. Table 2 Results on professionally simulated videos of the Inception V3 model trained solely on simulated sun kink images Actual\Predicted Positive Positive 6 Negative 1 Negative 0 5 Figure 6 Sun kink detected correctly on professionally simulated video CONCLUSION For the domain of rail data, training image generation for anomalies like vegetation and sun kink was attempted using conventional image processing techniques. It was observed that using pre-trained models and data augmentation brings down the number of training images required. The image simulation pipeline used was: track detection, region of interest selection, augmenting image for anomalies. A balanced dataset of 500 images containing healthy track as well as anomaly was generated. Two separate models were trained for vegetation overgrowth and sun kink. Inception V3 model pretrained on Imagenet dataset was finetuned for 2 class classification. For the case of vegetation overgrowth, the model generalized well on actual vegetation images, though it was trained and validated on simulated images which might have different distribution than the actual vegetation. Sun kink classifier could classify professionally simulated sun kink videos with a precision of 97.5%. REFERENCES 1. Routine inspection for railroad tracks http://www.railroadtrackinspection.com/track_problems.html 2. Machine Vision Approach for Automating Vegetation Detection on Railway Tracks, Siril Yella et al, Journal of Intelligent Systems 3. Automating Condition Monitoring of Vegetation on Railway Trackbeds and Embankments, Roger G. Nyberg https://www.diva-portal.org/smash/get/diva2:929179/FULLTEXT01.pdf 4. A Computational Approach to Edge Detection, John Canny, Morgan Kaufmann Publishers 5. Use of the Hough Transformation To Detect Lines and Curves in Pictures, https://www.cse.unr.edu/~bebis/CS474/Handouts/HoughTransformPaper.pdf 6. Going Deeper with Convolutions, Christian Szegedy et al. arXiv.org 

arXiv:1703.01150v1 [] 3 Mar 2017 On the intersection graph of ideals of Zm ∗ S. Khojasteh Department of Mathematics, Lahijan Branch, Islamic Azad University, Lahijan, Iran Abstract Let m > 1 be an integer, and let I(Zm )∗ be the set of all non-zero proper ideals of Zm . The intersection graph of ideals of Zm , denoted by G(Zm ), is a graph with vertices I(Zm )∗ and two distinct vertices I, J ∈ I(Zm )∗ are adjacent if and only if I ∩ J 6= 0. Let n > 1 be an integer and Zn be a Zm -module. In this paper, we introduce and study a kind of graph structure of Zm , denoted by Gn (Zm ). It is the undirected graph with the vertex set I(Zm )∗ , and two distinct vertices I and J are adjacent if and only if IZn ∩ JZn 6= 0. Clearly, Gm (Zm ) = G(Zm ). We obtain some graph theoretical properties of Gn (Zm ) and we compute some of its numerical invariants, namely girth, independence number, domination number, maximum degree and chromatic index. We also determine all integer numbers n and m for which Gn (Zm ) is Eulerian. 1 Introduction Let R be a commutative ring, and I(R)∗ be the set of all non-zero proper ideals of R. There are many papers on assigning a graph to a ring R, for instance see [1], [2], [5] and [6]. Also the intersection graphs of some algebraic structures such as groups, rings and modules have been studied by several authors, see [3, 9, 10]. In [9], the intersection graph of ideals of R, denoted by G(R), was introduced as the graph with vertices I(R)∗ and for distinct I, J ∈ I(R)∗ , the vertices I and J are adjacent if and only if I ∩ J 6= 0. Also in [3], the intersection graph of submodules of an R-module M , denoted by G(M ), is defined to be the graph whose vertices are the non-zero proper submodules of M and two ∗ Key words: intersection graph, independent set, dominating set, chromatic index, Eulerian graph. 2010 Mathematics Subject Classification: 05C15, 05C25, 05C45, 05C69. E-mail addresses: s [email protected] (S. Khojasteh). 1 distinct vertices are adjacent if and only if they have non-zero intersection. Let n, m > 1 be integers and Zn be a Zm -module. In this paper, we associate a graph to Zm , in which the vertex set is being the set of all non-zero proper ideals of Zm , and two distinct vertices I and J are adjacent if and only if IZn ∩ JZn 6= 0. We denote this graph by Gn (Zm ). Clearly, if n = m, then Gn (Zm ) is exactly the same as the intersection graph of ideals of Zm . This implies that Gn (Zm ) is a generalization of G(Zm ). As usual, Zm denotes the integers modulo m. Now, we recall some definitions and notations on graphs. Let G be a graph with the vertex set V (G) and the edge set E(G). Then we say the order of G is |V (G)| and the size of G is |E(G)|. Suppose that x, y ∈ V (G). If x and y are adjacent, then we write x — y. We denote by deg(x) the degree of a vertex x in G. Also, we denote the maximum degree of G by ∆(G). We recall that a path between x and y is a sequence x = v0 — v1 — · · · — vk = y of vertices of G such that for every i with 1 ≤ i ≤ k, the vertices vi−1 and vi are adjacent and vi 6= vj , where i 6= j. We say that G is connected if there is a path between any two distinct vertices of G. For vertices x and y of G, let d(x, y) be the length of a shortest path from x to y (d(x, x) = 0 and d(x, y) = ∞ if there is no path between x and y). The diameter of G, diam(G), is the supremum of the set {d(x, y) : x and y are vertices of G}. The girth of G, denoted by gr(G), is the length of a shortest cycle in G (gr(G) = ∞ if G contains no cycles). We use n-cycle to denote the cycle with n vertices, where n ≥ 3. Also, we denote the complete graph on n vertices by Kn . A null graph is a graph containing no edges. We use Kn to denote the null graph of order n. The disjoint union of two vertex-disjoint graphs G1 and G2 , which is denoted by G1 ∪ G2 , is a graph with V (G1 ∪ G2 ) = V (G1 ) ∪ V (G2 ) and E(G1 ∪ G2 ) = E(G1 ) ∪ E(G2 ). An independent set is a subset of the vertices of a graph such that no vertices are adjacent. The number of vertices in a maximum independent set of G is called the independence number of G and is denoted by α(G). A dominating set is a subset S of V (G) such that every vertex of V (G) \ S is adjacent to at least one vertex in S. The number of vertices in a smallest dominating set denoted by γ(G), is called the domination number of G. Recall that a k-edge coloring of G is an assignment of k colors {1, . . . , k} to the edges of G such that no two adjacent edges have the same color, and the chromatic index of G, χ′ (G), is the smallest integer k such that G has a k-edge coloring. In [9], the authors were mainly interested in the study of intersection graph of ideals of Zm . For instance, they determined the values of m for which G(Zm ) is connected, complete, Eulerian or has a cycle. In this article, we generalize these results to Gn (Zm ) and also, we find some new results. In Section 2, we compute its girth, independence 2 number, domination number and maximum degree. We also determine all integer numbers n and m for which Gn (Zm ) is a forest. In Section 3, we investigate the chromatic index of Gn (Zm ). In the last section, we determine all integer numbers n and m for which Gn (Zm ) is Eulerian. 2 Basic Properties of Gn (Zm) Let n, m > 1 be integers and Zn be a Zm -module. Clearly, Zn is a Zm -module if and only if n divides m. Throughout the paper, without loss of generality, we assume that m = pα1 1 · · · pαs s and n = pβ1 1 · · · pβs s , where pi ’s are distinct primes, αi ’s are positive integers, βi ’s are non-negative integers, and 0 ≤ βi ≤ αi for i = 1, . . . , s. Let S = {1, . . . , s}, S ′ = {i ∈ S : βi 6= 0}. The cardinality of S ′ is denoted by s′ . Also, we denote the least common multiple of integers a and b by [a, b]. We write a|b (a ∤ b) if a divides b (a does not divide b). We begin with a simple example. Example 1. Let m = 12. Then we have the following graphs. 2Z12 3Z12 2Z12 3Z12 2Z12 3Z12 2Z12 3Z12 4Z12 6Z12 4Z12 6Z12 4Z12 6Z12 4Z12 6Z12 G(Z12 ) G2 (Z12 ) G3 (Z12 ) G4 (Z12 ) Remark 1. It is easy to see that I(Zm ) = {dZm : d divides m} and |I(Zm )∗ | = Qs i=1 (αi + 1) − 2. Let Zn be a Zm -module. If n|d, then dZm is an isolated vertex of Gn (Zm ). Obviously, d1 Zm and d2 Zm are adjacent in Gn (Zm ) if and only if n ∤ [d1 , d2 ]. This implies that Gn (Zm ) is a subgraph of G(Zm ). By [3, Theorem 2.5], we have gr(G(Zm )) ∈ {3, ∞}. We extend this result to Gn (Zm ). Theorem 1. Let Zn be a Zm -module. Then gr(Gn (Zm )) ∈ {3, ∞}. Proof. With no loss of generality assume that S ′ = {1, . . . , s′ }. Clearly, if s′ ≥ 3, then p1 Zm — p2 Zm — p1 p2 Zm is a 3-cycle in Gn (Zm ). Therefore gr(Gn (Zm )) = 3. Now, consider two following cases: 3 Case 1. s′ = 2. If s ≥ 3, then p1 Zm — p3 Zm — p1 p3 Zm is a 3-cycle in Gn (Zm ). So we may assume that s = 2. If αi ≥ 3 for some i, i = 1, 2, then pi Zm — p2i Zm — p3i Zm is a 3-cycle in Gn (Zm ). Also, if βi ≥ 2 for some i, i = 1, 2, then p1 Zm — p2 Zm — p1 p2 Zm is a 3-cycle in Gn (Zm ). Now, assume that n = p1 p2 and α1 , α2 = 1, 2. It is easy to see that gr(Gn (Zm )) = ∞. Note that Gp1 p2 (Zp1 p2 ) ∼ = K2 , Gp1 p2 (Zp21 p2 ) ∼ = K2 ∪ K2 , and ∼ Gp p (Z 2 2 ) = K2 ∪ K2 ∪ K3 . p1 p2 1 2 Case 2. s′ = 1. If s ≥ 3, then p2 Zm — p3 Zm — p2 p3 Zm is a 3-cycle in Gn (Zm ). So assume that s = 2. If β1 ≥ 2, then p1 Zm — p2 Zm — p1 p2 Zm is a 3-cycle in Gn (Zm ). Also, if α2 ≥ 3, then p2 Zm — p22 Zm — p32 Zm is a 3-cycle in Gn (Zm ). Now, suppose that n = p1 and m = pα1 p2 or m = pα1 p2 . Then gr(Gn (Zm )) = ∞, since Gp (Z α1 ) ∼ = K2α 1 1 2 1 p1 p2 1 β and Gp1 (Zpα1 p2 ) ∼ = K3α1 −1 ∪ K2 . Finally, assume that n = p1 1 and m = pα1 1 . If β1 ≥ 4, 1 2 then p1 Zm — p21 Zm — p31 Zm is a 3-cycle in Gn (Zm ). It is easy to see that if β1 ≤ 3, then gr(Gpβ1 (Zpα1 )) = ∞. 1  1 As an immediate consequence of Theorem 1, we have the following corollary. Corollary 1. Let Zn be a Zm -module. Then Gn (Zm ) is a forest if and only if one of the following holds: (i) n = p1 p2 , m = pα1 1 pα2 2 and α1 , α2 ≤ 2. (ii) n = p1 , m = pα1 1 pα2 2 and α2 ≤ 2. (iii) n = pβ1 1 , m = pα1 1 and 1 ≤ β1 ≤ 3. By [3, Theorem 3.4], we find that G(Zm ) is a tree if and only if G(Zm ) is a star. Now, we have a similar result. Corollary 2. Let Zn be a Zm -module. Then Gn (Zm ) is a tree if and only if Gn (Zm ) is a star. In particular, Gn (Zm ) is a tree if and only if one of the following holds: (i) n = m = p21 . (ii) n = m = p31 . (iii) n = p1 and m = p21 . By Corollary 1, we can characterize the values of n and m for which Gn (Zm ) is a null graph. 4 Let Zn be a Zm -module. Then Gn (Zm ) is a null graph if and only if one Corollary 3. of the following holds: (i) n = p1 and m = pα1 1 p2 . (ii) n = p1 and m = pα1 1 . (iii) n = p21 and m = pα1 1 . (iv) n = m = p1 p2 . Throughout the rest of this paper, we use A to denote the set of all isolated vertices of Gn (Zm ). Lemma 1. Let Zn be a Zm -module. If Gn (Zm ) is not a null graph, then dZm is an isolated vertex of Gn (Zm ) if and only if n|d, except for the case n = p1 p2 , m = pα1 1 p2 and α1 ≥ 2, in which case Gn (Zm ) = Kα1 ∪ Kα1 . Proof. Clearly, if n|d, then dZm is an isolated vertex of Gn (Zm ). For the other side suppose that Gn (Zm ) is not a null graph, d = pr11 · · · prss is a divisor of m and n ∤ d. Since n ∤ d, we may assume that r1 < β1 . If s ≥ 3, then dZm is adjacent to one of pα2 2 · · · pαs s Zm or pα3 3 · · · pαs s Zm . Now, suppose that s = 2. If r1 6= 0, then dZm and pα2 2 Zm are adjacent. Hence r1 = 0 and d = pr22 . If r2 ≥ 2, then dZm and p2 Zm are adjacent. Therefore d = p2 . If β1 ≥ 2, then dZm and p1 Zm are adjacent. Thus β1 = 1. If α2 ≥ 2, then dZm and p22 Zm are adjacent. So α2 = 1. Then m = pα1 1 p2 and n = p1 or n = p1 p2 . If n = p1 , then by Corollary 3, Gn (Zm ) is a null graph, a contradiction. Therefore n = p1 p2 , m = pα1 1 p2 and α1 ≥ 2, in which case dZm is an isolated vertex of Gn (Zm ). Moreover, A = {pr11 p2 Zm : 0 ≤ r1 ≤ α1 − 1} and V (Gn (Zm )) \ A = {pr11 Zm : 1 ≤ r1 ≤ α1 }. Clearly, pr11 Zm and pr12 Zm are adjacent, where 1 ≤ r1 < r2 ≤ α1 . This implies that Gp1 p2 (Zpα1 p2 ) = Kα1 ∪ Kα1 , where α1 ≥ 2. Next, suppose that s = 1. Since Gn (Zm ) is 1 not a null graph, by Corollary 3, we conclude that β1 ≥ 3. Therefore dZm is adjacent to p1 Zm or p21 Zm . This completes the proof.  Lemma 2. Let Zn be a Zm -module. If Gn (Zm ) is not a null graph, then |A| = ( α1 , Qs i=1 (αi if n = p1 p2 , m = pα1 1 p2 and α1 ≥ 2; − βi + 1) − 1, otherwise. Proof. By Lemma 1, we know that if n = p1 p2 , m = pα1 1 p2 and α1 ≥ 2, then |A| = α1 . Otherwise, from Lemma 1, we find that dZm is an isolated vertex of Gn (Zm ) if and only 5 if n divides d. So A = {dZm : n divides d, d divides m, d 6= m} = {pr11 · · · prss Zm : βi ≤ Q ri ≤ αi } \ {0}. This implies that |A| = si=1 (αi − βi + 1) − 1 and the proof is complete. Corollary 4. Let Zn be a Zm -module. Then Gn (Zm ) contains no isolated vertex if and only if n = m 6= p1 , p21 , p1 p2 . Proof. One side is obvious. For the other side assume that Gn (Zm ) contains no isolated vertex. Hence n = m. By Corollary 3 and Lemma 2, it is clear that m 6= p1 , p21 , p1 p2 .  Theorem 2. Let Zn be a Zm -module. If Gn (Zm ) is not a null graph, then α(Gn (Zm )) = ( α1 + 1, if n = p1 p2 , m = pα1 1 p2 and α1 ≥ 2; Qs ′ i=1 (αi − βi + 1) − 1 + s , otherwise. Proof. By Lemma 1, it is clear that if n = p1 p2 , m = pα1 1 p2 and α1 ≥ 2, then α(Gn (Zm )) = α1 + 1. Otherwise, with no loss of generality we may assume that S ′ = o n Q β −1 Q αi αi ′ . Obviously, n ∤ pβj −1 {1, . . . , s′ }. Let B = A ∪ pj j p Z : 1 ≤ j ≤ s m j i6=j pi , i6=j i β −1 Q αi for every j, 1 ≤ j ≤ s′ . Hence by Lemma 1, pj j i6=j pi Zm is not an isolated vertex, for every j, 1 ≤ j ≤ s′ . Also, it is easy to see that B is an independent set and so Q α(Gn (Zm )) ≥ |A| + s′ = si=1 (αi − βi + 1) − 1 + s′ . Moreover, if C is an independent set of Gn (Zm ) \ A and |C| ≥ s′ + 1, then by Pigeonhole Principle we conclude that there exist Qs Qs ri′ ri ′ ′ i=1 pi Zm , i=1 pi Zm ∈ C such that rj , rj < βj , for some j, 1 ≤ j ≤ s . This implies ′ Qs Q r that i=1 pri i Zm and si=1 pi i Zm are adjacent, which is impossible. Therefore α(Gn (Zm )) = Qs i=1 (αi − βi + 1) − 1 + s′ .  We denote {i ∈ S : ri < βi } by Dd , where d = pr11 · · · prss is a divisor of m. Obviously, Dd ⊆ S ′ . Theorem 3. Let Zn be a Zm -module and d = pr11 · · · prss (6= 1, m) be a divisor of m. If n divides d, then deg(dZm ) = 0 and otherwise deg(dZm ) = s Y Y Y (αi + 1) − 2 − (αi + 1) (αi − βi + 1). i=1 i∈D / d 6 i∈Dd Proof. If n|d, then dZm is an isolated vertex and so deg(dZm ) = 0. Assume that n does not divide d. Then Dd is nonempty. Clearly, dZm and pt11 · · · ptss Zm are not adjacent if and only if ti ≥ βi for each i ∈ Dd . Thus the number of vertices not adjacent to Q Q dZm is i∈D i∈Dd (αi − βi + 1) − 1 and hence the number of its neighbors is / d (αi + 1) Q Qs Q  i∈Dd (αi − βi + 1). i=1 (αi + 1) − 2 − i∈D / d (αi + 1) Suppose that Zn is a Zm -module and Gn (Zm ) is not a null graph. If Q Q n = p1 · · · ps , then ∆(Gn (Zm )) = si=1 (αi + 1) − 2 − (α1 + 1) si=2 αi , where α1 ≥ · · · ≥ αs Q Q and otherwise ∆(Gn (Zm )) = si=1 (αi + 1) − 2 − si=1 (αi − βi + 1). Theorem 4. Proof. First, suppose that n = p1 · · · ps . By Theorem 3, if d = pr11 · · · prss is a divisor of Q Q Q m and n does not divide d, then deg(dZm ) = si=1 (αi + 1) − 2 − i∈D i∈Dd αi . / d (αi + 1) Q Q If α1 ≥ · · · ≥ αs , then deg(dZm ) = ∆(Gn (Zm )) if and only if i∈D i∈Dd αi = / d (αi + 1) Qs (α1 + 1) i=2 αi . Let {i ∈ S : αi = α1 } = {1, . . . , k} for some k, 1 ≤ k ≤ s. In fact, {pri i Zm : 1 ≤ i ≤ k, 1 ≤ ri ≤ αi } is the set of all vertices with maximum degree. So Q Q ∆(Gn (Zm )) = si=1 (αi + 1) − 2 − (α1 + 1) si=2 αi . Now, assume that there is an integer j such that βj 6= 1. We claim that there exist Q some vertices adjacent to all non-isolated vertices and so ∆(Gn (Zm )) = si=1 (αi + 1) − 2 − Qs i=1 (αi − βi + 1). If βj = 0, then pj Zm is adjacent to all non-isolated vertices. Otherwise, β −1 βj ≥ 2. With no loss of generality suppose that S ′ = {1, . . . , s′ }. Let d = p1β1 −1 · · · ps′s′ Then dZm is adjacent to all non-isolated vertices. The claim is proved. Theorem 5. .  Let Zn be a Zm -module. Then Gn (Zm ) has a vertex which is adjacent to all other vertices if and only if n = m and αj ≥ 2, for some j, 1 ≤ j ≤ s. Proof. If Gn (Zm ) has a vertex which is adjacent to all other vertices, then Gn (Zm ) has not any isolated vertex. This implies that n = m. By contradiction suppose that α1 = · · · = αs = 1. Let dZm be a vertex of Gn (Zm ) such that it is adjacent to all other vertices. With no loss of generality, we may assume that d = p1 · · · pt , where 1 ≤ t < s. Let d′ = pt+1 · · · ps . It is easy to see that dZm and d′ Zm are non-adjacent, a contradiction. Therefore αj ≥ 2, for some j, 1 ≤ j ≤ s. Conversely, if n = m and αj ≥ 2, for some j, 1 ≤ j ≤ s, then by Corollary 4, Gn (Zm ) has no isolated vertex. Also, in view of the proof of Theorem 4, we find that Gn (Zm ) has a vertex which is adjacent to all other vertices. The following corollary is a generalization of [9, Theorem 2.9]. 7 Corollary 5. Let Zn be a Zm -module. Then Gn (Zm ) is a complete graph if and only if n = m = pα1 1 and α1 ≥ 2. Proof. Suppose that Gn (Zm ) is a complete graph. By Theorem 5, we find that n = m Qs−1 αi pi Zm are two and αj ≥ 2, for some j, 1 ≤ j ≤ s. If s ≥ 2, then pαs s Zm and i=1 non-adjacent vertices which is a contradiction. Therefore n = m = pα1 1 and α1 ≥ 2. The other side is obvious.  Theorem 6. Let Zn be a Zm -module. If Gn (Zm ) is not a null graph, then γ(Gn (Zm )) = ( |A| + 1, if n 6= p1 · · · ps or n = p1 p2 , m = pα1 1 p2 and α1 ≥ 2; |A| + 2, otherwise. Proof. Suppose that n 6= p1 · · · ps . In view of the proof of Theorem 4, Gn (Zm ) has a vertex which is adjacent to every non-isolated vertices. This implies that γ(Gn (Zm )) = |A| + 1. Next, assume that n = p1 · · · ps . If s = 1, then Gn (Zm ) is a null graph. Now, assume that s ≥ 2. If n = p1 p2 , m = pα1 1 p2 and α1 ≥ 2, then by Lemma 1, γ(Gn (Zm )) = |A| + 1. Otherwise, let B = {p1 Zm , p2 · · · ps Zm }. If dZm is a non-isolated vertex, then n does not divide d. If p1 ∤ d, then dZm is adjacent to p2 · · · ps Zm . Otherwise, pj ∤ d, for some j, 2 ≤ j ≤ s. Therefore pj ∤ [d, p1 ]. This yields that dZm is adjacent to p1 Zm . Therefore B is a dominating set for Gn (Zm ) \ A. Now, we claim that Gn (Zm ) has not a vertex which is adjacent to every non-isolated vertices. By contradiction, suppose that dZm is adjacent to every non-isolated vertices. Since n ∤ d, we may assume that d = p1 · · · pt , where 1 ≤ t < s. Let d′ = pt+1 · · · ps . Clearly, dZm and d′ Zm are non-adjacent. Since n ∤ d, by Theorem 1, d′ Zm is not an isolated vertex, a contradiction. Thus γ(Gn (Zm )) = |A| + 2. 3  Chromatic Index of Gn (Zm) In this section, we study the chromatic index of Gn (Zm ). First, we need the following theorems: Theorem A. [8, Theorem 17.4] (Vizing’s Theorem) If G is a simple graph, then either χ′ (G) = ∆(G) or χ′ (G) = ∆(G) + 1. Theorem B. [7, Corollary 5.4] Let G be a simple graph. Suppose that for every vertex 8 u of maximum degree, there exists an edge u — v such that ∆(G) − deg(v) + 2 is more than the number of vertices with maximum degree in G. Then χ′ (G) = ∆(G). Theorem C. [11, Theorem D] If G has order 2k and maximum degree 2k − 1, then χ′ (G) = ∆(G). If G has order 2k + 1 and maximum degree 2k, then χ′ (G) = ∆(G) + 1 if and only if the size of G is at least 2k2 + 1. Theorem 7. Suppose that Zn is a Zm -module and Gn (Zm ) is not a null graph. If n = pβ1 1 , Q then χ′ (Gn (Zm )) = ∆(Gn (Zm )) if and only if β1 si=2 (αi + 1) is an odd integer. Proof. Clearly, the set of all non-isolated vertices of Gn (Zm ) is equal to {dZm : d = pr11 · · · prss , d 6= 1, 0 ≤ r1 ≤ β1 − 1 and 0 ≤ ri ≤ αi for i = 2, . . . , s}. So Gn (Zm ) \ A is a Q complete graph of order β1 si=2 (αi + 1) − 1. The result, now, follows from [4, Theorem 5.11]. Theorem 8.  Suppose that Zn is a Zm -module and Gn (Zm ) is not a null graph. If n = p1 p2 and m = pα1 1 pα2 2 , then χ′ (Gn (Zm )) = ∆(Gn (Zm )) if and only if max{α1 , α2 } is an even integer. Proof. If α1 = α2 = 1, then Gn (Zm ) is a null graph. If α1 = 1 and α2 ≥ 2, then r Gn (Zm ) ∼ = Kα2 ∪ Kα2 . If α2 , α2 ≥ 2, then assume that Bi = {pi i Zm : 1 ≤ ri ≤ αi }, for i = 1, 2. Clearly, B1 ∪ B2 is the set of all non-isolated vertices of Gn (Zm ), every Bi forms a complete graph and Gn (Zm ) \ A is the disjoint union of them. Therefore Gn (Zm ) \ A ∼  = Kα1 ∪ Kα2 . Now, [4, Theorem 5.11] completes the proof. Theorem 9. Let Zn be a Zm -module. If n = p1 · · · ps and s ≥ 3, then χ′ (Gn (Zm )) = ∆(Gn (Zm )). Proof. With no loss of generality assume that α1 ≥ · · · ≥ αs . Let {i ∈ S : αi = α1 } = {1, . . . , k} for some k, 1 ≤ k ≤ s. By Theorem 4, we know that {pri i Zm : 1 ≤ i ≤ k, 1 ≤ ri ≤ αi } is the set of all vertices with maximum degree. Hence Gn (Zm ) has kα1 vertices Q with maximum degree. On the other hand, we know that ∆(Gn (Zm )) = si=1 (αi + 1) − Q Q Qs−1 2 − (α1 + 1) si=2 αi and deg(p1 · · · ps−1 Zm ) = si=1 (αi + 1) − 2 − αs i=1 (αi + 1). It is easy to check that if s ≥ 4, then ∆(Gn (Zm )) − deg(p1 · · · ps−1 Zm ) + 2 is more than α1 + · · · + αs and so ∆(Gn (Zm )) − deg(p1 · · · ps−1 Zm ) + 2 is more than the number of 9 vertices with maximum degree. Note that deg(prss Zm ) = ∆(Gn (Zm )) (1 ≤ rs ≤ αs ) if and only if α1 = · · · = αs . Also, if s ≥ 4, then ∆(Gn (Zm )) − deg(p2 · · · ps Zm ) + 2 is more than sα1 . Therefore by Theorem B, we find that χ′ (Gn (Zm )) = ∆(Gn (Zm )). Now, consider s = 3. If α3 ≥ 2, then it is easy to see that ∆(Gn (Zm )) − deg(p1 · · · ps−1 Zm ) + 2 is more than α1 + α2 + α3 and the proof is complete. Therefore assume that α3 = 1. There are two following cases: Case 1. α2 = 1. If α1 = 1, then n = m and so Gn (Zm ) = G(Zm ). One can easily check that χ′ (G(Zp1 p2 p3 )) = ∆(G(Zp1 p2 p3 )) = 4 (see [9, Fig. 3]). If α1 ≥ 2, then Gn (Zm ) has α1 vertices with maximum degree. Also, ∆(Gn (Zm )) − deg(p1 p3 Zm ) + 2 is more than α1 and the result, follows from Theorem B. Case 2. α2 ≥ 2. In this case, it is easy to see that ∆(Gn (Zm )) − deg(p2 p3 Zm ) + 2 is more than α1 + α2 . Also, ∆(Gn (Zm )) − deg(p1 p3 Zm ) + 2 is more than α1 + α2 and the result, follows from Theorem B. Theorem 10.  Let Zn be a Zm -module. If n 6= p1 · · · ps and s ≥ 2, then χ′ (Gn (Zm )) = ∆(Gn (Zm )). Q Q Proof. By Theorem 4, we know that Gn (Zm ) has si=1 (αi + 1) − 1 − si=1 (αi − βi + 1) Q Q non-isolated vertices and ∆(Gn (Zm )) = si=1 (αi +1)−2− si=1 (αi −βi +1). Hence by TheQ Q orem C, we find that χ′ (Gn (Zm )) = ∆(Gn (Zm )), where si=1 (αi +1)−1− si=1 (αi −βi +1) Q Q is even. Next, assume that si=1 (αi + 1) − 1 − si=1 (αi − βi + 1) is odd. Since the size of a complete graph of order 2k + 1 is 2k2 + k, if we prove that Gn (Zm ), in this case,
Qs Qs losses at least k = i=1 (αi + 1) − 2 − i=1 (αi − βi + 1) /2 edges, then by Theorem Q C, χ′ (Gn (Zm )) = ∆(Gn (Zm )). Let d = si=1 pri i be a divisor of m such that n ∤ d and Q Q′ {i ∈ S ′ : ri ≥ βi } = 6 ∅. Let {i ∈ S ′ : ri ≥ βi } = {1, . . . , t}. Set d = si=t+1 pβi i si=s′ +1 pri i . It is easy to check that dZm and dZm are not adjacent. So we conclude that Gn (Zm )
Qs Qs Qs ′ Qs losses at least (α + 1) − (α − β + 1) − β (α + 1) /2 edges. We ′ i i i i i i=1 i=1 i=1 i=s +1 continue the proof in the following two cases: Case 1. βi ≥ 2, for some i, 1 ≤ i ≤ s′ . With no loss of generality we may aso n Q′ sume that βs′ ≥ 2. Suppose that B = pr11 si=2 pβi i Zm : 0 ≤ r1 ≤ β1 − 1 and C = o n Q pβ1 1 si=2 pri i Zm : 0 ≤ ri ≤ βi − 1, for 2 ≤ i ≤ s′ and otherwise 0 ≤ ri ≤ αi . Clearly, every element of B is not adjacent to every element of C. This implies that Gn (Zm ) losses at Q Q′ Q′ least si=1 βi si=s′ +1 (αi + 1) − ( si=2 βi + β1 − 1) new edges. Therefore Gn (Zm ) losses at Q Q′ Q Q Q′ least l = ( si=1 (αi + 1)− si=1 (αi − βi + 1)+ si=1 βi si=s′ +1 (αi + 1))/2− ( si=2 βi + β1 − 1) edges. It suffices we prove that k ≤ l. One can easily see that k ≤ l if and only if Q Q Q′ 2(β1 − 2) ≤ (β1 si=s′ +1 (αi + 1)− 2) si=2 βi . Since βs′ ≥ 2, so 2(β1 − 2) ≤ (β1 si=s′ +1 (αi + 10 1) − 2) Q s′ i=2 βi and hence k ≤ l. Case 2. β1 = · · · = βs′ = 1. Clearly, Qs ′ i=2 pi Zm and p1 Qs ri i=s′ +1 pi Zm are non- adjacent, where 0 ≤ ri ≤ αi , for i = s′ + 1, . . . , s . This implies that Gn (Zm ) losses at Q Q least si=s′ +1 (αi + 1) − 1 new edges. Therefore Gn (Zm ) losses at least l = ( si=1 (αi + 1) − Qs Qs i=1 (αi − βi + 1) + i=s′ +1 (αi + 1))/2 − 1 edges. In this case, k ≤ l is obvious and the proof is complete.  From the above theorems, we can deduce the next result. Corollary 6. Let Zn be a Zm -module. If Gn (Zm ) is not a null graph, then χ′ (Gn (Zm )) = ∆(Gn (Zm )), unless the following cases: (i) n = pβ1 1 and β1 Qs i=2 (αi + 1) is even. (ii) n = p1 p2 , m = pα1 1 pα2 2 and max{α1 , α2 } is odd. 4 Eulerian Tour in Gn (Zm) An Eulerian tour in a graph is a closed trail including all the edges of the graph. A graph is Eulerian if it has an Eulerian tour. By [8, Theorem 4.1], a simple connected graph is Eulerian if and only if it has no vertices of odd degree. In this section, we determine all integer numbers n and m for which Gn (Zm ) \ A is an Eulerian graph. We start with the following theorem. Theorem 11. Let Zn be a Zm -module. If Gn (Zm ) is not a null graph, then diam(Gn (Zm )\ A) ≤ 4, except for the case n = p1 p2 , m = pα1 1 p2α2 and α1 , α2 ≥ 2, in which case Gn (Zm ) \ A ∼ = Kα1 ∪ Kα2 . Proof. Assume that Gn (Zm ) is not a null graph. In view of the proof of Theorem 6, we conclude that if n 6= p1 · · · ps , then Gn (Zm ) \ A is connected and diam(Gn (Zm ) \ A) ≤ 2. Now, suppose that n = p1 · · · ps . Since Gn (Zm ) is not a null graph, so s 6= 1. First assume that s ≥ 3. By the proof of Theorem 6, we know that {p1 Zm , p2 · · · ps Zm } is a dominating set for Gn (Zm ) \ A. Since s ≥ 3, p2 Zm is adjacent to both p1 Zm and p2 · · · ps Zm . This implies that Gn (Zm ) \ A is connected and diam(Gn (Zm ) \ A) ≤ 4. Next, assume that s = 2. By Lemma 1, if n = p1 p2 , m = pα1 1 p2 and α1 ≥ 2, then diam(Gn (Zm ) \ A) = 1. Now, consider n = p1 p2 , m = pα1 1 pα2 2 and α1 , α2 ≥ 2. As we saw in the proof of Theorem 8, Gn (Zm ) \ A ∼  = Kα1 ∪ Kα2 and the proof is complete. 11 Now, we are in a position to generalize Theorem 5.1 of [9]. Theorem 12. Suppose that Zn is a Zm -module and Gn (Zm ) is not a null graph. Then Gn (Zm ) \ A is an Eulerian graph if and only if one of the following holds: (i) αi and βi are even integers for each i, 1 ≤ i ≤ s. (ii) αi is an odd integer and βi is an even integer for some i, 1 ≤ i ≤ s. (iii) n = p1 · · · ps and m = pα1 1 · · · pαs s , where αi ’ are odd integers and s ≥ 3. (iv) n = p1 p2 and m = pα1 1 p2 , where α1 > 1 is an odd integer. Proof. By Theorem 3, Gn (Zm ) \ A is an Eulerian graph if and only if for each nonQ Q Q isolated vertex dZm of Gn (Zm ) both si=1 (αi + 1) and i∈D i∈Dd (αi − βi + 1) / d (αi + 1) are even or odd integers. So the “if” part of theorem is obvious. For the converse, Q suppose that Gn (Zm ) \ A is an Eulerian graph. First assume that both si=1 (αi + 1) Q Q and i∈D i∈Dd (αi − βi + 1) are odd integers for a vertex dZm of Gn (Zm ) \ A. / d (αi + 1) Thus all αi ’ are even integers. If βi is an odd integer for some i, 1 ≤ i ≤ s, then there exists a vertex d′ Zm such that Dd′ = {i}. (Note that Gn (Zm ) is not a null graph.) So Q Q ′ i∈D / ′ (αi + 1) i∈D ′ (αi − βi + 1) is an even integer which implies that deg(d Zm ) is an d d odd integer, a contradiction. Therefore βi is an even integer for each i, 1 ≤ i ≤ s. Next, Q Q Q assume that both si=1 (αi + 1) and i∈D i∈Dd (αi − βi + 1) are odd integers for a / d (αi + 1) vertex dZm of Gn (Zm ) \ A. Hence αi is an odd integer for some i, 1 ≤ i ≤ s. With no loss of generality suppose that {i ∈ S : αi is an odd integer} = {1, . . . , t}, where 1 ≤ t ≤ s. Suppose that β1 , . . . , βt are odd integers. If n = p1 · · · ps and m = pα1 1 · · · pαs s , where αi ’ are odd integers, then we are done. (Note that Gn (Zm ) \ A is a connected graph.) Otherwise Q Q there exists a vertex d′ Zm such that Dd′ = {1, . . . , t}. So i∈D / d′ (αi +1) i∈Dd′ (αi −βi +1) is an odd integer which implies that deg(d′ Zm ) is an odd integer, a contradiction. Thus βi is an even integer for some i, 1 ≤ i ≤ t. The proof is complete.  References [1] S. Akbari, S. Khojasteh, Commutative rings whose cozero-divisor graphs are unicyclic or of bounded degree, Comm. Algebra, 42 (2014), 1594–1605. [2] S. Akbari, S. Khojasteh, A. Yousefzadehfard, The proof of a conjecture in Jacobson graph of a commutative ring, J. Algebra Appl., Vol. 14, No. 10 (2015) 1550107. 12 [3] S. Akbari, H.A. Tavallaee, S. Khalashi Ghezelahmad, Intersection graph of submodules of a module, J. Algebra Appl. 11 (2012), Article No. 1250019. [4] I. Anderson, A First Course in Discrete Mathematics, Springer-Verlag, 2001. [5] D.F. Anderson, P.S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra, 217 (1999), 434–447. [6] S.E. Atani, A. Yousefian Darani, E.R. Puczylowski, On the diameter and girth of ideal-based zero-divisor graphs, Publicationes Mathematicae Debrecen, 78:3-4 (2011) 607–612. [7] L.W. Beineke, B.J. Wilson, Selected Topics in Graph Theory, Academic Press Inc., London, 1978. [8] J.A. Bondy, U.S.R. Murty, Graph Theory, Graduate Texts in Mathematics, 244 Springer, New York, 2008. [9] I. Chakrabarty, S. Ghosh, T.K. Mukherjee, M.K. Sen, Intersection graphs of ideals of rings, Discrete Mathematics, 309 (2009), 5381–5392 [10] B. Csákány and G. Pollák, The graph of subgroups of a finite group (Russian), Czech. Math. J. 19 (1969), 241–247. [11] M.J. Plantholt, The chromatic index of graphs with large maximum degree, Discrete Math. 47 (1981), 91–96. 13 

1 Degrees of Freedom of the Broadcast Channel with Hybrid CSI at Transmitter and Receivers arXiv:1709.02884v1 [] 9 Sep 2017 Mohamed Fadel, Student Member, IEEE, and Aria Nosratinia, Fellow, IEEE Abstract In general the different links of a broadcast channel may experience different fading dynamics and, potentially, unequal or hybrid channel state information (CSI) conditions. The faster the fading and the shorter the fading block length, the more often the link needs to be trained and estimated at the receiver, and the more likely that CSI is stale or unavailable at the transmitter. Disparity of link fading dynamics in the presence of CSI limitations can be modeled by a multi-user broadcast channel with both non-identical link fading block lengths as well as dissimilar link CSIR/CSIT conditions. This paper investigates a MISO broadcast channel where some receivers experience longer coherence intervals (static receivers) and have CSIR, while some other receivers experience shorter coherence intervals (dynamic receivers) and do not enjoy free CSIR. We consider a variety of CSIT conditions for the above mentioned model, including no CSIT, delayed CSIT, or hybrid CSIT. To investigate the degrees of freedom region, we employ interference alignment and beamforming along with a product superposition that allows simultaneous but non-contaminating transmission of pilots and data to different receivers. Outer bounds employ the extremal entropy inequality as well as a bounding of the performance of a discrete memoryless multiuser multilevel broadcast channel. For several cases, inner and outer bounds are established that either partially meet, or the gap diminishes with increasing coherence times. Index Terms Broadcast channel, Channel state information, Coherence time, Coherence diversity, Degrees of freedom, Fading channel, Multilevel broadcast channel, Product superposition. The authors are with the Department of Electrical Engineering, University of Texas at Dallas, Richardson, TX 750830688 USA, E-mail: [email protected];[email protected]. This work was presented in part at the IEEE International Symposium on Information Theory (ISIT), Germany, June 2017 [1]. March 26, 2018 DRAFT 2 I. I NTRODUCTION The performance of a broadcast channel depends on both the channel dynamics as well as the availability and the quality of channel state information (CSI) on the two ends of each link [2]–[5]. The two issues of CSI and the channel dynamics are practically related. The faster the fading, the more often the channel needs training, thus consuming more channel resources, while a very slow fading link requires infrequent training, therefore slow fading models often assume that CSIR is available due to the cost of training being small when amortized over time. In practice, in a broadcast channel some links may fade faster or slower than others. Recently, it has been shown [6], [7], that the degrees of freedom of the broadcast channel are affected by the disparity of link fading speeds, but existing studies have focused on a few simple and uniform CSI conditions, e.g., neither CSIT nor CSIR were available in [6], [7] for any user. This paper studies a broadcast channel where the links experience both disparate fading conditions as well as non-uniform or hybrid CSI conditions. A review of the relevant literature is as follows. Under perfect instantaneous CSI, the degrees of freedom of a broadcast channel increase with the minimum of the transmit antennas and the total number of receivers antennas [8], [9]. However, due to the time-varying nature of the channel and feedback impairments, perfect instantaneous transmit-side CSI (CSIT) may not be available, and also receive-side CSI (CSIR) can be assumed for slow-fading channels only. Broadcast channel with perfect CSIR has been investigated under a variety of CSIT conditions, including imperfect, delayed, or no CSIT [2]–[4], [10]–[12]. In the absence of CSIT, Huang et al. [2] and Vaze and Varanasi [3] showed that the degrees of freedom collapse to that of the single-receiver, since the receivers are stochastically equivalent with respect to the transmitter. For a MISO broadcast channel Lapidoth et al. [4] conjectured that as long as the precision of CSIT is finite, the degrees of freedom collapse to unity. This conjecture was recently settled in the positive by Davoodi and Jafar in [10]. Moreover, for a MISO broadcast channel under perfect delayed CSIT Maddah-Ali and Tse in [11] showed using retrospective interference alignment that the degrees of freedom are 1 1 1+ 12 +...+ K > 1, where K is the number of the transmit antennas and also the number of receivers. A scenario of mixed CSIT was investigated in [12], where the transmitter has partial knowledge on the current channel in addition to delayed CSI. The potential variation between the quality of feedback links has led to the model of hybrid March 26, 2018 DRAFT 3 CSIT, where the CSIT with respect to different links may not be identical [10], [13]–[15]. A MISO broadcast channel with perfect CSIT for some receivers and delayed for the others was studied by Tandon et al. in [13] and Amuru et al. in [14]. Davoodi and Jafar in [10] showed that for a MISO two-receiver broadcast channel under perfect CSIT for one user and no CSIT for the other, the degrees of freedom collapse to unity. Tandon et al. in [15] considered a MISO broadcast channel with alternating hybrid CSIT to be perfect, delayed, or no CSIT with respect to different receivers. As mentioned earlier, investigation of broadcast channels under unequal link fading dynamics is fairly recent. An achievable degrees of freedom region for one static and one dynamic receiver was given in [16]–[19] via product superposition, producing a gain that is now known as coherence diversity. Coherence diversity gain was further investigated in [6], [7] for a K-receiver broadcast channel with neither CSIT nor CSIR. Also, a broadcast channel was investigated in [20], where the receivers MIMO fading links experience nonidentical spacial correlation. In this paper, we consider a multiuser model under a hybrid CSIR scenario where a group of receivers, denoted static receivers, are assumed to have CSIR, and another group with shorter link coherence time, denoted dynamic receivers, do not have free CSIR. We consider this model under a variety of CSIT conditions, including no CSIT, delayed CSIT, and two hybrid CSIT scenarios. In each of these conditions, we analyze the degrees of freedom region. A few new tools are introduced, and inner and outer bounds are derived that partially meet in some cases. The results of this paper are cataloged as follows. In the absence of CSIT, an outer bound on the degrees of freedom region is produced via bounding the rates of a discrete memoryless multilevel broadcast channel [21], [22] and then applying the extremal entropy inequality [23], [24]. Our achievable degrees of freedom region meets the outer bound in the limiting case where the coherence times of the static and dynamic receivers are the same. For delayed CSIT, we use the outdated CSI model that was used by Maddah-Ali and Tse [11] under i.i.d. fading and assuming global CSIR at all nodes. Noting that our model does not have uniform CSIR, we produced a technique with alignment over super-symbols to utilize outdated CSIT but merge it together with product superposition to reuse the pilots of the dynamic receivers for the purpose of transmission to static receivers. Moreover, we develop an outer bound that is suitable for block-fading channels with different coherence times, by appropriately enhancing March 26, 2018 DRAFT 4 static Tx dynamic Fig. 1. A broadcast channel with multiple static and multiple dynamic users the channel to a physically-degraded broadcast channel and then applying the extremal entropy inequality [23], [24]. For one static and one dynamic receiver, our achievable degrees of freedom partially meet our outer bound, and furthermore the gap decreases with the dynamic receiver coherence time T . Under hybrid CSIT, we analyze two conditions: First, we consider perfect CSIT for the static receivers and no CSIT with respect to the dynamic receivers. The achievable degrees of freedom in this case are obtained using product superposition with the dynamic receiver’s pilots reused and beamforming for the static receivers to avoid interference. Second, we consider perfect CSIT with respect to the static receivers and delayed CSIT with respect to the dynamic receivers. An achievable transmission scheme is proposed via a combination of beamforming, interference alignment, and product superposition methodologies. The outer bounds for the two hybrid-CSIT cases were based on constructing an enhanced physically degraded channel and then applying the extremal entropy inequality. For one static receiver with perfect CSIT and one dynamic receiver with delayed CSIT, the gap between the achievable and the outer sum degrees of freedom is 1 . T II. S YSTEM M ODEL Consider a broadcast channel with multiple single-antenna receivers and the transmitter is equipped with Nt antennas. The expressions “receiver” and “user” are employed without distinction throughout the paper, indicating the receiving terminals in the broadcast channel. The channels of the users are modeled as Rayleigh block fading where the channel coefficients remain constant over each block and change independently across blocks [25], [26]. As shown in Fig. 1, the users are partitioned into two sets based on channel availability and the length of the coherence interval: one set of dynamic users and another set of static users. The former contains March 26, 2018 DRAFT 5 TABLE I N OTATION Static Users Dynamic Users m′ m MISO channel gains g1 , . . . , gm′ h1 , . . . , hm received signals (continuous) ′ y1′ , . . . , ym ′ y1 , . . . , ym DMC receive variables ′ Y1′ , . . . , Ym ′ Y1 , . . . , Ym transmission rates ′ R1′ , . . . , Rm ′ R1 . . . , Rm ′ M1′ , . . . , Mm ′ M1 , . . . , Mm d′1 , . . . , d′m′ d1 , . . . , dm number of users messages degrees of freedom coherence time T ′ T General Variables X transmit signal ρ signal-to-noise ratio Ui , Vj , W auxiliary random variables H set of all channel gains Dx vertex of degrees of freedom region ei canonical coordinate vector m dynamic users having coherence time T and no free CSIR1 , and the latter contains m′ static users having coherence time T ′ and perfect instantaneous CSIR. We consider the transmitter is equipped with more antennas than the number of dynamic and static users, i.e., Nt ≥ m′ + m. The received signals yj′ (t), yi (t) at the static user j, and the dynamic user i, respectively, at time instant t are yj′ (t) = gj† (t)x(t) + zj′ (t), j = 1, . . . , m′ , yi (t) = h†i (t)x(t) + zi (t), i = 1, . . . , m, (1) where x(t) ∈ CNt is the transmitted signal, zj′ (t), zi (t) denote the corresponding additive i.i.d. Gaussian noise of the users, and gj (t) ∈ CNt , hi (t) ∈ CNt denote the channels of the static user j and the dynamic user i whose coefficients stay the same over T ′ and T time instances, 1 This means that the cost of knowing CSI at the receiver, e.g., by channel estimation, is not ignored. March 26, 2018 DRAFT 6 respectively. The distributions of gj and hi are globally known at the transmitter and at the users.2 Having CSIR, the value of gj (t) is available instantaneously and perfectly at the static user j. Furthermore, the static user j obtains an outdated version of the dynamic users channels hi , and also the dynamic user i obtains an outdated version of the static users channel gi (completely stale) [11]. CSIT for each user can take one of the following: • Perfect CSIT: the channel vectors, gj (t), hi (t), are available at the transmitter instantaneously and perfectly. • Delayed CSIT: the channel vectors, gj (t), hi (t), are available at the transmitter after they change independently in the following block (completely stale [11]). • No CSIT: the channel vectors, gj (t), hi (t), cannot not be known at the transmitter. We consider the broadcast channel with private messages for all users and no common ′ messages. More specifically, we assume that the independent messages Mj′ ∈ [1 : 2nRi (ρ) ], Mi ∈ [1 : 2nRi (ρ) ] associated with rates Rj′ (ρ), Ri (ρ) are communicated from the transmitter to the static user j and dynamic user i, respectively, at ρ signal-to-noise ratio. The degrees of freedom of the static and dynamic users achieving rates Rj′ (ρ), Ri (ρ) can be defined as Rj′ (ρ) , ρ→∞ log(ρ) Ri (ρ) di = lim , ρ→∞ log(ρ) j = 1, . . . , m′ , d′j = lim i = 1, . . . , m. (2) The degrees of freedom region is defined as  ′ +m ′ ∃(R1′ (ρ), . . . , Rm D = (d′1 , . . . , d′m′ , d1 , . . . , dm ) ∈ Rm ′ (ρ), R1 (ρ), . . . , Rm (ρ)) ∈ C(ρ), + Rj′ (ρ) Ri (ρ) , di = lim , j = 1, . . . , m′ , i = 1, . . . , m , ρ→∞ log(ρ) ρ→∞ log(ρ) d′j = lim (3) where C(ρ) is the capacity region at ρ signal-to-noise ratio. The sum degrees of freedom is defined as Csum (ρ) , ρ→∞ log(ρ) dsum = lim where ′ Csum (ρ) = max m X j=1 2 Rj′ (ρ) + m X (4) Ri (ρ). (5) i=1 Also, the coherence times of all channels are globally known at the transmitter and at the users. March 26, 2018 DRAFT 7 Y1’ X Y2’ p(y3’|y2’) p(ym’’|ym’-1’) p(y3|y2) p(ym|ym-1) Ym’’ p(y1’,y1|x) Y1 Fig. 2. p(y2’|y1’) p(y2|y1) Y2 Ym Discrete memoryless multiuser multilevel broadcast channel In the sequel, we study the degrees of freedom of the above MISO broadcast channel under different CSIT scenarios that could be perfect, delayed or no CSIT. III. N O CSIT FOR A LL U SERS In this section, we study the broadcast channel defined in Section II when there is no CSIT for all users. In particular, we give outer and achievable degrees of freedom regions in Section III-B and Section III-C, respectively. The outer degrees of freedom region is based on the construction of an outer bound on the rates of a multiuser multilevel discrete memoryless channel that is given in Section III-A. A. Multiuser Multilevel Broadcast Channel The multilevel broadcast channel was introduced by Borade et al. [21] as a three-user broadcast discrete memoryless broadcast channel where two of the users are degraded with respect to each other. The capacity of this channel under degraded message sets was established by Nair and El Gamal [22]. Here, we study a multiuser multilevel broadcast channel with two sets of degraded users (see Fig. 2). One set contains m′ users with Yj′ received signal at user j, and the other set contains m users with Yi received signal at user i. Therefore, X →Y1′ → Y2′ → · · · → Ym′ ′ X →Y1 → Y2 → · · · → Ym March 26, 2018 (6) DRAFT 8 form two Markov chains. We consider a broadcast channel with (m′ + m) private messages and no common message. An outer bound for the above multilevel broadcast channel is given in the following theorem. Theorem 1: The rate region of the multilevel broadcast channel with two sets of degraded users (Eq. (6)) is outer bounded by the intersection of R1 ≤I(Um′ , W ; Y1|V1 ) − I(W ; Ym′ ′ |Um′ ), (7) Ri ≤I(Vi−1 ; Yi |Vi ), (8) Rj′ ≤I(Uj−1 ; Yj′ |Uj ), i = 2, . . . , m, j = 1, . . . , m′ − 1, ′ ′ ′ ′ Rm ′ ≤I(W ; Ym′ |Um′ ) + I(X; Ym′ |Um′ , W ) − I(X; Ym′ |Um′ −1 ), (9) (10) and Ri ≤I(Ũi−1 ; Yi |Ũi ), i = 1, . . . , m − 1, (11) Rm ≤I(W̃ ; Ym |Ũm ) + I(X; Ym|Ũm , W̃ ) − I(X; Ym|Ũm−1 ), (12) R1′ ≤I(Ũm , W̃ ; Y1′ |Ṽ1 ) − I(W̃ ; Ym |Ũm ), (13) Rj′ ≤I(Ṽj−1; Yj′ |Ṽj ), (14) j = 2, . . . , m′ , for some pmf p(u1 , . . . , um′ , ũ1 , . . . , ũm , v1 , . . . , vm , ṽ1 , . . . , ṽm′ , w, w̃, x), (15) where Um′ → · · · → U1 →X → (Y1 , . . . , Ym, Y1′ , . . . , Ym′ ′ ) Vm → · · · → V1 → (W, Um′ ) →X → (Y1 , . . . , Ym, Y1′ , . . . Ym′ ′ ) Ũm → · · · → Ũ1 →X → (Y1 , . . . Ym , Y1′ , . . . , Ym′ ′ ) Ṽm′ → · · · → Ṽ1 → (W̃ , Ũm ) →X → (Y1 , . . . Ym , Y1′ , . . . , Ym′ ′ ) (16) forms Markov chains and U0 = Ũ0 , X. Proof: See Appendix I. Remark 1: Theorem 1 is an extension of the Körner-Marton outer bound [27, Theorem 5] to more than two users, and it recovers the Körner-Marton bound when m = m′ = 1. March 26, 2018 DRAFT 9 Remark 2: For the multiuser multilevel broadcast channel characterized by (6), we establish the capacity for degraded message sets in Appendix II, where one common message is communicated to all receivers and one further private message is communicated to one receiver. B. Outer Degrees of Freedom Region In the sequel, we give an outer bound on the degrees of freedom of the broadcast channel defined in Section II when there is no CSIT for all users. The outer bound development depends on the results of Theorem 1 in Section III-A. Theorem 2: An outer bound on the degrees of freedom region of the fading broadcast channel characterized by Eq. (1) without CSIT is, ′ m X d′j ≤ 1, m X di ≤ 1 − (17) j=1 i=1 m′ X j=1 d′j + m X i=1 di ≤   1  4 3 1 , T T = T ′ , ∆T = 0 (18) (19) otherwise, where ∆T is the offset between the two coherence intervals. Proof: Equations (17) and (18) are outer bounds for a broadcast channel whose users are either all homogeneously static or all homogeneously dynamic [7], [18]. The remainder of the proof is dedicated to establishing (19). We enhance the channel by giving all users global CSIR. When T ′ = T and ∆T = 0, (19) follows directly from [7], [18]. When T ′ 6= T or ∆T 6= 0, having no CSIT, the channel belongs to the class of multiuser multilevel broadcast channels in Section III-A. We then use the two outer bounds developed for the multilevel broadcast channels to generate two degrees of freedom bounds, and merge them to get the desired result. We begin with the outer bound described in (7)-(10); we combine these equations to obtain P P partial sum-rate bounds on the static ( Rj′ ) and dynamic ( Ri ) receivers: ′ m X Rj′ j=1 ≤ ′ −1 m X ′ ′ I(Uj−1 ; yj′ |Uj , H) + I(W ; ym ′ |Um′ , H) + I(x; ym′ |Um′ , W, H) j=1 ′ − I(x; ym ′ |Um′ −1 , H) March 26, 2018 DRAFT 10 = ′ −1 m X ′ ′ h(yj′ |Uj , H) − h(yj′ |Uj−1 , H) + I(W ; ym ′ |Um′ , H) + h(ym′ |Um′ , W, H) j=1 ′ − h(ym ′ |Um′ −1 , H) + o(log(ρ)) (20) ′ ′ =I(W ; ym ′ |Um′ , H) + h(ym′ |Um′ , W, H) + o(log(ρ)), (21) where H is the set of all channel vectors, (20) follows from the chain rule, h(yj′ |x, H) = o(log(ρ)), and (21) follows since the received signals of all static users, yj′ , have the same statistics [7], [18]. Also, using Theorem 1, m X Rj ≤I(Um′ , W ; y1|V1 , H) − ′ I(W ; ym ′ |Um′ , H) + m X I(Vj−1; yj |Vj , H) j=2 j=1 =h(y1 |V1 , H) − h(y1 |Um′ , W, H) − ′ I(W ; ym ′ |Um′ , H) + m X h(yj |Vj , H) j=2 − h(yj |Vj−1, H) (22) ′ = − h(y1 |Um′ , W, H) − I(W ; ym ′ |Um′ , H) + h(ym |Vm , H) + o(log(ρ)) (23) ′ ≤ − h(y1 |Um′ , W, H) − I(W ; ym ′ |Um′ , H) + log(ρ) + o(log(ρ)), (24) where (22) follows from the chain rule, (23) follows since yj have the same statistics, and (24) ′ follows since h(ym |Vm , H) ≤ n log(ρ) + o(log(ρ)). Define Yj,k to be the received signal of user j at time instance k. From (21) and (24), we can obtain the bound (27) on the rates. m′ m 1X ′ X 1 1 ′ ′ Rj + Rj ≤ I(W ; ym h(ym ′ |Um′ , H) + ′ |Um′ , W, H) − h(y1 |Um′ , W, H) 2 j=1 2 2 j=1 ′ − I(W ; ym ′ |Um′ , H) + log(ρ) + o(log(ρ)) 1 ′ = h(ym ′ |Um′ , W, H) − h(y1 |Um′ , W, H) + log(ρ) + o(log(ρ)) 2 1 ′ ≤ h(ym ′ , y1 |Um′ , W, H) − h(y1 |Um′ , W, H) + log(ρ) + o(log(ρ)) 2 n X 1 ′ ′ ′ ≤ h(ym ′ ,k , y1,k |Um′ , W, H, ym′ ,1 , . . . , ym′ ,k−1 , y1,1 , . . . , y1,k−1 ) 2 k=1 (25) ′ ′ − h(y1,k |Um′ , W, H, ym ′ ,1 , . . . , ym′ ,k−1 , y1,1 , . . . , y1,k−1 ) + log(ρ) + o(log(ρ)) ≤ March 26, 2018 max Tr{Σx }≤ρ,Σx <0 (26) EH 1 log |I + HΣx H† | − log(1 + h†1 Σx h1 ) + log(ρ) 2 DRAFT 11 + o(log(ρ)), (27) where (25) and (26) follow from the chain rule and that conditioning does not increase differential entropy, and (27) follows from extremal entropy inequality [23], [24], [28]. In order to bound (27), we use a specialization of [29, Lemma 3] as follows. Lemma 1: Consider two random matrices H1 ∈ CN1 ×Nt and H2 ∈ CN2 ×Nt , where N1 ≥ N2 . For a covariance matrix Σx , where Tr{Σx } ≤ ρ, we have 1 1 log |I + H1 Σx H†1 | − log |I + H2 Σx H†2 | ≤ o(log(ρ)). (28) Σx min{Nt , N1 } min{Nt , N2 } The proof of Lemma 1 is omitted as it directly follows from [29, Lemma 3]. Lemma 1 yields max the following outer bound on the degrees of freedom: m′ m 1X ′ X d + di ≤ 1. 2 j=1 j i=1 (29) We now repeat the exercise of bounding the sum rates and deriving degrees of freedom, this time starting from (11)-(14). By following bounding steps parallel to (21), (24), (27), ′ m X m d′j j=1 1X di ≤ 1. + 2 i=1 (30) Adding (29) and (30) yields the outer bound (19), completing the proof of Theorem 2. C. Achievable Degrees of Freedom Region Theorem 3: The fading broadcast channel described by Eq. (1) can achieve the following degrees of freedom without CSIT: m X di ≤ 1 − m X di ≤ 1. i=1 m′ X j=1 d′j + 1 , T (31) (32) i=1 Proof: The achievable scheme uses product superposition [17], [18], where the transmitter uses one antenna to send the super symbol to two users: one dynamic and one static, x† = xs x†d , (33) where xs ∈ C is a symbol intended for the static user, x†d = [xτ , x†δ ] March 26, 2018 (34) DRAFT 12 where xτ ∈ C is a pilot and xδ ∈ CT −1 is a super symbol intended for the dynamic user. Since degrees of freedom analysis is insensitive to the additive noise, we omit the noise component in the following. y† = hxs [xτ , x†δ ] = [hxτ , hx†δ ], (35) where h = hxs . The dynamic user estimates the equivalent channel h during the first time instance and then decodes xδ coherently based on the channel estimate. The static receiver only utilizes the received signal during the first time instance: y1′ = gxs . (36) Knowing its channel gain g, the static receiver can decode xs . The achievable degrees of freedom of the two users are, (d′ , d) = 1 1
. ,1 − T T (37) We now proceed to prove that the degrees of freedom region characterized by (31) and (32) can be achieved via a combination of two-user product superposition strategies that were outlined above, and single-user strategies. For clarity of exposition we refer to (31), which describes the degrees of freedom constraints of the dynamic receivers, as the non-coherent bound, and to (32) as the coherent bound. The non-negativity of degrees of freedom restricts them to the ′ m+m . The intersection of the coherent bound and the non-negative orthant non-negative orthant R+ is a (m′ + m)–simplex that has m + m′ + 1 vertices. The non-coherent bound is a hyperplane that partitions the simplex with m′ + 1 vertices on one side of the non-coherent bound and m on the other. Therefore the intersection of the simplex with the non-coherent bound produces a polytope with (m′ + 1)(m + 1) vertices.3 For illustration, see Fig. 3 showing the three-user degrees of freedom with two static users and Fig. 4 with one static user. We now verify that each of the (m′ + 1)(m + 1) vertices can be achieved with either a single-user strategy, or via a two-user product superposition strategy: • m′ vertices corresponding to single-user transmission to each static user j achieving one degree of freedom. 3 This can be verified with a simple counting exercise involving the number of edges of the simplex that cross the non-coherent bound. March 26, 2018 DRAFT 13 d d1 Non-coherent bound Non-coherent bound d2 Coherent bound d’2 d’ d’1 Fig. 3. Achievable degrees of freedom region of one dynamic and two static users • Coherent bound Fig. 4. Achievable degrees of freedom region of one static and two dynamic users m vertices corresponding to single-user transmission to each dynamic user i achieving (1 − T1 ) degrees of freedom. • m′ m vertices corresponding to product superposition applied to all possible pairs of static and dynamic users, achieving 1 T degrees of freedom for one static user and (1 − T1 ) degrees of freedom for one dynamic user. • One trivial vertex at the origin, corresponding to no transmission achieving zero degrees of freedom for all users. Hence, the number of the vertices is m′ + m + m′ m + 1 = (m + 1)(m′ + 1). This completes the achievability proof of Theorem 3. Remark 3: When the static and dynamic users have the same coherence time, the inner and outer bounds on degrees of freedom coincide. In this case it is degrees of freedom optimal to serve two-users at a time (one dynamic and one static). IV. D ELAYED CSIT FOR ALL USERS Under delayed CSIT, the transmitter knows each channel gain only after it is no longer valid. This condition is also known as outdated CSIT. We begin by proving inner and outer bounds March 26, 2018 DRAFT 14 when transmitting only to static users, only to dynamic users, and to one static and one dynamic user. We then synthesize this collection of bounds into an overall degrees of freedom region. A. Transmission to Static Users Theorem 4: The degrees of freedom region of the fading broadcast channel characterized by Eq. (1) with delayed CSIT and having m′ static users and no dynamic users is 1 j = 1, . . . , m′ . (38) 1 , 1 + + . . . + m′ Proof: The special case of fast fading (T ′ = 1) was discussed by Maddah-Ali and Tse d′j ≤ 1 2 in [11], where the achievability was established by retrospective interference alignment that aligns the interference using the outdated CSIT, and the converse was proved by generating an improved channel without CSIT having a tight degrees of freedom region against TDMA according to the results in [2], [3]. For T ′ ≥ 1, the achievability is established by employing retrospective interference alignment presented in [11] over super symbols each of length T ′ . The converse is proved by following the same procedures in [11] to generate a block-fading improved channel without CSIT and with identical coherence intervals of length T ′ . According to the results of [7], [18], TDMA is tight against the degrees of freedom region of the improved channel. B. Transmission to Dynamic Users Theorem 5: The fading broadcast channel characterized by Eq. (1), with delayed CSIT and having m dynamic users and no static users, can achieve the degrees of freedom di ≤ 1 1 + + ...+ 1 2 1 (1 m − m ), T i = 1, . . . , m. (39) An outer bound on the degrees of freedom region is di ≤1 − m X 1 , T (40) m . (41) 1 + + . . . + m1 i=1 Proof: The achievability part can be proved as follows. At the beginning of each super di ≤ 1 2 symbol, m pilots are sent for channel estimation. Then retrospective interference alignment in [11] over super symbols is employed during the remaining (T − m) instances, to achieve (39). March 26, 2018 DRAFT 15 For the converse part, (41) is proved by giving the users global CSIR, and then applying Theorem 4. Moreover, (40) is the single-user bound for each dynamic user that can be proved as follows. For a single user with delayed CSIT, feedback does not increase the capacity [30], and consequently the assumption of delayed CSIT can be removed. Hence, the single-user bound for each dynamic user with delayed CSIT is the same as the single-user bound without CSIT [26]. C. Transmission to One Static and One Dynamic User Theorem 6: The fading broadcast channel characterized by Eq. (1), with delayed CSIT and having one static and one dynamic user, can achieve the following degrees of freedom 2 1 2 2
(1 + ), (1 − ) , 3 T 3 T 1 1 D2 : (d′ , d) = ( , 1 − ). T T D1 : (d′ , d) = (42) (43) Furthermore, the achievable degrees of freedom region is the convex hull of the above degrees of freedom pairs. Proof: From Section III-C, product superposition achieves the pair (43) that does not require CSIT for any of the two users. The remainder of the proof is dedicated to the achievability of the pair (42). We provide a transmission scheme based on retrospective interference alignment [11] along with product superposition. 1) The transmitter first emits a super-symbol intended for the static user: X1 = [X1,1 , · · · , X1,ℓ ], where ℓ = T′ , T (44) and each X1,n ∈ C2×T occupies T time instances and has the following structure: X1,n = [Ūn , Ūn Un ], n = 1, . . . , ℓ, (45) both the diagonal matrix Ūn ∈ C2×2 and Un ∈ C2×(T −2) contain symbols intended for the ′ ′ ′ † † static user. The components of y1† = [y1,1 , · · · , y1,ℓ ] are: ′ † y1,n =[g1† Ūn , g1† Ūn Un ], † † =[g̃1,n , g̃1,n Un ], March 26, 2018 n = 1, . . . , ℓ (46) DRAFT 16 † = g1† Ūn . The static user by definition knows g1 so it can decode Ūn which where g̃1,n ′ yields 2 TT degrees of freedom. The remaining † Un involve (T − 2) observations in g̃1,n T′ T ′ 2 TT (T − 2) unknowns, so they require a further T′ T (T − 2) independent observations for reliable decoding. † † The components of y1† = [y1,1 , · · · , y1,ℓ ] are † =[h†1,n Ūn , h†1,n Ūn Un ], y1,n n = 1, . . . , ℓ =[h̃†1,n , h̃†1,n Un ], (47) where h̃†1,n = h†1,n Ūn is the equivalent channel estimated by the dynamic user. The dynamic user saves h̃†1,n Un for interference cancellation in the upcoming steps. 2) The transmitter sends a second super symbol intended for the dynamic user: X2 = [X2,1 , · · · , X2,ℓ ], (48) where X2,n = [Ũn , Ũn Vn ], n = 1, . . . , ℓ, (49) Ũn ∈ C2×2 is diagonal and includes 2 independent symbols intended for the static user, and Vn ∈ C2×(T −2) contains independent symbols intended for the dynamic user. The † † components of y2† = [y2,1 , · · · , y2,ℓ ] are † y2,n =[h†2,n Ũn , h†2,n Ũn Vn ], n = 1, . . . , ℓ =[h̃†2,n , h̃†2,n Vn ], (50) where h̃†2,n = h†2,n Ũn is the equivalent channel estimated by the dynamic user. The dynamic user saves h̃†2,n Vn which includes T′ T T′ (T − T ′† y2,ℓ ] are unknowns, and hence an additional ′ ′ † components of y2† = [y2,1 , ··· , ′ (T − 2) independent observations about 2 TT (T − 2) 2) observations are needed to decode Vn . The ′ y2,n =[g2† Ũn , g2† Ũn Vn ], n = 1, . . . , ℓ † † =[g̃2,n , g̃2,n Vn ], (51) † where g̃2,n = g2† Ũn is the equivalent channel estimated by the static user; the static user † saves g̃2,n Vn for the upcoming steps. Knowing g2 , the static user achieves 2 TT further ′ degrees of freedom from decoding Ũn . March 26, 2018 DRAFT 17 3) The transmitter emits a third super symbol consisting of a linear combination of the signals generated from the first and the second super symbols. X3 = [X3,1 , · · · , X3,ℓ ], (52) where † X3,n = [Ûn , Ûn (h̃†1,n Un + g̃2,n Vn )], n = 1, . . . , ℓ, (53) Ûn ∈ C2×2 is diagonal and contains 2 independent symbols intended for the static user, ′ and hence the static user achieves further 2 TT degrees of freedom. † The static user cancels g̃2,n Vn saved during the second super symbol and obtains h̃†1,n Un that includes the additional independent T′ T (T − 2) observations needed for decoding Un . T′ Therefore, the static user achieves 2 T (T − 2) further degrees of freedom. The dynamic user estimates the equivalent channel h̃†3,n = h†3,n Ûn , cancels h̃†1,n Un saved during the † Vn that contains the additional observations needed first super symbol, and obtains g̃2,n ′ for decoding Vn . Hence, the dynamic user achieves 2 TT (T − 2) degrees of freedom. In aggregate, over 3T ′ time instants, the static and dynamic user achieve the degrees of freedom d′ = 6 T′ T′ + 2 (T − 2), T T d=2 T′ (T − 2). T (54) This completes the proof of Theorem 6. Theorem 7: An outer bound on the degrees of freedom region of the fading broadcast channel characterized by Eq. (1), with one static and one dynamic user having delayed CSIT, is d′ + d ≤ 1, 2 d d′ + ≤ 1, 2 (55) (56) 1 . (57) T Proof: The inequality (57) represents the single-user outer bound [26]. We prove the d≤1− bound (55) as follows. We enhance the original channel by giving both users global CSIR. In addition, the channel output of the dynamic user, y(t), is given to the static user. Therefore, the channel outputs at time instant t are (y ′(t), y(t), H) at the static user, and (y(t), H) at the dynamic user. The enhanced channel is physically degraded [31], [32], hence, removing the delayed CSIT does not reduce the capacity [33]. Also, R′ ≤I(x(t); y ′(t), y(t)|U, H) = h(y ′(t), y(t)|U, H) − h(y ′ (t), y(t)|U, x(t), H) March 26, 2018 DRAFT 18 R ≤I(U; y(t)|H) = h(y(t)|H) − h(y(t)|U, H), (58) where U is an auxiliary random variable, and U → x → (y ′(t), y(t)) forms a Markov chain. Therefore, 1 R′ + R ≤h(y(t)|H) + h(y ′(t), y(t)|U, H) − h(y(t)|U, H) + o(log(ρ)) 2 2 1 (59) ≤ log(ρ) + h(y ′(t), y(t)|U, H) − h(y(t)|U, H) + o(log(ρ)) 2 1 ≤ log(ρ) + max EH log |I + HΣx H† | − log(1 + h† (t)Σx h(t)) + o(log(ρ)) Tr{Σx }≤ρ,Σx <0 2 (60) ≤ log(ρ) + o(log(ρ)), (61) where (59) follows since h(y(t)|H) ≤ log(ρ)+o(log(ρ)) [34], (60) follows from extremal entropy inequality [23], [24], [29], and (61) follows from Lemma 1. Hence, the bound (55) is proved. A similar argument, with the role of the two users reversed, leads to the bound (56). Remark 4: The inner and outer bounds obtained for the two-user case partially meet, with the gap diminishing with the coherence time of the dynamic user as shown in Fig. 5 and Fig. 6 for T = 15 and T = 30, respectively. D. Transmission to arbitrary number of static and dynamic users Theorem 8: The fading broadcast channel characterized by Eq. (1), with delayed CSIT, can achieve the multiuser degrees of freedom characterized by vectors Di , 1 D1 : 1 1 + 2 + ...+ ′ m X 1 m′ i=1 e†i , 1 2 2 2 D2 , . . . , Dmm′ +1 : (1 + )e†j + (1 − )e†m′ +i , j = 1, . . . , m′ , i = 1, . . . , m, 3 T 3 T m m † 1 m X † Dmm′ +2 , . . . , Dmm′ +m′ +2 : ej + e , j = 1, . . . , m′ , (1 − ) T T i=1 i 1 + 21 + . . . + m1 (62) (63) (64) where ej is the canonical coordinate vector. Their convex hull characterized an achievable degrees of freedom region. Proof: The achievability of (62) was proved in Section IV-A via multiuser transmission to static users. The achievability of (63) was proved in Section IV-C, via a two-user transmission to a dynamic-static pair. March 26, 2018 DRAFT 19 1.5 Achievable region Outer region 1 0.5 0 0 Fig. 5. 0.2 0.4 0.6 0.8 1 1.2 One static and one dynamic with delayed CSIT and T = 15 1.5 Achievable region Outer region 1 0.5 0 0 Fig. 6. 0.2 0.4 0.6 0.8 1 1.2 One static and one dynamic with delayed CSIT and T = 30 March 26, 2018 DRAFT 20 We now show the achievability of (64) via retrospective interference alignment [11] along with product superposition. Over a super symbol of length T , consider the following transmission: X = [U, UV], (65) where U ∈ Cm×m is diagonal and includes m independent symbols intended for the static user j, and V ∈ Cm×(T −m) is a super symbol containing independent symbols intended for the dynamic users according to retrospective interference alignment [11]. Therefore, the static user decodes U. Thus, over T time instants, the static user achieves m degrees of freedom and the dynamic users achieve 1 1 (T 1+ 12 +...+ m − m), hence (64) is achieved. Theorem 9: An outer bound on the degrees of freedom of the fading broadcast channel characterized by Eq. (1), with delayed CSIT, is ′ m X d′j di + ≤ 1, ′ m + m i=1 m (66) m m X d′j X di + ≤ 1, ′ ′ m m + m j=1 i=1 (67) m X j=1 ′ d′j ≤ 1, ∀j = 1, . . . , m′ , (68) 1 , ∀i = 1, . . . , m. (69) T Proof: The inequalities (68), (69) represent the single-user bounds on the static and the di ≤ 1 − dynamic users, respectively [26], [34]. The remainder of the proof is dedicated to establishing the bounds (66) and (67). We enhance the channel by providing global CSIR as well as allowing full cooperation among static users and full cooperation among dynamic users. The enhanced channel is equivalent to a broadcast channel with two users: one static equipped with m′ antennas and one dynamic ′ equipped with m antennas. Define Y ′ ∈ Cm and Y ∈ Cm to be the received signals of the static and the dynamic super-user, respectively, in the enhanced channel. We further enhance the channel by giving Y to the static user, generating a physically degraded channel since X → (Y ′ , Y) → Y forms a Markov chain. Feedback including delayed CSIT has no effect on capacity [33], therefore we remove it from consideration. Subsequently, we can utilize the Körner-Marton outer bound [27], ′ m X Rj′ ≤I(X; Y ′, Y|U, H) j=1 March 26, 2018 DRAFT 21 m X Ri ≤I(U; Y|H). (70) i=1 Therefore, from applying extremal entropy inequality [23], [29], [35] and Lemma 1, ′ m X j=1 m X Rj′ 1 1 Ri ′ + ≤ I(X; Y , Y|U, H) + I(U; Y|H) m′ + m i=1 m m′ + m m = m′ 1 1 1 h(Y ′, Y|U, H) + o(log(ρ)) + h(Y|H) − h(Y|U, H) +m m m ≤ log(ρ) + o(log(ρ)). (71) Therefore, the bound (66) is proved. Similarly, we can prove the bound (67) using the same steps after switching the roles of the two users in the enhanced channel. V. H YBRID CSIT: P ERFECT CSIT FOR THE S TATIC U SERS AND N O CSIT FOR THE DYNAMIC U SERS Theorem 10: The fading broadcast channel characterized by Eq. (1), with perfect CSIT for the static users and no CSIT for the dynamic users, can achieve the following multiuser degrees of freedom, ′ D1 : m X e†j , (72) j=1 ′ D2 , . . . , Dm+1 m 1 1X † ej + (1 − )e†i , : T j=1 T i = 1, . . . , m. (73) Therefore, their convex hull is also achievable. Proof: D1 is achieved by inverting the channels of the static users at the transmitter and then every static user achieves one degree of freedom. D2 , . . . , Dm+1 in (73) are achieved using product superposition along with channel inversion as follows. The transmitted signal over T instants is, X = [u, uv† ], where u = Pm′ j=1 bj uj , (74) uj is a symbol intended for the static user j, gj† bj = 0, and v ∈ CT −1 contain independent symbols intended for the dynamic user i. Each of the static users receive an interference-free signal during the first time instant achieving one degrees of freedom. The dynamic user estimates its equivalent channel during the first time instant and decodes v during the remaining (T − 1) time instants. March 26, 2018 DRAFT 22 Theorem 11: An outer bound on the degrees of freedom of the fading broadcast channel characterized by Eq. (1), with perfect CSIT for the static users and no CSIT for the dynamic users, is ′ m X j=1 m X d′j + di ≤ 1, m′ + 1 i=1 d′j ≤ 1, m X (75) ∀j = 1, . . . , m′ , (76) 1 . (77) T i=1 Proof: The inequalities (76) represent single-user bounds for the static users [34], and (77) di ≤ 1 − is a time-sharing outer bound for the dynamic users that was established in [7], [18]. It remains to prove (75), as follows. We enhance the channel by giving global CSIR to all users and allowing full cooperation between the static users. This gives rise to an equivalent static user with m′ antennas receiving Y ′ over an equivalent channel G and noise Z′ . At this point, we have a multi-user system where CSIT is available with respect to one user, but not others. We then bound the performance of this system with that of another (similar) system that has no CSIT. To do so, we use the local statistical equivalence property developed and used in [13], [15], [36]. First, we draw G̃, Z̃ according to the distribution of G, Z′ and independent of them. We enhance the channel by providing Ỹ = G̃X + Z̃ to the static receiver and G̃ to all receivers. Because we do not provide G̃ to the transmitter, there is no CSIT with respect to Ỹ. According to [36], we have h(Ỹ, Y ′|H) = h(Y ′ |H) + o(log(ρ)), where H = (G, G̃, h1 , . . . , hm ), therefore we can remove Y ′ from the enhanced channel without reducing its degrees of freedom. This new equivalent channel has one user with m′ antennas receiving (Ỹ, H), m single-antenna users receiving (yi , H), and no CSIT.4 Having no CSIT, the enhanced channel is in the form of a multilevel broadcast channel studied in Section III-A, and hence using Theorem 1, ′ m X Rj′ ≤I(W ; Ỹ|U, H) + I(X; Ỹ|U, W, H) j=1 4 In the enhanced channel after removal of Y ′ , the transmitter and receivers still share information about G, but this random variable is now independent of all (remaining) transmit and receive variables. March 26, 2018 DRAFT 23 R1 ≤I(U, W ; y1|V1 , H) − I(W ; Ỹ|U, H) Ri ≤I(Vi−1 ; yi |Vi , H), i = 2, . . . , m. (78) The dynamic receiver received signals have the same distribution. By following bounding steps parallel to (22), (23), (24), m X Ri ≤ log(ρ) + o(log(ρ)) − I(W ; Ỹ|U, H) − h(y1 |U, W, H). (79) j=1 Therefore, ′ m X j=1 m X Rj′ 1 h(Ỹ|U, W, H) + R ≤ log(ρ) + o(log(ρ)) + ( − 1)I(W ; Ỹ|U, H) + i m′ + 1 j=1 m′ + 1 m′ + 1 − h(y1 |U, W, H), ≤ log(ρ) + o(log(ρ)) + (80) h(Ỹ, y1 |U, W, H) − h(y1 |U, W, H) m′ + 1 ≤ log(ρ) + o(log(ρ)), (81) (82) where the last inequality follows from applying the extremal entropy inequality [23], [29], [35] and Lemma 1. This concludes the proof of the bound (75). VI. H YBRID CSIT: P ERFECT CSIT FOR THE S TATIC U SERS AND D ELAYED CSIT FOR THE DYNAMIC U SERS We begin with inner and outer bounds for one static and one dynamic user, then extend the result to multiple users. The transmitter knows the channel of the static users perfectly and instantaneously, and an outdated version of the channel of the dynamic users. A. Transmitting to One Static and One Dynamic User Theorem 12: For the fading broadcast channel characterized by Eq. (1) with one static and one dynamic user, with perfect CSIT for the static user and delayed CSIT for the dynamic user, the achievable degrees of freedom region is the convex hull of the vectors, 1 1 1 , − ), 2T 2 2T 1 1 D2 :(d′, d) = ( , 1 − ). T T D1 :(d′, d) = (1 − March 26, 2018 (83) (84) DRAFT 24 Proof: The degrees of freedom (84) can be achieved by product superposition as discussed in Section III, without CSIT. We proceed to prove the achievability of (83). 1) Consider [u1 , ··· , uT −1 ] to be a complex 2 × (T − 1) matrix containing symbols intended for the static user, [v1 , · · · , vT −1 ] intended for the dynamic user, and b ∈ C is a beamforming vector so that g† b = 0. In addition we define u0 = 0, v0 = 1. Using these components, the transmitter constructs and transmits a super-symbol of length T , whose value at time t is: x†1 (t) = ut + b vt . (85) Note that x1 (0) = b does not carry any information for either user, and serves as a pilot. The received super symbol at the static user is: ′ y1† = [0, g† u1 , · · · , g† uT −1 ]. (86) The received super symbol at the dynamic user y1† =[h†1 b, (h†1 u1 + h†1 bv1 ), · · · , (h†1 uT −1 + h†1 bvT −1 )]. (87) The dynamic user estimates its equivalent channel h†1 b from the received value in the first time instant. The remaining terms include symbols intended for the dynamic user plus some interference, whose cancellation is the subject of the next step. 2) The transmitter next sends a second super symbol of length T , x2 = [ū, ū(h†1 u1 ), · · · , ū(h†1 uT −1 )], (88) where ū ∈ C is a symbol intended for the static user. Hence, y2† = [h2 ū, h2 ū(h†1 u1 ), · · · , h2 ū(h†1 uT −1 )]. (89) The dynamic user estimates the equivalent channel h2 ū during the first time instant and then acquires h†1 ut , the interference in (87). Therefore, using y1 , y2 , the dynamic user solves for vt achieving (T − 1) degrees of freedom. Furthermore, ′ y2† = [g1 ū, g1 ū(h†1 u1 ), · · · , g1 ū(h†1 uT −1 )]. (90) The static user solves for ū achieving one degree of freedom and also uses h†1 ut to solve for ut achieving further 2 (T − 1) degrees of freedom. March 26, 2018 DRAFT 25 In summary, during 2T instants, the static user achieves (2T − 1) degrees of freedom and the dynamic user achieves (T − 1) degrees of freedom. This shows the achievability of (83). Theorem 13: For the fading broadcast channel characterized by Eq. (1) with one static and one dynamic user, where there is perfect CSIT for the static user and delayed CSIT for the dynamic user, an outer bound on the degrees of freedom region is, d′ + d ≤1, 2 d′ ≤1, (91) (92) 1 . (93) T Proof: The inequalities (92) and (93) represent the single-user outer bounds [26], [34]. It d ≤1 − only remains to prove the outer bound (91), as follows. 1) We enhance the channel by giving global CSIR to both users and also give y to the static user. The enhanced channel is physically degraded having (Y ′ , G) at the static user and (y, G) at the dynamic user, where Y ′ , (y ′, y) and G , (h, g). In a physically degraded channel, causal feedback (including delayed CSIT) does not affect capacity [33], so we can remove the delayed CSIT with respect to the dynamic user. 2) We now use another enhancement with the motivation to remove the remaining CSIT (noncausal, with respect to the static user). This is accomplished, similar to Theorem 11, via local statistical equivalence property [13], [15], [36] in the following manner. We create a channel G̃, and noise Z̃ with the same distribution but independently of the true channel and noise, and a signal Ỹ = G̃X + Z̃. A genie will give Ỹ to the static receiver and G̃ to both receivers. It has been shown [36] that h(Ỹ, Y ′ |H) = h(Y ′ |H) + o(log ρ), where H = (G, G̃), therefore we can remove Y ′ from the enhanced channel without reducing its degrees of freedom. 3) The enhanced channel is still physically degraded, therefore [31], [32] R′ ≤I(x; Ỹ|U, H) = h(Ỹ|U, H) + o(log(ρ)) R ≤I(U; y|H) = h(y|H) − h(y|U, H), (94) where U is an auxiliary random variable, and U → x → (y ′ , y) forms a Markov chain. Therefore, 1 1 ′ R + R ≤h(y|H) + h(Ỹ|U, H) − h(y|U, H) + o(log(ρ)) 2 2 March 26, 2018 DRAFT 26 1.5 Achievable region Outer region 1 0.5 0 0 Fig. 7. 0.2 0.4 0.6 0.8 1 1.2 One static and one dynamic user with hybrid CSIT and T = 15 ≤ log(ρ) + o(log(ρ)), (95) where the last inequality follows from extremal entropy inequality and Lemma 1 [23], [29], [35]. This concludes the proof of the bound (91). Remark 5: For the above broadcast channel with hybrid CSIT, the achievable sum degrees of freedom is dsum = 3 2 − T1 , and the outer bound on the sum degrees of freedom is dsum ≤ 32 . The gap decreases with the dynamic user coherence time (see Fig. 7 and 8). B. Multiple Static and Dynamic Users Theorem 14: The fading broadcast channel characterized by Eq. (1), with perfect CSIT for the static users and delayed CSIT for the dynamic users, can achieve the following degrees of freedom, ′ D1 : m X e†j , (96) j=1 D2 , . . . , Dmm′ +1 : (1 − March 26, 2018 1 1 † 1 † )ej + ( − )e , 2T 2 2T i j = 1, . . . , m′ , i = 1, . . . , m, (97) DRAFT 27 1.5 Achievable region Outer region 1 0.5 0 0 Fig. 8. 0.2 0.4 0.6 0.8 1 1.2 One static and one dynamic user with hybrid CSIT and T = 30 ′ Dmm′ +2 , . . . , Dmm′ +m+2 m′ Dmm′ +m+3 m 1 1X † ej + (1 − )e†i , : T j=1 T mX † 1 : ej + ( 1 T j=1 1 + 2 + ...+ i = 1, . . . , m, (98) m m X † (1 − )) e. 1 T i=1 i m (99) The achievable region consists of the convex hull of the above vectors. Proof: D1 is achieved by inverting the channel of the static users at the transmitter, providing one degree of freedom per static user. The achievability of D2 , . . . , Dmm′ +1 was established in Section VI-A, and that of Dmm′ +2 , . . . , Dmm′ +m+2 was proved in Section V without CSIT for the dynamic user, so it remains achievable with delayed CSIT. Dmm′ +m+3 is achieved by retrospective interference alignment [11] along with product superposition, as follows. The transmitted signal over T instants is X = [Ū, ŪV], (100) where Ū ∈ Cm×m contains independent symbols intended for the static users sent by inverting the channels of the static users. Therefore, during the first m time instants, each static user receives an interference-free signal and achieves m degree of freedom, and furthermore the dynamic users estimate their equivalent channels. During the remaining time instants, each dynamic receiver March 26, 2018 DRAFT 28 obtains coherent observations of (T − m) transmit symbols, which are pre-processed, combined and interference-aligned into super-symbols V according to retrospective interference alignment techniques of [11]. Accordingly, each dynamic receiver achieves 1 1 (1 1+ 12 +...+ m − m ) T degrees of freedom. Theorem 15: An outer bound on the degrees of freedom region of the fading broadcast channel characterized by Eq. (1), with perfect CSIT for the static users and delayed CSIT for the dynamic users, is ′ m X j=1 m X d′j di + ≤ 1, ′ m + m i=1 m m X i=1 di ≤ 1+ d′j ≤ 1, (101) 1 2 m + ...+ 1 m , j = 1, . . . , m′ , (102) (103) 1 , i = 1, . . . , m. (104) T Proof: The inequalities (103) and (104) represent the single-user outer bounds for the static di ≤ 1 − and dynamic users, respectively [26], [34]. According to Theorem 5, (102) represents an outer bound for the dynamic users. It only remains to prove (101) as follows. 1) The original channel is enhanced by giving the users global CSIR. Furthermore, we assume full cooperation between the static users and between the dynamic users. The resulting enhanced channel is a broadcast channel with two users: one static user equipped with m′ antennas, received signal Y ′ , channel G, and noise noise Z′ , and one dynamic user equipped with m antennas, received signal Y, channel H, and noise Z. 2) We further enhance the channel by giving Y to the static user, constructing a physically degraded channel. For the enhanced channel, the static receiver is equipped with m′ + m ′ antennas and has received signal Ŷ = [Y † , Y † ]† , channel Ĝ = [G† , H† ]† , and noise ′ Ẑ = [Z† , Z † ]† . Since any causal feedback (including delayed CSIT) does not affect the capacity of a physically degraded channel [33], the delayed CSIT for the dynamic receiver can be removed. 3) We now use another enhancement with the motivation to remove the remaining CSIT (non-causal, with respect to the static user). We create an artificial channel and noise, G̃, Z̃, with the same distribution but independent of Ĝ, Ẑ, and a signal Ỹ = G̃X + Z̃. A March 26, 2018 DRAFT 29 genie will give Ỹ to the static receiver and G̃ to both receivers. It has been shown [36] that h(Ỹ, Ŷ|H) = h(Ŷ|H) + o(log ρ), where H = (Ĝ, G̃), therefore we can remove Ŷ from the enhanced channel without reducing its degrees of freedom. 4) The enhanced channel is physically degraded without CSIT, therefore [31], [32], ′ m X Rj′ ≤ I(X; Ỹ|U, H) m X Ri ≤ I(U; Y|H). j=1 (105) i=1 Hence, ′ m X j=1 m X Rj′ 1 1 1 Ri + ≤ ′ h(Ỹ|U, H) + h(Y|H) − h(Y|U, H) + o(log(ρ)) ′ m + m j=1 m m + m m m ≤ log(ρ) + o(log(ρ)), (106) where the last inequality follows from the extremal entropy inequality [23], [29], [35] and Lemma 1 and since h(Y|H) ≤ m log(ρ) + o(log(ρ)) [34]. This concludes the proof of the bound (101). VII. C ONCLUSION A multiuser broadcast channel was studied where some receivers experience longer coherence intervals and have CSIR while other receivers experience a shorter coherence interval and do not enjoy free CSIR. The degrees of freedom were studied under delayed CSIT, hybrid CSIT, and no CSIT. Among the techniques employed were interference alignment and beamforming along with product superposition for the inner bounds. The outer bounds involved a bounding of the rate region of the multiuser (discrete memoryless) multilevel broadcast channel. Some highlights of the results are: for one static and one dynamic user with delayed CSIT, the achievable degrees of freedom region partially meets the outer bound. For one static user with perfect CSIT and one dynamic user with delayed CSIT, the gap between the achievable and the outer sum degrees of freedom is inversely proportional to the dynamic user coherence time. For each of the considered CSI conditions, inner and outer bounds were also found for arbitrary number of users. From these results we conclude that in the broadcast channel, coherence diversity delivers gains that are distinct from, and augment, the gains from beamforming and interference alignment. March 26, 2018 DRAFT 30 The authors anticipate that the tools and results of this paper can be helpful for future studies of hybrid CSIT/CSIR in other multi-terminal networks. A PPENDIX I P ROOF OF T HEOREM 1 Recall, Mj′ , Mi are the messages of users j = 1, . . . , m′ and i = 1, . . . , m, respectively. We ′ ′ enhance the channel by assuming that user j = 1, . . . , m′ knows the messages Mj+1 , . . . , Mm ′ and M1 , . . . , Mm and user i = 1, . . . , m knows the messages Mi+1 , . . . , Mm . Using Fano’s inequality, chain rule, and data processing inequality we can bound the rates of the static user j = 1, . . . , m′ , ′ ′ ′ ′ nRj′ ≤ I(Mj′ ; Yj,1 , . . . , Yj,n |Mj+1 , . . . , Mm ′ , M1 , . . . , Mm ) (107) = n X ′ I(Mj′ ; Yj,k |Uj,k ) (108) ≤ n X ′ ′ ′ I(Mj′ , Uj,k , Yj−1,1 , . . . , Yj−1,k−1 ; Yj,k |Uj,k ) (109) = n X ′ I(Uj−1,k ; Yj,k |Uj,k ) (110) k=1 k=1 k=1 where
′ ′ ′ ′ , . . . , Mm Uj,k = Mj+1 ′ , M1 , . . . , Mm , Yj,1 , . . . , Yj,k−1 , ′ denotes the received signal of user j at time instant k, Yj,k Um′ → · · · → U1 → X → (Y1′ , . . . , Ym′ ′ , Y1 , . . . , Ym) forms a Markov chain, and U0 = X. The rate of static user m′ can be bounded as n X ′ I(Um′ −1,k ; Ym′ ′ ,k |Um′ ,k ) nRm′ ≤ = ≤ = k=1 n X k=1 n X k=1 n X k=1 March 26, 2018 I(Xk ; Ym′ ′ ,k |Um′ ,k ) − n X I(Xk ; Ym′ ′ ,k |Um′ −1,k ) (111) (112) k=1 I(Xk , Y1,k+1, . . . , Y1,n ; Ym′ ′ ,k |Um′ ,k ) − n X I(Xk ; Ym′ ′ ,k |Um′ −1,k ) (113) k=1 I(Y1,k+1, . . . , Y1,n ; Ym′ ′ ,k |Um′ ,k ) + n X I(Xk ; Ym′ ′ ,k |Um′ ,k , Y1,k+1, . . . , Y1,n ) k=1 DRAFT 31 − n X I(Xk ; Ym′ ′ ,k |Um′ −1,k ) (114) k=1 = n X I(Wk ; Ym′ ′ ,k |Um′ ,k ) + n X I(Xk ; Ym′ ′ ,k |Um′ ,k , Wk ) I(Xk ; Ym′ ′ ,k |Um′ −1,k ), k=1 k=1 k=1 − n X (115) n where Wk = Y1,k+1 . Similarly, nRi ≤I(Mi ; Yi,1, . . . , Yi,n |Mi+1 , . . . , Mm ) = = = n X k=1 n X k=1 n X (116) I(Mi ; Yi,k |Vi,k ) (117) I(Mi , Vi,k , Yi−1,k+1, . . . , Yi−1,n ; Yi,k |Vi,k ) (118) I(Vi−1,k ; Yi,k |Vi,k ), (119) k=1 where we define Vi,k , (Mi+1 , . . . , Mm , Yi,k+1, . . . , Yi,n ), which leads to the Markov chain Vm → · · · → V1 → (Um′ , W ) → X → (Y1′ , . . . , Ym′ ′ , Y1 , . . . , Ym ) . Using the chain rule and Csiszár sum identity [37], we obtain the bound (125). R1 ≤ n X I(M1 , . . . , Mm ; Y1,k |V1,k ) (120) ≤ n X I(M1 , . . . , Mm , Y1,k+1, . . . , Y1,n ; Y1,k |V1,k ) (121) = n X I(M1 , . . . , Mm , Y1,k+1, . . . , Y1,n , Ym′ ′ ,1 , . . . , Ym′ ′ ,k−1 ; Y1,k |V1,k ) (122) k=1 k=1 k=1 − n X I(Ym′ ′ ,1 , . . . , Ym′ ′ ,k−1; Y1,k |M1 , . . . , Mm , Y1,k+1, . . . , Y1,n ) (123) k=1 = n X = n X I(Um′ ,k , Wk ; Y1,k |V1,k ) − n X I(Y1,k+1, . . . , Y1,n ; Ym′ ′ ,k |Um′ ,k ) (124) I(Um′ ,k , Wk ; Y1,k |V1,k ) − n X I(Wk ; Ym′ ′ ,k |Um′ ,k ). (125) k=1 k=1 k=1 k=1 By introducing a time-sharing auxiliary random variable, Q, [38] and defining X ,(XQ , Q), March 26, 2018 ′ Yj′ , (Yj,Q , Q) DRAFT 32 Yi ,(Yi,Q , Q), Ui , (Ui,Q , Q) Vj ,(Vj,Q , Q), W , (WQ , Q), (126) we establish (7)-(10). Similarly, we can follow the same steps to prove (11)-(14) after switching the role of the two sets of variables Y1′ , . . . , Ym′ ′ and Y1 , . . . , Ym . This completes the proof of Theorem 1. A PPENDIX II M ULTILEVEL B ROADCAST C HANNEL WITH D EGRADED M ESSAGE S ETS Here, we study the capacity of the multiuser multilevel broadcast channel that is characterized   by (6) with degraded message sets. In particular, M0 ∈ 1 : 2nR0 is to be communicated to   all receivers, and furthermore M1 ∈ 1 : 2nR1 is to be communicated to receiver Y1 .5 A threereceiver special case was studied by Nair and El Gamal [22] where the idea of indirect decoding was introduced, and the capacity is the set of rate pairs (R1 , R0 ) such that  R0 ≤ min I(U; Y2 ), I(V ; Y1′ ) , R1 ≤I(X; Y1|U), R0 + R1 ≤I(V ; Y1′ ) + I(X; Y1|V ), (127) for some pmf p(u, v)p(x|v). In the sequel, we give a generalization of Nair and El Gamal for multiuser multilevel broadcast channel. Theorem 16: The capacity of multiuser multilevel broadcast channel characterized by (6), with degraded message sets, is the set of rate pairs (R1 , R0 ) such that  R0 ≤ min I(U; Ym ), I(V ; Ym′ ′ ) , R1 ≤I(X; Y1|U), R0 + R1 ≤I(V ; Ym′ ′ ) + I(X; Y1 |V ), (128) for some pmf p(u, v)p(x|v). Proof: The converse parallels the proof of the converse of the three-receiver case studied by Nair and El Gamal in [22] after replacing Y2 , Y1′ with Ym , Ym′ ′ , respectively. In particular, U 5 For compactness of expression, here we refer to each receiver by the variable denoting its received signal. March 26, 2018 DRAFT 33 and V are defined as follows. Uk , (M0 , Y1,1, . . . , Y1,k−1, Ym,k+1, . . . , Ym,n ), Vk , (M0 , Y1,1, . . . , Y1,k−1, Ym′ ′ ,k+1, . . . , Ym′ ′ ,n ), k = 1, . . . , n, and let Q be a time-sharing random variable uniformly distributed over the set {1, . . . , n} and independent of X n , Y1n , Ym,1 , . . . , Ym,n , Ym′ ′ ,1 , . . . , Ym′ ′ ,n . We then set U = (UQ , Q), V = (VQ , Q), X = XQ , Y1 = Y1,Q , Ym = Ym,Q , and Ym′ ′ = Ym′ ′ ,Q . This completes the converse part of the proof. The achievability part uses superposition coding and indirect decoding as follows. • Rate splitting: divide the private message M1 into two independent messages M10 at rate R10 and M11 at rate R11 , where R1 = R10 + R11 . • Codebook generation: fix a pmf p(u, v)p(x|v) and randomly and independently generate   Q 2nR0 sequences un (m0 ), m0 ∈ 1 : 2nR0 , each according to nk=1 pU (uk ). For each m0 , randomly and conditionally independently generate 2nR10 sequences v n (m0 , m10 ), m10 ∈ [1 : Q 2nR10 ], each according to nk=1 pV |U (vk |uk (m0 )). For each pair (m0 , m10 ), randomly and conditionally independently generate 2nR11 sequences xn (m0 , m10 , m11 ), m11 ∈ [1 : 2nR11 ], Q each according to nk=1 pX|V (xk |vk (m0 , m10 )). • Encoding: to send the message pair (m0 , m1 ) = (m0 , m10 , m11 ), the encoder transmits xn (m0 , m10 , m11 ). • Decoding at the users Y2 , . . . , Ym : decoder i declares that m̂0i ∈ [1 : 2nR0 ] is sent if it is (n) the unique message such that (un (m̂0i ), yin) ∈ Tǫ . Hence, by law of large numbers and the packing lemma [38], the probability of error tends to zero as n → ∞ if R0 < min {I(U; Yi ) − δ(ǫ)}, 2≤i≤m = I(U; Ym ) − δ(ǫ), (129) where the last equality follows from applying data processing inequality on the Markov chain U → X → Y1 → Y2 → · · · → Ym . • Decoding at Y1 : decoder 1 declares that (m̂01 , m̂10 , m̂11 ) is sent if it is the unique message
triple such that un (m̂01 ), v n (m̂01 , m̂10 ), xn (m̂01 , m̂10 , m̂11 ), y1n ∈ [1 : 2nR0 ]. Hence, by law of large numbers and the packing lemma [38], the probability of error tends to zero as March 26, 2018 DRAFT 34 n → ∞ if R11 < I(X; Y1|V ) − δ(ǫ), R10 + R11 < I(X; Y1|U) − δ(ǫ), R0 + R10 + R11 < I(X; Y1) − δ(ǫ). • (130) Decoding at users Y1′ , . . . , Ym′ ′ : decoder j decodes m0 indirectly by declaring m̃0j is sent (n) if it is the unique message such that (un (m̃0j ), v n (m̃0j , m10 ), zjn ) ∈ Tǫ for some m10 ∈ [1 : 2nR0 ]. Hence, by law of large numbers and packing lemma, the probability of error tends to zero as n → ∞ if R0 + R10 < min {I(U, V ; Yj′ ) − δ(ǫ)}, min {I(V ; Yj′ ) − δ(ǫ)}, 1≤j≤m′ = 1≤j≤m′ = I(V ; Ym′ ′ ) − δ(ǫ), (131) where the last two equalities follow from applying the chain rule and data processing inequality on the Markov chain U → V → X → Y1′ → Y2′ → · · · → Ym′ ′ . 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Synchronization Patterns in Networks of Kuramoto Oscillators: A Geometric Approach for Analysis and Control arXiv:1709.06193v1 [math.OC] 18 Sep 2017 Lorenzo Tiberi, Chiara Favaretto, Mario Innocenti, Danielle S. Bassett, and Fabio Pasqualetti Abstract— Synchronization is crucial for the correct functionality of many natural and man-made complex systems. In this work we characterize the formation of synchronization patterns in networks of Kuramoto oscillators. Specifically, we reveal conditions on the network weights and structure and on the oscillators’ natural frequencies that allow the phases of a group of oscillators to evolve cohesively, yet independently from the phases of oscillators in different clusters. Our conditions are applicable to general directed and weighted networks of heterogeneous oscillators. Surprisingly, although the oscillators exhibit nonlinear dynamics, our approach relies entirely on tools from linear algebra and graph theory. Further, we develop a control mechanism to determine the smallest (as measured by the Frobenius norm) network perturbation to ensure the formation of a desired synchronization pattern. Our procedure allows us to constrain the set of edges that can be modified, thus enforcing the sparsity structure of the network perturbation. The results are validated through a set of numerical examples. I. I NTRODUCTION Synchronization of coupled oscillators is everywhere in nature [1], [2], [3] and in several man-made systems, including power grids [4] and computer networks [5]. While some systems require complete synchronization among all the parts to function properly [6], [7], others rely on cluster or partial synchronization [8], where subsets of nodes exhibit coherent behaviors that remain independent from the evolution of other oscillators in the network. For example, while partial synchronization patterns have been observed in healthy individuals [9], complete synchronization in neural systems is associated with degenerative diseases including Parkinson’s and Huntington’s diseases [10], [11], and epilepsy [12]. Cluster synchronization has received attention only recently, and several fundamental questions remain unanswered, including the characterization of the network features enabling the formation of a desired pattern, and the development of control mechanisms to enforce the emergence of clusters. In this paper we focus on networks of Kuramoto oscillators [13], and we characterize intrinsic and topological conditions that ensure the formation of desired clusters of oscillators. This material is based upon work supported by NSF award #BCS-1430279 and ARO award 71603NSYIP. Lorenzo Tiberi and Fabio Pasqualetti are with the Mechanical Engineering Department, University of California at Riverside, [email protected], [email protected]. Chiara Favaretto is with the Department of Information Engineering, University of Padova, [email protected]. Danielle S. Bassett is with the Departments of Bioengineering and Electrical and Systems Engineering, University of Pennsylvania, [email protected]. Mario Innocenti is with the Department of Electrical Systems and Automation, University of Pisa, [email protected]. Our network model is motivated by a large body of literature showing broad applicability of the Kuramoto model to virtually all systems that exhibit synchronization properties. Although Kuramoto networks exhibit nonlinear dynamics, we adopt tools from linear algebra and graph theory to characterize network conditions enabling the formation of a given synchronization pattern. Further, we design a control mechanism to perturb (a subset of) the network weights so as to enforce or prevent desired synchronization patterns. Related work Complete synchronization in networks of Kuramoto oscillators has been extensively studied, e.g., see [14], [15]. It has been shown that synchronization of all nodes emerges when the coupling strength among the agents is sufficiently larger than the heterogeneity of the oscillators’ natural frequencies. Partial synchronization and pattern formation have received considerably less attention, with the literature being composed of only few recent works. In [16] it is shown how symmetry of the interconnections may lead to partial synchronization. Methods based on graph symmetry have also been used to find all possible clusters in networks of Laplacian-coupled oscillators [17]. The relationship between clusterization and network topology has been studied in [18] for unweighted interconnections. In [19], the emergence and the stability of groups of synchronized agents within a network has been studied for different classes of dynamics, like delay-coupled laser models and neuronal spiking models. Here, the approach of master stability function has been used to characterize the results. In [20], [21] the idea is put forth to study an approximate notion of cluster synchronization in Kuramoto networks via tools from linear systems theory. It is quantitatively shown how cluster synchronization depends on strong intracluster and weak inter-cluster connections, similarity with respect to the natural frequencies of the oscillators within each cluster, and heterogeneity of the natural frequencies of coupled oscillators belonging to different subnetworks. With respect to this work, we focus on an exact notion of cluster synchronization, identify necessary and sufficient conditions on the network weights and oscillators’ natural frequencies for the emergence of a desired synchronization pattern, and exploit our analysis to design a structural control algorithm for the formation of a desired synchronization pattern. The work that is closer to this paper is [22], where the authors relate cluster synchronization to the notion of an external equitable partition in a graph. In fact, the notion of an external equitable partition can be interpreted in terms of invariant subspaces of the network adjacency matrix, a notion that we exploit in our development. However, the analysis in [22] is carried out with unweighted and undirected networks and, as we show in this paper, the conditions in [22] may not be necessary when dealing with directed and weighted networks. Further, our approach relies on simple notions from linear algebra, and leads to the development of our control algorithm for the formation of desired patterns. Paper contributions The contributions of this paper are twofold. First, we consider a notion of exact cluster synchronization, where the phases of the oscillators within each cluster remain equal to each other over time, and different from the phases of the oscillators in different clusters. We derive necessary and sufficient conditions for the formation of a given synchronization pattern in directed and weighted networks of Kuramoto oscillators. In particular we show that cluster synchronization is possible if and only if (i) the natural frequencies are equal within each cluster, and (ii) for each cluster, the sum of the weights of the edges from every separate group is the same for all nodes in the cluster. Second, we leverage our characterization of cluster synchronization to develop a control mechanism that modifies the network weights so as to ensure the formation of a desired synchronization pattern. Our control method is optimal, in the sense that it determines the smallest (measured by the Frobenius norm) network perturbation for a given synchronization pattern, and it guarantees the modification of only a desired subset of the edge weights. Paper organization The rest of this paper is organized as follows. Section II contains the problem setup and some preliminary definitions. Section III contains our characterization of cluster synchronization, and Section IV contains our structural control algorithm for the formation of a desired synchronization pattern. Section V concludes the paper. II. P ROBLEM SETUP AND PRELIMINARY NOTIONS Consider a network of heterogenous Kuramoto oscillators described by the digraph G = (V, E), where V = {1, . . . , n} denotes the set of oscillators and E ⊆ V × V their interconnections. Let A = [aij ] be the weighted adjacency matrix of G, where aij ∈ R if (i, j) ∈ E and aij = 0 otherwise. We assume that G is strongly connected [23]. Let θi ∈ R denote the phase of the i-th oscillator, whose dynamics reads as θ̇i = ωi + n X aij sin(θj − θi ), j=1 where ωi is the natural frequency of the i-th oscillator. The dynamics is a generalized version of the classic Kuramoto model [24]. Depending on the interconnection graph G, the adjacency matrix A, and the oscillators natural frequencies, different oscillatory patterns are possible corresponding to (partially) synchronized or chaotic states [25]. In this work we are particularly interested in the case where the phases of groups of oscillators evolve cohesively within each group, yet independently from the phases of oscillators in different groups. To formalize this discussion, let P = {P1 , . . . , Pm } be a partition of V, that is, V = ∪m i=1 Pi and Pi ∩ Pj = ∅ for all i, j ∈ {1, . . . , m} with i 6= j. We restrict our attention to the case m > 1. Throughout the paper we will assume 10 1 2 9 4 2 6 5 7 10 2 5 3 5 9 Fig. 1. A network of oscillators with partitions P1 = {1, 2, 3} and P2 = {4, 5, 6}. The sum of the weights of all edges (i, j) is equal for each node i of P1 (resp. P2 ), with j ∈ P2 (resp. j ∈ P1 ). In Section III we show that this is a necessary condition for phase synchronization of the partition P. without loss of generality that, given P = {P , . . . , Pm }, P1 i−1 the oscillators are labeled so that Pi = { j=1 |Pj | + Pi 1, . . . , j=1 |Pj |}, where |Pj | denotes the cardinality of the set Pj . While different notions of synchronization exist, we will use the following definitions. Definition 1: (Phase synchronization) For the network of oscillators G = (V, E), the partition P = {P1 , . . . , Pm } is phase synchronizable if, for some initial phases θ1 (0), . . . , θn (0), it holds θi (t) = θj (t), for all times t ∈ R≥0 and i, j ∈ Pk , with k ∈ {1, . . . , m}. Definition 2: (Frequency synchronization) For the network of oscillators G = (V, E), the partition P = {P1 , . . . , Pm } is frequency synchronizable if, for some initial phases θ1 (0), . . . , θn (0), it holds θ̇i (t) = θ̇j (t), for all times t ∈ R≥0 and i, j ∈ Pk , with k ∈ {1, . . . , m}. Clearly, phase synchronization implies frequency synchronization, while the converse statement typically fails to hold. Finally, we define the characteristic matrix associated with a partition P of the network nodes, which will be used to derive our synchronization conditions in Section III. Definition 3: (Characteristic matrix) For the network of oscillators G = (V, E) and the partition P = {P1 , . . . , Pm }, the characteristic matrix of P is VP ∈ Rn×m , where  VP = v1 v2 ··· 1 ··· {z  vm , and  viT = 0 | 0 ··· {z Pi−1 j=1 |Pj | 0 1 } | |Pi | 1 0 0 ··· } | P {z n j=i+1 |Pj |  0 . }  We conclude this section with an illustrative example. Example 1: (Setup and definitions) Consider the network of Kuramoto oscillators in Fig. 1, with graph G = (V, E), V = {1, 2, 3, 4, 5, 6} and partition P = {P1 , P2 }. The graph cases where this assumption is satisfied are presented in Fig. 2. A special case where Assumption (A1) is not satisfied is discussed at the end of this section. Theorem 3.1: (Cluster synchronization) For the network of oscillators G = (V, E), the partition P = {P1 , . . . , Pm } is phase synchronizable if and only if the following conditions are simultaneously satisfied: P (i) the network weights satisfy k∈P` aik − ajk = 0 for every i, j ∈ Pz and z, ` ∈ {1, . . . , m}, with z 6= `; (a) (ii) the natural frequencies satisfy ωi = ωj for every k ∈ {1, . . . , m} and i, j ∈ Pk . Proof: (If) Let θi = θj for all i, j ∈ Pk , k = 1, . . . , m. Let i, j ∈ P` , and notice that X X aik sin(θk − θi ) − ajk sin(θk − θj ) θ̇i − θ̇j = z6=` k∈Pz = X sz` z6=` (b) Fig. 2. For the network in Example 1 with natural frequencies ω = [30, 30, 30, 10, 10, 10]T Fig. (a) shows the frequencies of the oscillators in the clusters P1 = {1, 2, 3} and P2 = {4, 5, 6} as a function of time. Notice that Assumption A1 is satisfied over the entire time interval. In Fig. (b) we let the frequencies be more homogeneous, ω = [19, 19, 19, 10, 10, 10]T . Assumption A1 is satisfied within bounded time intervals, such as [t1 , t2 ]. G and the partition P are described by A and VP    1 0 0 0 0 0 10  1 0 0 0 5 0 5    1 0 0 0 0 10 0     A=  , VP = 0 9 0 0 0 0 0    0 0 9 0 0 0 0 0 0 7 2 2 0 0 as follows:  0 0  0 . 1  1 1  III. C ONDITIONS FOR CLUSTER SYNCHRONIZATION In this section we derive necessary and sufficient conditions ensuring phase (hence frequency) synchronization of a partition of oscillators. In particular, we show how synchronization of a partition depends both on the interconnection structure and weights, as well as the oscillators natural frequencies. We make the following technical assumption. (A1) For the partition P = {P1 , . . . , Pm } there exists an ordering of the clusters Pi and an interval of time [t1 , t2 ], with t2 > t1 , such that for all times t ∈ [t1 , t2 ]: max θ̇i > max θ̇i > · · · > max θ̇i . i∈P1 i∈P2 i∈Pm Assumption (A1) requires the phases of the oscillators in different clusters to evolve with different frequencies, at least in some interval of time. This assumption is in fact not restrictive, as this is typically the case when the oscillators in different clusters have different natural frequencies. Two X aik − ajk = 0, k∈Pz where we have used conditions (i) and (ii), and where szl = sin(θz −θ` ) depends on the clusters z and ` but not on i, j, k. Thus, when conditions (i) and (ii) are satisfied, θ ∈ Im(VP ) implies θ̇ ∈ Im(VP ), the image of VP . Im(VP ) is invariant and the network is phase synchronizable (θ(0) ∈ Im(VP )). (Only if) We first show that condition (i) is necessary for phase synchronization. Assume that the network is phase synchronized. Let i, j ∈ P` . At all times is must hold that X X 0 = θ̈i − θ̈j = aik cos(θk − θi )(θ̇k − θ̇i ) z6=` k∈Pz − X X ajk cos(θk − θj )(θ̇k − θ̇j ) (1) z6=` k∈Pz = X cz` vz` z6=` X aik − ajk , k∈Pz | {z } dz where cz` = cos(θz − θ` ) and vz` = θ̇z − θ̇` depend on the clusters z and `, but not on i, j, k. From (A1), possibly after reordering the clusters, in some nontrivial interval we have max θ̇i > max θ̇i > · · · > max θ̇i . i∈P1 i∈P2 i∈Pm Thus, (1) implies that, either dz = 0 for all z (thus implying condition (i)), or the functions cz` vz` must be linearly dependent at all times in the interval. Assume by contradiction that the functions czl vzl are linearly dependent at all times in the above interval. Then it must hold that X dn dz n cz` vz` = 0, dt z6=` n d for every nonnegative integer n, where dt n denotes n-times differentiation. In other words, not only the functions cz` vz` must be linearly dependent, but also all their derivatives, at some times in the above interval. Let d1 6= 0 (if d1 = 0, simply select the first nonzero coefficient), and i, j 6∈ P1 . Because of assumption (A1), there exists an integer n such dn dn that d1 dt n c1` v1`  dz dtn cz` vz` , for all z 6= 1. Thus, the functions cz` vz` cannot be linearly dependent. We conclude that statement (i) is necessary for phase synchronization. We now prove that, when the network is phase synchronized, statement (i) implies statement (ii). This shows that statement (ii) is necessary for phase synchronization. Let the network be phase synchronized, and let i, j ∈ P` . We have X X 0 = θ̇i − θ̇j = ωi − ωj + sz` aik − ajk , z6=` k∈Pz | {z =0 min ∆ s.t. } where sz` = sin(θz − θ` ) does not depend on the indices i, j, k (see above), and where we have used that statement (i) is necessary for phase synchronization. To conclude, ωi = ωj , and statement (ii) is also necessary for phase synchronization. Remark 1: (Necessity of assumption A1) Consider a network of oscillators with adjacency matrix   0 a12 0 0 a21 0 a23 0   A=  0 a32 0 a34  , 0 0 a43 0 and natural frequencies ωi = ω̄ for all i ∈ {1, . . . , 4}. Notice that condition (i) in Theorem 3.1 is not satisfied. Let θ1 (0) = θ2 (0) and θ3 (0) = θ4 (0) = θ1 (0) + π, and notice that θ̇i = ω̄ at all times and for all i ∈ {1, . . . , 4} (Assumption A1 is not satisfied). In other words, the partition P = {P1 , P2 }, with P1 = {1, 2} and P2 = {3, 4} is phase synchronized, independently of the interconnection weights among the oscillators. Thus, condition (i) in Theorem may not be necessary when Assumption A1 is not satisfied.  Let A B denote the Hadamard product between A and B [26], and Im(VP )⊥ the orthogonal subspace to Im(VP ). Corollary 3.2: (Matrix condition for synchronization) Condition (i) in Theorem 3.1 is equivalent to V̄PT ĀVP = 0, where V̄P ∈ Rn×(n−m) satisfies Im(V̄P ) = Im(VP )⊥ , and Ā = A − A typically not satisfied for arbitrary partitions and interconnection weights. In this section we develop a control mechanism to modify the oscillators’ interconnection weights so as to guarantee synchronization of a given partition. Specifically, we study the following minimization problem: VP VPT . (2) Proof: Let Ā = [āij ] and A = [aij ]. Notice that āij = aij when i and j belong to different clusters, and āij = 0 when i and j belong to the same cluster. Thus, (P / Pj , k∈Pj aik , if i ∈ [ĀVP ]ij = 0, if i ∈ Pj . v̄iT x Select V̄P so that V̄P = [v̄1 · · · v̄n−m ] and = xr − xs , with r, s ∈ P` , for a vector x of compatible dimension. Then, (P / Pj , k∈Pj ark − ask , r, s ∈ T [V̄P ĀVP ]ij = 0, r, s ∈ Pj , where r, s are the nonzero indices of v̄i . IV. C ONTROL OF CLUSTER SYNCHRONIZATION In the previous section we derive conditions on the network of oscillators to guarantee phase and frequency synchronization. These conditions are rather stringent, and are k∆k2F   V̄PT Ā + ∆ VP = 0 (3b) ∆∈H (3c) (3a) where k∆kF denotes the Frobenius norm of the matrix ∆, Ā is as in (2), and H encodes a desired sparsity pattern of the perturbation matrix ∆. For example, H may represent the set of matrices compatible with the graph G = (V, E), that is, H = {M : M ∈ R|V|×|V| and mij = 0 if (i, j) 6∈ E}. The constraint (3b) reflects the invariance condition in Corollary 3.2 and, together with condition (ii) in Theorem 3.1, ensures synchronization of the partition P. Thus, the minimization problem (3) determines the smallest perturbation of the interconnection weights that guarantees synchronization of a partition P and satisfies desired sparsity constraints. It should be observed that, given the solution ∆∗ to (3), the modified adjacency matrix is A + ∆∗ even if the constraint (3b) is expressed in terms of Ā. This follows from the fact that connections among nodes of the same cluster do not affect the synchronization properties of the partition P = {P1 , . . . , Pm } (see Corollary 3.2). To solve the minimization problem (3), we define the following minimization problem by including the sparsity constraints (3c) into the cost function: min ∆ s.t. k∆  T Hk2F (4a)  V̄P Ā + ∆ VP = 0 (4b) where denotes elementwise division, and H satisfies hij = 1 if there exists a matrix M ∈ H such that mij 6= 0, and hij = 0 otherwise. Clearly, the minimization problems (3) and (4) are equivalent, in the sense that ∆∗ is a (feasible) solution to (3) if and only if it has finite cost in (4). Theorem 4.1: (Synchronization via structured perturbation) Let T = [VP V̄P ], and let   Ã11 Ã12 = T −1 ĀT. Ã21 Ã22 The minimization problem (3) has a solution if and only if there exists a matrix Λ satisfying X = (V̄P ΛVPT ) H, and Ã21 = V̄PT XVP . Moreover, if it exists, a solution ∆∗ to (3) is  ∗  ˜ ˜∗ ∆ ∆ ∗ −1 11 12 ∆ = T ˜∗ ˜∗ T , ∆21 ∆ 22 ˜ ∗ = −V T XVP , ∆ ˜ ∗ = −V T X V̄P , ∆ ˜ ∗ = −Ã21 , where ∆ 11 12 21 P P ∗ T ˜ = −V̄ X V̄P . and ∆ 22 P Proof: We adopt the method of Lagrange multipliers to derive the optimality conditions for the problem (4). The Lagrangian is L(∆, Λ) = n X n X 2 −1 δij hij + i=1 j=1 m X T λT i V̄P (Ā + ∆)vi , i=1 where Λ = [λ1 , . . . , λm ] ∈ R(n−m)×m is a matrix collecting vectors of Lagrange multipliers, and vi ∈ Rn is the i-th column of VP . By equating the partial derivatives of L to zero we obtain the following optimality conditions: ∂L = 0 ⇒ V̄P (Ā + ∆)vi = 0, ∂λi m X ∂L T = 0 ⇒ 2δij h−1 + λT k v̄i vjk = 0, ij ∂δij (5a) ∆∗ = −V̄P V̄PT ĀVP VPT . Proof: Because hij = 1 for all i and j, the optimality condition (6b) becomes (5b) ∆ + V̄P ΛVPT = 0. k=1 where v̄i is the i-th row of V̄P and vjk is the entry (j, k) of the matrix VP . Finally, (5a) and (5b) can be rewritten as V̄PT (Ā + ∆)VP = 0, (6a) H + V̄P ΛVPT = 0, (6b) ∆ where the factor 2 of (5b) has been included into the Lagrange multipliers. Applying the change of coordinates ˜ −1 , with Id the T = [VP V̄P ], Ā = T ÃT −1 and ∆ = T ∆T identity matrix of dimension d, equation (6a) becomes ˜ −1 VP = V̄PT T (Ã + ∆)T  ˜ 11   Ã11 + ∆ 0 In−m ˜ 21 Ã21 + ∆ ˜ 12 Ã12 + ∆ ˜ 22 Ã22 + ∆ ˜ ∗ = −Ã21 . Equation (6b) is equivalent to which leads to ∆ 21 T ∆ + (V̄P ΛVP ) H = 0, which can be decomposed as    ˜ 11 ∆ ˜ 12 V T   ∆ P +(V̄ ΛV T ) VP V̄P ˜ H = 0. P P ˜ V̄PT | {z } ∆ 21 ∆22 {z } | T T −1 Consequently, (V̄P ΛVPT ) H = 0. We now pre- and post-multiply both sides of the above equality by V̄PT and VP , respectively, and obtain Λ = V̄PT ĀVP , ∆∗ = −V̄P V̄PT ĀVP VPT , where we have used (6a), VPT VP = I, and V̄PT V̄P = I. We now present an example where we modify the network weights to ensure synchronization of a desired partition. Example 2: (Enforcing synchronization of a partition) Consider the network in Fig. 3(a). The dashed edges and the solid edges represent constrained and uncostrained edges, respectively. The corresponding matrices Ā and H read as    Im = 0, 0 ˜ 11 VPT − V̄P Ã12 VPT + VP ∆ ˜ 12 V̄PT + V̄P ∆ ˜ 22 V̄PT )+ (VP ∆ Theorem 4.1 characterizes the smallest (measured by the Frobenius norm) structured network perturbation that ensures synchronization of a given partition. Without constraints, the optimal perturbation has a straightforward expression. Corollary 4.2: (Unconstrained minimization problem) Let H = {M : mij 6= 0 for all i and j}. The minimization problem (3) is always feasible, and its solution is (7) Let X = (V̄P ΛVPT ) H. Recall that V̄PT V̄P = In−m , = Im , and V̄PT VP = 0. By pre-multiplicating equation (7) by V̄PT and post-multiplicating it by VP , we obtain VPT VP −Ã21 + V̄PT XVP = 0, which is a system of linear equations, that can be solved with respect to the unknown Λ. Following the same reasoning of above, we can obtain the following other three equations that ˜ 11 , ∆ ˜ 12 , and ∆ ˜ 22 : entirely determine the solution ∆ ˜ 11 + V T XVP = 0, ∆ ˜ 12 + V T X V̄P = 0, and ∆ P P ˜ 22 + V̄PT X V̄P = 0. ∆ Finally, the optimal matrix ∆∗ , solution to the problem (4), is given in original coordinates as  ∗  ˜ ˜∗ ∆ ∆ ∗ −1 11 12 ∆ =T ˜∗ T . −Ã21 ∆ 22 0 0  0 Ā =  9 0 0 0 0 0 0 9 7 0 0 0 0 0 2 0 5 0 0 0 0 0 0 10 0 0 0   12 0  1 5   0 1 ,H =   0 0 1 0 0 1 1 0 1 1 1 0 1 1 0 1 1 0 0 0 0 0 1 1 0 1 1 1 0 1  0 0  1 . 1   1 0 Notice that H allows only a subset of interconnections to be modified, specifically, those corresponding to its unit entries. It can be shown that, because condition (i) in Theorem 3.1 is not satisfied (equivalently V̄PT ĀVP 6= 0), the network is not phase synchronizable (see Fig. 3(b) and 3(c) for an evolution of the oscillators’ phases and frequencies). From Theorem 4.1 we obtain the optimal perturbation that ensures synchronization, which leads to the network in Fig. 3(d). Notice that the network in Fig. 3(d) satisfies condition (i) in Theorem 3.1. In fact, when the natural frequencies are equal within each cluster (condition (ii) in Theorem 3.1), the clusters evolve cohesively; see Fig. 3(e) and 3(f).  V. C ONCLUSION In this work we study cluster synchronization in networks of Kuramoto oscillators. We derive necessary and sufficient conditions on the network interconnection weights and on the oscillators’ natural frequencies to guarantee that the phases of groups of oscillators evolve cohesively with one another, yet independently from the phases of oscillators belonging to different groups. Additionally, we develop a control mechanism to modify the edges of a network to ensure the formation of desired clusters. Our control method is optimal, as it determines the smallest perturbation (measured by the Frobenius norm) for a desired synchronization pattern that is compatible with a pre-specified set of structural constraints. 1 12 2 9 4 2 6 5 7 2 5 3 10 9 5 (a) 1 (c) (e) (f) 2 12 2 9 (b) 6 3 1 5 7 11 9 4 5 2 5 2 (d) Fig. 3. Fig. (a) shows the network in Example 2, where the dashed (resp. solid) edges correspond to the zero (resp. unit) entries of H. The partition P = {P1 , P2 }, with P1 = {1, 2, 3} and P2 = {4, 5, 6}, is not synchronizable because, for instance, the sum of the weights of the incoming edges to nodes 1 and 2 is different (see Theorem 3.1). Fig. (b) and (c) show the phases and frequencies of the oscillators as a function of time. Fig. (d) shows the modified network obtained from Theorem 4.1, which satisfies condition (i) in Theorem 3.1 and leads to a synchronizable partition P. 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Bayesian Loss-based Approach to Change Point Analysis arXiv:1702.05462v2 [stat.ME] 7 Jan 2018 Laurentiu Hinoveanu∗1 , Fabrizio Leisen†1 , and Cristiano Villa‡1 1 School of Mathematics, Statistics and Actuarial Science, University of Kent Abstract In this paper we present a loss-based approach to change point analysis. In particular, we look at the problem from two perspectives. The first focuses on the definition of a prior when the number of change points is known a priori. The second contribution aims to estimate the number of change points by using a loss-based approach recently introduced in the literature. The latter considers change point estimation as a model selection exercise. We show the performance of the proposed approach on simulated data and real data sets. Keywords: Change point; Discrete parameter space; Loss-based prior; Model selection. 1 Introduction There are several practical scenarios where it is inappropriate to assume that the distribution of the observations does not change. For example, financial data sets can exhibit alternate behaviours due to crisis periods. In this case it is sensible to assume changes in the underlying distribution. The change in the distribution can be either in the value of one or more of the parameters or, more in general, on the family of the distribution. In the ∗ [email protected] [email protected] ‡ [email protected] † 1 latter case, for example, one may deem appropriate to consider a normal density for the stagnation periods, while a Student t, with relatively heavy tails, may be more suitable to represent observations in the more turbulent stages of a crisis. The task of identifying if, and when, one or more changes have occurred is not trivial and requires appropriate methods to avoid detection of a large number of changes or, at the opposite extreme, seeing no changes at all. The change point problem has been deeply studied from a Bayesian point of view. Chernoff and Zacks (1964) focused on the change in the means of normally distributed variables. Smith (1975) looked into the single change point problem when different knowledge of the parameters of the underlying distributions is available: all known, some of them known or none of them known. Smith (1975) focuses on the binomial and normal distributions. In Muliere and Scarsini (1985) the problem is tackled from a Bayesian nonparametric perspective. The authors consider Dirichlet processes with independent base measures as underlying distributions. In this framework, Petrone and Raftery (1997) have showed that the Dirichlet process prior could have a strong effect on the inference and may lead to wrong conclusions in the case of a single change point. Raftery and Akman (1986) have approached the single change point problem in the context of a Poisson likelihood under both proper and improper priors for the model parameters. Carlin et al. (1992) build on the work of Raftery and Akman (1986) by considering a two level hierarchical model. Both papers illustrate the respective approaches by studying the well-known British coal-mining disaster data set. In the context of multiple change points detection, Loschi and Cruz (2005) have provided a fully Bayesian treatment for the product partitions model of Barry and Hartigan (1992). Their application focused on stock exchange data. Stephens (1994) has extended the Gibbs sampler introduced by Carlin et al. (1992) in the change point literature to handle multiple change points. Hannart and Naveau (2009) have used Bayesian decision theory, in particular 0-1 cost functions, to estimate multiple changes in homoskedastic normally distributed observations. Schwaller and Robin (2017) extend the product partition model of Barry and Hartigan (1992) by adding a graphical structure which could capture the dependencies between multivariate observations. Fearnhead and Liu (2007) proposed a filtering algorithm for the sequential multiple change points detection problem in the case of piecewise regression models. Henderson and Matthews (1993) introduced a partial Bayesian approach which involves the use of a profile likelihood, where the aim is to detect multiple changes in the mean of Poisson distributions with an application to haemolytic uraemic syndrome (HUS) data. The same data set was studied by Tian et al. (2009), who proposed a method which treats the change points as latent variables. Ko et al. (2015) have proposed an exten2 sion to the hidden Markov model of Chib (1998) by using a Dirichlet process prior on each row of the regime matrix. Their model is semiparametric, as the number of states is not specified in advance, but it grows according to the data size. Heard and Turcotte (2017) have proposed a new sequential Monte Carlo algorithm to infer multiple change points. Whilst the literature covering change point analysis from a Bayesian perspective is vast when prior distributions are elicited, the documentation referring to analysis under minimal prior information is limited, see Moreno et al. (2005) and Girón et al. (2007). The former paper discusses the single change point problem in a model selection setting, whilst the latter paper, which is an extension of the former, tackles the multivariate change point problem in the context of linear regression models. Our work aims to contribute to the methodology for change point analysis under the assumption that the information about the number of change points and their location is minimal. First, we discuss the definition of an objective prior for change point location, both for single and multiple changes, assuming the number of changes is known a priori. Then, we define a prior on the number of change points via a model selection approach. Here, we assume that the change point coincides with one of the observations. As such, given X1 , X2 , . . . , Xn data points, the change point location is discrete. To the best of our knowledge, the sole general objective approach to define prior distributions on discrete spaces is the one introduced by Villa and Walker (2015a). To illustrate the idea, consider a probability distribution f (x|m), where m ∈ M is a discrete parameter. Then, the prior π(m) is obtained by objectively measuring what is lost if the value m is removed from the parameter space, and it is the true value. According to Berk (1966), if a model is misspecified, the posterior distribution asymptotically accumulates on the model which is the most similar to the true one, where the similarity is measured in terms of the Kullback–Leibler (KL) divergence. Therefore, DKL (f (·|m)kf (·|m0 )), where m0 is the parameter characterising the nearest model to f (x|m), represents the utility of keeping m. The objective prior is then obtained by linking the aforementioned utility via the self-information loss:   0 π(m) ∝ exp min DKL (f (·|m)kf (·|m )) − 1, (1) 0 m 6=m where the Kullback–Leibler divergence (Kullback and Leibler, 1951) from the sampling distribution with density f (x|m) to the one with density f (x|m0 ) 3 is defined as: Z 0 DKL (f (·|m)kf (·|m )) =   f (x|m) f (x|m) · log dx. f (x|m0 ) X Throughout the paper, the objective prior defined in equation (1) will be referenced as the loss-based prior. This approach is used to define an objective prior distribution when the number of change points is known a priori. To obtain a prior distribution for the number of change points, we adopt a model selection approach based on the results in Villa and Walker (2015b), where a method to define a prior on the space of models is proposed. To illustrate, let us consider k Bayesian models: Mj = {fj (x|θj ), πj (θj )} j ∈ {1, 2, . . . , k}, (2) where fj (x|θj ) is the sampling density characterised by θj and πj (θj ) represents the prior on the model parameter. Assuming the prior on the model parameter, πj (θj ), is proper, the model prior probability Pr(Mj ) is proportional to the expected minimum Kullback– Leibler divergence from Mj , where the expectation is considered with respect to πj (θj ). That is:    Pr(Mj ) ∝ exp Eπj inf DKL (fj (x|θj )kfi (x|θi )) j = 1, . . . , k. (3) θi ,i6=j The model prior probabilities defined in equation (3) can be employed to derive the model posterior probabilities through: #−1 " k X Pr(Mj ) Bji , (4) Pr(Mi |x) = Pr(M ) i j=1 where Bji is the Bayes factor between model Mj and model Mi , defined as R fj (x|θj )πj (θj ) dθj Bji = R , fi (x|θi )πi (θi ) dθi with i 6= j ∈ {1, 2, . . . , k}. This paper is structured as follows: in Section 2 we establish the way we set objective priors on both single and multiple change point locations. Section 3 shows how we define the model prior probabilities for the number of change point locations. Illustrations of the model selection exercise are provided in Sections 4 and 5, where we work with simulated and real data, respectively. Section 6 is dedicated to final remarks. 4 2 Objective Prior on the Change Point Locations This section is devoted to the derivation of the loss-based prior when the number of change points is known a priori. Specifically, let k be the number of change points and m1 < m2 < . . . < mk their locations. We introduce the idea in the simple case where we assume that there is only one change point in the data set (see Section 2.1). Then, we extend the results to the more general case where multiple change points are assumed (see Section 2.2). A well-known objective prior for finite parameter spaces, in cases where there is no structure, is the uniform prior (Berger et al., 2012). As such, a natural choice for the prior on the change points location is the uniform (Koop and Potter, 2009). The corresponding loss-based prior is indeed the uniform, as shown below, which is a reassuring result as the objective prior for a specific parameter space, if exists, should be unique. 2.1 Single Change Point As mentioned above, we show that the loss-based prior for the single change point case coincides with the discrete uniform distribution over the set {1, 2, . . . , n− 1}. Let X(n) = (X1 , . . . , Xn ) denote an n-dimensional vector of random variables, representing the random sample, and m be our single change point location, that is m ∈ {1, 2, . . . , n − 1}, such that i.i.d. X1 , . . . , Xm |θ̃1 ∼ f1 (·|θ̃1 ) i.i.d. Xm+1 , . . . , Xn |θ̃2 ∼ f2 (·|θ̃2 ). (5) Note that we assume that there is a change point in the series, as such the space of m does not include the case m = n. In addition, we assume that θ̃1 6= θ̃2 when f1 = f2 . The sampling density for the vector of observations x(n) = (x1 , . . . , xn ) is: f (x (n) |m, θ̃1 , θ̃2 ) = m Y f1 (xi |θ̃1 ) i=1 n Y f2 (xi |θ̃2 ). (6) i=m+1 Let m0 6= m. Then, the Kullback–Leibler divergence between the model 5 parametrised by m and the one parametrised by m0 is: Z (n) (n) 0 DKL (f (x |m, θ̃1 , θ̃2 )kf (x |m , θ̃1 , θ̃2 )) = f (x(n) |m, θ̃1 , θ̃2 ) log f (x(n) |m, θ̃1 , θ̃2 ) f (x(n) |m0 , θ̃1 , θ̃2 ) ! dx(n) . (7) Without loss of generality, consider m < m0 . In this case, note that 0 m Y f (x(n) |m, θ̃1 , θ̃2 ) f2 (xi |θ̃2 ) = , f (x(n) |m0 , θ̃1 , θ̃2 ) i=m+1 f1 (xi |θ̃1 ) leading to DKL (f (x(n) |m, θ̃1 , θ̃2 )kf (x(n) |m0 , θ̃1 , θ̃2 )) = m0 Z X f2 (xi |θ̃2 ) log i=m+1 f2 (xi |θ̃2 ) f1 (xi |θ̃1 ) ! dxi . (8) On the right hand side of equation (8), we can recognise the Kullback–Leibler divergence from density f2 to density f1 , thus getting: DKL (f (x(n) |m, θ̃1 , θ̃2 )||f (x(n) |m0 , θ̃1 , θ̃2 )) = (m0 − m)DKL (f2 (·|θ̃2 )kf1 (·|θ̃1 )). (9) In a similar fashion, when m > m0 , we have that: DKL (f (x(n) |m, θ̃1 , θ̃2 )kf (x(n) |m0 , θ̃1 , θ̃2 )) = (m − m0 )DKL (f1 (·|θ̃1 )kf2 (·|θ̃2 )). (10) In this single change point scenario, we can consider m0 as a perturbation of the change point location m, that is m0 = m ± l where l ∈ N∗ , such that 1 ≤ m0 < n. Then, taking into account equations (9) and (10), the Kullback–Leibler divergence becomes: DKL (f (x(n) |m, θ̃1 , θ̃2 )kf (x(n) |m0 , θ̃1 , θ̃2 )) =   l · DKL (f2 (·|θ̃2 )kf1 (·|θ̃1 )),  l · DKL (f1 (·|θ̃1 )kf2 (·|θ̃2 )), 6 if m < m0 if m > m0 , and h i (n) (n) 0 min DKL (f (x |m, θ̃1 , θ̃2 )kf (x |m , θ̃1 , θ̃2 )) = 0 m 6=m = min {l · DKL (f2 (·|θ̃2 )kf1 (·|θ̃1 )), l · DKL (f1 (·|θ̃1 )kf2 (·|θ̃2 ))} 0 m 6=m = min {DKL (f2 (·|θ̃2 )kf1 (·|θ̃1 )), DKL (f1 (·|θ̃1 )kf2 (·|θ̃2 ))} · min {l} . m0 6=m m0 6=m | {z } 1 (11) We observe that equation (11) is only a function of θ̃1 and θ̃2 and does not depend on m. Thus, π(m) ∝ 1 and, therefore, π(m) = 1 n−1 m ∈ {1, . . . , n − 1}. (12) This prior was used, for instance, in an econometric context by Koop and Potter (2009) with the rationale of giving equal weight to every possible change point location. 2.2 Multivariate Change Point Problem In this section, we address the change point problem in its generality by assuming that there are 1 ≤ k < n change points. In particular, for the data x(n) = (x1 , . . . , xn ), we consider the following sampling distribution f (x (n) |m, θ̃) = m1 Y f1 (xi |θ̃1 ) i=1 j+1 k−1 Y mY n Y fj+1 (xi |θ̃j+1 ) j=1 i=mj +1 fk+1 (xi |θ̃k+1 ), i=mk +1 (13) where m = (m1 , . . . , mk ), 1 ≤ m1 < m2 < . . . < mk < n, is the vector of the change point locations and θ̃ = (θ̃1 , . . . , θ̃k , θ̃k+1 ) is the vector of the parameters of the underlying probability distributions. Schematically: X1 Xm1 +1 .. . , . . . , Xm1 |θ̃1 , . . . , Xm2 |θ̃2 . , . . . , .. i.i.d. ∼ f1 (·|θ̃1 ) ∼ f2 (·|θ̃2 ) .. .. ..... i.i.d. i.i.d. Xmk−1 +1 , . . . , Xmk |θ̃k ∼ fk (·|θ̃k ) i.i.d. Xmk +1 , . . . , Xn |θ̃k+1 ∼ fk+1 (·|θ̃k+1 ). If f1 = f2 = · · · = fk+1 , then it is reasonable to assume that some of the θ’s are different. Without loss of generality, we assume that θ̃1 6= θ̃2 6= · · · 6= 7 θ̃k 6= θ̃k+1 . In a similar fashion to the single change point case, we cannot assume mk = n since we require exactly k change points. In this case, due to the multivariate nature of the vector m = (m1 , . . . , mk ), the derivation of the loss-based prior is not as straightforward as in the one dimensional case. In fact, the derivation of the prior is based on heuristic considerations supported by the below Theorem 1 (the proof of which is in the Appendix). In particular, we are able to prove an analogous of equations (9) and (10) when only one component is arbitrarily perturbed. Let us define the following functions: d+1 j (θ̃) = DKL (fj+1 (·|θ̃j+1 )kfj (·|θ̃j )) d−1 j (θ̃) = DKL (fj (·|θ̃j )kfj+1 (·|θ̃j+1 )), where j ∈ {1, 2, . . . , k}. The following Theorem is useful to understand the behaviour of the loss-based prior in the general case. Theorem 1. Let f (x(n) |m, θ̃) be the sampling distribution defined in equation (13) and consider j ∈ {1, . . . , k}. Let m0 be such that m0i = mi for i 6= j, and let the component m0j be such that m0j 6= mj and mj−1 < m0j < mj+1 . Therefore, DKL (f (x(n) |m, θ̃)kf (x(n) |m0 , θ̃) = |m0j − mj |dSj (θ̃), where S = sgn(m0j − mj ). Note that, Theorem 1 states that the minimum Kullback–Leibler divergence is achieved when m0j = mj + 1 or m0j = mj − 1. This result is not surprising since the Kullback–Leibler divergence measures the degree of similarity between two distributions. The smaller the perturbation caused by changes in one of the parameters is, the smaller the Kullback–Leibler divergence between the two distributions is. Although Theorem 1 makes a partial statement about the multiple change points scenario, it provides a strong argument for supporting the uniform prior. Indeed, if now we consider the general case of having k change points, it is straightforward to see that the Kullback–Leibler divergence is minimised when only one of the components of the vector m is perturbed by (plus or minus) one unit. As such, the loss-based prior depends on the vector of parameters θ̃ only, as in the one-dimensional case, yielding the uniform prior for m. Therefore, the loss-based prior on the multivariate change point location is  −1 n−1 π(m) = , (14) k 8 where m = (m1 , . . . , mk ), 1 ≤ m1 < m2 < . . . < mk < n. The denominator in equation (14) has the above form because, for every number of k change points, we are interested
in the number of k-subsets from a set of n − 1 n−1 elements, which is k . The same prior was also derived in a different way by Girón et al. (2007). 3 Loss-based Prior on the Number of Change Points Here, we approach the change point analysis as a model selection problem. In particular, we define a prior on the space of models, where each model represents a certain number of change points (including the case of no change points). The method adopted to define the prior on the space of models is the one introduced in Villa and Walker (2015b). We proceed as follows. Assume 1 1 M1 n 𝜃̃1 M0 𝜃̃1 m1 n 𝜃̃2 1 1 M2 θ1 𝜃̃1 m1 1 M3 𝜃̃1 𝜃̃2 m2 n 𝜃̃3 2 θ1 m1 θ1 𝜃̃2 m2 𝜃̃3 m3 n 𝜃̃4 ⋮ 1 Mk 𝜃̃1 m1 𝜃̃2 m2 𝜃̃3 m3 𝜃̃4 ⋯ ⋯ ⋯ 𝜃̃𝑘 mk n 𝜃̃𝑘+1 Figure 1: Diagram showing the way we specify our models. The arrows indicate that the respective change point locations remain fixed from the previous model to the current one. we have to select from k + 1 possible models. Let M0 be the model with no change points, M1 the model with one change point and so on. Generalising, model Mk corresponds to the model with k change points. The idea is that the current model encompasses the change point locations of the previous model. As an example, in model M3 the first two change point locations will be the same as in the case of model M2 . To illustrate the way we envision our models, we have provided Figure 1. It has to be noted that the construction of the possible models from M0 to Mk can be done in a different way to one here described. Obviously, the approach to define the model priors stays 9 unchanged. Consistently with the notation used in Section 1,  θ̃1 , . . . , θ̃k+1 , m1 , . . . , mk if k = 1, . . . , n − 1 θk = θ̃1 if k = 0 represents the vector of parameters of model Mk , where θ̃1 , . . . , θ̃k+1 are the model specific parameters and m1 , . . . , mk are the change point locations, as in Figure 1. Based on the way we have specified our models, which are in direct correspondence with the number of change points and their locations, we state Theorem 2 (the proof of which is in the Appendix). Theorem 2. Let DKL (Mi kMj ) = DKL (f (x(n) |θi )kf (x(n) |θj )). For any 0 ≤ i < j ≤ k integers, with k < n, and the convention mj+1 = n, we have the following: DKL (Mi kMj ) = j h X i (mq+1 − mq ) · DKL (fi+1 (·|θ̃i+1 )kfq+1 (·|θ̃q+1 )) , q=i+1 and DKL (Mj kMi ) = j h X i (mq+1 − mq ) · DKL (fq+1 (·|θ̃q+1 )kfi+1 (·|θ̃i+1 )) . q=i+1 The result in Theorem 2 is useful when the model selection exercise is implemented. Indeed, the Villa and Walker (2015b) approach requires the computation of the Kullback–Leibler divergences in Theorem 2. Recalling equation (3), the objective model prior probabilities are then given by:    Pr(Mj ) ∝ exp Eπj inf DKL (Mj kMi ) j = 0, 1, . . . , k. (15) θi ,i6=j For illustrative purposes, in the Appendix we derive the model prior probabilities to perform model selection among M0 , M1 and M2 . It is easy to infer from equation (15) that model priors depend on the prior distribution assigned to the model parameters, that is on the level of uncertainty that we have about their true values. For the change point location, 10 a sensible choice is the uniform prior which, as shown in Section 2, corresponds to the loss-based prior. For the model specific parameters, we have several options. If one wishes to pursue an objective analysis, intrinsic priors (Berger and Pericchi, 1996) may represent a viable solution since they are proper. Nonetheles, the method introduce by Villa and Walker (2015b) does not require, in principle, an objective choice as long as the priors are proper. Given that we use the latter approach, here we consider subjective priors for the model specific parameters. Remark. In the case where the changes in the underlying sampling distribution are limited to the parameter values, the model prior probabilities defined in (15) follow the uniform distribution. That is, Pr(Mj ) ∝ 1. In the real data example illustrated in Section 5.1, we indeed consider a problem where the above case occurs. 3.1 A special case: selection between M0 and M1 Let us consider the case where we have to estimate whether there is or not a change point in a set of observations. This implies that we have to choose between model M0 (i.e. no change point) and M1 (i.e. one change point). Following our approach, we have:    Pr(M0 ) ∝ exp Eπ0 inf DKL (f1 (·|θ̃1 )kf2 (·|θ̃2 )) , (16) θ̃2 and    Pr(M1 ) ∝ exp Eπ1 (n − m1 ) · inf DKL (f2 (·|θ̃2 )kf1 (·|θ̃1 )) . (17) θ̃1 Now, let us assume independence between the prior on the change point location and the prior on the parameters of the underlying sampling distributions, that is π1 (m1 , θ̃1 , θ̃2 ) = π1 (m1 )π1 (θ̃1 , θ̃2 ). Let us further recall that, as per equation (14), π1 (m1 ) = 1/(n−1). As such, we observe that the model prior probability on M1 becomes:     n Eπ1 (θ̃1 ,θ̃2 ) inf DKL (f2 (·|θ̃2 )kf1 (·|θ̃1 )) . (18) Pr(M1 ) ∝ exp 2 θ̃1 We notice that the model prior probability for model M1 is increasing when the sample size increases. This behaviour occurs whether there is or not a 11 change point in the data. We propose to address the above problem by using a non-uniform prior for m1 . A reasonable alternative, which works quite well in practice, would be the following shifted binomial as prior:   m −1  n−m1 −1 n−2 n−1 1 1 π1 (m1 ) = , 1 ≤ m1 ≤ n − 1. (19) m1 − 1 n n To argument the choice of (19), we note that, as n increases, the probability mass will be more and more concentrated towards the upper end of the support. Therefore, from equations (17) and (19) follows:     2n − 2 Pr(M1 ) ∝ exp Eπ1 (θ̃1 ,θ̃2 ) inf DKL (f2 (·|θ̃2 )kf1 (·|θ̃1 )) . (20) n θ̃1 For the more general case where we consider more than two models, the problem highlighted in equation (18) vanishes. 4 Change Point Analysis on Simulated Data In this section, we present the results of several simulation studies based on the methodologies discussed in Sections 2 and 3. We start with a scenario involving discrete distributions in the context of the one change point problem. We then show the results obtained when we consider continuous distributions for the case of two change points. The choice of the underlying sampling distributions is in line with Villa and Walker (2015b). 4.1 Single sample Scenario 1. The first scenario concerns the choice between models M0 and M1 . Specifically, for M0 we have: i.i.d. X1 , X2 , . . . , Xn |p ∼ Geometric(p) and for M1 we have: i.i.d. X1 , X2 , . . . , Xm1 |p ∼ Geometric(p) i.i.d. Xm1 +1 , Xm1 +2 , . . . , Xn |λ ∼ Poisson(λ) Let us denote with f1 (·|p) and f2 (·|λ) the probability mass functions of the Geometric and the Poisson distributions, respectively. The priors for the parameters of f1 and f2 are p ∼ Beta(a, b) and λ ∼ Gamma(c, d). 12 In the first simulation, we sample n = 100 observations from model M0 with p = 0.8. To perform the change point analysis, we have chosen the following parameters for the priors on p and λ: a = 2, b = 2, c = 3 and d = 1. Applying the approach introduced in Section 3, we obtain Pr(M0 ) ∝ 1.59 and Pr(M1 ) ∝ 1.81. These model priors yield the posterior distribution probabilities (refer to equation (4)) Pr(M0 |x(n) ) = 0.92 and Pr(M1 |x(n) ) = 0.08. As expected, the selection process strongly indicates the true model as M0 . Table 1 reports the above probabilities including other information, such as the appropriate Bayes factors. The second simulation looked at the opposite setup, that is we sample n = 100 observations from M1 , with p = 0.8 and λ = 3. We have sampled 50 data points from the Geometric distribution and the remaining 50 data points from the Poisson distribution. In Figure 2, we have plotted the simulated sample, where it is legitimate to assume a change in the underlying distribution. Using the same prior parameters as above, we obtain Pr(M0 |x(n) ) = 0.06 and Pr(M1 |x(n) ) = 0.94. Again, the model selection process is assigning heavy posterior mass to the true model M1 . These results are further detailed in Table 1. 8 6 4 2 0 1 20 40 60 80 100 Figure 2: Scatter plot of the data simulated from model M1 in Scenario 1. Scenario 2. In this scenario we consider the case where we have to select among three models, that is model M0 : i.i.d. X1 , X2 , . . . , Xn |λ, κ ∼ Weibull(λ, κ), (21) model M1 : i.i.d. X1 , X2 , . . . , Xm1 |λ, κ ∼ Weibull(λ, κ) i.i.d. Xm1 +1 , Xm1 +2 , . . . , Xn |µ, τ ∼ Log-normal(µ, τ ), 13 (22) Pr(M0 ) Pr(M1 ) B01 B10 Pr(M0 |x(n) ) Pr(M1 |x(n) ) True model M0 M1 0.47 0.47 0.53 0.53 12.39 0.08 0.08 12.80 0.92 0.06 0.08 0.94 Table 1: Model prior, Bayes factor and model posterior probabilities for the change point analysis in Scenario 1. We considered samples from, respectively, model M0 and model M1 . with 1 ≤ m1 ≤ n−1 being the location of the single change point, and model M2 : i.i.d. X1 , X2 , . . . , Xm1 |λ, κ ∼ Weibull(λ, κ) i.i.d. Xm1 +1 , Xm1 +2 , . . . , Xm2 |µ, τ ∼ Log-normal(µ, τ ) i.i.d. Xm2 +1 , Xm2 +2 , . . . , Xn |α, β ∼ Gamma(α, β), (23) with 1 ≤ m1 < m2 ≤ n − 1 representing the locations of the two change points, such that m1 corresponds exactly to the same location as in model M1 . Analogously to the previous scenario, we sample from each model in turn and perform the selection to detect the number of change points. Let f1 (·|λ, κ), f2 (·|µ, τ ) and f3 (·|α, β) represent the Weibull, Log-normal and Gamma densities, respectively, with θ̃1 = (λ, κ), θ̃2 = (µ, τ ) and θ̃3 = (α, β). We assume a Normal prior on µ and Gamma priors on all the other parameters as follows: λ ∼ Gamma(1.5, 1) κ ∼ Gamma(5, 1) τ ∼ Gamma(16, 1) α ∼ Gamma(10, 1) µ ∼ Normal(0.05, 1), β ∼ Gamma(0.2, 0.1). In the first exercise, we have simulated n = 100 observations from model M0 , where we have set λ = 1.5 and κ = 5. We obtain the following model priors: Pr(M0 ) ∝ 1.09, Pr(M1 ) ∝ 1.60 and Pr(M2 ) ∝ 1.37, yielding the posteriors Pr(M0 |x(n) ) = 0.96, Pr(M1 |x(n) ) = 0.04 and Pr(M2 |x(n) ) = 0.00. We then see that the approach assigns high mass to the true model M0 . Table 2 reports the above probabilities and the corresponding Bayes factors. The second simulation was performed by sampling 50 observations from a Weibull with parameter values as in the previous exercise, and the remaining 14 Pr(M0 ) Pr(M1 ) Pr(M2 ) B01 B02 B12 Pr(M0 |x(n) ) Pr(M1 |x(n) ) Pr(M2 |x(n) ) M0 0.27 0.39 0.34 36.55 1.84 × 103 50.44 0.96 0.04 0.00 True model M1 M2 0.27 0.27 0.39 0.39 0.34 0.34 3.24 × 10−4 4.65 × 10−40 0.02 1.27 × 10−45 55 2.72 × 10−6 0.00 0.00 0.98 0.00 0.02 1.00 Table 2: Model prior, Bayes factor and model posterior probabilities for the change point analysis in Scenario 2. We considered samples from, respectively, model M0 , model M1 and model M2 . 50 observations from a Log-normal density with location parameter µ = 0.05 and scale parameter τ = 16. The data is displayed in Figure 3. 2.5 2 1.5 1 0.5 0 1 20 40 60 80 100 Figure 3: Scatter plot of the observations simulated from model M1 in Scenario 2. The model posterior probabilities are Pr(M0 |x(n) ) = 0.00, Pr(M1 |x(n) ) = 0.98 and Pr(M2 |x(n) ) = 0.02, which are reported in Table 2. In this case as well, we see that the model selection procedure indicates M1 as the true model, as expected. 15 12 10 8 6 4 2 0 1 20 40 60 80 100 Figure 4: Scatter plot of the observations simulated from model M2 in Scenario 2. Finally, for the third simulation exercise we sample 50 and 20 data points from, respectively, a Weibull and a Log-normal with parameter values as defined above, and the last 30 observations are sampled from a Gamma distribution with parameters α = 10 and β = 2. From Table 2, we note that the posterior distribution on the model space accumulates on the true model M2 . 4.2 Frequentist Analysis In this section, we perform a frequentist analysis of the performance of the proposed prior by drawing repeated samples from different scenarios. In particular, we look at a two change points problem where the sampling distributions are Student-t with different degrees of freedom. In this scenario, we perform the analysis with 60 repeated samples generated by different densities with the same mean values. Then, we repeat the analysis of Scenario 2 by selecting 100 samples for n = 500 and n = 1500. We consider different sampling distributions with the same mean and variance. In this scenario, where we added the further constraint of the equal variance, it is interesting to note that the change in distribution is captured when we increase the sample size, meaning that we learn more about the true sampling distributions. We also compare the performances of the loss-based prior with the uniform prior when we analyse the scenario with different sampling distributions. Namely, Weibull/Log-normal/Gamma. It is interesting to note that the uniform prior is unable to capture the change in distribution even for a large sample size. On the contrary, the loss-based prior is able to detect the number of change points when n = 1500. Furthermore, for n = 500, even though 16 both priors are not able to detect the change points most of the times, the loss-based prior has a higher frequency of success when compared to the uniform prior. Scenario 3. In this scenario, we consider the case where the sampling distributions belong to the same family, that is Student-t, where the true model has two change points. In particular, let f1 (·|ν1 ), f2 (·|ν2 ) and f3 (·|ν3 ) represent the densities of three standard t distributions, respectively. We assume that ν1 , ν2 and ν3 are positive integers strictly greater than one so to have defined mean for each density. Note that this allows us to compare distributions of the same family with equal mean. The priors assigned to the number of degrees of freedom assume a parameter space of positive integers strictly larger than 1. As such, we define them as follows: ν1 ∼ 2 + Poisson(30) ν2 ∼ 2 + Poisson(3) ν3 ∼ 2 + Poisson(8). In this experiment, we consider 60 repeated samples, each of size n = 300 and with the following structure: • X1 , . . . , X100 from a Student-t distribution with ν1 = 30, • X101 , . . . , X200 from a Student-t distribution with ν2 = 3, • X201 , . . . , X300 from a Student-t distribution with ν3 = 8. Table 3 reports the frequentist results of the simulation study. First, note that P (M1 ) = P (M2 ) = P (M3 ) = 1/3 as per the Remark in Section 3. For all the simulated samples, the loss-based prior yields a posterior with the highest probability assigned to the true model M2 . We also note that the above posterior is on average 0.75 with a variance 0.02, making the inferential procedure extremely accurate. Pr(M0 |x(n) ) Pr(M1 |x(n) ) Pr(M2 |x(n) ) Mean posterior 0.01 0.24 0.75 Variance posterior 3.84 × 10−4 0.0160 0.0190 Freq. true model 0/60 0/60 60/60 Table 3: Average model posterior probabilities, variance and frequency of true model for the Scenario 3 simulation exercise. Scenario 4. In this scenario, we perform repeated sampling from the setup described in scenario 2 above, where the true model has two change points. In 17 particular, we draw 100 samples with n = 500 and n = 1500. For n = 500, the loss-based prior probabilities are P (M0 ) = 0.18, P (M1 ) = 0.16 and P (M2 ) = 0.66. For n = 1500, the loss-based prior probabilities are P (M0 ) = 0.015, P (M1 ) = 0.014 and P (M2 ) = 0.971. The simulation results are reported, respectively, in Table 4 and in Table 5. The two change point locations for n = 500 are at the 171st and 341st observations. For n = 1500, the first change point is the 501st observation, while the second is at the 1001st observation. We note that there is a sensible improvement in detecting the true model, using the loss-based prior, when the sample size increases. In particular, we move from 30% to 96%. (n) Pr(M0 |x ) Pr(M1 |x(n) ) Pr(M2 |x(n) ) Mean posterior 9.88 × 10−4 0.63 0.37 Variance posterior 2.60 × 10−5 0.0749 0.0745 Freq. true model 0/100 70/100 30/100 Table 4: Average model posterior probabilities, variance and frequency of true model for the Scenario 4 simulation exercise with n = 500 and the loss-based prior. (n) Pr(M0 |x ) Pr(M1 |x(n) ) Pr(M2 |x(n) ) Mean posterior 1.33 × 10−13 0.08 0.92 Variance posterior 1.76 × 10−24 0.0200 0.0200 Freq. true model 0/100 4/100 96/100 Table 5: Average model posterior probabilities, variance and frequency of true model for the Scenario 4 simulation exercise with n = 1500 and the loss-based prior. To compare the loss-based prior with the uniform prior we have run the simulation on the same data samples used above. The results for n = 500 and n = 1500 are in Table 6 and in Table 7, respectively. Although we can observe an improvement when the sample size increases, the uniform prior does not lead to a clear detection of the true model for both sample sizes. Pr(M0 |x(n) ) Pr(M1 |x(n) ) Pr(M2 |x(n) ) Mean posterior 16 × 10−4 0.82 0.18 Variance posterior 7.15 × 10−5 0.0447 0.0443 Freq. true model 0/100 91/100 9/100 Table 6: Average model posterior probabilities, variance and frequency of true model for the Scenario 4 simulation exercise with n = 500 and the uniform prior. 18 Mean posterior 8.64 × 10−12 0.501 0.499 (n) Pr(M0 |x ) Pr(M1 |x(n) ) Pr(M2 |x(n) ) Variance posterior 7.45 × 10−21 0.1356 0.1356 Freq. true model 0/100 49/100 51/100 Table 7: Average model posterior probabilities, variance and frequency of true model for the Scenario 4 simulation exercise with n = 1500 and the uniform prior. 0.3 Weibull Log-normal Gamma 0.25 Pdfs 0.2 0.15 0.1 0.05 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Figure 5: The densities of Weibull(λ, κ), Log-normal(µ, τ ) and Gamma(α, β) with the same mean (equal to 5) and the same variance (equal to 2.5). Finally, we conclude this section with a remark. One may wonder why the change point detection requires an increasing in the sample size, and the reply can be inferred from Figure 5, which displays the density functions of the distributions employed in this scenario. As it can be observed, the densities are quite similar, which is not surprising since these distributions have the same means and the same variances. The above similarity can also be appreciated in terms of Hellinger distance, see Table 8. In other words, from Figure 5 we can see that the main differences in the underlying distributions are in the tail areas. It is therefore necessary to have a relatively large number of observations in order to be able to discern differences in the 19 densities, because in this case only we would have a sufficient representation of the whole distribution. Weibull(λ, κ) Log-normal(µ, τ ) Hellinger distances Weibull(λ, κ) Log-normal(µ, τ ) Gamma(α, β) 0.1411996 0.09718282 0.04899711 Table 8: Hellinger distances between all the pairs formed from a Weibull(λ, κ), Lognormal(µ, τ ) and Gamma(α, β). The six hyperparameters are such that the distributions have the same mean=5 and same variance=2.5. 5 Change Point Analysis on Real Data In this section, we illustrate the proposed approach applied to real data. We first consider a well known data set which has been extensively studied in the literature of the change point analysis, that is the British coal-mining disaster data (Carlin et al., 1992). The second set of data we consider refers to the daily returns of the S&P 500 index observed over a period of four years. The former data set will be investigated in Section 5.1, while the latter in Section 5.2. 5.1 British Coal-Mining Disaster Data The British coal-mining disaster data consists of the yearly number of deaths for the British coal miners over the period 1851-1962. It is believed that the change in the working conditions, and in particular, the enhancement of the security measures, led to a decrease in the number of deaths. This calls for a model which can take into account a change in the underlying distribution around a certain observed year. With the proposed methodology we wish to detect if the assumption is appropriate. In particular, if a model with one change point is more suitable to represent the data than a model where no changes in the sampling distribution are assumed. Figure 6 shows the number of deaths per year in the British coal-mining industry from 1851 to 1962. As in Chib (1998), we assume a Poisson sampling distribution with a possible change in the parameter value. That is i.i.d. X1 , X2 , . . . , Xm |φ1 ∼ Poisson(φ1 ) i.i.d. Xm+1 , Xm+2 , . . . , Xn |φ2 ∼ Poisson(φ2 ), 20 (24) 7 6 5 4 3 2 1 0 1851 1871 1891 1911 1931 1951 Year Figure 6: Scatter plot of the British coal-mining disaster data. where m is the unknown location of the single change point, such that 1 ≤ m ≤ n, and a Gamma(2, 1) is assumed for φ1 and φ2 . The case m = n corresponds to the scenario with no change point, that is model M0 . The case m < n assumes one change point, that is model M1 . Let f1 (·|φ1 ) and f2 (·|φ2 ) be the Poisson distributions with parameters φ1 and φ2 , respectively. Then, the analysis is performed by selecting between model M0 , that is when the sampling distribution is f1 , and model M1 , where the sampling distribution is f1 up to a certain m < n and f2 from m + 1 to n. As highlighted in the Remark at the end of Section 3, the prior on the model space is the discrete uniform distribution, that is Pr(M0 ) = Pr(M1 ) = 0.5. The proposed model selection approach leads to the Bayes factors B01 = 1.61 × 10−13 and B10 = 6.20 × 1012 , where it is obvious that the odds are strongly in favour of model M1 . Indeed, we have Pr(M1 |x(n) ) ≈ 1. 5.2 Daily S&P 500 Absolute Log-Return Data The second real data analysis aims to detect change points in the absolute value of the daily logarithmic returns of the S&P500 index observed from the 14/01/2008 to the 31/12/2011 (see Figure 7). As underlying sampling distributions we consider the Weibull and the Log-normal (Yu, 2001), and the models among which we select are as follows. M0 is a Weibull(λ, κ), M1 is formed by a Weibull(λ, κ) and a Log-normal(µ1 , τ1 ) and, finally, M2 is formed by a Weibull(λ, κ), a Log-normal(µ1 , τ1 ) and a Log-normal(µ2 , τ2 ). An interesting particularity of this problem is that we will consider a scenario where the changes are in the underlying distribution or in the parameter values of the same distribution. As suggested in Section 4.1.3 of Kass and Raftery (1995), due to the large sample size of the data set, we could approximate 21 the Bayes factor by using the Schwarz criterion. Therefore, in this case the specification of the priors for the parameters of the underlying distributions is not necessary. From the results in Table 9, we see that the model indicated 0.12 0.1 0.08 0.06 0.04 0.02 0 14/01/08 29/10/08 14/08/09 31/05/10 16/03/11 30/12/11 Figure 7: Absolute daily log-returns of the S&P500 index from 14/01/08 to 30/12/11. by the proposed approach is M2 . In other words, there is very strong indication that there are two change points in the data set. From Table 9, we note Pr(M0 ) Pr(M1 ) Pr(M2 ) B01 B02 B12 Pr(M0 |x(n) ) Pr(M1 |x(n) ) Pr(M2 |x(n) ) 0.36 0.32 0.32 7.72 × 1018 3.30 × 10−3 4.28 × 10−22 0.00 0.00 1.00 Table 9: Model prior, Bayes factor and model posterior probabilities for the S&P500 change point analysis. that the prior on model M1 and M2 assigned by the proposed method are the same. This is not surprising as the only difference between the two models is an additional Log-normal distribution with different parameter values. 6 Conclusion Bayesian inference in change point problems under the assumption of minimal prior information has not been deeply explored in the past, as the limited literature on the matter shows. 22 We contribute to the area by deriving an objective prior distribution to detect change point locations, when the number of change points is known a priori. As a change point location can be interpreted as a discrete parameter, we apply recent results in the literature (Villa and Walker, 2015a) to make inference. The resulting prior distribution, which is the discrete uniform distribution, it is not new in the literature (Girón et al., 2007), and therefore can be considered as a validation of the proposed approach. A second major contribution is in defining an objective prior on the number of change points, which has been approached by considering the problem as a model selection exercise. The results of the proposed method on both simulated and real data, show the strength of the approach in estimating the number of change points in a series of observations. A point to note is the generality of the scenarios considered. Indeed, we consider situations where the change is in the value of the parameter(s) of the underlying sampling distribution, or in the distribution itself. Of particular interest is the last real data analysis (S&P 500 index), where we consider a scenario where we have both types of changes, that is the distribution for the first change point and on the parameters of the distribution for the second. The aim of this work was to set up a novel approach to address change point problems. In particular, we have selected prior densities for the parameters of the models to reflect a scenario of equal knowledge, in the sense that model priors are close to represent a uniform distribution. Two remarks are necessary here. First, in the case prior information about the true value of the parameters is available, and one wishes to exploit it, the prior densities will need to reflect it and, obviously, the model prior will be impacted by the choice. Second, in applications it is recommended that some sensitivity analysis is performed, so to investigate if and how the choice of the parameter densities affects the selection process. 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Scandinavian Journal of Statistics 42 (4), 947– 966. Yu, J., 2001. Chapter 6 - Testing for a Finite Variance in Stock Return Distributions. In: Knight, J., Satchell, S. E. (Eds.), Return Distributions in Finance (Quantitative Finance). Butterworth-Heinemann, Oxford, pp. 143 – 164. 26 Appendix A Model prior probabilities to select among models M0 , M1 and M2 Here, we show how model prior probabilities can be derived for the relatively simple case of selecting among scenarios with no change points (M0 ), one change point (M1 ) or two change points (M2 ). First, by applying the result in Theorem 2, we derive the Kullback–Leibler divergences between any two models. That is: • the prior probability for model M0 depends on the following quantities: DKL (M0 kM1 ) =(n − m1 ) · DKL (f1 (·|θ̃1 )kf2 (·|θ̃2 )) DKL (M0 kM2 ) =(m2 − m1 ) · DKL (f1 (·|θ̃1 )kf2 (·|θ̃2 )) + (n − m2 ) · DKL (f1 (·|θ̃1 )kf3 (·|θ̃3 )) • the prior probability for model M1 depends on the following quantities: DKL (M1 kM2 ) =(n − m2 ) · DKL (f2 (·|θ̃2 )kf3 (·|θ̃3 )) DKL (M1 kM0 ) =(n − m1 ) · DKL (f2 (·|θ̃2 )kf1 (·|θ̃1 )) • the prior probability for model M2 depends on the following quantities: DKL (M2 kM1 ) =(n − m2 ) · DKL (f3 (·|θ̃3 )kf2 (·|θ̃2 )) DKL (M2 kM0 ) =(m2 − m1 ) · DKL (f2 (·|θ̃2 )kf1 (·|θ̃1 )) + (n − m2 ) · DKL (f3 (·|θ̃3 )kf1 (·|θ̃1 )) The next step is to derive the minimum Kullback–Leibler divergence computed at each model: 27 • for model M0 :    inf DKL (M0 kM1 ) = inf (n − m1 ) · inf DKL (f1 (·|θ̃1 )kf2 (·|θ̃2 )) θ1 m1 6=n θ̃2 | {z }  1 = inf DKL (f1 (·|θ̃1 )kf2 (·|θ̃2 )) θ̃  2   inf DKL (M0 kM2 ) = inf (m2 − m1 ) · inf DKL (f1 (·|θ̃1 )kf2 (·|θ̃2 )) θ2 m1 6=m2 θ̃2 {z } | 1     + inf (n − m2 ) · inf DKL (f1 (·|θ̃1 )kf3 (·|θ̃3 )) m2 6=n θ̃3 | {z } 1 = inf DKL (f1 (·|θ̃1 )kf2 (·|θ̃2 )) + inf DKL (f1 (·|θ̃1 )kf3 (·|θ̃3 )) θ̃2 θ̃3 • for model M1 :    inf DKL (M1 kM2 ) = inf (n − m2 ) · inf DKL (f2 (·|θ̃2 )kf3 (·|θ̃3 )) θ2 m2 6=n θ̃3 | {z }  1 = inf DKL (f2 (·|θ̃2 )kf3 (·|θ̃3 )) θ̃3 inf DKL (M1 kM0 ) =(n − m1 ) · inf DKL (f2 (·|θ̃2 )kf1 (·|θ̃1 )) θ0 =θ̃1 θ̃1 • for model M2 : inf DKL (M2 kM1 ) =(n − m2 ) · inf DKL (f3 (·|θ̃3 )kf2 (·|θ̃2 )) θ1 θ̃2 inf DKL (M2 kM0 ) =(m2 − m1 ) · inf DKL (f2 (·|θ̃2 )kf1 (·|θ̃1 )) θ0 =θ̃1 θ̃1 + (n − m2 ) · inf DKL (f3 (·|θ̃3 )kf1 (·|θ̃1 )) θ̃1 Therefore, the model prior probabilities can be computed through equation (15), so that: • the model prior probability Pr(M0 ) is proportional to the exponential of the minimum between:     Eπ0 inf DKL (f1 (·|θ̃1 )kf2 (·|θ̃2 )) , Eπ0 inf DKL (f1 (·|θ̃1 )kf2 (·|θ̃2 )) θ̃2 θ̃2  + inf DKL (f1 (·|θ̃1 )kf3 (·|θ̃3 )) θ̃3 28 • the model prior probability Pr(M1 ) is proportional to the exponential of the minimum between:    Eπ1 inf DKL (f2 (·|θ̃2 )kf3 (·|θ̃3 )) , θ̃   3 Eπ1 (n − m1 ) · inf DKL (f2 (·|θ̃2 )kf1 (·|θ̃1 )) θ̃1 • the model prior probability Pr(M2 ) is proportional to the exponential of the minimum between:    Eπ2 (n − m2 ) · inf DKL (f3 (·|θ̃3 )kf2 (·|θ̃2 )) , θ̃2  Eπ2 (m2 − m1 ) · inf DKL (f2 (·|θ̃2 )kf1 (·|θ̃1 )) + (n − m2 ) θ̃1  · inf DKL (f3 (·|θ̃3 )kf1 (·|θ̃1 )) θ̃1 29 B Proofs Proof of Theorem 1 We distinguish two cases: S = +1 and S = −1. When S = +1, equivalent to mj < m0j : ! Z (n) f (x |m, θ̃) DKL (f (x(n) |m, θ̃)kf (x(n) |m0 , θ̃)) = f (x(n) |m, θ̃) · ln dx(n) f (x(n) |m0 , θ̃)  0 ! mj Z X fj+1 (xi |θ̃j+1 )  (n) = f (x(n) |m, θ̃) ·  dx ln fj (xi |θ̃j ) i=mj +1 0 = mj Z X " f (x(n) |m, θ̃) · ln i=mj +1 = m0j ( X 1n−1 · fj+1 (xi |θ̃j+1 ) fj (xi |θ̃j ) " Z fj+1 (xi |θ̃j+1 ) · ln i=mj +1 !# dx(n) fj+1 (xi |θ̃j+1 ) fj (xi |θ̃j ) !# ) dxi m0 = j X DKL (fj+1 (xi |θ̃j+1 )kfj (xi |θ̃j )) i=mj +1 =(m0j − mj ) · DKL (fj+1 (·|θ̃j+1 )kfj (·|θ̃j )) =(m0j − mj ) · d+1 j (θ̃). (25) When S = −1, equivalent to mj > m0j , in a similar fashion, we get DKL (f (x(n) |m, θ̃)kf (x(n) |m0 , θ̃)) = (mj − m0j ) · d−1 j (θ̃) (26) From equations (25) and (26), we get the result in Theorem 1. Proof of Theorem 2 We recall that the model parameter θi is the vector (m1 , m2 , . . . , mi , θ̃1 , θ̃2 , . . . , θ̃i+1 ), where i = 0, 1, . . . , k. Here, θ̃1 , θ̃2 , . . . , θ̃i+1 represent the parameters of the underlying sampling distributions considered under model Mi and m1 , m2 , . . . , mi are the respective i change point locations. In this setting, f (x (n) |θi ) = m1 Y r=1 f1 (xr |θ̃1 ) i−1 m t+1 Y Y ft+1 (xr |θ̃t+1 ) t=1 r=mt +1 30 n Y r=mi +1 fi+1 (xr |θ̃i+1 ) (27) We proceed to the computation of DKL (Mi kMj ), that is the Kullback–Leibler divergence introduced in Section 3. Similarly to the proof of Theorem 1, we obtain the following result. ! mi+2 X Z f (x | θ̃ ) i+1 r i+1 dx(n) DKL (Mi kMj ) = f (x(n) |θi ) ln f (x | θ̃ ) i+2 r i+2 r=mi+1 +1 ! Z mi+3 X f (x | θ̃ ) i+1 r i+1 + f (x(n) |θi ) ln dx(n) + fi+3 (xr |θ̃i+3 ) r=mi+2 +1 ! Z n X f (x | θ̃ ) i+1 r i+1 ... + f (x(n) |θi ) ln dx(n) . f (x | θ̃ ) j+1 r j+1 r=mj +1 Given equation (27), if we integrate out the variables not involved in the logarithms, we obtain DKL (Mi kMj ) =(mi+2 − mi+1 ) · DKL (fi+1 (·|θ̃i+1 )kfi+2 (·|θ̃i+2 )) + (mi+3 − mi+2 ) · DKL (fi+1 (·|θ̃i+1 )kfi+3 (·|θ̃i+3 ))+ . . . + (n − mj ) · DKL (fi+1 (·|θ̃i+1 )kfj+1 (·|θ̃j+1 )). In a similar fashion, it can be shown that DKL (Mj kMi ) =(mi+2 − mi+1 ) · DKL (fi+2 (·|θ̃i+2 )kfi+1 (·|θ̃i+1 )) + (mi+3 − mi+2 ) · DKL (fi+3 (·|θ̃i+3 )kfi+1 (·|θ̃i+1 ))+ . . . + (n − mj ) · DKL (fj+1 (·|θ̃j+1 )kfi+1 (·|θ̃i+1 )) 31 

1 arXiv:1705.09704v1 [] 26 May 2017 Lock-step simulation is child’s play Experience Report – Extended Version JOACHIM BREITNER, University of Pennsylvania CHRIS SMITH, Google Implementing multi-player networked games by broadcasting the player’s input and letting each client calculate the game state – a scheme known as lock-step simulation – is an established technique. However, ensuring that every client in this scheme obtains a consistent state is infamously hard and in general requires great discipline from the game programmer. The thesis of this report is that in the realm of functional programming – in particular with Haskell’s purity and static pointers – this hard problem becomes almost trivially easy. We support this thesis by implementing lock-step simulation under very adverse conditions. We extended the educational programming environment CodeWorld, which is used to teach math and programming to middle school students, with the ability to create and run interactive, networked multi-user games. Despite providing a very abstract and high-level interface, and without requiring any discipline from the programmer, we can provide consistent lock-step simulation with client prediction. ACM Reference format: Joachim Breitner and Chris Smith. 2017. Lock-step simulation is child’s play. Proc. ACM Program. Lang. 1, 1, Article 1 (January 2017), 18 pages. DOI: 10.1145/nnnnnnn.nnnnnnn Networked multi-user games must tackle the challenge of ensuring that all participating players on a network with potentially significant latency still see the same game state. In some circumstances, an appealing choice is lock-step simulation. In this scheme, which dates back to the age of Doom, the state of the game itself is never transmitted over the network. Instead, the clients exchange information about their player’s interactions – as abstract game moves or just the actual user input events – and each client independently calculates the state of the game. Of course, this only works as intended if all clients end up with the same state. The technique is fraught with danger if the programmer is not very careful and disciplined about managing that state. Terrano and Bettner (2001), who implemented the network code for the real time strategy games Age of Empires 1 & 2, report: As much as we check-summed the world, the objects, the pathfinding, targeting and every other system – it seemed that there was always one more thing that slipped just under the radar. [. . . ] Part of the difficulty was conceptual – programmers were not used to having to write code that used the same number of calls to random within the simulation. More drastic words were voiced by Smith (2011), also a video game software engineer: One of the most vile bugs in the universe is the desync bug. They’re mean sons of bitches. The grand assumption to the entire engine architecture is all players being fully synchronous. What happens if they aren’t? What if the simulations diverge? Chaos. Anger. Suffering. © 2017 ACM. This is the author’s version of the work. It is posted here for your personal use. Not for redistribution. The definitive Version of Record was published in Proc. ACM Program. Lang., http://dx.doi.org/10.1145/nnnnnnn.nnnnnnn. Proc. ACM Program. Lang., Vol. 1, No. 1, Article 1. Publication date: January 2017. 1:2 Joachim Breitner and Chris Smith The pitfalls facing a programmer implementing lockstep simulation include reading the system clock, querying the random number generator, other I/O, uninitialized memory, and local or hidden statefulness. In short: side effects! What if we chose a programming language without such side-effects? Would these problems disappear? Intuitively, we expect that pure functional programming makes lock-step simulation easy. This experience report corroborates our expectation. We have implemented lock-step simulation in Haskell under very adverse conditions. The authors of the quotes above are professional programmers working on notable games. They can be expected to maintain a certain level of programming discipline, and to tolerate additional complexity. Our implementation is part of CodeWorld1 , Fig. 1. The Snake game an educational, web-based programming environment used to teach mathematics and coding to students as early as middle school. These children, who are just learning to code, can write and run arbitrary game logic, using a simple API, without adhering to any additional requirements or coding discipline. Nevertheless, we still guarantee consistent lock-step simulation and avoid the dreaded desync bug. The main contributions of this experience report are: • With a bold disregard for pesky implementation detail, we design a natural extension to CodeWorld’s existing interfaces that can describe multi-user interactive programs in as straightforward, simple and functional a manner as possible (Section 2.1). • We identify a complication – unwanted capture of free variables – which can thwart consistency of such a program. We solve it using either using the module system (Section 2.2) or the Haskell language extension static pointers (Section 2.3). • We explain how to implement this interface. Despite its abstractness, we present an eventually consistent implementation that works for arbitrary client code, and includes client prediction to react immediately to local input while still reconciling delayed input from other users (Section 3). • We share lessons learned in stress-testing the system (Section 4.1). Testing was successful, but we identified an inconsistency in floating point transcendental functions. Replacing these with deterministic approximations recovers the consistency that we rely upon (Section 4.2). • We show that, even with no knowledge of the structure of the program’s state, our approach still allows us to smooth out artifacts that arise due to network latency (Section 4.3). • Overall, we show that pure functional programming makes lock-step simulation easy. 1 https://code.world/haskell Proc. ACM Program. Lang., Vol. 1, No. 1, Article 1. Publication date: January 2017. Lock-step simulation is child’s play 1 1:3 CODEWORLD In this section, we give a brief overview of how students interact with the CodeWorld environment, the programming interfaces that are provided by CodeWorld and how student programs are executed. Many of the figures illustrating this paper are created by students. These and more can be found in the CodeWorld gallery at https://code.world/gallery.html. To ease deployment, students need only a web browser to use CodeWorld. They write their code with an integrated editor inside the browser. Programs are written in Haskell, which the CodeWorld server compiles to JavaScript using GHCJS (Stegeman and Mackenzie 2017) and sends that back to browser to execute in a canvas beside the editor. These programs are always graphical: students create static pictures, then animations, and finally interactive games and other activities. 1.1 Two flavors of Haskell The standard Haskell language is not an ideal vessel for the children in CodeWorld’s target audience. Therefore, CodeWorld by default provides a specially tailored educational environment. In this mode, a custom prelude is used to help students avoid common obstacles. Graphics primitives are available without an import, to create appealing visual programs. Functions of multiple arguments are not curried but rather take their arguments in a tuple, both to improve error messages and match mathematical notation that students are already learning. Finally, a single Number type (isomorphic to Double) is provided to avoid the need for type classes, and the RebindableSyntax language extension makes literals monomorphic. Compiler error messages are post-processed to make them more intelligible to the students. Nevertheless, the code students write is still Haskell, and is accepted by GHC. However, at https://code.world/haskell instead of https://code.world/, one finds a standard Haskell environment, with full access to the standard library. In this paper we focus on the latter variant. 1.2 API design principles An important principle of CodeWorld is to provide students with the simplest possible abstraction for a given task. This allows them to concentrate on the ideas they want to express and think clearly about the meaning of their code, and hides as many low-level details as possible. The first and simplest task that students face is to produce a static drawing. This is done with the abstract data type Picture, with a simple compositional API (Figure 3) which was heavily inspired by the Gloss library (Lippmeier 2017). Complex pictures are built by combining and transforming simple geometric objects. The entry point used for this has the very simple type drawingOf :: Picture → IO () This function takes care of the details of displaying the Fig. 2. Smiley student’s picture on the screen, redrawing upon window size changes and so on. So all it takes for a student to get the computer to smile like in Figure 2 is to write import CodeWorld smiley = translated (−4) 4 (solidCircle 2) & translated 4 4 (solidCircle 2) & thickArc 2 (−pi) 0 6 & colored yellow (solidCircle 10) main = drawingOf smiley Proc. ACM Program. Lang., Vol. 1, No. 1, Article 1. Publication date: January 2017. 1:4 Joachim Breitner and Chris Smith data Picture –– abstract –– Various geometric shapes (circle, rectangle, arc, polygon etc.) are –– available, and can either be filled or equipped with a thickness. E.g.: –– Parameter: radius solidCircle :: Double → Picture –– Parameters: thickness, radius, start angle and end angle thickArc :: Double → Double → Double → Double → Picture –– Pictures can be transformed and overlaid colored :: Color → (Picture → Picture) translated :: Double → Double → (Picture → Picture) rotated :: Double → (Picture → Picture) scaled :: Double → Double → (Picture → Picture) (&) :: Picture → Picture → Picture Fig. 3. An excerpt of CodeWorld’s Picture API As a next step, the students can create animations and simulations to make their pictures move, before eventually making their programs react to user input in interactions. The game in Figure 4 is a typical interaction, where the player saves flying Grandma from various obstacles by attaching balloons or parachutes to her wheelchair. These are created by calling the following interface: interactionOf :: world → (Double → world → world) → (Event → world → world) → (world → Picture) → IO () In a typical call main = interactionOf start step handle draw Fig. 4. Yo Grandma, by Sophia (6th grade) the student passes four arguments, namely: (1) an initial state, start, (2) a time step function, step, which calculates an updated state as time passes, (3) an event handler function, handle, which calculates an updated state when the user interacts with the program and (4) a visualization function, draw, to depict the current state as a Picture. The Event type, shown in Figure 5, is a simple algebraic data type that describes the press or release of a key or mouse button, or a movement of the mouse pointer. The type of the state, world, is chosen by the user and consists of the domain-specific data needed by the program. The world type is completely unconstrained, and this will be an important factor influencing our design. It need not even be serializable, nor comparable for equality. In particular, the state may contain first-class functions and infinite lazy data structures. One way that students commonly make use of this capability is by defining infinite lazy lists of anticipated future events, based on a random number source fetched before the simulation begins. Proc. ACM Program. Lang., Vol. 1, No. 1, Article 1. Publication date: January 2017. Lock-step simulation is child’s play 1:5 type Point = (Double, Double) data Event = KeyPress Text | KeyRelease Text | MousePress MouseButton Point | MouseRelease MouseButton Point | MouseMovement Point data MouseButton = LeftButton | MiddleButton | RightButton Fig. 5. The Event type 2 AN INTERFACE FOR MULTI-PLAYER GAMES We would like students to extend their programming to networked multi-user programs, so that they can invite their friends to join over the internet and collaborate together on a drawing, fight each other in a fierce duel of Snake, or interact in any other way the student designs and implements. In this section, we turn our attention to choosing an API for such a task. 2.1 Wishful thinking Let us apply “API design by wishful thinking”, and ask: What is the most convenient abstract model of a multi-player game we can hope for, independent of implementation concerns or constraints? As experienced programmers, our thoughts might drift to network protocols or message passing between independent program instances, each with its own local state. Our students, though, care about none of this, and ideally we would not burden them with it. In fact, motivated students have already implemented games to be played with classmates, using different keys on the same device. An example is shown in Figure 6, where the red player uses the keys W A S D and the blue player the keys ↑ ← ↓ → , in a race to consume more dots. Their games, which they have already designed, are described in terms of one shared global state. Why should the programming model change drastically simply because of one detail – that the code will now run on multiple nodes communicating over a network? Fig. 6. Dot Grab, by Adrian (7th grade) We conclude, then, that an interactive multi-user program is a generalization of an interactive single-user program, and the centerpiece of the API is still a single, global state, which is mutually acted upon by all players. Basing the API on interactionOf, we make only minimal changes to adapt to the new environment: • A new first parameter specifies the number of players. • The parameters start and step remain as they are. • The handle parameter, though, ought to know which user pressed a certain button or moved their mouse, so it receives the player number (a simple Int) as an additional parameter. • Different players may also see different views of the state, so the draw function also receives the player number for which it should render the screen – but it is free to ignore that parameter, of course. Proc. ACM Program. Lang., Vol. 1, No. 1, Article 1. Publication date: January 2017. 1:6 Joachim Breitner and Chris Smith All together, we arrive at the following “ideal” interface that we call collaborations, which allows students to build networked multi-player games and other activities: collaborationOf :: Int → world → (Double → world → world) → (Int → Event → world → world) → (Int → world → Picture) → IO () A small example will clarify how this interface is used. The following code traces the mouse movements of two players using colored, fading circles, and Figure 7 shows this program in action. The green player is a bot that simply mirrors the red player’s movements. import CodeWorld type World = [(Color, Double, Double, Double)] step :: Double → World → World Fig. 7. Two players’ mouse movements step dt dots = [(c, exp (−dt) ∗ r, x, y) | (c, r, x, y) ← dots, r > 0.1] handle :: Int → Event → World → World handle 0 (MouseMovement (x, y)) dots = (red, 1, x, y) : dots handle 1 (MouseMovement (x, y)) dots = (green, 1, x, y) : dots handle dots = dots draw :: Int → World → Picture draw dots = mconcat [translated x y (colored c (solidCircle r)) | (c, r, x, y) ← dots] main :: IO () main = collaborationOf 2 [ ] step handle draw A collaboration begins with a lobby, featuring buttons to create or join a game. Upon creating a new game, the player is given a four-letter code to be shared with friends. Those friends may enter the four-letter code to join the game. Once enough players have joined, the game begins. 2.2 Solving random problems with the module system Like interactionOf before it, the parameters of collaborationOf provide enough information to completely determine the behavior of the program from the sequence of time steps and UI events that occur. Unlike interactionOf, however, a collaboration involves more than one use of the collaborationOf API, as the function is executed by each participating player. To ensure that there is a single, well-defined behavior, it is essential that all players run collaborationOf with the same arguments. Obviously, we need to ensure that all clients run the same program, and the CodeWorld server does so. But even with the same code, the arguments to collaborationOf can differ from client to client: main = do r ← randomRIO (0, 1) collaborationOf numPlayers start step (handle r) draw The event handling function now depends on I/O – specifically, the choice of a random number – and it is very unlikely that all clients happen to pick the same random number. Despite sharing the same code, the clients will disagree about the correct behavior of the system. Proc. ACM Program. Lang., Vol. 1, No. 1, Article 1. Publication date: January 2017. Lock-step simulation is child’s play 1:7 The problem is not the use of random numbers per se, but rather the unconstrained flow of client-specific state resulting from any I/O into the collaborationOf API via free variables in its parameters. Since most of the parameters to collaborationOf have function types, we cannot just compare them to establish consistency at runtime. We solve this problem in two ways: one in the educational environment, and the other in the standard Haskell environment. In the former, we have tight control over the set of library functions available to the student. No packages are exposed except for a custom standard library with a heavily customized Prelude module, and this library simply does not provide any functions to compose IO operations, such as as the monadic bind operators (>>=, >>). This also rules out the use of Haskell’s do-notation, which under the regime of RebindableSyntax requires an operator called (>>=) to be in scope. A valid Haskell program requires a top-level function main :: IO (), and since the only available way to obtain an IO () is through our API entry points (drawingOf, interactionOf, and so on), we know that all CodeWorld collaborations are of essentially the form main = collaborationOf . . . In particular, no I/O can be executed prior to the collaboration, and hence no client-dependent behavior is possible. 2.3 Solving random problems syntactically This solution is not suitable for the standard Haskell environment, where we do not want to restrict the user’s access to the standard library. We can still prevent the user from using the results of client-specific I/O in arguments to collaborationOf. To accomplish this, we creatively use the work of Epstein et al. (2011), who sought to bring Erlang-like distributed computing to Haskell. They had to exchange functions over the network, which is possible by passing code references, as long as no potentially unserializable values are captured from the environment. To guarantee that, they introduced a Haskell language extension, static pointers, which introduces: • a new type constructor StaticPtr a, which wraps values of type a, • a new syntactic construct static foo, such that for any expression foo of type a, the expression static foo has type StaticPtr a, but is only valid if foo does not contain any locally bound free variables, • a pure function deRefStaticPtr :: StaticPtr a → a, to unwrap the static pointer, and • a pure function staticKey :: StaticPtr a → StaticKey which produces a key that – within one program – uniquely identifies a static pointer. The requirement that StaticPtr values cannot have locally bound free variables turns out to be exactly what we need to prevent programs from smuggling client-specific state obtained with I/O actions into collaborations. We therefore further refine the API to require its arguments to be static pointers: collaborationOf :: Int → StaticPtr world → StaticPtr (Double → world → world) → StaticPtr (Int → Event → world → world) → StaticPtr (Int → world → Picture) → IO () The mouse tracing program in Figure 7 must now change its definition of main to main = collaborationOf 2 (static [ ]) (static step) (static handle) (static draw). On the other hand, writing static (handle r) to smuggle in a randomly drawn number r, as in the example above, will fail at compile time. Requiring the static keyword here admittedly muddies the clarity of the API a bit. We believe that the target audience of CodeWorld’s standard Haskell Proc. ACM Program. Lang., Vol. 1, No. 1, Article 1. Publication date: January 2017. 1:8 Joachim Breitner and Chris Smith mode can handle this. Beginners working within the educational mode need not deal with this slight complication. A somewhat more clever attempt, though, still causes problems: main = do coinFlip ← randomIO let step = if coinFlip then static step1 else static step2 collaborationOf 2 (static [ ]) step (static handle) (static draw) This program is accepted by the compiler because the arguments to collaborationOf are indeed StaticPtr values of the right types, yet it raises the same questions when clients disagree on the choice of step function. While we cannot prevent this case at compile time, we can at least detect it at runtime. Static pointers can be serialized using the function staticKey :: StaticPtr a → StaticKey. Before a game starts, the participating clients compare the keys of their arguments to check that they match. This is a subtly different use of static pointers from the original intent of sending functions over a network in a message-passing protocol. We need not actually receive the original values on the remote end of our connections, but instead use the serialized keys only to check for consistency. With this check in place – short of using unsafe features such as unsafePerformIO – we are confident that every client is indeed running the same functions. However, this forces our games to be entirely deterministic. This is a problem, since many games involve an element of chance! To restore the possibility of random behavior, we supply a random number source to use in building the initial state, with a consistent seed in all clients. The type of the start parameter is now StaticPtr (StdGen → world). (This is not entirely new: CodeWorld’s educational environment has never exported a random number generator, and its simulations and interactions have always been initialized with an infinite list of random numbers.) This completes our derivation of collaborationOf, which in its final form is collaborationOf :: Int → StaticPtr (StdGen → world) → StaticPtr (Double → world → world) → StaticPtr (Int → Event → world → world) → StaticPtr (Int → world → Picture) → IO () 3 FROM WISHFUL THINKING TO RUNNING CODE How can we implement this interface? It turns out that our implementation options are severely narrowed down by the following requirements: (1) We need to handle any code using the API. Given the educational setting of CodeWorld, we cannot require any particular discipline. (2) The players need to see an eventually consistent state. They may have different ideas about the state of the world, but only until everybody receives information about everybody’s interactions. (3) The effects of a player’s own interactions are immediately visible to that player. Even a “local” interaction, such as selecting a piece in a game of Chess, will have to represented in the game state, and any latency here would make the user interface sluggish. The first requirement in particular implies that the game state is completely opaque to us. This already rules out the usual client-server architecture, where only the central server manages the game state and the clients send abstract moves (e.g., “white moves the knight to e8”) and render Proc. ACM Program. Lang., Vol. 1, No. 1, Article 1. Publication date: January 2017. Lock-step simulation is child’s play 1:9 the game state that they receive from the server. We have neither insight into what constitutes an abstract move, nor how to serialize and transmit the game state. We could avoid this problem by sending the raw UI Event instead of an abstract move to the server, and letting the server respond to each client with the Picture to show. This “dumb terminal” approach however would run afoul of our third requirement, as every user interaction would be delayed by the time it takes messages to travel to the server and back. The requirement of immediate responsiveness implies that every client needs to manage its own copy of the game state, and being abstract in the game state implies that there is nothing else but the UI events that the clients can transmit to synchronize the state. In other words, lock-step simulation is the only way for us. This approach assumes the integrity of client code. Since all clients track the entire game state, malicious players could trick CodeWorld into running a modified version of the program which, among other things, could then reveal hidden parts of the game state. Given the educational goals of CodeWorld, we are willing to trade this security for a cleaner API. 3.1 Types and messages We seek, then, to implement the API by exchanging UI events between clients. For the purposes of this paper, it does not matter how events are transmitted from client to client. The CodeWorld implementation uses a very simple relay server that broadcasts messages from one client to the others via WebSockets (a full-duplex server-client protocol for web applications), but peer-to-peer communication using WebRTC (a peer-to-peer protocol for web applications) or other methods would work equally well, as long as they deliver events reliably and in order. Every such message obviously needs to contain the actual Event and the player number. In addition, it must contain a timestamp, so that each client applies the event at the same time despite differences in network latency. Otherwise – assuming a time-sensitive game with a non-trivial step function – the various clients would obtain different views of the world. Timestamps are Double values, measured in seconds since the start of the game. type Timestamp = Double type Player = Int type Message = (Timestamp, Player, Event) 3.2 Resettable state Having fixed the message type still leaves open the question of what to do with these messages, which is non-trivial due to the network latency. Assume that 23.5 seconds into a real-time strategy game, I send my knights to attack the other player. My client sends the corresponding message (23.500, 0, MousePress LeftButton (20, 30)) to the other player. The message arrives, say, 100ms later. As mentioned before, the other player cannot simply let my knights set out a bit later. What else? The classical solution (Terrano and Bettner 2001) is to not act on local events immediately, but add a delay of, say, 200ms. The message would be (23.700, 0, MousePress LeftButton (20, 30)), and assuming it reaches all other players in time, all are able to apply the event at precisely the same moment. This solution works well if the UI can somehow respond to the user’s actions immediately, e.g. by letting the knight audibly confirm the command, so hide this delay from the user. The luxury of such a separation is not available to us – according to the third requirement, each client must immediately apply its own events – and the message really has to have the timestamp 23.500. This leaves the other player, when it receives the message 100ms later, with no choice but to roll back the game state to time 23.500, apply my event, and replay the following 100ms. While Proc. ACM Program. Lang., Vol. 1, No. 1, Article 1. Publication date: January 2017. 1:10 Joachim Breitner and Chris Smith rollback and replay are hard to implement in imperative programming paradigms, where every piece of data can have local mutable state, they are easy in Haskell, where we know that the value of type world really holds all relevant bits of the program’s state. One way of allowing such recalculation is to simply not store the state at all, and re-calculate it every time we draw the game screen. The function to do so would expect the game specification, the current time and the list of messages that we have seen so far, including the locally generated ones, and would calculate the game state. Its type signature would thus be currentState :: Game world ⇒ Timestamp → [Message] → world where the hypothetical type class Game captures the user-defined game logic; we introduce it here to avoid obscuring the following code listings by passing it explicitly around as an argument: class Game world where start :: world step :: Double → world → world handle :: Player → Event → world → world Assume, for a short while, that there was no step function, i.e. the game state changes only when there is an actual event. Then the timestamps are only required to put the events into the right order and to disregard events which are not yet to be applied (which can happen if the player’s game time started at slightly different points in time): currentState :: Game world ⇒ Timestamp → [Message] → world currentState now messages = applyEvents to_apply start where to_apply = takeWhile (λ(t, , ). t 6 now) (sortMessages messages) sortMessages :: [Messages] → [Messages] sortMessages = sortOn (λ(t, p, ). (t, p)) messages applyEvents :: Game world ⇒ [Message] → world → world applyEvents messages w = foldl apply w messages where apply w ( , p, e) = handle p e w Eventually, every client receives the same list of messages, up to the interleaving of events from different players. After a stable sort by timestamp and player, the lists of events will be identical, so all clients will calculate the same game state. 3.3 A few more steps This is nice and simple, but ignores the step function, which models the evolution of the state as time passes. Clearly, we have to call step before each event, and again at the end. In order to calculate the time passed since the last event, we also have to keep track of which timestamp a snapshot of the game state corresponds to: currentState :: Game world ⇒ Timestamp → [Message] → world currentState now messages = step (now − t) world where to_apply = takeWhile (λ(t, , ). t 6 now) (sortMessages messages) (t, world) = applyEvents to_apply (0, start) applyEvents :: Game world ⇒ [Message] → (Timestamp, world) → (Timestamp, world) applyEvents messages ts = foldl apply ts messages where apply (t0, world) (t1, p, e) = (t1, handle p e (step (t1 − t0) world))) Unfortunately, students would not be quite happy with this implementation. The step function is commonly used to calculate a single step in a physics simulation, which requires that it is called often enough to achieve a decent simulation frequency. Proc. ACM Program. Lang., Vol. 1, No. 1, Article 1. Publication date: January 2017. Lock-step simulation is child’s play 1:11 Player 1: 1 t 3 Player 2: 2 4 t Fig. 8. The evolution of a simple multi-player program with latency in the messages For instance, when simulating a projectile, a common technique is to adjust the position linearly along the velocity vector, and the velocity linearly according to forces like gravity or drag. The result is a stepwise-linear approximation, the precision of which depends on the sampling frequency. Another common technique is to do collision detection only once per time step, and again the result depends on the frequency of steps. It is important, then, that the step function is called at a reasonably high frequency. We could leave students to resolve this themselves, by dividing time steps into multiple finer steps, if necessary, in their step implementation. However, imposing that burden would violate our first requirement: not requiring any discipline from the user. Therefore, we have to ensure that the step function is called often enough, even if there is no user event for a while. In simulations and interactions, the implemented behavior is to evaluate the step function as quickly as possible between animation frames. Thus, simulations running on faster computers may take smaller steps and be more accurate. The need for eventual consistency precludes this strategy here. Instead, the desired step length for collaborationOf is defined globally and set to one-sixteenth of a second: gameRate :: Double gameRate = 1 / 16 We can obtain the desired resolution by wrapping the student’s step function in one that iterates step on time steps larger than the desired rate: gameStep :: Game world ⇒ Double → world → world gameStep dt world | dt 6 0 = world | dt > gameRate = gameStep (dt − gameRate) (step gameRate world) | otherwise = step dt world Replacing step with gameStep in the implementation of currentState and applyEvents above yields a correct solution. To see this code in action, we construct the following program: As the time passes, a column grows on the screen, from bottom to top. Initially, it is gray. When a player presses a number key, the column begins to grow in a different color. Additionally, whenever step is called, this current height of the column is marked with a black line. Because the program output is one-dimensional, we can use the horizontal dimension to show in Figure 8 how the players’ displays evolves over time. The dashed arrows indicate the transfer Proc. ACM Program. Lang., Vol. 1, No. 1, Article 1. Publication date: January 2017. 1:12 Joachim Breitner and Chris Smith of each packet to the other player, which is not instant. When a message from the other player arrives, the state is updated to reflect this change. Because this game essentially records its history, these delayed updates result in a “flicker” as the client updates the state. In many cases the effect will be less noticeable than it is here. We can see that the algorithm achieved eventual consistency, as the right edge of the drawing looks identical for both clients. 3.4 Limiting time travel In the course of a game, quite a large number of events occur. As time goes by, the cost of calculating the current state from scratch grows without bound, and will eventually become too large to be completed between each frame, and animations will stop being smooth. Clearly, some of that computation is quite pointless to repeat. Our message transport guarantees that messages from each client are delivered in order, so that when we receive a message, we know that we have seen all messages from the sender up to that timestamp. If we call this the client’s commit time, then we know that no new events will be received before the earliest commit time of any client, which we call the commit horizon. We can now precompute the game state up to the commit horizon, forget all older state and events, and use this as the basis for future state recalculations. In the following we will explain the data structure and associated operations that CodeWorld uses to keep track of the committed state, the pending events and each player’s commit time. The main data type is data Log world = Log { committed :: (Timestamp, world), events :: [Message], latest :: [(Player, Timestamp)] } Initially, there are no events, and everything is at timestamp zero: initLog :: Game world ⇒ [Player] → Log world initLog ps = Log (0, start) [ ] [(p, 0) | p ← ps] When an event comes in, the message is added to events via the public addEvent function. addEvent :: Game world ⇒ Message → Log world → Log world addEvent (t, p, e) log = recordActivity t p (log { events = events’ }) where events’ = sortMessages (events log ++ [(t, p, e)]) Then, the client’s commit time in latest is updated. recordActivity :: Game world ⇒ Timestamp → Player → Log world → Log world recordActivity t p log | t < t_old = error "Messages out of order" | otherwise = advanceCommitted (log { latest = latest’ }) where latest’ = (p, t) : delete (p, t_old) (latest log) Just t_old = lookup p (latest log) This might have moved the commit horizon, and if some of the messages from the list events are from before the commit horizon, we can integrate them into the committed state. advanceCommitted :: Game world ⇒ Log world → Log world advanceCommitted log = log { events = to_keep, committed = applyEvents to_commit (committed log) } where (to_commit, to_keep) = span (λ(t, , ). t < commitHorizon log) (events log) commitHorizon :: Log world → Timestamp commitHorizon log = minimum [t | (p, t) ← latest log] The final public function is used to query the current state of the game. Starting from the committed state, it applies the pending events. Proc. ACM Program. Lang., Vol. 1, No. 1, Article 1. Publication date: January 2017. Lock-step simulation is child’s play 1:13 currentState :: Game world ⇒ Timestamp → Log world → world currentState now log | now < commitHorizon log = error "Cannot look into the past" currentState now log = gameStep (now − t) world where past_events = takeWhile (λ(t, , ). t 6 now) (events log) (t, world) = applyEvents past_events (committed log) This algorithm, printed in Figure 9 in its entirety, relies on these assumptions: (1) The list of players provided to initLog is correct. (2) For each player, events are added in order, with monotonically increasing timestamps. (3) The state is never queried at a time that lies before commitHorizon. The first assumption is ensured by the CodeWorld framework. The second is ensured by using a monotonic time source to create the timestamps, and by using an order-preserving communication channel. The third follows from the fact that every client’s own timestamps are always in that player’s past, and therefore the argument to currentState is later than the commit horizon. If one of the players were to stop interacting with the program, that client would not send any messages. In this case, no events can be committed and the list of events to be processed by currentState would again grow without bound. To avoid this, each client sends empty messages (“pings”) whenever the user has not produced input for a certain amount of time. When such a ping is received, the addPing function advances the latest field without adding a new event: addPing :: Game world ⇒ (Timestamp, Player) → Log world → Log world addPing (t, p) log = recordActivity t p log This way, the number of events in the events field is bound by max input rate×(max network delay+max time between events or pings)×(number of players−1) which is independent of how long the game has been running. More tweaks are possible. In the CodeWorld implementation, we also cache the current state, so that querying the current state again, when no new events were received, is much cheaper. When an input event from another player comes in, we discard this cached value and recalculate it based on the committed state and the stored events. The main property of the code in Figure 9 is: No matter the interleaving of events from the various players, the result of currentState is the same. To increase our confidence that this property holds we used the QuickCheck library to randomly generate pairs of lists of events with monotonically increasing timestamps, considered all possible interleavings and checked that the resulting Log world data structure is identical. 4 EXPERIENCES AND DISCUSSION The interface from Section 2 allows the creation of multiuser applications with great ease, and with the algorithms in Section 3, CodeWorld can provide a smooth user experience. The reader may wonder, though, how well this works in practice, and what the drawbacks are for this approach. 4.1 Early experience For a first practical evaluation of the system, the second author organized a stress test, involving four colleagues, a selection of games with different styles, and small prizes for winners. During the event, participants play-tested Fig. 10. The tank game Proc. ACM Program. Lang., Vol. 1, No. 1, Article 1. Publication date: January 2017. 1:14 Joachim Breitner and Chris Smith type Timestamp = Double type Player = Int type Message = (Timestamp, Player, Event) data Log world = Log { committed :: (Timestamp, world), events :: [Message], latest :: [(Player, Timestamp)] } –– Public interface initLog :: Game world ⇒ [Player] → Log world initLog ps = Log (0, start) [ ] ([(p, 0) | p ← ps]) addEvent :: Game world ⇒ Message → Log world → Log world addEvent (t, p, e) log = recordActivity t p (log { events = events’ }) where events’ = sortMessages (events log ++ [(t, p, e)]) addPing :: Game world ⇒ (Timestamp, Player) → Log world → Log world addPing (t, p) log = recordActivity t p $ log currentState :: Game world ⇒ Timestamp → Log world → world currentState now log | now < commitHorizon log = error "Cannot look into the past" currentState now log = gameStep (now − t) world where past_events = takeWhile (λ(t, , ). t 6 now) (events log) (t, world) = applyEvents past_events (committed log) –– Internal functions gameRate :: Double gameRate = 1 / 16 sortMessages :: [Message] → [Message] sortMessages = sortOn (λ(t, p, ). (t, p)) recordActivity :: Game world ⇒ Timestamp → Player → Log world → Log world recordActivity t p log | t < t_old = error "Messages out of order" | otherwise = advanceCommitted (log { latest = latest’ }) where latest’ = (p, t) : delete (p, t_old) (latest log) Just t_old = lookup p (latest log) advanceCommitted :: Game world ⇒ Log world → Log world advanceCommitted log = log { committed = applyEvents to_commit (committed log) , events = to_keep } where (to_commit, to_keep) = span (λ(t, , ). t < commitHorizon log) (events log) commitHorizon :: Log world → Timestamp commitHorizon log = minimum [t | (p, t) ← latest log] applyEvents :: Game world ⇒ [Message] → (Timestamp, world) → (Timestamp, world) applyEvents messages ts = foldl apply ts messages where apply (t0, world) (t1, p, e) = (t1, handle p e (gameStep (t1 − t0) world)) gameStep :: Game world ⇒ Double → world → world gameStep dt world | dt 6 0 = world | dt > gameRate = gameStep (dt − gameRate) (step gameRate world) | otherwise = step dt world Fig. 9. The complete client prediction code discussed in Section 3.4 Proc. ACM Program. Lang., Vol. 1, No. 1, Article 1. Publication date: January 2017. Lock-step simulation is child’s play 1:15 the games, hoping to uncover any bugs or unexpected quirks of the format. The games involved, which can be played at https://code.world/gallery-icfp17.html, include: • The Dot Grab game (Figure 6), which was originally written by a student as a singlecomputer interaction. Since the API for games is a straightforward extension of the one for interactions, it was trivial to make this game networked. • The game “Snake” (Figure 1), where a player has to move across the playing field while avoiding the other player’s trails and the walls. • A tank game (Figure 10) where each player steers a tank using the keyboard, aims using the mouse and fires bullets that explode after a certain time. Here the game evolves over time and manages a larger number of moving parts – tanks, bullets, and explosions. Manual testing showed that the system is nicely responsive and that the artifacts due to network latency are noticeable, but not irritating. The system handled the more complex tank game well. A separate test using a high latency satellite connection remained playable, but with more pronounced latency-related artifacts, as expected. We plan to introduce the API to students in the Spring semester of 2017. 4.2 Floating point calculation A dominant concern in the implementation in Section 3 was to guarantee eventual consistency of all clients, so that game states would always converge over time. We achieve that requirement, on the assumption that the code passed to collaborationOf consists of pure functions. This result relies on a strong notion of pure function, though, which requires that outputs are predictable even between instances of the code running on different machines, operating systems, and runtime environments. In this sense, even functions in Haskell may not always be pure! A notable source of nondeterminism in Haskell is underspecified floating point operations. The Double type in Haskell is implementation-defined, and “should cover IEEE double-precision” (Marlow 2010). Our interest is limited to the Haskell-to-JavaScript compiler GHCJS (Stegeman and Mackenzie 2017), which inherits the floating point operation semantics from JavaScript. The ECMA standard (ECMA International 2015) specifies a JavaScript number to be a “double-precision 64-bit binary format IEEE 754-2008 value” – which is luckily already a quite specific specification. We are optimistic that the basic arithmetic operations are deterministic, and this optimism is supported by anecdotal reports from a game developer with Gas Powered Games (Emerson 2009) We have never had a problem with the IEEE standard across any PC CPU, AMD and Intel, with this approach. None of our [. . . ] customers have had problems with their machines either, and we are talking over 1 million customers here. We would have heard if there was a problem with the FPU not having the same results as replays or multi-player mode wouldn’t work at all. However, transcendental functions (exp, sin, cos, log, etc.) are not completely specified by IEEE-754, and different browser/system combinations are allowed to yield slightly different results here. We tested this with a double pendulum simulation, which makes heavy use of sin and cos in every simulation step. The double pendulum is a well-known example of a chaotic system, and we expect it to quickly magnify any divergence in state. Indeed, after running the program on two different browsers (Firefox and Chrome, on the same Linux machine) for several minutes, the simulations take different paths, confirming the worries about these functions. If, however, we use a custom implementation of sin – based on a quadratic curve approximation – the simulation runs consistently. We tested this variant on multiple JavaScript engines (Chrome, Firefox, and Microsoft Edge), on different operating systems (Windows, Linux, Android, and ChromeOS) and on different CPUs (Intel and ARM), and did not uncover any more consistency Proc. ACM Program. Lang., Vol. 1, No. 1, Article 1. Publication date: January 2017. 1:16 Joachim Breitner and Chris Smith Player 1: 1 t 3 Player 2: 2 4 t Fig. 11. Smoothing the effect of late events issues. The tests confirm again that, apart from inconsistent implementations of transcendental functions, basic floating point operations are reliably deterministic in practice. We can deploy a fix to transcendental functions in two ways. In CodeWorld’s educational mode, where we have implemented a custom standard library, it is easy to just substitute new implementations of these functions. In the plain Haskell variant, however, we would like to allow the programmer to make use of existing libraries, which may use standard floating point functions. To achieve this, we can instead replace these operations at the JavaScript level, ensuring that even third-party Haskell libraries are deterministic. In the future, we also plan to automate checks for synchronization problems like this. We cannot directly compare program states in our implementation, since they are of arbitrary type. However, we can compare the generated pictures – or a hash thereof – to achieve essentially the same effect. 4.3 Interpolating the effects of delayed messages Another trick in the game programming toolbox is interpolation to smooth out artifacts that result from corrections to the game state. These artifacts can be clearly seen in Figure 8: The moment the message 2 reaches the first player, the top segment of the growing column abruptly changes from green to red. Similarly, in a game like the tank-fighting game in Figure 10, an opponent can appear to teleport to a new location. In this situation, many games would instead interpolate the position smoothly over a fraction of a second. This can introduce new anomalies of its own, such as characters passing through walls, or tanks moving sideways, but in most cases, it is hoped the result will appear more realistic than the alternative. By providing an API that is completely abstract in the game state, it seems that we have shut the door on implementing this trick. We lack the ability to look inside the state and adjust positions. Surprisingly, though, a form of interpolation is possible. All that is needed is a sort of change of coordinates. While we cannot interpolate in space, we can interpolate in time! When a delayed event arrives, we initially treat it as if its timestamp is “now” and then slide it backward in time over a short interpolation period until it reaches its actual time. Usually, the step function is approximately continuous, and as a result, moving an event backwards in time gives a smooth interpolation in the state as well. This can be seen in Figure 11: After the message 2 arrives at Player 1, the column smoothly changes its color from green to red, from the tip downwards, until the correct state is reached. Like all interpolation, though, anomalies can Proc. ACM Program. Lang., Vol. 1, No. 1, Article 1. Publication date: January 2017. Lock-step simulation is child’s play 1:17 still happen. This scheme introduces abrupt artifacts as we slide a delayed message past another event with a non-commuting effect. In Figure 11 the second player smoothly integrates the delayed 3 message, and the top of the column changes color from blue to yellow. But the moment this event is pushed before the local event 4 , the column abruptly changes its color back to blue. This is an elegant trick to recover the ability to do interpolation. However, it is not clear if interpolation is always the best experience, and a jerky, abrupt update may be preferred for certain games. 4.4 Irreversible updates In some cases, the visual artifacts due to delayed messages, whether smooth or jerky, pose a serious problem. Consider, for example, a card game in which both players click to draw cards from the same deck. Suppose player 1 clicks to draw a card first, but the message from player 1 to player 2 arrives after player 2 clicks as well. For a brief moment before the message is received, player 2 sees the top card, even though it ultimately ends up in the first player’s hand! This is an example of a case where eventual consistency in the game state is not good enough. This problem is hard to avoid, given our constraints and the third requirement of responding immediately to local events. It can be mitigated by the game programmer, by adding a short delay before major events such as those that reveal secrets. The delay can sometimes be creatively hidden by animations or effects. This trick dodges the problem as long as network latency is shorter than this delay, but it provides no guarantee. A complete solution to this problem must involve the programmer in a way that is undesirable in our setting, since only the programmer understands which state changes represent a significant enough event to postpone. 4.5 Lock-step simulation and CRDTs Our approach to lock-step simulation may remind some readers of conflict-free replicated data types (CRDTs), introduced by Shapiro et al. (2011) as a lightweight approach to providing strong consistency guarantees in distributed systems, even in the face of network failure, partition or out-of-order event delivery. These data types come in two forms: “convergent” replicated data types (CvRDTs) are based on transmitting state directly, while “commutative” replicated data types (CmRDTs), are based on transmitting operations that act on that state. Despite the similarity, our game state does not form a CmRDT, as these require that update operations on the game state are commutative. This limits the types of data that can used in such an approach and is inconsistent with our first requirement of supporting arbitrary game state. We find, however, that type Log type defined in Figure 9 forms a CmRDT. The addEvent events from different players commute, as both just add the event to the set. The theory of CRDTs hence provides another argument that the resulting game state is eventually consistent (in fact, strongly so). 5 CONCLUSIONS By implementing lock-step simulation with client prediction generically in the educational programming environment CodeWorld, we have demonstrated once more that that pure functional programming excels at abstraction and modularity. In addition, this work will directly support the education of our next generation of programmers. Proc. ACM Program. Lang., Vol. 1, No. 1, Article 1. Publication date: January 2017. 1:18 Joachim Breitner and Chris Smith ACKNOWLEDGMENTS The first author has been supported by the National Science Foundation under Grant No. CCF1319880 and Grant No. 14-519. We thank Stephanie Weirich, Zach Kost-Smith, Sterling Stein and Justin Hsu for helpful comments on a draft of this paper, as well as the reviewers for their comments. REFERENCES ECMA International. 2015. ECMAScript Language Specification (6th ed.). Geneva. http://www.ecma-international.org/ ecma-262/6.0/ECMA-262.pdf Elijah Emerson. 2009. How to make Box2D more deterministic? (2009). http://www.box2d.org/forum/viewtopic.php?t= 1800&start=10#p16662 Jeff Epstein, Andrew P. Black, and Simon L. Peyton Jones. 2011. Towards Haskell in the cloud. In Proceedings of the 4th ACM SIGPLAN Symposium on Haskell, Haskell 2011, Tokyo, Japan, 22 September 2011, Koen Claessen (Ed.). ACM, 118–129. DOI: http://dx.doi.org/10.1145/2034675.2034690 Ben Lippmeier. 2017. gloss. http://gloss.ouroborus.net/. (2017). Simon Marlow (Ed.). 2010. Haskell 2010 Language Report. Marc Shapiro, Nuno Preguiça, Carlos Baquero, and Marek Zawirski. 2011. A comprehensive study of Convergent and Commutative Replicated Data Types. Research Report RR-7506. Inria – Centre Paris-Rocquencourt ; INRIA. 50 pages. https://hal.inria.fr/inria-00555588 Forrest Smith. 2011. Synchronous RTS Engines and a Tale of Desyncs. (2011). https://blog.forrestthewoods.com/ synchronous-rts-engines-and-a-tale-of-desyncs-9d8c3e48b2be Luite Stegeman and Hamish Mackenzie. 2017. GHCJS. https://github.com/ghcjs/ghcjs. (2017). Mark Terrano and Paul Bettner. 2001. 1500 Archers on a 28.8: Network Programming in Age of Empires and Beyond. In Proceedings of the 15th Games Developers Conference. http://www.gamasutra.com/view/feature/3094/1500 archers on a 288 network .php Proc. ACM Program. Lang., Vol. 1, No. 1, Article 1. Publication date: January 2017. 

Some asymptotic results for fiducial and confidence distributions Piero Veronese 1 and Eugenio Melilli Bocconi University, via Röntgen, 1, 20136, Milan, Italy arXiv:1612.04288v2 [] 17 Oct 2017 [email protected], [email protected] Abstract. Under standard regularity assumptions, we provide simple approximations for specific classes of fiducial and confidence distributions and discuss their connections with objective Bayesian posteriors. For a real parameter the approximations are accurate at least to order O(n−1 ). For the mean parameter µ = (µ1 , . . . , µk ) of an exponential family, our fiducial distribution is asymptotically normal and invariant to the importance ordering of the µi ’s. Keywords: ancillary statistic, confidence curve, coverage probability, natural exponential family, matching prior, reference prior. 1 Introduction Confidence and fiducial distributions, often confused in the past, have recently received a renewed attention by statisticians thanks to several contributions which clarify the concepts within a purely frequentist setting and overcome the lack of rigor and completeness typical of the original formulations. For a wide and comprehensive presentation of the theory of confidence distributions and a rich bibliography we refer the reader to the book by Schweder & Hjort (2016) and to the review paper by Xie & Singh (2013). This latter also highlights the importance of this theory in meta-analysis, see also Liu et al. (2015). For what concerns fiducial distributions Hannig and his coauthors, starting from the original idea of Fisher, have developed in several papers a generalized fiducial inference which is suitable for a large range of situations; see Hannig et al. (2016) for a complete review on the topic and updated references. Given a random vector S (representing the observations or a sufficient statistic) with distribution indexed by η = (θ, λ), where θ is the real parameter of interest, a confidence distribution (CD) for θ is a function C of S and θ such that: i) C(s, ·) is a distribution function on R for any fixed realization s of S and ii) C(S, θ) has a uniform distribution on (0, 1), whatever the true value of η. The second condition is crucial because it implies that the coverage of the intervals derived from C is exact. If it is satisfied only for the sample size tending to infinity, C is an asymptotic CD and the coverage is correct only approximately. Given a CD, it is 1 Corresponding author 1 possible to define the confidence curve ccs (θ) = |1 − 2C(s, θ)|, which displays the confidence intervals induced by C for all levels, see Schweder & Hjort (2016, Sec. 1.6). A fiducial distribution (FD) for a parameter θ has been obtained by several authors starting from a datagenerating equation S = G(U, θ), with U a random vector with known distribution, which allows to transfer randomness from S to θ. In particular Hannig (2009, 2016) derives an explicit expression for the density of a FD which coincides with that originally proposed by Fisher (1930), namely hs (θ) = |∂Fθ (s)/∂θ|, when both θ and S are real and G(U, θ) = Fθ−1 (U ), with Fθ distribution function of S and U uniform in (0, 1). In this paper we consider the specific definition of FD given in Veronese & Melilli (2016), recalled in Section 3, which for a real parameter and a continuous S again simplifies to the Fisher’s formula. In particular, we assume that the FD function is Hs (θ) = 1 − Fθ (s) = Prθ (S > s) (1) with Fθ (s) decreasing and differentiable in θ and with limits 0 and 1 when θ tends to the boundaries of its parameter space. This conditions are always true, for example, if Fθ belongs to a regular real natural exponential family (NEF). This FD is also a CD (asymptotically in the discrete case). For the multi-parameter case a peculiar aspect of our FD is its dependence on the inferential importance ordering of the parameters, similarly to what happens for the objective Bayesian posterior obtained from a reference prior. The connections between our definition and Hannig’s setup are discussed in Veronese & Melilli (2016). In Section 2.1, extending a result proved in Veronese & Melilli (2015) for a NEF, we give a second order asymptotic expansion of our FD/CD in the real parameter case based only on the maximum likelihood estimator (MLE). This expansion does not require any other regularity conditions than the standard ones usually assumed in maximum likelihood asymptotic theory. Furthermore, we show that it coincides with the expansion of the Bayesian posterior induced by the Jeffrey prior. This fact establishes a connection with objective Bayesian inference, whose aim is to produce posterior distributions free of any subjective prior information. In Section 2.2, starting from the well known p∗ -formula of Barndorff-Nielsen (1980, 1983), we propose and discuss a FD/CD which, using an ancillary statistic in addition to the MLE, has good asymptotic behavior. Higher order asymptotics for generalized fiducial distributions have been discussed, at our knowledge, only in the unpublished paper Pal Majumder & Hannig (2016). However, its focus is different being devoted to identify data generating equation with desirable properties. In Section 3 we consider a NEF with a multidimensional parameter and show that, without any further regularity conditions, the asymptotic FD of the mean parameter is normal, it does no longer depend on the inferential ordering of the parameters and coincides with the cor- 2 responding asymptotic Bayesian posterior. Some examples illustrate the good properties and performances of the various proposed FD/CD with emphases on coverage and expected length of confidence intervals. Finally, the Appendix includes the proofs of all the theorems and propositions stated in the paper. 2 Asymptotics for fiducial and confidence distributions: the real parameter case 2.1 An expansion with error of order O(n−1 ) In Veronese & Melilli (2015) an Edgeworth expansion with an error of order O(n−1 ) of the FD/CD for the mean parameter of a real NEF was derived. Here we generalize this result to an arbitrary regular model. Let X = (X1 , . . . , Xn ) be an i.i.d. sample of size n from a density (with respect to the Lebesgue measure) parameterized by θ belonging to an open set Θ ⊆ R. Let θ̂ be the MLE of θ based on X and denote by pθ (θ̂) its density. Let ℓ(θ) = n−1 log pθ (θ̂) and let ℓ′′ (θ̂) and ℓ′′′ (θ̂) be the second and the third derivative of ℓ(θ) with respect to θ, evaluated in θ̂. Then the expected and observed Fisher information of θ̂ (per unit) are I(θ) = −n−1 Eθ (∂ 2 log pθ (θ̂)/∂θ2 ) and −ℓ′′ (θ̂), respectively. Let b = b(θ̂) = −1/ℓ′′ (θ̂). Consider now p Z = n/b (θ − θ̂), which is an approximate standardized version of θ in the FD/CD-setup, and let Hn,θ̂ (z) be its FD/CD derived from the sampling distribution of θ̂. If θ̂ is sufficient, Hn,θ̂ (z) is exact, otherwise it is a natural approximation of the exact one, see e.g. Schweder & Hjort (2016). To prove our result we resort to the expansion of the frequentist probability Prθ (Z ≤ z) provided in Datta & Ghosh (1995) or in Mukerjee & Ghosh (1997). Thus we need the regularity assumptions used in these papers, see also Ghosh (1994, Ch. 8) and Bickel & Ghosh (1990) for a precise statement. Notice that the conditions required for the frequentist expansion of the distribution of the MLE are rarely reported in a rigorous way in books and papers. However, what is important here is that, in order to prove our result, we do not need any further assumption and this fact allows an immediate and fair comparison between MLE- and FD/CD-asymptotic theory. Theorem 1 Let X be an i.i.d. sample of size n from a density pθ , θ ∈ Θ ⊆ R. Then, under the regularity p assumptions cited above, the distribution function Hn,θ̂ (z) of the FD/CD for Z = n/b(θ−θ̂) has the expansion  1 3/2 ′′′ 2 Hn,θ̂ (z) = Φ(z) − φ(z) b ℓ (θ̂)(z − 1) n−1/2 + O(n−1 ). 6  (2) If pθ also satisfies the conditions for the expansion of a Bayesian posterior, see e.g. Johnson (1970, Theorem 2.1, with K = 1), we have the following 3 Corollary 1 In the same setting of Theorem 1, let π J (θ) ∝ I(θ)1/2 be the Jeffreys prior for θ. If π J is improper, assume that there exists an n0 ≥ 1 such that the posterior distribution π J (θ|z) of θ is proper for n ≥ n0 , almost surely for all θ. Then the expansion of π J (θ|z) coincides with that of Hn,θ̂ (z) given in (2). Theorem 1 and Corollary 1 confirm the idea that the Jeffreys posterior is really free of any subjective prior information. Furthermore, they naturally establish a connection between FD/CD-theory and matching priors, i.e. priors that ensure approximate frequentist validity of posterior credible sets. More precisely, a prior π for which Prθ (θ ≤ q1−α (X, π)) = 1 − α + o(n−r/2 ), where q1−α (X, π) denotes the (1 − α)th posterior quantile of θ, is called a matching prior of order r, see Datta & Mukerjee (2004) for a general review and references. For a regular model indexed by a real parameter it is well known, see Datta & Mukerjee (2004, Theorem 2.5.1), that the Jeffreys prior π J is the unique first order matching prior and is also a second order matching prior if and only if the model satisfies the following condition: I(θ)−3/2 Eθ [(∂ℓ(θ)/∂θ)3 ] is a constant free of θ. (3) Veronese & Melilli (2015) study the existence of a prior (named fiducial prior ) which induces a Bayesian posterior coinciding with the FD, extending a result given by Lindley (1958) for a continuous univariate sufficient statistic, see also Taraldsen & Lindqvist (2015) for a generalization to multivariate group models. Because a FD/CD realizes the exact matching, we immediately have the following Corollary 2 If a fiducial prior π F exists, then it coincides with the Jeffreys prior π J . Furthermore, the condition (3) is necessary for the existence of π F . Notice that for a model belonging to a NEF, with mean parameter µ and variance function V (µ), condition (3) becomes: “2V ′ (µ)V (µ)−1/2 is constant”. The solution of this differential equation is V (µ) = (c1 µ + c2 )2 , i.e. a fiducial prior for a parameter of a NEF may exist only if its variance function is quadratic. This result was found for the first time in Veronese & Melilli (2015), using a totally different approach. Example 1 (Exponential distribution). Let (X1 , . . . , Xn ) be an i.i.d. sample from an exponential distribution with mean µ. The MLE µ̂ of µ is the sample mean and b = b(µ̂) = µ̂2 . Then the expansions of the √ √ FD/CD for Z = ( n/µ̂)(µ − µ̂) in (2) and that of the standardized MLE W = −Z = ( n/µ̂)(µ̂ − µ), see (A.3), are respectively   Φ(z) − φ(z) 2(z 2 − 1)/3 n−1/2 + O(n−1 ),   Φ(w) − φ(w) −(2w2 + 1)/3 n−1/2 + O(n−1 ). 4 (4) (5) 10 8 6 2 4 Expected length 12 0.92 0.90 Coverage 0.88 0 0.86 0 5 10 15 0 µ 5 10 15 µ Figure 1: Coverages and expected lengths for the 90% intervals with n = 15 based on: exact FD/CD (red), Normal approximation (green), expansions of FD/CD (black) and of MLE (blue). It follows that the confidence intervals obtained from (4) and (5) are different, contrary to what happens for those based only on the normal approximation. Their coverages and expected lengths are reported in Figure 1 for a sample of size n = 15 and confidence level 0.9. Notice that the coverage of the FD/CD-intervals is much closer to the nominal level than that of the intervals based on the MLE, while the expected lengths are quite similar. For the sake of comparison Figure 1 reports also the coverage and the expected length of the intervals based on the exact FD/CD, which is an inverse-gamma(n, nµ̂), see Veronese & Melilli (2015, Tab.1). The latter intervals are clearly exact, but wider. Finally, by Corollary 1, the expansion (4) coincides with that of the Jeffreys posterior. It is easy to verify, according to Corollary 2, that the fiducial prior exists and coincides with π J (µ) ∝ 1/µ. ⋄ Example 2 (Fisher’s gamma hyperbola-Nile problem). Fisher (1973, Sec.VI.9) considers a sample of size n from a curved exponential family obtained by two independent gamma distributions with means constrained on an hyperbole. Following Efron & Hinkley (1978), we directly start with the sufficient statistic S = (S1 , S2 ), with S1 and S2 distributed according to ga(n, e−η ) and ga(n, eη ), respectively. Here ga(α, β) denotes a gamma distribution with shape parameter α and mean α/β. It follows that the likelihood of the model is Lη (s) = exp{−e−η s1 − eη s2 } and that the MLE of η is η̂ = (1/2) log(S1 /S2 ). Even if an exact inference on η cannot be performed using only η̂, the minimal sufficient statistic S is indeed bivariate, an asymptotic FD/CD for η, based on η̂, can be easily obtained from Theorem 1. Since ℓ′′′ (η̂) = 0, it follows from (2) that, in this case, the √ normal distribution N(η̂, b/n), with b = −1/ℓ′′(η̂) = n/(2 s1 s2 ), is an approximate FD/CD of η with error of order O(n−1 ). Figure 2 reports the plot of its density compared with the exact FD/CD based on S, which will be derived in the next section. It shows the goodness of the approximation even for a very small sample size (n = 5 in the plot). Finally, it is easy to check that Eη [(∂ℓ(η)/∂η)3 ] = 0, thus condition (3) holds and, by Corollary 2, a fiducial prior might exist for this model. Indeed it exists and we will find it in the next section.⋄ Another criterium to define matching priors studied in the Bayesian literature is based directly on the distribution functions, see Datta & Mukerjee (2004, Sec. 3.2). Because Hn,θ̂ (z) is stochastic in a frequentist setup, as it occurs for a posterior distribution, we can consider the matching between Eθ (Hn,θ̂ (z)) = 5 Figure 2: Approximate (black) and exact(red) fiducial densities for η in the Fisher’s gamma hyperbola for a sample size n = 5, s1 = 17.321, s2 = 0.116. np  np o o n/b(θ − θ̂) ≤ z n/b(θ − θ̂) ≤ z . Clearly quantiles and distribution functions are Eθ Prθ̂ and Prθ strongly connected and thus it is not surprising that the conditions for the existence of matching priors in the two criteria are related. Indeed, the first order matching conditions are the same, while this is not true for the second order ones. Notice that the matching in terms of quantiles is obtained using the quantity Z which can be seen as an approximate pivotal quantity. This is meaningful in an asymptotic setting, but it is not appropriate for small sample sizes. In this case, the FD/CD realizes an exact matching if we replace Z with the pivotal quantity given by the distribution function of θ̂, namely Fθ (θ̂). Indeed, we have  
Eθ Prθ̂ {Fθ (θ̂) ≤ z} = Eθ Prθ̂ {1 − Hθ̂ (θ) ≤ z} =

Eθ Prθ̂ {Hθ̂ (θ) ≥ 1 − z} = 1 − Eθ Prθ̂ {Hθ̂ (θ) ≤ 1 − z} = 1 − Eθ (1 − z) = z, and because Prθ {Fθ (θ̂) ≤ z} = z, the exact matching for distribution functions holds. However, an exact FD/CD does not always exist and thus it is natural to look for approximations which have nice asymptotic properties. Furthermore, in a multiparameter case quantiles are not well defined and thus the study of the frequentist properties of a multivariate FD/CD can be conducted along the lines developed for matching distribution functions. 2.2 An approximation based on the Barndorff-Nielsen p∗ -formula Consider a sample X whose distribution depends on a real parameter θ. In the previous section we have obtained an approximate FD/CD for θ starting from the distribution of the MLE θ̂. However, if θ̂ is not sufficient, the approximation of the FD/CD can be improved adding the remaining information included in the sample. This can be done resorting to the “conditionality resolution” of the statistical model, i.e. the construction of an ancillary statistic A and of an approximate conditional distribution of θ̂ given A = a. We refer to Barndorff-Nielsen (1980, 1983) for a detailed discussion on the topic and recall here only some useful facts. His well known approximate distribution of θ̂ given A = a is p∗θ (θ̂|a) = c(a, θ)|j(θ̂)|1/2 L(θ; x)/L(θ̂; x), 6 (6) where L(θ; x) is the likelihood function, j(θ̂) is the observed Fisher information and c(a, θ) is the normalizing constant which does not depend on θ in many important cases. Formula (6) is quite simple, is generally accurate to order O(n−1 ), or even O(n−3/2 ), and exact in specific cases. Here the term approximation refers to one of the two following situations: i) there exists an ancillary statistic A, but it is not possible to construct the exact conditional distribution of θ̂ given A = a; ii) an exact ancillary statistic does not exist and an approximate one is used. It is worth to remark that formula p∗ is invariant to reparameterizations and is exact for transformation models. Furthermore, under repeated sampling from a real NEF, where no conditioning is involved, p∗ is often of order O(n−3/2 ) and is exact for normal (known variance), gamma (known shape) and inverse-gaussian (known shape) distributions. If Fθ∗ (θ̂|a) denotes the distribution function corresponding to p∗θ (θ̂|a) and satisfies the conditions reported after (1), we can derive an approximate FD/CD for θ as h∗x (θ) = |∂Fθ∗ (θ̂|a)/∂θ|. This construction of a FD/CD is not different in essence from the widespread procedure used to derive a Bayesian posterior starting from an approximate (e.g. profile, pseudo or composite) likelihood. A similar approach based on approximate likelihood is used also by Schweder & Hjort (2016) to construct a CD. The next result concerning a real NEF is useful when the exact distribution of θ̂ is difficult to obtain. Proposition 1 If θ̂ is the MLE of θ based on an i.i.d. sample from a real regular NEF, with density pθ (x) = exp{θx − M (θ)}, then h∗θ̂ (θ) = |∂Fθ∗ (θ̂)/∂θ| is an exact FD/CD for θ based on p∗θ (θ̂). It is an approximate FD/CD based on the whole sample and its order of approximation depends on that of p∗θ (θ̂). The following examples, concerning curved exponential families, i.e. NEFs in which a constraint on the natural parameter space is imposed, illustrate another typical case in which formula (6) can be fruitfully applied to construct a FD/CD. Example 2 ctd. As previously observed, the MLE η̂ is not sufficient and thus the exact FD/CD can be √ obtained starting from the conditional distribution of η̂ given the ancillary statistic A = S1 S2 /n, proposed by Fisher (1973, Sec. VI.10-11). After some calculations, one obtains pη (η̂|a) = exp{−2na cosh(η̂ − η)}/(2K0 (2na)), where K0 (w) = R∞ 0 (7) exp{−w cosh(z)}dz is the modified Bessel function of the second order evaluated in (0, w). As observed by Efron & Hinkley (1978), it is easy to see from (7) that this example involves a translation (and thus a transformation) model, so that pη (η̂|a) = p∗η (η̂|a). Thus the exact FD for η is hη̂,a (η) = −∂Fη∗ (η̂|a)/∂η and, because η is a location parameter, it equals the posterior obtained from the Jeffreys prior π J (η) ∝ 1, see Veronese & Melilli (2016, Prop.8). The nature of the parameter η also implies that inferences based on MLE 7 Figure 3: Confidence curves for a sample size n = 15, generated from ρ = 0.3 with s1 = 19.248 and s2 = 4.827, r = 0.414 and ρ̂ = 0.209. Left graph: ccr (green), ccrstab (brown), cc0 (blue) and cc1 (red). Right graph: cc1 (red), cc∗ (black), ccJ (orange). The horizontal line identifies the 95% confidence intervals. and hη̂,a (η) coincide. ⋄ Example 3 (Bivariate normal model). Consider an i.i.d. sample (Xi , Yi ), i = 1, . . . , n, from a bivariate normal distribution with expectations 0, variances 1 and correlation coefficient ρ. This is a simple curved Pn Pn exponential model with sufficient statistics S1 = i=1 (Xi2 + Yi2 )/2 and S2 = i=1 Xi Yi , but the inference on ρ is a challenging problem as shown in Fosdick & Raftery (2012) and Fosdick & Perlman (2016). Both Efron & Hinkley (1978) and Barndorff-Nielsen (1980) use this example to illustrate the construction of an approximate ancillary statistic in a conditional inference setting. Their proposals essentially coincide and lead p to consider the “affine” ancillary A = (S1 − n)/ n(1 + ρ̂2 ), where ρ̂ is the MLE of ρ. To discuss the performance of h∗ obtained starting from p∗ , we compare it with other possible asymptotic FDs and with the Bayesian posterior obtained from the Jeffreys prior π J (ρ) ∝ (ρ2 + 1)1/2 /(1 − ρ2 ). In particular, we consider the following FDs: hr and hrstab obtained from the sample correlation coefficient r and its stabilizing transformation which, as well known, improves the inferential performance of r, see also Schweder & Hjort (2016, pag. 209 and 224); h0 and h1 obtained considering the first one or the first two terms of (2), respectively. We assume a sample size n = 15 because a larger value of n, e.g. 50, produces essentially the same (good) results for all choices. The left graph of Figure 3 reports an example of the confidence curves ccr , ccrstab , cc0 and cc1 corresponding to the previous FDs. The curves present different behaviors because they are based on the two estimators r and ρ̂ of ρ, which assume quite different values in the sample. The right graph compares cc1 with cc∗ and ccJ obtained from h∗ and Jeffreys posterior, respectively. As expected, the last two curves, both based on the sufficient statistics S1 and S2 , are very similar and induce confidence intervals narrower than those induced by cc1 . To better appreciate the good behavior of h∗ , we compare the corresponding coverage and expected length with those of hr , hrstab and the Jeffreys posterior. Figure 4 confirms the very bad inferential performance of hr . The intervals corresponding to h∗ have the coverage closest to the nominal one, while those obtained by hrstab present an over-coverage. However, these latter intervals have a uniformly larger expected length. Finally, Bayesian intervals show an intermediate behavior in terms of both coverage and expected length. The same example is discussed by Pal Majumder & Hannig 8 1.0 0.8 0.6 expected length 0.4 0.2 coverage 0.86 0.88 0.90 0.92 0.94 0.96 0.98 0.0 0.2 0.4 0.6 0.8 0.0 0.2 ρ 0.4 0.6 0.8 ρ Figure 4: Coverages and expected lengths of the 95% intervals with n = 15 based on: h∗ (black), hrstab (brown), π J (orange) and hr (green). (2016), but they have a different aim and consider different FDs. 3 Asymptotics for fiducial distributions: the multidimensional parameter case For a parameter θ in Rd , inspired by the step-by-step procedure proposed by Fisher (1973), Veronese & Melilli (2016) give a simple and quite general definition of FD, which we summarize here. We refer to the latter paper for details, examples, relationships with objective Bayesian analysis performed using reference priors and a comparison with Hannig’s fiducial approach. Notice that for a multidimensional parameter there is not a unique definition of CD, see Schweder & Hjort (2016, Ch.9), so that in the following we refer only to FDs. Given a random vector S, representing the sample or a sufficient statistic, with dimension m ≥ d and density pθ , consider the partition S = (S[d] , S−[d] ), where S[d] = (S1 , . . . , Sd ) and S−[d] = (Sd+1 , . . . , Sm ), and suppose that S−[d] is ancillary for θ. Clearly, if d = m, S−[d] disappears. Thus, the density pθ of S can be written as pθ (s[d] |s−[d] )p(s−[d] ) and the information on θ provided by the whole sample is included in the conditional distribution of S[d] given S−[d] . Assume now that there exists a one-to-one smooth reparameterization from θ to φ = (φ1 , . . . , φd ), with the φi ’s ordered with respect to their inferential importance, such that pφ (s[d] |s−[d] ) = d Y k=1 pφd−k+1 (sk |s[k−1] , s−[d] ; φ[d−k] ), (8) with obvious meaning for s[0] and φ[0] . If, for each k, the one-dimensional conditional distribution function of Sk is monotone and differentiable in φk and has limits 0 and 1 when φk tends to the boundaries of its parameter space (this is always true, for example, if this distribution belongs to a regular real NEF), it is possible to define the joint fiducial density of φ as hs (φ) = d Y k=1 hs[k] ,s−[d] (φd−k+1 |φ[d−k] ), 9 (9) where hs[k] ,s−[d] (φd−k+1 |φ[d−k] ) = ∂ Fφ (sk |s[k−1] , s−[d] ; φ[d−k] ) ∂φd−k+1 d−k+1 (10) is inspired by the definition of the FD for a real parameter. Some remarks useful in the sequel follow. i) When m = d = 1, so that an ancillary statistic is not needed, formulas (9) and (10) reduce to hs (φ) = |∂Fφ (s)/∂φ|, the original proposal of Fisher (1930). ii) When d > 1 but the parameter of interest is φ1 only, it follows from (9) that its FD is simply given by hs (φ1 ) = ∂ Fφ (sd |s[d−1] , s−[d] ) , ∂φ1 1 which is based on the whole sample and is also a CD. A typical choice for Sd is given by the MLE φb1 of φ1 and thus, when φb1 is not sufficient, one has to consider the distribution of φb1 given the ancillary statistic s−[d] as done in Section 2.2. iii) The FD in (9) is generally not invariant under a reparameterization of the model unless the transformation from φ to λ = (λ1 , . . . , λd ) say, maintains the same increasing order of importance in the components of the two vectors and λk is a function of φ1 , . . . , φk , for each k = 1, . . . , d, i.e. φ(λ) is a lower triangular transformation. In Veronese & Melilli (2015) it is shown that the univariate FD/CD for a real NEF is asymptotically normal. Because the multivariate FD defined in (9) is a product of one-dimensional conditional FDs, it is quite natural to expect that also the FD for a d-dimensional NEF is asymptotically normal. Theorem 2 Let X = (X1 , . . . , Xn ) be an i.i.d. sample from a regular NEF on Rd with Xi having density Pd pθ (xi ) = exp{ k=1 θk xk − M (θ)}, mean vector µ = µ(θ) and variance function V(µ) = Varµ (Xi ). FurtherPn more, let x̄ be the observed value of the sample mean X̄ = n−1 i=1 Xi . If Xi admits bounded density with respect to the Lebesgue measure or is supported by a lattice, then the fiducial distribution of µ is asymptotically order-invariant and asymptotically normal with mean x̄ and covariance matrix V(x̄)/n. Since V(x̄) coincides with the reciprocal of both the observed and the estimated expected Fisher information matrix, recalling standard results about asymptotic Bayesian posterior distributions, see e.g. Johnson & Ladalla (1979), the following corollary immediately holds. Corollary 3 Consider the statistical model specified in Theorem 2. If we assume a positive prior for µ having continuous first partial derivatives, then the asymptotic Bayesian posterior for µ coincides with the asymptotically normal fiducial distribution. The asymptotic normality for multidimensional generalized fiducial distributions has been proved by Sonderegger & Hannig 10 (2014) under a set of regularity assumptions. We remark that the previous two results are specific for our definition of FD and hold for NEFs without any extra regularity condition. Furthermore, the proof of Theorem 2, given in the Appendix, is completely different from the standard ones used to show asymptotic normality in frequentist, Bayesian or generalized fiducial settings. It is based on the convergence of the conditional distributions determined by the importance ordering of the parameters, it heavily relies on the properties of the mixed parametrization of the NEF and consequently the result is given in terms of the mean parameter, which is more interpretable than the natural one. Consider now a parameter λ = g(µ), with g a one-to-one lower triangular continuously differentiable function. From Veronese & Melilli (2016, Prop. 1), it follows that the FD for λ can be obtained from that for µ by the standard change of variable technique and thus we can construct the asymptotic FD in the same way. However, Theorem 2 states that the asymptotic FD for µ is order invariant and hence it could be interesting to investigate if this is true also for an arbitrary parameter. This conjecture might be reasonable looking at what happens in the Bayesian theory where the asymptotic (reference) posteriors do not depend on the order of the parameter components. The following example illustrate this point. Example 4. Consider a sample of size n from a multinomial experiment with outcome probability vector P P p = (p1 , . . . , pd ), with dk=1 pk ≤ 1. Then, the vector of counts S = (S1 , . . . , Sd ), with dk=1 Sk ≤ n, is distributed according to a multinomial distribution with parameters n and p. Using the step-by-step procedure described above, Veronese & Melilli (2016, formula 25) have proved that the FD for p is a generalized Dirichlet distribution which depends on the specific fixed ordering of the pi ’s. Assume now d = 2 and consider the transformation φ1 = p1 /p2 and φ2 = p2 which is not lower triangular. The FD of φ = (φ1 , φ2 ) in this order is s −1/2 hs (φ) ∝ φ11 s +s2 −1/2 (1 + φ1 )−1/2 φ21 (1 − (1 + φ1 )φ2 )n−s1 −s2 −1/2 . (11) This latter is different from the FD induced by that of p but coincides with the posterior distribution obtained from the reference prior for φ, see Veronese & Melilli (2016, Sec. 5.4). Consider now the asymptotic setting. From Theorem 2 it follows that the asymptotic FD of p = (p1 , . . . , pd ) is N(x̄, V(x̄)/n), with x̄ = s/n and where the elements of V(x̄) are vkk = x̄k (1 − x̄k ) and vkr = −x̄k x̄r , k 6= r. It is easy to verify that for d = 2 it induces on φ a normal distribution with means x̄1 /x̄2 , x̄2 , variances x̄1 (x̄1 + x̄2 )/(nx̄32 ), x̄2 (1− x̄2 ) and covariance −x̄1 /x̄2 . This distribution coincides with the asymptotic distribution corresponding to (11) ( derived for example using standard results on Bayesian theory) and this fact supports our conjecture that asymptotic FDs are invariant to the importance ordering of the parameters and can always been derived through the standard delta method. 11 Appendix Proof of Theorem 1. For the sake of clearness, in this proof we denote by Θ̂ the MLE of a parameter θ and by θ̂ the corresponding estimate. If Fθ (θ̂) is the distribution function of Θ̂, assumed decreasing in θ, let 1 − Fθ (θ̂) be the FD for θ. If Fθ (θ̂) is increasing the proof is similar with 1 − Fθ (θ̂) replaced by Fθ (θ̂). Then Hn,θ̂ (z) = Prθ̂ where θn = z np o n/b(θ − θ̂) ≤ z = Prθ̂ {θ ≤ θn } = 1 − Prθn {Θ̂∗n ≤ θ̂}, (12) p ∗ b/n + θ̂ and Θ̂∗n is the MLE based on n i.i.d. random variables Xn,i , i = 1, . . . , n, belonging to the same family of distributions of Xi , but with parameter θn . Note that θn converges to θ for n → ∞, because θ̂ converges to the “true” value θ for almost all sequences (x1 , x2 , . . .) and Θ is an open interval. Thus θn belongs to Θ for n large enough and for each z ∈ R. Starting from (12), we can also write p p Hn,θ̂ (z) = 1 − Prθn { n/b(Θ̂∗n − θn ) ≤ n/b(θ̂ − θn )} p p = Prθn { n/b(Θ̂∗n − θn ) ≥ −z} = Prθn { n/b(θn − Θ̂∗n ) ≤ z}. Thus, the asymptotic expansion of Hn,θ̂ (z) can be derived by expanding the frequentist distribution function of p ∗ n/b(θn − Θ̂∗n ). This expansion can be directly obtained by standard results, even if {Xn,i , i = 1, 2, . . . , n; n = 1, 2, . . . , } is a triangular array because we consider only random variables and a first order approximation, see p e.g. Garcı́a-Soidán (1998) and Petrov (1995, Theorem 5.22). The frequentist expansion of Z = n/b(θ − Θ̂) has been provided in several papers about matching priors under a set of regularity assumptions. Using formula (3.2.3) in Datta & Mukerjee (2004) with θ = θn and recalling that Θ̂∗n is the MLE of θn , we obtain Prθn {     p 1 I ′ (θn ) 1 ℓ′′′ (θn ) 1 1 2 √ + O( ). n/b(θn − Θ̂∗n ) ≤ z} = Φ(z) − φ(z) E (z + 2) + θ 2 I(θn )3/2 6 n (−ℓ′′ (θn ))3/2 n n Now, because −ℓ′′ (θ̂)−I(θ̂) = Op (n−1/2 ) (see e.g. Severini, 2000, Sec. 3.5.3) and θn − θ̂ = z (13) p b/n = Op (n−1/2 ), we have I(θn ) = −ℓ′′ (θ̂) + Op (n−1/2 ) = 1/b + Op (n−1/2 ). Moreover, applying the delta method to the expectation in (13), this expansion becomes   p 1 1 Prθn { n/b(θn − Θ̂∗n ) ≤ z} = Φ(z) − φ(z) − b3/2 ℓ′′′ (θ̂) + b3/2 ℓ′′′ (θ̂)(z 2 + 2) n−1/2 + O(n−1 ) 2 6   1 = Φ(z) − φ(z) b3/2 ℓ′′′ (θ̂)(z 2 − 1) n−1/2 + O(n−1 ), 6 and the theorem is proved. ⋄ 12 Proof of Corollary 1. The result follows immediately using the expansion of the posterior distribution provided by Johnson (1970, Theorem 2.1 and formulae (2.25) and (2.26)), assuming π J (θ) ∝ I(θ)1/2 as prior. Notice that under the stated conditions on the posterior, this result can be used even if the prior is improper, as observed in Ghosh et al. (2006, pag. 106). ⋄ Proof of Proposition 1. Recalling that for a real NEF x̄ = M ′ (θ̂), we can write p∗θ (θ̂) = exp{n(θM ′ (θ̂) − M ∗ (θ))}, R where M ∗ (θ) = log( exp{n(θM ′ (θ̂))}dν(θ̂)), with ν(θ̂) denoting the dominating measure of the density of θ̂. Thus p∗θ (θ̂) belongs to a regular real NEF and the result follows immediately by Veronese & Melilli (2015, Theorem 1). ⋄ Proof of Theorem 2. Given a square d × d matrix A, we use Ak[r] to denote the vector of the first r elements of the k-th row of A and A[k][k] to denote the matrix identified by the first k rows and columns of A. Moreover, AT denotes the transpose of A.and In order to determine the asymptotic FD of µ we apply the step-by-step procedure introduced in Section 3 to the conditional distribution of X̄k given X̄[k−1] = x̄[k−1] for each k. Clearly for k = 1, we have the marginal distribution of X̄1 . Since the covariance matrix V(µ) of Xi is finite, by the central limit theorem X̄ is asymptotically N(µ, n−1 V(x̄)) and thus the marginal distribution of X̄[k] is also asymptotically normal with E(X̄[k] ) = µ[k] and V ar(X̄[k] ) = n−1 V(x̄)[k][k] . Let  −1 λk = µk + V(x̄)k[k−1] V(x̄)[k−1][k−1] (x̄[k−1] − µ[k−1] ) (14)  −1  T qk = V(x̄)kk − V(x̄)k[k−1] V(x̄)[k−1][k−1] V(x̄)k[k−1] . (15) and Using known results about the convergence of conditional distributions, see Steck (1957, Theorem 2.4) or Barndorff-Nielsen & Cox (1979, Sec.4), it follows that the conditional distribution of X̄k given X̄[k−1] = x̄[k−1] is asymptotically N(λk , n−1 qk ). Now recall that for a NEF it is always possible to consider the so called “mixed parameterization” (µ[k] , θ−[k] ) which is one-to-one with the natural parameter θ, see e.g. Brown (1986, ch. 3). For θ−[k] fixed, the distribution of X̄[k] belongs to a NEF with parameter θ [k] and thus the conditional distribution of X̄k given X̄[k−1] = x̄[k−1] depends only on θk . The same must be true of course for the corresponding asymptotic distribution, so that its mean parameter λk depends only on θk and hence only on µk . Considering now 13 the alternative mixed parameter (µ[k−1] , θk , θ −[k] ), it follows that there exists a one-to-one correspondence between θk and µk , for µ[k−1] and θ−[k] fixed. As a consequence µ[k−1] can be fixed arbitrarily in the mixed parameterizations (µ[k−1] , θk , θ−[k] ) with no effect on the conditional distribution and we specifically assume µ[k−1] = x̄[k−1] . Using the parameter (x̄[k−1] , µk , θ−[k] ), we have that λk coincides with µk , see (14). Summing up, each of the three parameters λk , θk and µk represents a possible parameterization of the asymptotic conditional distribution of X̄k given X̄[k−1] = x̄[k−1] , for fixed x̄[k−1] and θ−[k] . Thus we can find the asymptotic FD of λk . Consider now a random vector X̄∗ with distribution belonging to the same family of that of X̄, √ with mixed parameter (x̄[k−1] , µ∗k , θ −[k] ), where µ∗k = x̄k + zk / n, with zk ∈ R, as in the proof of Theorem 1. Notice that the marginal distributions of X̄∗[k−1] and of X̄[k−1] are equal. Such a µ∗k is well defined for large n since (x̄[k−1] , x̄k , θ−[k] ) is a possible value for the mixed parameter in the distribution of the whole vector, because the NEF is regular and thus the parameter space is open. For n varying and fixed k, the sequence of marginal sample means X̄∗[k] derives from random vectors whose mean parameter depends on n, so that it forms a triangular array. In order to determine the FD of λk , we can √ consider the quantity n(λk − x̄k ), which is a sort of standardization of λk in our fiducial context. Using (1), similarly to what done in (12), we can write Prx̄k   zk λk ≤ x̄k + √ X̄∗ [k−1] = x̄[k−1] , θ−[k] n
∗ (16) = 1 − Prλ∗k X̄k ≤ x̄k |X̄∗ [k−1] = x̄[k−1] , θ−[k] ,
√ n(λk − x̄k ) ≤ zk |X̄∗ [k−1] = x̄[k−1] , θ−[k] = Prx̄k √ where λ∗k = x̄k + zk / n. Since V ar(X̄∗[k] ) is a continuous function of µ∗ = E(X̄∗ ), it converges to a positive definite matrix for each k when µ∗ converges to the “true” value of µ, for n → ∞. Then, using the result on the convergence of a conditional distribution presented at the beginning of the proof with µ replaced by µ∗ , we have that X̄k∗ given X̄[k−1] = x̄[k−1] is asymptotically N(λ∗k , qk /n). Notice that from the existence of the second moment of each component of X̄∗[k−1] , it follows that the condition required by Steck (1957, Theorem 2.4, formula (28)), for the case of triangular arrays, is satisfied. Thus, the asymptotic normality of X̄k∗ given X̄[k−1] = x̄[k−1] implies, for n → +∞,
sup Prλ∗k X̄k∗ ≤ x̄k |X̄[k−1] = x̄[k−1] , θ −[k] − Φ zk r  n (x̄k − λ∗k ) → 0 a.s. qk Recalling the expression of λ∗k , we obtain
√ sup Prx̄k +zk /√n X̄k ≤ x̄k |X̄[k−1] = x̄[k−1] , θ−[k] − Φ (−zk / qk ) → 0 zk 14 a.s. which, using (16), gives sup Prx̄k zk
√ √ n(λk − x̄k ) ≤ zk |X̄[k−1] = x̄[k−1] , θ−[k] − Φ (zk / qk ) → 0 a.s. We can conclude that the conditional FD of λk given θ−[k] is asymptotically normal with mean x̄k and variance n−1 qk , and thus it does not depend on θ−[k] . Recalling the one-to-one correspondence between θk and λk , for fixed θ−[k] , and in particular that λd is a one-to-one function of θd , it follows that λ1 , λ2 , . . . , λd are asymptotically independent, so that the full vector λ = (λ1 , λ2 , . . . , λd ) is asymptotically N(x̄, n−1 Q(x̄)), where Q(x̄) is the diagonal matrix with k-th element qk . To obtain the asymptotic FD of µ we consider the one-to-one lower-triangular transformation µ = g(λ), with λ = g−1 (µ) given by (14) for k = 1, . . . , d. Consider now the lower d × d triangular matrix A = A(x̄) whose k-th row is made up by the vector −V(x̄)k[k−1] [V(x̄)[k−1][k−1] ]−1 , in the first k−1 positions, 1 in the k-th position and 0 elsewhere. Thus we can write λ = Aµ + (I − A)x̄ and µ = A−1 λ + (I − A−1 )x̄, with I denoting the identity matrix of order d. By applying the Cramér delta method it follows that µ is asymptotically normal with (asymptotic) mean and covariance matrix A−1 x̄ + (I − A−1 )x̄ = x̄ and n−1 A−1 Q(x̄)A−1 T , respectively. We now show that A−1 Q(x̄)A−1 T = V(x̄) or, equivalently, Q(x̄) = AV(x̄)AT . By direct computation it is easy to see that the (k, h)-th element of AV(x̄), k, h = 1, 2, . . . , d, is T V(x̄)kh − V(x̄)k[k−1] [V(x̄)[k−1][k−1] ]−1 V(x̄)h[k−1] . (17) Notice that (17) is 0 for k > h because the product of its last two factors gives a (k −1)-dimensional vector with 1 in the h-th position and 0 otherwise. The matrix AV(x̄)AT is of course symmetric, so that it is sufficient to proceed only for k ≥ h. On its diagonal we have T V(x̄)kk − V(x̄)k[k−1] [V(x̄)[k−1][k−1] ]−1 V(x̄)k[k−1] , k = 1, . . . , d, (18) because the only nonzero element in the product of the k-th row of AV(x̄) and the k-th column of AT is the product of (17), with h = k, and 1. For k > h, the (k, h)-th element of AV(x̄)AT is 0, because the first k − 1 components of the k-th row of AV(x̄) and the last d − h components of the h-th column of AT are zero. Thus the matrix AV(x̄)AT coincides with Q(x̄) and this completes the proof of the theorem. 15 ⋄ Acknowledgments This research was supported by grants from Bocconi University. References Barndorff-Nielsen, O. (1980). Conditionality resolutions. Biometrika, 67, 293–310. Barndorff-Nielsen, O. & Cox, D. R. (1979). Edgeworth and saddle-point approximation with statistical applications. J. R. Stat. Soc. Ser. B, 41, 279–312. Bickel, P. J. & Ghosh, J. (1990). A decomposition for the likelihood ratio statistic and the bartlett correction–a bayesian argument. 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A Generalization of the maximal-spacings in several dimensions and a convexity test. Catherine Aarona , Alejandro Cholaquidisb , Ricardo Fraimanb a Université Blaise-Pascal Clermont II. b Universidad de la República. arXiv:1411.2482v2 [] 5 May 2016 Abstract The notion of maximal-spacing in several dimensions was introduced and studied by Deheuvels (1983) for data uniformly distributed on the unit cube. Later on, Janson (1987) extended the results to data uniformly distributed on any bounded set, and obtained a very fine result, namely, he derived the asymptotic distribution of different maximal-spacings notions. These results have been very useful in many statistical applications. We extend Janson’s results to the case where the data are generated from a Hölder continuous density that is bounded from below and whose support is bounded. As an application, we develop a convexity test for the support of a distribution. Key words:maximal spacing, convexity test, non-parametric density estimation. 1 Introduction The notion of spacings, which for one dimensional data are just the differences between two consecutive order statistics, have been extensively studied in the one dimensional setting; see, e.g., the review papers [18, 19]. Many important applications to testing and estimation problems have been derived from the study of the asymptotic behaviour of the spacings. Applications to testing problems date back to [20], who address the asymptotic theory of a class of tests for Increasing Failure Rate. For estimation problems, [21] propose the maximum spacing estimation method to estimate the parameters of a univariate statistical model. In the multidimensional case, several different notions of maximal-spacing have been proposed. Most of them are based on the nearest-neighbors balls (see for instance [14] or [2]) or on the Voronoi tessellation [16], (a comparison can be found in [22]), but they do not capture the key idea of ‘largest set missing the observations’. In contrast, this is the case with the different and global notion proposed in [7], and generalized in [12]. In [7], the notion of maximal-spacing is defined and studied Q for iid data uniformly distributed in [0, 1]d as the maximal length a of a cube C = [xi , xi + a], included in [0, 1]d that does not contain any of the observations. This notion has been extended in [12], in which the uniformity assumption remains but the support of the distribution is no longer assumed to be [0, 1]d but may be any compactum S. Moreover, C is allowed to be any compact and convex set. Finally, while in [7] only bounds are given, in [12] the asymptotic distribution for the maximal spacing is provided. The notion of maximal multivariate spacing, and in particular Janson’s result, has been used to solve different statistical problems. In set estimation (see, for instance, [3] 1 and [4] ), it is used to prove the optimality of the rates of convergence. The aim of this paper is to extend Janson’s result to Hölder continuous densities, and develop, using that extension, a test to decide whether the support is convex or not. It is organized as follows: Section 2 is devoted to the extension of Janson’s results. A new definition (which includes Janson’s as a particular case) is given and the associated theoretical results are presented. The proofs of these results are given in Appendix A. Section 3 is dedicated to the problem of testing the convexity of the support. The corresponding proofs are given in Appendix B. Just to mention some application of this test, let us recall that when dealing with support estimation, if the support is known to be convex, the convex hull of the observations provides a consistent and well studied (see for instance [23],[24] and [27]) estimator of S which does not require any smoothing parameter. Also, a convexity test can be used to, a posteriori, select a tuning parameter. In [6] a test for convexity is also proposed, and applied to choose the parameter of the ISOMAP (see [26]) method for dimensionality reduction. The convexity test based on Janson’s extension allows us to provide an estimation of the p-value, whereas, in [6], it has to be estimated via the Monte Carlo method. 2 Main definitions and results We start by fixing some notation that will be used throughout the paper. Given a set S ⊂ Rd , we denote by ∂S, S̊, S, diam(S), |S|, and H(S), the boundary, interior, closure, diameter, Lebesgue measure, and convex hull of S, respectively. We denote by k · k and h·, ·i the euclidean norm and the inner product respectively. We write N (S, ε) for the inner covering number of S (i.e.: the minimum number of balls of radius ε and centred in S required to cover S). Recall that if S is compact, there exists CS such that N (S, ε) ≤ CS ε−d . We denote by B(x, ε) the closed ball in Rd , of radius ε, centered at x. We set ωd = |B(0, 1)|. Given λ ∈ R, A, C ⊂ Rd , we set λA = {λa : a ∈ A}, A ⊕ C = {a + c : a ∈ A, c ∈ C}, and A C = {x : {x} ⊕ C ⊂ A}. For the sake of simplicity, we use the notation x + C, instead of {x} ⊕ C. If λ ≥ 0, we set Aλ = A ⊕ λB(0, 1) and A−λ = A λB(0, 1). Given A, C ⊂ Rd two non-empty compact sets, the Hausdorff (or Pompeiu–Hausdorff) distance between them is given by   dH (A, C) = max max d(a, C), max d(c, A) , a∈A c∈C where d(a, C) = inf{ka − ck : c ∈ C}. Let A ⊂ Rd be a compact convex set with |A| = 1, let v be a vector of Rd , and let αA (v) be the constant defined in equation (2.4) of [12]: Z Z
d 1 αA (v) = ... Det n(yi ) i=1 dω(y1 ) . . . dω(yd ), (1) d! where ω denotes the d − 1 dimensional Hausdorff measure, and for y ∈ ∂A, n(y) denotes the exterior unit normal vector to A at y. The integral in (1) is over all y1 , . . . , yd ∈ ∂A 2 such that v is a linear combination of n(y1 ), . . . , n(yd ) with positive coefficients, and Det(n(yi ))di=1 is the determinant of the vectors n(yi ) in an orthonormal basis. Corollary 7.4 in [13] proves that αA (v) is almost everywhere independent of v, so that there can be defined an αA such that αA (v) = αA almost everywhere. If A is the unit cube, then αA = 1; while if B is the unit ball, then 1 αB = d! √ πΓ Γ d 2 +1
d+1 2
!d−1 .
Lastly, let U be a random variable such that P(U ≤ t) = exp − exp(−t) . 2.1 Janson’s result and its extension Let S ⊂ Rd be a bounded set with |S| = 1 and |∂S| = 0. Let ℵn = {X1 , . . . , Xn } be iid random vectors uniformly distributed on S, and A a bounded convex set. In [12], the maximal-spacing is defined as n o ∆∗ (ℵn ) = sup r : ∃x such that x + rA ⊂ S \ ℵn . To generalize the results of [12] to the non-uniform case, we need to extend the definition of maximal-spacing. When the sample is drawn according to a probability measure PX , we consider the probability measure of the largest set λA missing ℵn . If |A| = 1, ∆∗ (ℵn )d is the Lebesgue measure of the largest set x + rA ⊂ S \ ℵn . When the sample is drawn from a non-uniform probability measure, is natural to use the same definition, replacing the Lebesgue measure by the true underling distribution PX . If PX has continuous density f , then PX (x + rA) ∼ f (x)rd for sufficiently small r, so one can define the maximal-spacing as the largest r such that there exists x with x + f (x)r 1/d A ⊂ S \ ℵn . Definition 1. Let ℵn = {X1 , . . . , Xn } be an iid random sample of points in Rd , drawn according to a density f with bounded support S. Let A ⊂ Rd be a convex and compact set such that |A| = 1 and its barycentre is the origin of Rd . We define n o r ∆(ℵn ) = sup r : ∃x such that x + A ⊂ S \ ℵ , n f (x)1/d V (ℵn ) = ∆d (ℵn ), and
U (ℵn ) = n∆d (ℵn ) − log(n) − (d − 1) log log(n) − log(αA ). The following result can be found in [12]. Theorem 1. Let S ⊂ Rd be a bounded set such that |S| = 1 and |∂S| = 0. Let ℵn = {X1 , . . . , Xn } be iid random vectors uniformly distributed on S. Then, 3 i) L U (ℵn ) −→ U when n → ∞, ii) lim inf n→+∞ nV (ℵn ) − log(n) = d − 1 a.s., log(log(n)) iii) lim sup n→+∞ nV (ℵn ) − log(n) = d + 1 a.s. log(log(n)) A rescaling extends these results to the case where |S| = 6 1. Corollary 1. Let S ⊂ Rd be a bounded set such that |∂S| = 0 and |S| > 0. Let ℵn = {X1 , . . . , Xn } be iid random vectors uniformly distributed on S. Then, i) L U (ℵn ) −→ U when n → ∞, ii) lim inf n→+∞ nV (ℵn ) − log(n) = d − 1 a.s., log(log(n)) iii) lim sup n→+∞ nV (ℵn ) − log(n) = d + 1 a.s. log(log(n)) Janson’s result does not require any condition on the shape of the support S, while in our extension it will be required that the inside covering number of ∂S is such that there exists C∂S > 0 and κ < d satisfying N (∂S, ) ≤ C∂S −κ . Note that this is a very mild hypothesis: if ∂S is smooth enough (for instance, a C1 (d − 1)−dimensional manifold), it is fulfilled for κ = d − 1. More generally, it also holds for any set S with finite Minkowski content of the boundary, for κ = d − 1 (see, for instance [15]). With respect to the distribution of the sample, we require that the density is Hölder continuous on S (i.e. there exists Kf and β ∈ (0, 1] such that for all x, y ∈ S, |f (x) − f (y)| ≤ Kf kx − ykβ ) and bounded from below on it support, by a positive constant f0 . This is stated in our main theorem, given below. Theorem 2. Let ℵn = {X1 , . . . , Xn } be iid random vectors distributed according to a distribution PX whose density f with respect to Lebesgue measure is Hölder continuous and bounded from below on its support S. Let us assume that S is compact, and there exists κ < d and C∂S > 0 such that N (∂S, ε) ≤ C∂S ε−κ Then, we have that L U (ℵn ) −→ U lim inf n→+∞ when n → ∞, nV (ℵn ) − log(n) ≥ d − 1 a.s., log(log(n)) 4 (2) (3) lim sup n→+∞ nV (ℵn ) − log(n) ≤ d + 1 a.s. log(log(n)) (4) The proof is given in Appendix A. 3 3.1 A new test for convexity The semi-parametric case In this section we propose, using the concept of maximal-spacing defined in Section 2, a consistent hypothesis test based on an iid sample {X1 , . . . , Xn } uniformly distributed on a compact set S, to decide whether S is convex or not. The main idea is the following: if S is convex and the sample is uniformly distributed on S, then H(ℵn ) is a good approximation to S and |H(ℵn )|−1 IH(ℵn ) is a good approximation of the uniform law. As a result, n o ˜ n ) = sup r : ∃x such that x + r|H(ℵn )|1/d B(0, 1) ⊂ H(ℵn ) \ ℵn ∆(ℵ is a plug-in estimator of the maximal spacing and should converge to 0. On the other ˜ n ) is expected to converge to a positive constant (that hand, if S is not convex, ∆(ℵ depends on the shape of S). In order to unify notation, let us first define the maximal inner radius. Definition 2. Let S ⊂ Rd be a bounded set satisfying S̊ 6= ∅. We define the maximal inner radius of S as  R(S) = sup r : ∃x ∈ S such that B(x, r) ⊂ S . 1/d ˜ n ) = R(H(ℵn )\ℵn )ω 1/d |H(ℵn )|1/d . Remark 1. We have ∆(ℵn ) = R(S\ℵn )ωd |S|1/d and ∆(ℵ d ˜ n ), we When testing the convexity of the support using a test statistic based on ∆(ℵ only obtain, in general, an upper asymptotic bound on the test level. However, if the boundary of the support is smooth enough, we have a converging estimation of the level. The regularity condition is the following. Condition (P): For all x ∈ ∂S there exists a unique vector ξ = ξ(x) with kξk = 1, such that hy, ξi ≤ hx, ξi for all y ∈ S, and kξ(x) − ξ(y)k ≤ lkx − yk ∀ x, y ∈ ∂S, where l is a constant. We will denote by CP the class of convex subsets that satisfy condition (P). The convexity test and its asymptotic behaviour is given in the following theorem. 5 Theorem 3. Let S ⊂ Rd be compact with non-empty interior. Let ℵn = {X1 , . . . , Xn } be a set of iid random vectors uniformly distributed on S. For the following decision problem, ( H0 : the set S is convex (5) H1 : the set S is not convex,
d the test based on the statistic Ṽn = |H(ℵn )|ωd R H(ℵn ) \ ℵn with critical region given by  RC = Ṽn > cn,γ , where cn,γ = 

1 − log − log(1 − γ) + log(n) + (d − 1) log log(n) + log(αB ) , n and αB is the constant defined in (1), is asymptotically of level less than or equal to γ. Moreover, if S ∈ CP , the asymptotic level is γ. If S is not convex, the test has power one for all sufficiently large n. The proof of Theorem 3 is given in Appendix B. 3.2 The non-parametric case We now assume that we have a sample ℵn = {X1 , . . . , Xn } of iid random vectors in Rd drawn according to an unknown density f . As in the semi-parametric case, the idea is to estimate the maximal-spacing and use this estimation as a test statistic. As before, H(ℵn ) is proposed as an estimator of S. To ensure that the test proposed in Theorem 3 allows determining whether the support is convex or not, the density estimator should have a non-conventional behaviour: it is expected to converge toward the unknown density when the support is convex, but not when the support is not convex. That is why we propose the following density estimator.  Definition 3. Let Vor(Xi ) be the Voronoi cell of the point Xi (i.e. Vor(X R i) = x : K is a kernel function (i.e. K ≥ 0, K = 1 and kx R − Xi k = miny∈ℵn kx − yk ). 1 IfP uK(u)du = 0) and fn (x) = nhd K((x − Xi )/hn ) denotes the usual kernel density n estimator, we define fˆn (x) = max fn (Xi )Ix∈H(ℵn ) . (6) i:x∈Vor(Xi ) We propose to test the convexity using the following plug-in estimator of ∆(ℵn ):  
r δ̂ H(ℵn ) \ ℵn = sup r : ∃x such that x + A ⊂ H(ℵn ) \ ℵn , fˆn (x)1/d −1/d with A = ωd B(0, 1), and reject H0 (the support is convex) if δ̂(H(ℵn ) \ ℵn ) is sufficiently large. The proof of Theorem 4 makes use of Theorem 2.3 in [11]. In order to apply that result, we will introduce some technical hypotheses on the kernel function. 6 Definition 4. Let K be the set of kernel functions K(u) = φ(p(u)), where p is a polynomial and φ a is bounded real function of bounded variation, such that cK = R kukK(u)du < ∞, K ≥ 0 and there exists rK and c0K > 0 such that K(x) ≥ c0K for all x ∈ B(0, rK ). Note that, for example, the Gaussian and the uniform kernel are in K. Definition 5. A set S is standard if there exist positive numbers r0 , cS such that |B(x, r) ∩ S| ≥ cS ωd rd for all r ≤ r0 . We write C for the class of compact convex sets with non-empty interior, and A for the class of all compact standard sets. Finally it is also necessary to impose some conditions on the density. Condition (B): A density f with support S fulfils condition B if its restriction to S is Lipschitz continuous (i.e. there exists kf such that ∀x, y ∈ S, |f (x)−f (y)| ≤ kf kx−yk) and there exists f0 > 0 such that f (x) ≥ f0 for all x ∈ S. We denote f1 = maxx∈S f (x). Remark 2. The condition f (x) ≥ f0 > 0 for all x ∈ S is a necessary condition to test convexity, as indeed is mentioned in [6]: ‘...an assumption like the density being bounded away from zero on its support is necessary for consistent decision rules.’ Theorem 4. Let K ∈ K, and let fˆn be as defined in (3). Assume that hn = O(n−β ) for some 0 < β < 1/d. Assume also that the unknown density fulfils condition B. For the following decision problem, ( H0 : S ∈ C (7) H1 : S ∈ / C, a) the test based on the statistic V̂n = δ̂ H(ℵn )\ℵn cn,γ }, where cn,γ =
d with critical region RC = {V̂n ≥  1 − log(− log(1 − γ)) + log(n) + (d − 1) log(log(n)) + log(αB ) , n has an asymptotic level less than γ. b) Moreover, if S ∈ A is not convex, the power is 1 for sufficiently large n. Remark 3. Notice that the ‘optimal’ kernel sequence size, hn = h0 n1/(d+4) , satisfies the hypothesis of our theorem, so that any bandwidth selection method should be suitable for testing for convexity. However, in the semi-parametric case it is possible to derive the asymptotic behaviour for the level under regularity conditions on the support. In this more general setup, we will not have a convergent level estimation but only a bound for the level (the price to pay for estimating the density). The proof of Theorem 4 is given in Appendix B. 7 3.3 Simulations We have performed two simulation studies to assess the behaviour of our test in the scenarios described in Sections 3.1 and 3.2. For the first study, the data were drawn uniformly from sets S ⊂ R2 , and we will perform the test defined in Section 3.1 to obtain estimations of the power and the level. In the second study, the non-parametric case, the data can be not uniformly drawn drawn, and we estimate the density using the estimator given by (6). In this case, we consider the same sets and density as in [6]. 3.3.1 Semi-Parametric case The data were generated uniformly from the sets Sϕ = [0, 1]2 \ Tϕ , where Tϕ is the isosceles triangle with height 1/2 (see Figure 1) whose angle at the vertex (1/2, 1/2) is equal to ϕ. If we have a random sample from Sϕ , it is clear that as ϕ increases, it should be easier to detect the non-convexity of the set. The results of the simulations are summarized in Table 1. ϕ = π/4 n β̂ 100 .4 130 .636 160 .835 200 .946 300 .997 ϕ = π/6 n β̂ 200 .565 250 .787 300 .926 400 .996 500 1 ϕ = π/8 n β̂ 300 .543 350 .679 400 .846 500 .976 600 .997 Table 1: Power estimated over 1000 replications, for different values of ϕ, when the sample is uniformly distributed on [0, 1]2 \ Tϕ , where Tϕ is an isosceles triangle, (see Figure 1). φ Figure 1: [0, 1]2 \ Tϕ where Tϕ is an isosceles triangle with height 1/2. 3.3.2 Non-parametric case We performed a simulation study for the same sets used in [6]. Consider the curves γR,θ = R(cos(θ), sin(θ)) with θ ∈ [ 3π(R−1) , 32 π] and the reflections of those curves along 2R the y axis (which will be denoted by ζR,θ ). We consider ΓR = T(0,R) (γR,θ ) ∪ T(0,−R) (ζR,θ ) with θ ∈ [ 3π(R−1) , 23 π], where Tv is the translation along the vector v. It is easy to 2R 8 see that the length of every ΓR is 32 π. We will consider, for different values of R, the S-shaped sets (see the first row in Figure 2)  SR = T(0,R)   [  γr,θ  ∪ T(0,−R)  R−0.6≤r≤R+0.6  [ ζr,θ  . R−0.6≤r≤R+0.6 Observe that when R approaches infinity, the sets S converge to a rectangle (which corresponds to the convex case). We have generated the data according to two different densities. The first one is the same as that considered in [6]: that is, along the orthogonal direction of ΓR , we choose a random variable with normal density (with zero mean and standard deviation σ = 0.15) truncated to 0.6 (the truncation is performed to ensure that we obtain a point in the set SR ). In the second case, we consider a random variable along the orthogonal direction of ΓR but uniformly distributed on [−0.6, 0.6]. In Tables 2 and 3, we have summarized the results of the simulations, for different sample sizes (we performed the test B = 100 times). The results are quite encouraging and slightly better that those obtained in [6] since the non-convexity is better detected (see Fig. 7 in [6] for comparison) with no need for the decision rule to be calibrated. R 1 1.5 3 6 12 24 ∞ N=100 np unif .13 .44 .98 1 .38 .24 .08 .09 .01 .05 0 .07 0 .04 N=250 np unif .55 .99 1 1 1 1 .41 .66 .02 .08 .01 .05 0 .09 N=500 np unif 1 1 1 1 1 1 1 1 .39 .68 0 .09 0 .04 N=1000 np unif 1 1 1 1 1 1 1 1 .98 1 .07 .48 .01 .05 Table 2: Power estimated over B replications, for different values of R, when the sample is uniformly distributed along the orthogonal direction of ΓR . R 1 1.5 3 6 12 24 ∞ N=100 np unif 1 1 1 1 1 .99 .67 .41 .25 .19 .1 .30 0 .33 N=250 np unif 1 1 1 1 1 1 .99 1 .62 .98 .30 .92 .04 .92 N=500 np unif 1 1 1 1 1 1 1 1 .85 1 .38 1 .06 1 N=1000 np unif 1 1 1 1 1 1 1 1 .94 1 .48 1 .04 1 Table 3: Power estimated over B replications, for different values of R, when the sample is drawn according to a truncated normal distribution along the orthogonal direction of ΓR . 9 Figure 2: SR for different values of R together with the sample drawn with a uniform radial noise (top) and with a truncated Gaussian noise (bottom) . 4 Appendix A In Appendix A we proof the main result on the generalization of the maximal-spacing, given in Theorem 2. First we settle some preliminary lemmas, then we prove a weaker version of Theorem 2, for the case of piecewise constant densities on disjoint sets. We continue by considering piecewise constant densities, and finally we prove the result for Hölder continuous densities. 4.1 Preliminary Lemmas As we mentioned before, the proof of Corollary 1 follows from a simple rescaling in the lemmas stated in [12], used to prove Theorem 1. In particular the following rescaled lemma will be used in this section. Lemma 1. Let S ⊂ Rd be a bounded set, |S| > 0, |∂S| = 0, and ℵn = {X1 , . . . , Xn } iid random vectors with uniform distribution on S. Then, there exists aS− = aS− (w, n), aS+ = aS+ (w, n) such that aS− → αA and aS+ → αA as w → ∞ and w/n → 0, such that if n d−1 −w γ = |S| w e ,
exp(−γaS+ |S|) ≤ P nV (ℵn ) < w ≤ exp(−γaS− |S|). (8) The functions aS+ and aS− only depend on the “shape” of S (i.e. are invariant by similarity transformations). Without loss of generality they can be chosen such that, for all w0 ≥ w and n0 ≥ n: aS+ (w0 , n0 ) ≤ a+ (w, n) and aS( − w0 , n0 ) ≥ a− (w, n). Next we settle two lemmas whose proofs are quite similar. The first one (Lemma 2) gives a first rough upper bound for the maximal spacing. The second one (Lemma 3) bounds a sort “constrained” maximal-spacing (the centre x of the largest set x + rA missing the observation is constrained to be in a “small” subset of the support). Recall first (see [1]) that, since A is convex, there exist ε0 > 0 such that, for all ε ≤ ε0 , A−ε 6= ∅ and |A−ε | = |A| − ε|∂A|d−1 + o(ε). 10 (9) It can also be proved easily that, for all r > 0, and for all x ∈ B(0, ε0 /r), x + (rA)−kxk ⊂ rA. (10) Lemma 2. Let ℵn = {X1 , . . . , Xn } be iid random vectors in Rd , with common density f . Assume that f has bounded support S and there exist 0 < f0 < f1 < ∞ such that f0 ≤ f (x) ≤ f1 for all x ∈ S. Then, for all rf > 0 such that rfd > 2f1 /f0 we have,  ∆(ℵn ) ≤ rf log(n) n 1/d eventually almost surely. Proof. First observe that S can be covered with N (S, n−1/d ) ≤ CS n balls of radius n−1/d 1/d  with rfd > 2f1 /f0 . centered at some points {x1 , . . . , xνn } ⊂ S. Denote wn = rf log(n) n First observe that ∆(ℵn ) ≥ wn ⇔ ∃x ∈ S, such that x + wn f (x)−1/d A ⊂ S \ ℵn , then −1/d ∆(ℵn ) ≥ wn ⇒ ∃x ∈ S, such that x+wn f1 A ⊂ S \ℵn . Applying (10), for sufficiently large n (that is possible because n−1/d  wn ) we get −1/d ∆(ℵn ) ≥ wn ⇒ ∃xi such that xi + (wn f1 1/d A)−1/n ⊂ S \ ℵn . (11) Next notice that,  −1/d P xi + wn f1 A
−1/n1/d  ⊂ S \ ℵn = ≤ ≤ 1 − PX  −1/d
−1/n1/d wn f1 A 1 − f0 1−  −1/d
−1/n1/d xi + wn f1 A f 0 f1 wnd − f0 d−1 !n !n wnd−1 n−1/d (1  + o(1)) f1 d The last inequality is obtained using (9). Since wn  n−1/d , we finally get n    f0 d −1/d
−1/n1/d P xi + wn f1 A ⊂ S \ ℵn ≤ 1 − wn (1 + o(1)) . f1 From this inequality and (11) it follows that,   n
1/d  f0 P ∆(ℵn ) ≥ rf log(n)n/n ≤ N (S, n−1/d ) 1 − wnd (1 + o(1)) f1  f  0 ≤ CS n exp − nwnd (1 + o(1)) , f1 and therefore,   d P ∆(ℵn ) ≥ rf (log(n)/n)1/d ≤ CS n1−rf f0 /f1 +o(1) . 11 !n . 
P Finally, since rfd > 2f1 /f0 we have P ∆(ℵn ) ≥ rf (log(n)/n)1/d < ∞. Thus, the Borel-Cantelli Lemma entails that ∆(ℵn ) ≤ rf (log(n)/n)1/d eventually almost surely. Lemma 3. Let ℵn = {X1 , . . . , Xn } be iid random vectors in Rd with common distribution distribution PX supported on a compact set S and density f continuous on S. Assume that there exists f0 > 0 such that f (x) ≥ f0 ∀x ∈ S. Let Gn be a sequence of sets included in S, with the following property: there exist C such that N (Gn , n−1/d ) ≤ Cn1−a (log(n))b for some a > 0 and b > 0. Let A be a compact and convex set with |A| = 1 such that its barycenter is the origin of Rd . Let us denote n o r ∆(ℵn , Gn ) = sup r : ∃x ∈ Gn such that x + A ⊂ S \ ℵ , n f (x)1/d V (ℵn , Gn ) = ∆d (ℵn , Gn ), U (ℵn , Gn ) = nV (ℵn , Gn ) − log(n) − (d − 1) log(log(n)) − log(αA ).
Then, P U (ℵn , Gn ) ≥ − log(log(n)) → 0. Proof. Let us first cover Gn with νn = N (Gn , n−1/d ) balls of radius n−1/d , centred at A ) 1/d ) some points {x1 , . . . , xνn } belonging to S, and choose wn = ( log(n)+(d−2) log(log(n))+log(α n (observe that wn  (1/n)1/d ). As in the proof of Lemma 2 we have, ∆(ℵn ) ≥ wn ⇔ ∃x ∈ Gn , such that x + wn f (x)−1/d A ⊂ S \ ℵn . which implies, ∆(ℵn ) ≥ wn ⇒ ∃xi ∃x ∈ B(xi , n−1/d ), such that xi + (wn f (x)−1/d A)−1/n 1/d ⊂ S \ ℵn . Therefore,  P xi + (wn f (x)−1/d A) −1/n1/d  ⊂ S \ ℵn = 1 − PX 
−(1/n1/d )  xi + wn f (x)−1/d A !n . With rough bounds on the density, 
−1/n1/d  mint∈S∩(xi +wn f1−1/d A) f (t) d PX xi + wn f (x)−1/d A ≥ w (1 + o(1)). maxt∈S∩B(xi ,n−1/d ) f (t) n Since f is uniformly continuous on S, A is bounded, wn → 0 and n−1/d → 0, for all c < 1 there exist nc such that for all n ≥ nc    n 1/d P xi + (wn f (x)−1/d A)−1/n ⊂ S \ ℵn ≤ 1 − cwnd (1 + o(1)) , then,
P ∆(ℵn , Gn ) ≥ wn ≤ Cn1−a (log n)b (1 − cwnd (1 + o(1)))n . 12 Taking c = 1 − a/2, we finally get that
−1+a/2 −a/2 P U (ℵn , Gn ) ≥ − log(log(n)) ≤ CαA n (log(n))b−(1−a/2)(d−2) (1 + o(1)) → 0. The next lemma relates the behaviour of the maximal-spacing for two different densities having the same support. Lemma 4. Let us consider f and h, two densities with compact support S such that, h(x) > h0 for all x ∈ S and maxx∈S |f (x) − h(x)| ≤ εh0 for a given ε ∈ (0, 1/2). Denote by n0 = bn(1 − 2ε)c and n1 = dn(1 + 2ε)e the floor and ceiling of n(1 − 2ε) and n(1 + 2ε) −1 ) respectively. For any w ∈ R, let us define wn,0 = w(1−2ε−n and wn,1 = w(1−ε) 1+2ε . Then, (1+ε) P n0 V (Yn0 ) ≤ wn,0 and
 1−
1 − ε ≤ P nV (ℵn ) ≤ w , nε

1 + 2ε + n−1 −1 , P nV (ℵn ) ≤ w ≤ P n1 V (Yn1 ) ≤ wn,1 1 − (nε + 1)(1 + ε) (12) (13) where Yn0 = {Y1 , . . . , Yn0 } and Yn1 = {Y1 , . . . , Yn1 } are iid random vectors on Rd , with density h, and ℵn = {X1 , . . . , Xn } are iid random vectors with density f . Proof. We first prove (12). Observe that X can be generated from the following mixture: with probability p = 1 − ε, X is drawn with density h, and, with probability 1 − p, X IS (x). Let us denote is drawn with the law given by the density g(x) = f (x)−h(x)(1−ε) ε ∗ by N0 the number of points drawn according to h on S and ℵN0 = {Y1 , . . . , YN0 } the associated sample. Let us recall that o n r A ⊂ S \ ℵ . ∆(ℵn ) = sup r : ∃x such that x + n f (x)1/d Observe that sup h0 |f (x)/h(x) − 1| ≤ sup h(x)|f (x)/h(x) − 1| ≤ h0 ε, x x so f (x)/h(x) ≤ 1 + ε. Then we have, n ∆(ℵn ) ≤ (1 + ε)1/d sup r : ∃x such that x + From the inclusion ℵ∗N0 ⊂ ℵn we get n 1/d ∆(ℵn ) ≤ (1 + ε) sup r : ∃x such that x + o r A ⊂ S \ ℵ . n h(x)1/d o r ∗ A ⊂ S \ ℵ N0 , h(x)1/d and therefore ∆(ℵn ) ≤ (1 + ε)1/d ∆(ℵ∗N0 ), which entails that V (ℵn ) ≤ (1 + ε)V (ℵ∗N0 ). Then, for all w > 0,

P nV (ℵn ) ≤ w ≥ P (1 + ε)nV (ℵ∗N0 ) ≤ w , 13 and 


 P nV (ℵn ) ≤ w ≥ P (1 + ε)nV (ℵ∗N0 ) ≤ w ∩ N0 ≥ n0 . For N0 ≥ n0 , let us denote by Yn0 = {Y1 , . . . , Yn0 } the n0 first values of ℵ∗N0 . Clearly we have V (ℵ∗N0 ) ≤ V (Yn0 ) so,  
PN0 ≥n0 (1 + ε)nV (ℵ∗N0 ) ≤ w ≥ P (1 + ε)nV (Yn0 ) ≤ w , where PN0 ≥n0 denotes the conditional probability given N0 ≥ n0 . Therefore, 
wn0  P nV (ℵn ) ≤ w ≥ P n0 V (Yn0 ) ≤ P(N0 ≥ n0 ). (1 + ε)n On the other hand, since N0 ∼ Bin(n, 1 − ε), we obtain,   P(N0 < n0 ) = P N0 − (1 − ε)n < n0 − (1 − ε)n ≤ P(N0 − (1 − ε)n ≤ −εn). Since n0 = bn(1 − 2ε)c, n0 − n(1 − ε) ≤ −εn, which together with Chebyshev’s inequality entails that nε(1 − ε) (1 − ε) = , P(N0 < n0 ) ≤ 2 2 n ε nε and then, (1 − ε) P(N0 ≥ n0 ) ≥ 1 − . nε −1 ) Let us denote by wn,0 = w(1−2ε−n (1+ε) from where it follows that
. Since n(1 − 2ε) − 1 ≤ n0 we have wn,0 ≤  P nV (ℵn ) ≤ w ≥ P n0 V (Yn0 ) ≤ wn,0  1−ε 1− nε wn0 (1+ε)n ,  . Equation (13) is proved in the same way. We just provide a sketch of the proof. The key point for the proof of (12) was to think the law of a random variable Y drawn with the 1 density h as the following mixture: with probability p = 1+ε , Y as a random variable with (x) density f , and, with probability 1−p, Y is drawn with density g(x) = h(x)(1+ε)−f IS (x). ε Next, we consider a sample Yn1 = {Y1 , . . . , Yn1 } of iid copies of Y , (that follows a law given by h). Denote by N the number of the points that drawn according to the density f and Y∗N = {X1 , . . . XN } these points. The rest of the proof follows using the same argument to prove (12). 4.2 Uniform mixture on disjoint supports Proposition 1. Let E1 , . . . , Ek be subsets of Rd such that for all i 6= j ⇒ Ei ∩ Ej = ∅, and 0 < |Ei | < ∞ for all i. Let ℵn = {X1 , . . . , Xn } be iid random vectors in S = ∪i Ei with density k X f (x) = pi IEi (x), i=1 14 where p1 , . . . , pk are positive real numbers. Then, L U (ℵn ) −→ U when n → ∞. Proof. First let us introduce some notation, for i = 1, . . . , k • Ni = #{ℵn ∩ Ei } denotes the number of data points in Ei . Notice that Ni ∼ Bin(n, pi |Ei |). • ℵiNi = {Xi1 , . . . , XiNi } denotes the subsample of ℵn that falls in Ei . Observe that they all are uniformly distributed. P • ai = P pi |Ei | for i = 1, . . . , k, that fulfils ai = 1, a0 = mini ai , A0 = maxi ai and 1−ai C= . ai • εn,i = Ni −ai n nai . Since the support of f is ∪i Ei , and by assumption i 6= j ⇒ Ei ∩ Ej = ∅, we have n o r ∆(ℵn ) = sup r : ∃x∃i such that x + 1/d A ⊂ Ei \ ℵn , pi so while o n r|Ei |1/d i ∆(ℵn ) = max sup r : ∃x such that x + A ⊂ E \ ℵ i Ni , i (|Ei |pi )1/d (14) o n ∆(ℵiNi ) = sup r0 : ∃x ∈ Ei such that x + r0 |Ei |1/d A ⊂ Ei \ ℵiNi . (15) From (14) and (15) we derive that n o n o ∆(ℵn ) = max (|Ei |pi )1/d ∆(ℵiNi ) and V (ℵn ) = max |Ei |pi V (ℵiNi ) , i i which entails that  Y  wNi i P(nV (ℵn ) ≤ w) = P Ni V (ℵNi ) ≤ . ai n i − → Let P n (A) = P(A|N1 = n1 , . . . , Nk = nk ) stand for the conditional probability given the number of points that fall in each Ei . We have that, P − → n
nV (ℵn ) ≤ w = k Y P − → n n|Ei |pi V (ℵini )
≤w = i=1 Now, taking wn,i =  exp − k X wni n|Ei |pi , k Y  P ni V (ℵini ) ≤ − → n i=1 γn,i =  i γn,i aE + |Ei | d−1 −wn,i ni wn,i e |Ei | − → n ≤P and applying Lemma 1 we obtain,
nV (ℵn ) ≤ w ≤ exp i=1  − k X i=1 15 wni n|Ei |pi  . i γn,i aE − |Ei |  . On the other hand, k X i γn,i aE + |Ei | = i=1 = k X i=1 k X  ni wni n|Ei |pi d−1   wni i aE exp − + n|Ei |pi i ni wd−1 (1 + εi )d−1 exp(−w(1 + εi ))aE + (wn,i , ni ). i=1 E Let εn = maxi |εn,i | and εa+ = maxi k X |a+i (wn,i ,ni )−αA | , αA then we have d−1 i γn,i aE exp(−w)αA (1 + εn )d−1 exp(wε)(1 + εa+ ). + |Ei | ≤ nw i=1 Taking w = wn = x + log(n) + (d − 1) log(log(n)) + log(αA ), we obtain that nV ≤ w ⇔ U ≤ x, which implies that  
− → P n U (ℵn ) ≤ x ≥ exp −e−x (1 + ε)d−1 exp(log(n)εn )(1 + εa+ )(1 + on (1)) . In the same way it can be proved,  
− → P n U (ℵn ) ≤ x ≤ exp −e−x (1 − εn )d−1 exp(− log(n)εn )(1 − εa− )(1 + on (1)) . Ei (wn,i ,ni )−α |a | A where we denoted εa− = maxi − . αA 2 Suppose that εn = max |εn,i | ≤ 1/ log(n) , then, if n ≥ 5, a0 n/2 ≤ Ni ≤ n for all i, which imply that for all i,
wn,i ≥ log(n)a0 /(2A0 ) → ∞ and wn,i /n ≤ x + log(n) + (d − 1) log(log(n)) + log(αA ) /(na0 ) → 0. Then εa− and εa+ converges to 0, according to Lemma 4. Therefore
2 Pεn ≤1/ log(n) U (ℵn ) ≤ x → exp(− exp(−x)) when n → ∞. (16) Since  P max |εn,i | ≥ i k [ 1  X  1  1  = P |ε | ≥ ≤ P |ε | ≥ , n,i n,i log(n)2 log(n)2 log(n)2 i i=1 from Chebyshev’s inequality we obtain  1  P |εn,i | ≥ ≤ log(n)4 V(ε2n,i ) log(n)2 and therefore where V(ε2n,i ) =
4 log(n) 1  ≤C . P εn ≥ log(n)2 n  Finally, from equations (16) and (17) we get,
P U (ℵn ) ≤ x → exp(− exp(−x)) when n → ∞, which concludes the proof. 16 1 − ai , nai (17) 4.3 Uniform mixture Proposition 2. Let E1 , . . . , Ek be subsets of Rd such that, 1) i 6= j ⇒ |Ei ∩ Ej | = 0. 2) 0 < |Ei | < ∞ for i = 1, . . . , k. 3) There exists a constant C > 0 such that, for all i, for all ε > 0, N (∂Ei , ε) ≤ Cε−d+1 . Let ℵn = {X1 , . . . , Xn } be iid random vectors with density f (x) = k X pi IE̊i , i=1 where p1 , . . . , pk are positive real numbers. If there exist constants r0 > 0 and c > 1−1/d such that, for all r ≤ r0 and all x ∈ ∪i E̊i , mint∈S∩B(x,r) f (t) ≥ c, maxt∈S∩B(x,r) f (t) then, L U (ℵn ) −→ U when n → ∞. Proof. We start by introducing some definitions and notation. n o r ˚ n ) = sup r : ∃x∃i, such that x + ∆(ℵ A ⊂ E̊ \ ℵ , i n f (x)1/d ˚ d (ℵn ), V̊ (ℵn ) = ∆
Ů (ℵn ) = nV̊ (ℵn ) − log(n) − (d − 1) log log(n) − log(αA ) With the same ideas used to prove Proposition 1 (and the fact that |Ei | = |E̊i |) it
L


follows that Ů ℵn −→ U . Let Fn (x) = P Ů ℵn ≤ x . Clearly U (ℵn ) ≥ Ů ℵn , and therefore
P U (ℵn ) ≤ x ≤ Fn (x) → exp(− exp(−x)). (18) In order to prove the other inequality let us define
S • G = i,j Ei ∩ Ej . • p0 = mini pi . • ρA = maxx∈A kxk.
1/d
1/d 1/d with rf such that ∆(ℵn ) ≤ rf log(n)/n even• ρn = (rf ρA /p0 ) log(n)/n tually almost surely (whose existence follows from Lemma 2). Notice that condition 3) ensures that N (Gρn , n−1/d ) = O(n1−1/d (log(n))1/d ). 17 o r A ⊂ S \ ℵ . n f (x)1/d o r such that x + A ⊂ S \ ℵ n . f (x)1/d n
• ∆ ℵn , S \ Gρn = sup r : ∃x ∈ S \ Gρn such that x + n
• ∆ ℵn , Gρn = sup r : ∃x ∈ Gρn Clearly we have that, n

o ∆(ℵn ) = max ∆ ℵn , S \ Gρn , ∆ ℵn , Gρn . (19) For the chosen ρn , we are going to prove that,

˚ ℵn ≥ ∆ ℵn , S \ Gρn eventually almost surely. ∆ (20)
Let us suppose first that ∆ ℵn , S \ Gρn ≤ rf (log(n)/n)1/d (which holds, e.a.s., due to Lemma 2) then
∆ ℵn , S \ Gρn − ε ρn for all ε > 0 there exists xε ∈ S \ G such that xε + A ⊂ S \ ℵn , f (xε )1/d and
  ∆ ℵ n , S \ Gρ n − ε rf (log(n)/n)1/d − ε xε + A ⊂ B xε , ρA ⊂ B(xε , ρn ). 1/d f (xε )1/d p 0
S ∆ ℵn ,S\Gρn −ε A ⊂ i E̊i \ ℵn . Then, for all ε > 0, From d(xε , G) ≥ ρn we get xε + f (xε )1/d

˚ ℵn ≥ ∆ ℵn , S \ Gρn − ε e.a.s., which concludes the proof of (20). ∆ By (19), introducing U (ℵn , Gρn ) = n∆(ℵn , Gρn )d − log(n) − (d − 1) log(log(n)) − log(αA ), one can bound P(U (ℵn ) ≥ x) for all x, as follows:  P(U (ℵn ) ≥ x) ≤ P(Ů (ℵn ) ≥ x) + P(U (ℵn , Gρn ) ≥ − log(log(n))) if x ≥ − log(log(n)) P(U (ℵn ) ≥ x) ≤ 1 if x ≤ − log(log(n)) By Lemma 3 we obtain:  P(U (ℵn ) ≤ x) ≥ Fn (x) + o(1) P(U (ℵn ) ≤ x) ≥ 0 if x ≥ − log(log(n)) (21) if x ≤ − log(log(n)).
Finally using (18), (21) and that P U ≤ − log(log(n)) = 1/n → 0 we conclude the proof. Proof of Theorem 2 1 Let cn = (log(n)/n) 3d . Take a “mesh” of Rd with small squares of side cn , d h i Y ki cn , (ki + 1)cn i=1 18 with ki ∈ N, and denote by mn ≤ |S|c−d n the number of these squares {C1 , . . . , Cmn } that are included in S. Like in the proof of Proposition 2 let us denote, n o r ˚ n ) = sup r : ∃x∃i, such that x + ∆(ℵ A ⊂ C̊ \ ℵ , i n f (x)1/d ˚ d (ℵn ), V̊ (ℵn ) = ∆
Ů (ℵn ) = nV̊ (ℵn ) − log(n) − (d − 1) log log(n) − log(αA ).  
S n From the inclusion m C̊ ⊂ S it follows that, P U (ℵ ) ≤ x ≤ P Ů (ℵ ) ≤ x . i n n i=1 Like in the proof of Proposition 1 let us denote, for i = 1, . . . , mn . • Ni = #{ℵn ∩ Ci }, R P 1−ai • ai = Ci f (t)dt; a0 = mini ai ; A0 = maxi ai and C = ai . Observe that P d ai = 1 and a0 ≥ f0 cn . • ℵiNi = {Xi1 , . . . , XiNi }, the subsample of ℵn that belongs to Ci . Observe that Xij
for j = 1, . . . , Ni has density fi (x) = f (x)/ai ICi (x). • εn,i = Ni −ai n nai , εn = maxi εi . We start with some asymptotic properties about εn,i and εn . If we bound |ai | ≥ f0 |cn |d and apply Hoeffding’s inequality we get   P(log(n)|εn,i | ≥ t) ≤ 2 exp − 2t2 f02 (log(n))−4/3 n1/3 , and, P(log(n)|εn | ≥ t) ≤   2|S|n1/3 2 2 −4/3 1/3 . exp − 2t f (log(n)) n 0 (log n)1/3 a.s. Borel-Cantelli Lemma entails (log(n))εn → 0. Then, with probability 1, for n large enough, f0 (log n)1/3 n2/3 ≤ Ni ≤ 2f1 (log n)1/3 n2/3 for i = 1, . . . , mn . 2 In what follows n is large enough so that (22) is fulfilled. Proceeding exactly as in the proof of Proposition 1 we can derive that n ˚ n ) = max sup r : ∃x such that x + ∆(ℵ i 1/d o rai i A ⊂ C̊ \ ℵ i Ni , (ai fi (x))1/d and therefore n o ˚ n ) = max a1/d ∆(ℵiN ) ∆(ℵ i i i and 19 n o V (ℵn ) = max ai V (ℵiNi ) . i (22) First we to bound P(Un (ℵn ) ≥ x) from above. As in Proposition 1  mn 

Y wn Ni d P nV (ℵn ) ≤ wn ≤ P nV̊ (ℵn ) ≤ wn = P Ni ∆ (ℵNi ) ≤ . ai n (23) i=1 At any of the small squares Ci , by Hölder continuity, the density is close to the uniform density, that will allow us to apply Lemma 4 with h = 1/|Ci |ICi . More precisely: for all i an for all y ∈ Ci , Z Z Z 1 f (y) 1 |Ci | − 1 = fi (y)|Ci | − 1 = f (y)dt − f (t)dt ≤ Kf |y − t|β dt. ai ai Ci ai Ci Ci √ Since |y − t| ≤ dcn , if we denote Af = Kf f0−1 dβ/2 we derive that fi (y)|Ci | − 1 ≤ √ β 1 Kf d cd+β ≤ Af cβn n ai Let Ni0 = dNi (1 + 2Af cβn )e, wn0 = wn 1−Af cβ n 1+2Af cβ n ∀y ∈ Ci . and YNi0 a sample of Ni0 variables uniformly drawn on Ci , then Lemma 4 implies that !−1     1 + 2Af cβn + Ni−1 wn0 Ni0 wn Ni 0 d d ≤ P Ni ∆ (YNi0 ) ≤ 1− . P Ni ∆ (ℵNi ) ≤ ai n ai n (Ni Af cβn )(1 + Af cβn ) (24) On the other hand, by (22), with probability one, for n large enough we have that, !−1  −1 1 + 2Af cβn + Ni−1 2 1 ≤ 1− 1− for all i, f0 Af (log n)1/3+β/3d n2/3−β/3d (1 + o(1)) (Ni Af cβn )(1 + Af cβn ) (25) and, !−1  −1 1 + 2Af cβn + Ni−1 1 1 + o(1) 1− ≥ 1− for all i. 2f1 Af (log n)1/3+β/3d n2/3−β/3d (Ni Af cβn )(1 + Af cβn ) (26) Let us prove that, !−1 mn Y 1 + 2Af cβn + Ni−1 a.s. 1− → 1. (27) β β (Ni Af cn )(1 + Af cn ) i=1 Since the right hand side of (25) can be express, for n large enough, as   exp − C(log n)1/3+β/3d n2/3−β/3d (1 + o(1)) , being C a positive constant, we get, for n large enough, !−1 mn   Y 1 + 2Af cβn + Ni−1 1/3+β/3d 2/3−β/3d 1− ≤ exp −Cm (log n) n (1+o(1)) → 1, n (Ni Af cβn )(1 + Af cβn ) i=1 20 where the limit follows from mn (log n)−1/3−β/3d n−2/3+β/3d ≤ (log n)−β/3d n−1/3+β/3d → 0. (27) is obtained doing the same with (26).  0 N0 Q n  0 d wn i 0 P N ∆ (Y ) ≤ Now let us study the asymptotic behaviour of m . If we Ni i i=1 ai n apply Lemma 1, (observe that the functions aS− only depends on the shape of S) we get,   0 0 d−1   0 0  ! mn mn  0 N0 0ω0 X Y d N ω N ω w N [0,1] i n i n Ni0 P Ni0 ∆d (YNi0 ) ≤ n i ≤ exp − exp − i n a− , Ni0 . ai n ai n ai n ai n i=1 i=1 Let, ε0n,i = 0 Ni wn ai n wn 0 n − 1 for i = 1, . . . , mn . Observe that ε0n,i = (1 + εn,i ) w wn − 1 = a.s. 1−Af cβ n − 1 and ε0n = maxi |ε0n,i |. Since (log(n))εn → 0, ε0n fulfils 1+2Af cβ n all i, Ni0 ≥ Ni . The previous equation together with (23) entails, (1 + εn,i ) and for a.s. log(n)ε0n → 0 
[0,1]d
 P(nV (ℵn ) ≤ wn ) ≤ exp −nωnd−1 (1−ε0n )d−1 exp −wn (1+ε0n ) a− (min wn (1−ε0n ), min Ni ) . i (28) If we choose wn = x + n log(n) + (d − 1) log(log(n)) + log(αA ) in (28) we get, P(U (ℵn ) ≤ x) ≤ exp(− exp(−x)) + o(1). (29) In order to conclude the proof of (2) we have to bound P(U (ℵn ) ≤ x) from below. We provide just a sketch of the proof since the arguments are similar to those in Proposition 2, using Lemma 4 as in the proof of (29). Let us denote, • ρn = rf ρA log(n)
1/d 1/d n f with ρA = maxx∈A kxk. 0 n • Gn = ∪m i6=j (Ci ∩ Cj ). cn n • Hn = S \ (∪m i Ci ), notice that Hn ⊂ ∂S . Proceeding as in Proposition 2 we have n o
U (ℵn ) ≤ max Ů ℵn , U (ℵn , Gρnn ), U (ℵn , Hn ) eventually almost surely, and P(Ů (ℵn ) ≤ x) ≥ exp(− exp(−x)) + o(1). Then, reasoning as in Proposition 2 we get P(U (ℵn ) ≤ x) ≥ exp(− exp(−x)) + o(1). Finally, in order to conclude the proof of (2) it suffices to prove that Gρnn and Hn satisfies the hypothesis of Lemma 3. Gn is the union of less than mn 2d (d − 1)−dimensional cubes of size cn . Each −d+1 balls of radius ρ (with a a positive of them can be cover by less than a1 cd−1 n 1 n ρn 21 S constant), centered at some points {xij }i,j . Since Gρnn ⊂ i,j B(xij , 2ρn ), and every B(xij , 2ρn ) can be covered by a2 ρdn n balls of radius n−1/d , Gρnn can be covered by less 2 2 −d+1 a ρd n = O(n1− 3d (log n) 3d ) balls of radius n−1/d . than νn = mn a1 cd−1 2 n n ρn d+κ d+κ In the same way it can be proved that Hn can be covered with O(n1− 3d log(n) 3d ) balls of radius n−1/d . Indeed, cover ∂S with O(c−κ n ) balls of radius cn , apply triangular inequality to obtain that the union of the balls with the same centre but with a √ √ dc n radius cn d covers ∂S and then covers Hn , finally cover every of these balls by O((cn n1/d )−d ) balls of radius n−1/d . Proof of (3) In Equation (28), let √ us choose wn = log(n) + c log(log(n)) with c < (d − 1) and introduce Nk = dexp( k)e, like in equation (3.12) in [12], we obtain for k large enough, d−1−c P(Nk V (ℵNk ) ≤ wNk ) ≤ exp(−αA k 2 /2) and, Borel-Cantelli Lemma implies that, with probability one, for k large enough Nk V (ℵNk ) ≤ wNk . The rest of the proof is exactly the same as in Lemma 5 in [12] Proof of (4): For any u > 0 let us introduce the sequence rn = 1 log(n) log(log(n)) .  log n+(d+1+u) log(log(n)) n 1/d and εn = −d Cover S with νn ≤ CS ε−d n rn balls of radius εn rn . Reasoning like in [12] we get    n nV (ℵn ) − log(n) P ≥ d + 1 + u = P(∆n ≥ rn ) ≤ νn 1 − rnd (1 − εn )d (1 − 2Kf rn diam(A)) , log(log(n)) that implies,  nV (ℵn ) − log(n) (log(log(n)))d P ≥ d + 1 + u ≤ CS . log(log(n)) (log(n))2+u+o(1) √ From Borel-Cantelli Lemma, taking Nk = dexp( k)e it follows that, with probability  Nk V (ℵNk )−log(Nk ) ≤ d + 1 + u. Now log(log(Nk )) Nk V (ℵNk )−log(Nk ) ≤ d + 1 + u. We have, log(log(Nk )) one, for k large enough and suppose that Nk+1 nV (ℵn ) ≤ V (ℵNk ) ≤ exp Nk  take n ≥ Nk and n ≤ Nk+1  1 √ (1 + o(1)) (log(Nk ) + (d + 1 + u) log(log(Nk ))), 2 k that entails,  nV (ℵn ) ≤ 1 +  1 (1 + o(1))) (log(n) + (d + 1 + u) log(log(n))). 2 log(n) Finally for n large enough, nV (ℵn ) ≤ log(n) + (d + 1 + u) log(log(n)) + 1, 22 so that nV (ℵn ) − log(n) 1 ≤d+1+u+ , log(log(n)) log(log n) which concludes the proof of (4). 5 Appendix B 5.1 Proof of Theorem 3 The proof make use of the following two propositions. The first one gives conditions under which the maximal-spacing of two compact sets are close. The second one shows that if the set S is not convex, then R(H(S) \ S) > 0. Proposition 3. Let A and B be bounded and non-empty subsets of Rd . If dH (A, B) ≤ ε and dH (∂A, ∂B) ≤ ε. Then R(A) − R(B) ≤ 2ε.  Proof. First we introduce A0 = x ∈ A : d(x, ∂A) > 2ε and prove that A0 ⊂ B by contradiction. Suppose that there exists x ∈ A such that d(x, ∂A) > 2ε and x ∈ / B. Since dH (A, B) ≤ ε we have A ⊂ B ε , then x ∈ B ε \ B, so d(x, ∂B) ≤ ε. Now, as dH (∂A, ∂B) ≤ ε, by the triangular inequality, d(x, ∂A) ≤ 2ε. which is a contradiction. From A0 ⊂ B it follows that R(A0 ) ≤ R(B). Now for all r < R(A) there exist x ∈ A such that B(x, r) ⊂ A so that B(x, r − 2ε) ⊂ A0 which entails R(A0 ) ≥ R(A) − 2ε and, finally, R(B) ≥ R(A) − 2ε. Proceeding in the same way, we get R(A) ≥ R(B) − 2ε that conclude the proof. d Proposition
4. Let S ⊂ R be a non–convex, closed set with non-empty interior. Then, R H(S) \ S > 0. Proof. Since S is closed and non-convex there exists x ∈ H(S) \ S with d(x, S) = r > 0. ˚ so that, for all ε > 0, there exists By Corollary 7.1 in [10] we know that H(S) = H(S) xε and νε > 0 such that |xε − x| ≤ ε and B(xε , ν
ε ) ⊂ H(S). Taking ε = r/2 and ρ = min(νr/2 , r/2) > 0 we conclude that R H(S) \ S ≥ ρ > 0. Now we can prove Theorem 3.

First observe that, if S is convex R H(ℵn ) \ ℵn ≤ R S \ ℵn and |H(ℵn )| ≤ |S| so
that V˜n ≤ V (ℵn ). Then, from Corollary 1 we obtain that P V˜n > cn,γ ≤ γ + o(1), and the test is asymptotically of level smaller than γ.
Now we prove that if S ∈ CP , then PH0 Ṽn > cn,γ → γ. Recall that, if S ⊂ Rd is convex and ℵn = {X1 , . . . , Xn } is an iid random sample, uniformly drawn on S ∈ CP , in 23 [9] it is proved that, almost surely:  

2/(d+1) 

2/(d+1)  dH H(ℵn ), S = O log(n)/n and dH ∂H(ℵn ), ∂S = O log(n)/n . (30) 


2/(d+1)  Thus, by Proposition 3, we have that R H(ℵn )\ℵn −R S\ℵn = O log(n)/n almost surely. Therefore  1/d 
2/(d+1)  ˜ n (ℵn ) = |H(ℵn )| ∆ ∆(ℵ ) + O log(n)/n a.s. n |S|1/d 
2/(d+1)  The second equation in (30) also provides that |H(ℵn )| = |S|+O log(n)/n almost surely. Finally we have,  
2/(d+1)  1+O Ṽn = V (ℵn ) 1 + O log(n)/n
2/(d+1) !!d log(n)/n a.s. ∆(ℵn ) Observe that from Corollary 1 ii) and iii) ∆(ℵn ) = (log(n)/n)1/d (1 + o(1)) almost surely, then 
1+ d−1  d(d+1) a.s. Ṽn = V (ℵn ) + O log(n)/n This, together with cn,γ = O(log(n)/n) entails that P(Ṽn ≥ cn,γ ) → γ, as desired. To conclude the proof of the theorem consider now that S is not convex. First we prove that, if εn = dH (S, ℵn ), then dH (H(S), H(ℵn )) ≤ 2εn . Indeed, for all x ∈ H(S) there exist x1 ∈ S, x2 ∈ S and λ ∈ [0, 1] such that x = λx1 + (1 − λ)x2 . Since εn = dH (S, ℵn ) there exist Xi ∈ ℵn and Xj ∈ ℵn such that kx1 −Xi k ≤ εn and kx2 −Xj k ≤ εn so that y = λXi +(1−λ)Xj belongs to H(ℵn ). By the triangular inequality kx−yk ≤ 2εn . Since H(ℵn ) ⊂ H(S) we also have that dH (∂H(S), ∂H(ℵn )) ≤ 2εn , which implies that kH(ℵn )|−|H(S)k ≤ O(εn ). On the other hand we have dH (H(S)\ℵn , H(ℵn )\ℵn ) ≤ 2εn and dH (∂(H(S) \ ℵn ), ∂(H(ℵn ) \ ℵn )) ≤ 2εn . By Proposition 3 we get, R(H(ℵn ) \ ℵn ) ≥ R(H(S) \ ℵn ) − 2εn . Since H(S) \ S ⊂ H(S) \ ℵn it follows, R(H(ℵn ) \ ℵn ) ≥ R(H(S) \ S) − 2εn . a.s. (31) Since εn → 0 almost surely, |H(ℵn )| → |H(S)| and R(H(ℵn ) \ ℵn ) ≥ R(H(S) \ S) eventually almost surely. Then, there exists CS a positive constant that depends on S such that, Ṽn ≥ CS eventually almost surely. Finally, since cn,γ → 0 we conclude the proof. 5.2 Proof of Theorem 4 The proof of Theorem 4 is based on the following lemma 24 Lemma 5. Assume that the unknown density f fulfils condition B and S ∈ A. Take K ∈ K, and hn = O(n−β ) with β ∈ (0, 1/d). Let fˆn (x) be the density estimator introduced in Definition 3. Then, + (i) there exists a sequence ε+ n such that log(n)εn → 0 and for all x ∈ S, 1− ε+ n  f (x) fˆn (x) 1/d ≥ e.a.s. (ii) there exist a sequence ε− n → 0 and a constant λ0 > 0 such that for all x ∈ H(ℵn ), (fˆn (x))1/d ≥ λ0 − ε− . e.a.s. n Proof. We start the proof by establishing some useful preliminary results. First notice that, for S ∈ A, with exactly the same kind of calculation we did to prove Lemma 2,  1/d n choosing ρn = f04clog we have, S ωd n P(dH (ℵn , S) ≥ ρn ) ≤ CS n−2 for n large enough. (32) Notice that, since K ∈ K, S ∈ A, and K is bounded from below on a neighbourhood of the origin, there exist c00K > 0 and rK > 0 such that, Z 0 K((u − x)/r)du ≥ c00K rd for all x ∈ S and r ≤ rK . (33) S We have, for all x ∈ S, Z Efn (x) = K(u)f (x + uhn )du. {u:x+uhn ∈S} Using that f is Lipschitz and Z Efn (x) ≤ R Rd K(u)du = 1, we get, for all x ∈ S
K(u) f (x) + kf kukhn du ≤ f (x) + kf hn cK . (34) {u:x+uhn ∈S} From (33) and the condition f (x) > f0 for all x ∈ S, it follows that, Efn (x) ≥ f0 c0K for all x ∈ S. We start by proving (i). The triangular inequality entails that,

max fˆn (x) − f (x) ≤ sup fˆn (x) − Efˆn (x) + sup Efˆn (x) − f (x) . x∈S (35) (36) x∈S x∈S In order to deal with the first term on the right hand side of (36), observe that, since K ∈ K and hn = O(n−β ) with β ∈ (0, 1/d) we can apply Theorem 2.3 in [11]. Then, there exists a constant C1 such that, with probability one, for n large enough, s nhdn sup fn (x) − Efn (x) ≤ C1 . − log(hn ) x∈Rd 25 Thus s nhdn sup fn (x) − Efn (x) ≤ C1 , − log(hn ) x∈ℵn and therefore, s nhdn sup fˆn (x) − Efˆn (x) ≤ C1 . − log(hn ) x∈S (37) Now let us bound supx∈S (Efˆn (x) − f (x)) from above. For all x ∈ S we have, E(fˆn (x)) = E(fˆn (x)|dH (ℵn , S) ≤ ρn )P(dH (ℵn , S) ≤ ρn )+ E(fˆn (x)|dH (ℵn , S) > ρn )P(dH (ℵn , S) > ρn ). (38) Since {(x, y) ∈ S 2 , kx − yk ≤ hn } is compact, the Lebesgue dominate convergence theorem entails that there exist y0 ∈ S such that kx − y0 k ≤ ρn , and a sequence yk with yk → y0 , kyk − y0 k ≤ ρn , such that for n large enough, with probability one  E(fˆn (x)|dH (S, ℵn ) ≤ ρn ) ≤ sup E x∈S lim sup  fn (y) = y∈S:kx−yk≤ρn   sup E lim fn (yk ) = sup lim (E(fn (yk ))) ≤ sup x∈S yk →y0 x∈S yk →y0 sup E(fn (y)). x∈S y∈S:kx−yk≤ρn Now applying (34) and the Lipschitz continuity of f we obtain, E(fˆn (x)|dH (S, ℵn ) ≤ ρn ) ≤ max y∈S,kx−yk≤ρn {f (y) + kf hn cK } ≤ f (x) + kf ρn + kf hn cK . (39) With the same kind of argument it can be proved that, E(fˆn (x)|dH (ℵn , S) ≥ ρn ) ≤ sup Efn (y) ≤ f1 + kf hn cK . (40) y∈S From equations (38),(39),(40) and (32) we get, sup(E(fˆn (x)) − f (x)) ≤ kf ρn + kf hn cK + (f1 + kf hn cK )CS n−2 . (41) x∈S  1/2 nhdn Take now εn = kf ρn + kf hn cK + (f1 + kf hn cK )CS n−2 + C1 − log(h . Which n) fulfils log(n)εn → 0. From equations (36), (37), (41), we obtain that, with probability one, for n large enough,
max fˆn (x) − f (x) ≤ εn . x∈S Then, for all x ∈ S, fˆn (x) − f (x) ≤ f (x)εn /f0 , and thus, lently, !1/d   εn −1/d f (x) ≥ 1+ . f0 fˆn (x) 26 fˆn (x) f (x) ≤ 1+ εn f0 , or equiva- Finally, taking ε+ = (1 − (1 + εn /f0 )−1/d ) ∼ εn /(df0 ) (observe that we have ε+ n log(n) → n 1/d 0) then maxx∈S ˆf (x) ≥ 1 − ε+ n eventually almost surely, which concludes the proof fn (x) of (i). In order to prove (ii), observe that min fˆn (x) ≥ min Efˆn (x) − max Efˆn (x) − fˆn (x) . x∈Rd x∈Rd x∈Rd Since we have already proved that maxx∈Rd Efˆn (x)− fˆn (x) → 0 a.s., it remains to prove that minx∈Rd Efˆn (x) is bounded from below by a positive constant. From minx∈Rd Efˆn (x) = minx∈ℵn Efn (x) and (35), we get min Efˆn (x) ≥ min Efn (x) ≥ f0 c0K . x∈S x∈Rd Now we are ready to prove Theorem 4 . Proof of Theorem 4 a) Recall that  δ̂ H(ℵn ) \ ℵn = sup r : ∃x such that x +
 r fˆn (x)1/d A ⊂ H(ℵn ) \ ℵn , −1/d with A the ball B(O, ωd ). Under H0 (S is convex),  δ̂ H(ℵn ) \ ℵn ≤ sup r : ∃x such that x +
r fˆn (x)1/d  A ⊂ S \ ℵn . If we apply Lemma 5 (i), (notice that all convex sets are in A) we get,  r + (1 − εn )A ⊂ S \ ℵn . δ̂ H(ℵn ) \ ℵn ≤ sup r : ∃x such that x + f (x)1/d
Equivalently, ∆(ℵn ) ≥ (1−ε+ n )δ̂ H(ℵn )\ℵn , and therefore P(V̂n ≥ cn,γ ) ≤ P V (ℵn ) ≥
d (1 − ε+ n ) cn,γ , from where it follows that P(V̂n ≥ cn,γ ) can be majorized by, 


 d + d P U (ℵn ) ≥ −(1−ε+ n ) log −log(1−γ) + (1−εn ) −1 log(n)+(d−1) log(log(n))+log(αB ) .
 Therefore, by Theorem 2, (using that log(n)ε+ n → 0) we get that,
P(V̂n ≥ cn,γ ) ≤ P U (ℵn ) ≥ − log(− log(1 − γ)) + o(1) → γ. 27 Proof of Theorem 4 b) From Lemma 5 (ii), we have that

δ̂ H(ℵn ) \ ℵn ≥ (λ0 − ε− n )R H(ℵn ) \ ℵn , where ε− n → 0 a.s. Then, under H1 (S is not convex), from (31), we obtain


δ̂ H(ℵn ) \ ℵn ≥ (λ0 − ε− n ) R H(S) \ S − 2dH (S, ℵn ) .
Since S ∈ A, dH (S, ℵn ) → 0 a.s. (see [4]), and R H(S) \ S > 0 (see Proposition 4) then with probability one, for n large enough

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Multimodal Deep Learning for Robust RGB-D Object Recognition Andreas Eitel Jost Tobias Springenberg Luciano Spinello Martin Riedmiller Wolfram Burgard arXiv:1507.06821v2 [] 18 Aug 2015 input Abstract— Robust object recognition is a crucial ingredient of many, if not all, real-world robotics applications. This paper leverages recent progress on Convolutional Neural Networks (CNNs) and proposes a novel RGB-D architecture for object recognition. Our architecture is composed of two separate CNN processing streams – one for each modality – which are consecutively combined with a late fusion network. We focus on learning with imperfect sensor data, a typical problem in real-world robotics tasks. For accurate learning, we introduce a multi-stage training methodology and two crucial ingredients for handling depth data with CNNs. The first, an effective encoding of depth information for CNNs that enables learning without the need for large depth datasets. The second, a data augmentation scheme for robust learning with depth images by corrupting them with realistic noise patterns. We present stateof-the-art results on the RGB-D object dataset [15] and show recognition in challenging RGB-D real-world noisy settings. I. I NTRODUCTION RGB-D object recognition is a challenging task that is at the core of many applications in robotics, indoor and outdoor. Nowadays, RGB-D sensors are ubiquitous in many robotic systems. They are inexpensive, widely supported by open source software, do not require complicated hardware and provide unique sensing capabilities. Compared to RGB data, which provides information about appearance and texture, depth data contains additional information about object shape and it is invariant to lighting or color variations. In this paper, we propose a new method for object recognition from RGB-D data. In particular, we focus on making recognition robust to imperfect sensor data. A scenario typical for many robotics tasks. Our approach builds on recent advances from the machine learning and computer vision community. Specifically, we extend classical convolutional neural network networks (CNNs), which have recently been shown to be remarkably successful for recognition on RGB images [13], to the domain of RGB-D data. Our architecture, which is depicted in Fig. 1, consists of two convolutional network streams operating on color and depth information respectively. The network automatically learns to combine these two processing streams in a late fusion approach. This architecture bears similarity to other recent multi-stream approaches [21], [23], [11]. Training of the individual stream networks as well as the combined architecture follows a stage-wise approach. We start by separately training the networks for each modality, followed by a third training stage in which the two streams are jointly finetuned, together with a fusion network that performs the final All authors are with the Department of Computer Science, University of Freiburg, Germany. This work was partially funded by the DFG under the priority programm “Autonomous Learning” (SPP 1527). {eitel, springj, riedmiller, spinello, burgard}@cs.uni-freiburg.de 227 conv-1 conv-2 RGB conv-3 fc6 conv-4 fc7 fc1-fus class conv-5 227 3 96 256 256 384 384 256 384 384 256 depth 4096 4096 4096 51 96 3 Fig. 1: Two-stream convolutional neural network for RGBD object recognition. The input of the network is an RGB and depth image pair of size 227 × 227 × 3. Each stream (blue, green) consists of five convolutional layers and two fully connected layers. Both streams converge in one fully connected layer and a softmax classifier (gray). classification. We initialize both the RGB and depth stream network with weights from a network pre-trained on the ImageNet dataset [19]. While initializing an RGB network from a pre-trained ImageNet network is straight-forward, using such a network for processing depth data is not. Ideally, one would want to directly train a network for recognition from depth data without pre-training on a different modality which, however, is infeasible due to lack of large scale labeled depth datasets. Due to this lack of labeled training data, a pre-training phase for the depth-modality – leveraging RGB data – becomes of key importance. We therefore propose a depth data encoding to enable re-use of CNNs trained on ImageNet for recognition from depth data. The intuition – proved experimentally – is to simply encode a depth image as a rendered RGB image, spreading the information contained in the depth data over all three RGB channels and then using a standard (pre-trained) CNN for recongition. In real-world environments, objects are often subject to occlusions and sensor noise. In this paper, we propose a data augmentation technique for depth data that can be used for robust training. We augment the available training examples by corrupting the depth data with missing data patterns sampled from real-world environments. Using these two techniques, our system can both learn robust depth features and implicitly weight the importance of the two modalities. We tested our method to support our claims: first, we report on RGB-D recognition accuracy, then on robustness with respect to real-world noise. For the first, we show that our work outperforms the current state of the art on the RGB-D Object dataset of Lai et al. [15]. For the second, we show that our data augmentation approach improves object recognition accuracy in a challenging real-world and noisy environment using the RGB-D Scenes dataset [16]. II. R ELATED W ORK Our approach is related to a large body of work on both convolutional neural networks (CNNs) for object recognition as well as applications of computer vision techniques to the problem of recognition from RGB-D data. Although a comprehensive review of the literature on CNNs and object recognition is out of the scope of this paper, we will briefly highlight connections and differences between our approach and existing work with a focus on recent literature. Among the many successful algorithms for RGB-D object recognition a large portion still relies on hand designed features such as SIFT in combination with multiple shape features on the depth channel [15], [14]. However, following their success in many computer vision problems, unsupervised feature learning methods have recently been extended to RGB-D recognition settings. Blum et al. [3] proposed an RGB-D descriptor that relies on a K-Means based feature learning approach. More recently Bo et al. [5] proposed hierarchical matching pursuit (HMP), a hierarchical sparsecoding method that can learn features from multiple channel input. A different approach pursued by Socher et al. [22] relies on combining convolutional filters with a recursive neural network (a specialized form of recurrent neural network) as the recognition architecture. Asif et al. [1] report improved recognition performance using a cascade of Random Forest classifiers that are fused in a hierarchical manner. Finally, in recent independent work Schwarz et al. [20] proposed to use features extracted from CNNs pre-trained on ImageNet for RGB-D object recognition. While they also make use of a two-stream network they do not fine-tune the CNN for RGB-D recognition, but rather just use the pre-trained network as is. Interestingly, they also discovered that simple colorization methods for depth are competitive to more involved preprocessing techniques. In contrast to their work, ours achieves higher accuracy by training our fusion CNN end-to-end: mapping from raw pixels to object classes in a supervised manner (with pre-training on a related recognition task). The features learned in our CNN are therefore by construction discriminative for the task at hand. Using CNNs trained for object recognition has a long history in computer vision and machine learning. While they have been known to yield good results on supervised image classification tasks such as MNIST for a long time [17], recently they were not only shown to outperform classical methods in large scale image classification tasks [13], object detection [9] and semantic segmentation [8] but also to produce features that transfer between tasks [7], [2]. This recent success story has been made possible through optimized implementations for high-performance computing systems, as well as the availability of large amounts of labeled image data through, e.g., the ImageNet dataset [19]. While the majority of work in deep learning has focused on 2D images, recent research has also been directed towards using depth information for improving scene labeling and object detection [6], [10]. Among them, the work most similar to ours is the one on object detection by Gupta et al. [10] who introduces a generalized method of the R-CNN detector [9] that can be applied to depth data. Specifically, they use large CNNs already trained on RGB images to also extract features from depth data, encoding depth information into three channels (HHA encoding). Specifically, they encode for each pixel the height above ground, the horizontal disparity and the pixelwise angle between a surface normal and the gravity direction. Our fusion network architecture shares similarities with their work in the usage of pre-trained networks on RGB images. Our method differs in both the encoding of depth into color image data and in the fusion approach taken to combine information from both modalities. For the encoding step, we propose an encoding method for depth images (’colorizing’ depth) that does not rely on complicated preprocessing and results in improved performance when compared to the HHA encoding. To accomplish sensor fusion we introduce additional layers to our CNN pipeline (see Fig. 1) allowing us to automatically learn a fusion strategy for the recognition task – in contrast to simply training a linear classifier on top of features extracted from both modalities. Multi-stream architectures have also been used for tasks such as action recognition [21], detection [11] and image retrieval [23]. An interesting recent overview of different network architectures for fusing depth and image information is given in Saxena et al. [18]. There, the authors compared different models for multimodal learning: (1) early fusion, in which the input image is concatenated to the existing image RGB channels and processed alongside; (2) an approach we denote as late fusion, where features are trained separately for each modality and then merged at higher layers; (3) combining early and late fusion; concluding that late fusion (2) and the combined approach perform best for the problem of grasp detection. Compared to their work, our model is similar to the late fusion approach but widely differs in training – Saxena et al. [18] use a layer-wise unsupervised training approach – and scale (the size of both their networks and input images is an order of magnitude smaller than in our settings). III. M ULTIMODAL ARCHITECTURE FOR RGB-D OBJECT RECOGNITION An overview of the architecture is given in Fig. 1. Our network consists of two streams (top-blue and bottom-green part in the figure) – processing RGB and depth data independently – which are combined in a late fusion approach. Each stream consists of a deep CNN that has been pretrained for object classification on the ImageNet database (we use the CaffeNet [12] implementation of the CNN from Krizhevsky et al. [13]). The key reason behind starting from original size 0 0 0 0 0 1 1 0 0 1 1 our approach 0 warped Fig. 2: Different approaches for color encoding of depth images. From left to right: RGB, depth-gray, surface normals [5], HHA [10], our method. 1 1 1 0 1 1 Fig. 3: CNNs require a fixed size input. Instead of the widely used image warping approach (middle), our method (bottom) preserves shape information and ratio of the objects. We rescale the longer side and create additional image context, by tiling the pixels at the border of the longer side, e.g., 1. We assume that the depth image is already transformed to three channels using our colorization method. a pre-trained network is to enable training a large CNN with millions of parameters using the limited training data available from the Washington RGB-D Object dataset (see, e.g., Yosinski et al. [25] for a recent discussion). We first pre-process data from both modalities to fully leverage the ImageNet pre-training. Then, we train our multimodal CNN in a stage-wise manner. We fine-tune the parameters of each individual stream network for classification of the target data and proceed with the final training stage in which we jointly train the parameters of the fusion network. The different steps will be outlined in the following sections. A. Input preprocessing To fully leverage the power of CNNs pre-trained on ImageNet, we pre-process the RGB and depth input data such that it is compatible with the kind of original ImageNet input. Specifically, we use the reference implementation of the CaffeNet [12] that expects 227 × 227 pixel RGB images as input which are typically randomly cropped from larger 256 × 256 RGB images (see implementation details on data augmentation). The first processing step consists of scaling the images to the appropriate image size. The simplest approach to achieve this is to use image warping by directly rescaling the original image to the required image dimensions, disregarding the original object ratio. This is depicted in Fig. 3 (middle). We found in our experiments that this process is detrimental to object recognition performance – an effect that we attribute to a loss of shape information (see also Section IV-C). We therefore devise a different preprocessing approach: we scale the longest side of the original image to 256 pixels, resulting in a 256 × N or an N × 256 sized image. We then tile the borders of the longest side along the axis of the shorter side. The resulting RGB or depth image shows an artificial context around the object borders (see Fig. 3). The same scaling operation is applied to both RGB and depth images. While the RGB images can be directly used as inputs for the CNNs after this processing step, the rescaled depth data requires additional steps. To realize this, recall that a network trained on ImageNet has been trained to recognize objects in images that follow a specific input distribution (that of natural camera images) that is incompatible with data coming from a depth sensor – which essentially encodes distance of objects from the sensor. Nonetheless, by looking at a typical depth image from a household object scene (c.f., Fig. 4) one can conclude that many features that qualitatively appear in RGB images – such as edges, corners, shaded regions – are also visible in, e.g., a grayscale rendering of depth data. This realization has previously led to the idea of simply using a rendered version of the recorded depth data as an input for CNNs trained on ImageNet [10]. We compare different such encoding strategies for rendering depth to images in our experiments. The two most prevalent such encodings are (1) rendering of depth data into grayscale and replicating the grayscale values to the three channels required as network input; (2) using surface normals where each dimension of a normal vector corresponds to one channel in the resulting image. A more involved method, called HHA encoding [10], encodes in the three channels the height above ground, horizontal disparity and the pixelwise angle between a surface normal and the gravity direction. We propose a fourth, effective and computationally inexpensive, encoding of depth to color images, which we found to outperform the HHA encoding for object recognition. Our method first normalizes all depth values to lie between 0 and 255. Then, we apply a jet colormap on the given image that transforms the input from a single to a three channel image (colorizing the depth). For each pixel (i, j) in the depth image d of size W × H, we map the distance to color values ranging from red (near) over green to blue (far), essen- tially distributing the depth information over all three RGB channels. Edges in these three channels often correspond to interesting object boundaries. Since the network is designed for RGB images, the colorization procedure provides enough common structure between a depth and an RGB image to learn suitable feature representations (see Fig. 2 for a comparison between different depth preprocessing methods). object boundaries occlusion B. Network training noise min N X WD ,θ D

L softmax WD g D (di ; θD ) , y i , (1) k noise patterns Fig. 5: We create synthetic training data by inducing artificial patterns of missing depth information in the encoded image. log likelihood min Wf ,θ I ,θ D ,θ F N X

L softmax Wf f ([gI , gD ]; θF ) , y i , i=1 (2) where gI = g I (xi ; θI ), gD = g D (di ; θD ). Note that in this stage training can also be performed by optimizing only the weights of the fusion network (effectively keeping the weights from the individual stream training intact). C. Robust classification from depth images i=1 where W D are the weights of the ping from g(·) to RM , the softmax softmax(z) = P exp(z)/kzk1 and the L(s, y) = − yk log sk . Training the Fig. 4: Kitchen scene in the RGB-D Scenes dataset showing objects subjected to noise and occlusions. noised image Let D = {(x1 , d1 , y1 ), . . . , (xN , dN , yN )} be the labeled data available for training our multimodal CNN; with xi , di denoting the RGB and pre-processed depth image respectively and yi corresponding to the image label in one-hot encoding – i.e., yi ∈ RM is a vector of dimensionality M (the number of labels) with yki = 1 for the position k denoting the image label. We train our model using a three-stage approach, first training the two stream networks individually followed by a joint fine-tuning stage. 1) Training the stream networks: We first proceed by training the two individual stream networks (c.f., the blue and green streams in Fig. 1). Let g I (xi ; θI ) be the representation extracted from the last fully connected layer (fc7) of the CaffeNet – with parameters θI – when applied to an RGB image xi . Analogously, let g D (di ; θD ) be the representation for the depth image. We will assume that all parameters θI and θD (the network weights and biases) are initialized by copying the parameters of a CaffeNet trained on the ImageNet dataset. We can then train an individual stream network by placing a randomly initialized softmax classification layer on top of f D and f I and minimizing the negative log likelihood L of the training data. That is, for the depth image stream network we solve softmax layer mapfunction is given by loss is computed as RGB stream network then can be performed by an analogous optimization. After training, the resulting networks can be used to perform separate classification of each modality. 2) Training the fusion network: Once the two individual stream networks are trained we discard their softmax weights, concatenate their – now fine-tuned – last layer responses g I (xi ; θI ) and g D (di ; θD ) and feed them through an additional fusion stream f ([g I (xi ; θI ), g D (di ; θD )]; θF ) with parameters θF . This fusion network again ends in a softmax classification layer. The complete setup is depicted in Fig. 1, where the two fc7 layers (blue and green) are concatenated and merge into the fusion network (here the inner product layer fc1-fus depicted in gray). Analogous to Eq. (1) the fusion network can therefore be trained by jointly optimizing all parameters to minimize the negative Finally, we are interested in using our approach in real world robotics scenarios. Robots are supposed to perform object recognition in cluttered scenes where the perceived sensor data is subject to changing external conditions (such as lighting) and sensor noise. Depth sensors are especially affected by a non-negligible amount of noise in such setups. This is mainly due to the fact that reflective properties of materials as well as their coating, often result in missing depth information. An example of noisy depth data is depicted in Fig. 4. In contrast to the relatively clean training data from the Washington RGB-D Object dataset, the depicted scene contains considerable amounts of missing depth values and partial occlusions (the black pixels in the figure). To achieve robustness against such unpredictable factors, we propose a new data augmentation scheme that generates new, noised training examples for training and is tailored specifically to robust classification from depth data. Our approach utilizes the observation that noise in depth data often shows a characteristic pattern and appears at object boundaries or object surfaces. Concretely, we sampled a representative set of noise patterns P = {P1 , . . . , PK } that occur when recording typical indoor scenes through a Kinect sensor. For sampling the noise patterns we used the RGB-D SLAM dataset [24]. First, we extract 33,000 random noise patches of size 256 × 256 from different sequences at varying positions and divide them into five groups, based on the number of missing depth readings they contain. Those noise patches are 2D binary masks patterns. We randomly sample pairs of noise patches from two different groups that are randomly added or subtracted and optionally inverted to produce a final noise mask pattern. We repeat this process until we have collected K = 50, 000 noise patterns in total. Examples of the resulting noise patterns and their application to training examples are shown in Fig. 5. Training the depth network with artificial noise patterns then proceeds by minimizing the objective from Equation Eq. (1) in which each depth sample di is randomly replaced with a noised variant with probability 50%. Formally, ( p ∼ B{0.5} di if p = 1 i with (3) d = i k ∼ U{1, K}, Pk ◦ d else where ◦ denotes the Hadamard product, B the Bernoulli distribution and U the discrete uniform distribution. IV. E XPERIMENTS We evaluate our multimodal network architecture on the Washington RGB-D Object Dataset [15] which consists of household objects belonging to 51 different classes. As an additional experiment – to evaluate the robustness of our approach for classification in real-world environments – we considered classification of objects from the RGB-D Scenes dataset whose class distribution partially overlaps with the RGB-D Object Dataset. A. Experimental setup All experiments were performed using the publicly available Caffe framework [12]. As described previously we use the CaffeNet as the basis for our fusion network. It consists of five convolutional layers (with max-pooling after the first, second and fifth convolution layer) followed by two fully connected layers and a softmax classification layer. Rectified linear units are used in all but the final classification layer. We initialized both stream networks with the weights and biases of the first eight layers from this pre-trained network, discarding the softmax layer. We then proceeded with our stage-wise training. In the first stage (training the RGB and depth streams independently) the parameters of all layers were adapted using a fixed learning rate schedule (with initial learning rate of 0.01 that is reduced to 0.001 after 20K iterations and training is stopped after 30K iterations). In the second stage (training the fusion network, 20k iterations, mini-batch size of 50) we experimented with fine-tuning all weights but found that fixing the individual stream networks (by setting their learning rate to zero) and only training the fusion part of the network resulted in the best performance. The number of training iterations were chosen based on the validation performance on a training validation split in TABLE I: Comparisons of our fusion network with other approaches reported for the RGB-D dataset. Results are recognition accuracy in percent. Our multi-modal CNN outperforms all the previous approaches. Method Nonlinear SVM [15] HKDES [4] Kernel Desc. [14] CKM Desc. [3] CNN-RNN [22] Upgraded HMP [5] CaRFs [1] CNN Features [20] Ours, Fus-CNN (HHA) Ours, Fus-CNN (jet) RGB 74.5 ± 3.1 76.1 ± 2.2 77.7 ± 1.9 N/A 80.8 ± 4.2 82.4 ± 3.1 N/A 83.1 ± 2.0 84.1 ± 2.7 84.1 ± 2.7 Depth 64.7 ± 2.2 75.7 ± 2.6 78.8 ± 2.7 N/A 78.9 ± 3.8 81.2 ± 2.3 N/A N/A 83.0 ± 2.7 83.8 ± 2.7 RGB-D 83.9 ± 3.5 84.1 ± 2.2 86.2 ± 2.1 86.4 ± 2.3 86.8 ± 3.3 87.5 ± 2.9 88.1 ± 2.4 89.4 ± 1.3 91.0 ± 1.9 91.3 ± 1.4 a preliminary experiment. A fixed momentum value of 0.9 and a mini-batch size of 128 was used for all experiments if not stated otherwise. We also adopted the common data augmentation practices of randomly cropping 227 × 227 sub-images from the larger 256 × 256 input examples and perform random horizontal flipping. Training of a single network stream takes ten hours, using a NVIDIA 780 graphics card. B. RGB-D Object dataset The Washington RGB-D Object Dataset consists of 41,877 RGB-D images containing household objects organized into 51 different classes and a total of 300 instances of these classes which are captured under three different viewpoint angles. For the evaluation every 5th frame is subsampled. We evaluate our method on the challenging category recognition task, using the same ten cross-validation splits as in Lai et al. [15]. Each split consists of roughly 35,000 training images and 7,000 images for testing. From each object class one instance is left out for testing and training is performed on the remaining 300−51 = 249 instances. At test time the task of the CNN is to assign the correct class label to a previously unseen object instance. Table I shows the average accuracy of our multi-modal CNN in comparison to the best results reported in the literature. Our best multi-modal CNN, using the jet-colorization, (Fus-CNN jet) yields an overall accuracy of 91.3 ± 1.4% when using RGB and depth (84.1 ± 2.7% and 83.8 ± 2.7% when only the RGB or depth modality is used respectively), which – to the best of our knowledge – is the highest accuracy reported for this dataset to date. We also report results for combining the more computationally intensive HHA with our network (Fus-CNN HHA). As can be seen in the table, this did not result in an increased performance. The depth colorization method slightly outperforms the HHA fusion network (Fus-CNN HHA) while being computationally cheaper. Overall our experiments show that a pretrained CNN can be adapted for recognition from depth data using our depth colorization method. Apart from the results reported in the table, we also experimented with different fusion architectures. Specifically, performance slightly drops to 91% when the intermediate fusion layer (fc1-fus) is removed from the network. Adding additional fusion layers 1.0 0.8 0.6 0.4 0.2 lime notebook flashlight banana glue stick cereal box hand towel sponge marker lemon scissors toothbrush instant noodles food can water bottle toothpaste comb food cup onion rubber eraser binder food bag pliers dry battery greens soda can garlic shampoo tomato kleenex lightbulb keyboard stapler bell pepper cap bowl apple orange coffee mug food jar plate cell phone food box calculator potato pear camera ball pitcher peach mushroom 0.0 Fig. 6: Per-class recall of our trained model on all test-splits. The worst class recall belongs to mushrooms and peaches. also did not yield an improvement. Finally, Fig. 6 shows the per-class recall, where roughly half of the objects achieve a recall of ≈ 99%. C. Depth domain adaptation for RGB-D Scenes To test the effectiveness of our depth augmentation technique in real world scenes, we performed additional recognition experiments on the more challenging RGB-D Scenes dataset. This dataset consists of six object classes (which overlap with the RGB-D Object Dataset) and a large amount of depth images subjected to noise. For this experiment we trained two single-stream depthonly networks using the Object dataset and used the Scenes dataset for testing. Further, we assume that the groundtruth bounding box is given in order to report only on recognition performance. The first “baseline” network is trained by following the procedure described in Section III-B.1, with the total number of labels M = 6. The second network is trained by making use of the depth augmentation outlined in III-C. The results of this experiment are shown in Table II (middle and right column) that reports the recognition accuracy for each object class averaged over all eight video sequences. As is evident from the table, the adapted network (right column) trained with data augmentation outperforms the baseline model for all classes, clearly indicating that additional domain adaptation is necessary for robust recognition in real world scenes. However, some classes (e.g., cap, bowl, soda can) benefit more from noise aware training than others (e.g., flashlight, coffe mug). The kitchen scene depicted in Fig. 4 gives a visual intuition for this result. On the one hand, some objects (e.g., soda cans) often present very noisy object boundaries and surfaces, thus they show improved recognition performance using the adapted approach. On the other hand, small objects (e.g. a flashlight), which are often captured lying on a table, are either less noisy or just small, hence susceptible to be completely erased by the noise from our data augmentation approach. Fig. 7 shows several exemplary noisy depth images from the test set that are correctly classified by the domain-adapted network while the baseline network labels them incorrectly. We also tested the effect of different input image rescaling techniques – previously described in Fig. 3 – in this setting. As shown in the left column of Table II, standard image warping performs poorly, which supports our intuition that shape information TABLE II: Comparison of the domain adapted depth network with the baseline: six-class recognition results (in percent) on the RGB-D Scenes dataset [16] that contains everyday objects in real-world environments. Class flashlight cap bowl soda can cereal box coffee mug class avg. Ours, warp. 93.4 62.1 57.4 64.5 98.3 61.9 73.6 ± 17.9 Ours, no adapt. 97.5 68.5 66.5 66.6 96.2 79.1 79.1 ± 14.5 Ours, adapt. 96.4 77.4 69.8 71.8 97.6 79.8 82.1 ± 12.0 gets lost during preprocessing. D. Comparison of depth encoding methods Finally, we conducted experiments to compare the different depth encoding methods described in Fig. 2. For rescaling the images, we use our proposed preprocessing method described in Fig. 3 and tested the different depth encoding. Two scenarios are considered: 1) training from scratch using single channel depth images 2) for each encoding method, only fine-tuning the network by using the procedure described in Section III-B.1. When training from scratch, the initial learning rate is set to 0.01, then changed to 0.001 after 40K iterations thus stopped after 60K iterations. Training with more iterations did not further improve the accuracy. From the results, presented in Table III, it is clear that training the network from scratch – solely on the RGB-D Dataset – is inferior to fine-tuning. In the latter setting, the results suggest that the simplest encoding method (depth-gray) performs considerably worse than the other three methods. Among these other encodings (which all produce colorized images), surface normals and HHA encoding require additional image preprocessing – meanwhile colorizing depth using our depthjet encoding has negligible computational overhead. One potential reason why the HHA encoding underperforms in this setup is that all objects are captured on a turntable with the same height above the ground. The height channel used in the HHA encoding therefore does not encode any additional information for solving the classification task. In this experiment, using surface normals yields slightly better performance than the depth-jet encoding. Therefore, we tested the fusion architecture on the ten splits of the RGBD Object Dataset using the surface normals encoding but this did not further improve the performance. Specifically, the recognition accuracy on the test-set was 91.1 ± 1.6 which is comparable to our reported results in Table I. V. C ONCLUSION We introduce a novel multimodal neural network architecture for RGB-D object recognition, which achieves state of the art performance on the RGB-D Object dataset [15]. Our method consists of a two-stream convolutional neural network that can learn to fuse information from both RGB and depth automatically before classification. We make use of an effective encoding method from depth to image data that allows us to leverage large CNNs trained for object recognition on the ImageNet dataset. We present a novel [7] J. Donahue, Y. Jia, O. Vinyals, J. Hoffman, N. Zhang, E. Tzeng, and T. Darrell, “Decaf: A deep convolutional activation feature for generic visual recognition,” arXiv preprint arXiv:1310.1531, 2013. [8] C. Farabet, C. Couprie, L. Najman, and Y. LeCun, “Learning hierarchical features for scene labeling,” TPAMI, pp. 1915–1929, 2013. [9] R. Girshick, J. Donahue, T. Darrell, and J. 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Fig. 7: Objects from the RGB-D Scenes test-set for which the domain adapted CNN predicts the correct label, while the baseline (no adapt.) CNN fails. Most of these examples are subject to noise or partial occlusion. [14] L. Bo, X. Ren and D.Fox, “Depth kernel descriptors for object recognition,” in Proc. of the IEEE/RSJ Int. Conf. on Intelligent Robots and Systems (IROS), 2011. TABLE III: Comparison of different depth encoding methods on the ten test-splits of the RGB-D Object dataset. [16] K. Lai, L. Bo, X. R. Ren, and D. Fox, “Detection-based object labeling in 3d scenes,” in Proc. of the IEEE Int. Conf. on Robotics & Automation (ICRA), 2012. a) bowl b) cap c) soda can Depth Encoding Depth-gray (single channel), from scratch Depth-gray Surface normals HHA Depth-jet encoding Accuracy 80.1 ± 2.6 82.0 ± 2.8 84.7 ± 2.3 83.0 ± 2.7 83.8 ± 2.7 depth data augmentation that aims at improving recognition in noisy real-world setups, situations typical of many robotics scenarios. 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Learning to Detect Violent Videos using Convolutional Long Short-Term Memory arXiv:1709.06531v1 [] 19 Sep 2017 Swathikiran Sudhakaran1,2 and Oswald Lanz2 1 2 University of Trento, Trento, Italy Fondazione Bruno Kessler, Trento, Italy {sudhakaran,lanz}@fbk.eu Abstract Developing a technique for the automatic analysis of surveillance videos in order to identify the presence of violence is of broad interest. In this work, we propose a deep neural network for the purpose of recognizing violent videos. A convolutional neural network is used to extract frame level features from a video. The frame level features are then aggregated using a variant of the long short term memory that uses convolutional gates. The convolutional neural network along with the convolutional long short term memory is capable of capturing localized spatio-temporal features which enables the analysis of local motion taking place in the video. We also propose to use adjacent frame differences as the input to the model thereby forcing it to encode the changes occurring in the video. The performance of the proposed feature extraction pipeline is evaluated on three standard benchmark datasets in terms of recognition accuracy. Comparison of the results obtained with the state of the art techniques revealed the promising capability of the proposed method in recognizing violent videos. 1. Introduction Nowadays, the amount of public violence has increased dramatically. This can be a terror attack involving one or a number of persons wielding guns to a knife attack by a single person. This has resulted in the ubiquitous usage of surveillance cameras. This has helped authorities in identifying violent attacks and take the necessary steps in order to minimize the disastrous effects. But almost all the systems nowadays require manual human inspection of these videos for identifying such scenarios, which is practically infeasible and inefficient. It is in this context that the proposed study becomes relevant. Having such a practical system that can automatically monitor surveillance videos and identify the violent behavior of humans will be of immense help and assistance to the law and order establishment. In this work, we will be considering aggressive human behavior as violence rather than the presence of blood or fire. The development of several deep learning techniques, brought about by the availability of large datasets and computational resources, has resulted in a landmark change in the computer vision community. Several techniques with improved performance for addressing problems such as object detection, recognition, tracking, action recognition, caption generation, etc. have been developed as a result. However, despite the recent developments in deep learning, very few deep learning based techniques have been proposed to tackle the problem of violence detection from videos. Almost all the existing techniques rely on hand-crafted features for generating visual representations of videos. The most important advantage of deep learning techniques compared to the traditional hand-crafted feature based techniques is the ability of the former to achieve a high degree of generalization. Thus they are able to handle unseen data in a more effective way compared to handcrafted features. Moreover, no prior information about the data is required in the case of a deep neural network and they can be inputted with raw pixel values without much complex pre-processing. Also, deep learning techniques are not application specific unlike the hand-crafted feature based methods since a deep neural network model can be easily applied for a different task without any significant changes to the architecture. Owing to these reasons, we choose to develop a deep neural network for performing violent video recognition. Our contributions can be summarized as follows: • We develop an end-to-end trainable deep neural network model for performing violent video classification • We show that a recurrent neural network capable of encoding localized spatio-temporal changes generates a better representation, with less number of parameters, for detecting the presence of violence in a video • We show that a deep neural network trained on the frame difference performs better than a model trained on raw frames • We experimentally validate the effectiveness of the proposed method using three widely used benchmarks for violent video classification The rest of the document is organized as follows. Section 2 discusses some of the relevant techniques for performing violent video recognition followed by a detailed explanation of the proposed deep neural network model in Section 3. The details about the various experiments conducted as part of this research are given in Section 4 and the document is concluded in Section 5. 2. Related Works Several techniques have been proposed by researchers for addressing the problem of violence detection from videos. These include methods that use the visual content [21, 2], audio content [23, 12] or both [33, 1]. In this section, we will be concentrating on methods that use the visual cues alone since it is more related to the proposed approach and moreover audio data is generally unavailable with surveillance videos. All the existing techniques can be divided into two classes depending on the underlying idea 1. Inter-frame changes: Frames containing violence undergo massive variations because of fast motion due to fights [28, 5, 4, 8] 2. Local motion in videos: The motion change patterns taking place in the video is analyzed [6, 3, 7, 21, 15, 32, 20, 24, 13, 2, 11, 34] Vasconcelos and Lippman [28] used the tangent distance between adjacent frames for detecting the inter-frame variations. Clarin et al. improves this method in [5] by finding the regions with skin and blood and analyzing these regions for fast motion. Chen et al. [4] uses the motion vector encoded in the MPEG-1 video stream for detecting frames with high motion content and then detects the presence of blood for classifying the video as violent. Deniz et al. [8] proposes to use the acceleration estimate computed from the power spectrum of adjacent frames as an indicator of fast motion between successive frames. Motion trajectory information and the orientation of limbs of the persons present in the scene is proposed as a measure for detecting violence by Datta et al. [6]. Several other methods follow the techniques used in action recognition, i.e., to identify spatio-temporal interest points and extract features from these points. These include Harris corner detector [3], Space-time interest points (STIP) [7], motion scale-invariant feature transform (MoSIFT) [21, 32]. Hassner et al. [15] introduces a new feature descriptor called violent flows (ViF), which is the flow magnitude over time of the optical flow between adjacent frames, for detecting violent videos. This method is improved by Gao et al. [11] by incorporating the orientation of the violent flow features resulting in oriented violent flows (OViF) features. Substantial derivative, a concept in fluid dynamics, is proposed by Mohammadi et al. [20] as a discriminative feature for detecting violent videos. Gracia et al. [13] proposes to use the blob features, obtained by subtracting adjacent frames, as the feature descriptor. The improved dense trajectory features commonly used in action recognition is used as a feature vector by Bilinski et al. in [2]. They also propose an improved Fisher encoding technique that can encode spatiotemporal position of features in a video. Zhang et al. [34] proposes to use a modified version of motion Weber local descriptor (MoIWLD) followed by sparse representation as the feature descriptor. The hand-crafted feature based techniques used methods such as bag of words, histogram, improved Fisher encoding, etc. for aggregating the features across the frames. Recently various models using long short term memory (LSTM) RNNs [16] have been developed for addressing problems involving sequences such as machine translation [27], speech recognition [14], caption generation [31, 29] and video action recognition [9, 26]. The LSTM was introduced in 1997 to combat the effect of vanishing gradient problem which was plaguing the deep learning community. The LSTM incorporates a memory unit which contains information about the inputs the LSTM unit has seen and is regulated using a number of fully-connected gates. The same idea of using LSTM for feature aggregation is proposed by Dong et al. in [10] for violence detection. The method consisted of extracting features using a convolutional neural network from raw pixels, optical flow images and acceleration flow maps followed by LSTM based encoding and a late fusion. Recently, Xingjian et al. [30] replaced the fullyconnected gate layers of the LSTM with convolutional layers and used this improved model for predicting precipitation nowcasting from radar images with improved performance. This newer model of the LSTM is named as convolutional LSTM (convLSTM). Later, it has been used for predicting optical flow images from videos [22] and for anomaly detection in videos [18]. By replacing the fully-connected layers in the LSTM with convolutional layers, the convLSTM model is capable of encoding spatiotemporal information in its memory cell. 3. Proposed method The goal of the proposed study was to develop an endto-end trainable deep neural network model for classifying Frame 1 ConvLSTM Frame 2 Frame 2 96 256 384 384 256 96 256 384 384 256 96 256 384 384 256 Hidden State (256) Frame 3 Frame N-1 Frame N 10 2 256 1000 Figure 1. Block diagram of the proposed model. The model consists of alternating convolutional (red), normalization (grey) and pooling (blue) layers. The hidden state of the ConvLSTM (green) at the final time step is used for classification. The fully-connected layers are shown in brown colour. videos in to violent and non-violent ones. The block diagram of the proposed model is illustrated in figure 1. The network consists of a series of convolutional layers followed by max pooling operations for extracting discriminant features and convolutional long short memory (convLSTM) for encoding the frame level changes, that characterizes violent scenes, existing in the video. 3.1. ConvLSTM Videos are sequences of images. For a system to identify if a fight is taking place between the humans present in the video, it should be capable of identifying the locations of the humans and understand how the motion of the said humans are changing with time. Convolutional neural networks (CNN) are capable of generating a good representation of each video frame. For encoding the temporal changes a recurrent neural network (RNN) is required. Since we are interested in changes in both the spatial and temporal dimensions, convLSTM will be a suitable option. Compared to LSTM, the convLSTM will be able to encode the spatial and temporal changes using the convolutional gates present in them. This will result in generating a better representation of the video under analysis. The equations of the convLSTM model are given in equations 1-6. it = σ(wxi ∗ It + whi ∗ ht−1 + bi ) (1) ft = σ(wxf ∗ It + whf ∗ ht−1 + bf ) c̃t = tanh(wxc̃ ∗ It + whc̃ ∗ ht−1 + bc̃ ) (2) (3) ct = c̃t ⊙ it + ct−1 ⊙ ft ot = σ(wxo ∗ It + who ∗ ht−1 + bo ) (4) (5) ht = ot ⊙ tanh(ct ) (6) In the above equations, ‘*’ represents convolution operation and ‘⊙’ represents the Hadamard product. The hidden state ht , the memory cell ct and the gate activations it , ft and ot are all 3D tensors in the case of convLSTM. For a system to identify a video as violent or non-violent, it should be capable of encoding localized spatial features and the manner in which they change with time. Handcrafted features are capable of achieving this with the downside of having increased computational complexity. CNNs are capable of generating discriminant spatial features but existing methods use the features extracted from the fullyconnected layers for temporal encoding using LSTM. The output of the fully-connected layers represents a global descriptor of the whole image. Thus the existing methods fail to encode the localized spatial changes. As a result, they resort to methods involving addition of more streams of data such as optical flow images [10] which results in increased computational complexity. It is in this context that the use of convLSTM becomes relevant as it is capable of encoding the convolutional features of the CNN. Also, the convolutional gates present in the convLSTM is trained to encode the temporal changes of local regions. In this way, the whole network is capable of encoding localized spatiotemporal features. 3.2. Network Architecture Figure 1 illustrates the architecture of the network used for identifying violent videos. The convolutional layers are trained to extract hierarchical features from the video frames and are then aggregated using the convLSTM layer. The network functions as follows: The frames of the video under consideration are applied sequentially to the model. Once all the frames are applied, the hidden state of the convLSTM layer in this final time step contains the representation of the input video frames applied. This video representation, in the hidden state of the convLSTM, is then applied to a series of fully-connected layers for classification. In the proposed model, we used the AlexNet model [17] pre-trained on the ImageNet database as the CNN model for extracting frame level features. Several studies have found Table 1. Classification accuracy obtained with the hockey fight dataset for different models Input Video Frames (random initialization) Video Frames (ImageNet pre-trained) Difference of Video Frames (random initialization) Difference of Video Frames (ImageNet pre-trained) Classification Accuracy 94.1±2.9% 96±0.35% 95.5±0.5% 97.1±0.55% out that networks trained on the ImageNet database is capable of having better generalization and results in improved performance for tasks such as action recognition [25] [19]. In the convLSTM, we used 256 filters in all the gates with a filter size of 3 × 3 and stride 1. Thus the hidden state of the convLSTM consists of 256 feature maps. A batch normalization layer is added before the first fully-connected layer. Rectified linear unit (ReLU) non-linear activation is applied after each of the convolutional and fully-connected layers. In the network, instead of applying the input frames as such, the difference between adjacent frames are given as input. In this way, the network is forced to model the changes taking place in adjacent frames rather than the frames itself. This is inspired by the technique proposed by Simonyan and Zisserman in [25] to use optical flow images as input to a neural network for action recognition. The difference image can be considered as a crude and approximate version of optical flow images. So in the proposed method, the difference between adjacent video frames are applied as input to the network. As a result, the computational complexity involved in the optical flow image generation is avoided. The network is trained to minimize the binary cross entropy loss. rithm. Since the number of videos present in the datasets are limited, data augmentation techniques such as random cropping and horizontal flipping are used during training stage. During each training iteration, a portion of the frame of size 224 × 224 is cropped, from the four corners or from the center, and is randomly flipped before applying to the network. Note that the same augmentation technique is followed for all the frames present in a video. The network is run for 7500 iterations during the training stage. In the evaluation stage, the video frames are resized to 224 × 224 and are applied to the network for classifying them as violent or non-violent. All the training video frames in a dataset are normalized to make their mean zero and variance unity. 4.2. Datasets To evaluate the effectiveness of the proposed approach in classifying violent videos, three benchmark datasets are used and the classification accuracy is reported. The performance of the proposed method is evaluated on three standard public datasets namely, Hockey Fight Dataset [21], Movies Dataset [21] and Violent-Flows Crowd Violence Dataset [15]. They contain videos captured using mobile phones, CCTV cameras and high resolution video cameras. Hockey Fight Dataset: Hockey fight dataset is created by collecting videos of ice hockey matches and contains 500 fighting and non-fighting videos. Almost all the videos in the dataset have a similar background and subjects (humans). 20 frames from each video are used as inputs to the network. Movies Dataset: This dataset consists of fight sequences collected from movies. The non-fight sequences are collected from other publicly available action recognition datasets. The dataset is made up of 100 fight and 100 nonfight videos. As opposed to the hockey fight dataset, the videos of the movies dataset is substantially different in its content. 10 frames from each video are used as inputs to the network. Violent-Flows Dataset: This is a crowd violence dataset as the number of people taking part in the violent events are very large. Most of the videos present in this dataset are collected from violent events taking place during football matches. There are 246 videos in this dataset. 20 frames from each video are used as inputs to the network. 4.1. Experimental Settings 4.3. Results and Discussions The network is implemented using the Torch library. From each video, N number of frames equally spaced in time are extracted and resized to a dimension of 256 × 256 for training. This is to avoid the redundant computations involved in processing all the frames, since adjacent frames contain overlapping information. The number of frames selected is based on the average duration of the videos present in each dataset. The network is trained using RMSprop algorithm with a learning rate of 10−4 and a batch size of 16. The model weights are initialized using Xavier algo- Performance evaluation is done using 5-folds cross validation scheme, which is the technique followed in existing literature. The model architecture selection was done by evaluating the performance of the different models on the hockey fight dataset. The classification accuracies obtained for the two cases, video frames as input and difference of frames as input, is listed in table 1. From the table, it can also be seen that using a network that is pre-trained on the ImageNet dataset (we used BVLC AlexNet from Caffe model zoo) results in better performance compared to us- 4. Experiments and Results Table 2. Comparison of classification results Method MoSIFT+HIK[21] ViF[15] MoSIFT+KDE+Sparse Coding[32] Deniz et al.[8] Gracia et al.[13] Substantial Derivative[20] Bilinski et al.[2] MoIWLD[34] ViF+OViF[11] Three streams + LSTM[10] Proposed Hockey Dataset 90.9% 82.9±0.14% 94.3±1.68% 90.1±0% 82.4±0.4% 93.4 96.8±1.04% 87.5±1.7% 93.9 97.1±0.55% ing a network that is randomly initialized. In this way, we decided to use frame difference as the input and to use a pre-trained network in the model. Table 2 gives the classification accuracy values obtained for the various datasets considered in this study and is compared against 10 state of the art techniques. From the table, it can be seen that the proposed method is able to better the results of the existing techniques in the case of hockey fights dataset and movies dataset. As mentioned earlier, this study considers aggressive behavior as violent. The biggest problem of considering this definition occurs in the case of sports. For instance, in the hockey dataset, the fight videos consists of players colliding against each other and hitting one another. So one easy way to detect violent scenes is to check if one player moves closer to another. But the non-violent videos also consist of players hugging each other or doing high fives as part of a celebration. It is highly likely that these videos could be mistaken as violent. But the proposed method is able to avoid this which suggests that it is capable of encoding motion of localized regions (motion of limbs, reaction of involved persons, etc.). However, in the case of violent-flows dataset, the proposed method is not able to best the previous state of the art technique (it came second in terms of accuracy). Analyzing the dataset, it is found that in most of the violent videos, only a small part of the crowd is found to be involved in aggressive behavior while a large part remained as spectators. This forces the network to mark such videos as non-violent since majority of the people present in it is found to behave normally. Further studies are required for devising techniques to alleviate this problem involved with crowd videos. One technique that can be considered is to divide the frame in to sub-regions and predict the output of the regions separately and mark the video as violent if any of the regions is outputted by the network as violent. In order to compare the advantage of convLSTM over traditional LSTM, a different model that consists of LSTM is trained and tested on the hockey fights dataset. The new Movies Dataset 89.5% 98.0±0.22% 97.8±0.4% 96.89±0.21% 99 100±0% Violent-Flows Dataset 81.3±0.21% 89.05±3.26% 85.43±0.21% 96.4 93.19±0.12% 88±2.45% 94.57±2.34% Table 3. Comparison between convLSTM and LSTM models in terms of classification accuracy obtained in the hockey fights dataset and number of parameters Model convLSTM LSTM Accuracy 97.1±0.55% 94.6±1.19% No. of Parameters 9.6M(9619544) 77.5M(77520072) model consists of the AlexNet architecture followed by an LSTM RNN layer. The output of the last fully-connected layer (fc7) of AlexNet is applied as input to an LSTM with 1000 units. The rest of the architecture is similar to the one that uses convLSTM. The results obtained with this model and the number of trainable parameters associated with it are compared against the proposed model in table 3. The table clearly shows the advantages of using convLSTM over LSTM and the capability of convLSTM in generating useful video representation. It is also worth mentioning that the number of parameters that are required to be optimized, in the case of convLSTM, is very much less compared to LSTM (9.6M vs 77.5M). This helps the network to generalize better without overfitting in the case of limited data. The proposed model is capable of processing 31 frames per second on an NVIDIA K40 GPU. 5. Conclusions This work presents a novel end-to-end trainable deep neural network model for addressing the problem of violence detection in videos. The proposed model consists of a convolutional neural network (CNN) for frame level feature extraction followed by feature aggregation in the temporal domain using convolutional long short term memory (convLSTM). The proposed method is evaluated on three different datasets and resulted in improved performance compared to the state of the art methods. It is also shown that a network trained to model changes in frames (frame difference) performs better than a network trained using frames as inputs. A comparative study between the tradi- tional fully-connected LSTM and convLSTM is also done and the results show that the convLSTM model is capable of generating a better video representation compared to LSTM with less number of parameters, thereby avoiding overfitting. References [1] E. Acar, F. Hopfgartner, and S. Albayrak. Breaking down violence detection: Combining divide-et-impera and coarseto-fine strategies. Neurocomputing, 208:225–237, 2016. 2 [2] P. Bilinski and F. Bremond. Human violence recognition and detection in surveillance videos. In AVSS, 2016. 2, 5 [3] D. Chen, H. Wactlar, M.-y. Chen, C. Gao, A. Bharucha, and A. Hauptmann. Recognition of aggressive human behavior using binary local motion descriptors. In International Conference of the IEEE Engineering in Medicine and Biology Society (EMBS), 2008. 2 [4] L.-H. 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Logical Methods in Computer Science Vol. 12(1:7)2016, pp. 1–32 www.lmcs-online.org Submitted Published Sep. 10, 2015 Mar. 31, 2016 HISTORY-REGISTER AUTOMATA RADU GRIGORE a AND NIKOS TZEVELEKOS b a University of Kent e-mail address: [email protected] b Queen Mary University of London e-mail address: [email protected] Abstract. Programs with dynamic allocation are able to create and use an unbounded number of fresh resources, such as references, objects, files, etc. We propose History-Register Automata (HRA), a new automata-theoretic formalism for modelling such programs. HRAs extend the expressiveness of previous approaches and bring us to the limits of decidability for reachability checks. The distinctive feature of our machines is their use of unbounded memory sets (histories) where input symbols can be selectively stored and compared with symbols to follow. In addition, stored symbols can be consumed or deleted by reset. We show that the combination of consumption and reset capabilities renders the automata powerful enough to imitate counter machines, and yields closure under all regular operations apart from complementation. We moreover examine weaker notions of HRAs which strike different balances between expressiveness and effectiveness. 1. Introduction Program analysis faces substantial challenges due to its aim to devise finitary methods and machines which are required to operate on potentially infinite program computations. A specific such challenge stems from dynamic generative behaviours such as, for example, object or thread creation in Java, or reference creation in ML. A program engaging in such behaviours is expected to generate a possibly unbounded amount of distinct resources, each of which is assigned a unique identifier, a name. Hence, any machine designed for analysing such programs is expected to operate on an infinite alphabet of names. The latter need has brought about the introduction of automata over infinite alphabets in program analysis, starting from prototypical machines for mobile calculi [27] and variable programs [23], and recently developing towards automata for verification tasks such as equivalence checks of ML programs [29, 30], context-bounded analysis of concurrent programs [10, 3] and runtime program monitoring [20]. The literature on automata over infinite alphabets is rich in formalisms each based on a different approach for tackling the infiniteness of the alphabet in a finitary manner (see e.g. [36] for an overview). A particularly intuitive such model is that of Register 2012 ACM CCS: [Theory of computation]: Formal languages and automata theory. Key words and phrases: register automata, automata over infinite alphabets, infinite systems reachability, freshness, counter automata. l LOGICAL METHODS IN COMPUTER SCIENCE c DOI:10.2168/LMCS-12(1:7)2016 CC R. Grigore and N. Tzevelekos Creative Commons 2 R. GRIGORE AND N. TZEVELEKOS q0 P ∅, 1 1, 2 O O P The automaton starts at state q0 with empty history and nondeterministically makes a transition to state P or Q, accepting the respective symbol. From state P , it accepts any input name a which does not appear in any of its histories (this is what ∅ stands for), puts it in history number 1, and moves back to q0 . From state O, it accepts any input name a which appears in history number 1, puts it in history number 2, and moves back to q0 . Figure 1: History-register automaton accepting L0 . Automata (RA) [23, 31], which are machines built around the concept of an ordinary finitestate automaton attached with a fixed finite amount of registers. The automaton can store in its registers names coming from the input, and make control decisions by comparing new input names with those already stored. Thus, by talking about addresses of its memory registers rather than actual names, a so finitely-described automaton can tackle the infinite alphabet of names. Driven by program analysis considerations, register automata have been recently extended with the feature of name-freshness recognition [38], that is, the capability of the automaton to accept specific inputs just if they are fresh — they have not appeared before during computation. Those automata, called Fresh-Register Automata (FRA), can account for languages like the following, L0 = {a1 . . . an ∈ N ∗ | ∀i 6= j. ai 6= aj } which captures the output of a fresh-name generator (N is a countably infinite set of names). FRAs are expressive enough to model, for example, finitary fragments of languages like the π-calculus [38] or ML [29]. The freshness oracle of FRAs administers the automata with perhaps too restricted an access to the full history of the computation: it allows them to detect name freshness, but not non-freshness. Consider, for instance, the following simple language, L0 = {w ∈ ({O, P } × N )∗ | each letter of w appears exactly once in it ∧ each (O, a) in w is preceded by some (P, a) } where the alphabet is made of pairs containing an element from the set {O, P } and a name (O and P can be seen as different processes or agents exchanging names). The language L0 represents a paradigmatic scenario of a name generator P coupled with a name consumer O: each consumed name must have been created first, and no name can be consumed twice. It can capture e.g. the interaction of a process which creates new files with one that opens them, where no file can be opened twice. The inability of FRAs to detect non-freshness, as well as the fact that names in their history cannot be removed from it, do not allow them to express L0 . More generally, the notion of re-usage or consumption of names is beyond the reach of those machines. Another limitation of FRAs is the failure of closure under concatenation, interleaving and Kleene star. Aiming at providing a stronger theoretical tool for analysing computation with names, in this work we further capitalise on the use of histories by effectively upgrading them to the status of registers. That is, in addition to registers, we equip our automata with a fixed number of unbounded sets of names (histories) where input names can be stored and compared with names to follow. As histories are internally unordered, the kind of HISTORY-REGISTER AUTOMATA 3 FRA unary HRA HRA RA non-reset HRA DA / CMA Figure 2: Expressiveness of history-register automata compared to previous models (in italics). The inclusion M −→ M0 means that for each A ∈ M we can effectively construct an A0 ∈ M0 accepting the same language as A. All inclusions are strict. name comparison we allow for is name belonging (does the input name belong to the i-th history? ). Moreover, names can be selected and removed from histories, and individual histories can be emptied/reset. We call the resulting machines History-Register Automata (HRA). For example, L0 is accepted by the HRA with 2 histories depicted in Figure 1, where by convention we model pairs of symbols by sequences of two symbols.1 The strengthening of the role of histories substantially increases the expressive power of our machines. More specifically, we identify three distinctive features of HRAs: (1) the capability to reset histories; (2) the use of multiple histories; (3) the capability to select and remove individual names from histories. Each feature allows us to express one of the paradigmatic languages below, none of which are FRA-recognisable. L1 = {a0 w1 . . . a0 wn ∈ N ∗ | ∀i. wi ∈ N ∗ ∧ a0 wi ∈ L0 } for given a0 L2 = {a1 a01 . . . an a0n ∈ N ∗ | a1 . . . an , a01 . . . a0n ∈ L0 } L3 = {a1 . . . an a01 . . . a0n0 ∈ N ∗ | a1 . . . an , a01 . . . a0n0 ∈ L0 ∧ ∀i.∃j. a0i = aj } Apart from the gains in expressive power, the passage to HRAs yields a more well-rounded automata-theoretic formalism for generative behaviours as these machines enjoy closure under all regular operations apart from complementation. On the other hand, the combination of features (1-3) above enables us to use histories as counters and simulate counter machines. We therefore obtain non-primitive recursive bounds for checking language emptiness. Given that language containment and universality are undecidable already for register automata [31], HRAs are fairly close to the decidability boundary for properties of languages over infinite alphabets. Nonetheless, starting from HRAs and weakening them in each of the first two factors (1,2) we obtain automata models which are still highly expressive but computationally more tractable. Overall, the expressiveness hierarchy of the machines we examine is depicted in Figure 2 (weakening in (1) and (2) respectively occurs in the second column of the figure). Motivation and related work. The motivation for this work stems from semantics and verification. In semantics, the use of names to model resource generation originates in the work of Pitts and Stark on the ν-calculus [32] and Stark’s PhD [37]. Names have subsequently been incorporated in the semantics literature (see e.g. [21, 4, 1, 24]), especially after the advent 1Although, technically speaking, the machines we define below do not handle constants (as e.g. O, P ), the latter are encoded as names appearing in initial registers, in standard fashion. 4 R. GRIGORE AND N. TZEVELEKOS of Nominal Sets [18], which provided formal foundations for doing mathematics with names. Moreover, recent work in game semantics has produced algorithmic representations of game models using extensions of fresh-register automata [29, 30, 28], thus achieving automated equivalence checks for fragments of ML and Java. In a parallel development, a research stream on automated analysis of dynamic concurrent programs has developed essentially the same formalisms, this time stemming from trace-based operational techniques [10, 3]. This confluence of different methodologies is exciting and encourages the development of stronger automata for a wider range of verification tasks, and just such an automaton we examine herein. Although our work is driven by program analysis, the closest existing automata models to ours come from XML database theory and model checking. Research in the latter area has made great strides in the last years on automata over infinite alphabets and related logics (e.g. see [36] for an overview from 2006). As we show in this paper, history-register automata fit very well inside the big picture of automata over infinite alphabets (cf. Figure 2) and in fact can be seen as closely related to Data Automata (DA) [7] or, equivalently, Class Memory Automata (CMA) [5]. A crucial difference lies in the reset capabilities of our machines, which allow us to express languages like L1 that cannot be expressed by DA/CMAs. On the other hand, the local termination conditions of DA/CMAs allow them to express languages that HRAs cannot capture. We find the correspondence between HRAs and DAs particularly pleasing as it relates two seemingly very different kind of machines, with distant operational descriptions and intuitions. A recent strand of research in foundations of atom-based computation [6, 9, 8] has examined nominal variants of classical machine models, ranging from finite-state automata to Turing machines. Finally, since the publication of the conference version of this paper [39], there has been work in nested DA/CMAs [14, 13], which can be seen as extensions of non-reset HRAs whereby the histories satisfy some nesting relations. The latter are a clean extension of our machines, leading to higher reachability complexities. This article is the journal version of [39], with strengthened results and with full proofs. Section 4 is new: it collects all results that show how registers can be simulated by other means. Many upper bounds are tighter: Propositions 4.1, 4.4, 5.4, 6.2, 6.8. Some results are new: Propositions 4.5, 4.7, 5.6, 6.10. Example 4.3 is new. Most proofs have been revised. Section 7 is new: it collects in one place the main properties of HRAs. Also, we relate our work to what has been done after the conference version was published. Overview. In Section 2 we introduce HRAs and their basic properties. In Section 3 we examine regular closure properties of HRAs. In Section 4 we explain how registers can be simulated by other means, such as histories. In Section 5 we prove that emptiness is Ackermann-complete. In Section 6 we introduce weaker models, and study their properties. In Section 7 we summarize the main properties of HRAs. In Section 8 we connect HRAs to existing automata formalisms. We conclude by discussing future directions which emanate from this work. 2. Definitions and first properties We start by fixing some notation. Let N be a countably infinite alphabet of names, over which we range by a, b, c, etc. For any pair of natural numbers i ≤ j, we write [i, j] for the set {i, i+1, . . . , j}, and for each i we let [i] be the set {1, . . . , i}. For any set S, we write |S| HISTORY-REGISTER AUTOMATA 5 for the cardinality of S, we write P(S) for the powerset of S, we write Pfn (S) for the set of finite subsets of S, and we write P6=∅ (S) for the set of nonempty subsets of S. We write id : S → S for the identity function on S, and img(f ) for the image of f : S → T . We define automata which are equipped with a fixed number of registers and histories where they can store names. Each register is a memory cell where one name can be stored at a time; each history can hold an unbounded set of names. We use the term place to refer to both histories and registers. Transitions are of two kinds: name-accepting transitions and reset transitions. Those of the former kind have labels of the form (X, X 0 ), for sets of places X and X 0 ; and those of the latter carry labels with single sets of places X. A transition labelled (X, X 0 ) means: • accept name a if it is contained precisely in places X, and • update places in X and X 0 so that a be contained precisely in places X 0 after the transition (without touching other names). By a being contained precisely in places X we mean that it appears in every place in X, and in no other place. In particular, the label (∅, X 0 ) signifies accepting a fresh name (one which does not appear in any place) and inserting it in places X 0 . On the other hand, a transition labelled by X resets all the places in X; that is, it updates each of them to the empty set (registers are modelled as sets with at most one element). Reset transitions do not accept names; they are -transitions from the outside. Note that the label (X, ∅) has different semantics from the label X: the former stipulates that a name appearing precisely in X be accepted and then removed from X; whereas the latter clears all the contents of places in X, without accepting anything. 2.1. Definitions. Formally, let us fix positive integers m and n which will stand for the default number of histories and registers respectively in the machines we define below. The set Asn of assignments and the set Lab of labels are: Asn = { H : [m + n] → Pfn (N ) | ∀i > m. |H(i)| ≤ 1 } Lab = P([m + n])2 ∪ P([m + n]) For example, {(i, ∅) | i ∈ [m+n]} is the empty assignment.2 We range over elements of Asn by H and variants, and over elements of Lab by ` and variants. Let H ∈ Asn be an assignment, let a ∈ N be a name, let S ⊆ N be a set of names, and let X ⊆ [m + n] be a set of places. We introduce the following notation: • We set H@X to be the T S set of names which appear preciselySin places X in H; that is, H@X = i∈X H(i) \ i∈X / H(i). In particular, H@ ∅ = N \ i H(i) is the set of names which do not appear in H. • H[X 7→ S] is the update H 0 of H so that all places in X are mapped to S; that is, H 0 = {(i, H(i)) | i 6∈ X} ∪ {(i, S) | i ∈ X}. E.g., H[X 7→ ∅] resets all places in X. • H[a in X] is the update of H which removes name a from all places and inserts it back in X; that is, H[a in X] is the assignment: {(i, H(i) ∪ {a}) | i ∈ X ∩ [m]} ∪ {(i, {a}) | i ∈ X \ [m]} ∪ {(i, H(i) \ {a}) | i ∈ / X} Note above that operation H[a in X] acts differently in the case of histories (i ≤ m) and registers (i > m) in X: in the former case, the name a is added to the history H(i), while in the latter the register H(i) is set to {a} and its previous content is cleared. 2We represent functions as sets of pairs. 6 R. GRIGORE AND N. TZEVELEKOS We can now define our automata. Definition 2.1. A history-register automaton (HRA) of type (m, n) is a tuple A = hQ, q0 , H0 , δ, F i where: • Q is a finite set of states, q0 is the initial state, F ⊆ Q are the final ones, • H0 ∈ Asn is the initial assignment, and • δ ⊆ Q × Lab × Q is the transition relation. For brevity, we shall call A an (m, n)-HRA. X,X 0 X We write transitions in the forms q −→ q 0 and q −→ q 0 , for each kind of transition label. In diagrams, we may unify different transitions with common source and target, for example X,X 0 X,X 0 / Y,Y 0 Y,Y 0 q −→ q 0 and q −→ q 0 may be written q −−−−−−−→ q 0 ; moreover, we shall lighten notation and write i for the singleton {i}, and ij for {i, j}. We already gave an overview of the semantics of HRAs. This is formally defined by means of configurations representing the current computation state of the automaton. A configuration of A is a pair (q, H) ∈ Q̂, where: Q̂ = Q × Asn From the transition relation δ we obtain the configuration graph of A as follows. Definition 2.2. Let A be an (m, n)-HRA as above. Its configuration graph (Q̂, −→),
x where −→ ⊆ Q̂ × N ∪ {} × Q̂, is constructed by setting (q, H) −→ (q 0 , H 0 ) if and only if one of the following conditions is satisfied. X,X 0 • x = a ∈ N and there is q −→ q 0 ∈ δ such that a ∈ H@X and H 0 = H[a in X 0 ]. X • x =  and there is q −→ q 0 ∈ δ such that H 0 = H[X 7→ ∅]. The language accepted by A is w L(A) = { w ∈ N ∗ | (q0 , H0 ) −→ −→ (q, H) and q ∈ F } x1 ...x x x k 1 k where −→ −→ is the reflexive transitive closure of −→ (i.e. q̂ −−−−→ → q̂ 0 if q̂ −→ · · · −→ q̂ 0 ). Note that we use  both for the empty sequence and the empty transition so, in particular, when writing sequences of the form x1 . . . xk we may implicitly consume ’s. It is worth noting here that our formulation follows M-automata [23] in that multiple places can be mentioned at each transition. Example 2.3. The language L1 of the Introduction is recognised by the following (1, 1)HRA (leftmost below), with initial assignment {(1, ∅), (2, a0 )}. The automaton starts by accepting a0 , leaving it in register 2, and moving to state q1 . There, it loops accepting fresh names (appearing in no place) which it stores in history 1. From q1 it goes back to q0 by resetting its history. ∅,1 / 2,12 ∅,1 q0 2,2 1 q1 q0 1,∅ ∅,1 q1 q0 1,∅ q1 ∅,2 / 1,12 We can also see that the other two HRAs, of type (2, 0) and (1, 0), accept the languages L2 and L3 respectively. Both automata start with empty assignments. HISTORY-REGISTER AUTOMATA 7 Finally, the automaton we drew in Figure 1 is, in fact, a (2,2)-HRA where its two registers initially contain the names O and P respectively. The transition label O corresponds to (3, 3), and P to (4, 4). As mentioned in the introductory section, HRAs build upon (Fresh) Register Automata [23, 31, 38]. The latter can be defined within the HRA framework as follows.3 Definition 2.4. A Register Automaton (RA) of n registers is a (0, n)-HRA with no reset transitions. A Fresh-Register Automaton (FRA) of n registers is a (1, n)-HRA S A = hQ, q0 , H0 , δ, F i such that H0 (1) = i H0 (i) and: • for all (q, `, q 0 ) ∈ δ, there are X, X 0 such that ` = (X, X 0 ) and 1 ∈ X 0 ; • for all (q, {1}, X 0 , q 0 ) ∈ δ, there is also (q, ∅, X 0 , q 0 ) ∈ δ. Thus, in an FRA all the initial names must appear in its history, and the same holds for all the names the automaton accepts during computation (1 ∈ X 0 ). As, in addition, no reset transitions are allowed, the history effectively contains all names of a run. On the other hand, the automaton cannot recognise non-freshness: if a name appearing only in the history is to be accepted at any point then a totally fresh name can be also be accepted in the same way. Now, from [38] we have the following. Lemma 2.5. The languages L1 , L2 and L3 are not FRA-recognisable. Proof. L1 was explicitly examined in [38]. For L2 and L3 we use a similar argument as the one for showing that L0 ∗ L0 is not FRA-recognisable [38] . 2.2. Bisimulation. Bisimulation equivalence, also called bisimilarity, is a useful tool for relating automata, even from different paradigms. It implies language equivalence and is generally easier to reason about than the latter. We will be using it avidly in the sequel. Definition 2.6. Let Ai = hQi , q0i , H0i , δi , Fi i be (m, n)-HRAs, for i = 1, 2. A relation R ⊆ Q̂1 × Q̂2 is called a simulation on A1 and A2 if, for all (q̂1 , q̂2 ) ∈ R,   • if q̂1 −→ −→ q̂10 and π1 (q̂10 ) ∈ F1 then q̂2 −→ −→ q̂20 for some π1 (q̂20 ) ∈ F2 , where π1 is the first projection function;  a  a • if q̂1 −→ −→ · −→ q̂10 then q̂2 −→ −→ · −→ q̂20 for some (q̂10 , q̂20 ) ∈ R. R is called a bisimulation if both R and R−1 are simulations. We say that A1 and A2 are bisimilar, written A1 ∼ A2 , if ((q01 , H01 ), (q02 , H02 )) ∈ R for some bisimulation R. Lemma 2.7. If A1 ∼ A2 then L(A1 ) = L(A2 ). 2.3. Determinism. We close our presentation here by describing the deterministic class of HRAs. We defined HRAs in such a way that, at any given configuration (q, H) and for any input symbol a, there is at most one set of places X that can match a, i.e. such that a ∈ H@X. As a result, the notion of determinism in HRAs can be ensured by purely syntactic means. X1 Xn 0 X Below we write q −→ −→ q 0 ∈ δ if there is a sequence of transitions q −→ · · · −→ q in δ such Sn ∅ that X = i=1 Xi . In particular, q −→ −→ q ∈ δ. 3The definitions given in [23, 31, 38] are slightly different but can routinely be shown equivalent. 8 R. GRIGORE AND N. TZEVELEKOS Definition 2.8. We say that an HRA A is deterministic when, for any reachable config a  a uration q̂ and any name a, if q̂ −→ −→ · −→ q̂1 and q̂ −→ −→ · −→ q̂2 then q̂1 = q̂2 . We say that an HRA A is strongly deterministic when, for any state q and any sets X, X1 , X2 , Y1 , Y2 , Y X\Y1 ,X1 Y X\Y2 ,X2 1 2 if q −→ −→ · −−−−−→ q1 ∈ δ and q −→ −→ · −−−−−→ q2 ∈ δ then q1 = q2 , Y1 = Y2 and X1 = X2 . Even if A is deterministic, it is still possible to have multiple paths in the configuration graph that are labeled by the same word. However, such paths may only differ in their -transitions. In the definition of ‘strongly deterministic’, the set X is guessing where the name a occurs. Lemma 2.9. If A is strongly deterministic then it is deterministic. 3. Closure properties History-register automata enjoy good closure properties with respect to regular language operations. In particular, they are closed under union, intersection, concatenation and Kleene star, but not closed under complementation. In fact, the design of HRAs is such that the automata for union and intersection come almost for free through a straightforward product construction which is essentially an ordinary product for finite-state automata, modulo reindexing of places to account for duplicate labels (cf. [23]). The constructions for Kleene star and concatenation are slightly more involved as there is need for passing through intermediate automata which do not touch their initial names. We shall need the following technical gadget. Given an (m, n)-HRA A and a sequence w of k distinct names, we construct a bisimilar (m, n+k)-HRA, denoted A fix w, in which the names of w appear exclusively in the additional k registers, which, moreover, remain unchanged during computation. The construction will allow us, for instance, to create feedback loops in automata ensuring that after each feedback transition the same initial configuration occurs. Lemma 3.1. Let A be an (m, n)-HRA with initial assignment H0 and w = a1 . . . ak a sequence of distinct names. We can effectively construct an (m, n+k)-HRA A fix w with initial assignment H00 such that A fix w ∼ A and: • H00 (m+n+i) = ai for all i ∈ [k], and H00 (i) = H0 (i) \ {a1 , . . . , ak } for all i ∈ [m+n]; • for all reachable configurations (q, H) of A fix w and all i > m+n, H(i) = H00 (i). Proof. We construct A fix w = hQ0 , q00 , H00 , δ 0 , F 0 i as follows. First, we insert/move all names of w to the new registers (places [m+n+1, m+n+k]), i.e. we set H00 (i) = H0 (i) \ {a1 . . . ak } for all i ∈ [m+n], and H00 (m+n+i) = {ai } for each i ∈ [k]. The role of the new registers is to constantly store the names in w and act on the behalf of other places when the latter intend to use those names: during computation, whenever an ai is captured by a transition of the initial automaton A, in A fix w it will be instead simulated by a transition involving the new registers. In order for the simulation to be accurate, we shall inject inside states information specifying the intended location of the ai s in the places of A. Thus, the states of the new automaton are pairs (q, f ), where q ∈ Q and f is a function recording, for each of the new registers, where would the name of the register appear in the original automaton A. That is, Q0 = Q × {f : [k] → P([m+n]) | ∀j 6= j 0. f (j) ∩ f (j 0 ) ⊆ [m]} HISTORY-REGISTER AUTOMATA 9 while q00 = (q0 , {(i, {j | ai ∈ H0 (j)}) | i ∈ [k]}) and F 0 = {(q, f ) ∈ Q0 | q ∈ F }. Finally, δ 0 operates just like δ albeit taking into account the f ’s of states to figure out the intended positions of the ai s and, at the same time, update the f ’s after each transition. We therefore include in δ 0 precisely the following transitions. Below we write i◦ for m+n+i. For each X,X 0 (q, f ) ∈ Q0 and q −→ q 0 ∈ δ, X,X 0 • add a transition (q, f ) −→ (q 0 , f ); {i◦ },{i◦ } • if f (i) = X for some i then add (q, f ) −−−−−→ (q 0 , f 0 ) where f 0 = f [i 7→ X 0 ]; X X Moreover, for each q −→ q 0 ∈ δ include (q, f ) −→ (q 0 , f 0 ) where f 0 = {(j, f (j) \ X) | j ∈ [k]}. Following the above line of reasoning, we can show that the relation {((q, H), (q, f, H 0 )) | ∀i ∈ [m+n]. H(i) = H 0 (i) ∪ {aj | i ∈ f (j)}} with (q, H), (q, f, H 0 ) reachable configurations, is a bisimulation. We write L ◦ L0 for concatenation of languages, and L∗ for Kleene closure of a language. We use the same definitions as the standard ones for languages over finite alphabets: L ◦ L0 is { ww0 | w ∈ L ∧ w0 ∈ L0 }, and L∗ is the least fixed-point of the equations  ∈ L∗ and L∗ ◦ L ⊆ L∗ , where  is the empty word. Proposition 3.2. Languages recognised by HRAs are closed under union, intersection, concatenation and Kleene star. Proof. We show concatenation and Kleene star only. For the former, consider HRAs Ai = hQi , q0i , H0i , δi , Fi i, i = 1, 2, and assume wlog that they have common type (m, n). Let w be an enlistment of all names in H02 and construct A0i = Ai fix w, for i = 1, 2. Then, the concatenation L(A1 ) ◦ L(A2 ) is the language recognised by connecting A01 and A02 serially, that is, the automaton obtained by connecting each final state of A01 to the initial state of A02 with a transition labelled [m+n], and with initial/final states those of A01 /A02 respectively. Finally, given an (m, n)-HRA A and an enlistment w of its initial names, we construct an automaton A0 by connecting the final states of A fix w to its initial state with a transition labelled [m+n]. We can see that L(A0 ) = L(A)∗ . As we shall next see, while universality is undecidable for HRAs, their emptiness problem can be decided by reduction to coverability for transfer-reset vector addition systems. In combination, these results imply that HRAs cannot be effectively complemented. In fact, there are HRA-languages whose complements are not recognisable by HRAs. This can be shown via the following example, adapted from [26]. Lemma 3.3. HRAs are not closed under complementation. Example 3.4. Consider L4 = {w ∈ N ∗ | not all names of w occur exactly twice in it }, which is accepted by the (2, 0)-HRA below. ∅,1 / 1,1 ∅,1 / 1,1 q0 ∅,2 q1 ∅,1 / 1,1 ∅,1 / 1,1 2,2 q2 2,1 q3 The automaton non-deterministically selects an input name which either appears only once in the input or at least three times. We claim that L4 , the language of all words whose names occur exactly twice in them, is not HRA-recognisable. 10 R. GRIGORE AND N. TZEVELEKOS Proof. Suppose it were recognisable (wlog, Proposition 4.1) by an (m, 0)-HRA A with k states. Then, A would accept the word w = a1 . . . ak a1 . . . ak where all ai ’s are distinct and do not appear in the initial assignment of A. Let p = p1 p2 be the path in A through which w is accepted, with each pi corresponding to one of the two halves of w. Since all ai s are fresh for A, the non-reset transitions of p1 must carry labels of the form (∅, X), for some sets X. Let q be a state appearing twice in p1 , say p1 = p11 (q)p12 (q)p13 . Consider now the path p0 = p01 p2 where p01 is the extension of p1 which repeats p12 , that is, p01 = p11 (q)p12 (q)p12 (q)p13 . We claim that p0 is an accepting path in A. Indeed, by our previous observation on the labels of X,Y p1 , the path p01 does not block, i.e. it cannot reach a transition q1 −−→ q2 , with X 6= ∅, in some configuration (q1 , H1 ) such that H1 @X = ∅. We need to show that p2 does not block either (in p0 ). Let us denote (q, H1 ) and (q, H2 ) the configurations in each of the two visits of q in the run of p on w; and let us write (q, H3 ) for the third visit in the run of p01 , given that for the other two visits we assume the same configurations as in p. Now observe that, for each nonempty X ⊆ [m], repeating p12 cannot reduce the number of names appearing precisely in X, therefore |H2 @X| ≤ |H3 @X|. The latter implies that, since p does not block, p0 does not block either. Now observe that any word accepted by w0 is not in L4 , as p01 accepts more than k distinct names, a contradiction. 4. Removing Registers Although registers are convenient for expressing some languages (Example 4.2 and Example 4.3), when reasoning about HRAs it is more convenient to focus on histories only. In this section, we show that this approach is sound and in particular we present three ways of removing registers: • we can construct a bisimilar automaton if we are allowed to use extra histories and resets; • we can preserve language if we are allowed to use extra histories (but no resets); • we can preserve emptiness if we are allowed to use extra states (but no histories nor resets). Each of these constructions will be useful in the sequel either for devising emptiness checks and, more generally, they demonstrate that registers can be considered as a derivative notion. 4.1. Simulating Registers with Histories and Resets. The semantics of registers is very similar to that of histories. The main difference is that registers are forced to contain at most one name. To simulate registers with histories, we reset histories before inserting names. Resetting histories might cause the automaton to forget names that are necessary for deciding how to proceed. The solution is to use two histories for each register: one holds the old name, the other holds the new name. Only the history where the new name will be written needs to be reset. Proposition 4.1. Let A = hQ, q0 , H0 , δ, F i be an (m, n)-HRA. We can construct an (m + 0 n 2n, 0)-HRA A0 = hQ0 , q00 , H00 , δ 0 , F 0 i that is bisimilar to A. We have |Q
| ∈ O(2 |Q|) and 0 n 0 0 |δ | ∈ O(2 |δ|). The construction can be done in O (m + n)(|Q | + |δ |) time. Proof. For each q in Q, we include 2n states (q, f ) in Q0 , where f : [n] → [2n] is such that f (i) ∈ {i, i + n} for all i. The name that A holds in register i will be found in history m + f (i) of A0 . We set f¯ to be the complement of f ; that is, f¯(i) , n + 2i − f (i). Let f † (i) be i if i ∈ [m] and m + f (i) otherwise. Moreover, q00 = (q0 , id), F 0 = {(q, f ) | q ∈ F }, and H00 is HISTORY-REGISTER AUTOMATA 11 H0 extended so that H00 (i) = ∅ for all i > m + n. Finally, we include in δ 0 precisely the following transitions. X,X 0 Y f † (X),f¯† (X 0 ) • For each q −→ q 0 ∈ δ, add (q, f ) −→ · −−−−−−−−→ (q 0 , f 0 ) where Y = [m+1, m+2n] \ img(f ), and f 0 is given by: f 0 (i) = f¯(i) if i ∈ X ∪ X 0 and f 0 (i) = f (i) otherwise. (Note that we need a few extra states in Q0 , which remain nameless in this proof.) f † (X) X • For each q −→ q 0 ∈ δ, add (q, f ) −−−−→ (q 0 , f ). The relation { ((q, H), ((q, f ), H 0 )) | H = H 0 ◦ f † } witnesses bisimilarity. 4.2. Replacing Registers with Histories using Colours. The result of Proposition 4.1 comes at the cost of introducing reset transitions even if the original automaton does not have such transitions. Reset transitions are undesirable because they increase the complexity of the emptiness problem (Section 5). We can avoid introducing reset transitions by using the colouring technique of [5]. The construction is more involved and the resulting automaton is not bisimilar to the original, but it is language equivalent. Before we proceed with the proof, let us illustrate the technique on two examples. Example 4.2 illustrates how to check that a name is not in the simulated register; Example 4.3 illustrates also how to check that a name is in the simulated register. Example 4.2. The language L5 = { a1 . . . an | ai 6= ai+1 for all i } is recognized by both of the automata below (with histories initially empty): ∅,1 / 2,1 q0 ∅,1 ∅,2 / 2,2 q0 ∅,2 / 1,2 q1 ∅,1 / 1,1 The one on the left is a (0, 1)-HRA, and we can straight away see its accepted language is L5 . The one on the right is a (2, 0)-HRA for which it is less clear why it accepts L5 . The reason is the following invariant: • H(1) ] H(2) is a partition of all names seen so far; and • if the state is qk then the last seen name is in H(k + 1), for k ∈ {0, 1}; and • all possible partitions of the names are realisable. The first two points are easy to check. Because all transitions have labels of the form (X, {i}), all seen names are remembered in precisely one history. Because all transitions incoming into qk have labels the form (X, {k + 1}), the last seen name is remembered in H(k + 1). The consequence of these first two points is that the automaton on the right accepts a name only if it is different from the last seen name. Indeed, all transitions outgoing from qk have labels of the form (X, X 0 ) with k + 1 ∈ / X. Thus, the first two points should be seen as lemmas which let us establish that all words accepted by the automaton on the right belong to L5 . Informally, the third point is the key lemma that lets us establish the converse, that all words in L5 are accepted. However, to see why this is so, we need to rephrase it in a more formal way. Given a word a1 . . . an ∈ L5 , we consider an arbitrary partition H(1) ] H(2) of the set {a1 , . . . , an }. Without loss of generality, assume an ∈ H(1). The claim is that 12 R. GRIGORE AND N. TZEVELEKOS there exists a run of the automaton on the right that accepts the word a1 . . . an and ends in configuration (q0 , H). We can prove this by induction. If an−1 ∈ H(k + 1) then the previous state in the run must have been qk , for k ∈ {0, 1}. Suppose an−1 ∈ H(2); the other case is symmetric. Then, by the induction hypothesis, we know that there is a run that accepts the word a1 . . . an−1 and ends in configuration (q1 , H 0 ), where we choose ( H(1) \ {an } if an ∈ / {a1 , . . . , an−1 } H 0 (1) , H 0 (2) , H(2) \ {an } H(1) ∪ {an } if an ∈ {a1 , . . . , an−1 } It is clear that H 0 (1) ] H 0 (2) is a partition of {a1 , . . . , an−1 }, as required. Finally, we need an to show that (q1 , H 0 ) −→ (q0 , H) belongs to the configuration graph. If an ∈ / {a1 , . . . , an−1 }, ∅,1 then this is true because of q1 −→ q0 ; if an ∈ {a1 , . . . , an−1 }, then this is true because of 1,1 q1 −→ q0 . Example 4.3. The language L6 = { a1 . . . an | ai = ai+1 iff i is odd } is recognized by both of the following automata: ∅,1 / 3,1 q2 ∅,1 q0 q1 q0 ∅,1 q1 1,2 1,3 1,1 ∅,1 / 2,1 q3 The one on the left is a (0, 1)-HRA, while the one on the right is a (3, 0)-HRA. As in the previous example, the fact that the (3, 0)-HRA accepts L6 is not immediately clear. Informally, the reason is the following invariant: • H(1) ] H(2) ] H(3) is a partition of the names seen so far; • if the state is qk then the last seen name is in H(k), for k ∈ {1, 2, 3}; • |H(1)| = 1 in q1 , and |H(1)| = 0 otherwise; and • for each partition H(1) ] H(2) ] H(3) that is compatible with the previous constraints, there is a nondeterministic run that realises it. The first two points hold for the same reasons as in Example 4.2. The third point holds because all incoming transitions of q1 insert a name in H(1), all outgoing transitions of q1 remove a name from H(1), and all runs alternate between state q1 and some other state. After an odd number of names was processed, the automaton is in state q1 and H(1) contains only the last name seen. All outgoing transitions from q1 accept a name only if it is in H(1) — in other words, if it equals the last seen name. After an even and positive number of names was processed, the automaton is in state q2 or q3 . All outgoing transitions from q2 accept a name only if it is not in H(2), where the last seen name is; q3 acts symmetrically. Thus, the first three points let us establish that all words accepted by the automaton on the right belong to L6 . As in Example 4.2, the last point lets us establish that all word in L6 are accepted. Given that the proof of the last point is very similar to the one in Example 4.2, let us only sketch it. Given a word a1 . . . an ∈ L6 with n > 0, we consider an arbitrary partition H(1) ] H(2) ] H(3) of the set {a1 , . . . , an } such that H(1) ⊆ {an }. Let k be such that an ∈ H(k). Formally, the claim of the last point is that there exists a run of the automaton HISTORY-REGISTER AUTOMATA 13 on the right that is labelled by the word a1 . . . an and ends in the configuration (qk , H). The case n odd and > 1 is similar to the previous example: We pick k 0 such that an−1 ∈ H(k 0 ) and ( H(5 − k 0 ) \ {an } if an ∈ / {a1 , . . . , an−1 } 0 0 0 0 0 0 H (1) , ∅ H (k ) , H(k ) \ {an } H (5 − k ) , 0 H(5 − k ) ∪ {an } if an ∈ {a1 , . . . , an−1 } Then we invoke the induction hypothesis to show there is a run that accepts a1 . . . an−1 and ends in configuration (qk0 , H 0 ). In the case n even, we pick H 0 (1) , {an } H 0 (2) , H(2) \ {an } H 0 (3) , H(3) \ {an } and then invoke the induction hypothesis to show that there is a run that accepts a1 . . . an−1 and ends in configuration (q1 , H 0 ). We skip the case n = 1 in this proof sketch. We are now ready for the general result, which we prove in two steps. The main construction will be presented first and only applies to HRAs with initially empty registers. (The correctness of the main construction requires that certain graphs are 2-colourable. The arguments in the previous two examples can be seen as giving explicit colouring algorithms that work in special cases.) At a second stage, we show how to initially simulate nonempty registers at the expense of some more additional histories. Proposition 4.4. Let A = hQ, q0 , H0 , δ, F i be an (m, n)-non-reset-HRA with registers initially empty. We can construct an (m + 3n, 0)-non-reset-HRA A0 = hQ0 , q00 , H00 , δ 0 , F 0 i that accepts the same language as A. We have |Q0 | ∈ O(22n · |Q|) and |δ 0 | ∈ O(23.3n · |δ|). Proof. Each register i will be simulated by three histories, named iR , iB and iY respectively.4 Each state q ∈ Q will be simulated by several states (q, f ) ∈ Q0 , where f : [m + 1, m + n] → {∅, R, B, Y}. The construction will ensure the following invariant: • H(iR ) ] H(iB ) ] H(iY ) is a partition of the names that have been written to register i and have subsequently been rewritten by other names or are still in register i;5 • |H(iR )| = 1 if f (i) = R, and |H(iR )| = 0 otherwise; and • if f (i) 6= ∅ then the current name of register i is in H(if (i) ); register i is empty otherwise. Thus, according to the last point above, f records in which of histories iR , iB , iY has the current name of register i been stored. We shall instrument A0 in such a way that it will only store that name, say a, in iR if the next time register i is being invoked by A is for reading a. This way we shall ensure that iR never contains more than one name. Otherwise, i.e. if A next invokes i for overwriting its contents, f will be mapping i to one of iB , iY . These can be seen as garbage collecting histories: they contain all names that have passed from register i and will not be immediately read from it. The reason why we need two of these, iB and iY , is to be able to reuse old names of register i without running the risk of confusing them with its current name a. X,X 0 To simulate one transition q −→ q 0 , accepting say a name a, we shall use several Z,Z 0 transitions of the form (q, f ) −→ (q 0 , f 0 ) where f and f 0 agree outside X ∪ X 0 . Let us consider an arbitrary such pair (f, f 0 ), and see how to pick Z and Z 0 . On histories, X and Z coincide: X ∩ [m] = Z ∩ [m]. For each register i ∈ X \ [m], we need that a be equal to the 4B and Y are the black and yellow colours of [5]; R stands for ‘read’. 5note that a name can also be transferred out of register i (via transition with label X, Y where i ∈ X \ Y ), instead of being directly rewritten, in which case we would not store it in H(iR ) ] H(iB ) ] H(iY ). 14 R. GRIGORE AND N. TZEVELEKOS name currently written in register i. But, we know the current name in register i only if Z,Z 0 f (i) = R. That is why we include a transition (q, f ) −→ (q 0 , f 0 ) only if X \ [m] ⊆ f −1 (R), and we include { iR | i ∈ X \ [m] } in Z. For each register i ∈ [m + 1, m + n] \ X, we must ensure that a is not equal to the current name in register i, which resides in H(if (i) ). Hence, we pick all Z such that Z = (X ∩ [m]) ] Z1 ] Z0 where Z1 = { iR | i ∈ X \ [m] } and Z0 ⊆ { ix | i ∈ [m + 1, m + n] \ X ∧ x ∈ {B, Y} ∧ x = 6 f (i) } is such that, for all i, |Z0 ∩ {iB , iY }| ≤ 1. Now we must write the current name to histories X 0 ∩ [m], and we must simulate writing the current name to registers X 0 \ [m]. For each i ∈ X 0 \ [m] we shall nondeterministically write the current name to one of iR , iB , iY by guessing whether register i will be used next for reading or not. The place where we write is given by f 0 (i), that is, Z 0 = (X 0 ∩ [m]) ] { if 0 (i) | i ∈ X 0 \ [m] }. Finally, we make sure that all i ∈ ([m + 1, m + n] ∩ X) \ X 0 in f 0 are mapped to ∅, as these registers are now empty, i.e. we impose f 0 (([m + 1, m + n] ∩ X) \ X 0 ) ⊆ {∅}. Thus, in summary, we take Q0 = Q × ([m + 1, m + n] → {∅, R, B, Y}) and: 0 • q0 = (q0 , {(i, ∅) | i ∈ [m + 1, m + n]}) and F 0 = F × ([m + 1, m + n] → {∅, R, B, Y}); • H00 = H0 ∪ {(i, ∅) | i ∈ [m + n + 1, m + 3n]}; Z,Z 0 X,X 0 • we include (q, f ) −→ (q 0 , f 0 ) in δ 0 just if there is some q −→ q 0 in δ such that: – X \ [m] ⊆ f −1 (R), – for all i ∈ [m + 1, m + n] \ (X ∪ X 0 ), f 0 (i) = f (i), – for all i ∈ ([m + 1, m + n] ∩ X) \ X 0 , f 0 (i) = ∅, – Z = (X ∩ [m]) ] {iR | i ∈ X \ [m]} ] Z0 with Z0 ⊆ {ix | i ∈ [m + 1, m + n] \ X ∧ x 6= f (i)}, 0 – Z = (X 0 ∩ [m]) ] { if 0 (i) | i ∈ X 0 \ [m] }. Let us now see what is the size of the HRA A0 so constructed. We have |Q0 | ∈ O(4n · |Q|). X,X 0 Z,Z 0 For each transition q −→ q 0 in A, we introduce several transitions (q, f ) −→ (q 0 f 0 ) in A0 . Let us count how many. There are ≤ 4n choices for f ; there are ≤ 3n choices for Z0 because for each i we pick iB , or iY , or none of them; and there are ≤ 3n choices for f 0 because f 0 (X 0 ) ⊆ {iR , iB , iY } and f 0 is uniquely determined outside X 0 . In summary, |δ 0 | ∈ O(22(1+log2 3)n · |δ|). Finally, we show that L(A) = L(A0 ). Let first w ∈ L(A0 ) have an accepting transition Zk ,Z 0 path p0 in A0 with edges (qk , fk ) −→k (qk+1 , fk+1 ), for k = 1, . . . , N . Reading the definition Xk ,X 0 of δ 0 backwards, this yields an accepting transition path p in A with edges qk −→k qk+1 where • Xk = (Zk ∩ [m]) ∪ { i | iR ∈ Zk }, • Xk0 = (Zk0 ∩ [m]) ∪ { i | {iR , iB , iY } ∩ Zk0 6= ∅ }. To see that p accepts w, suppose that p0 yields a sequence of configurations ((qk , fk ), Hk0 ). Then, by induction, we can show that p yields a sequence of configurations (qk , Hk ), where: • for all i ∈ [m], Hk (i) = Hk0 (i); • for all i ∈ [m + 1, m + n], if fk (i) 6= ∅ then Hk (i) = {a} for some a ∈ Hk0 (ifk (i) ), otherwise Hk (i) = ∅; • for all names a, if a ∈ Hk0 @Zk then a ∈ Hk @Xk . HISTORY-REGISTER AUTOMATA 15 Hence, w ∈ L(A). Conversely, let w = a1 · · · aN ∈ L(A) have an accepting transition path p in A with edges Xk ,X 0 qk −→k qk+1 for k = 1, . . . , N . We construct a corresponding accepting path p0 in A0 with Zk ,Z 0 edges (qk , fk ) −→k (qk+1 , fk+1 ) as follows. We have that f0 = {(i, ∅) | i ∈ [m]}. Moreover, Zk = (Xk ∩ [m]) ∪ Wk and Zk0 = (Xk0 ∩ [m]) ∪ Wk0 where: (a) For each position k such that the previous appearance of ak in w is some ak0 = ak with k 0 < k, we set Wk = Wk0 0 . If there is no previous appearance, we set Wk = ∅. (b) For each position k and i ∈ Xk0 \ [m] such that the next appearance of ak in w is some ak0 with i ∈ Xk0 , we include iR in Wk0 . (c) For each position k and i ∈ Xk0 \ [m] such that the next appearance of ak in w is some ak0 with i ∈ / Xk0 , we include in Wk0 one of iB , iY . We do the same also if there is no next appearance of ak in w. The above specifications determine the values of all Wk , Wk0 , modulo the choice between B and Y in case (c). Clearly, if the path p0 can be so constructed then A0 accepts w. It remains to show that p0 can indeed be implemented in A0 . The form of the fk ’s is derived from (a-c) according to the definition of δ 0 . But note that the definition of δ 0 imposes the following condition: (d) For each position k and iY ∈ Wk0 such that the next appearance of any ix in p0 is in some Wk0 (with k < k 0 ), we must have iB ∈ Wk0 . Dually if iB ∈ Wk0 . For example, if iY ∈ W20 but none of iB , iY , iR occurs in any of W3 , W30 , W4 , W40 , W5 , W50 , then {iB , iY , iR } ∩ W6 ⊆ {iB }. This condition stems from the interdiction to include if (i) in Wk0 when f (i) 6= iR . The consequence of condition (d) is that in case (c) above we cannot pick B and Y arbitrarily. We need to show that a choice of “colours” (B and Y) satisfying both (c) and (d) can be made. We achieve this by applying a graph colouring argument. Let us define a labelled graph G with: • Vertices (k, i) and (k, i)0 for each k ∈ [0, N − 1] and i ∈ [m + 1, m + n]; • For each i and k < k 0 as in (c) above, an edge between (k, i)0 and (k 0 , i) labelled with “=”. • For each i and k < k 0 as in (d) above, an edge between (k, i)0 and (k 0 , i) labelled with “6=”. Then, a valid choice of colours can be made as long as G can be coloured with B and Y in such a way that =-connected vertices have matching colours, while 6=-connected vertices have different colours. For the latter, it suffices to show that the graph obtained by merging =-connected vertices can be 2-coloured, for which it is enough to show that G contains no cycles. Suppose G contained a cycle. Then, by definition of the edge relation of the graph, it must be the case that the leftmost vertex (i.e. the one with the least k index) in the cycle be some (k, i)0 . The vertex (k, i)0 has two outgoing edges, one for each label. The 6=-edge in particular connects to some (k 0 , i) such that k < k 0 , obtained from condition (d). Since (k 0 , i) is part of the cycle, it must have an outgoing =-edge to some vertex (k 00 , i)0 with k 00 < k 0 . But note that condition (d) stipulates that there is no mention of i between k and k 0 in p0 , and therefore k 00 ≤ k. Moreover, k 00 = k is not an option as it would imply that register i was not rewritten between steps k and k 0 in p, in which case k and k 0 would fall under case (b) above. Hence, k 00 < k which contradicts our assumption that (k, i)0 was the leftmost vertex in the cycle. 16 R. GRIGORE AND N. TZEVELEKOS Note that the above result can be extended to handle the case in which registers are not initially empty by simply making use of Lemma 3.1. However, the construction in that lemma leads to a doubly exponential blow-up in size, which we can be avoided by the alternative approach that follows. Proposition 4.5. Let A = hQ, q0 , H0 , δ, F i be an (m, n)-non-reset-HRA. We can construct a bisimilar (m + n, n)-non-reset-HRA A0 = hQ0 , q00 , H00 , δ 0 , F 0 i such that, for all i ∈ [m + n + 1, m + 2n], H0 (i) = ∅. Moreover, we have |Q0 | ∈ O(2n · |Q|) and |δ 0 | ∈ O(22n · |δ|). Proof. The main idea behind the construction of A0 is to use the additional n histories to store just the initial names of the registers in A. Once these names have been used in the computation, they are transferred to their actual registers (if any). We will also need to track which of the registers in A are still simulated by histories in A0 . Thus, we set Q0 = Q × ([m + 1, m + n] → {0, 1}) and q00 = (q0 , {(i, 1) | i ∈ [m + 1, m + n]}), H00 = H0 ∪ {(m + n + i, ∅) | i ∈ [n]} and X,X 0 F 0 = F × ([m + 1, m + n] → {0, 1}). Moreover, for each q −→ q 0 in δ and map f , we include Y,Y 0 in δ 0 a transition (q, f ) −→ (q 0 , f 0 ) where: • Y = (X ∩ [m]) ] Y1 ] Y0 , where Y1 = { i ∈ X | f (i) = 1 } ∪ { n + i | i ∈ X ∧ f (i) = 0 } and Y0 ⊆ { i ∈ [m + 1, m + n] | i ∈ / X ∧ f (i) = 0 }; • Y 0 = (X 0 ∩ [m]) ∪ { n + i | i ∈ X 0 \ [m] }; • f 0 = f [i 7→ 0 | i ∈ X ∪ X 0 ]. Then, taking R to be the relation: R = { ((q, H), ((q, f ), H 0 ) | H  [m] = H 0  [m] ∧ ∀i ∈ [m + 1, m + n]. ∧ f (i) = 1 =⇒ H 0 (n + i) = ∅ ∧ H 0 (i) = H(i) ∧ f (i) = 0 =⇒ H 0 (n + i) = H(i) ∧ ∀j ∈ [m + 1, m + n]. H 0 (i) ∩ H 0 (n + j) = ∅ } we can show that R is a bisimulation. Let us now see what is the size of the HRA A0 we constructed. We have |Q0 | ∈ O(2n ·|Q|). X,X 0 Y,Y 0 For each transition q −→ q 0 in A, we introduce several transitions (q, f ) −→ (q 0 f 0 ) in A0 : there are ≤ 2n choices for f ; and there are ≤ 2n choices for Y0 . In summary, |δ 0 | ∈ O(22n ·|δ|). Hence, the general case follows. Corollary 4.6. Let A = hQ, q0 , H0 , δ, F i be an (m, n)-non-reset-HRA. We can construct an (m + 4n, 0)-non-reset-HRA A0 = hQ0 , q00 , H00 , δ 0 , F 0 i that accepts the same language as A. We have |Q0 | ∈ O(23n · |Q|) and |δ 0 | ∈ O(25.3n · |δ|). HISTORY-REGISTER AUTOMATA 17 4.3. Simulating Registers Symbolically. So far we saw how to simulate registers using histories. If we are interested only in emptiness/reachability rather than language equivalence, we can actually simulate the behaviour of registers without the inclusion of additional histories. This alternative is going to be crucial in Section 6.2, where the number of histories will be fixed to just one. We next describe how this simulation can be done. Given an assignment H with m histories and n registers, we can represent H symbolically as follows: • we map each name stored in the registers of H to a number from the set [n]; • we subsequently replace in H all these names by their number. For example, consider the assignment  1 7→ {a, b, c}, 2 7→ {d}, 3 7→ ∅, 4 7→ {a}, 5 7→ {d} of a (1, 4)-HRA. We can simulate it symbolically by mapping d to 1, and a to 2. This results to a symbolic representation:  1 7→ {2, b, c}, 2 7→ 1, 3 7→ ∅, 4 7→ 2, 5 7→ 1 where the nominal part has been curtailed to the fact that H(1) contains the names b and c. We can now employ this representation technique to represent configurations of (m, n)HRAs by corresponding ones belonging to (m, 0)-HRAs. In particular, given the configuration    q, 1 7→ {a, b, c}, 2 7→ {d}, 3 7→ ∅, 4 7→ {a}, 5 7→ {d} of a (1, 4)-HRA, we map it to the configuration   
 q, 1 7→ {2}, 2 7→ 1, 3 7→ ∅, 4 7→ 2, 5 7→ 1 , 1 → 7 {b, c} of a (1, 0)-HRA which incorporates the non-nominal part of our representation scheme in its state. Clearly, the state space of the new automaton in this simulation will experience an exponential blowup, as the next result shows. However, no additional histories will be needed, which is the main target here. Proposition 4.7. Let A = hQ, q0 , H0 , δ, F i be an (m, n)-HRA. We can construct an (m, 0)HRA A0 = hQ0 , q00 , H00 , δ 0 , F 0 i that is empty if and only if A is empty. We have |Q0 | ∈ O(2mn nBn |Q|) and |δ 0 | ∈ O(2mn nBn |δ|), where Bn is the nth Bell number. Moreover, A0 contains reset transitions if and only if A contains reset transitions. Proof. Each state q ∈ Q will be simulated by several states (q, f ) ∈ Q0 , where f : [m + n] → P([n]) will be called an assignment skeleton. Such a skeleton f is valid when: • |f (i)| ≤S1 for registers i ∈ [m + 1, m + n]; • f (i) ⊆ nj=1 f (m + j) for histories i ∈ [m]; S • for all k ∈ [n] there is a k 0 ∈ [k] such that ki=1 f (m + i) = [k 0 ]. The latter condition essentially stipulates that f is a partition function on the set [m+1, m+n]: the elements of the set are uniquely assigned numbers which can be seen as class indices — two elements are assigned the same number iff they belong to the same class. There is also a special class in this partition, namely of all elements of [m + 1, m + n] to which f assigns ∅. We can now define the rest of A0 . First, we let q00 = (q0 , f0 ), where (f0 , H00 ) is the symbolic representation of H0 . In order to construct δ 0 we define a transition relation on skeletons, which is very similar to the configuration graph of HRAs (Definition 2.2) except X,X 0 that it allows symbols to be permuted after the transition is taken. We write f −→ f 0 when 18 R. GRIGORE AND N. TZEVELEKOS there exists a permutation π on [n] and a k ∈ [n] such that k ∈ f @X and f 0 = π ◦ (f [k in X 0 ]). X We write f −→ f 0 when there exists a permutation π such that f 0 = π ◦ (f [X 7→ ∅]). X,X 0 To simulate one transition of the form q −→ q 0 from δ, we use several transitions of the ` form (q, f ) −→ (q 0 , f 0 ) in δ 0 . Let us consider an arbitrary pair (f, f 0 ) of valid skeletons, and see how to pick `. There are four cases, depending on whether X and X 0 mention or not registers. ∅,∅ • Case X ⊆ [m] and X 0 ⊆ [m]. It must be that f −→ f 0 , and we pick ` = (X, X 0 ). ∅,X 0 • Case X ⊆ [m] and X 0 6⊆ [m]. It must be that f −→ f 0 , and we pick ` = (X, ∅). X,∅ • Case X 6⊆ [m] and X 0 ⊆ [m]. It must be that f −→ f 0 , and we pick ` = (∅, X 0 ). X,X 0 • Case X 6⊆ [m] and X 0 6⊆ [m]. It must be that f −→ f 0 , and we pick ` = (∅, ∅). X Similarly, each reset transition q −→ q 0 from δ yields several transitions of the form Z (q, f ) −→ (q 0 , f 0 ) in δ 0 . Given an arbitrary pair (f, f 0 ) of valid skeletons, we pick Z as follows. X If X ⊆ [m] then it must be that f 0 = f and we pick Z = X. Otherwise, f −→ f 0 and we pick Z = ∅. To estimate |Q0 | it suffices to count how many valid skeletons there are. The values f (m + 1), . . . , f (m + n) of a valid skeleton correspond to a partition of the registers and a selection of a class (if any) whose registers are empty. There are Bn possible partitions and ≤ (n + 1) possible selections, which gives ≤ (n + 1)Bn cases. For the values f (1), . . . , f (m) of a valid skeleton there are ≤ 2mn possibilities. In total, |Q0 | ≤ 2mn (n + 1)Bn |Q|. To estimate |δ 0 |, note that once f is fixed in the construction above, the constraints on 0 f determine it uniquely. So, the number of transitions increases by the same factor as the number of states.
Since log Bn ∈ Θ(n log n), we have that log 2mn (n + 1)Bn ∈ Θ(mn + n log n). 5. Emptiness and Universality 5.1. Emptiness. Here we show that deciding emptiness is Ackermann-complete. We work by reducing from and to state reachability problems in counter systems (similarly e.g. to [7, 5]). For the upper bound, we reduce nonemptiness of HRAs to control-state reachability of TVASSs. For the lower bound, we reduce control-state reachability of R-VASSs to nonemptiness of HRAs. Recall that the nonemptiness problem for HRAs asks, given an HRA A with w initial state q0 and initial assignment H0 , whether (q0 , H0 ) −→ −→ (qF , HF ) for some word w, final state qF and assignment HF . The configurations of a k-dimensional TR-VASS (Transfer–Reset Vector Addition System with States) have the form (q, ~v ), where q is a state from a finite set, and ~v is a k-dimensional vector of nonnegative counters. A VASS has moves that shift the counter vector, changing ~v into ~v + ~v 0 , where ~v 0 comes from some finite and fixed subset of Zk . An R-VASS also has moves that reset a counter, changing ~v into ~v [i 7→ 0] for some counter i. A T-VASS also has moves that transfer the content of one counter into another counter, changing ~v into ~v [j 7→ ~v (i) + ~v (j)][i 7→ 0] for some i = 6 j. A TR-VASS is the obvious combination of the above, and is formally defined as follows. HISTORY-REGISTER AUTOMATA 19 Definition 5.1 (TR-VASS). A k-dimensional Transfer-Reset Vector Addition System with States A is a pair hQ, δi, where Q is a finite set of states, and δ ⊆ Q×(Zk ][k]2 ][k])×Q is a transition relation. A configuration of A is a pair (q, ~v ) of a state q and a vector ~v ∈ Nk of counter values. The configuration graph of A is constructed by including an arc (q, ~v ) → (q 0 , ~v 0 ) when one of the following holds: • there is some (q, ~v 00 , q 0 ) ∈ δ such that ~v 0 = ~v + ~v 00 • there is some (q, (i, j), q 0 ) ∈ δ such that ~v 0 = ~v [i 7→ 0][j 7→ ~v (i) + ~v (j)] and i 6= j • there is some (q, (i, i), q 0 ) ∈ δ and ~v 0 = ~v • there is some (q, i, q 0 ) ∈ δ such that ~v 0 = ~v [i 7→ 0]. The control-state reachability problem for A asks whether, given states q0 , qF and initial vector ~v0 , is there some ~vF such that (q0 , ~v0 ) −→ −→ (qF , ~vF ). The reduction from a (m, 0)-HRA to a T-VASS of dimension 2m − 1 is done by mapping e of the corresponding T-VASS. Then, each nonempty set of histories X into a counter X name-accepting transitions are mapped into counter decreases and increases, while resets
result in transfers between the counters. Let e· : P [m] → [0, 2m − 1] be a bijection such i−1 . Further, given an assignment H, e ,P that e ∅ = 0; for instance, one could take X i∈X 2 m −1 2 e denote the vector (h1 , . . . , h2m −1 ) ∈ N let H such that hXe = |H@X|, for all nonempty X ⊆ [m]; that is, hXe counts how many names occur in exactly the histories indexed by X. Lemma 5.2. Given an (m, 0)-HRA A it is possible to construct a T-VASS A0 of dimension 2m − 1 such that, for all q, q 0 , H, H 0 , w f0 ). e → ∃w, (q, H) −→ −→A (q 0 , H 0 ) if and only if (q, H) →A0 (q 0 , H Let Q and δ be the states and the transitions of A, and let Q0 and δ 0 be the transitions of A0 . We have that |Q0 | ∈ O(2m |Q|) and |δ 0 | ∈ O(2m |δ|). Moreover, the construction takes O(|Q0 | + m|δ 0 |) time. If there are no reset transitions in A, then A0 is a |δ|-dimensional VASS with Q0 = Q and |δ 0 | = |δ| that uses only increments and decrements. Let ~0 be the all-zero vector (0, . . . , 0). Let ~δi be ~0[i 7→ 1] for i ∈ [m], and ~δ0 be ~0. ~ δ e 0 −~ δe X,X 0 X −−−X → q 0 in Proof. For each transition q −→ q 0 of the HRA, we construct a transition q −− X the T-VASS. For each transition q −→ q 0 of the HRA, we construct a path 1,j1 2,j2 3,j3 2m −2,j2m −2 2m −1,j2m −1 q −→ · −→ · −→ · · · −−−−−−−−→ · −−−−−−−−→ q 0 in the T-VASS such that jYe = Y^ \ X. To construct such a path we iterate through 2m − 1 nonempty sets Y , and for each we compute Y \ X in O(m) time. Lemma 5.2 implies that nonemptiness of a HRA reduces to control-state reachability of a T-VASS. We shall describe an algorithm that solves control-state reachability for the T-VASS constructed in Lemma 5.2. The analysis of this algorithm depends on the so-called Length Function Theorem, which is phrased in terms of the Fast Growing Hierarchy and bad sequences. We define these next. The Fast Growing Hierarchy consists of classes F0 , F1 , F2 , . . . of functions, where F0 = F1 contain the linear functions, F2 contains the elementary functions, primitive 20 R. GRIGORE AND N. TZEVELEKOS recursive functions are in Fk for some finite k, and Fω is the Ackermann complexity class. The classes Fk are defined in terms of the following functions: Fn+1 (x) , (Fn ◦ · · · ◦ Fn )(x) = Fnx+1 (x) {z } | F0 (x) , x + 1 Fω (x) , Fx (x) x + 1 times f ∈ O(Fkn ) For k ≥ 2, (a) f ∈ Fk if and only if for some n; and (b) a nondeterministic algorithm using space bounded by some function in Fk can be transformed into a deterministic algorithm using time bounded by some (other) function in Fk . Let X be a partially ordered set with some size function |·| : X → N. We say that a sequence x0 , x1 , x2 , . . . of elements of X is a bad sequence when xi 6≤ xj for all i < j. Given a strictly increasing function g : N → N, we say that the sequence is controlled by g when |xi+1 | ≤ g(|xi |) for all i. We will consider such sequences of VASS configurations, where the order is given by (q, ~v ) ≤ (q 0 , ~v 0 ) iff q = q 0 ∧ ~v (1) ≤ ~v 0 (1) ∧ ~v (2) ≤ ~v 0 (2) ∧ ~v (3) ≤ ~v 0 (3) ∧ . . . Lemma 5.3 (Length Function Theorem [34]). Let q̂0 , q̂1 , q̂2 , . . . be a bad sequence of kdimensional VASS configurations. If the sequence is controlled by some function g ∈ Fγ with γ ≥ 1, then its length is bounded by f (|q̂0 |) for some function f ∈ Fγ+k . We can now describe and analyze an algorithm for deciding emptiness of a (m, 0)-HRA. Proposition 5.4. The emptiness problem for (m, 0)-HRAs is in F2m when m > 0. Thus, the emptiness problem is in Fω when m is part of the input. Proof. Let A be the given HRA, and let A0 be the T-VASS constructed as in Lemma 5.2. We use the backward coverability algorithm [34, Sections 1.2.2 and 2.2.2], which explores all bad sequences (q0 , ~v0 ), (q1 , ~v1 ), . . . , (qL , ~vL ) such that • (q0 , ~v0 ) is a minimal final configuration, and • (qk , ~vk ) is a minimal configuration out of those that can reach a configuration ≥ (qk−1 , ~vk−1 ). The constraint that (q0 , ~v0 ) is a minimal final configuration simply means that q0 is final and ~v0 = ~0. To construct such sequences, we need an effective way of generating all possible (qk , ~vk ), given a fixed (qk−1 , ~vk−1 ). For this, we enumerate all transitions of A0 that go to qk−1 . ~ δi −~ δj i,j ~ δi −~ δj There are two types of such transitions: qk −→ qk−1 and qk −→ qk−1 . For qk −→ qk−1 , i,j we let ~vk be max(~vk−1 − ~δi + ~δj , ~0), where max is taken pointwise. For qk −→ qk−1 with i,j i = j, we take ~vk to equal ~vk−1 . For qk −→ qk−1 with i 6= j, we may have multiple choices for ~vk . Assuming ~vk−1 (i) = 0, it could be that ~vk (i) is any of 0, 1, . . . , ~vk−1 (j); otherwise, if ~vk−1 (i) 6= 0, the transition could not have been taken. In all the cases from above, we keep only those choices of ~vk that ensure the sequence is bad. To show that the sequences so constructed are finite, we use Lemma 5.3. Let |(q, ~v )| be the number bits in a concrete representation of (q, ~v ): we encode q with ∼ log2 |Q0 | = m log2 |Q| bits, then we write each ~v (i) in binary, precede each of its bits by 1 and mark the end with 0. (For example, we represent 5 by 1110110.) For the sequences constructed as in the previous paragraph, we have |(qk , ~vk )| ≤ 2 · |(qk−1 , ~vk−1 )|. Thus, the sequences are controlled by g(x) = 2x, which is a function in F1 . As A0 has dimension 2m − 1, Lemma 5.3 gives us that the length L of the sequence is bounded by some function in F2m . A nondeterministic algorithm can repeatedly guess the correct successor in the sequence, using 2L · |(q0 , ~0)| space. If m ≥ 1, then this is bounded by f (m log |Q|) for some function HISTORY-REGISTER AUTOMATA 21 f ∈ F2m , and we are in a situation where the distinctions time/space and deterministic/ nondeterministic are irrelevant. It is possible to modify the algorithm described in the previous proof so that it works directly on the HRA representation, without appealing to Lemma 5.2. Similarly, it is possible to extend the algorithm described in the previous proof to handle registers directly, without appealing to Proposition 4.1. Such improvements may be worthwhile in an implementation, but the complexity upper bound remains Ackermannian. Doing the opposite reduction we show that deciding emptiness is Ackermann-hard even for strongly deterministic HRAs. In this direction, each R-VASS of dimension m can be simulated by an (m, 0)-HRA so that the value of each counter i of the former is the same as the number of names appearing precisely in history i of the latter. In order to extend the bound to strongly deterministic HRAs one can choose to reduce from a restricted class of R-VASSs, so that the image of the reduction can be made strongly deterministic, or resolve nondeterminacy at the level of HRAs by appropriate obfuscation. We follow the latter, simpler solution. Proposition 5.5. The emptiness problem for strongly deterministic HRAs is Ackermannhard. Proof. Let A be an m-dimensional R-VASS whose additive transitions only increment or ~v decrement single counters: for each transition q −→ q 0 , we have ~v = ±~δi for some i. By [35], control-state reachability for such R-VASSs is Ackermann-hard. We construct an (m, 0)δ ∅,{i} −δ i i HRA A0 with the same states as A, and we map: each q − → q 0 to q −→ q 0 , each q −−→ q 0 to {i},∅ i {i} q −→ q 0 , and each q → − q 0 to q −→ q 0 . We can see that A0 simulates the behaviour of A by storing the value of each counter i as |H@{i}|. Hence, L(A0 ) is nonempty if and only if qF is reachable, from (q0 , ~v0 ), in the R-VASS A. We observe that A0 may not be strongly deterministic. Suppose that the size of the transition function of A is n. We can then impose strong determinacy on A0 by enriching it with n registers and preluding each transition of the above translation with a transition reading from one of the additional registers. We thus obtain an (m, n)-HRA that is strongly deterministic and simulates A as above. Proposition 5.6. The emptiness problem of HRAs is Ackermann-complete. Proof. Combine Proposition 4.1 with Proposition 5.4 and Proposition 5.5. 5.2. Universality. We finally consider universality and language containment. Note first that our machines inherit undecidability of these properties from register automata [31]. However, these properties are decidable in the deterministic case. In order to simplify our analysis, we shall be reducing HRAs to the following compact form where -transitions are incorporated inside name-accepting ones. As we show below, no expressiveness is lost by this packed form. A packed (m, 0)-HRA is a tuple A = hQ, q0 , δ, H0 , F i defined exactly as an (m, 0)-HRA, with the exception that now: δ ⊆ Q × P([m]) × P([m]) × P([m]) × Q 22 R. GRIGORE AND N. TZEVELEKOS Y ;X,X 0 We shall write q −−−−−→ q 0 for (q, Y, X, X 0 , q 0 ) ∈ δ. The semantics of such a transition is Y X,X 0 the same as that of a pair of transitions q −→ · −→ q 0 of an ordinary HRA. Formally, configurations of packed HRAs are pairs (q, H), like in HRAs, and the configuration graph a of a packed HRA A like the above is constructed as follows. We set (q, H) −→ (q, H 0 ) if Y ;X,X 0 there is some q −−−−−→ q 0 in δ such that, setting HY = H[Y 7→ ∅], we have a ∈ HY @X and H 0 = HY [a in X 0 ]. Lemma 5.7. Let A be an (m, 0)-HRA. There is a packed (m, 0)-HRA A0 such that A ∼ A0 . Proof. Let A = hQ, q0 , δ, H0 , F i. We set A0 = hQ, q0 , δ 0 , H0 , F 0 i where: Y F 0 = {q 0 ∈ Q | ∃q ∈ F, Y. q 0 −→ −→ q ∈ δ} Y X,X 0 δ 0 = {(q, Y, X, X 0 , q 0 ) | q −→ −→ · −→ q 0 ∈ δ} Bisimilarity of A and A0 is witnessed by the identity on configurations, which means that R = { ((q, H), (q, H)) | q ∈ Q ∧ H ∈ Asn } is a bisimulation. We shall decide language containment via complementation. In particular, given a deterministic packed HRA A, the automaton A0 accepting the language N ∗ \ L(A) can be constructed in the analogous way as for deterministic finite-state automata, namely by obfuscating the automaton with all missing transitions and swapping final with non-final states. Lemma 5.8. Deterministic packed HRAs are closed under complementation. Proof. Let A = hQ, q0 , δ, H0 , F i be a packed (m, 0)-HRA. Following the above rationale, we construct a packed (m, 0)-HRA A0 = hQ ] {qF }, q0 , δ ∪ δ 0 , H0 , F 0 i, where F 0 = {qF } ∪ (Q \ F ) Y ;X\Y,X 0 and δ 0 is given as follows. For each q ∈ Q and all X such that there is no q −−−−−−→ q 0 add ∅;X,∅ [m];∅,∅ a transition q −−−−→ qF in δ 0 . In addition, δ 0 contains a transition qF −−−−−→ qF . We claim that L(A0 ) = N ∗ \ L(A). Indeed, if s ∈ L(A0 ) and s is accepted at a state in Q \ F then, since A is deterministic, we have s ∈ / L(A). Otherwise, if s = s0 as00 with a the point where a transition to the sink state is taken then, upon acceptance of s0 by A, a appears precisely in some histories X such that A has no transition to accept a at that point. Thus, s∈ / L(A). Conversely, if s ∈ N ∗ \ L(A) then either s induces a configuration in A which does not end in a final state, or s = s0 as00 where s0 is accepted by A but at that point a is not a possible transition. We can see that, in each case, s ∈ L(A0 ). Proposition 5.9. Language containment and universality are undecidable for (general) HRAs and Ackermann-complete for strongly deterministic HRAs. Proof. Undecidability in the general case is inherited from RAs. Now consider two HRAs A and A0 such that we can compute the complement of A0 . Then, we can decide the language containment L(A) ⊆ L(A0 ) by checking whether the product of A with the complement of A0 is empty. The product construction is polynomial, and the emptiness check is in Ackermann (Proposition 5.4). Thus, language containment is in Ackermann if computing the complement of A0 is in Ackermann. This is the case because (a) removing registers can be done while preserving determinism with only an exponential increase in size (Proposition 4.1), and (b) complementing deterministic HRAs HISTORY-REGISTER AUTOMATA 23 without registers takes polynomial time (Lemma 5.8). For hardness, note that emptiness and universality are equally hard in the deterministic case (Lemma 5.8), and emptiness is Ackermann-hard (Proposition 5.5). We showed that language containment is in Ackermann and universality is Ackermann-hard. Finally, note that there is a trivial reduction from universality to language containment. 6. Weakening HRAs Since the complexity of HRAs is substantially high, e.g. for deciding emptiness, it is useful to seek for restrictions thereof which allow us to trade expressiveness for efficiency. As the encountered complexity stems from the fact that HRAs can simulate computations of R-VASSs, our strategy for producing weakenings is to restrict the functionalities of the corresponding R-VASSs. We follow two directions: (a) We remove reset transitions. This corresponds to removing counter transfers and resets and drops the complexity of control-state reachability to exponential space. (b) We restrict the number of histories to just one. We thus obtain polynomial space complexity as the corresponding counter machines are simply one-counter automata. This kind of restriction is also a natural extension of FRAs with history resets. Observe that each of the aspects of HRAs targeted above corresponds to features (1,2) we identified in the Introduction, witnessed by the languages L1 and L2 respectively. We shall see that each restriction leads to losing the corresponding language. 6.1. Non-reset HRAs. We first weaken our automata by disallowing resets. We show that the new machines retain all their closure properties apart from Kleene-star closure. The latter is concretely manifested in the fact that language L1 of the Introduction is lost. On the other hand, the emptiness problem reduces in complexity to exponential space. Definition 6.1. A non-reset HRA of type (m, n) is an (m, n)-HRA A = hQ, q0 , H0 , δ, F i X such that there is no q −→ q 0 ∈ δ. Closure properties. Of the closure constructions of Section 3 we can see that union and intersection readily apply to non-reset HRAs, while the construction for concatenation needs some amendments. More specifically, of the two constructions presented in the proof of Proposition 3.2, the one for concatenation can be adapted to non-reset HRAs as follows. We add empty transitions from the final states of A01 to the initial state of a version of A02 which keeps the places used by A01 untouched and uses its own separate copy of places, obfuscating its own transitions so as to capture accidental matchings of the legacy names of A01 . This solution cannot be used for Kleene closure as in each loop the automaton needs to find a fresh copy of its initial configuration, and be able to use it (in the previous construction, the final assignment of A01 is lost). On the other hand, using an argument similar to that of [5, Proposition 7.2], we can show that the language L1 is not recognised by non-reset HRAs and, hence, the latter are not closed under Kleene star. Finally, note that the HRA constructed for the language L4 24 R. GRIGORE AND N. TZEVELEKOS in Example 3.4 is a non-reset HRA, which implies that non-reset HRAs are not closed under complementation. Emptiness. In the general case we saw an upper bound of F2m (Proposition 5.4), by a reduction to T-VASS followed by the backward coverability algorithm. For a non-reset HRA, the same reduction yields a VASS, without transfers. In the absence of transfers, better bounds are known for the backward coverability algorithm [11]. More generally, it has been known for some time that coverability for VASS is ExpSpace-complete [12, 25]. The following result refers to the number N of bits used to represent an HRA. Of course, N depends on the exact representation being used. Still, we do not make this representation explicit because the result holds for a wide variety of possible representations. We only require that the representation obeys m, n, |δ|, log |Q| ∈ O(N ). Proposition 6.2. The emptiness problem for a non-reset HRA is in ExpSpace. More
precisely, it is in NSpace 2O(N log N ) , where N is the number of bits used to represent the HRA. Proof. We start with an (m, n)-non-reset-HRA A = hQ, q0 , δ, H0 , F i. We use Proposition 4.7 to construct an (m, 0)-non-reset-HRA A0 = hQ0 , q00 , δ 0 , H00 , F 0 i that preserves emptiness. Moreover, log |Q0 | ∈ O(mn + n log n + log |Q|). Using Lemma 5.2, we reduce A0 to a (|δ| + 1)dimensional VASS with |Q0 | states that uses only increments/decrements. (Lemma 5.2 creates an m-dimensional VASS where m is the number of sets labelling transitions. Proposition 4.7 puts in A0 only sets that already occurred in A, with the possible exception of ∅.) Now we apply the backward coverability algorithm, as described in the proof of Proposition 5.4. By [11, Theorem 2], the algorithm will only consider counter values less than some V = O(|δ| log |δ|) (3|Q0 |)2 . In the nondeterministic version of the algorithm, we guess the next configuration, which means we only need space O((|δ| + 1) log V ) to store a couple of configurations. We have log log V = O(|δ| log |δ| + log log |Q0 |) = O(|δ| log |δ| + log(mn + n log n + log |Q|)) Since m, n, |δ|, log |Q| ∈ O(N ), we conclude log |δ| + log log V ∈ O(N log N ). This implies that a nondeterministic version of the backward coverability algorithm works in
O(N log N ) NSpace 2 . The previous proposition has a couple of obvious consequences. First, emptiness is also in
O(N log N )
DSpace 2O(N log N ) , by Savitch’s theorem. Second, emptiness is also in DTime 22 , by a standard easy argument [19, Theorem 5.3]. In fact, one can show that the time bound applies to the backward coverability algorithm, without invoking generic constructions from complexity theory: By [11, Theorem 2], the runtime of the backward coverability algorithm — O(|δ| log |δ|)
like the counter values — is also upper bounded by some T = O (3|Q0 |)2 . The rest of the argument is as in the proof of Proposition 6.2. Proposition 6.3. The emptiness problem for non-reset HRAs is ExpSpace-hard. Proof. By [25], the control-state reachability problem for VASS is ExpSpace-hard even if all the transitions are restricted to have labels of the form ±~δi . (More precisely, Lipton proves that certain parallel programs of size poly(k) can simulate any Turing machine that uses < 2k space. Then, [25, Lemma 2] asserts that reachability in these programs reduces to reachability in VAS, the full proof being: ‘We omit a detailed proof of this lemma. It HISTORY-REGISTER AUTOMATA 25 should, however, be clear that parallel programs can be encoded as vector addition systems.’ Similarly, we claim without proof, that it should be clear how Lipton’s programs reduce to the control-state reachability problem for VASSs whose transitions only increment/decrement single counters.) We shall reduce the control-state reachability problem for such VASSs to the emptiness problem for non-reset HRAs. Let m be the dimension of the VASS. We construct a HRA with m0 histories, where 0 0 m is the smallest integer such that m ≤ 2m − 1. As a result, there exists an injection φ : [m] → P6=∅ ([m0 ]); we fix arbitrarily one such injection. +~ δ ∅,φ(i) −~ δ φ(i),∅ • For each transition q −→i q 0 in the VASS, we include a transition q −→ q 0 in the HRA. • For each transition q −→i q 0 in the VASS, we include a transition q −→ q 0 in the HRA. This construction maintains the invariant |H@φ(i)| = ~v (i). To establish the invariant, we pick the initial history assignment H0 accordingly. Finally, we set as final the state in whose reachability we are interested. The reduction described above is clearly polynomial, from which it follows that emptiness of non-reset HRAs (even without registers) is ExpSpace-hard. Proposition 6.4. The emptiness problem for non-reset HRAs is ExpSpace-complete. Proof. Immediate from Proposition 6.2 and Proposition 6.3. 6.2. Unary HRAs. Our second restriction concerns allowing resets but bounding the number of histories to just one. Thus, these automata are closer to the spirit of FRAs and, in fact, extend them by rounding up their history capabilities. We show that these automata require polynomial space complexity for emptiness and retain all their closure properties apart from intersection. The latter is witnessed by failing to recognise L2 from the Introduction. We can see that extending this example to multiple interleavings we can show that intersection is in general incompatible with bounding the number of histories. Definition 6.5. A (1, n)-HRA is called unary HRA of n registers. In other words, unary HRAs are extensions of FRAs where names can be selectively inserted or removed from the history and, additionally, the history can be reset. These capabilities give us in fact a strict extension. Example 6.6. The automata used in Example 2.3 for L1 and L3 were unary HRAs. Note that neither of those languages is FRA-recognisable. On the other hand, in order to recognise L2 , an HRA would need to use at least two histories: one history for the odd positions of the input and another for the even ones. We can formalise an argument to show that L2 is not recognisable by unary HRAs as follows. Proof. Suppose L2 = L(A) for some unary HRA A of n registers and let w = a1 b1 . . . ak bk b1 a1 · · · bk ak for k = n + 1 and some pairwise distinct names a1 , b1 , . . . , ak , bk . As w ∈ L2 , there is a path, say p, in A which accepts w. We divide p as p1 p2 with p2 accepting the second half of w. Let p̂ = p̂1 p̂2 be the corresponding configuration path and let (q 0 , H 0 ) be the first configuration in p̂2 . We set S = {a1 , b1 , · · · , ak , bk } \ {a | a ∈ H 0 (i) ∧ i > 1} and do a case analysis on the labels of the form (X, X 0 ) which appear in p2 and accept names from S. Since names in S 26 R. GRIGORE AND N. TZEVELEKOS do not appear in any H 0 (i), for i > 0, it must be that each such X is either {1} or ∅. We have the following cases. • There are two such labels, say ({1}, Xi ) and ({1}, Xj ), accepting names ai and bj respectively. But this would imply that A also accepts w0 , where w0 is w with these occurrences of ai and bj swapped, contradicting L(A) = L2 (as w0 ∈ / L2 ). • There are two such labels, say (∅, Xi ) and (∅, Xj ), accepting names ai and bj respectively. In order for A not to accept w0 (w0 as above), it is necessary that a reset transition with label Y 3 1 occurs between the two transitions. Suppose i < j. Then, since k > n, there is a name ai0 which does not appear in any place after clearing Y . Thus, (∅, Xj ) can accept ai0 and complete the path p by accepting a word w0 ∈ / L2 . Dually if j ≤ i. • Each ai ∈ S is accepted by a label ({1}, X 0 ), and each bj ∈ S by a label (∅, X 0 ). Let ai ∈ S be the last such accepted in p2 . This means that the rest of the path has length at most 2n. Therefore, since k > n, there is a bj ∈ S accepted in p2 before ai . Let (q, H) be the configuration just before accepting bj . In order for A not to accept any ai0 at that point, it must be that all ai0 ∈ S appear in H. Since |S| > n + 1, there exists ai0 ∈ H(1) ∩ S such that ai0 6= ai . But then, the transition accepting ai can accept ai0 instead and lead to acceptance of a word w0 6∈ L2 . We therefore reach a contradiction in every case. Closure properties. The closure constructions of Section 3 readily apply to unary HRAs, with one exception: intersection. For the latter, we can observe that L2 = L(A1 ) ∩ L(A2 ), where L(A1 ) = {a1 a01 . . . an a0n ∈ N ∗ | a1 . . . an ∈ L0 } and L(A2 ) = ∅,1 q0 q1 {a1 a01 . . . an a0n ∈ N ∗ | a01 . . . a0n ∈ L0 }, and A1 and A2 are the unary (1, 0)-HRAs on the side, with empty initial assignments. On ∅,∅ / 1,1 the other hand, unary HRAs are not closed under complementation ∅,∅ / 1,1 as well, as one can construct unary HRAs accepting L(A1 ) and q0 q1 L(A2 ), and then take their union to obtain a unary HRA for L2 . ∅,1 Emptiness. In the case of just one history, the results on TR-VASS reachability [35, 16] from Section 5 provide rather rough bounds. It is therefore useful to do a direct analysis. We reduce nonemptiness for unary HRAs to control-state reachability for one dimensional R-VASSs. Our analysis below shows that the minimal path has length at most quadratic, from which it follows that nonemptiness has polynomial complexity. The following result applies to any R-VASS representation for which |Q| log |Q| ∈ O(N ), where |Q| is the number of states of the R-VASS, and N is the number of bits used to represent the R-VASS. Note that the condition is true if all the states are listed in the R-VASS representation, something all reasonable representations would do. Lemma 6.7. Control-state reachability for one dimensional R-VASSs is in NL, provided that non-reset transitions increase and decrease the counter by at most 1. Proof. Let A = hQ, δi be an R-VASS of dimension 1. The proof relies on two observations: Fact 1: If (q, i) −→ −→ (q 0 , i0 ) is a configuration path of A then, for each k > 0, there is a path (q, i+k) −→ −→ (q 0 , i00 ) of the same length. Fact 2: If (q, i) −→ −→ (q 0 , i0 ) is a configuration path of A in which there are no reset transitions and the counter never becomes less than some k > 0, then there is a path (q, i−k) −→ −→ (q 0 , i00 ) of the same length. HISTORY-REGISTER AUTOMATA 27 Consider an instance (A, q0 , i0 , qF ) of the control-state reachability problem: Is the state qF reachable in A starting from configuration (q0 , i0 )? Let p be a configuration path of minimal length from (q0 , i0 ) to some configuration whose state is qF . Let us see if a state can appear repeatedly in p. By Fact 1, p is non-decreasing: any path segment (q, i) −→ −→ (q, i0 ) 0 can be circumvented if i ≥ i . Now suppose that p contains a segment (q, i) −→ −→ (q, i + k) for some k > 0. Consider the segment (q, i + k) −→ −→ (q 00 , i00 ) that follows, uses only non-reset transitions, and is maximal. By Fact 2, if the counter never becomes < k in the latter segment, then there exists a path (q, i) −→ −→ (q 00 , i00 ) of the same length. Since p is minimal, this is a contradiction, and therefore the counter must become < k, somewhere after (q, i + k). Let p0 be the segment (q, i + k) −→ −→ (q 0 , k − 1). Since non-reset transitions decrease the counter by ≤ 1, it must be that all the values i + k, i + k − 1, . . . , k − 1 occur in p0 . When one of these values is reached for the first time, it must be paired with a state that was not used for the bigger values. It follows that (i + k) − (k − 1) + 1 ≤ |Q|, and so i ≤ |Q| − 2. This gives us a bound on the counter value of any state that can be repeated in p. Thus, each state can appear in p at most |Q| times. This implies that the length of p is at most |Q|2 and that in p the counter does not exceed the value i0 + |Q|2 . We can therefore answer the instance (A, q0 , i0 , qF ) of the control-state reachability problem as follows. Note first that, by Facts 1 and 2 and because the length of minimal reaching path is ≤ |Q|2 , we can replace i0 by min(i0 , |Q|2 ). Because we only consider initial counter values ≤ |Q|2 and because the minimal path has length ≤ |Q2 |, we can store one configuration on the minimal path using O(log |Q|) bits. Since |Q| log |Q| ∈ O(N ), we have log |Q| ∈ O(log N ), and therefore O(log N ) bits suffice to represent a configuration of the minimal path. Finally, we note that a nondeterministic algorithm can guess the next configuration on the minimal path. We remark that an NL upper bound follows from an analysis of the backward coverability algorithm as well. However, the proof from above has the advantage that it is self-contained. We now give an upper bound for the emptiness problem of unary HRAs. The result holds for all representations that obey several weak requirements. Let hQ, q0 , δ, H0 , F i be a unary HRA with n registers, represented with N bits. We require that • n ∈ O(N ), which is justified because there are > 2n possible labels on transitions; • |δ| ∈ O(N ), which is justified because we expect each transition to require at least a bit; • |Q| log |Q| ∈ O(N ), which is justified because we expect each state to be mentioned at least once in the representation. (This last point implies that log |Q| ∈ O(log N ).) Proposition 6.8. The emptiness problem for unary HRAs is in PSpace. More precisely, it is in NSpace(N log N ), where N is the number of bits used to represent the HRA. Proof. Let A = hQ, q0 , δ, H0 , F i be the given unary HRA. Using Proposition 4.7, we build a (1, 0)-HRA A0 that preserves emptiness, has O(Bn 2n n log |δ|) transitions, and has O(Bn 2n n log |Q|) states. Using the construction from Lemma 5.2, we reduce the emptiness of A0 to control-state reachability in an R-VASS A00 . Specialized to our case, the construction says that ∅,{1} +1 {1},∅ −1 • for each transition q −→ q 0 in A0 , we include a transition q −→ q 0 in A00 ; • for each transition q −→ q 0 in A0 , we include a transition q −→ q 0 in A00 ; and {1} reset • for each transition q −→ q 0 in A0 , we include a transition q −→ q 0 in A00 . 28 R. GRIGORE AND N. TZEVELEKOS According to Lemma 6.7, the control-state reachability problem for A00 is in NSpace(log N 00 ), where N 00 is the number of bits used to represent A00 . Thus, it remains to compute N 00 as a function of N . For this, we pick one particular representation of A00 , namely
a list of transitions. For such a representation we have N 00 = O Bn 2n n|δ| · log(Bn 2n n|Q|) . Thus, log N 00 = O(n log n + log |δ| + log(n log n + log |Q|)) = O(N log N ) The last step assumes that n, |Q| log |Q|, |δ| ∈ O(N ). We require the representation of A to satisfy these assumptions. Proposition 6.9. The emptiness problem for unary HRAs is PSpace-hard. Proof. By [15, Theorem 5.1a], the nonemptiness problem of register automata is PSpacehard. Register automata are a special case of unary HRAs. Proposition 6.10. The emptiness problem for unary HRAs is PSpace-complete. Proof. Immediate from Proposition 6.8 and Proposition 6.9. 7. Summary of Main Results The theorems in this section summarize the main results proved in the previous sections. Theorem 7.1. Languages recognised by HRAs are closed under union, intersection, concatenation, and Kleene star, but not under complementation. Also, • if resets are banned, then closure under Kleene star is lost; • if the number of histories is bounded, then closure under intersection is lost. Proof. Immediate from Proposition 3.2, Lemma 3.3, and the closure results of Section 6. Theorem 7.2. Deciding emptiness of an (m, n)-HRA has the following complexity: (a) NL-complete if m = n = 0; (b) NP-complete if m = 0 and all sets labelling transitions are singletons; (c) PSpace-complete if m ≤ 1; (d) ExpSpace-complete if there are no reset transitions; and (e) Ackermann-complete in the general case. Proof. (a) When m = n = 0, nonemptiness is equivalent to reachability in a directed graph, which is a standard NL-complete problem. (b) In this case, HRAs are equivalent to RAs that disallow repetitions of values in registers, as they were originally defined [23]. For such RAs, nonemptiness is known to be NP-complete [33, Theorem 4]. (c) Proposition 6.10. (d) Proposition 6.4. (e) Proposition 5.6. For universality and language inclusion, see Proposition 5.9. HISTORY-REGISTER AUTOMATA 29 8. Connections with existing formalisms We have already seen that HRAs strictly extend FRAs. In this section, we compare HRAs with CMAs (class memory automata). Like HRAs and FRAs, CMAs work on infinite alphabets, and have a decidable nonemptiness problem. Implicitly, we also compare with other formalisms: CMAs have been shown to express the same languages as data automata [5, Proposition 3.7]; and data automata have been shown to express the same languages as the two-variable fragment of existential monadic second order logic with data equality, position successor, and class successor [7, Proposition 14]. Definition 8.1. A Class Memory Automaton (CMA) is a tuple A = hQ, q0 , φ0 , δ, F1 , F2 i where Q is a finite set of states, q0 ∈ Q is initial, F1 ⊆ F2 ⊆ Q are sets of final states and the transition relation is of type δ ⊆ Q × (Q ∪ {⊥}) × Q. Moreover, φ0 is an initial class memory function, that is, a function φ : N → Q ∪ {⊥} with finite domain ({ a | φ(a) 6= ⊥ } is finite). The semantics of a CMA A is given as follows. Configurations of A are pairs of the form (q, φ), where q ∈ Q and φ a class memory function. The configuration graph of A is a constructed by setting (q, φ) −→ (q 0 , φ0 ) just if there is (q, φ(a), q 0 ) ∈ δ and φ0 = φ[a 7→ q 0 ]. The initial configuration is (q0 , φ0 ), while a configuration (q, φ) is accepting just if q ∈ F1 and, for all a ∈ N , φ(a) ∈ F2 ∪ {⊥}. Thus, CMAs resemble HRAs in that they store input names in “histories”, only that histories are identified with states: for each state q there is a corresponding history q (note notation overloading), and a transition which accepts a name a and leads to a state q must store a in the history q. Moreover, each name appears in at most one history (hence the type of φ), while the finality conditions for configurations allow us to impose that, at the end, all names must appear in specific histories, if they appear in any. For instance, the language L4 of Example 3.4, which we know cannot be recognized by HRAs (Lemma 3.3), can be recognized by the following CMA on the left (with F1 = F2 = {q0 }). ∅,1 ⊥ q0 q1 q1 q0 ⊥ 1,2 q1 q1 1,2 ∅,1 Each name is put in history q1 when seen for the first time, and in history q0 when seen for the second time. The automaton accepts if all its names are in q0 . This latter condition is what makes the essential difference to HRAs, namely the capability to check where the names reside for acceptance. For example, the HRA on the right above would accept the same language it we were able to impose S the condition that accepting configurations (q, H) satisfy a ∈ H@{2} for all names a ∈ i H(i). Note though that, extending HRAs with such finality conditions would render their nonemptiness problem reducible from reachability of R-VASS (i.e. the question whether a specific state and counter content can be reached), a problem known to be undecidable [2]. The above example proves that HRAs cannot express the same languages as CMAs. Conversely, as shown in [5, Proposition 7.2], the fact that CMAs lack resets does not allow them to express languages like, for example, L1 . As a result, the languages expressed by CMAs are closed under intersection, union and concatenation, but not under Kleene star. In the latter sections of [5] several extensions of CMAs are considered, one of which does 30 R. GRIGORE AND N. TZEVELEKOS involve resets. However, the resets considered there do not seem directly comparable to the reset capability of HRAs. On the other hand, a direct comparison can be made with non-reset HRAs. We already saw in Proposition 4.4 that, in the latter idiom, histories can be used for simulating register behaviour. In the absence of registers, CMAs differ from non-reset HRAs solely in their constraint of relating histories to states (and their termination behaviour, which is more expressive). As the latter can be easily counterbalanced by obfuscating the set of states, we obtain the following. Proposition 8.2. For each non-reset HRA A there is a CMA A0 such that L(A) = L(A0 ). 9. Further directions Our goal is to apply automata with histories in static and runtime verification. For static verification, the complexity results derived in this paper may seem discouraging at first. However, they are based on very specific representations of hard problems; in practice, we expect programs to yield automata of simpler complexities. Experience with tools based on coverability of TR-VASSs, like e.g. BFC [22], positively testify in that respect. Another solution, already pursued herein, is to explore constrained versions of our machines. A specific such variant we envisage to consider is one with restricted resets, in analogy to e.g. [17]. In a related direction, we aim to look at abstractions that would allow us to attack the model-checking problem for these automata, and also look at temporal logics that capture part or all of the expressivity of HRAs. In this work we examined nondeterministic automata but did not look at alternating variants. This is justified by the undecidability of universality already at the level of register automata. However, if one is willing to restrict the number of registers and histories, there may still be room for decidability. In the case of register automata, it has been shown [15] that alternating register automata with one register are decidable for emptiness, and become undecidable at two registers. While these automata cannot capture languages that inherently require more than one register, they can use alternation to express name freshness and e.g. capture the languages L0 , L2 of the Introduction, and also a variant of L1 which uses constants for tokenizing the input (instead of a0 ). It would be useful to examine whether a similar restriction can yield decidable alternating HRAs, and what would their expressivity be. Finally, a problem left open here is decidability and complexity of bisimilarity. (In a private communication, Piotrek Hofman sketched a proof that bisimilarity is decidable.) References [1] S. Abramsky, D. R. Ghica, A. S. Murawski, C.-H. L. Ong, and I. D. B. Stark. Nominal games and full abstraction for the nu-calculus. In Logic in Computer Science (LICS), 2004. [2] T. Araki and T. Kasami. Some decision problems related to the reachability problem for Petri nets. Theoretical Computer Science (TCS), 1976. [3] M. Faouzi Atig, A. Bouajjani, and S. Qadeer. Context-bounded analysis for concurrent programs with dynamic creation of threads. Logical Methods in Computer Science (LMCS), 2011. [4] N. Benton and B. Leperchey. Relational reasoning in a nominal semantics for storage. In Typed Lambda Calculi and Applications (TLCA), 2005. [5] H. Björklund and T. Schwentick. On notions of regularity for data languages. Theoretical Computer Science (TCS), 2010. HISTORY-REGISTER AUTOMATA 31 [6] M. Bojanczyk, L. Braud, B. Klin, and S. Lasota. Towards nominal computation. In Principles of Programming Languages (POPL), 2012. [7] M. Bojanczyk, C. David, A. Muscholl, T. Schwentick, and L. Segoufin. Two-variable logic on data words. Transactions on Computational Logic (TOCL), 2011. [8] M. Bojanczyk, B. Klin, and S. Lasota. Automata theory in nominal sets. Logical Methods in Computer Science (LMCS), 2014. [9] M. Bojanczyk, B. Klin, S. Lasota, and S. Torunczyk. Turing machines with atoms. In Logic in Computer Science (LICS), 2013. [10] A. Bouajjani, S. Fratani, and S. Qadeer. Context-bounded analysis of multithreaded programs with dynamic linked structures. In Computer Aided Verification (CAV), 2007. [11] L. Bozzelli and P. Ganty. Complexity analysis of the backward coverability algorithm for VASS. In Reachability Problems (RP), 2011. [12] Rackoff C. The covering and boundedness problems for vector addition systems. Theoretical Computer Science (TCS), 1978. [13] C. Cotton-Barratt, A. S. Murawski, and C.-H. Luke Ong. Weak and nested class memory automata. In Language and Automata Theory and Applications (LATA), 2015. [14] N. Decker, P. Habermehl, M. Leucker, and D. Thoma. Ordered navigation on multi-attributed data words. In Concurrency Theory (CONCUR), 2014. [15] S. Demri and R. Lazić. LTL with the freeze quantifier and register automata. Transactions on Computational Logic (TOCL), 2009. [16] D. Figueira, S. Figueira, S. Schmitz, and P. Schnoebelen. Ackermannian and primitive-recursive bounds with Dickson’s lemma. In Logic in Computer Science (LICS), 2011. [17] A. Finkel and A. Sangnier. Mixing coverability and reachability to analyze VASS with one zero-test. In Current Trends in Theory and Practice of Computer Science (SOFSEM), 2010. [18] M. J. Gabbay and A. M. Pitts. A new approach to abstract syntax with variable binding. Formal Aspects of Computing, 2002. [19] O. Goldreich. Computational Complexity: A Conceptual Perspective. Cambridge University Press, 2008. [20] R. Grigore, D. Distefano, R. L. Petersen, and N. Tzevelekos. Runtime verification based on register automata. In Tools and Algorithms for the Construction and Analysis of Systems (TACAS), 2013. [21] A. Jeffrey and J. Rathke. Towards a theory of bisimulation for local names. In Logic in Computer Science (LICS), 1999. [22] A. Kaiser, D. Kroening, and T. Wahl. Efficient coverability analysis by proof minimization. In Concurrency Theory (CONCUR), 2012. [23] M. Kaminski and N. Francez. Finite-memory automata. 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Schwentick, and V. Vianu. Finite state machines for strings over infinite alphabets. Transactions on Computational Logic (TOCL), 2004. [32] A. M. Pitts and I. Stark. On the observable properties of higher order functions that dynamically create local names, or: What’s new? In Mathematical Foundations of Computer Science (MFCS), 1993. [33] H. Sakamoto and D. Ikeda. Intractability of decision problems for finite-memory automata. Theoretical Computer Science (TCS), 2000. 32 R. GRIGORE AND N. TZEVELEKOS [34] S. Schmitz and P. Schnoebelen. Algorithmic aspects of WQO theory. Lecture Notes hcel-00727025v2i, 2012. [35] P. Schnoebelen. Revisiting Ackermann-hardness for Lossy Counter Machines and Reset Petri Nets. In Mathematical Foundations of Computer Science (MFCS), 2010. [36] L. Segoufin. Automata and logics for words and trees over an infinite alphabet. In Computer Science Logic (CSL), 2006. [37] I. D. B. Stark. Names and Higher-Order Functions. PhD thesis, University of Cambridge Computing Laboratory, 1995. [38] N. Tzevelekos. Fresh-register automata. In Principles of Programming Languages (POPL), 2011. [39] N. Tzevelekos and R. Grigore. History-register automata. In Foundations of Software Science and Computation Structures (FoSSaCS), 2013. This work is licensed under the Creative Commons Attribution-NoDerivs License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nd/2.0/ or send a letter to Creative Commons, 171 Second St, Suite 300, San Francisco, CA 94105, USA, or Eisenacher Strasse 2, 10777 Berlin, Germany 

1 Energy Storage Sharing in Smart Grid: A Modified Auction Based Approach arXiv:1512.07700v1 [] 24 Dec 2015 Wayes Tushar, Member, IEEE, Bo Chai, Chau Yuen, Senior Member, IEEE, Shisheng Huang, Member, IEEE, David B. Smith, Member, IEEE, H. Vincent Poor Fellow, IEEE, and Zaiyue Yang, Member, IEEE Abstract—This paper studies the solution of joint energy storage (ES) ownership sharing between multiple shared facility controllers (SFCs) and those dwelling in a residential community. The main objective is to enable the residential units (RUs) to decide on the fraction of their ES capacity that they want to share with the SFCs of the community in order to assist them storing electricity, e.g., for fulfilling the demand of various shared facilities. To this end, a modified auction-based mechanism is designed that captures the interaction between the SFCs and the RUs so as to determine the auction price and the allocation of ES shared by the RUs that governs the proposed joint ES ownership. The fraction of the capacity of the storage that each RU decides to put into the market to share with the SFCs and the auction price are determined by a noncooperative Stackelberg game formulated between the RUs and the auctioneer. It is shown that the proposed auction possesses the incentive compatibility and the individual rationality properties, which are leveraged via the unique Stackelberg equilibrium (SE) solution of the game. Numerical experiments are provided to confirm the effectiveness of the proposed scheme. Index Terms—Smart grid, shared energy storage, auction theory, Stackelberg equilibrium, strategy-proof, incentive compatibility. I. I NTRODUCTION E NERGY storage (ES) devices are expected to play a significant role in the future smart grid due to their capabilities of giving more flexibility and balance to the grid by providing a back-up to the renewable energy [1]–[9]. ES can improve the electricity management in a distribution network, reduce the electricity cost through opportunistic demand response, and improve the efficient use of energy [10]. The distinct features of ES make it a perfect candidate to assist in W. Tushar and C. Yuen are with Singapore University of Technology and Design (SUTD), 8 Somapah Road, Singapore 487372 (Email: {wayes tushar, yuenchau}@sutd.edu.sg). B. Chai is with the State Grid Smart Grid Research Institute, Beijing, 102211, China (Email: [email protected]). S. Huang is with the Ministry of Home Affairs, Singapore (Email: [email protected]). D. B. Smith is with the National ICT Australia (NICTA), ACT 2601, Australia and adjunct with the Australian National University (Email: [email protected]). H. V. Poor is with the School of Engineering and Applied Science at Princeton University, Princeton, NJ, USA (Email: [email protected]). Z. Yang is with the State Key Laboratory of Industrial Control Technology at Zhejiang University, Hangzhou, China (Email: [email protected]). This work is supported in part by the Singapore University of Technology and Design (SUTD) through the Energy Innovation Research Program (EIRP) Singapore NRF2012EWT-EIRP002-045 and IDC grant IDG31500106, and in part by the U.S. National Science Foundation under Grant ECCS-1549881. D. B. Smith’s work is supported by NICTA, which is funded by the Australian Government through the Department of Communications and the Australian Research Council. residential demand response by altering the electricity demand due to the changes in the balance between supply and demand. Particularly, in a residential community setting, where each household is equipped with an ES, the use of ES devices can significantly leverage the efficient flows of energy within the community in terms of reducing cost, decarbonization of the electricity grid, and enabling effective demand response (DR). However, energy storage requires space. In particular for large consumers like shared facility controllers (SFCs) of large apartment buildings [11], the energy requirements are very high, which consequently necessitates the actual installment of very large energy storage capacity. The investment cost of such storage can be substantial whereas due to the random usage of the facilities (depending on the usage pattern of different residents) some of the storage may remain unused. Furthermore, the use of ESs for RUs is very limited for two reasons [10]: firstly, the installation cost of ES devices is very high and the costs are entirely borne by the users. Secondly, the ESs are mainly used to save electricity costs for the RUs rather than offer any support to the local energy authorities, which further makes their use economically unattractive. Hence, there is a need for solutions that will capture both the problems related to space and cost constraints of storage for SFCs and the benefit to RUs for supporting third parties. To this end, numerous recent studies have focused on energy management systems with ES devices as we will see in the next section. However, most of these studies overlook the potential benefits that local energy authorities such as SFCs can attain by jointly sharing the ES devices belonging to the RUs. Particularly due to recent cost reduction of small-scale ES devices, sharing of ES devices installed in the RUs by the SFCs has the potential to benefit both the SFCs and the RUs of the community as we will see later. In this context, we propose a scheme that enables joint ES ownership in smart grid. During the sharing, each RU leases the SFCs a fraction of its ES device to use, and charges and discharges from the rest of its ES capacity for its own purposes. On the contrary, each SFC exclusively uses its portion of ES devices leased from the RUs. This work is motivated by [10], in which the authors discussed the idea of joint ownership of ES devices between domestic customers and local network operators, and demonstrated the potential system-wide benefits that can be obtained through such sharing. However, no policy has been developed in [10] to determine how the fraction of battery capacity, which is shared by the network operators and the domestic users, is decided. Note that, as an owner of an ES device, each RU can decide IEEE Trans. Smart Grid, Dec. 2015 II. S TATE - OF -T HE A RT whether or not to take part in the joint ownership scheme with the SFCs and what fraction of the ES can be shared with the SFCs. Hence, there is a need for solutions that can capture this decision making process of the RUs by interacting with the SFCs of the network. In this context, we propose a joint ES ownership scheme in which by participating in storage sharing with the SFCs, both the RUs and SFCs benefit economically. Due to the interactive nature of the problem, we are motivated to use auction theory to study this problem [12]. Exploiting the two-way communications aspects, auction mechanisms can exchange information between users and electricity providers, meet users’ demands at a lower cost, and thus contribute to the economic and environmental benefits of smart grid1 [13]. In particular, 1) we modify the Vickrey auction technique [14] by integrating a Stackelberg game between the auctioneer and the RUs and show that the modified scheme leads to a desirable joint ES ownership solution for the RUs and the SFCs. To do this, we modify the auction price derived from the Vickrey auction, to benefit the owner of the ES, through the adaptation of the adopted game as well as keep the cost savings to the SFCs at the maximum; 2) We study the attributes of the technique, and show that the proposed auction scheme possesses both the incentive compatibility and the individual rationality properties leveraged by the unique equilibrium solution of the game; 3) We propose an algorithm for the Stackelberg game that can be executed distributedly by the RUs and the auctioneer, and the algorithm is shown to be guaranteed to reach the desired solution. We also discuss how the proposed scheme can be extended to the time varying case; and 4) Finally, we provide numerical examples to show the effectiveness of the proposed scheme. The importance and necessity of the proposed study with respect to actual operation of smart grid lies in assisting the SFCs of large apartment buildings in smart communities to reduce space requirements and investment costs of large energy storage units. Furthermore, by participating in storage sharing with the SFCs, the RUs can benefit economically, which can consequently influence them to efficiently schedule their appliances and thus reduce the excess use of electricity. We stress that multi-agent energy management schemes are not new in the smart grid paradigm and have been discussed in [11], [15] and [16]. However, the scheme discussed in the paper differs from these existing approaches in terms of the considered system model, chosen methodology and analysis, and the use of the set of rules to reach the desired solution. The remainder of the paper is organized as follows. We provide a comprehensive literature review of the related work in Section II followed by the considered system model in Section III. Our proposed modified auction-based mechanism is demonstrated in Section IV where we also discuss how the scheme can be adopted in a time varying environment. The numerical case studies are discussed in Section V, and finally we draw some concluding remarks in Section VI. In the recent years, there has been an extensive research effort to understand the potential of ES devices for residential energy management [17]. This is mainly due to their capabilities in reducing the intermittency of renewable energy generation [18] as well as lowering the cost of electricity [19]. The related studies can be divided into two general categories. The first category of studies consisting of [20], [21], which assume that the ESs are installed within each RU premises and are used solely by the owners in order to perform different energy management tasks such as optimal placement, sizing and control of charging and discharging of storage devices. The second type of studies deal with ES devices that are not installed within the RUs but located in a different location such as in electric vehicles (EVs). Here, the ESs of EVs are used to provide ancillary services for RUs [22]–[24] and local energy providers [25]–[27]. Furthermore, another important impact of ES devices on residential distribution grids is studied in [28] and [29]. In particular, these studies focus on how the use of ES devices can bring benefits for the stakeholders in external energy markets. In [28], the authors propose a multi-objective optimization method for siting and sizing of ESs of a distribution grid to capture the trade-offs between the storage stakeholders and the distribution system operators. Furthermore, in [29], optimal storage profiles for different stakeholders such as distribution grid operators and energy traders are derived based on case studies with real data. Studies of other aspects of smart grid can be found in [30]–[36]. As can be seen from the above discussion, the use of ES devices in smart grid is not only limited to address the intermittency of renewable generation [18] and assisting users to take part in energy management to reduce their cost of electricity [19], [21] but also extends to assisting the grid (or, other similar energy entities such as an SFC) [37] and generating revenues for stakeholders [28], [29]. However, one similarity between most of the above mentioned literature is that only one entity owns the ES and uses it according to its requirements. Nonetheless, this might not always be the case if there are large number of RUs2 in a community. In this regard, considering the potential benefits of ES sharing, as discussed in [10], this paper investigates the case in which the SFCs in a smart community are allowed to share some fraction of the ESs owned by the RUs through a third party such as an auctioneer or a community representative. The proposed modified auction scheme differs from the existing techniques for multi-agent energy management such as those in [11], [15], [16] in a number of ways. Particularly, in contrast to these studies, the proposed auction scheme captures the interaction between the SFCs and the RUs, whereby the decision on the auction price is determined via a Stackelberg game. By exploiting auction rules including the determination rule, payment rule, and allocation rule the interaction between the SFCs and RUs is greatly simplified. For instance, the determination rule can easily identify the number of RUs that are participating in the auction process, which further leverage 1 Please note that such a technique can be applied in the real distribution network such as in electric vehicle charging stations by using the two-way information and power flow infrastructure of smart grids [3]. 2 Each RU may participate as a single entity or as a group where RUs connected via an aggregator [38]. 2 IEEE Trans. Smart Grid, Dec. 2015 sicap either due to the fact that some SFCs do not have their own ESs [11] or that the ESs of the SFCs are not large enough to store all the excess energy at that time. It is important to note that the ES requirement of the SFCs can stem from any type of intermittent generation profile that the SFCs or RUs can adopt. For example, one can consider that the proposed scheme is based on a hybrid generation profile comprising both solar and wind generation. However, the proposed technique is equally suitable for other types of intermittent generation as well. We assume that there are N = |N | RUs, where N is the set of all RUs in the system, that are willing to share some parts of their ES with the SFCs of the network. The battery cap capacity of each RU i ∈ N is si , and each RU i wants to put xi fraction of its ES in the market to share with the SFCs, where
(1) xi ≤ bi = scap i − di . sicap : Total capacity of ES device of RU i. di : The amount that RU i does not sell. bi : The maximum ES space the RU i might sell to the SFCs. bi = sicap − di di Sharing price pt x1 ≤ b1 x2 ≤ b2 Each i decides ri ri < pt Yes No x N ≤ bN Leaves the et sharing market q1 q2 Yes Each m decides am qM Take ke part in ES sharing am > pt No Sharing price pt Here, bi is the maximum amount of battery space that the RU can share with the SFCs if the cost-benefit tradeoff for the sharing is attractive for it. di is the amount of ES that the RU does not want to share, rather uses for its own needs, e.g., to run the essential loads in the future if there is any electricity disruption within the RU or if the price of electricity is very high. Fig. 1: The fraction of the ES capacity that an RU i is willing to share with the SFCs of the community. the determination of the auction price via the Stackelberg game in the payment rule. Furthermore, on the one hand the work here complements the existing works focusing on the potential of ES for energy management in smart grid. On the other hand, the proposed work has the potential to open new research opportunities in terms of control of energy dispatch from ES, the size of ES, and exploring other interactive techniques such as cooperative games and bi-level optimization for ES sharing. To this end, to offer an ES space xi , on the one hand, each RU i decides on an reservation price ri per unit of energy. Hereinafter, we will use ES space and energy interchangeably to refer to the ES space that each RU might share with the SFCs. However, if the price pt , which each RU received for sharing its ES, is lower than ri , the RU i removes its ES space xi from the market as the expected benefit from the joint sharing of ES is not economically attractive for it. On the other hand, each SFC m ∈ M, that needs to share ES space with the RUs to store their energy, decides a reservation bid am , which represents the maximum unit price the SFC m is willing to pay for sharing per unit of ES with the RUs in the smart community, to enter into the sharing market. And, if am > pt , the SFC removes its commitment of joint ES ownership with RUs from the market due to the same reason as mentioned for the RU. A graphical representation of the concept of ES sharing and their decision making process of sharing the ES space of each RU i with the SFCs are shown in Fig. 1. Please note that to keep the formulation simple, we do not include any specific storage model in the scheme. However, by suitably modeling some related parameters such as the storage capacity scap i , and parameters like di and bi , the proposed scheme can be adopted for specific ES devices. III. S YSTEM M ODEL Let us consider a smart community that consists of a large number of RUs. Each RU can be an individual home, a single unit of a large apartment complex, or a large number of units connected via an aggregator that acts as a single entity [38]–[40]. Each RU is equipped with an ES device that the RU can use to store electricity from the main grid or its renewable energy sources, if there are any, or can perform DR management according to the real-time price offered by the grid. The ES device can be a storage device installed within each RU premises or can be the ES used for the RU’s electric vehicles. The entire community is considered to be divided into a number of blocks, where each block consists of a number of RUs and an SFC. Each SFC m ∈ M, where M is the set of all SFCs and M = |M|, is responsible for controlling the electrical equipment and machines such as lifts, parking lot lights and gates, water pumps, and lights in the corridor area of a particular block of the community, which are shared and used by the residents of that block on regular basis. Each SFC is assumed to have its own renewable energy generation and is also connected to the main electricity grid with appropriate communication protocols. Considering the fact that the nature of energy generation and consumption is highly sporadic [41], let us assume that the SFCs in the community need some extra ESs to store their electricity after meeting the demand of their respected shared facilities at a particular time of the day. This can be The interaction that arises from the choice of ES sharing price between the SFCs and RUs as well as the need of the SFCs to share the ES space to store their energy and the profits that the RUs can reap from allowing their ESs to be shared give rise to a market of ES sharing between the RUs and the SFCs in the smart grid. In this market, the involved N RUs and M SFCs will interact with each other to decide as to how many of them will take part in sharing the ESs between themselves, and also to agree on the ES sharing parameters such as the trading price pt and the amount of ES space to 3 IEEE Trans. Smart Grid, Dec. 2015 IV. AUCTION BASED ES OWNERSHIP Vickrey auction is a type of sealed-bid auction scheme, where the bidders submit their written bids to the auctioneer without knowing the bids of others participating in the auction [14]. The highest bidder wins the auction but pays the second highest bid price. Nevertheless, in this paper, we modify the classical Vickrey auction [14] to model the joint ES ownership scheme for a smart community consisting of multiple customers (i.e., the SFCs) and multiple owners of ES devices (i.e., the RUs). The modification is motivated by the following factors: 1) unlike the classical Vickrey auction, the modified scheme would enable the multiple owners and customers to decide simultaneously and independently whether to take part in the joint ES sharing through the determination rule of the proposed auction process, as we will see shortly; 2) the modification of the auction provides each participating RU i with flexibility of choosing the amount of ES space that they may want to share with the SFCs in cases when the auction price5 pt is lower than their expected reservation price ri ; and 3) finally, the proposed auction scheme provides solutions that satisfy both the incentive compatibility and individual rationality properties, as we will see later, which are desirable in any mechanism that adopts auction theory [41]. To this end, the proposed auction process, as shown in Fig. 2, consists of three elements: Fig. 2: Energy management in a smart community through auction process consisting of multiple RUs with ES devices, an auctioneer and a number of SFCs. be shared. In the considered model, the RUs not only decide on the reservation prices ri , but also on the amount of ES space xi that they are willing to share with the SFCs. The amount of xi is determined by the trade-off between between the economic benefits that the RU i expects to obtain from giving the SFCs the joint ownership of its ES device and the associated reluctance αi of the RU for such sharing. The reluctance to share ESs may arise from the RUs due to many factors. For instance, sharing would enable frequent charging and discharging of ESs that reduce the lifespan3 of an ES device [42]. Hence, an RU i may set its αi higher so as to increase its reluctance to participate in the ES sharing. However, if the RU is more interested in earning revenue rather than increasing ES life time, it can reduce its αi and thus get more net benefits from sharing its storage. Therefore, for a given set of bids am , ∀m and storage requirement qm , ∀m by the SFCs, the maximum amount of ES xi that each RU i will decide to put for sharing is strongly affected by the trading price pt and the reluctance parameter4 αi of each RU i ∈ N during the sharing process. In this context, we develop an auction based joint ES ownership scheme in the next section. We understand that the proposed scheme involves different types of users such as auctioneers, SFCs, and RUs. Therefore, the communication protocol used by them could be asynchronous. However, in our study we assume that the communication between different entities of the system are synchronous. This is mainly due to the fact that we assume our algorithm is executed once in a considered time slot, and the duration of this time slot can be one hour [43]. Therefore, synchronization is not a significant issue for the considered case and the communication complexity is affordable. For example, the auctioneer can wait for five minutes until it receives all the data from SFCs and the RUs and then the algorithm, which is proposed in Section IV-C, can be executed. 1) Owner: The RUs in set N , that own the ES devices, and expect to earn some economic benefits, e.g., through maximizing a utility function, by letting the SFCs to share some fraction of their ES spaces. 2) Customer: The SFCs in set M, that are in need of ESs in order to store some excess electricity at a particular time of the day. The SFCs offer the RUs a price with a view to jointly own some fraction of their ES devices. 3) Auctioneer: A third party (e.g., estate or building manager), that controls the auction process between the owners and the customers according to some predefined rules. The proposed auction policies consist of A) determination rule, B) payment rule and C) storage allocation rule. Here, determination rule allows the auctioneer to determine the maximum limit for the auction price pmax and the number t of SFCs and RUs that will actively take part in the ES sharing scheme once the auction process is initiated. The payment rule enables the auctioneer to decide on the price that the customer needs to pay to the owners for sharing their ES devices, which allows the RUs to decide how much storage space they will be putting into the market to share with the SFCs. Finally, the auctioneer allocates the ES spaces for sharing for each SFC following the allocation rule of the proposed auction. It is important to note that although both the customers and owners do not have any access to others private information such as the amount of ES to be shared by an RU or the required energy space by any SFC, the rules of auction are known to all the participants of the joint ownership process. 3 Please note that the life time degradation due to charging and discharging may not true for all electromechanical systems such as redox-flow system. 4 Reluctance parameter refers to the opposite of preference parameter [38]. 5 Hereinafter, p will be used to refer to auction price instead of sharing or t trading price. 4 IEEE Trans. Smart Grid, Dec. 2015 of the SFCs and RUs in the network: 90 Cosumers participating in auction Owner Customer am < ri ; ∀m ∈ M/{1, 2, . . . , K}, ∀i ∈ N /{1, 2, . . . , J}. (4) 80 Hence, the joint ownership of ES would be a detrimental choice for the RUs and the SFCs within the set N /{1, 2, . . . , J} and M/{1, 2, . . . , K} respectively, which consequently remove them from the proposed auction process. Now, one desirable property of any auction mechanism is that no participating agents in the auction mechanism will cheat once the payment and allocation rules are being established. To this end, we propose that, once J and K are determined, K − 1 SFCs and J − 1 RUs will be engaged in the joint ES sharing process, which is a necessary condition for matching total demand and supply while maintaining a truthful auction scheme [44]. Nevertheless, if truthful auction is not a necessity, SFC K and RU J can also be allowed to participate in the joint ES ownership auction. Price 70 60 Maximum auction price pmax t Vickrey price pmin t 50 40 Owners participating in auction 30 0 50 100 150 Storage amount Fig. 3: Determination of the Vickrey price, the maximum auction price, and the number of participating RUs and SFCs in the auction process. The proposed scheme initially determines the set of SFCs ⊂ M and RUs ⊂ N that will effectively take part in the auction mechanism once the upper bound of the auction price pmax is determined. Eventually, the payment and the allocation t rules are executed in the course of the auction plan. B. Payment Rule We note that the intersection of the demand and supply curves demonstrates the highest reservation price pmax for the t participating J − 1 RUs. According to the Vickrey auction mechanism [14], the auction price for sharing the ES devices would be the second highest reservation price, i.e., the Vickrey price, which will be indicated as pmin hereinafter. However, we t note that this second highest price might not be considerably beneficial for all the participating RUs in the auction scheme. In contrast, if pt is set to pt = pmax t , the price could be detrimental for some of the SFCs. Therefore, to make the auction scheme attractive and beneficial to all the participating RUs and, at the same time, to be cost effective for all the SFCs, we strike a balance between the pmax and pmin t t . To do so, we propose a scheme for deciding on both the auction price pt and the amount of ESs xi that RUs will put into the market for sharing according to pt . In particular, we propose a Stackelberg game between the auctioneer that decides on the auction price pt to maximize the average cost savings to the SFCs as well as satisfying their desirable needs of ESs, and the RUs, that decide on the vector of the amount of ES x = [x1 , x2 , . . . , xJ−1 ] that they would like to put into the market for sharing such that their benefits are maximized. Please note that the solution of the proposed problem formulation can also be solved following other distributed algorithms, e.g., algorithms designed via the bi-level optimization technique [45]. Stackelberg game: Stackelberg game is a multi-level decision making process, in which the leader of the game takes the first step to choose its strategy. The followers, on the other hand, choose their strategy in response to the decision made by the leader. In the proposed game, we assume the auctioneer as the leader and the RUs as the followers. Hence, it can be seen as a single-leader-multiple-follower Stackelberg game (SLMFSG). We propose that the auctioneer, as a leader of the SLMFSG Γ, will take the first step to choose a suitable min auction price pt from the range [pmin t , pt ]. Meanwhile, each RU i ∈ {1, 2, . . . , J − 1}, as a follower of the game, will play its best strategy by choosing a suitable xi ∈ [0, bi ] in response to the price pt offered by the auctioneer. The best A. Determination Rule The determination rule of the proposed scheme is executed by the following steps (inspired from [44]): i) The RUs of set N , i.e., the owners of the ESs, declare their reservation price ri , ∀i in an increasing order, which we can consider, without loss of generality, as: r1 < r2 < . . . < rN . (2) The RUs submit the reservation price along with the amount xi of ES that they are interested to share with the SFCs to the auctioneer. ii) The SFCs’ bidding prices, i.e., am , ∀m, are arranged in a decreasing order, i.e., a1 > a2 > . . . > aM . (3) The SFCs submit to the auctioneer along with the quantity qm ∀m of ES that they require. iii) Once the auctioneer receives the ordered information from the RUs and the SFCs, it generates the aggregated supply (reservation price of the RUs versus the amount of ES the RUs interested to share) and demand curves (reservation bids am verses the quantity of ES qm needed) using (2) and (3) respectively. iv) The auctioneer determines the number of of participating SFCs K and RUs J that satisfies aK ≥ rJ from the intersection of the two curves using any standard numerical method [44]. As soon as the SFC K ≤ M and RU J ≤ N are determined from the intersection point, as shown in Fig. 3, an important aspect of the auction mechanism is to determine the number of SFCs and RUs, which will take part in the joint ownership of ESs. We note that once the number of SFCs K and RUs J are determined, the following relationship holds for the rest 5 IEEE Trans. Smart Grid, Dec. 2015 response strategy of each RU i will stem from a utility function Ui , which captures the benefit that an RU i can gain from deciding on the amount of ES xi to be shared for the offered price. Whereas the auctioneer chooses the price pt with a view to maximize the average cost savings Z of the SFCs in the network. Now to capture the interaction between the auctioneer and the RUs, we formally define the SLMFSG Γ as Γ = {{{1, 2, . . . , J − 1}, {Auctioneer}}, {Ui }i∈{1,2,...,J−1} , {Xi }i∈{1,2,...,J−1} , Z, pt }, which consists of: i) the set of RUs {1, 2, . . . , J − 1} participating in the auction scheme and the auctioneer; ii) the utility Ui that each RU i reaps from choosing a suitable strategy xi in response to the price pt announce by the auctioneer; iii) the strategy set Xi of each RU i ∈ {1, 2, . . . , J − 1}; iv) the average cost savings Z that incurred to each SFC m ∈ {1, 2, . . . , K − 1} from the strategy  by the  chosen max of the auctioneer, and v) the strategy pt ∈ pmin t , pt auctioneer. In the proposed approach, each RU i iteratively responses to the strategy pt chosen by the auctioneer independent of other RUs in set {1, 2, . . . , J − 1}/{i}. The response of i is affected by the offered price pt , its reluctance parameter αi and the initial reservation price ri . However, we note that the auctioneer does not have any control over the decision making process of the RUs. It only sets the auction price pt with a view to maximize the cost savings Z, with respect to the cost with the initial bidding price, for the SFCs. To this end, the target of auctioneer is (5)assumed to maximize the average cost savings ! J−1 PK−1 X (a − p ) m t m=1 Z= xi (8) K −1 i=1 by choosing an appropriate price p to offer to each RU from PK−1 t min max m=1 (am −pt ) is the average savthe range [pt , pt ]. Here, K−1 ing in auction price that the SFCs pay to the RUs for sharing P the ESs and i xi is the total amount of ES that all the SFCs share from the RUs. From Z, we note that the cost savings will be more if pt is lower for all m ∈ {1, 2, . . . , K − 1}. However, this is conflicted by that fact that a lower pt may lead to the choice of lower xi ∀i ∈ {1, 2, . . . , J − 1} by the RUs, which in turn will affect the cost to the SFCs. Hence, to reach a desirable solution set (x∗ , p∗t ), the auctioneer and the RUs continue to interact with each other until the game reaches a Stackelberg equilibrium (SE). Now, the utility function Ui , which defines the benefits that an RU i can attain from sharing xi amount of its ES with the SFCs, is proposed to be Ui (xi ) = (pt − ri )xi − αi x2i , xi ≤ bi , (6) where, αi is the reluctant parameter of RU i, and ri is the reservation price set by RU i. Ui mainly consists of two parts. The first part (pi − ri )xi is the utility in terms of its revenue that an RU i obtains from sharing its xi portion of ES device. The second part αi x2i , on other hand, is the negative impact in terms of liability on the RU i stemming from sharing its ES with the SFC. This is mainly due to the fact that once an RU decides to share its xi amount of storage space with an SFC, the RU can only use scap i − xi amount of storage for its own use. The term αi x2i captures this restriction of the RU on the usage of its own ES. In (6), the reluctance parameter αi is introduced as a design parameter to measure the degree of unwillingness of an RU to take part in energy sharing. In particular, a higher value of αi refers to the case when an RU i is more reluctant to take part in the ES sharing, and thus, as can be seen from (6), even with the same ES sharing attains a lower net benefit. Thus, Ui can be seen as a net benefit to RU i for sharing its ES. The utility function is based on the assumption of a non-decreasing marginal utility, which is suitable for modeling the benefits of power consumers, as explained in [46]. In addition, the proposed utility function also possesses the following properties: i) the utility of any RU increases as the amount of price pt paid to it for sharing per unit of ES increases; ii) as the reluctance parameter αi increases, the RU i becomes more reluctant to share its ES, and consequently the utility decreases; and iii) for a particular price pt , the more an RU shares with the SFCs, the less interested it becomes to share more for the joint ownership. To that end, for a particular price pt and reluctance parameter αi , the objective of RU i is   (7) max (pt − ri )xi − αi x2i , xi ≤ bi . Definition 1. Let us consider the game Γ as described in (5), where the utility of each RU i and the average utility per SFC are described via Ui and Z respectively. Now, Γ will reach a SE (x∗ , p∗t ), if and only if the solution of the game satisfies the following set of conditions: Ui (x∗i , x∗−i , p∗t ) ≥ Ui (xi , x∗−i , p∗t ), ∀i ∈ {1, 2, . . . , J − 1}, max ∀xi ∈ Xi , p∗t ∈ [pmin t , pt ], (9) and PK−1 − p∗t ) X ∗ xi ≥ K −1 i m=1 (am PK−1 − pt ) X ∗ xi , K −1 i m=1 (am (10) where x−i = [x1 , x2 , . . . , xi−1 , xi+1 , . . . , xJ−1 ]. Hence, according to (9) and (10), both the RUs and the SFCs achieve their best possible outcomes at the SE. Hence, neither the RUs nor the auctioneer will have any incentive to change their strategies as soon as the game Γ reaches the SE. However, achieving an equilibrium solution in pure strategies is not always guaranteed in non-cooperative games [38]. Therefore, we need to investigate whether the proposed Γ possesses an SE or not. Theorem 1. There always exists a unique SE solution for the proposed SLMFSG Γ between the auctioneer and the participating RUs in set {1, 2, . . . , J − 1}. Proof: Firstly, we note that the strategy set of the auction max . eer is non-empty and continuous within the range pmin t , pt Hence, there will always be a non-empty strategy for the auctioneer that will enable the RUs to offer some part of their xi 6 IEEE Trans. Smart Grid, Dec. 2015 ES, within their limits, to the SFCs. Secondly, for any price pt , the utility function Ui in (6) is strictly concave with respect 2 of xi , ∀i ∈ {1, 2, . . . , J − 1}, i.e., δδxU2i < 0. Hence, for any i  min max  price pt ∈ pt , pt , each RU will have a unique xi , which will be chosen from a bounded range [0, bi ] and maximize Ui . Therefore, it is evident that as soon as the scheme will find a unique p∗t such that the average utility Z per SFC attains a maximum value, the SLMFSG Γ will consequently reach its unique SE. To this end, first we note that the amount of ES x∗i , at which the RU i achieves its maximum utility in response to a price pt can be obtained from (6), pt − ri x∗i = . (11) 2αi Algorithm 1: Algorithm for SLMFSG to reach the SE ∗ 1: Initialization: p∗t = pmin t , Z = 0. 2: for Auction price pt from pmin to pmax do t t 3: for Each RU i ∈ {1, 2, . . . , (J − 1)} do 4: RU i adjusts its amount of ES xi to share according to x∗i = arg 5: 6: 7: 8: Now, replacing the value of x∗i in (8) and doing some simple arithmetics, the auction price p∗t , which maximizes the average cost savings to the SFCs can be found as P  P  P J−1 1 K−1 J−1 a + i=1 ri (K−1) m i=1 2αi m=1 2αi ∗ pt = , (12) PJ−1 K−1 i=1 max 0≤xi ≤bi [(pt − ri )xi − αi x2i ]. (13) end for The auctioneer computes the average cost savings to SFCs ! J −1 PK−1 X ∗ m=1 (am − pt ) xi . (14) Z= K−1 i=1 if Z ≥ Z ∗ then The auctioneer record the desirable price and maximum average cost savings p∗t = pt , Z ∗ = Z. (15) 9: end if 10: end for The SE (x∗ , p∗t ) is achieved. αi guaranteed to reach SE of the proposed SLMFSG Γ. where am for any m ∈ {1, 2, . . . , K − 1} and αi for any i ∈ {1, 2, . . . , J − 1} is exclusive. Therefore, p∗t is unique for Γ, and thus Theorem 1 is proved. Proof: In the proposed algorithm, we note that the choice of strategies by the RUs emanate from the choice pt of the auctioneer, which as shown in (12) will always attain a nonempty single value p∗t at the SE due to its bounded strategy set max [pmin t , pt ]. On the other hand, as the Algorithm 1 is designed, in response to the p∗t , each RU i will choose its strategy xi from the bounded range [0, bi ] in order to maximize its utility function Ui . To that end, due to the bounded strategy set and continuity of Ui with respect to xi , it is confirmed that each RU i will always reach a fixed point x∗i for the given p∗t . Therefore, the proposed Algorithm 1 is always guaranteed to reach the unique SE of the SLMFSG. C. Algorithm for payment To attain the SE, the auctioneer, which has the information of am , m = {1, 2, 3, . . . , K − 1}, needs to communicate with each RU. It is considered that the auctioneer does not have any knowledge of the private information of the RUs such as αi , ∀i. In this regard, in order to decide on a suitable auction price pt that will be beneficial for both the RUs and the SFCs, the auctioneer and the RUs interact with one another. To capture this interaction, we design an iterative algorithm, which can be implemented by the auctioneer and the RUs in a distributed fashion to reach the unique SE of the proposed SLMFSG. The algorithm initiates with the auctioneer who sets the auction price pt to pmin and the optimal average t cost saving per SFC Z ∗ to 0. Now, in each iteration, after having the information on the offered auction price by the auctioneer, each RU i plays its best response xi ≤ bi and submits its choice to the auctioneer. The auctioneer, on other hand, receives the information on x = [x1 , x2 , . . . , xJ−1 ] from all the participating RUs and determines the average cost savings per SFC Z from its knowledge on the reservation bids [a1 , a2 , . . . , aK−1 ] and using (8). Then, the auctioneer compares the Z with Z ∗ . If Z > Z ∗ , the auctioneer updates the optimal auction price to the one recently offered and sends a new choice of price to the RUs in the next iteration. However, if Z ≤ Z ∗ , the auctioneer keeps the same price and offers another new price to the RUs in the next iteration. The iteration process continues until the conditions in (9) and (10) are satisfied, and hence the SLMFSG reaches the SE. We show the step-by-step process of the proposed algorithm in Algorithm 1. D. Allocation Rule Now, once the the amount of ES x∗i that each RU i ∈ {1, 2, . . . , J − 1} decides to put into the market for sharing in response to the auction price p∗t is determined, the auctioneer allocates the quantity Qi to be jointly shared by each RU i and the SFCs according to following rule [44]: ( PK−1 PJ−1 x∗i if i=1 x∗i ≤ m=1 qm , Qi (x) = (16) P P J−1 ∗ (x∗i − ηi )+ if i=1 xi > K−1 m=1 qm , where (f )+P = max(0, Pf ) and ηi is the allotment of the J−1 ∗ excess ES i=1 xi − K−1 m=1 qm that an RU i must endure. Essentially, the rule in (16) emphasizes that if the requirements of the SFCs exceed the available ES space from the RUs, each RU i will allow the SFCs to share all of the ES xi that it put into the market. However, if the available ES exceeds the total demand by the SFCs, then RU P i will have to share a Peach J−1 K−1 fraction of the oversupply i=1 x∗i − m=1 qm . Nonethless, this burden, if there is any, can be distributed in different ways among the participating RUs. For instance, the burden can be distributed either proportionally to the amount of ES x∗i that each RU i shared with the SFCs or proportionally to Theorem 2. The algorithm proposed in Algorithm 1 is always 7 IEEE Trans. Smart Grid, Dec. 2015 the reservation price6 ri of each RU. Alternatively, the total burden can also be shared equally by the RUs in the auction scheme [44]. 1) Proportional allocation: In proportional allocation [47], a fraction of the total burden ηi is allocated to each RU i in to the reservation price ri (or, x∗i ) such that PJ−1 PK−1 P proportion ∗ i ηi = i=1 xi − m=1 qm , which can be implemented as follows: P PK−1  J−1 ∗ x − i=1 i m=1 qm ri P ηi = , i = [1, 2, . . . , J − 1]. (17) i ri it is clear that all the participants in the proposed auction scheme are individually rational, which leads to the following Corollary 1. Corollary 1. The proposed auction technique possesses the individual rationality property, in which the J − 1 rational owners and K − 1 rational customers actively participate in the mechanism to gain the higher utility. Theorem 3. The proposed auction mechanism is incentive compatible, i.e., truthful auction is the best strategy for any RU i ∈ {1, 2, . . . , J − 1} and SFC m ∈ {1, 2, . . . , K − 1}. By replacing ri with x∗i in (17), the burden allocation can be determined in proportion to the shared ES by each RU. 2) Equal allocation: According to equal allocation [44], each RU bears an equal burden ! K−1 J−1 X X 1 ∗ ηi = x − qm , i = [1, 2, . . . , J − 1] (18) J − 1 i=1 i m=1 Proof: To validate Theorem 3, first we note that the choice of strategies by the RUs always guaranteed to converge to a unique SE, i.e., x∗ = [x∗1 , x∗2 , . . . , x∗J−1 ] as proven in Theorem 1 and Theorem 2, which confirms the stability of their selections. Now, according to [44], once the owners of an auction process, i.e., the RUs in this proposed case, decide on a stable amount of commodity, i.e., x∗i ∀i ∈ {1, 2, . . . , J − 1}, to supply to or to share with the customers, the auction process always converges to a strategy-proof auction if the allocation of commodity is conducted according to the rules described in (16) and (18). Therefore, neither any RU nor any SFC will have any intention to falsify their allocation once they adopt (16) and (18) [44] for sharing the storage space of the RUs from their SE amount. Therefore, the auction process is incentive compatible, and thus Theorem 3 is proved. of the oversupply. Here it is important to note that, although proportional allocation allows the distribution of oversupply according to some properties of the RUs, equal allocation is more suitable to make the auction scheme strategy proof [44]. Strategy proofness is important for designing auction mechanisms as it encourages the participating players not to lie about their private information such as reservation price [41], which is essential for the acceptability and sustainability of such mechanisms in energy markets. Therefore, we will use equal allocation of (18) for the rest of the paper. F. Adaptation to Time-Varying Case To extend the proposed scheme to a time-varying case, we assume that the ES sharing scheme works in a time-slotted fashion where each time slot has a suitable time duration based on the type of application, e.g., 1 hour [43]. It is considered that in each time slot all the RUs and SFCs take part in the proposed ES sharing scheme to decide on the parameters such as the auction price and the amount of ESs that needs to be shared. However, in a time-varying case, the amount of ES that an RU shares at time slot t may be affected by the burden that the RU needed to bear in the previous time slot t − 1. To this end, first we note that once the number of participating RUs and SFCs is decided for a particular time slot via the determination rule, the rest of the procedures, i.e., the payment and allocation rules are executed following the descriptions in Section IV-B and IV-D respectively for the respective time slot. Now, if the total number of RUs and SFCs is fixed, the RUs and SFCs that participate in the modified auction scheme in any time slot is determined by their respective reservation and bidding prices for that time slot. Further, the proposed auction process may evolve across different time slots based on the change of the amount of ES that each participating RU i may want to share and the change in the total amount of ES required for the SFCs in different time slots. Now, before discussing how the proposed modified auction scheme can be extended to a time-varying environment7, first we define the E. Properties of the Auction Process We note that once the auction process is executed, there is always a possibility that the owners of the ES might cheat on the amount of storage that they wanted put into the market during auction [13]. In this context, we need to investigate whether the proposed scheme is beneficial enough, i.e., individually rational, for the RUs such that they are not motivated to cheat, i.e., incentive compatible, once the auction is executed. Now for the individual rationality property, first we note that all the players, i.e., the RUs and the auctioneer on behalf of the SFCs, take part in the SLMFSG to maximize their benefits in terms of their respected utility from their choice of strategies. The choice of the RUs is to determine vector of ES x∗ such that each of the RU can be benefitted at its maximum. On the other hand, the strategy of the auctioneer is to choose a price pt to maximize the savings of the SFCs. Accordingly, once both the RUs and the auctioneer reach such a point of the game when neither the owners nor the customers can be benefitted more from choosing another strategy, the SLMFSG reaches the SE. To this end, it is already proven in Theorem 1 that the proposed Γ in this auction process must possesses a unique SE. Therefore, as a subsequent outcome of the Theorem 1, 7 Certain loads such as lifts and water pumps in large apartment buildings are not easy to schedule as they are shared by different users of the buildings. Hence, we focus on the time variation of the storage sharing process by the RUs of the considered system. 6 Please note that the reservation price ri indicates how much each RU i wants to be paid forPsharing its ES with the SFCs, and thus affects the determination of total x∗i and the total burden. 8 IEEE Trans. Smart Grid, Dec. 2015 which households are equipped with a dedicated battery to sell the stored electricity to the grid [48]. Nonetheless, xi,t is also affected by the amount of burden ηi,t−1 that an RU needed to bear due to an oversupply of ES spaces, if there was any, in the previous time slot. To this end, the amount of ES space that an RU i can offer to the SFCs at t can be defined as ( xi,t−1 if i ∈ / Jt−1 xi,t = . (21) max(bi,t − (xi,t−1 − ηi,t−1 ), 0) otherwise following parameters: t: index of time slot. T : total number of time slot. ri,t : the reservation price of RU i ∈ N at time slot t. ri = [ri,1 , ri,2 , . . . , ri,T ]: is the reservation price vector for RU i ∈ N . xi,t : the fraction of ES space that the RU i wants to shares with the SFCs at time slot t. xi = [xi,1 , xi,2 , . . . , xi,T ]: the vector of ES space shared by RU i with the SFC during the total considered times. bi,t : maximum available ES of RU i for sharing at time slot t. am,t : the bidding price of each SFC m ∈ M at time slot t. am = [am,1 , am,2 , . . . , am,T ]: is the reservation price vector for SFC m ∈ N . qm,t : the required ES space by each SFC m at time slot t. pt,t : the auction price at time slot t. Ui,t : the benefit that each RU i achieves at time slot t. Zt : the average cost saving per SFC at time slot t. ηi,t : the burden that is shared by each participating RU at time slot t. Kt : number of participating SFCs in the modified auction scheme at time slot t. Jt : number of participating RUs in the modified auction scheme at time slot t. To this end, the utility function Ui,t of each RU i and the average cost savings Zt per SFC at time slot t can be defined as Ui,t (xi,t ) = (pt,t − ri,t )xi,t − αi x2i,t , The SFC m, on the other hand, decides on the amount of ES qm,t that it needs to share from the RUs at t based on the random requirement of the shared facilities at t, the available shared ES space qm,t−1 from time slot t − 1, and the random generation of renewable energy sources, where appropriate. Hence, qm,t = f (qm,t−1 , renewables, facility requirement). Now, if we assume that the fraction of shared ES available from previous time slot is negligible, i.e., qm,t−1 ≈ 0, the requirement qm,t can be assumed to be random for each time slot t considering the random nature of both renewable generation and energy requirement of shared facilities. Note that this assumption is particularly valid if the SFC uses all its shared ESs from the previous time slot for meeting the demand of the shared facilities and cannot use them in considered time slot. Nonetheless, please note that this assumption does not imply that the inter-temporal relationship between the auction process across different time slots is non-existent. The auction process in one time slot still depends on other time slots due to the inter-temporal dependency of xi,t via (21). To this end, for the modeled xi,t ∀i ∈ N and qm,t ∀m ∈ M, the proposed modified auction scheme studied in Section IV can be adopted in each time slot t = 1, 2, . . . , T with a view to maximize (19) and (20) ∀t. It is important to note that the reservation price vector ri of each RU i ∈ N and the bidding price vector am of each SFC m ∈ M can be modeled through any existing time-varying pricing schemes such as time-of-use price [3]. Now, p∗t = [p∗t,1 , p∗t,2 , . . . , p∗t,T ] and x∗ = [x∗1 , x∗2 , . . . , x∗N ] constitute the solutions of the proposed modified auction scheme in a time-varying condition, if the x∗ comprises the solution vector of all ES spaces shared by the participating RUs in each time slot t = 1, 2, . . . , T for the auction price vector p∗t . Further, all the auction rules adopted in each time slot of the proposed time-varying case will be similar to the rules discussed in Section IV. Hence, the solution of the proposed modified auction scheme for a timevarying environment also possesses the incentive compatibility and individual rationality properties for each time slot. (19) and Zt = PKt −1 (am,t − pt,t ) Kt − 1 m=1 ! J−1 X xi,t (22) (20) i=1 respectively8. Now, at time slot t, the determination rule of the proposed scheme determines the number of participating RUs and SFCs based on their reservation and bidding prices for that time slot. The number of participation is also motivated by the available ES space of each RU and the requirement of each SFC. However, unlike the static case, in a time-varying environment the offered ES space by an RU at time slot t is influenced by its contribution to the auction process in the previous time slot. For instance, if an RU i receives a burden ηi,t−1 in time slot t − 1, its willingness to share ES space xi,t at time slot t may reduce. xi,t is also affected by the maximum amount of ES bi,t available to RU i at t. For simplicity, we assume that bi,t and αi,t do not change over different time slots. Therefore, an RU i can offer to share the same amount of ES space xi,t to the SFCs at time slot t if it did not share any amount in time slot t − 1. An analogous example of such arrangement can be found in FIT scheme with ES device in V. C ASE S TUDY For numerical case studies, we consider a number of RUs at different blocks in a smart community that are interested in allowing the SFCs of the community to jointly share their ES devices. We stress that when there are a large number of RU and SFCs in the system, the reservation and bidding prices will vary significantly from one another. Therefore, it will be difficult to find an intersection point to determine the highest 8 Please note that in each time slot t, (19) and (20) are related with each other in a similar manner as (7) and (8) are related for the static case. However, unlike the static case, the execution of the auction process in each time slot t is affected by the value of parameters such as xi,t and pt for that particular time slot. 9 ES shared by each RU IEEE Trans. Smart Grid, Dec. 2015 TABLE I: Change of average utility achieved by each SFC and each RU in the network (according to Algorithm 1) due to the change of the reluctance of each RU for sharing one kWh ES with the SFC. RU5 300 RU3 RU2 RU4 Reluctance Parameter ! 0.001 0.01 0.1 1 200 RU1 100 0 0 5 10 RU6, RU7, RU8 15 20 Number of iteration 25 30 Average utility per RU (Net benefit) 3450.6 2883.6 (-16.43%) 1117.3 (-67.6%) 142.37 (-95.3%) Average utility for SFC (Average cost savings) 7500 4578.3 (-38.9%) 1671.7 (-77.7%) 259.03 (-96%) 4 Average utility for SFC 8 x 10 6 RU to put into the market for sharing. As can be seen from the figure, on the one hand, RU 1, RU 2, and RU 3 reach the SE much quicker than RU 4 and RU 5. On the other hand, no interest for sharing any ES is observed for RU 4, 5 and 6. This is due to the fact that as the interaction between the auctioneer and the RUs continues, the auction price pt is updated in each iteration. In this regard, once the auction price for any RU becomes larger than its reservation price, it put all its reserve ES to the market with an intention to be shared by the SFCs. Due to this reason, RU 1, RU 2, and RU 3 put their ESs in the market much sooner, i.e., after the 2nd iteration, than RU 4 and RU 5 with higher reservation prices, whose interest for sharing ES reaches the SE once the auction price is encouraging enough for them to share their ESs after the 5th and 20th iterations. Unfortunately, the utilities of RU 6, 7, and 8 are not convenient enough to take part in the auction process, and therefore their shared ES fractions are 0. We note that the demonstration of the convergence of the SLMFSG to a unique SE subsequently demonstrates the proofs of Theorem 1, Theorem 2, Theorem 3 and Corollary 1, which are strongly related to the SE as explained in the previous section. Now, we would like to investigate how the reluctance parameters of the RUs may affect their average utility from Algorithm 1, and thus affecting their decisions to share ES. To this end, we first determine the average utility that is experienced by each RU and SFC for a reluctance parameter of αi = 0.001 ∀i. Then considering the outcome as a benchmark, we show the effect of different reluctance parameters on the achieved average benefits of each SFC and RU in Table I. The demonstration of this property is necessary in order to better understand the working principle of the designed technique for ES sharing. According to Table I, as the reluctance of each RU increases, it becomes more uncomfortable, i.e., lower utility, to put its ES in the market to be jointly owned by the SFCs. As a consequence, it also affects the average utility achieved by each SFC. As shown in Table I, the reduction in average utilities per RU are 16.73%, 67.6% and 95.3% respectively compared to the average utility achieved by an RU at αi = 0.001 for every ten times reduction in the reluctance parameter. For similar settings, the reduction of average utility for the SFCs are 38.9%, 77.7% and 96% at αi = 0.01, 0.1 and 1 respectively. Therefore, the proposed scheme will enable the RUs to put more storage in the auction market if the related reluctance for this sharing is small. Note that although the current investment cost of batteries is very high compared to their relative short life times, it is expected that battery costs will go down in the near future [10] and become very popular for addressing * Z 4 2 0 0 5 10 15 20 Number of iteration 25 30 Fig. 4: Convergence of Algorithm 1 to the SE. At SE, the average utility per SFC reaches its maximum and the ES that each RU wants to put into the market for share reaches a steady state level that maximize their benefits. reservation price pmax according to the determination rule. So, t in this paper, we limit ourself to around 6 − 10 RUs. However, having 6-10 RUs can in fact cover a large community, e.g., through aggregation such as discussed in [38], [39]. Here, each RU is assumed to be a group of [5, 25] households, where each household is equipped with a battery of capacity 25 kilo-Watt hour (kWh) [49]. The reluctance parameter of all RUs are assumed to be similar, which is taken from range of [0, 0.1]. It is important to note that αi is considered as a design parameter in the proposed scheme, which we used to map the reluctance of each RU to share its ES with the SFCs. Such reluctance of sharing can be affected by parameters like ES capacity, the condition of the environment (if applicable) and the RU’s own requirement. Now, considering the different system parameters in our proposed scheme, we capture these two extremes with 0 (not reluctant) and 0.1 (highly reluctant). The required electricity storage for each SFC is assumed to be within the range of [100, 500] kWh. Nevertheless, the required ES for sharing could be different if the usage pattern by the users changes. Since, the type of ESs (and their associated cost) used by different RUs can vary significantly [50], the choices of reservation price to share their ESs with the SFCs can vary considerably as well. In this context, we consider that the reservation price set by each RU and SFC is taken from a range of [20, 70]. It is important to note that all chosen parameter values are particular to this study only, and may vary according the availability and number of RUs, requirements of SFCs, trading policy, time of the day/year and the country. Now, we first show the convergence of Algorithm 1 to the SE of the SLMFG in Fig. 4. For this case study, we assume that there are five SFCs in the smart grid community that are taking part in an auction process with eight RUs. From Fig. 4, first we note that the proposed SLMFG reaches the SE after 20 interations when the average cost savings per SFC reaches its maximum. Hence, the convergence speed, which is just few seconds, is reasonable. Nonetheless, an interesting property can be observed when we examine the choice of ES by each 10 IEEE Trans. Smart Grid, Dec. 2015 1100 for the SFCs, it would put a higher burden on the RUs to carry. As a consequence, the relative utility from auction is lower. Nevertheless, if the requirement of the SFCs is higher, the sharing brings significant benefits to the RUs as can be seen from Fig. 5. On the other hand, for higher reluctance, RUs tend to share a lower ES amount, which then enables them to endure a lower burden in case of lower demands from the SFCs. This consequently enhances their achieved utility. Nonetheless, if the requirement is higher from the SFCs, their utility reduces subsequently compared to the RUs with lower reluctance parameters. Thus, from observing the effects of different αi ’s on the average utility per RU in Fig. 5, we understand that, if the total required ES is smaller, RUs with higher reluctance benefit more and vice versa. This illustrates the fact that even RUs with high unwillingness to share their ESs can be beneficial for SFCs of the system if their required ESs are small. However, for a higher requirement, SFCs would benefit more from having RUs with lower reluctances as they will be interested in sharing more to achieve higher average utilities. Now, we discuss the computational complexity of the proposed scheme, which is greatly reduced by the determination rule of the modified auction scheme as this rule determines the actual number of participating RUs and SFCs in the auction. We also note that after determining the number of participating SFCs and RUs, the auctioneer iteratively interacts with each of the RUs and sets the auction price with a view to increase the average savings for the SFC. Therefore, the main computational complexity of the modified auction scheme stems from the interactions between the auctioneer and the participating RUs to decide on the auction price. In this context, the computational complexity of the problem falls within a category of that of a single leader multiple follower Stackelberg game, whose computational complexity, which can be approximated to increase linearly with the number of followers [38], and is shown to be reasonable in numerous studies such as in [11] and [38]. Hence, the computational complexity is feasible for adopting the proposed scheme. Having an insight into the properties of the proposed auction scheme, we now demonstrate how the technique can benefit the RUs of the smart network compared to existing ES allocation schemes such as equal distribution (ED) [38] and FIT schemes [48]. ED is essentially an allocation scheme that allows the SFCs to meet their total storage requirements by sharing the total requirement equally from each of the participating RUs. We assume that if the shared ES amount exceeds the total amount of reservation storage that an RU puts into the market, the RU will share its full reservation amount. In FIT, which is a popular scheme for energy trading between consumers and the grid, we assume that each RU prefers to sell the same storage amount of energy to the grid at an FIT price rate, e.g., 22 cents/kWh [52] instead of sharing the same fraction of storage with the SFC. To this end, the resulting average utilities that each RU can achieve from sharing its ES space with the SFCs by adopting the proposed, ED, and FIT schemes are shown in Table II. From Table II, first we note that as the amount of required ES by the SFCs increases the average utility achieved per α = 0.001 (more willing to share) Average utility achieved by the RUs 1000 α = 0.01 (less willing to share) 900 800 700 600 500 400 300 η0.001 > 0 η η0.01 > 0 η Supply > Demand 200 100 150 200 0.001 0.01 η >0 0.001 η =0 0.01 =0 =0 Supply < Demand 250 300 350 400 450 500 Required battery space by the SFCs (kWh) 550 600 Fig. 5: Effect of change of required ES amount by the SFCs on the achieved average utility per RU. intermittency of renewables [51]. We have foreseen such a near future when our proposed scheme will be applicable to gain the benefit of storage sharing and thus motivate the RUs to keep their αi ∀i small. According to the observation from Table I, it can further be said that if the reluctance parameters of RUs change over either different days or different time slots, the performance of the system in terms of average utility per RU and average cost savings per SFC will change accordingly for the given system parameters. Once all the participating RUs put their ES amount into the auction market, they are distributed according to the allocation rule described in (16) and (18). In this regard, we investigate how the average utility of each RU is altered as the total storage amount required by the SFCs changes from in the network. For this particular case, the considered total ES requirement of the SFCs is assumed to be 100, 150, 200, 250, 300, 350, 400, 450, 500, 550 and 600. In general, as shown in Fig. 5, the average utility of each RU initially increases with the increase required by the SFCs and eventually becomes saturated to a stable value. This is due to the fact that as the required amount of ES increases, the RU can share more of its reserved ES that it put into the market with the SFCs with the determined auction price from the SLMFSG. Hence, its utility increases. However, each RU has a particular fixed ES amount that it puts into the market to share. Consequently, once the shared ES amount reaches its maximum, even with the increase of requirement by the SFCs the RU cannot share more, i.e., ηi = 0. Therefore, its utility becomes stable without any further increment. Interestingly, the proposed scheme, as can be seen in Fig. 5, favors the RUs with higher reluctance more when the ES requirement by the SFCs is relatively lower and favors the RUs with lower reluctance during higher demands. This is due to the way we have designed the proposed allocation scheme, which is dictated by the burden in (18) and the allocation of ES through (16). We note that, according to (11), if αi is lower, the RU i will put a higher amount of ES in the market to share. However, if the total required amount of ES is lower 11 IEEE Trans. Smart Grid, Dec. 2015 TABLE II: Comparison of the change of average utility per RU in the smart grid system as the required total amount of energy storage required by the SFCs varies. Required ES space by the SFCs Average utility (net benefit) of RU for equal distribution (ED) scheme Average utility (net benefit) of RU for FIT scheme Average utility (net benefit) of RU for proposed scheme Percentage improvement (%) compared to ED scheme Percentage improvement (%) compared to FIT scheme ES shared by each RU (kWh) ES available to share (kWh) RU also increases for all the cases. The reason for this increment is explained in Fig. 5. Also, in all the studied cases, the proposed scheme shows a considerable performance improvement compared to the ED and FIT schemes. An interesting trend of performance improvement can be observed if we compare the performance of the proposed scheme with the ED and FIT performances for each of the ES requirements. In particular, the performance of the proposed scheme is higher as the requirement of the ES increases from 200 to 350. However, the improvement is relatively less significant as the ES requirement switches from 400 to 450. This change in performance can be explained as follows: In the proposed scheme, as we have seen in Fig. 5, the amount of ES shared by each participating RU is influenced by their reluctance parameters. Hence, even the demand of the SFCs could be larger, the RUs may choose not to share more of their ES spaces once their reluctance is limited. In this regard, the RUs in the current case study increase their share of ES as the requirement by the SFCs increases, which in turn produces higher revenue for the RUs. Furthermore, once the RUs choice of ESs reach the saturation, the increase in demand, i.e., from 200 to 350 in this case, does not affect their share. As a consequence, their performance improvement is not as noticeable as the previous four cases. Nonetheless, for all the considered cases, the auction process performs superior to the ED scheme with an average performance improvement of 34.76%, which clearly shows the value of the proposed methodology to adopt joint ES sharing in smart grid. The performance improvement with respect to the FIT scheme, which is 34.34% on average, is due to the difference between the determined auction price and the price per unit of energy for the FIT scheme. Finally, we show how the decision making process of each RU in the system is affected by its decision in the previous time slot and the total storage requirement by the SFCs. The total number of time slots that are considered to show this performance analysis is four. In this context, we assume that there are five RUs in the system with ES of 100, 200, 300, 200 and 200 kWh respectively to share with the SFCs. The total ES requirements of the SFCs for considered four time slot are 500, 250, 500, and 100. Please note that these numbers are considered for this case study only and may have different values for different scenarios. Now, in Fig. 6, we show the available ES to each of the RUs at the begining of each time slot and how much they are going to share if the modified auction scheme is adopted in each time slot. For a simple analysis, we assume that once an RU shares its total available ES, it cannot share its ES for the remaining of the time slots. 200 536.52 537.83 629.82 17.4 17.1 250 581.85 583.16 789.82 35.74 35.43 300 624.52 626.83 944.26 51.19 50.63 350 669.85 673.16 960.09 43.32 42.61 400 715.19 717.50 960.09 34.24 33.81 450 757.85 759.16 960.09 26.68 26.46 300 RU1 RU2 RU3 RU4 RU5 200 100 0 1 2 3 Number of time slot 4 300 RU1 RU2 RU3 RU4 RU5 200 100 0 1 2 3 Number of time slot 4 Fig. 6: Demonstration of how the proposed modified auction scheme can be extended to time varying system. The reservation ES amount varies by the RUs varies between different time slots based on their sharing amount in the previous time slot. The total required storage by the SFCs is chosen randomly due to the reasons explained in Section IV-F. The reservation prices are considered to change from one time to the next based on a predefined time of use price scheme. Now, as can be seen from Fig. 6, in time slot 1, RU1 and RU2 share all their available ESs with the SFC, whereby other RUs do not share their ESs due to the reasons explained in Fig. 4. Since, the total requirement is 500, therefore neither of RU1 and RU2 needs to carry any burden. In time slot 3, only RU3 shares its ESs of 300 to meet the requirement. As the SFC’s requirement is lower than the supply, RU3 needs to carry a burden of 50 kWh. Similarly, in time slot 3 and 4, all of RU3, RU4 and RU5 take part in the energy auction scheme as they have enough ES to share with the SFC. However, the ES to share in time slot 4 stems from the burden of oversupply from time slot 3. The scheme is not shown for more than time slot 4 as the available ES from all RUs is already shared by the SFCs by the end of time slot 4. Thus, the proposed modified auction scheme can successfully capture the time variation if the scheme is modified as given in Section IV-F. VI. C ONCLUSION In this paper, we have modeled a modified auction based joint energy storage ownership scheme between a number of residential units (RUs) and shared facility controllers (SFCs) in smart grid. We have designed a system and discussed the determination, payment and allocation rule of the auction, where the payment rule of this scheme is facilitated by a Single-leader-multiple-follower Stackelberg game (SLMFSG) 12 IEEE Trans. Smart Grid, Dec. 2015 between the auctioneer and the RUs. The properties of the auction scheme and the SLMFSG have been studied, and it has been shown that the proposed auction possesses the individual rationality and the incentive compatibility properties leveraged by the unique Stackeberg equilibrium of the SLMFSG. We have proposed an algorithm for the SLMFSG, which has been shown to be guaranteed to reach the SE and that also facilitates the auctioneer and the RUs to decide on the auction price as well as the amount of ES to be put into the market for joint ownership. A compelling extension of the proposed scheme would be to study of the feasibility of scheduling of loads such as lifts and water machines in shared space. Another interesting research direction would be to determine how a very large number of SFCs or RUs with different reservation and bidding prices can take part in such a modified auction scheme. One potential way to look at this problem can be from a cooperative game-theoretic point-of-view in which the SFCs and RUs may cooperate to decide on the amount of reservation ES and bidding price they would like to put into the market so as to participate in the auction and benefit from sharing. 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Published as a conference paper at ICLR 2018 M ODELING L ATENT ATTENTION W ITHIN N EURAL N ETWORKS arXiv:1706.00536v2 [] 30 Dec 2017 Christopher Grimm [email protected] Department of Computer Science & Engineering University of Michigan Dilip Arumugam, Siddharth Karamcheti, David Abel, Lawson L.S. Wong, Michael L. Littman {dilip_arumugam, siddharth_karamcheti, david_abel, lsw, michael_littman}@brown.edu Department of Computer Science Brown University Providence, RI 02912 A BSTRACT Deep neural networks are able to solve tasks across a variety of domains and modalities of data. Despite many empirical successes, we lack the ability to clearly understand and interpret the learned mechanisms that contribute to such effective behaviors and more critically, failure modes. In this work, we present a general method for visualizing an arbitrary neural network’s inner mechanisms and their power and limitations. Our dataset-centric method produces visualizations of how a trained network attends to components of its inputs. The computed “attention masks” support improved interpretability by highlighting which input attributes are critical in determining output. We demonstrate the effectiveness of our framework on a variety of deep neural network architectures in domains from computer vision and natural language processing. The primary contribution of our approach is an interpretable visualization of attention that provides unique insights into the network’s underlying decision-making process irrespective of the data modality. 1 I NTRODUCTION Machine-learning systems are ubiquitous, even in safety-critical areas. Trained models used in self-driving cars, healthcare, and environmental science must not only strive to be error free but, in the face of failures, must be amenable to rapid diagnosis and recovery. This trend toward realworld applications is largely being driven by recent advances in the area of deep learning. Deep neural networks have achieved state-of-the-art performance on fundamental domains such as image classification (Krizhevsky et al., 2012), language modeling (Bengio et al., 2000; Mikolov et al., 2010), and reinforcement learning from raw pixels (Mnih et al., 2015). Unlike traditional linear models, deep neural networks offer the significant advantage of being able to learn their own feature representation for the completion of a given task. While learning such a representation removes the need for manual feature engineering and generally boosts performance, the resulting models are often hard to interpret, making it significantly more difficult to assign credit (or blame) to the model’s behaviors. The use of deep learning models in increasingly important application areas underscores the need for techniques to gain insight into their failure modes, limitations, and decision-making mechanisms. Substantial prior work investigates methods for increasing interpretability of these systems. One body of work focuses on visualizing various aspects of networks or their relationship to each datum they take as input Yosinski et al. (2015); Zeiler & Fergus (2015). Other work investigates algorithms for eliciting an explanation from trained machine-learning systems for each decision they make Ribeiro et al. (2016); Baehrens et al. (2010); Robnik-Šikonja & Kononenko (2008). A third line of work, of which our method is most aligned, seeks to capture and understand what networks focus on and what they ignore through attention mechanisms. Attention-based approaches focus on network architectures that specifically attend to regions of their input space. These “explicit” attention mechanisms were developed primarily to improve 1 Published as a conference paper at ICLR 2018 network behavior, but additionally offer increased interpretability of network decision making through highlighting key attributes of the input data (Vinyals et al., 2015; Hermann et al., 2015; Oh et al., 2016; Kumar et al., 2016). Crucially, these explicit attention mechanisms act as filters on the input. As such, the filtered components of the input could be replaced with reasonably generated noise without dramatically affecting the final network output. The ability to selectively replace irrelevant components of the input space is a direct consequence of the explicit attention mechanism. The insight at the heart of the present work is that it is possible to evaluate the property of “selective replaceability” to better understand a network that lacks any explicit attention mechanism. An architecture without explicit attention may still depend more on specific facets of its input data when constructing its learned, internal representation, resulting in a “latent” attention mechanism. In this work, we propose a novel approach for indirectly measuring latent attention mechanisms in arbitrary neural networks using the notion of selective replaceability. Concretely, we learn an auxiliary, “Latent Attention Network” (LAN), that consumes an input data sample and generates a corresponding mask (of the same shape) indicating the degree to which each of the input’s components are replaceable with noise. We train this LAN by corrupting the inputs to a pre-trained network according to generated LAN masks and observing the resulting corrupted outputs. We define a loss function that trades off maximizing the corruption of the input while minimizing the deviation between the outputs generated by the pre-trained network using the true and corrupted inputs, independently. The resultant LAN masks must learn to identify the components of the input data that are most critical to producing the existing network’s output (i.e. those regions that are given the most attention by the existing network.) We empirically demonstrate that the LAN framework can provide unique insights into the inner workings of various pre-trained networks. Specifically, we show that classifiers trained on a Translated MNIST domain learn a two-stage process of first localizing a digit within the image before determining its class. We use this interpretation to predict regions on the screen where digits are less likely to be properly classified. Additionally, we use our framework to visualize the latent attention mechanisms of classifiers on both image classification (to learn the visual features most important to the network’s prediction), and natural language document classification domains (to identify the words most relevant to certain output classes). Finally, we examine techniques for generating attention masks for specific samples, illustrating the capability of our approach to highlight salient features in individual members of a dataset. 2 R ELATED W ORK We now survey relevant literature focused on understanding deep neural networks, with a special focus on approaches that make use of attention. Attention has primarily been applied to neural networks to improve performance Mnih et al. (2014); Gregor et al. (2015); Bahdanau et al. (2014). Typically, the added attention scheme provides an informative prior that can ease the burden of learning a complex, highly structured output space (as in machine translation). For instance, Cho et al. (2015) survey existing content-based attention models to improve performance in a variety of supervised learning tasks, including speech recognition, machine translation, image caption generation, and more. Similarly, Yang et al. (2016) apply stacked attention networks to better answer natural language questions about images, and Goyal et al. (2016) investigate a complementary method for networks specifically designed to answer questions about visual content; their approach visualizes which content in the image is used to inform the network’s answer. They use a strategy similar to that of attention to visualize what a network focuses on when tasked with visual question answering problems. Yosinski et al. (2015) highlight an important distinction for techniques that visualize aspects of networks: dataset-centric methods, which require a trained network and data for that network, and network-centric methods, which target visualizing aspects of the network independent of any data. In general, dataset-centric methods for visualization have the distinct advantange of being network agnostic. Namely, they can treat the network to visualize entirely as a black box. All prior work for visualizing networks, of both dataset-centric and network-centric methodologies, is specific to particular network architectures (such as convolutional networks). For example, Zeiler & Fergus (2015) introduce a visualization method for convolutional neural networks (CNNs) that illustrates which input patterns activate feature maps at each layer of the network. Their core methodology 2 Published as a conference paper at ICLR 2018 x A A(x) ⌘ ⌦ x̃ F F F (x̃) F (x) Figure 1: Diagram of the Latent Attention Network (LAN) framework. is to project activations of nodes at any layer of the network back to the input pixel space using a Deconvolutional Network introduced by Zeiler et al. (2011), resulting in highly interpretable feature visualizations. An exciting line of work has continued advancing these methods, as in Nguyen et al. (2016); Simonyan et al. (2013), building on the earlier work of Erhan et al. (2009) and Berkes & Wiskott (2005). A different line of work focuses on strategies for eliciting explanations from machine learning systems to increase interpretability Ribeiro et al. (2016); Baehrens et al. (2010); Robnik-Šikonja & Kononenko (2008). Lei et al. (2016) forces networks to output a short “rationale” that (ideally) justifies the network’s decision in Natural Language Processing tasks. Bahdanau et al. (2014) advance a similar technique in which neural translation training is augmented by incentivizing networks to jointly align and translate source texts. Lastly, Zintgraf et al. (2017) describe a method for eliciting visualizations that offer explanation for decisions made by networks by highlighting regions of the input that are considered evidence for or against a particular decision. In contrast to all of the discussed methods, we develop a dataset-centric method for visualizing attention in an arbitrary network architecture. To the best of our knowledge, the approach we develop is the first of its kind in this regard. One similar class of methods is sensitivity analysis, introduced by Garson (1991), which seeks to understand input variables’ contribution to decisions made by the network Wang et al. (2000); Gedeon (1997); Gevrey et al. (2006). Sensitivity analysis has known limitations Mazurowski & Szecowka (2006), including failures in highly dependent input spaces and restriction to ordered, quantitative input spaces Montano & Palmer (2003). 3 M ETHOD A key distinguishing feature of our approach is that we assume minimal knowledge about the network to be visualized. We only require that the network F : Rd 7→ R` be provided as a black-box function (that is, we can provide input x to F and obtain output F (x)) through which gradients can be computed. Since we do not have access to the network architecture, we can only probe the network either at its input or its output. In particular, our strategy is to modify the input by selectively replacing components via an attention mask, produced by a learned Latent Attention Network (LAN). 3.1 L ATENT ATTENTION N ETWORK F RAMEWORK A Latent Attention Network is a function A : Rd 7→ [0, 1]d that, given an input x (for the original network F ), produces an attention mask A(x) of the same shape as x. The attention mask seeks to identify input components of x that are critical to producing the output F (x). Equivalently, the attention mask determines the degree to which each component of x can be corrupted by noise while minimally affecting F (x). To formalize this notion, we need two additional design components: LF : R` × R` 7→ R d H : R 7→ R a loss function in the output space of F , a noise probability density over the input space of F . 3 (1) Published as a conference paper at ICLR 2018 We can now complete the specification of the LAN framework. As illustrated in Figure 1, given an input x, we draw a noisy vector η ∼ H and corrupt x according to A(x) as follows: x̃ = A(x) · η + (1 − A(x)) · x, (2) where 1 denotes a tensor of ones with the same shape as A(x), and all operations are performed element-wise. Under this definition of x̃, the components of A(x) that are close to 0 indicate that the corresponding components of x represent signal/importance, and those close to 1 represent noise/irrelevance. Finally, we can apply the black-box network F to x̃ and compare the output F (x̃) to the original F (x) using the loss function LF . An ideal attention mask A(x) replaces/corrupts as many input components as possible (it has A(x) components close to 1), while minimally distorting the original output F (x), as measured by LF . Hence we train the LAN A by minimizing the following training objective for each input x: h i LLAN (x) = Eη∼H LF (F (x̃), F (x)) − βA(x) , (3) where A(x) denotes the mean value of the attention mask for a given input, x̃ is a function of both η and A(x) as in Equation 2, and β > 0 is a hyperparameter for weighting the amount of corruption applied to the input against the reproducibility error with respect to LF , for more information about this trade-off see Section E in the Appendix. 3.2 L ATENT ATTENTION N ETWORK D ESIGN To specify a LAN, we provide two components: the loss function LF and the noise distribution H. The choice of these two components depends on the particular visualization task. Typically, the loss function LF is the same as the one used to train F itself, although it is not necessary. For example, if a network F was pre-trained on some original task but later applied as a black-box within some novel task, one may wish to visualize the latent attention with respect to the new task’s loss to verify that F is considering expected parts of the input. The noise distribution H should reflect the expected space of inputs to F , since input components’ importance is measured with respect to variation determined by H. In the general setting, H could be a uniform distribution over Rd ; however, we often operate in significantly more structured spaces (e.g. images, text). In these structured cases, we suspect it is important to ensure that the noise vector η lies near the manifold of the input samples. Based on this principle, we propose two methods of defining H via the generating process for η: • Constant noise η const : In domains where input features represent quantities with default value c (e.g. 0 word counts in a bag of words, 0 binary valued images), set η = c1, where 1 is a tensor of ones with the appropriate shape and c ∈ R. • Bootstrapped noise η boot : Draw uniform random samples from the training dataset. We expect that the latter approach is particularly effective in domains where the data occupies a small manifold of the input space. For example, consider that that the set of natural images is much smaller than the set of possible images. Randomly selecting an image guarantees that we will be near that manifold, whereas other basic forms of randomness are unlikely to have this property. 3.3 S AMPLE -S PECIFIC L ATENT ATTENTION M ASKS In addition to optimizing whole networks that map arbitrary inputs to attention masks, we can also directly estimate that attention-scheme of a single input. This sample-specific approach simplifies a LAN from a whole network to just a single, trainable variable that is the same shape as the input. This translates to the following optimization procedure: h i LxSSL = Eη∼H LF (F (x̃), F (x)) − βA(x) (4) where A(x) represents the attention mask learned specifically for sample x and x̃ is a function of η, A and x defined in Eq. (2). 4 Published as a conference paper at ICLR 2018 Figure 2: Visualization of attention maps for different translated MNIST digits. For each pair of images, the original translated MNIST digit is displayed on the top, and a visualization of the attention map is displayed on the bottom (where warmer colors indicate more important regions to the pre-trained classifier). Notice the blobs of network importance around each digit, and the seemingly constant “griding” pattern present in each of the samples. 4 E XPERIMENTS To illustrate the wide applicability of the LAN framework, we conduct experiments in a variety of typical learning tasks, including digit classification and object classification in natural images. The goal of these experiments is to demonstrate the effectiveness of LANs to visualize latent attention mechanisms of different network types. Additionally, we conduct an experiment in a topicmodeling task to demonstrate the flexibility of LANs across multiple modalities. While LANs can be implemented with arbitrary network architectures, we restrict our focus here to fully-connected LANs and leave investigations of more expressive LAN architectures to future work. More specifically, our LAN implementations range from 2–5 fully-connected layers each with fewer than 1000 hidden units. At a high level, these tasks are as follows (see supplementary material for training details): Translated MNIST Data : A dataset of 28 × 28 grayscale images with MNIST digits, scaled down to 12 × 12, are placed in random locations. No modifications are made to the orientation of the digits. Task : We train a standard deep network for digit classification. CIFAR-10 Data : A dataset of 3-channel 32 × 32 color images of objects or animals, each belonging to one of ten unique classes. The images are typically centered around the classes they depict. Task : We train a standard CNN for object detection. Newsgroup-20 Data : A dataset consisting of news articles belonging to one of twenty different topics. The list of topics includes politics, electronics, space, and religion, amongst others. Task : We train a bag-of-words neural network, similar to the Deep Averaging Network (DAN) of Iyyer et al. (2015) to classify documents into one of the twenty different categories. For each experiment, we train a network F (designed for the given task) to convergence. Then, we train a Latent Attention Network, A on F . For all experiments conducted with image data, we used bootstrapped noise while our exploratory experiment with natural language used constant noise. Since LANs capture attention in the input space, the result of the latter training procedure is to visualize the attention mechanism of F on any sample in the input. For a detailed description of all experiments and associated network architectures, please consult the supplementary material. 5 5.1 R ESULTS T RANSLATED MNIST R ESULTS Results are shown in Figure 2. We provide side-by-side visualizations of samples from the Translated MNIST dataset and their corresponding attention maps produced by the LAN network. In these 5 Published as a conference paper at ICLR 2018 attention maps, there are two striking features: (1) a blob of attention surrounding the digit and (2) an unchanging grid pattern across the background. This grid pattern is depicted in Figure 3a. In what follows, we support an interpretation of the grid effect illustrated in Figure 3a. Through subsequent experiments, we demonstrate that our attention masks have illustrated that the classifier network operates in two distinct phases: 1. Detect the presence of a digit somewhere in the input space. 2. Direct attention to the region in which the digit was found to determine its class. Under this interpretation, one would expect classification accuracy to decrease in regions not spanned by the constant grid pattern. To test this idea, we estimated the error of the classifier on digits centered at various locations in the image. We rescaled the digits to 7 × 7 pixels to make it easier to fit them in the regions not spanned by the constant grid. Visualizations of the resulting accuracies are displayed in Figure 3b. Notice how the normalized accuracy falls off around the edges of the image (where the constant grid is least present). This effect is particularly pronounced with smaller digits, which would be harder to detect with a fixed detection grid. To further corroborate our hypothesis, we conducted an additional experiment with a modified version of the Translated MNIST domain. In this new domain, digits are scaled to 12 × 12 pixels and never occur in the bottom right 12 × 12 region of the image. Under these conditions, we retrained our classifier and LAN, obtaining the visualization of the constant grid pattern and probability representation presented in Figure 3(c-d). Notice how the grid pattern is absent from the bottom right-hand corner where digits never appeared at training time. Consequently, the accuracy of the classifier falls off if tested on digits in this region. Through these results, we showcase the capability of LANs to produce attention masks that not only provide insights into the inner workings of a trained network but also serve as a diagnostic for predicting likely failure modes. (a) (b) (c) (d) Figure 3: (a) Constant grid pattern observed in the attention masks on Translated MNIST. (b) Accuracy of the pre-trained classifier on 7 × 7 digits centered at different pixels. Each pixel in the images is colored according to the estimated normalized accuracy on digits centered at that pixel where warmer colors indicate higher normalized accuracy. Only pixels that correspond to a possible digit center are represented in these images, with other pixels colored dark blue. (c–d) Duplicate of (a) and (b) for a pre-trained network on a modified Translated MNIST domain where no digits can appear in the bottom right hand corner. 5.2 CIFAR-10 CNN In Figure 4, we provide samples of original images from the CIFAR-10 dataset alongside the corresponding attention masks produced by the LAN. Notice that, for images belonging to the same class, the resulting masks capture common visual features such as tail feathers for birds or hulls/masts for ships. The presence of these features in the mask suggests that the underlying classifier learns a canonical representation of each class to discriminate between images and to confirm its classification. We further note that, in addition to revealing high level concepts in the learned classifier, the LAN appears to demonstrate the ability to compose those concepts so as to discriminate between classes. This property is most apparent between the horse and deer classes, both of which show extremely similar regions of attention for capturing legs while deviating in their structure to confirm the presence of a heads or antlers, respectively. 6 Published as a conference paper at ICLR 2018 Figure 4: Each frame pairs an input image (left) with its LAN attention mask (right). Each column represents a different category: horse, plane, truck, bird, ship, and deer. 5.3 N EWSGROUP -20 D OCUMENT C LASSIFICATION R ESULTS Tables 1 and 2 contrast words present in documents against the 15 most important words, as determined by the corresponding attention mask, for topic classification. We note that these important words generally tend to be either in the document itself (highlighted in yellow) or closely associated with the category that the document belongs to. The absence of important words from other classes is explained by our choice of η0 -noise, which produces more visually appealing attention-masks, but doesn’t penalize the LAN for ignoring such words. We suspect that category-associated words not present in the document occur due to the capacity limitations on the fully-connected LAN architecture on a high dimensional and poorly structured bag-of-words input space. Future work will further explore the use of LANs in natural language tasks. Document Topic Document Words (Unordered) 15 Most Important Words comp.sys.mac.hardware ralph, rutgers, rom, univ, mac, gonzalez, gandalf, work, use, you, phone, drives, internet, camden, party, floppy, science, edu, roms, drive, upgrade, disks, computer mac, drive, computer, problem, can, this, drives, disk, use, controller, UNK memory, for, boot, fax Table 1: A visualization of the attention mask generated for a specific document in the Newsgroup20 Dataset. The document consists of the words above, and is labeled under the category “comp.sys.mac.hardware” which consists about topics relating to Apple Macintosh computer hardware. Note the top 15 words identified by the LAN Mask, and how they seem to be picking important words relevant to the true class of the given document. 5.4 S AMPLE -S PECIFIC ATTENTION M ASKS In all of the previous results, there is a strong sense in which the resultant attention masks are highly correlated with the pre-trained network outputs and less sensitive to variations in the individual input samples. Here we present results on the same datasets (see Figures 5, 3 and 4) using the sample specific objective defined in Eq. (4). We notice that these learned attention masks are more representative of nuances present in each invididual sample. This increase in contained information seems reasonable when considering the comparative ease of optimizing a single attention mask for a single sample rather than a full LAN that must learn to map from all inputs to their corresponding attention masks. 6 C ONCLUSION As deep neural networks continue to find application to a growing collection of tasks, understanding their decision-making processes becomes increasingly important. Furthermore, as this space of tasks 7 Published as a conference paper at ICLR 2018 Document Topic Document Words (Unordered) 15 Most Important Words soc.religion.christian UNK, death, university, point, complaining, atheists, acs, isn, since, doesn, never, that, matters, god, incestuous, atterlep, rejection, forever, hell, step, based, talk, vela, eternal, edu, asked, worse, you, tread, will, not, and, rochester, fear, opinions, die, faith, fact, earth oakland, lot, don, christians, alan, melissa, rushing, angels, comparison, heaven, terlep UNK, clh, jesus, this church, christians, interested, lord, christian, answer, will, heaven, find, worship, light Table 2: Another visualization of the attention mask generated for a specific document in the Newsgroup-20 Dataset. This document consists of the words above, and is labeled under the category “soc.religion.christian”, which consists of topics relating to Christianity. The presence of UNK as an important word in this religious documents could be attributed to a statistically significant number of references to people and places from Abrahamic texts which are converted to UNK due to their relative uncommonness in the other document classes. Figure 5: Each image pair contains a CIFAR-10 image and its corresponding sample-specific attention mask. Each column contains images from a different category: car, cat deer, dog, horse and ship. Notice how these sample-specific attention masks retain the class specific features mentioned in Section 5.2 while more closely tracking the subjects of the images. Document Topic Document Words (Unordered) 15 Most Important Words comp.sys.ibm.pc.hardware UNK, video, chip, used, color, card, washington, drivers, name, edu, driver, chipset, suffice, functions, for, type, cica card, chip, video, drivers, driver, type, used, cica, edu, washington, bike, functions, time, sale, color Table 3: A visualization of the sample specific attention mask generated for a specific document in the Newsgroup-20 Dataset. The document consists of the words above and is labeled under the category “comp.sys.ibm.pc.hardware” which consists of topics relating to personal computing and hardware. Words that are both in the document and detected by the sample specific attention mask are highlighted in yellow. grows to include areas where there is a small margin for error, the ability to explore and diagnose problems within erroneous models becomes crucial. 8 Published as a conference paper at ICLR 2018 Document Topic Document Words (Unordered) 15 Most Important Words talk.religion.misc newton, jesus, spread, died, writes, truth, ignorance, bliss, sandvik, not, strength, article, that, good, apple, kent ignorance, died, sandvik, kent, newton, bliss, jesus, truth, good, can, strength, for, writes, computer, article Table 4: A visualization of the sample specific attention mask generated for a specific document in the Newsgroup-20 Dataset. The document consists of the words above and is labeled under the category “talk.religion.misc” which consists of topics relating to religion. Words that are both in the document and detected by the sample specific attention mask are highlighted in yellow. In this work, we proposed Latent Attention Networks as a framework for capturing the latent attention mechanisms of arbitrary neural networks that draws parallels between noise-based input corruption and attention. We have shown that the analysis of these attention measurements can effectively diagnose failure modes in pre-trained networks and provide unique perspectives on the mechanism by which arbitrary networks perform their designated tasks. We believe there are several interesting research directions that arise from our framework. First, there are interesting parallels between this work and the popular Generative Adversarial Networks (Goodfellow et al., 2014). It may be possible to simultaneously train F and A as adversaries. Since both F and A are differentiable, one could potentially exploit this property and use A to encourage a specific attention mechanism on F , speeding up learning in challenging domains and otherwise allowing for novel interactions between deep networks. Furthermore, we explored two types of noise for input corruption: η const and η boot . It may be possible to make the process of generating noise a part of the network itself by learning a nonlinear transformation and applying it to some standard variety of noise (such as Normal or Uniform). Since our method depends on being able to sample noise that is similar to the “background noise” of the domain, better mechanisms for capturing noise could potentially enhance the LAN’s ability to pick out regions of attention and eliminate the need for choosing a specific type of noise at design time. Doing so would allow the LAN to pick up more specific features of the input space that are relevant to the decision-making process of arbitrary classifier networks. R EFERENCES David Baehrens, Timon Schroeter, Stefan Harmeling, Motoaki Kawanabe, Katja Hansen, and KlausRobert MÞller. How to explain individual classification decisions. Journal of Machine Learning Research, 11(Jun):1803–1831, 2010. Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio. Neural machine translation by jointly learning to align and translate. arXiv preprint arXiv:1409.0473, 2014. Yoshua Bengio, Réjean Ducharme, Pascal Vincent, and Christian Janvin. A neural probabilistic language model. In Journal of Machine Learning Research, 2000. Pietro Berkes and Laurenz Wiskott. Slow feature analysis yields a rich repertoire of complex cell properties. Journal of vision, 5(6):9–9, 2005. Kyunghyun Cho, Aaron Courville, and Yoshua Bengio. Describing multimedia content using attention-based encoder-decoder networks. 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Volodymyr Mnih, Koray Kavukcuoglu, David Silver, Andrei A. Rusu, Joel Veness, Marc G. Bellemare, Alex Graves, Martin A. Riedmiller, Andreas Fidjeland, Georg Ostrovski, Stig Petersen, Charles Beattie, Amir Sadik, Ioannis Antonoglou, Helen King, Dharshan Kumaran, Daan Wierstra, Shane Legg, and Demis Hassabis. Human-level control through deep reinforcement learning. Nature, 518 7540:529–33, 2015. JJ Montano and A Palmer. Numeric sensitivity analysis applied to feedforward neural networks. Neural Computing & Applications, 12(2):119–125, 2003. Anh Mai Nguyen, Jason Yosinski, and Jeff Clune. Multifaceted feature visualization: Uncovering the different types of features learned by each neuron in deep neural networks. CoRR, abs/1602.03616, 2016. Junhyuk Oh, Valliappa Chockalingam, Satinder P. Singh, and Honglak Lee. Control of memory, active perception, and action in minecraft. In ICML, 2016. Marco Tulio Ribeiro, Sameer Singh, and Carlos Guestrin. 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Grammar as a foreign language. In NIPS, 2015. Wenjia Wang, Phillis Jones, and Derek Partridge. Assessing the impact of input features in a feedforward neural network. Neural Computing & Applications, 9(2):101–112, 2000. Zichao Yang, Xiaodong He, Jianfeng Gao, Li Deng, and Alex Smola. Stacked attention networks for image question answering. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 21–29, 2016. Jason Yosinski, Jeff Clune, Anh Nguyen, Thomas Fuchs, and Hod Lipson. Understanding neural networks through deep visualization. arXiv preprint arXiv:1506.06579, 2015. Eliezer Yudkowsky. Artificial intelligence as a positive and negative factor in global risk. Global catastrophic risks, 1(303):184, 2008. Matthew Zeiler and Rob Fergus. Visualizing and Understanding Convolutional Networks. pp. 1–11, 2015. ISSN 16113349. Matthew D Zeiler, Graham W Taylor, and Rob Fergus. Adaptive deconvolutional networks for mid and high level feature learning. In Computer Vision (ICCV), 2011 IEEE International Conference on, pp. 2018–2025. IEEE, 2011. Luisa M Zintgraf, Taco S Cohen, Tameem Adel, and Max Welling. Visualizing deep neural network decisions: Prediction difference analysis. arXiv preprint arXiv:1702.04595, 2017. 11 Published as a conference paper at ICLR 2018 In the following experiment subsections we describe network architectures by sequentially listing their layers using an abbreviated notation: Conv (hNum Filtersi, hStridei, hFilter Dimensionsi, hActivation Functioni) ConvTrans (hNum Filtersi, hStridei, hFilter Dimensionsi, hActivation Functioni) FC (hNum Hidden Unitsi, hActivation Functioni) (5) for convolutional, convolutional-transpose and fully connected layers respectively. In all network architectures, `-ReLU denotes the leaky-ReLU Maas et al. (2013). We now describe each experiment in greater detail. A T RANSLATED MNIST H ANDWRITTEN D IGIT C LASSIFIER Here we investigate the attention masks produced by a LAN trained on a digit classifier. We show how LANs provide intuition about the particular method a neural network uses to complete its task and highlight failure-modes. Specifically, we construct a “translated MNIST” domain, where the original digits are scaled down from 28 × 28 to 12 × 12 and positioned in random locations in the original 28 × 28 image. The network F is a classifier, outputting the probability of each digit being present in a given image. The pre-trained network, F has the following architecture: Conv(10, 2, (4 × 4), `-ReLU), Conv(20, 2, (4 × 4), `-ReLU), FC(10, softmax). F is trained P with the Adam Optimizer for 100, 000 iterations with a learning rate of 0.001 and with LF = − i yi log F (x)i where y ∈ R10 is a one-hot vector indicating the digit class. The latent attention network, A has the following architecture: FC(100, `-ReLU), FC(784, sigmoid), with its output being reshaped to a 28 × 28 image. A is trained with the Adam Optimizer for 100, 000 iterations with a learning rate of 0.0001. We use β = 5.0 and η = ηboot for this experiment. B CIFAR-10 CNN In this experiment we demonstrate that the LAN framework can illuminate the decision making of classifier (based on the Alexnet architecture) on natural images. To avoid overfitting, we augment the CIFAR-10 dataset by applying small random affine transformations to the images at train time. We used β = 5.0 for this experiment. The pre-trained network, F has the following architecture: Conv(64, 2, (5 × 5), `-ReLU), Conv(64, 2, (5 × 5), `-ReLU), Conv(64, 1, (3 × 3), `-ReLU), Conv(64, 1, (3 × 3), `-ReLU), Conv(32, 2, (3 × 3), `-ReLU), FC(384, tanh), FC(192, tanh), FC(10, softmax), where dropout and local response normalization is applied at each layer. F is trainedPwith the Adam Optimizer for 250, 000 iterations with a learning rate of 0.0001 and with LF = − i yi log F (x)i where y ∈ R20 is a one-hot vector indicating the image class. The latent attention network, A has the following architecture: FC(500, `-ReLU), FC(500, `-ReLU), FC(500, `-ReLU), FC(1024, sigmoid), with its output being reshaped to a 32 × 32 × 1 image and tiled 3 times on the channel dimension to produce a mask over the pixels. A is trained with the Adam Optimizer for 250, 000 iterations with a learning rate of 0.0005. We used β = 7.0 and η = ηboot for this experiment. C 20 N EWSGROUPS D OCUMENT C LASSIFICATION In this experiment, we extend the LAN framework for use on non-visual tasks. Namely, we show that it can be used to provide insight into the decision-making process of a bag-of-words document classifier, and identify individual words in a document that inform its predicted class label. To do this, we train a Deep Averaging Network (DAN) (Iyyer et al., 2015) for classifying documents from the Newsgroup-20 dataset. The 20 Newsgroups Dataset consists of 18,821 total documents, partioned into a training set of 11,293 documents, and a test set of 7,528 documents. Each document belongs to 1 of 20 different categories, including topics in religion, sports, computer hardware, and 12 Published as a conference paper at ICLR 2018 politics, to name a few. In our experiments, we utilize the version of the dataset with stop words (common words like “the”, “his”, “her”) removed. The DAN Architecture is very simple - each document is represented with a bag-of-words histogram vector, with dimension equal to the number of unique words in the dataset (the size of the vocabulary). This bag of words vector is then multiplied with an embedding matrix and divided by the number of words in the document, to generate a low-dimension normalized representation. This vector is then passed through two separate hidden layers (with dropout), and then a final softmax layer, to produce a distribution over the 20 possible classes. In our experiments we use an embedding size of 50, and hidden layer sizes of 200 and 150 units, respectively. We train the model for 1,000,000 mini-batches, with a batch size of 32. Like with our previous experiments, we utilize the Adam Optimizer Kingma & Ba (2014), with a learning rate of 0.00005. The latent attention network, A has the following architecture: FC(100, `-ReLU), FC(1000, `-ReLU), FC(vocab-size, sigmoid). A is trained with the Adam Optimizer for 100, 000 iterations with a learning rate of 0.001. We used β = 50.0 and η = ηconst with a constant value of 0. D S AMPLE S PECIFIC E XPERIMENTS In the sample specific experiments the same pre-trained networks are used as in the standard CIFAR10 and Newsgroup-20 experiments. To train the sample specific masks, we used a learning rate of 0.001 and 0.05 for the Newsgroup-20 and CIFAR-10 experiments respectively. Both experiments used the Adam Optimizer and each mask is trained for 10, 000 iterations. We used β = 50.0 and η = ηboot for both experiments. E B ETA H YPERPARAMETERS In this section, we illustrate the role of the β hyperparameter in the sample specific experiments. As stated earlier, β controls the trade-off between the amount of corruption in the input and the similarity of the corrupted and uncorrupted inputs (measured as a function of the respective pretrained network outputs). Based on the loss functions presented in equations 3 and 4, high values of β encourage a larger amount of input corruption and weight the corresponding term more heavily in the loss computation. Intuitively, we would like to identify the minimal number of dimensions in the input space that most critically affect the output of the pre-trained network. The small amount of input corruption that corresponds to a low value of β would make the problem of reconstructing pre-trained network outputs too simple, resulting in attention masks that deem all or nearly all input dimensions as important; conversely, a heavily corrupted input from a high value of β could make output reconstruction impossible. We illustrate this for individual images from the CIFAR-10 dataset below: 0.25 2 0.75 13 4 5 Published as a conference paper at ICLR 2018 F I MAGENET R ESULTS To demonstrate the capacity of our technique to scale up, we visualize attention masks learned on top of the Inception network architecture introduced in Szegedy et al. (2015) and trained for the ILSVRC 2014 image classification challenge. We utilize a publicly available Tensorflow implementation of the pre-trained model1 . In these experiments we learned sample-specific attention masks for different settings of β on the images shown below. For our input corruption, we used uniformly sampled RGB noise: η ∼ Uniform(0, 1). We note that the attention masks produced on this domain seem to produce much richer patterns of contours than in the CIFAR-10 experiments. We attribute this difference to both the increased size of the images and the increased number of classes between the two problems(1000 vs 10). 1.5 G 3 5 M OTIVATING E XAMPLE : TANK D ETECTION In this section, we motivate our approach by demonstrating how it identifies failure modes within a real-world application of machine learning techniques. There is a popular and (allegedly) apocryphal story, relayed in Yudkowsky (2008), that revolves around the efforts of a US military branch to use neural networks for the detection of camouflaged tanks within forests. Naturally, the researchers tasked with this binary classification problem collected images of camouflaged tanks and empty forests in order to compile a dataset. After training, the neural network model performed well on the testing data and yet, when independently evaluated by other government agencies, failed to achieve performance better than random chance. It was later determined that the original dataset only collected positive tank examples on cloudy days leading to a classifier that discriminated based on weather patterns rather than the presence of camouflaged tanks. We design a simple domain for highlighting the effectiveness of our approach, using the aforementioned story as motivation. The problem objective is to train an image classifier for detecting the presence of tanks in forests. Our dataset is composed of synthetic images generated through the 1 https://github.com/tensorflow/models/tree/master/research/slim 14 Published as a conference paper at ICLR 2018 random placement of trees, clouds and tanks. A representative image from this dataset is provided below on the left with its component objects highlighted and displayed on the right: As in the story, our training and testing datasets are generated such that clouds are only present in positive examples of camouflaged tanks. A simple convolutional neural network architecture is then trained on this data and treated as the pre-trained network in our LAN framework. Unsurprisingly, this classifier suffers from the same problems outlined in Yudkowsky (2008); despite high accuracy on testing data, the classifier fails to detect tanks without the presence of clouds. We now observe the sample-specific attention masks trained from this classifier: The resulting attention mask (β = 1.0 with bootstrapped noise η = ηboot ) assigns high importance to the pixels associated with the cloud while giving no importance to the region of the image containing the tank. With this example, we underscore the utility of our methods in providing a means of visualizing the underlying “rationale” of a network for producing a given output. Our attention mask help recognize that the classifier’s basis for detecting tanks is incorrectly based on the presence of clouds. 15 

arXiv:1708.07818v2 [] 1 Sep 2017 ON BOUNDARIES OF RELATIVELY HYPERBOLIC RIGHT-ANGLED COXETER GROUPS MATTHEW HAULMARK, HOANG THANH NGUYEN, AND HUNG CONG TRAN Abstract. We give “visual descriptions” of cut points and non-parabolic cut pairs in the Bowditch boundary of relatively hyperbolic right-angled Coxeter groups. We also prove necessary and sufficient conditions for a relatively hyperbolic right-angled Coxeter group whose defining graph has a planar flag complex with minimal peripheral structure to have the Sierpinski carpet or the 2-sphere S2 as its Bowditch boundary. We apply these results to the problem of quasi-isometry classification of right-angled Coxeter groups. Additionally, we study right-angled Coxeter groups with isolated flats whose CAT(0) boundaries are Menger curve. 1. Introduction For each finite simplicial graph Γ the associated right-angled Coxeter group GΓ has generating set S equal to the vertices of Γ, relations s2 = 1 for each s in S and relations st = ts whenever s and t are adjacent vertices. In geometric group theory, groups acting on CAT(0) cube complexes are fundamental objects and right-angled Coxeter groups provide a rich source of these such groups. The coarse geometry of right-angled Coxeter groups was studied by Caprace [Cap09, Cap15], Dani-Thomas [DT15, DT], Dani-StarkThomas [DST], Behrstock-Hagen-Sisto [BHS17], Levcovitz [Lev] and others. In this paper, we will study boundaries of relatively hyperbolic right-angled Coxeter groups. The notion of a relatively hyperbolic group was introduced by Gromov [Gro87] to generalize both word hyperbolic and geometrically finite Kleinian groups. Introduced by Bowditch [Bow12] there is a boundary for relatively hyperbolic groups. The Bowditch boundary generalizes the Gromov boundary of a word hyperbolic group and the limit set of a geometrically finite Kleinian group. Under modest hypotheses on the peripheral subgroups, the homeomorphism type of the Bowditch boundary is known to be a quasiisometry invariant of the group (see Groff [Gro13]). Combining the work of Groff [Gro13] and Behrstock-Druţu-Mosher [BDM09], we elaborate on the homeomorphism between Bowditch boundaries induced by a quasi-isometry between two relatively hyperbolic groups whose peripheral subgroups are not relatively hyperbolic (see Theorem 2.12). Date: September 4, 2017. 2000 Mathematics Subject Classification. 20F67, 20F65. 1 ON BOUNDARIES OF RIGHT-ANGLED COXETER GROUPS 2 In [Cap09, Cap15], Caprace gives necessary and sufficient conditions on the defining graph Γ for the associated right-angled Coxeter group GΓ to be relatively hyperbolic with respect to a collection of finitely generated subgroups. Behrstock-Hagen-Sisto [BHS17] develop further the work of Caprace to describe the minimal peripheral structure of a relatively hyperbolic right-angled Coxeter group (see Definition 2.8). In Sections 3 and 4, we study Bowditch boundaries of relatively hyperbolic right-angled Coxeter groups with minimal peripheral structure. We use our results concerning the Bowditch boundary to study quasi-isometry classification of certain classes of right-angled Coxeter groups. Lastly, we will also study CAT(0) boundaries of right-angled Coxeter groups with isolated flats. 1.1. Visual descriptions of cut points and non-parabolic cut pairs. Cut points and non-parabolic cut pairs are topological features of the Bowditch boundary of a relatively hyperbolic group. In relatively hyperbolic rightangled Coxeter groups, we are able to visualize these features via defining graphs. Our first main result of Section 3 is the following theorem relating cut points in the Bowditch boundary of a relatively hyperbolic right-angled Coxeter group to the defining graph. Theorem 1.1. Let Γ be a simplicial graph and J be a collection of induced proper subgraphs of Γ. Assume that the right-angled Coxeter groups GΓ is one-ended, hyperbolic relative to the collection P = { GJ | J ∈ J }, and suppose each subgroup in P is also one-ended. Then each parabolic point vgGJ0 is a global cut point if and only if some induced subgraph of J0 separates the graph Γ. Therefore, we obtain a new quasi-isometry invariant among all relatively hyperbolic right-angled Coxeter groups. Corollary 1.2. Let Γ1 and Γ2 be simplicial graphs such that GΓ1 and GΓ2 are both one-ended. Assume that each graph Γi has a peripheral structure Ji that consists of proper subgraphs of Γi such that each subgroup in Pi = { GJ | J ∈ Ji } is one-ended and not relatively hyperbolic. If GΓ1 and GΓ2 are quasi-isometric, then for each graph K ∈ J1 there is a graph L ∈ J2 such that GK and GL are quasi-isometric and vice versa. Moreover, if K has an induced subgraph that separates Γ1 , then L also has an induced subgraph which separates Γ2 . We refer the reader to Example 3.7 for an illustration of the application of Theorem 1.1 to the quasi-isometry classification of right-angled Coxeter groups. In Section 3 we also visualize non-parabolic cut pairs of the Bowditch boundary of a relatively hyperbolic group via its defining graph. Theorem 1.3. Let Γ be a simplicial graph and J be a collection of induced proper subgraphs of Γ. Assume that the right-angled Coxeter groups GΓ is one-ended and hyperbolic relative to the collection P = { GJ | J ∈ J }, and suppose each subgroup in P is one-ended. If the Bowditch boundary ∂(GΓ , P) ON BOUNDARIES OF RIGHT-ANGLED COXETER GROUPS 3 has a non-parabolic cut pair, then Γ has a separating complete subgraph suspension. Moreover, if Γ has a separating complete subgraph suspension whose non-adjacent vertices do not lie in the same subgraph J ∈ J, then the Bowditch boundary ∂(GΓ , P) has a non-parabolic cut pair. We remark that Theorem 1.3 can help us prove that Bowditch boundaries of two distinct groups GΓ1 and GΓ2 are not homeomorphic in certain cases by detecting non-parabolic cut pairs in their Bowditch boundaries. Therefore, we can conclude these groups are not quasi-isometric if their Bowditch boundaries are based on their minimal peripheral structures (see Example 3.10). 1.2. Relatively hyperbolic right-angled Coxeter groups with Sierpinski carpet or sphere Bowditch boundary. In Section 4, we also give necessary and sufficient conditions for a relatively hyperbolic right-angled Coxeter group whose defining graph has planar flag complex to have the Sierpinski carpet or S2 as its Bowditch boundary. We refer the reader to the beginning of Section 4 for definitions of the graph theoretic terms used in the statement of the following theorem. Theorem 1.4. Let Γ be a graph whose flag complex is planar. Assume that Γ has a non-trivial peripheral structure J and let P = { GJ | J ∈ J }. The Bowditch boundary ∂(GΓ , P) is S2 or the Sierpinski carpet if and only if Γ is inseparable and each graph in J is a strongly non-separating 4-cycle extension graph. In addition, the Bowditch boundary ∂(GΓ , P) is the Sierpinski carpet if and only if Γ contains a strongly non-separating induced n–cycle extension K (n ≥ 5) satisfying the following properties: (1) K does not contain any nonadjacent vertices of an induced 4-cycle extension; (2) No vertex outside K is adjacent to two non-adjacent vertices of K. A graph is inseparable if it is connected, has no separating complete subgraph, no cut pair, and no separating complete subgraph suspension. We note that if the defining graph Γ is inseparable, distinct from a complete graph, and has planar flag complex distinct from a triangulation of S 2 as in Theorem 1.4, then the CAT(0) boundary of a Davis complex ΣΓ is a Sierpinski carpet (see Świa̧tkowski [Ś]). With the additional assumption that Γ has a non-trivial peripheral structure, the group GΓ becomes a relatively hyperbolic group and Theorem 1.4 gives a classification of such right-angled Coxeter groups. More importantly, the classification in Theorem 1.4 also contributes to the problem of quasi-isometry classification of right-angled Coxeter groups as we demonstrate in Example 4.15. We now restrict Theorem 1.4 to the case of 2–dimensional right-angled Coxeter groups GΓ (i.e. Γ is triangle free and has at least one edge) and ON BOUNDARIES OF RIGHT-ANGLED COXETER GROUPS 4 we obtain a slightly simpler characterization of relatively hyperbolic rightangled Coxeter groups whose boundaries are the Sierpinski carpet or the sphere. Corollary 1.5. Let Γ be a triangle free, planar graph. Assume that Γ has a non-trivial peripheral structure J and let P = { GJ | J ∈ J }. The Bowditch boundary ∂(GΓ , P) is S2 or the Sierpinski carpet if and only if Γ is inseparable and each graph in J is a strongly non-separating 4-cycle. In addition, the Bowditch boundary ∂(GΓ , P) is the Sierpinski carpet if and only if Γ contains a strongly non-separating induced n–cycle K (n ≥ 5) satisfying the following properties: (1) K does not contain any nonadjacent vertices of an induced 4-cycle; (2) No vertex outside K is adjacent to two non-adjacent vertices of K. We remark that in [DT] Dani-Thomas study 2–dimensional hyperbolic right-angled Coxeter groups. The above corollary can be considered an extension of their work to relatively hyperbolic right-angled Coxeter groups. We also note that the above corollary is no longer true if we drop the planar condition on the defining graph (see Remark 5.9). 1.3. Non-hyperbolic right-angled Coxeter groups with Menger curve boundary. Hyperbolic groups with 1-dimensional boundary, generically have a boundary which is homeomorphic to the Menger curve [KK00, DGP11]. For non-hyperbolic groups, examples of groups with Menger curve boundary are scarce. In fact, prior to [Haua] there were no known techniques for constructing non-hyperbolic groups with Menger curve CAT(0) boundary. The first example of a non-hyperbolic group with Menger curve boundary is constructed in [DHW] using 3-manifolds. In Section 5 we use the main result of [Haua] and a theorem of Mihalik-Tschantz [MT09] to prove the following: Proposition 1.6. Let Γ be a triangle free, non-planar graph with a nontrivial peripheral structure J that consists of induced 4-cycles. If the graph Γ is inseparable and the CAT(0) boundary ∂ΣΓ of the right-angled Coxeter Davis complex ΣΓ is not a Sierpinski carpet, then ∂ΣΓ is a Menger curve. Inspired by the examples in Dani-Haulmark-Walsh [DHW], we use Proposition 1.6 to construct examples of non-hyperbolic right-angled Coxeter groups with Menger curve CAT(0) boundary. We then use the Bowditch boundary to show that the constructed examples are not quasi-isometric (see Lemma 5.7). Acknowledgments. First, all three authors would like to thank Chris Hruska for suggestions and insights. The first author would also like thank Genevieve Walsh for many helpful conversations. Lastly, the authors would like to thank Jason Behrstock for a correction to Corollary 1.2. ON BOUNDARIES OF RIGHT-ANGLED COXETER GROUPS 5 2. Preliminaries In this section, we review some concepts in geometric group theory: CAT(0) spaces, δ–hyperbolic spaces, CAT(0) spaces with isolated flats, relatively hyperbolic groups, CAT(0) boundaries, Gromov boundaries, Bowditch boundaries, and peripheral splitting of relatively hyperbolic groups. We also use the work of Behrstock-Druţu-Mosher [BDM09] and Groff [Gro13] to prove that Bowditch boundary is a quasi-isometry invariant among relatively hyperbolic groups with non-relatively hyperbolic peripheral subgroup structures. We review right-angled Coxeter groups and discuss the work of Caprace [Cap09, Cap15] and Behrstock-Hagen-Sisto [BHS17] on peripheral structures of relatively hyperbolic right-angled Coxeter groups. 2.1. CAT(0) spaces, δ–hyperbolic spaces, and relatively hyperbolic groups. We first discuss on CAT(0) spaces, δ–hyperbolic spaces, Gromov boundaries, and CAT(0) boundaries. We refer the reader to the book [BH99] for more details. Definition 2.1. We say that a geodesic triangle ∆ in a geodesic space X satisfies the CAT (0) inequality if d(x, y) ≤ d(x, y) for all points x, y on the edges of ∆ and the corresponding points x, y on the edges of the comparison triangle ∆ in Euclidean space E2 . Definition 2.2. A geodesic space X is said to be a CAT (0) space if every triangle in X satisfies the CAT(0) inequality. If X is a CAT(0) space, then the CAT (0) boundary of X, denoted ∂X, is defined to be the set of all equivalence classes of geodesic rays in X, where two rays c and c′ are equivalent if the Hausdorff distance between them is finite. We note that for any x ∈ X and ξ ∈ ∂X there is a unique geodesic ray αx,ξ : [0, ∞) → X with αx,ξ = x and [αx,ξ ] = ξ. The CAT(0) boundary has ′ a natural topology with
basis given by the sets U (x, ξ, R, ǫ) = { ξ ∈ ∂X | d αx,ξ (R), αx,ξ ′ (R) ≤ ǫ }, where x ∈ X, ξ ∈ ∂X, R > 0 and ǫ > 0. Definition 2.3. A geodesic metric space (X, d) is δ–hyperbolic if every geodesic triangle with vertices in X is δ–thin in the sense that each side lies in the δ–neighborhood of the union of other sides. If X is a δ–hyperbolic space, then we could build the Gromov boundary of X, denoted ∂X, in the same way as for a CAT(0) space. That is, the Gromov boundary of X is defined to be the set of all equivalence classes of geodesic rays in X, where two rays c and c′ are equivalent if the Hausdorff distance between them is finite. However, the topology on it is slightly different from the topology on the boundary of a CAT(0) space (see for example [BH99, Section III.3] for details). We now review relatively hyperbolic groups and related concepts. ON BOUNDARIES OF RIGHT-ANGLED COXETER GROUPS 6 Definition 2.4 (Combinatorial horoball [GM08]). Let T be any graph with the vertex set V . We define the combinatorial horoball based at T , H(= H(T )) to be the following graph: (1) H(0) = V × {{0} ∪ N}. (2) H(1) = {((t, n), (t, n + 1))} ∪ { ((t1 , n), (t2 , n)) | dT (t1 , t2 ) ≤ 2n }. We call edges of the first set vertical and of the second horizontal. Remark 2.5. In [GM08], the combinatorial horoball is described as a 2complex, but we will only require we only require the 1-skeleton for the horoball in this paper. Definition 2.6 ([GM08]). Let H be the horoball based at some graph T . Let D : H → [0, ∞) be defined by extending the map on vertices (t, n) → n linearly across edges. We call D the depth function for H and refer to vertices v with D(v) = n as vertices of depth n or depth n vertices. Because T × {0} is homeomorphic to T , we identify T with D −1 (0). Definition 2.7 (Cusped space [GM08]). Let G be a finitely generated group and P a finite collection of finitely generated subgroups of G. Let S be a finite generating set of G such that S ∩ P generates P for each P ∈ P. For each left coset gP of subgroup P ∈ P let H(gP ) be the horoball based at a copy of the subgraph TgP with vertex set gP of the Cayley graph Γ(G, S). The cusped space X(G, P, S) is the union of Γ(G, S) with H(gP ) for every left coset of P ∈ P, identifying the subgraph TgP with the depth 0 subset of H(gP ). We suppress mention of S and P when they are clear from the context. Definition 2.8 (Relatively hyperbolic group [GM08]). Let G be a finitely generated group and P a finite collection of finitely generated proper subgroups of G. Let S be a finite generating set of G such that S ∩ P generates P for each P ∈ P. If the cusped space X(G, P, S) is δ–hyperbolic then we say that G is hyperbolic relative to P or that (G, P) is a relatively hyperbolic. Collection P is a peripheral structure, each group P ∈ P is a peripheral subgroup and its left cosets are peripheral left cosets. The peripheral structure P is minimal if for any other peripheral structure Q on G, each P ∈ P is conjugate into some Q ∈ Q. Remark 2.9. Replacing S for some other finite generating set S ′ may change the value of δ, but does not affect the hyperbolicity of the cusped space for some δ′ (see [GM08]). Consequently, the concept of relatively hyperbolic group does not depend on the choice of finite generating set. We say that a finitely generated group is not relatively hyperbolic if it is not relatively hyperbolic with respect to any collection of proper subgroups. 2.2. The Bowditch boundary. We now discuss the Bowditch boundary of a relatively hyperbolic group and prove that it is a quasi-isometry invariant. We also recall peripheral splitting of relatively hyperbolic groups. ON BOUNDARIES OF RIGHT-ANGLED COXETER GROUPS 7 Definition 2.10 (Bowditch boundary [Bow12]). Let (G, P) be a finitely generated relatively hyperbolic group. Let S be a finite generating set of G such that S ∩ P generates P for each P ∈ P. The Bowditch boundary, denoted ∂(G, P), is the Gromov boundary of the associated cusped space, X(G, P, S). Remark 2.11. There is a natural topological action of G on the Bowditch boundary ∂(G, P) that satisfies certain properties (see [Bow12]). Bowditch has shown that the Bowditch boundary does not depend on the choice of finite generating set (see [Bow12]). More precisely, if S and T are finite generating sets for G as in the above definition, then the Gromov boundaries of the cusped spaces X(G, P, S) and X(G, P, T ) are G–equivariantly homeomorphic (see [Bow12]). For each peripheral left coset gP the limit set of the associated horoball H(gP ) consists of a single point in ∂(G, P), called parabolic point vgP . The stabilizer of the point vgP is the subgroup gP g−1 . We call each infinite subgroup of gP g−1 a parabolic subgroup and subgroup gP g−1 a maximal parabolic subgroup. The homeomorphism type of the Bowditch boundary was already known to be a quasi-isometry invariant of the group (see Groff [Gro13]) under modest hypotheses on the peripheral subgroups. However, we combine the work of Groff [Gro13] and Behrstock-Druţu-Mosher [BDM09] to elaborate the homeomorphism between Bowditch boundaries induced by a quasi-isometry between two relatively hyperbolic groups whose peripheral subgroups are not relatively hyperbolic. Theorem 2.12. Let (G1 , P1 ) and (G2 , P2 ) be finitely generated relatively hyperbolic groups such that all peripheral subgroups of both (G1 , P1 ) and (G2 , P2 ) are not relatively hyperbolic. If G1 and G2 are quasi-isometric, then there is a homeomorphism f from ∂(G1 , P1 ) to ∂(G2 , P2 ) that maps the set of parabolic points of ∂(G1 , P1 ) bijectively onto the set of parabolic points of ∂(G2 , P2 ). Moreover, if a parabolic point v of ∂(G1 , P1 ) is labelled by some peripheral left coset g1 P1 in G1 and the parabolic point f (v) of ∂(G2 , P2 ) is labelled by some peripheral left coset g2 P2 in G2 , then P1 and P2 are quasi-isometric. Proof. Fix generating sets S1 and S2 as in Definition 2.10 for G1 and G2 respectively, then there is a quasi-isometry q : Γ(G1 , S1 ) → Γ(G2 , S2 ). By Theorem 4.1 in [BDM09], the map q takes a peripheral left coset g1 P1 of G1 to within a uniform bounded distance of the corresponding peripheral left coset g2 P2 of G2 . In particular, P1 and P2 are quasi-isometric. Using the proof of Theorems 6.3 in [Gro13], we can extend q to the quasi-isometry q̂ : X(G1 , P1 , S1 ) → X(G2 , P2 , S2 ) between cusped spaces such that q̂ restricts to a quasi-isometry embedding on each individual horoball of X(G1 , P1 , S1 ) and the image of the horoball lies in some neighborhood of a horoball of X(G2 , P2 , S2 ). Therefore, there is a homeomorphism f induced by q̂ from ON BOUNDARIES OF RIGHT-ANGLED COXETER GROUPS 8 ∂(G1 , P1 ) to ∂(G2 , P2 ) that maps the set of parabolic points of ∂(G1 , P1 ) bijectively onto the set of parabolic points of ∂(G2 , P2 ). Moreover, if a parabolic point v of ∂(G1 , P1 ) is labelled by some peripheral left coset g1 P1 in G1 and the parabolic point f (v) of ∂(G2 , P2 ) is labelled by some peripheral left coset g2 P2 in G2 , then by the above observation P1 and P2 are quasiisometric.  Definition 2.13 ([Bow01]). Let G be a group. By a splitting of G, over a given class of subgroups, we mean a presentation of G as a finite graph of groups, where each edge group belongs to this class. Such a splitting is said to be relative to another class P of subgroups if each element of P is conjugate into one of the vertex groups. A splitting is said to be trivial if there exists a vertex group equal to G. Assume G is hyperbolic to a collection P. A peripheral splitting of (G, P) is a representation of G as a finite bipartite graph of groups, where P consists precisely of the (conjugacy classes of) vertex groups of one color. Obviously, any peripheral splitting of (G, P) is relative to P and over subgroups of elements of P. Peripheral splittings of (G, P) are closely related to cut points in the Bowditch boundary ∂(G, P) ([Bow01]). Definition 2.14. Given a compact connected metric space X, a point x ∈ X is a global cut point (or just simply cut point) if X − {x} is not connected. If {a, b} ⊂ X contains no cut points and X − {a, b} is not connected, then {a, b} is a cut pair. A point x ∈ X is a local cut point if X − {x} is not connected, or X − {x} is connected and has more than one end. 2.3. CAT(0) spaces with isolated flats. In this section, we discuss the work of Hruska-Kleiner [HK05] on CAT(0) spaces with isolated flats. Definition 2.15. A k–flat in a CAT(0) space X is an isometrically embedded copy of Euclidean space Ek for some k ≥ 2. In particular, note that a geodesic line is not considered to be a flat. Definition 2.16. Let X be a CAT(0) space, G a group acting geometrically on X, and F a G–invariant set of flats in X. We say that X has isolated flats with respect to F if the following two conditions hold. (1) There is a constant D such that every flat F ⊂ X lies in a D– neighborhood of some F ′ ∈ F. (2) For each positive r < ∞ there is a constant ρ = ρ(r) < ∞ so that
for any two distinct flats F, F ′ ∈ F we have diam Nr (F ) ∩ Nr (F ) < ρ. We say X has isolated flats if it has isolated flats with respect to some G–invariant set of flats. Theorem 2.17 ([HK05]). Suppose X has isolated flats with respect to F. For each F ∈ F the stabilizer StabG (F ) is virtually abelian and acts cocompactly on F . The set of stabilizers of flats F ∈ F is precisely the set of maximal virtually abelian subgroups of G of rank at least two. These stabilizers lie in only finitely many conjugacy classes ON BOUNDARIES OF RIGHT-ANGLED COXETER GROUPS 9 Theorem 2.18 ([HK05]). Let X have isolated flats with respect to F. Then G is relatively hyperbolic with respect to the collection of all maximal virtually abelian subgroups of rank at least two. The previous theorem also has the following converse. Theorem 2.19 ([HK05]). Let G be a group acting geometrically on a CAT(0) space X. Suppose G is relatively hyperbolic with respect to a family of virtually abelian subgroups. Then X has isolated flats A group G that admits an action on a CAT(0) space with isolated flats has a “well-defined” CAT(0) boundary, often denoted by ∂G, by the following theorem. Theorem 2.20 ([HK05]). Let G act properly, cocompactly, and isometrically on CAT(0) spaces X and Y . If X has isolated flats, then so does Y , and there is a G–equivariant homeomorphism ∂X → ∂Y . 2.4. Right-angled Coxeter groups and their relatively hyperbolic structures. In this section, we review the concepts of right-angled Coxeter groups and Davis complexes. We also review the work of Caprace [Cap09, Cap15] and Behrstock-Hagen-Sisto [BHS17] on peripheral structures of relatively hyperbolic right-angled Coxeter groups. Definition 2.21. Given a finite simplicial graph Γ, the associated rightangled Coxeter group GΓ is generated by the set S of vertices of Γ and has relations s2 = 1 for all s in S and st = ts whenever s and t are adjacent vertices. Let S1 be a subset of S. The subgroup of GΓ generated by S1 is a rightangled Coxeter group GΓ1 , where Γ1 is the induced subgraph of Γ with vertex set S1 (i.e. Γ1 is the union of all edges of Γ with both endpoints in S1 ). The subgroup GΓ1 is called a special subgroup of GΓ . Definition 2.22. Given a finite simplicial graph Γ, the associated Davis complex ΣΓ is a cube complex constructed as follows. For every k–clique, T ⊂ Γ, the special subgroup GT is isomorphic to the direct product of k copies of Z2 . Hence, the Cayley graph of GT is isomorphic to the 1–skeleton of a k–cube. The Davis complex ΣΓ has 1–skeleton the Cayley graph of GΓ , where edges are given unit length. Additionally, for each k–clique, T ⊂ Γ, and coset gGT , we glue a unit k–cube to gGT ⊂ ΣΓ . The Davis complex ΣΓ is a CAT(0) space and the group GΓ acts properly and cocompactly on the Davis complex ΣΓ (see [Dav08]). Theorem 2.23 (Theorem A’ in [Cap09, Cap15]). Let Γ be a simplicial graph and J be a collection of induced subgraphs of Γ. Then the right-angled Coxeter groups GΓ is hyperbolic relative to the collection P = { GJ | J ∈ J } if and only if the following three conditions hold: (1) If σ is an induced 4-cycle of Γ, then σ is an induced 4-cycle of some J ∈ J. ON BOUNDARIES OF RIGHT-ANGLED COXETER GROUPS 10 (2) For all J1 , J2 in J with J1 6= J2 , the intersection J1 ∩ J2 is empty or J1 ∩ J2 is a complete subgraph of Γ. (3) If a vertex s commutes with two non-adjacent vertices of some J in J, then s lies in J. Theorem 2.24 (Theorem B in [Cap09, Cap15]). Let Γ be a simplicial graph. If GΓ is relatively hyperbolic with respect to finitely generated subgroups H1 , · · · , Hm , then each Hi is conjugate to a special subgroup of GΓ . Theorem 2.25 (Theorem I in [BHS17]). Let T be the class consisting of the finite simplicial graphs Λ such that GΛ is strongly algebraically thick. Then for any finite simplicial graph Γ either: Γ ∈ T , or there exists a collection J of induced subgraphs of Γ such that J ⊂ T and GΓ is hyperbolic relative to the collection P = { GJ | J ∈ J } and this peripheral structure is minimal. Remark 2.26. In Theorem 2.25 we use the notion of strong algebraic thickness which is introduced in [BD14] and is a sufficient condition for a group to be non-hyperbolic relative to any collection of proper subgroups. We refer the reader to [BD14] for more details. The following theorem from [BHS17] characterizes all strongly algebraically thick right-angled Coxeter groups and it will prove useful for studying peripheral subgroups of relatively hyperbolic right-angled Coxeter groups. Theorem 2.27 (Theorem II in [BHS17]). Let T be the class of finite simplicial graphs whose corresponding right-angled Coxeter groups are strongly algebraically thick. Then T is the smallest class of graphs satisfying the following conditions: (1) The 4-cycle lies in T . (2) Let Γ ∈ T and let Λ ⊂ Γ be an induced subgraph which is not a complete graph. Then the graph obtained from Γ by coning off Λ is in T . (3) Let Γ1 , Γ2 ∈ T and suppose there exists a graph Γ, which is not a complete graph, and which arises as a subgraph of each of the Γi . Then the union Λ of Γ1 , Γ2 along Γ is in T , and so is any graph obtained from Λ by adding any collection of edges joining vertices in Γ1 − Γ to vertices of Γ2 − Γ. Definition 2.28. The graphs in the collection T from Theorems 2.25 and 2.27 are called thick graphs. Let Γ be a simplicial, non-thick graph. The collection J of induced subgraphs of Γ from Theorem 2.25 is called a peripheral structure of Γ and each graph in J is called a peripheral subgraph of Γ. A peripheral structure J of a graph Γ is non-trivial if J is non-empty and does not contain Γ. A graph Γ has a non-trivial peripheral structure if and only if GΓ is non-trivially relative hyperbolic group but GΓ is not hyperbolic. ON BOUNDARIES OF RIGHT-ANGLED COXETER GROUPS 11 3. Bowditch boundaries of relatively hyperbolic right-angled Coxeter groups In this section, we give “visual” descriptions of cut points and nonparabolic cut pairs of Bowditch boundaries of relatively hyperbolic rightangled Coxeter groups. The Bowditch boundary is a quasi-isometry invariant. So, these results can be applied, in certain cases, to differentiate two relatively hyperbolic RACGs in terms of quasi-isometry equivalence. In [Tra13], the third author investigates the connection between the Bowditch boundary of a relatively hyperbolic group (G, P) and the boundary of a CAT(0) space X on which G acts geometrically. For relatively hyperbolic right-angled Coxeter groups, the relevant result from [Tra13] can be stated as follows: Theorem 3.1 (Tran [Tra13]). Let Γ be a finite simplicial graph. Assume that the right-angled Coxeter group GΓ is relatively hyperbolic with respect to a collection P of its subgroups. Then the Bowditch boundary ∂(GΓ , P) is obtained from the CAT(0) boundary ∂ΣΓ by identifying the limit set of each peripheral left coset to a point. Moreover, this quotient map is GΓ -invariant. We now introduce some definitions concerning defining graphs of rightangled Coxeter groups that we will use to visualize cut points and nonparabolic cut points in the Bowditch boundary. Definition 3.2. Let Γ1 and Γ2 be two graphs, the join of Γ1 and Γ2 , denoted Γ1 ∗Γ2 , is the graph obtained by connecting every vertex of Γ1 to every vertex of Γ2 by an edge. If Γ2 consists of distinct vertices u and v, then the join Γ1 ∗ {u, v} is the suspension of Γ1 . Definition 3.3. Let Γ be a simplicial graph. A pair of non-adjacent vertices {a, b} in Γ is called a cut pair if {a, b} separates Γ. An induced subgraph Γ1 of Γ is a complete subgraph suspension if Γ1 is a suspension of a complete subgraph σ of Γ. If σ is a single vertex, then Γ1 is a vertex suspension. An induced subgraph Γ1 of Γ is separating if Γ1 separates Γ. By this way, we can also consider a cut pair as a separating complete subgraph suspension which is a suspension of the empty graph. We will need the following lemma in order to visualize cut points in the Bowditch boundary of a relatively hyperbolic right-angled Coxeter group. Lemma 3.4. Let Γ be a simplicial graph and J a collection of induced proper subgraphs of Γ. Assume that the right-angled Coxeter groups GΓ is oneended, hyperbolic relative to the collection P = { GJ | J ∈ J }, and suppose each subgroup in P is one-ended. Let J0 be an element in J such that some induced subgraph of J0 separates the graph Γ. Then we can write Γ = Γ1 ∪Γ2 such that the following conditions hold: (1) Γ1 , Γ2 are both proper induced subgraphs of Γ; (2) Γ1 ∩ Γ2 is an induced subgraph of J0 . ON BOUNDARIES OF RIGHT-ANGLED COXETER GROUPS 12 (3) For each J in J, J lies completely inside either Γ1 or Γ2 Proof. Since each subgroup in P is one-ended, each graph J in J is connected and J has no separating complete subgraph. Let L be an induced subgraph of J0 that separates the graph Γ. Let Γ′1 be L together with some of the components of Γ − L, and let Γ′2 be L together with the remaining components of Γ − L. Then, Γ′1 , Γ′2 are both proper induced subgraphs of Γ, L = Γ′1 ∩ Γ′2 , and Γ = Γ′1 ∪ Γ′2 . Since J0 is a proper subgraph of Γ, Γ′1 − J0 6= ∅ or Γ′2 − J0 6= ∅ (say Γ′1 − J0 6= ∅). Let Γ1 = Γ′1 , Γ2 = Γ′2 ∪ J0 . Then, Γ1 is an induced proper subgraph of Γ. We now prove that Γ2 is also an induced proper subgraph. Choose a vertex w ∈ Γ1 − J0 = Γ′1 − J0 . Since Γ1 ∩ Γ2 = Γ′1 ∩ (Γ′2 ∪ J0 ) ⊂ J0 , the vertex w does not belong to Γ2 . Therefore, Γ2 is a proper subgraph of Γ. We now prove that Γ2 is induced. Let e be an arbitrary edge with endpoints u and v in Γ2 . If e is an edge of Γ′2 , then e is the edge of Γ2 . Otherwise, e is an edge of Γ′1 = Γ1 , because Γ = Γ′1 ∪ Γ′2 . In particular, u and v are also the vertices of Γ1 . Again, Γ1 ∩ Γ2 is a subgraph of J0 . Then u and v are vertices of J0 . Therefore, e is an edge of J0 , because J0 is an induced subgraph. Thus, e is also an edge of Γ2 . Thus, Γ2 is an induced subgraph. This implies that Γ1 ∩ Γ2 is an induced subgraph. We already checked that Γ1 ∩ Γ2 is a subgraph of J0 , and by construction Γ = Γ1 ∪ Γ2 . We now prove that for each J in J, J lies completely inside either Γ1 or Γ2 . By the construction, J0 is a subgraph of Γ2 . Therefore, we only need to check the case where J 6= J0 . It suffices to show that J lies completely inside either Γ′1 or Γ′2 . By Theorem 2.23, for each J 6= J0 in J the intersection J ∩J0 is empty or it is a complete subgraph of Γ. Also the intersection J ∩ L is an induced subgraph of J ∩ J0 if J ∩ L 6= ∅. Therefore, J ∩ L is empty or it is a complete subgraph of Γ. Recall that each graph in J is connected and has no separating complete subgraph. Therefore, J − L = J − (J ∩ L) is connected for each J 6= J0 in J. By the construction J − L lies completely inside either Γ′1 or Γ′2 . Thus, J also lies completely inside either Γ′1 or Γ′2 . Therefore, J lies completely inside either Γ1 or Γ2 .  We can now give the visual description of cut points in the Bowditch boundary of relatively hyperbolic right-angled Coxeter groups. Theorem 3.5. Let Γ be a simplicial graph and J a collection of induced proper subgraphs of Γ. Assume that the right-angled Coxeter groups GΓ is one-ended, hyperbolic relative to the collection P = { GJ | J ∈ J }, and suppose each subgroup in P is also one-ended. Then each parabolic point vgGJ0 is a global cut point if and only if some induced subgraph of J0 separates the graph Γ. Proof. We first assume that some induced subgraph of J0 separates the graph Γ. By Lemma 3.4, we can write Γ = Γ1 ∪ Γ2 such that the following conditions hold: (1) Γ1 , Γ2 are both proper induced subgraphs of Γ; ON BOUNDARIES OF RIGHT-ANGLED COXETER GROUPS 13 (2) K = Γ1 ∩ Γ2 is an induced subgraph of J0 . (3) For each J in J, J lies completely inside either Γ1 or Γ2 This implies that GΓ = GΓ1 ∗GK GΓ2 , GΓ1 6= GΓ , and GΓ2 6= GΓ . Since J lies completely inside either Γ1 or Γ2 for each J ∈ J, each peripheral subgroup in P must be a subgroup of GΓ1 or GΓ2 . Therefore, GΓ splits non-trivially relative to P over the parabolic subgroup GK ≤ GJ0 . By the claim following Theorem 1.2 of [Bow01] the parabolic point vGJ0 labelled by GJ0 is a global cut point of ∂(G, P). Also, the group GΓ acts topologically on ∂(G, P) and gvGJ0 = vgGJ0 . Thus, each parabolic point vgGJ0 is also a global cut point. We now assume that some parabolic point vgGJ0 is a global cut point. Again, the group GΓ acts topologically on ∂(G, P) and gvGJ0 = vgGJ0 . Therefore, vGJ0 is also a global cut point. So, by Theorem 3.3 of [Haub] GΓ splits non-trivially over a subgroup H of GJ0 . Theorem 1 of [MT09] implies that there is some induced subgraph K of Γ which separates Γ such that GK is contained in some conjugate of H. Therefore, GK is also contained in some conjugate of the peripheral subgroup GJ0 . Moreover, GK and GJ0 are both special subgroups of GΓ . Thus, K is an induced subgraph of J0 . (This is a standard fact, and we leave the details to the reader.)  The following corollary follows directly from Theorems 2.12 and 3.5. This corollary can be used to distinguish between quasi-isometry classes of certain relatively hyperbolic right-angled Coxeter groups. Corollary 3.6. Let Γ1 and Γ2 be simplicial graphs such that GΓ1 and GΓ2 are both one-ended. Assume that each graph Γi has a peripheral structure Ji that consists of proper subgraphs of Γi such that each subgroup in Pi = { GJ | J ∈ Ji } is one-ended and not relatively hyperbolic. If GΓ1 and GΓ2 are quasi-isometric, then for each graph K ∈ J1 there is a graph L ∈ J2 such that GK and GL are quasi-isometric and vice versa. Moreover, if K has some induced subgraph that separates Γ1 , then L also has some induced subgraph that separates Γ2 . We now discuss a few examples related to cut points in Bowditch boundaries of relatively hyperbolic right-angled Coxeter groups. These examples illustrate an application of Theorem 3.5 to the problem of quasi-isometry classification of right-angled Coxeter groups. Example 3.7. Let Γ1 , Γ2 , and Γ3 be the graphs in Figure 1. Observe that all groups GΓi are one-ended. We will prove that groups GΓ1 , GΓ2 , and GΓ3 are not pairwise quasi-isometric by investigating their peripheral structures. (1) (2) In Γ1 , let K1 and K1 be an induced subgraphs generated by {a1 , a2 , a3 , a4 , a5 } and {a6 , a7 , a8 , a9 , a10 }, respectively. It is easy to see that Γ1 has only six (1) (2) induced 4-cycles which are not subgraphs of K1 and K1 . Denote these (i) (i) (j) cycles by L1 (i = 1, 2, · · · , 6). Let J1 be the set of all graphs L1 and K1 . By Theorems 2.23 and 2.25, J1 is a peripheral structure of Γ1 . Moreover, ON BOUNDARIES OF RIGHT-ANGLED COXETER GROUPS 14 Γ2 Γ1 a2 Γ3 c2 b2 a3 a1 a5 b3 b1 a4 c4 b4 a7 c7 b7 a8 a6 c5 c1 b5 a10 b8 b6 a9 b10 b9 c8 c6 c9 Figure 1. The three groups GΓ1 , GΓ2 , and GΓ3 are pairwise not quasi-isometric because the Bowditch boundaries with respect to their minimal peripheral structures are pairwise not homeomorphic. no induced subgraph of a graph in J1 separates Γ1 . Therefore by Theorem 3.5, GΓ1 is hyperbolic relative to the collection P1 = { GJ | J ∈ J1 } and the Bowditch boundary ∂(GΓ1 , P1 ) has no global cut point. (1) (2) Similarly, let K2 and K2 be an induced subgraphs of Γ2 generated by {b1 , b2 , b3 , b4 , b5 } and {b6 , b7 , b8 , b9 , b10 }, respectively. It is easy to see that Γ2 (i) has only four induced 4-cycles, denoted L2 (i = 1, 2, · · · , 4), such that each (1) (2) of them is not a subgraph of K2 and K2 . Let J2 be the set of all graphs (i) (j) L2 and K2 . Then by Theorems 2.23 and 2.25, J2 is a peripheral structure (1) (2) of Γ2 . Moreover, K2 and K2 are the only graphs in J2 which contain induced subgraphs that separate Γ2 . Therefore, GΓ2 is hyperbolic relative to the collection P2 = { GJ | J ∈ J2 }, the Bowditch boundary ∂(GΓ2 , P2 ) has global cut points and each of them is labelled by some left coset of GK (1) 2 or GK (2) by Theorem 3.5. 2 (1) Finally, let K3 be an induced subgraph of Γ3 generated by {c6 , c7 , c8 , c9 , c10 }. (i) It is easy to see that Γ3 has only three induced 4-cycles, denoted L3 (1) (i = 1, 2, 3), such that each of them is not a subgraph of K3 . Assume (1) that L3 is the induced 4-cycle generated by {c1 , c2 , c5 , c4 }. Let J3 be the (i) (1) set of all graphs L3 and K3 . Again, by Theorem 2.23 and 2.25 we have (1) (1) that J3 is a peripheral structure of Γ3 . Moreover, K3 and L3 are the only graphs in J3 which contain induced subgraphs that separate Γ3 . Therefore, GΓ3 is hyperbolic relative to the collection P3 = { GJ | J ∈ J3 }, the Bowditch boundary ∂(GΓ3 , P3 ) has global cut points and each of them is labelled by some left coset of GK (1) or GL(1) by Theorem 3.5. 3 3 c10 ON BOUNDARIES OF RIGHT-ANGLED COXETER GROUPS 15 Note that all the groups in Pi are one-ended. The Bowditch boundary ∂(GΓ1 , P1 ) has no global cut point, but the Bowditch boundaries ∂(GΓ2 , P2 ) and ∂(GΓ3 , P3 ) do. So, GΓ1 cannot be quasi-isometric to GΓ2 and GΓ3 . Additionally, the Bowditch boundary ∂(GΓ3 , P3 ) has global cut points labelled by some left coset of GL(1) . Meanwhile, no global cut point of the 3 Bowditch boundary ∂(GΓ2 , P2 ) is labelled by the left coset of a peripheral subgroup that is quasi-isometric to GL(1) . Therefore, GΓ2 and GΓ3 are not 3 quasi-isometric. In the remainder of this section, we work on the description of nonparabolic cut pairs in Bowditch boundaries of relatively hyperbolic rightangled Coxeter groups in terms of their defining graphs. Proposition 3.8. Let Γ be a simplicial graph and J be a collection of induced proper subgraphs of Γ. Assume that the right-angled Coxeter groups GΓ is one-ended, hyperbolic relative to the collection P = { GJ | J ∈ J }, and suppose each subgroup in P is also one-ended. If Γ has a separating complete subgraph suspension whose non-adjacent vertices do not lie in the same subgraph J ∈ J, then the CAT(0) boundary ∂ΣΓ has a cut pair and the Bowditch boundary ∂(GΓ , P) has a non-parabolic cut pair. Proof. Let K be a separating complete subgraph suspension of Γ whose nonadjacent vertices u and v do not both lie in the same subgraph J ∈ J. Let T be the set of all vertices of Γ which are both adjacent to u and v. Then T is a vertex set of a complete subgraph σ of Γ. Otherwise, the two vertices u and v both lie in the same 4-cycle. Thus, u and v lie in the same subgraph J ∈ J, a contradiction. Let K = σ ∗ {u, v}. We can easily verify the following properties of K. (1) For each J ∈ J the intersection K ∩ J is empty or a complete subgraph. (2) No vertex outside K is adjacent to the unique pair of nonadjacent vertices {u, v} of K. (3) K is an induced subgraph of K. Therefore, the collection J = J ∪ {K} satisfies all the conditions of Theorem 2.23, which implies that GΓ is hyperbolic relative to the collection P = { GJ | J ∈ J }. Using an argument similar to that of Lemma 3.4, we can write Γ = Γ1 ∪Γ2 such that the following conditions hold: (1) Γ1 , Γ2 are both proper induced subgraphs of Γ; (2) Γ1 ∩ Γ2 is an induced subgraph L of K. (3) For each J in J, J lies completely inside either Γ1 or Γ2 Therefore, we can prove that the Bowditch boundary ∂(G, P) has a global cut point vGK stabilized by the subgroup GK by using an argument similar to the one in Theorem 3.5. ON BOUNDARIES OF RIGHT-ANGLED COXETER GROUPS 16 By Theorem 3.1, the Bowditch boundary ∂(GΓ , P) is obtained from the CAT(0) boundary ∂ΣΓ by identifying the limit set of each peripheral left coset of a subgroup in P to a point. Let f be this quotient map. Since GK is two-ended, its limit set consists of two points w
1 and w2 in ∂ΣΓ . Therefore, f (w1 ) = f (w2 ) = vGK and f ∂ΣΓ − {w1 , w2 } = ∂(GΓ , P) − {vGK }. Since ∂(GΓ , P) − {vGK } is not connected, the space ∂ΣΓ − {w1 , w2 } is also not connected. This implies that {w1 , w2 } is a cut pair of the CAT(0) boundary ∂ΣΓ . Again, by Theorem 3.1 the Bowditch boundary ∂(GΓ , P) is obtained from the CAT(0) boundary ∂ΣΓ by identifying limit set of each peripheral left coset of a subgroup in P to a point. Let g be this quotient map. The two points w1 and w2 do not lie in limit sets of peripheral left cosets of subgroups in P. Therefore, g(w1 ) 6= g(w2 ) and they are non-parabolic
points in the Bowditch boundary ∂(GΓ , P). Moreover, g ∂ΣΓ − {w1 , w2 } = ∂(GΓ , P) − {g(w1 ), g(w2 )} and limit set of each peripheral left coset of a subgroup in P lies completely inside ∂ΣΓ − {w1 , w2 }. We observe that for any two points s1 , s2 ∈ ∂ΣΓ − {w1 , w2 } satisfying g(s1 ) = g(s2 ) the two points s1 and s2 both lie in some limit set C of a peripheral left coset of a subgroup in P. Also, each subgroup in P is oneended, so C is connected. Therefore, s1 and s2 lie in the same connected component of ∂ΣΓ − {w1 , w2 }. This implies that if U and V are different components of ∂ΣΓ −{w1 , w2 }, then g(U )∩g(V ) = ∅. Therefore, ∂(GΓ , P)− {g(w1 ), g(w2 )} is not connected. This implies that {g(w1 ), g(w2 )} is a nonparabolic cut pair of the Bowditch boundary ∂(GΓ , P).  Theorem 3.9. Let Γ be a simplicial graph and J be a collection of induced proper subgraphs of Γ. Assume that the right-angled Coxeter groups GΓ is one-ended and hyperbolic relative to the collection P = { GJ | J ∈ J }, and suppose each subgroup in P is one-ended. If the Bowditch boundary ∂(GΓ , P) has a non-parabolic cut pair, then Γ has a separating complete subgraph suspension. Moreover, if Γ has a separating complete subgraph suspension whose non-adjacent vertices do not lie in the same subgraph J ∈ J, then the Bowditch boundary ∂(GΓ , P) has a non-parabolic cut pair. Proof. Since GΓ is one-ended, the Bowditch boundary ∂(GΓ , P) is connected. If the Bowditch boundary ∂(GΓ , P) is a circle, then GΓ is virtually a surface group, and the peripheral subgroups are the boundary subgroups of that surface by Theorem 6B in [Tuk88]. This is a contradiction because each peripheral subgroup is one-ended. Therefore, the Bowditch boundary ∂(GΓ , P) is not a circle. We now assume that the Bowditch boundary ∂(GΓ , P) has a non-parabolic cut pair {u, v}. Then u is obviously a non-parabolic local cut point. Therefore by Theorem 1.1 in [Haub], GΓ splits over a two-ended subgroup H. Since GΓ splits over a two-ended subgroup H, there is an induced subgraph K of Γ which separates Γ such that GK is contained in some conjugate of H by Theorem 1 in [MT09]. Because the group GΓ is one-ended, the group ON BOUNDARIES OF RIGHT-ANGLED COXETER GROUPS u Γ1 17 v Γ2 Figure 2. Graph Γ1 has no separating complete subgraph suspension while graph Γ2 has cut pair (u, v) such that u and v do not lie in the same 4-cycle. GK is two-ended. This implies that K is a complete subgraph suspension. The remaining conclusion is obtained from Proposition 3.8.  Example 3.10. Let Γ1 and Γ2 be the graphs in Figure 2. Then GΓ1 and GΓ2 are both one-ended. Let J1 and J2 be the sets of all induced 4-cycles of Γ1 and Γ2 , respectively. By Theorems 2.23 and 2.25, the collection Ji is a peripheral structure of Γi for each i. Also, subgroups in each Pi = { GJ | J ∈ Ji } are virtually Z2 and thus one-ended. Moreover, for each i no induced subgraph of a graph in Ji separates Γi . Thus by Theorem 3.5, both Bowditch boundaries ∂(GΓ1 , P1 ) and ∂(GΓ2 , P2 ) have no cut points. So in this case, we cannot use cut points to differentiate GΓ1 and GΓ2 up to quasi-isometry. However, the graph Γ1 has no separating complete subgraph suspension, so by Theorem 3.9 the Bowditch boundary ∂(GΓ1 , P1 ) has no non-parabolic cut pair. Meanwhile, Γ2 has a cut pair (u, v) such that u and v do not lie in the same subgraph in J2 . Again, by Theorem 3.9 the Bowditch boundary ∂(GΓ2 , P2 ) has a non-parabolic cut pair. By Theorem 2.12, GΓ1 and GΓ2 are not quasi-isometric. 4. Relatively hyperbolic right-angled Coxeter groups with Sierpinski carpet or sphere Bowditch boundary In this section, we give a necessary and sufficient conditions for a relatively hyperbolic right-angled Coxeter group whose defining graph has planar flag complex to have the Sierpinski carpet or sphere as its Bowditch boundary. We begin by recalling topological descriptions of the Sierpinski carpet and Menger curve due to Whyburn [Why58] and Anderson [And58a, And58b], respectively. Anderson’s result will be used in Section 5 to study nonhyperbolic right-angled Coxeter groups with Menger curve CAT(0) boundary. ON BOUNDARIES OF RIGHT-ANGLED COXETER GROUPS 18 Theorem 4.1 ([Why58], [And58a, And58b]). A compact space Σ is homeomorphic to the Sierpinski carpet if and only if it is 1–dimensional, planar, connected, locally connected, and has no local cut points. Let Σ be the Sierpinski carpet, then the non-separating embedded circles (called peripheral circles) are pairwise disjoint, and form an infinite countable set. A compact space Σ is homeomorphic to the Menger curve if and only if it is 1–dimensional, connected, locally connected, has no local cut points, and no non-empty open subset of Σ is planar. We begin by introducing some terminology that we will use to characterize a certain class of relatively hyperbolic right-angled Coxeter groups whose boundaries are the Sierpinski carpet or 2-sphere. Some of these terms will also be used in Section 5. Definition 4.2. Let Γ be a finite simplicial graph. The flag complex of Γ is the simplicial complex with 1-skeleton Γ and any complete subgraph of Γ is the 1-skeleton of some simplex. Obviously, if Γ is triangle free, the flag complex of Γ is Γ itself. Definition 4.3. A graph is inseparable if it is connected, has no separating complete subgraph, no cut pair, and no separating complete subgraph suspension. Obviously, a triangle free graph is inseparable if and only if it is connected, has no separating vertex, no separating edge, no cut pair, and no separating vertex suspension. Definition 4.4. A graph K is an n–cycle extension if K is an n–cycle or K is a join graph between an n–cycle and a complete graph. Note that if K is an n–cycle extension subgraph of a triangle free graph, then K is an n–cycle. An induced n–cycle extension subgraph K is strongly non-separating if no induced subgraph of K separates the ambient graph. Theorem 4.5 (Theorem A.2 in [DT]). Let K be a finite simplicial graph. Then the right-angled Coxeter group GK is a hyperbolic virtual surface group if and only if K is an n–cycle extension (n ≥ 5). The following lemma is an easy exercise and is left to the reader. Lemma 4.6. Let K be a finite simplicial graph. Then the right-angled Coxeter group GK is a virtually Z2 group if and only if K is a 4–cycle extension. We first prove the sufficient conditions of the characterization (see Theorem 1.4). We recall Theorem 3.1 that the Bowditch boundary of a relatively hyperbolic right-angled Coxeter group GΓ is obtained from the CAT(0) boundary of the Davis complex ΣΓ by identifying the limit set of each peripheral left coset to a point. Therefore, we recall the following result due to Świa̧tkowski [Ś] about CAT(0) boundaries of right-angled Coxeter groups. We then investigate the limit set of each peripheral left coset in the CAT(0) boundary and compute the quotient space ∂(GΓ , P). ON BOUNDARIES OF RIGHT-ANGLED COXETER GROUPS 19 Theorem 4.7 (Corollary 1.4 in [Ś]). Let Γ be a graph whose flag complex is planar. Then the CAT(0) boundary ∂ΣΓ is the Sierpinski carpet if and only if Γ is inseparable, distinct from a complete graph, and the flag complex of Γ is distinct from a triangulation of S 2 . Lemma 4.8. Let Γ be a graph whose flag complex is planar. Assume that the graph Γ is inseparable and Γ has a non-trivial peripheral structure J that consists of induced 4-cycle extension subgraphs. Let K be a strongly nonseparating induced 4-cycle extension or K a strongly non-separating induced n–cycle extension (n ≥ 5) with the following properties: (1) K does not contain any nonadjacent vertices of an induced 4-cycle extension; (2) No vertex outside K is adjacent to two non-adjacent vertices of K. Then the limit set of the subgroup GK is a peripheral circle in the Sierpinski carpet ∂ΣΓ . Proof. Let J = J ∪ {K} and P = { GJ | J ∈ J }. By Theorem 2.23, the group GΓ is relatively hyperbolic with respect to the collection P. We assume that the limit set of the subgroup GK is a separating circle C of the Sierpinski carpet ∂ΣΓ . By Theorem 3.1, the Bowditch boundary ∂(GΓ , P) is obtained from the CAT(0) boundary ∂ΣΓ by identifying the limit set of each peripheral left coset of a subgroup in P to a point. Let f be this quotient map. Let vGK be the point in ∂(GΓ , P) that is image of C under the map f . Since no induced subgraph of K separates Γ, the point vGK is not a global cut point of ∂(GΓ , P) by Theorem 3.5. We observe that for any two points u, v ∈ ∂ΣΓ − C satisfying f (u) = f (v) the two points u and v both lie in some circle C1 ⊂ ∂ΣΓ − C that is the limit set of some peripheral subgroup. Therefore, u and v lie in the same connected component of ∂ΣΓ − C. This implies that if U and V are different components of ∂ΣΓ −C, then f (U )∩f (V ) = ∅. Therefore, vGK is the global cut point which is a contradiction. Thus, the limit set of the subgroup GK is a peripheral circle in the Sierpinski carpet ∂ΣΓ .  Proposition 4.9. Let Γ be a graph whose flag complex is planar. Assume that Γ is inseparable and Γ has a non-trivial peripheral structure J that consists of strongly non-separating induced 4-cycle extension graphs. Then the group GΓ is relatively hyperbolic with respect to the collection P = { GJ | J ∈ J } and the Bowditch boundary ∂(GΓ , P) is S2 or the Sierpinski carpet. Proof. By Theorem 4.7, the CAT(0) boundary ∂ΣΓ of the Davis complex ΣΓ is the Sierpinski carpet. Also, by Lemma 4.8 the limit set of each peripheral left coset is a peripheral circle of the Sierpinski carpet ∂ΣΓ . By Theorem 3.1, the Bowditch boundary ∂(GΓ , P) is obtained from the CAT(0) boundary ∂ΣΓ by identifying the limit set of each peripheral left coset to a point. We now consider two cases. If all peripheral circles of the Sierpinski ∂ΣΓ are limit sets of peripheral left cosets, then the Bowditch boundary ∂(GΓ , P) ON BOUNDARIES OF RIGHT-ANGLED COXETER GROUPS 20 is S2 . Otherwise, it is clear that the Bowditch boundary ∂(GΓ , P) is still the Sierpinski carpet.  We now work on the necessary conditions for the characterization (see Theorem 1.4). Theorem 4.10 (Theorem 0.3 and Corollary 0.4 in [Dah05]). If (G, P) is a relatively hyperbolic group whose boundary is the Sierpinski carpet or S2 , then each subgroup in P is virtually a surface group. Moreover, if (G, P) is a minimal relatively hyperbolic structure, then each subgroup in P is virtually Z2 . Proposition 4.11. Let Γ be a graph whose flag complex is planar. Assume that Γ has a non-trivial peripheral structure J and let P = { GJ | J ∈ J }. If the Bowditch boundary ∂(GΓ , P) is S2 or the Sierpinski carpet, then Γ is inseparable and each graph in J is a strongly non-separating 4-cycle extension graph. Proof. If the graph Γ is not connected or Γ has a separating complete subgraph, then GΓ is not one-ended. Therefore, the Bowditch boundary ∂(GΓ , P) is not connected which is a contradiction. Thus, the graph Γ is connected and has no separating complete subgraph. By Theorem 4.10, each subgroup in P is virtually Z2 . Therefore by Lemma 4.6, each subgraph in J must be a 4-cycle extension. Also, ∂(GΓ , P) has no global cut point. Then each graph in J is strongly non-separating. Also, there is no cut pair and no separating complete subgraph suspension of the ambient graph Γ which is a subgraph of a graph in J. By way of contradiction, we assume that graph Γ has a cut pair or a separating complete subgraph suspension. Then, the non-adjacent vertices of the cut pair (or of the separating complete subgraph suspension) does not lie in the same J ∈ J. Therefore by Proposition 3.8, the Bowditch boundary ∂(GΓ , P) has a non-parabolic cut pair, a contradiction. Thus, Γ is inseparable.  In the following proposition, we further differentiate between the cases of Sierpinski carpet and S2 Bowditch boundaries. Proposition 4.12. Let Γ be a graph whose flag complex is planar. Assume that Γ is inseparable graph and Γ has a non-trivial peripheral structure J that consists of strongly non-separating induced 4-cycle extension graphs. Let P = { GJ | J ∈ J }. Then the Bowditch boundary ∂(GΓ , P) is the Sierpinski carpet if and only if Γ contains a strongly non-separating induced n–cycle extension K (n ≥ 5) satisfying the following properties: (1) K does not contain any nonadjacent vertices of an induced 4-cycle extension; (2) No vertex outside K is adjacent to two non-adjacent vertices of K. S Proof. Let S be the vertex set of Γ and P = P ∈P P . We first assume that Γ contains a strongly non-separating induced n–cycle extension K (n ≥ 5) ON BOUNDARIES OF RIGHT-ANGLED COXETER GROUPS 21 with all the properties as above. We will prove that the Bowditch boundary ∂(GΓ , P) is the Sierpinski carpet. By Theorem 4.7 and Lemma 4.8, the CAT(0) boundary ∂ΣΓ is the Sierpinski carpet and the limit set of the subgroup GK is a peripheral circle in ∂ΣΓ . This implies that not all peripheral circles of the Sierpinski carpet ∂ΣΓ are limit sets of peripheral left cosets. Therefore, the Bowditch boundary ∂(GΓ , P) is still the Sierpinski carpet as in the proof of Proposition 4.9. We now assume that the Bowditch boundary ∂(GΓ , P) is the Sierpinski carpet. As in the proof of Proposition 4.9, there is some peripheral circle C in ∂ΣΓ which is not the limit set of a peripheral left coset. Therefore, this circle C still survives after identifying the limit set of each peripheral left coset in ∂ΣΓ to a point to obtain the Bowditch boundary ∂(GΓ , P). Moreover, the quotient map is GΓ –equivariant (see Theorem 3.1), so the stabilizers of C via the actions of GΓ on ∂ΣΓ and ∂(GΓ , P) are the same. Call this group Stab(C). By the proof of Proposition 3.7 in [Dah05], Stab(C) acts as a uniform convergence group on C, which is clearly its limit set. Moreover, g Stab(C)g −1 ∩ Stab(C) is finite for each g ∈ / Stab(C). Therefore by Theorem 9.9 of [Hru10], Stab(C) is generated by a finite set T and the inclusion Stab(C) ֒→ GΓ in
duces a quasi-isometric embedding Γ Stab(C), T ֒→ Γ(GΓ , S ∪P). By Theorem 1.5 in [Osi06], GΓ is hyperbolic relative to the collection P∪{Stab(C)}. Therefore by Theorem 2.24, Stab(C) is the conjugate of some special subgroup GK of GΓ . Thus, the group GΓ is relatively hyperbolic with respect to the collection P = P ∪ {GK }, and GK is the stabilizer of some translation C1 of C. Since the group GΓ is relatively hyperbolic with respect to the collection P = P ∪ {GK }, the subgraph K satisfies Condition (1) and (2) of the proposition. Also, the limit set of GK is the peripheral circle C1 in ∂ΣΓ . Arguing as in Proposition 4.9, we have that the Bowditch boundary ∂(GΓ , P) is either S2 or the Sierpinski carpet. In particular, ∂(GΓ , P) has no global cut point. Therefore, no induced subgraph of K separates Γ. Also, GK is virtually a surface group by Theorem 4.10, and GK is not virtually Z2 . Therefore by Theorem 4.5, K is a strongly non-separating n–cycle (n ≥ 5).  The following theorem is obtained from Propositions 4.9, 4.11, and 4.12. With planar condition imposed on the flag complex of defining graphs, this theorem gives a complete characterization of relatively hyperbolic rightangled Coxeter groups whose Bowditch boundaries are the Sierpinski carpet or S2 . Theorem 4.13. Let Γ be a graph whose flag complex is planar. Assume that Γ has a non-trivial peripheral structure J and let P = { GJ | J ∈ J }. The Bowditch boundary ∂(GΓ , P) is S2 or the Sierpinski carpet if and only if Γ is inseparable and each graph in J is a strongly non-separating 4-cycle extension graph. ON BOUNDARIES OF RIGHT-ANGLED COXETER GROUPS Γ1 Γ2 22 Γ3 Figure 3. The three groups GΓ1 , GΓ2 , and GΓ3 all have homeomorphic CAT(0) boundaries (Sierpinski carpet), but they have different Bowditch boundaries with respect to their minimal peripheral structures. In addition, the Bowditch boundary ∂(GΓ , P) is the Sierpinski carpet if and only if Γ contains a strongly non-separating induced n–cycle extension K (n ≥ 5) satisfying the following properties: (1) K does not contain any nonadjacent vertices of an induced 4-cycle extension; (2) No vertex outside K is adjacent to two non-adjacent vertices of K. We now restrict Theorem 4.13 to the case of 2–dimensional right-angled Coxeter groups GΓ (i.e. Γ is triangle free and has at least one edge) to obtain a slightly simpler characterization of relatively hyperbolic right-angled Coxeter groups whose boundaries are the Sierpinski carpet or S2 . Corollary 4.14. Let Γ be a triangle free, planar graph. Assume that Γ has a non-trivial peripheral structure J and let P = { GJ | J ∈ J }. The Bowditch boundary ∂(GΓ , P) is S2 or the Sierpinski carpet if and only if Γ is inseparable and each graph in J is a strongly non-separating 4-cycle. In addition, the Bowditch boundary ∂(GΓ , P) is the Sierpinski carpet if and only if Γ contains an induced strongly non-separating n–cycle K (n ≥ 5) satisfying the following properties: (1) K does not contain any nonadjacent vertices of an induced 4-cycle; (2) No vertex outside K is adjacent to two non-adjacent vertices of K. We now discuss some examples of Bowditch boundaries of relatively hyperbolic right-angled Coxeter groups whose defining graphs are triangle free and planar. Example 4.15. Let Γ1 , Γ2 , and Γ3 be the graphs in Figure 3. Let J1 , J2 , and J3 be the sets of all induced 4-cycles of Γ1 , Γ2 , and Γ3 , respectively. By Theorems 2.23 and 2.25, for each i the collection Ji is a peripheral structure of Γi . Therefore, each group GΓi is hyperbolic relative to the collection Pi = { GJ | J ∈ Ji }. Moreover, each Davis complex ΣΓi is a CAT(0) space with isolated flats and its CAT(0) boundary is the Sierpinski carpet by Theorem 4.7. However, their Bowditch boundaries are pairwise distinct. ON BOUNDARIES OF RIGHT-ANGLED COXETER GROUPS 23 In fact, the Bowditch boundary ∂(GΓ1 , P1 ) is S2 and the Bowditch boundary ∂(GΓ2 , P2 ) is the Sierpinski carpet by Theorem 4.13. In particular, this implies that GΓ1 and GΓ2 are not quasi-isometric. Also, the graph Γ3 has a separating induced 4-cycle. Then, the Bowditch boundary ∂(GΓ3 , P3 ) has global cut points. Therefore, GΓ3 is not quasi-isometric to the other two groups. 5. Non-hyperbolic right-angled Coxeter groups with Menger curve boundary In this section, we study the CAT(0) boundary of 2–dimensional rightangled Coxeter groups with isolated flats. In particular, we give conditions which guarantee that the CAT(0) boundary of such a group will be the Menger curve (see Proposition 5.3). We conclude the section with a pair of examples of right-angled Coxeter groups with Menger Curve CAT(0) boundary and a show in Lemma 5.7 that these examples have non-homeomorphic Bowditch boundaries. Our work in this section is mainly based on the following results of the first author in [Haua] and Mihalik-Tschantz [MT09]. Theorem 5.1 (Theorem 1.2 in [Haua]). Let Γ be a group acting geometrically on a CAT(0) space X with isolated flats. Assume ∂X is 1–dimensional. If Γ does not split over a virtually cyclic subgroup then one of the following holds: (1) ∂X is a circle (2) ∂X is a Sierpinski carpet (3) ∂X is a Menger curve. Theorem 5.2 ([MT09]). Let Γ be a triangle free, connected graph with no separating vertex and no separating edge. The right-angled Coxeter group GΓ splits over a two-ended subgroup H if and only if Γ has a cut pair {u, v} or has a separating vertex suspension σ. Moreover, the special subgroup generated by the cut pair {u, v} or by the separating vertex suspension σ is contained in some conjugate of H. We are now ready for the main proposition of this section. Proposition 5.3. Let Γ be a triangle free, non-planar graph with a nontrivial peripheral structure J that consists of induced 4-cycles, and assume the Γ is inseparable. Let ΣΓ be the Davis complex of the right-angled Coxeter group defined by Γ. If the CAT(0) boundary ∂ΣΓ is not a Sierpinski carpet, then ∂ΣΓ is a Menger curve. Proof. We first observe that ΣΓ is a CAT(0) space with isolated flats and the group GΓ is hyperbolic relative to the collection P = { GJ | J ∈ J }. The Davis complex is one-ended and 2–dimensional, because Γ is triangle free, connected with no separating vertex, and has no separating edge. Therefore, ∂ΣΓ is connected and 1–dimensional (see [GO07]). The graph Γ has no cut ON BOUNDARIES OF RIGHT-ANGLED COXETER GROUPS 24 Ω2 Ω1 Figure 4. The two groups GΩ1 and GΩ2 have homeomorphic CAT(0) boundaries (Menger curve), but they have different Bowditch boundaries with respect to their standard peripheral structures. pair and no separating vertex suspension so by Theorem 5.2 the group GΓ does not split over a two-ended subgroup. Because Γ is inseparable and cannot contain a separating clique, we also have that GΓ does not split over a finite group. Therefore by Theorem 5.1, ∂ΣΓ must be a circle, a Sierpinski carpet, or a Menger curve. Since Γ is not a 4–cycle and Γ has non-trivial peripheral structure J, the CAT(0) boundary ∂ΣΓ contains infinitely many disjoint circles. Thus, ∂ΣΓ is not a circle. By hypothesis ∂ΣΓ is not a Sierpinski carpet, so ∂ΣΓ must be a Menger curve.  We now provide two examples illustrating Proposition 5.3. These examples were inspired by an example from [DHW]. Dani-Haulmark-Walsh take a triple of a 3–manifold glued along a common boundary component to construct the first example of a non-hyperbolic group whose CAT(0) boundary is the Menger curve. We apply this idea to right-angled Coxeter groups, thereby constructing the first examples of non-hyperbolic right-angled Coxeter groups whose CAT(0) boundaries are Menger curve. Construction 5.4. Let K1 , K2 , and K3 be copies of the graph Γ1 in Figure 3. In each graph Ki we fix an induced 4-cycle Ci and glue all 4-cycles Ci to a 4-cycle C to obtain the graph Ω1 (see Figure 4). Let J1 be the collection of all induced 4-cycles of Ω1 . The graph Ω1 is triangle free and inseparable. Using Theorems 2.23 and 2.25, it is not hard to check J1 is a peripheral structure of Ω1 . Next we show that the CAT(0) boundary of ΣΩ1 cannot be the Sierpinski carpet by showing the ∂ΣΩ1 contains a non-planar subspace. ON BOUNDARIES OF RIGHT-ANGLED COXETER GROUPS 25 Lemma 5.5. Let Ω1 be the graph constructed above. Then the CAT(0) boundary ∂ΣΩ1 is the Menger curve. Proof. For each graph Ki let ΣKi be the associated Davis complex in ΣΩ1 that contains the identity e. Then all spaces ΣKi share the Davis complex ΣC . We remind the reader that C is the 4-cycle on which each 4-cycle Ci of the graph Ki is glued, and point out that limit set of ΣC is a circle. Therefore, all limit sets ∂ΣKi meet in the limit set ∂ΣC . By Theorem 4.7 and Lemma 4.8, each limit set ∂ΣKi is a Sierpinski carpet and ∂ΣC is a peripheral circle of ∂ΣKi . Therefore, the union of all limit sets ∂ΣKi is nonplanar (see [DHW]). Since ∂ΣΩ1 contains a non-planar subspace, it is not the Sierpinski carpet. Thus by Proposition 5.3, ∂ΣΩ1 is a Menger curve.  We now construct a variation Ω2 of the graph Ω1 such that the CAT(0) boundaries of the two right-angled Coxeter groups GΩ1 and GΩ2 are homeomorphic, but their Bowditch boundaries with respect to minimal peripheral structures are not. Construction 5.6. Let L1 , L2 , and L3 be copies of the graph Γ2 in Figure 3. In each graph Li we fix a 5-cycle Di and we glue all 5-cycles Di to a 5-cycle D to obtain the graph Ω2 (see Figure 4). Let J2 be the collection of all induced 4-cycles of Ω2 . Again, it is not hard to check that J2 is a peripheral structure of Ω2 and that the graph Ω2 satisfies all the conditions in Proposition 5.3. We now prove that the CAT(0) boundaries of the right-angled Coxeter groups GΩ1 and GΩ2 are the same (the Menger curve), but their Bowditch boundaries ∂(GΩ1 , P1 ) and ∂(GΩ2 , P2 ) are not homeomorphic, where Pi = { GJ | J ∈ Ji }. Lemma 5.7. Let Ω1 and Ω2 be the graphs constructed above. Then, the CAT(0) boundaries of the two right-angled Coxeter groups GΩ1 and GΩ2 are the Menger curve, but their Bowditch boundaries ∂(GΩ1 , P1 ) and ∂(GΩ2 , P2 ) are not homeomorphic where each peripheral structure Pi of GΩi is constructed above. Moreover, the Bowditch boundary ∂(GΩ2 , P2 ) is not homeomorphic to S2 or the Sierpinski carpet. Proof. For each graph Li let ΣLi be the associated Davis complex in ΣΩ2 that contains the identity e, and let ΣD be the associated Davis complex for the common 5-cycle D. The proof that ∂ΣΩ2 is a Menger curve is similar to the proof of Lemma 5.5. We note that by Lemma 4.8 the limit set ∂ΣD is a peripheral circle in each Sierpinski carpet ∂ΣLi . However, the limit set ∂ΣD is not the limit set of a peripheral left coset because D is not an induced 4–cycle of Ω2 . We now claim that the Bowditch boundaries ∂(GΩ1 , P1 ) and ∂(GΩ2 , P2 ) are not homeomorphic, where Pi = { GJ | J ∈ Ji }. Indeed, the Bowditch boundary ∂(GΩ1 , P1 ) has a global cut point since the induced 4-cycle C separates the graph Ω1 . Meanwhile, there is no induced subgraph of an ON BOUNDARIES OF RIGHT-ANGLED COXETER GROUPS 26 induced 4–cycle in Ω2 that separates Ω2 . This implies the Bowditch boundary ∂(GΩ2 , P2 ) has no global cut point. Therefore, the Bowditch boundaries ∂(GΩ1 , P1 ) and ∂(GΩ2 , P2 ) are not homeomorphic. We now prove that the Bowditch boundary ∂(GΩ2 , P2 ) is not homeomorphic to S2 or the Sierpinski carpet. By Theorem 3.1, the Bowditch boundary ∂(GΩ2 , P2 ) is obtained from the CAT(0) boundary ∂ΣΩ2 by identifying the limit set of each peripheral left coset to a point, let f be the quotient map. We will prove that f (∂ΣL1 ) ∪ f (∂ΣL2 ) ∪ f (∂ΣL3 ) is also a triple of the Sierpinski carpets along a peripheral circle. We first observe that for each 4-cycle J ∈ J2 the intersection J ∩ L1 is a single edge or the whole 4-cycle J. Therefore, the intersection between the limit set ∂ΣL1 and the limit set ∂(gGJ ) of a peripheral left coset gGJ is empty or the whole limit set ∂(gGJ ). Moreover, the second case occurs if and only if J is a 4-cycle in L1 and g is an element in GL1 . The image f (∂ΣL1 ) is obtained from ∂ΣL1 by identifying each peripheral circle on ∂ΣL1 which is the limit set of a peripheral left coset gGJ , where g ∈ GL1 and J ⊂ L1 , to a point. Therefore, the image f (∂ΣL1 ) is still a Sierpinski carpet. Similarly, the images of f (∂ΣL2 ) and f (∂ΣL3 ) are also a Sierpinski carpets. Lastly, all limit sets ∂ΣLi share the limit set ∂ΣD as a common subspace. The limit set ∂ΣD is a peripheral circle in each Sierpinski carpet ∂ΣLi , but it is not the limit set of a peripheral left coset. Therefore, the images f (∂ΣL1 ), f (∂ΣL2 ), and f (∂ΣL3 ) pairwise intersect in the circle ∂ΣD . This implies that the union f (∂ΣL1 ) ∪ f (∂ΣL2 ) ∪ f (∂ΣL3 ) is a triple of the Sierpinski carpet along a peripheral circle. Since f (∂ΣL1 ) ∪ f (∂ΣL2 ) ∪ f (∂ΣL3 ) cannot be topologically embedded into S2 or the Sierpinski carpet, the Bowditch boundary ∂(GΩ2 , P2 ) is not homeomorphic to S2 or the Sierpinski carpet.  Corollary 5.8. 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1 Boosted KZ and LLL Algorithms arXiv:1703.03303v4 [] 10 Oct 2017 Shanxiang Lyu and Cong Ling, Member, IEEE Abstract—There exist two issues among popular lattice reduction (LR) algorithms that should cause our concern. The first one is Korkine-Zolotarev (KZ) and Lenstra–Lenstra–Lovász (LLL) algorithms may increase the lengths of basis vectors. The other is KZ reduction suffers much worse performance than Minkowski reduction in terms of providing short basis vectors, despite its superior theoretical upper bounds. To address these limitations, we improve the size reduction steps in KZ and LLL to set up two new efficient algorithms, referred to as boosted KZ and LLL, for solving the shortest basis problem (SBP) with exponential and polynomial complexity, respectively. Both of them offer better actual performance than their classic counterparts, and the performance bounds for KZ are also improved. We apply them to designing integer-forcing (IF) linear receivers for multiinput multi-output (MIMO) communications. Our simulations confirm their rate and complexity advantages. Index Terms—lattice reduction, KZ, LLL, shortest basis problem, integer-forcing I. I NTRODUCTION L ATTICE reduction (LR) is a process that, given a lattice basis as input, to ascertain another basis with short and nearly orthogonal vectors [1]. Their applications in signal processing include global positioning system (GPS) [2], color space estimation in JPEG images [3], and data detection/precoding in wireless communications [4], [5]. Recent advances in LR algorithms are mostly made in wireless communications and cryptography [6], [7]. Popular LR algorithms with exponential complexity include Korkine-Zolotarev (KZ) [8], [9] and Minkowski reductions [10], which have set the benchmarks for the best possible performance in LR aided successive interference cancellation (SIC) and zero-forcing (ZF) detectors for multi-input multi-output (MIMO) systems [10]. In MIMO detection problems, KZ and Minkowski reductions are preferable when the channel coefficients stay fixed for a long time frame so that their high complexity can be shared across time. In the part of polynomial or fixed complexity algorithms, the celebrated Lenstra–Lenstra–Lovász (LLL) [11] algorithm has been well studied and many new variants have been proposed. Typical variants of LLL in wireless communications can be summarized into two types: either sacrificing the execution of full size reductions [12], [13], or controlling the implementation order of swaps and size reductions [14], [15], [16], [17]. The reason to establish the first type variants is that a full size reduction has little influence on the performance of LR aided SIC detectors. Variants of the second type, e.g., fixed complexity LLL [14], [15] and greedy LLL [16], [17], serve the purpose of enhancing the system This work was supported in part by the Royal Society and in part by the China Scholarship Council. S. Lyu and C. Ling are with the Department of Electrical and Electronic Engineering, Imperial College London, London SW7 2AZ, United Kingdom (e-mail: [email protected], [email protected]). performance especially when the number of LLL iterations is restrained. It is also noteworthy to introduce the block KZ (BKZ) reduction [18] as a tradeoff between KZ and LLL. BKZ is scarcely probed in MIMO but more often in cryptography. Many records in the shortest vector problem (SVP) challenge hall of fame [19] are set by using BKZ although no good upper bound on the complexity of BKZ is known. In this work, we point out two issues among popular LR algorithms which were rarely investigated before. The first one is that KZ and LLL may elongate basis vectors. This issue was discovered when we applied LLL to Gaussian random matrices of dimensions higher than 40. The second one is KZ reduction practically suffers much worse performance than Minkowski reduction in terms of providing short basis vectors, while Nguyen and Stehle conjectured in [20, P. 46:7] that KZ may be stronger than Minkowski in high dimensions because theoretically all vectors of a KZ reduced basis are known to be closer to the successive minima than Minkowski’s. So engineers may be quite confused about the discrepancies between theory and practice. The contributions of this work are twofold. First, we propose improved algorithms to address the above limitations of KZ and LLL, and they are in essence suitable for any application that needs to solve the shortest basis problem (SBP). Second, we show that our algorithms can be applied to the design of integer forcing (IF) linear MIMO receivers [21] to obtain some gains in rates. The first algorithm is referred to as boosted KZ. It harnesses the strongest length reduction every time after the shortest vector in a projected lattice basis has been found, and such an operation is proved to be valid. We improve analysis on the best known bounds for the lengths of basis vectors and Gram-Schmidt vectors via boosted KZ. After choosing sphere decoding as subroutines, the total complexity of boosted KZ is shown to be closed to that of conventional KZ. In the second algorithm called boosted LLL, it also dumps conventional size reduction conditions in LR while deploying a flexible effort to perform length reduction. In order to maintain the Siegel condition [22], two criteria for doing length reductions before/after testing the necessity of swaps are proposed, which guarantee the basis potential is decreasing after swaps and the lengths of vectors shrink at the largest extent. With our scheme, bounds on basis lengths and orthogonal defects can also be obtained. An optimal principle of choosing Lovász constants is proposed as well. The complexity of this algorithm is of O(Ln4+c ln n) if the condition number of the input basis is of O(ln n), where L is the number of routes in boosted LLL and c > 1 is a constant. IF is a new MIMO receiver architecture that attempts to decode an integer combination of lattice codes [21]. It can be thought of as a special case of compute and forward 2 [23] because this design has full cooperation among receive antennas. We apply our algorithms to IF because it represents the kind of applications that need to find multiple short lattice vectors, as opposed to LR aided SIC receivers [24], lattice Gaussian samplers [25], and those searching the shortest or closest vectors [26], [27]. This receiver is more general than LR aided minimum mean square error (MMSE) receiver in that it allows concise evaluation on rates, owning to lattice coding and dithering. In [21], the performance of IF receiver is shown to outperform conventional ZF and MMSE receivers, and the optimality in diversity-multiplexing gain tradeoff (DMT) is also proved. We will elaborate on the IF architecture and the SBP interface where boosted KZ and LLL turn out to be beneficial. Simulations will verify the advantages of our algorithms in terms of rates and complexity. The rest of this paper is organized as follows. Backgrounds about lattices and lattice reduction algorithms are reviewed in Section II. After that, we provide a motivating example to indicate the drawback of KZ and LLL. The boosted KZ and LLL algorithms are subsequently constructed and analyzed in Sections III and IV, respectively. After introducing the IF framework, exemplary simulation results are then shown in Section V to emphasize that the proposed algorithms can deliver higher rates. We mention some open questions Section VI. Notation: Matrices and column vectors are denoted by uppercase and lowercase boldface letters. For a matrix D, Di:j,i:j denotes the submatrix of D formed by rows and columns i, i+1, . . . , j. When referring to the (i, j)th element of D, we simply write di,j . In and 0n denote the n×n identity matrix and n × 1 zero vector, and the operation (·)⊤ denotes transposition. For an index set Γi = {1, . . . , i − 1}, DΓi denotes the columns of D indexed by Γi . span(DΓi ) denotes the vector space spanned by vectors in DΓi . πDΓi (x) and ⊥ πD (x) denote the projection of x onto span(DΓi ) and the Γi orthogonal complement of span(DΓi ). ⌊x⌉ denotes rounding x to the nearest integer, |x| denotes getting the absolute value of x, and kxk denote the Euclidean norm of vector x. The set of all n × n matrices with determinant ±1 and integer coefficients will be denoted by GLn (Z). II. P RELIMINARIES A. Lattices A full rank n-dimensional lattice L is a discrete additive subgroup in Rn . The lattice generated by a basis D = [d1 , . . . , dn ] ∈ Rn×n can be written as     X c i di ; c i ∈ Z ; L(D) = v | v =   i∈[n] its dual lattice L(D̃) has a basis D̃ = D−⊤ . If the lattice basis is clear from the context, we omit D and simply write L. Definition 1. SBP is, given a lattice basis D0 of rank n, find min D, L(D)=L(D0 ) l(D) where l(D) = maxi kdi k, D ranges over all possible bases of L(D0 ), and l(D) is referred to as basis length. The Gram-Schmidt orthogonalization (GSO) vectors of a ⊥ basis D can be found by: d∗1 = d1 , d∗i = πD (di ) = Γi Pi−1 ∗ µ d , for i = 2, . . . , n, where µ = di − i,j j=1 i,j j ∗ ∗ 2 hdi , dj i/ dj . In matrix notations, GSO vectors can be written as D = [d∗1 , . . . , d∗n ][µi,j ]⊤ , where [µi,j ] is a lowertriangular matrix with unit diagonal elements. In relation to the QR decomposition, let Λ be a diagonal matrix with diagonal entries kd∗1 k , . . . , kd∗n k, then we have [d∗1 , . . . , d∗n ]Λ−1 = Q and Λ[µi,j ]⊤ = R whose diagonal elements reflect the lengths of GSO vectors. The ith successive minimum of an n dimensional lattice L(D) is the smallest real number r such that L contains i linearly independent vectors of length at most r: λi = inf {r | dim(span((L ∩ B(0, r))) ≥ i} , in which B(t, r) denotes a ball centered at t with radius r. We also write λi as λi (D) to distinguish different lattices. Hermite’s constant γn is defined by γn = λ1 (D)2 . |det(D)|2/n sup D∈Rn×n Exact values for γn are known for n ≤ 8 and n = 24. With Minkowski’s convex body theorem, we can obtain γn ≤ π4 Γ(1 + n/2)2/n , which yields γn ≤ 2n 3 for n ≥ 2 [28]. It also follows from the work of Blichfeldt [29] that n . γn ≤ π2 Γ(2 + n/2)2/n , whose asymptotic value is πe The open Voronoi cell of lattice L with center v is the set Vv (D) = {x ∈ Rn | kx − vk < kx − v − v′ k , ∀v′ ∈ L} , in which the outer radius of the Voronoi cell centered at the origin is denoted as “covering radius”, i.e., ρ(D) = maxt∈span(L) dist(t, L). The orthogonality defect (OD), ξ(D), can alternatively quantify the goodness of a basis: Qn kdi k ξ(D) = p i=1 . (1) det(D⊤ D) It has a lower bound ξ(D) ≥ 1 in accordance with Hadamard’s inequality. B. Lattice reduction algorithms In this subsection, we review three popular LR metrics where the lengths of basis vectors can be upper bounded by scaled versions of the successive minima. Operations/transforms to reach these metrics are referred to as the corresponding algorithms. Let R be the R matrix of a QR decomposition on D, with elements ri,j ’s. Definition 2. A basis D is called LLL reduced if the following two conditions hold [11]: 1. |ri,j /ri,i | ≤ 21 , 1 ≤ i ≤ n, j > i. (Size reduction conditions) 2 ⊥ 2. δ πD (di ) ≤ Γi (Lovász conditions) ⊥ πD (di+1 ) Γ i 2 , 1 ≤ i ≤ n − 1. In the definition, δ ∈ (1/4, 1] is called the Lovász constant. If D is LLL reduced, it has [11] kdi k ≤ β n−1 λi (D), 1 ≤ i ≤ n, (2) 3 p √ in which β = 1/ δ − 1/4 ∈ (2/ 3, ∞). Definition 3. A basis D is called KZ reduced if it satisfies the ⊥ size reduction conditions, and πD (di ) is the shortest vector Γi ⊥ of the projected lattice πDΓ ([di , . . . , dn ]) for 1 ≤ i ≤ n [28]. i (Projection conditions) For a KZ reduced basis, it satisfies [28] √ i+3 kdi k ≤ λi (D), 1 ≤ i ≤ n. (3) 2 Definition 4. A lattice basis D is called Minkowski reduced if for any integers c1 , ..., cn such that ci , ..., cn are altogether coprime, it has kd1 c1 + · · · + dn cn k ≥ kdi k for 1 ≤ i ≤ n [10]. For a Minkowski reduced basis, it satisfies [10] o n kdi k ≤ max 1, (5/4)(i−4)/2 λi (D), 1 ≤ i ≤ n. (4) When n ≤ 4, Minkowski reduction is optimal as it reaches all the successive minima. Its bounds on lengths are however exponential for n > 4. III. B OOSTED KZ In this section, we propose to improve KZ by abandoning its size reduction conditions, as well as employing the exact closest vector problem (CVP) oracles to reduce di with L(DΓi ) after the projection condition has been met at each time i. Better theoretical results can be obtained via boosted KZ, and the implication is using CVP for LR can be better than solely relying on SVP. This should not be a surprise because CVP is generally believed to be harder than SVP [1]. A. Replacing size reduction with CVP We first show that imposing size reduction conditions in KZ and LLL may lengthen basis vectors, and thus enlarging OD’s. Proposition 1. There always exist real value bases of rank n, n ≥ 3, such that KZ and LLL algorithms lengthen basis vectors. 1 Proof: We prove this by constructing examples in dimension n = 3 because bases of higher ranks can be built by concatenating another identity matrix in the diagonal direction. Consider the following matrix   1 c1 0 R =  0 1 c2  , (5) 0 0 1 where |c1 | < 1/2, |c2 | > 1/2. Since |r1,2 /r1,1 | < 1/2 and |r2,3 /r2,2 | > 1/2, it follows from the definition of KZ or LLL that r1 and r2 will remain unchanged, while size reducing r3 by r2 yields a new vector r′3 = [−c1 ⌊c2 ⌉, c2 − ⌊c2 ⌉, 1]⊤ . If ⌊c2 ⌉ = ±1, then r′3 cannot be further reduced by r1 . So we 2 2 can assume kr′3 k > kr3 k and
solve this inequality about c2 , which yields |c2 | < 1 + c21 /2. Therefore, there exist at least matrices like (5) with |c1 | < 1/2 and 1/2 < |c2 | <
1 + c21 /2 such that KZ/LLL lengthens basis vectors. 1 This proposition is inspired by [20, Lem. 2.2.3]. To avoid such problems in KZ, we shall review the process of the KZ reduction algorithm [10]. In the beginning, the projection conditions are met by finding the shortest lattice vectors of the projected lattices and carrying them to the lattice basis. The size reduction conditions are subsequently addressed by using Babai points v’s in L(DΓi ) to reduce di by di ← di − v for all i. Concerning the above procedure, what we try to ameliorate are the size reduction operations. The “di ← di − v” step is redefined as length reduction, in which the optimal update needs to solve a CVP. Definition 5. CVP is a problem that, given a vector y ∈ Rn and a lattice basis D of rank n, find a vector v ∈ L(D) such 2 2 that ky − vk ≤ ky − wk , ∀ w ∈ L(D). An algorithm solving CVP, which quantizes any input to a lattice point, is denoted as v = QL(D) (y). It is evident that QL(DΓi ) (πDΓi (di )) = QL(DΓi ) (di ) . To obtain explicit properties from the length reductions, we first establish Proposition 2 to show di − QL(DΓi ) (πDΓi (di )) < kdi k if QL(DΓi ) (πDΓi (di )) 6= 0 for all i. The proof is given in Appendix A. Proposition 2. If πDΓi (di ) lies outside the Voronoi region V0 (DΓi ), i.e., v , QL(DΓi ) (πDΓi (di )) 6= 0, then we can replace di with di − v because kdi − vk < kdi k. Together with the case of QL(DΓi ) (πDΓi (di )) = 0, we conclude that di − QL(DΓi ) (πDΓi (di )) ≤ kdi k (6) for all i, which means, during the length reductions, all solutions provided by CVP can be treated as effective updates. We call them effective because each di is the shortest vector that can be extended to a basis for L([DΓi , di ]), and the length reductions never increase the lengths of di ’s. After executing these length reduction operations as di ← di − QL(DΓi ) (πDΓi (di )), all πDΓi (di )’s must lie inside the Voronoi regions V0 (DΓi )’s, so that 2 ⊥ kdi k ≤ πD (di ) Γi 2 + ρ(DΓi )2 (7) for all i, where ρ(DΓi ) is the covering radius of L(DΓi ). B. Algorithm description The concrete steps of boosted KZ are presented in Algorithm 1. This algorithm can be briefly explained as follows. In line 4, the Schnorr and Euchner (SE) enumeration algorithm [18] is applied to solve SVP over L(Ri:n,i:n ), in that if Ri:n,i ⊥ is the shortest vector of L(Ri:n,i:n ), then πD (di ) is the Γi ⊥ shortest vector of the projected lattice πDΓ ([di , . . . , dn ]). i Lines 5 to 7 are designed to plug new vectors found into the lattice basis, and the basis expansion method in [10] can do this efficiently. Other basis expansion methods include using LLL reduction [30] or employing the Hermite normal form of the coefficient matrix [1, Lem. 7.1], but both of them have higher complexity than the one in [10]. Lines 8 to 10 restore 4 the upper triangular property of R, and these be alternatively implemented by performing another QR decomposition. Line 11 is the unique new design of boosted KZ, i.e., to reduce R1:n,i by using its closest vector in L(R1:n,1:i−1 ).2 A direct application of the above proposition also shows a boosted KZ reduced basis has length √ n+2 l(D) ≤ λn (D). (10) 2 Algorithm 1: The boosted KZ algorithm. Input: original lattice basis D ∈ Rn×n , Lovász constant δ. Output: reduced basis D, unimodular matrix T 1 [Q, R] = qr(D); ⊲ The QR decomposition of D; 2 T = I; 3 for i = 1 : n do 4 find the shortest vector Ri:n,i:n c1 in L(Ri:n,i:n ) by LLL aided SE enumeration; ⊲ SVP subroutine; 5 construct a (n − i + 1) × (n − i + 1) unimodular matrix U whose first column is c1 ; 6 R1:n,i:n ← R1:n,i:n U; 7 T1:n,i:n ← T1:n,i:n U; 8 define G as a unitary matrix that can restore the upper triangular property of R; 9 R ← GR; 10 Q ← QG⊤ ; 11 find the closest vector R1:n,1:i−1 c2 in L(R1:n,1:i−1 ) to R1:n,i with SE enumeration; ⊲ CVP subroutine; 12 R1:n,i ← R1:n,i − R1:n,1:i−1 c2 ; 13 T1:n,i ← T1:n,i − T1:n,1:i−1 c2 ; Remark 1. Our results of (8), (9) and (10) are better than those of KZ and Minkowski reductions. If we assume all the successive minima are available, then there exists a polynomial time√transformation that generates a basis with l(D) ≤ max {1, n/2} λn (D) [1, Lem. 7.1]. 14 D ← QR. Proposition 4. Suppose a basis D is boosted KZ reduced, then this basis satisfies λ1 (D)2 ≤ 8i 2 (i − 1)ln(i−1)/2 ri,i , 9   2i 2 2 kdi k ≤ 1 + (i − 1)1+ln(i−1)/2 ri,i 9 (11) (12) for 1 ≤ i ≤ n. C. Properties of boosted KZ Based on Algorithm 1, a lattice basis D is called boosted ⊥ KZ reduced if πD (di ) is the shortest vector of the projected Γi ⊥ lattice πDΓ ([di , . . . , dn ]), and πDΓi (di ) ∈ V0 (DΓi ) for all i i. In boosted KZ, all length reductions are the strongest, and they can help us to deliver better bounds for the lengths of basis vectors, as given in Proposition 3. The is given n proof √ o n in Appendix B. We have kdn k ≤ max 1, 2 λn (D), √ outperforming the kdn k ≤ n+3 2 λn (D) bound in [28, Thm. 2.1] which was conjectured not tight in their work. Proposition 3. Suppose a basis D is boosted KZ reduced, then this basis satisfies ) ( √ ) (√ i+3 i λi (DΓi+1 ) λi (D), max 1, kdi k ≤ min 2 2 (8) for 1 ≤ i < n, and  √  n kdn k ≤ max 1, λn (D). (9) 2 2 Since The lengths of GSO vectors in this new algorithm remains the same as those of KZ reduction, so readily we can claim that 2 2 λ1 (D)2 ≤ i1+ln(i) ri,i and kdi k2 ≤ i2+ln(i) ri,i as given in [28, Prop. 4.2], where ri,i denotes the (i, i)th entry of the R matrix of a QR decomposition on D. As another contribution, now we show these two bounds can be improved in Proposition 4, whose proof is given in Appendix C. we only modify the size reduction steps in KZ, one may employ any improved KZ implementation, e.g., [9], to make the boosted KZ faster. We adhere to the current version for making a fair complexity comparison with Minkowski’s reduction which employs similar subroutines [10]. The relaxed versions of (11) and (12) can be read as 2 2 2 λ1 (D)2 ≤ i1+ln(i)/2 ri,i and kdi k ≤ i2+ln(i)/2 ri,i . Proposition 4 can be either applied to bound the complexity of boosted KZ, or to achieve the best explicit bounds for the proximity factors of lattice reduction aided decoding, i.e., updating Eqs. (41) and (45) of [24]. Moreover, (12) leads to an alternative bound for OD, ξ(D) ≤ n Y i=1 i1+ln(i)/4 ≤ nn+ln(n!)/4 . A better bound on ξ(D) comes after applying Minkowski’s second theorem [31, P. 202] to (8) and (9), √ n ξ(D) ≤ 2 n−1 Y i=1 !  √ n/2 2 i+3 n . 2 3 (13) Remark 2. The properties of |ri,j /ri,i | ≤ 12 for 1 ≤ i ≤ n, j > i, are no longer guaranteed in boosted KZ. Of independent interests, we have another attribute in Proposition 5 that each pair (d1 , di ) of the boosted KZ reduced basis is Lagrange reduced [7, P. 41] for all i, which may not hold in the conventional KZ. The proof can be found in Appendix D. Proposition 5. Suppose a basis D is boosted KZ reduced, then this basis satisfies |r1,i /r1,1 | ≤ 12 , and kd1 k, kdi k reaches the first and second successive minima of L([d1 , di ]) for 2 ≤ i ≤ n. 5 D. Implementation and complexity The complexity of boosted KZ is dominated by its SVP and CVP subroutines, in which the SE enumeration algorithm [18] will be adopted for our implementations and complexity analysis. The total complexity is assessed by counting the number of floating-point operations (flops). 1) Complexity of CVP subroutines: It suffices to discuss the complexity of the most time-consuming nth round of reducing R1:n,n , which represents an n − 1 dimensional CVP problem. First of all, the complexity of SE is directly proportional to the number of nodes in the search tree. In the kth layer of the enumeration, the number of nodes is n Nk (s) = |xk:n−1 | | xk:n−1 ∈ Zn−k , o kRk:n−1,n − Rk:n−1,k:n−1 xk:n−1 k2 ≤ s2 where s refers to the radius of a specified sphere and R1:n−1,n is the projection of R1:n,n onto span(R1:n−1,1:n−1 ). From Vn−k (1)sn−k [32], Nk (s) can be estimated by Nk (s) ≈ |rk,k |···|rn−1,n−1 |
n/2 1 π n/2 √ ∼ 2eπ where Vn (1) = Γ(1+n/2) stands for the n πn volume of an n dimensional unit ball. Since limn→∞ Vn (1) = 0, we can cancel this term in the asymptotic analysis. By summing the nodes from layer 1 to n − 1, the total number of nodes in the n − 1 dimensional CVP can be given as NCVP,n−1 (s) = n−1 X k=1 Vn−k (1)sn−k . |rk,k | · · · |rn−1,n−1 | ≤ n−1 X k=1 n−1 X Vn−k (1)s Qn−1 j=k j 1/2+ln(j)/4 (2k + 7) λ1 (D)n−k , (14) in which Proposition 4 has been used to get the inequality. Then we present a general strategy to choose s. It starts from s = |r1,1 |/2 which equals to the packing radius because qP k 1 2 λ1 (R1:n,1:k−1 ) = |r1,1 |, and improves s to 2 j=1 rj,j with k = 2, . . . , n−1 gradually until at least one node can be found inside the searching sphere. For a random basis, one may expect s = |r1,1 |/2 to work well with high probability, qP though n−1 2 1 we can also use the worst case criterion of s = 2 j=1 rj,j (i.e., larger than the covering radius). In the worst case, let √ n−1λn (D) C1 = 2λ1 (D) . Since Vn (1)(2n+7) also vanishes for large n, then (14) can be written as FCVP,n−1 (C1 λ1 (D)) ≤ nC1n nn/2+ln(n!)/4 k=1 λ1 (D)n−k+1 . (16) A practical principle for choosing s is to set s = kR1:n,1 k, which is no smaller than λ1 (D). It follows from an LLL reduced basis property of kR1:n,1 k ≤ β n−1 λ1 (D) that (16) becomes FSVP,n (β n−1 λ1 (D)) ≤ nβ n(n−1) β n(n−1)/2 = nβ 3n(n−1)/2 . (17) 3) Total complexity in flops: The complexity of other operations in Algorithm 1 can be counted as well. In the ith round, Lines 5 to 10 being implemented by the method described in [10, Fig. 3] costs O(n(n−i)). The total complexity of boosted KZ is therefore upper bounded as  FboostKZ ≤ (n−1) FCVP,n−1 (C1 λ1 (D))+FSVP,n (β n−1 λ1 (D))  4 + O(n2 − n) + n3 + (2n − 1)n2 . (18) 3 By plugging (15) and (17) inside (18), we can explicitly obtain 2 Nk (s)(2k + 7) k=1 n−k FSVP,n (s) ≤ n X Vn−k+1 (1)sn−k+1 β (n+k−2)(n−k+1)/2 (2k + 7) 2 FboostKZ ≤ C1n nn/2+ln(n!)/4+O(ln n) + β 3n(n−1)/2+O(ln n) . For the visit of each node, the operations of updating the residual and outer radius etc. cost around 2k + 7 flops in layer k, so the complexity of the n − 1 dimensional CVP problem, FCVP,n−1 (s), can be accessed: FCVP,n−1 (s) = β (n+k−2)(n−k+1)/2 /λ1 (D)n−k+1 , so the number of flops spent by the first round SVP subroutine can be similarly bounded as (15) 2) Complexity of SVP subroutines: Among the SVP subroutines of boosted KZ, its first round of finding λ1 (D) is the most difficult one. By invoking the SiegelQcondition of n |ri−1,i−1 | ≤ β|ri,i | due to LLL [22], we have 1/ j=k |rj,j | ≤ A few comments are made regarding the above analysis. Firstly, it provides a worst case analysis for strong lattice reductions like KZ and boosted KZ, which is broad enough to include many applications. A byproduct of our analysis is that we can replace the term FCVP,n−1 (C1 λ1 (D)) in (18) with O(n3 ) to get the worst case complexity of KZ, i.e., 2 FKZ ≤ β 3n(n−1)/2+O(ln n) . This compensates the expected complexity analysis in [10, Sec. III.C] which hinges on Gaussian lattice bases. Secondly, we can observe from (18) that how much harder the boosted KZ has become by using CVP. If √ λ1 (D) is of the same order as λn (D), we can put into (18) to conclude that boosted KZ is not much C1 ≈ n−1 2 more complicated than KZ. Actually, if the lattices are random (see [33, Sec. 2] for more details about random lattices), then the Gaussian heuristic implies λ1 (D) ≈ · · · ≈ λn (D) [34]. In the application to IF, our claim that boosted KZ is not much harder than KZ will be supported by simulations in Section V. IV. B OOSTED LLL In the same spirit of extending size reductions to length reductions, we will revamp LLL towards better performance in this section. In a nutshell, the boosted LLL algorithm implements its length reduction via the parallel nearest plane (PNP) algorithm [35, Sec. 4] and rejection. PNP can be regarded as a compromise between Babai’s nearest plane algorithm and the CVP oracle. If PNP has a route number L = 1, then it becomes equivalent to Babai’s algorithm, while setting L infinitely large solves CVP. The complexity of boosted LLL is about L times as large as the that of LLL. In our algorithm, setting L = 1 means only imposing a rejection operation. 6 A. Replacing size reduction with PNP and rejection First of all, the classic LLL algorithm consists of two sequential phases, i.e., size reductions by using Babai points, and swaps based on testing the Lovász conditions. To reduce di with DΓi , the sharpest reduction should utilize the closest vector of πDΓi (di ) in L(DΓi ) as shown in Proposition 2. In order to devote flexible efforts to these length reductions, we shall investigate the success probability of a Babai point being optimum. Generally, assume πDΓi (di ) is uniformly distributed over L(DΓi ), then the probability of a Babai point being the closest vector is R I(x ∈ P(D∗Γi ))dx |V0 (DΓi ) ∩ P(D∗Γi )| , |V0 (DΓi )| |V0 (DΓi )| o(19) nP i−1 ∗ ∗ is in which P(DΓi ) = k=1 ck dk | − 1/2 ≤ ck ≤ 1/2  ∗ ∗ the parallelepiped of GSO vectors d1 , . . . , di−1 , and I(·) denotes an indicator function. One evident observation from Eq. (19) is, by updating d∗1 ← p1 d∗1 , . . . , d∗i−1 ← pi−1 d∗i−1 , the probability in Eq. (19) rises if we choose some constants p1 > 1, . . . , pi−1 > 1. Another implication from Eq. (19) is, if πDΓi (di ) belongs to both the external of P(D∗Γi ) and internal of V0 (DΓi ), then a Babai point should be rejected; otherwise it elongates di . An example of i = 3 is shown in Fig. 1. x∈V0 (DΓi ) = total number of routes it consists of, and (pi−1 , . . . , p1 ) ∈ (Z+ )i−1 . Then each route of PNP can be marked by a label (qi−1 , . . . , q1 ) where (qi−1 , . . . , q1 ) ∈ {1, . . . , pi−1 } × · · · × {1, . . . , p1 }. From layer i−1 of each route, let ci−1,(qi−1 ,...,q1 ) be the qi−1 th closest integer to ⌊ri−1,i /ri−1,i−1 ⌉, and ri,(qi−1 ,...,q1 ) = ri , we set ri,(qi−1 ,...,q1 ) ← ri,(qi−1 ,...,q1 ) − ci−1,(qi−1 ,...,q1 ) ri−1 and repeat this process down to layer 1, resulting in L pairs of coefficient vectors c(qi−1 ,...,q1 ) = [−c1,(qi−1 ,...,q1 ) , . . . , −ci−1,(qi−1 ,...,q1 ) , 1] and residuals ri,(qi−1 ,...,q1 ) = RΓi+1 c(qi−1 ,...,q1 ) . We also mark the old ri by ri,(0,··· ,0) = ri , and c(0,··· ,0) = [0, . . . , 0, 1]. At this stage, it can choose the shortest vector among all the L + 1 ∗ candidates as the reduced version of ri , i.e., ri = ri,(zi−1 ,...,z1∗ ) where  ∗ (zi−1 , ..., z1∗ ) = arg min ri,(zi−1 ,...,z1 ) . (20) (zi−1 ,...,z1 ) If one also intends to export the unimodular transformation matrix T, then it can be simultaneously updated inside PNP, ∗ ∗ which means ti,(zi−1 ,...,z1∗ ) , T1:n,1:i = ,...,z1∗ ) = TΓi+1 c(zi−1 ∗ ∗ ti,(zi−1 . ,...,z1 ) Since |rk,i /rk,k | < 1/2 for k < i is no longer guaranteed, together with the Lovász condition they may destroy the Siegel 2 2 for 2 ≤ i ≤ n with some condition [22] ri−1,i−1 ≤ ( 43 + ε)ri,i small ε > 0. For this reason, we should relax the Lovász condition to the diagonal reduction (DR) condition [13]. Definition 6 (DR condition [13]). An upper triangular lattice basis R satisfies the DR condition with parameter δ (1/2 < δ < 1) if it has 2 2 δri−1,i−1 ≤ ri,i + (ri−1,i − ⌊ri−1,i /ri−1,i−1 ⌉ri−1,i−1 )2 (21) for all 2 ≤ i ≤ n, where δ is still referred to as the Lovász constant. If (21) holds, the Siegel condition must be true, so we let i ← i + 1 and safely go to the next iteration. However, if (21) fails, one should also investigate whether a swap can tweak such cases. Consider the sublattice L(RΓi+1 ) generated by the first i vectors and define the potential of basis R [11] as n n Y Y 2(n−i+1) . (22) ri,i det(L(RΓi+1 ))2 = Pot(R) = i=1 i=1 Fig. 1. The rejected region (orange dots) of the possible projections πDΓ (d3 ) in L(DΓ3 ) with respect to V0 (DΓ3 ) (red hexagon) and 3 P(d∗1 , d∗2 ) (black rectangle), where the size reduction of LLL elongates d3 . The four blue triangles are the region whose Babai point is the origin and size reductions cannot alter a suboptimal d3 . With the above demonstrations, we propose to amplify the success probabilities of Babai points with minimal efforts and to reject operations that elongate current basis vectors. Either using lattice Gaussian sampling [25] or PNP suffices the first objective, but we shall adhere to PNP because it is deterministic and this feature will be employed by (27). We detail the length reductions in boosted LLL as follows. Assume we are working on the R matrix of a QR decomposition and trying to reduce ri by Q ri−1 , . . . , r1 . Let PNP i−1 be abstracted by a parameter L = k=1 pk indicating the ⊥ If the DR condition fails in πR (R1:n,i−1:i ) and we swap Γi ri−1 and ri , then the potential of the basis should be decreasing for the sake of bounding the number of iterations. After the swap, Ri−1:i,i−1:i becomes   ri−1,i ri−1,i−1 . (23) ri,i 0 Let G be a 2 × 2 unitary matrix   √ 2ri−1,i2 √ 2 ri,i 2 ri,i +ri−1,i ri,i +ri−1,i  ; r √ 2ri−1,i2 − √ 2 i,i 2 ri,i +ri−1,i (24) ri,i +ri−1,i clearly, GRi−1:i,1:n can restore the upper triangular property of (23), which transforms to   q ri−1,i ri−1,i−1 2 + r2 √ ri,i 2 +r 2 i−1,i ri,i .  i−1,i (25) r r 0 − √i,i2 i−1,i−1 2 ri,i +ri−1,i 7 From (22) and (25), one can obtain the potential ratio between two consecutive bases R′ and R as 2(n−i+1) 2(n−i+2)  q ri,i ri−1,i−1 2 + r2 √ ri,i ′ 2 2 i−1,i ri,i +ri−1,i Pot(R ) = 2(n−i+2) 2(n−i+1) Pot(R) ri−1,i−1 ri,i ≤ 2 2 δ(ri,i + ri−1,i ) , 2 + (r 2 − ⌊r /r ri,i i−1,i i−1,i i−1,i−1 ⌉ri−1,i−1 ) (26) where′ the last inequality comes from (21). Based on (26), Pot(R ) Pot(R) ≤ δ if and only if ⌊ri−1,i /ri−1,i−1 ⌉ = 0. As ∗ ∗ a result, preparing the pairs ti,(zi−1 ,...,z1∗ ) and ri,(zi−1 ,...,z1∗ ) based on (20) is only suitable for reductions before checking the DR conditions. In case that this condition fails, we should also prepare ti,(z′ ,...,z′ ) and ri,(z′ ,...,z′ ) that make 1 1 i−1 i−1 ⌊ri−1,i /ri−1,i−1 ⌉ = 0: ′ ′ (zi−1 , ..., z1 ) = arg min (zi−1 ,...,z1 ) n ri,(zi−1 ,...,z1) , o s.t. ⌊ri−1,i,(zi−1 ,...,z1 ) /ri−1,i−1 ⌉ = 0 , (27) in which ri−1,i,(zi−1 ,...,z1) denotes the (i − 1)th component of ri,(zi−1 ,...,z1 ) . In such a manner, if a vector is swapped to the front, it is not only a short vector, but also the one that decreases the basis potential so that this kind of swaps cannot happen too many times. and the rejection operation marking r3 is r3,(0,0) = [0, 0.52, 1]⊤ . Eq. (20) would choose the shortest among the above four routes. Let it be r3,(2,1) (or r3,(0,0) ), which is employed by Line 5. Eq. (27) can only choose from r3,(1,1) . Then we test the DR condition and it succeeds, so the while loop stops. Algorithm 2: The boosted LLL algorithm. Input: original lattice basis D ∈ Rn×n , Lovász constant δ, list number L. Output: reduced basis D, unimodular matrix T 1 [Q, R] = qr(D); ⊲ The QR decomposition of D; 2 i = 2, T = I; 3 while i ≤ n do ∗ ∗ 4 use (20) to get [ri,(zi−1 ,...,z1∗ ) , ti,(zi−1 ,...,z1∗ ) ] and use (27) to get [ri,(z′ ,...,z′ ) , ti,(z′ ,...,z′ ) ]; 1 1 i−1 i−1 ∗ ∗ 5 R1:n,i = ri,(zi−1 ,...,z1∗ ) ,T1:n,i = ti,(zi−1 ,...,z1∗ ) ; 6 if condition (21) fails then 7 R1:n,i = ri,(z′ ,...,z′ ) ,T1:n,i = ti,(z′ ,...,z′ ) ; 1 1 i−1 i−1 8 define G as in (24); 9 swap R1:n,i and R1:n,i−1 , T1:n,i and T1:n,i−1 ; 10 Ri−1:i,1:n ← GRi−1:i,1:n ; 11 Q1:n,i−1:i ← Q1:n,i−1:i G⊤ ; 12 i ← max(i − 1, 2); 13 14 B. Algorithm description Combing the length reduction process above, the procedure of boosted LLL is given in Algorithm 2. Inside the loops, it employs a fixed structure column traverse strategy rather than using a parallel traversing [12], [13], such that a theoretical O(n) factor in bounding the number of loops can be saved. In line 4, the PNP algorithm and rejection prepare two versions ∗ of reduced vectors. The stronger version ri,(zi−1 ,...,z1∗ ) is used before testing the DR condition (line 6), so that the new ri is the shortest candidate among the L routes of PNP and the old ri . If it cannot pass this test, a weaker version ri,(z′ ,...,z′ ) 1 i−1 is used in line 7, who has identical value in the first layer as the Babai point and a variety of pi−2 × · · · × p1 routes in the remaining layers. Lastly, line 10 restores the upper triangular feature of R via a lightweight 2×2 Givens rotation matrix and line 11 balances the unitary matrix. The toy example below may help to understand our algorithm. Example 1. Suppose we are  1 R= 0 0 reducing a matrix  0.4 0 1 0.52  0 1 in round i = 3 and executing from Line 4 of Algorithm 2. For the PNP algorithm, we set the L = 3 routes as p2 × p1 = 3 × 1. The three nearest integers to r2,3 /r2,2 are 1, 0, 2, so the corresponding PNP routes are  ⊤  r3,(1,1) = [−0.4, −0.48, 1] , r3,(2,1) = [0, 0.52, 1]⊤ ,   r3,(3,1) = [−0.8, −1.48, 1]⊤ , 15 else i ← i + 1; D ← QR. In essence, this algorithm attempts to minimize the basis length while keeping the Siegel condition, and the PNP algorithm offers flexible efforts to do so. If we replace lines 4 and 5 by using the Babai point and delete line 7, Algorithm 2 degrades to the classic LLL algorithm [11]. C. Properties of boosted LLL 2 When the boosted LLL algorithm terminates, δri−1,i−1 − ri−1,i 2 2 2 ( ri−1,i−1 −⌊ ri−1,i−1 ⌉) ri−1,i−1 ≤ ri,i holds for all 2 ≤ i ≤ n, which ensure the Siegel properties hold: ri−1,i |ri−1,i−1 | ≤ β|ri,i |. (28) Assume the PNP algorithm has parameters (pk−1 , . . . , p1 ) for 2 ≤ k ≤ n, then πRΓi (ri ) is contained in V0 (RΓi ) ∪ P( q1 r∗1 , . . . , qi−1 r∗i−1 ), where r∗1 , . . . , r∗i−1 are the GSO vectors of RΓi . Though this region can be much larger than  P( r∗1 , . . . , r∗i−1 ), we have 1X 2 r (29) 4 j<i j,j   / if πRΓi (ri ) ∈ P( r∗1 , . . . , r∗i−1 ). If πRΓi (ri ) ∈ P( r∗1 , . . . , r∗i−1 ), we can always find the Babai point P 2 2 2 2 due to (20) r′i such that kri k < kr′i k ≤ ri,i + 41 j<i rj,j and (27), so condition (29) always holds in boosted LLL. With (28) and (29), classical properties of LLL can be trivially proved: 2 2 kri k ≤ ri,i + 8 l(D) ≤ β n−1 λn (D), (30) ξ(D) ≤ β n(n−1)/2 . (31) Since we have devoted much effort to implement the length reductions, (30) and (31) are the least bounds that we should expect from boosted LLL. However, moving any step forward seems difficult because even using CVP as length reduction still fails to generate a better explicit bound than (29). The difficulty of improving bounds on lengths exists in all variants of LLL, including the LLL with deep insertions (LLL-deep) [18]. In this regard, boosted LLL only serves as an ameliorated practical algorithm. D. Implementation and Complexity The total number of loops K in Algorithm 2 equals to the number testing condition (21), whose number of positive and negative tests are denoted as K + and K − , respectively. The total number of negative test is   1 Pot(D0 ) Pot(D0 ) − , (32) = ×ln K ≤ log1/δ Pot(DK ) ln(1/δ) Pot(DK ) where D0 , DK are the initial basis and the basis after K loops, respectively. In the fixed traversing strategy, we also have K + ≤ K − + n − 1. We first show how to choose δ such that the boosted LLL algorithm has the best performance 1 remains to be a polynomial number. After that, while ln(1/δ) Pot(D0 ) Pot(DK ) is evaluated to complete our complexity analysis. 1) Optimal δ: Among literature, δ is often chosen arbitrarily close to 1 while explanations are lacking. In Micciancio’s book [1, Lem. 2.9], it is shown if δ = 1/4 + (3/4)n/(n−1) , 1 ≤ nc for all c > 1. More generally, we can define then ln(1/δ) an optimal principle of choosing δ, i.e., 1 δ(a , n) = ∗ + a ∗  a∗ − 1 a∗ n/(n−1) where a∗ = 1−e1 −1 . With such settings, three distinctive properties exist: 1 c ln(1/δ) ≤ n for all n; δ is asymptotically close to 1 so that the algorithm has the best performance; and it is the smallest value satisfying the previous two attributes (the fastest one among the class of best performance). Proposition 6 justifies these claims and the proof is given in Appendix E. Proposition 6. For arbitrary constants a > 1, c > 1, if
n/(n−1) δ(a, n) = a1 + a−1 , then for all n, a 1 ≤ nc . ln(1/δ) (33) Let a = limn→∞ arg mina δ(a, n) be defined as the universal good constant, then 1 . 1 − e−1 0) With reference to [36], we have Ψ(D ψ(D0 ) ≤ κ(D0 ), where κ(D0 ) is the condition number of D0 . So if the condition number of the input basis satisfies ln κ(D0 ) = O(ln n), then the number of iterations in boosted LLL is K ≤ 2K − +n−1 = O(nc+2 ln n), where c > 1 is a constant arbitrarily close to 1. By further counting the number of flops inside and outside the loop of Algorithm 2, the total complexity of boosted LLL is O(Ln4+c ln n). Remark 3. The complexity analysis above is quite general. For instance, if D0 is Gaussian, then it follows from [36] that E(ln κ(D0 )) ≤ ln n + 2.24. In the application to IF [21], we can also take a detour to employ this property of Gaussian matrices. Firstly, the condition number of the input basis D0 would increase if the signal to noise ratio (SNR) rises, so it suffices to investigate the case for infinite SNR. The target then becomes the dual of a Gaussian random matrix that has the same condition number, so E(ln κ(D0 )) ≤ ln n+2.24 also holds in IF. V. A PPLICATION TO INTEGER FORCING In the context of optimizing the achievable rates of IF, some results based on LR have been presented in [37], where the difference between KZ and Minkowski is not obvious because the system size is small (2 × 2 or 4 × 4). Since we have improved the classic KZ and LLL, we will verify our boosted algorithms in IF by showing their performance about ergodic rates, orthogonal defects (inversely proportional to sum-rates), and complexity in flops. A. IF and SBP In this subsection, the IF transceiver architecture will be reviewed by using real value representations for simplicity. In a MIMO system with size n × n, each antenna has a message wi = [wi (1), wi (2), . . . , wi (k)]⊤ , where i ∈ {1, . . . , n}, wi ∈ Fkp , and Fp is a finite field with size p. As the conversion from message layer to physical layer, an encoder Ei : Fkp → Rn maps the length-k message wi into a lattice codeword xi = [xi (1), xi (2), . . . , xi (T )]⊤ , ∗ a∗ = 2) Total complexity in flops: Further define ψ(D) = 2 2 mini ri,i and Ψ(D) = maxi ri,i . Since our length reduction does not change ri,i while (25) shows that any swap can narrow the gap between ri−1,i−1 and ri,i , the number of negative tests between the initial basis D0 to the final basis DK is   Pot(D0 ) K − ≤ nc ln Pot(DK )   Ψ(D0 ) nc+1 (n + 1) . (35) ln ≤ 2 ψ(D0 ) (34) 2 where kxi k ≤ T P , T stands for code length and P stands for SNR. All encoders operate at the same lattice with the same rate: k RTX = log2 p. T 9 Let xi (j) be the jth symbol of xi , we may write the transmitted vector across all antennas in time j as x[j] = [x1 (j), . . . , xn (j)]⊤ . An observation y[j] ∈ Rn can be subsequently written as y[j] = Hx[j] + z[j] (36) in which H ∈ Rn×n denotes the MIMO channel matrix and z[j] is the additive white Gaussian noise (AWGN) with zi [j] ∼ N (0, 1). Let Y, X, and Z be the concatenated y[j], x[j] and z[j] from time slots 1 to T . In a linear receiver architecture, the receiver will project Y with a matrix B = [b1 , . . . , bn ]⊤ ∈ Rn×n to get the useful information AX for further decoding, BY = AX |{z} useful information + (BH − A)X + BZ. | {z } (37) effective noise ⊤ We choose A = [a1 , . . . , an ] ∈ Zn×n because these lattice codewords are closed under integer combinations. A should also be full rank to avoid losing information. For a preprocessing matrix B, the following computation rate can be obtained in the ith effective channel if the coding lattices satisfy goodnesses for channel coding and quantization [21] ! P 1 + , (38) R(H, ai , bi ) = log2 2 2 2 kbi k + P kH⊤ bi − ai k where log+ 2 (x) = max {log2 (x), 0}. The first step towards maximizing the rates is to set o n 2 2 /∂bi = 0 for a fixed IF co∂ kbi k + P H⊤ bi − ai efficient matrix A, which leads to 1 bi = (HH⊤ + I)−1 Hai . P Plug this into (38) and use Woodbury matrix identity for the inverse of a matrix, we have ! 1 P + R(H, ai ) = log2 , (39) 2 2 kDai k 1 where D = Λ− 2 V⊤ and VΛV⊤ = H⊤ H + 1/P I is the eigendecomposition. Achieving the optimum rate is therefore equivalent to solving SIVP on lattice L(D): arg min A∈Zn×n , rank(A)=n 2 max kDai k , i (40) 2 in which minA∈Zn×n , rank(A)=n maxi kDai k = λn (D). Now we explain how to obtain the estimations of messages. Upon quantizing BY to the fine lattice and modulo the coarse lattice in a row-wise manner [23], a converter Di : Rn → Fkp then maps the physical layer codeword to a message under finite field representations, i.e., ûi = [W⊤ ai ] mod p, W = [w1 , . . . , wn ]⊤ . These combinations are then collected, so as to decode the messages as ⊤ [ŵ1 , . . . , ŵn ]⊤ = A−1 p [û1 , . . . , ûn ] , where Ap is a full rank matrix over Zp and A−1 p is taken over the same field. With the above demonstrations in mind, there should be at least two reasons for us to restrain SIVP to SBP arg min 2 A∈GLn (Z) max kDai k . i (41) The first reason is about flexibility. With SBP, we can choose among lattice reduction algorithms from polynomial to exponential complexity with guaranteed properties, and these algorithms are still efficient when SNR is high. The second reason is about complexity, where the inverse of A over finite fields is much easier to calculate when A ∈ GLn (Z), and algorithms for SIVP or the successive minima problem (SMP) are generally more complicated than those of SBP [37], [38]. For instance, we can observe that for the enumeration routines of SMP, Minkowski reduction and boosted KZ reduction, one needs to verify the linear independence of a new vector with previous lattice vectors for SMP, while Minkowski reduction only needs to check the greatest common divisor of the enumerated coefficients [10] and boosted KZ does not require such inspections. B. Simulation results This subsection examines the rates and complexity performance when applying the proposed boosted KZ and boosted LLL algorithms for IF receivers. We show the achievable rates rather than the bit error rates of IF MIMO receivers, since the latter depend on which capacity-approaching code for the AWGN channel is used at the transmitter. All simulations are performed on real matrices with random entries drawn from i.i.d. Gaussian distributions N (0, 1). Results in the figures are all averaged from 103 Monte Carlo runs. The boosted LLL algorithm is referred as “b-LLL-L” with L being the total number of branches in the PNP algorithm, i.e., L = pi−1 pi−2 · · · p1 remains unchanged for different columns i’s. If L = 1, this version means only adding a rejection operation to the classic LLL algorithm [11]. When L = 3 or L = 9, we expand 3 branches in the first or first two layers of the PNP algorithm. Regarding other typical variants of LLL, such as the effective LLL [12] and greedy LLL [17], they all boil down to the same performance as LLL if we implement a full size reduction at the end of their algorithms, so we omit comparing our algorithms with these variants. The boosted KZ algorithm (“b-KZ”) is implemented as described in Algorithm 1. To ensure a fair comparison, the KZ algorithm follows the same routine as Algorithm 1 except replacing the “CVP subroutine” with a size reduction. Minkowski reduction is also included as our reference with the label “Minkow”, whose implementation follows [10, Sec. V]. 1) Achievable rate: The actual achievable rate of the IF receivers can be quantitatively evaluated by the ergodic rate defined by [37]   RE = E n min R(H, ai ) , i where the expectation is taken over different realizations of H, and R(H, ai ) was defined in (39). In Fig. 2, we have plotted the rate performance of different LR algorithms in a 20 × 20 real-valued MIMO channel, in which the channel capacity 21 log2 (det(I + P HH⊤ )) serves 10 70 1100 1000 60 900 50 700 LLL b−LLL−1 b−LLL−3 b−LLL−9 KZ b−KZ Minkow Capacity 40 30 20 0 5 10 OD Ergodic rate 800 600 LLL b−LLL−1 b−LLL−3 b−LLL−9 KZ b−KZ Minkow 500 400 300 200 100 0 15 5 SNR/dB Fig. 2. SNR versus ergodic rate for different LR algorithms. 6 10 120 110 4 100 LLL b−LLL−1 b−LLL−3 b−LLL−9 KZ b−KZ Minkow Capacity 90 80 70 60 10 OD Ergodic rate 15 Fig. 4. SNR versus OD for different LR algorithms. 130 50 15 10 SNR/dB LLL b−LLL−1 b−LLL−3 b−LLL−9 KZ b−KZ Minkow 2 10 0 20 25 30 dimension n 10 5 10 15 20 dimension n 25 30 Fig. 3. Dimension versus ergodic rate for different LR algorithms. Fig. 5. Dimension versus OD for different LR algorithms. as an upper bound. In the figure, the b-LLL-1 algorithm has higher rates than KZ and LLL, and the improvements after we increase the list number to L = 3, 9 can still be spotted in this crowded figure. The b-KZ method attains almost the same rates as those of Minkowski reduction. KZ reduction does not offer better rates than LLL because KZ only guarantees to yield a basis with the smallest potential, and both of them are under the curse Proposition 1. In Fig. 3, we fixed the SNR to be 20dB and study how the size of the system is affecting their ergodic rates. From this graph, the differences of rates among different LR methods amplify as dimension n increases, and their mutual relations are the same as those of Fig. 2. 2) Orthogonal defect: The ergodic rate RE is only determined by the basis length l(D). To evaluate the sum-rates for all data streams, OD’s can be employed which are proportional to the length products of basis vectors. Such a quantity can reveal the gaps between different algorithms more vividly. In Figs. 4 (fixed size of 20 × 20) and 5 (fixed SNR of 20dB), we have plotted the SNR versus OD, and dimension versus OD relations for distinct lattice reduction algorithms. From these two figures, several phenomenons can be observed. Boosted KZ cannot surpass Minkowski reduction but remains close to it. The performance improvements from b-LLL-1, to b-LLL-3, b-LLL-9 are approximately proportional to the increment in the list size L. One interesting thing to observe from Fig. 4 is, the performance gaps between boosted and non-boosted algorithms are becoming larger as P rises. Since 1 D = Λ− 2 V⊤ and VΛV⊤ = H⊤ H + 1/P I, the increment of P is, intrinsically, changing the goodness of the corresponding minimal basis. It also says that the possibility of size reduction being suboptimal would increase if the lattice bases tend to be more random. Lastly, Fig. 5 shows an evident “Minkow < bKZ < b-LLL-9 < b-LLL-3 < b-LLL-1 < KZ < LLL” relation about OD, and their OD values are much better than their theoretical bounds (see e.g., Eqs. (13) and (31)). 3) Complexity: In addition to our theoretical analysis on the complexity of the proposed algorithms, we further compare their empirical costs by the expected number of flops, which are clearly shown in Fig. 6. Not surprisingly, the b- 11 KZ algorithm spends about 1.5 times the efforts of KZ in the dimensions depicted in Fig. 6, and the b-LLL-1, 3, 9 algorithms costs around 1, 1.5, and 3 times the efforts of LLL. Both b-KZ and KZ reductions have dramatically lower number of flops than Minkowski reduction. Moreover, the boosted LLL algorithms have much smaller complexity than KZ while reducing the bases more effectively as Figs. 2 to 5 have revealed. To sum up, concerning the complexity-performance tradeoffs as well as the theoretical bounds, the boosted KZ and LLL algorithms can be the ideal candidates for reducing lattice bases in IF with exponential and polynomial complexity, respectively. x 10 LLL b−LLL−1 b−LLL−3 b−LLL−9 KZ b−KZ Minkow complexity in flops 6 5 4 A PPENDIX B P ROOF OF PROPOSITION 3 Proof: Regarding (9), first recall a fact that we cannot produce n independent vectors by using a lattice of rank n − 1, then among λ1 (D), . . . , λn (D), at P least one of them, n−1 xi di where e.g., λi′ (D), corresponds to v = xn dn + i=1 Pn−1 i=1 xi di ∈ L(DΓn ), xn ∈ Z \ 0. With QR decomposition [Q, R] = qr(D),   i−1 n X X rj,i qj  xi ri,i qi + v = 6 7 there are two lattice points (0 and v) inside Vv (DΓi ), which contradicts the basic property of a Voronoi cell, i.e., there can be only one lattice point inside a Voronoi region. We proceed to prove (42). Since πDΓi (di ) is quantized to v, their difference πDΓi (di ) − v lies inside the Voronoi cell 2 V0 (DΓi ), which yields hπDΓi (di ) − v, wi < kwk /2 for all w ∈ L(DΓi ) and w 6= 0. As v−πDΓi (di ) ∈ V0 (DΓi ), choose 2 an instance of w = v 6= 0 for hv − πDΓi (di ), wi < kwk /2, then (42) is obtained. = 2 n−1 X i=1 1 0 j=1 i=1 3 5 10 dimension n 15 20 |rn,n |2 ≤ QUESTIONS We have only demonstrated the theoretical superiority of boosted KZ over KZ and Minkowski, while Minkowski reduction still yields shorter vectors in our simulations. One interesting open question is whether there exist better performance bounds for Minkowski reduction. It is also of sufficient interests to improve the performance analysis on boosted LLL. A PPENDIX A P ROOF OF PROPOSITION 2 2 Proof: First of all, the reducible condition kdi − vk < 2 < kdi k can be reformulated as πDΓi (di ) − v 2 πDΓi (di ) , which becomes equivalent to 2 kvk2 /2 < hπDΓi (di ), vi. xi ri,i qi + i−1 X j=1  rj,i qj  + xn rn,n qn . Notice that −v also corresponds to λi′ (D), so we can confine xn > 0 and consider the following two cases. 1) If xn > 1, it is  observed that kvk = Pi−1 Pn−1  ≥ xn |rn,n | r q x i,i i + j=1 rj,i qj + xn rn,n qn i=1 i because the qi ’s are orthogonal, which yields Fig. 6. Dimension versus complexity for different LR algorithms. VI. O PEN  (42) It is necessary to show hπDΓi (di ), vi > 0 for v 6= 0 so that the inequality we pursuit makes sense. We give a proof by contradiction. Suppose that θ(πDΓi (di ), v) > π/2, due to symmetricity of the Voronoi cell Vv (DΓi ), there exists a symmetric point d′i of πDΓi (di ) on v, such that θ(d′i , v) > π/2. Define the half-space of v as Hv = n {x convex combination among  ∈ R | hv, xi > 0}, then the Vv (DΓi ) ∩ Hv , πDΓi (di ), d′i must include the origin. Then 1 1 λi′ (D)2 ≤ λn (D)2 . x2n 4 (43) We then proceed to bound the last term in (7). For 1 ≤ i ≤ n, the covering radius of lattice L(DΓn ) is ρ(DΓn ) = ≤ max dist (x, L(DΓn )) x v un−1 uX 2 , rk,k 1/2t (44) k=1 where the inequality is obtained after choosing x as a “deep hole” [39, P. 33] and solving this CVP by applying Babai’s nearest plane algorithm [40]. Since boosted KZ still assures ⊥ |rk,k | = λ1 (πD ([dk , . . . , dn ])), and the projection of the Γk kth successive minimum in D onto the orthogonal complement of DΓk must have a least one non-zero coefficient for [dk , . . . , dn ], we have |rk,k | ≤ λk (D); plug this into (44), v un−1 uX λk (D)2 . (45) ρ(DΓn ) ≤ 1/2t k=1 Put (43) and (45) into (7), then kdn k2 ≤ 14 λn (D)2 + Pn−1 1 n 2 2 k=1 λk (D) ≤ 4 λn (D) . 4 2) If xn = 1, recall that our length reduction by CVP (line 11) ensures dn is the shortest vector among the set 12 n o Pn−1 dn + i=1 zi di | ∀ zi ∈ Z , so kdn k ≤ λi′ (D) ≤ λn (D) in such a scenario. Combining 1) and 2) proves (9).  As for (8), since all sublattices L(DΓi+1 ) | 1 ≤ i ≤ n are also boosted KZ it follows from the proved (9) n reduced, √ o i that kdi k ≤ max 1, 2 λi (DΓi+1 ). With |ri,i | ≤ λi (D) and √ the bound for the covering radius, we also have kdi k ≤ i+3 2 λi (D) for all i. So choosing the minimum among them yields (8). A PPENDIX C P ROOF OF PROPOSITION 4 Proof: Since |ri,i | = λ1 (Ri:n,i:n ), we apply Minkowski’s second theorem [31, P. 202] to lattices L(Ri:n,i:n ) with 2 1 ≤ i ≤ n − 1, then we have rn−j+1,n−j+1 ≤ Q 1/j j 2 γj . As those of [28, Prop. 4.2], we k=1 rn−k+1,n−k+1 cancel duplicated terms in this inequality and use induction from L(Rn−1:n,n−1:n ) to L(R1:n,1:n ), then 2 rn−j+1,n−j+1 ≤ γj j Y 1/(k−1) γk k=2 2j 3 , ! 2 rn,n . (46) Qj
2k 1/(k−1) 3 As γj ≤ we define g(j) = k=2 evaluate this term. Let z = k − 1, it can be shown that  !  j−1 X 8j 2z + 2 1 g(j) = exp ln 9 3 z z=2 ! j−1 X (a) ln(z) 8j exp ≤ 9 z z=2   Z j−1 (b) 8j ln(z) ≤ exp dz 9 z 1 8j = (j − 1)ln(j−1)/2 , 9 8j 2 (j − 1)ln(j−1)/2 rn,n , 9 2 ≤ ≤  i X 8j 1 2 1 + (j − 1)ln(j−1)/2  ri,i 4 j=2 9   2i 2 1 + (i − 1)1+ln(i−1)/2 ri,i 9 for 1 ≤ i ≤ n, so (12) is proved. (48) All bases can be evaluated via Eq. (48) because either [u, v] or [u, −v] must be acute. In the boosted KZ algorithm, if we cannot reduce the length of di with only d1 , then |cd1 |/(d21 + d22 ) < 1/2. In the other direction, reducing d1 by di is also impossible because d1 is already the shortest, so |d1 /c| < 1/2. By combining these two non-reducible conditions and the acute condition of d1 /c ≥ 0, requirements in (48) can be met. A PPENDIX E P ROOF OF PROPOSITION 6 Proof: The proof of (33) follows those in [1, Lem. 2.9]. To prove (33), it suffices to prove n→∞ and 1 − e(−(1/n) ) a−1 a−1 n/(n−1) ≤ 1, a −( a ) (49) where its l.h.s. is an indeterminate form. Replace n by another variable x as n = x+1 x , then by using L’Hospital’s Rule, the l.h.s. of (49) becomes c c 1 x c−1 −1/(1+1/x) ) c( x+2 1 − e−1/(1+1/x) (x+1)2 e lim a−1 = lim =0 a−1 x+1 x+1 ln( a−1 ) x→0 x→0 −( a−1 a −( a ) a ) a and thus (49) is proved. n) n 1 1/(n−1) = (n−1)a − As for (34), let ∂δ(a, 2 (1 − a ) ∂a 1 ′ a2 = 0, we obtain the stationary point of δ(a, n) as a = n) ∂δ(a, n) 1 < 0 if a ∈ (1, a′ ) and ∂δ(a, > 1−(1−1/n)n−1 , where ∂a ∂a 0 if a ∈ (a′ , ∞). Notice that (1 − 1/n)n−1 = e after using L’Hospital’s Rule again, we have ln(1−1/n) 1/(n−1) , then 1 1 1 = = lim . n−1 −1+1/n n→∞ 1 − e n→∞ 1 − (1 − 1/n) 1 − e−1 lim (47) for 2 ≤ j ≤ n, and this is the condition in boosted KZ that corresponds to the Siegel condition in LLL. Let j = n and apply (47) to each of L(R1:n,1:i ) for 2 ≤ i ≤ n, then we obtain ln(j−1)/2 2 rj,j . 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Sparse CCA: Adaptive Estimation and Computational Barriers ∗ Chao Gao1, Zongming Ma2 , and Harrison H. Zhou1 arXiv:1409.8565v4 [stat.ME] 4 Apr 2016 1 2 Yale University University of Pennsylvania Abstract Canonical correlation analysis is a classical technique for exploring the relationship between two sets of variables. It has important applications in analyzing high dimensional datasets originated from genomics, imaging and other fields. This paper considers adaptive minimax and computationally tractable estimation of leading sparse canonical coefficient vectors in high dimensions. First, we establish separate minimax estimation rates for canonical coefficient vectors of each set of random variables under no structural assumption on marginal covariance matrices. Second, we propose a computationally feasible estimator to attain the optimal rates adaptively under an additional sample size condition. Finally, we show that a sample size condition of this kind is needed for any randomized polynomial-time estimator to be consistent, assuming hardness of certain instances of the Planted Clique detection problem. The result is faithful to the Gaussian models used in the paper. As a byproduct, we obtain the first computational lower bounds for sparse PCA under the Gaussian single spiked covariance model. Keywords. Convex programming, group-Lasso, Minimax rates, Computational complexity, Planted Clique, Sparse CCA (SCCA), Sparse PCA (SPCA) 1 Introduction Canonical correlation analysis (CCA) [23] is a classical and important tool in multivariate statistics [1, 31]. For two random vectors X ∈ Rp and Y ∈ Rm , at the population level, CCA finds successive vectors uj ∈ Rp and vj ∈ Rm (called canonical coefficient vectors) that solve max a′ Σxy b, a,b subject to a′ Σx a = b′ Σy b = 1, a′ Σx ul = b′ Σy vl = 0, ∀0 ≤ l ≤ j − 1, (1) where Σx = Cov(X), Σy = Cov(Y ), Σxy = Cov(X, Y ), u0 = 0, and v0 = 0. Since our primary interest lies in the covariance structure among X and Y , we assume that their means are zeros from here on. Then the linear combinations (u′j X, vj′ Y ) are the j-th pair of canonical variates. This technique has been widely used in various scientific fields to explore the relationship between two sets of variables. In practice, one does not have knowledge about ∗ The research of C. Gao and H. H. Zhou is supported in part by NSF Career Award DMS-0645676 and NSF FRG Grant DMS-0854975. The research of Z. Ma is supported in part by NSF Career Award DMS-1352060. 1 b x, the population covariance, and Σx , Σy , and Σxy are replaced by their sample versions Σ b y , and Σ b xy in (1). Σ Recently, there have been growing interests in applying CCA to analyzing high-dimensional datasets, where the dimensions p and m could be much larger than the sample size n. It has been well understood that classical CCA breaks down in this regime [25, 4, 18]. Motivated by genomics, neuroimaging and other applications, people have become interested in seeking sparse leading canonical coefficient vectors. Various estimation procedures imposing sparsity on canonical coefficient vectors have been developed in the literature, which are usually termed sparse CCA. See, for example, [42, 43, 33, 22, 27, 38, 3]. The theoretical aspect of sparse CCA has also been investigated in the literature. A useful model for studying sparse CCA is the canonical pair model proposed in [14]. In particular, suppose there are r pairs of canonical coefficient vectors (and canonical variates) among the two sets of variables, then the model reparameterizes the cross-covariance matrix as Σxy = Σx U ΛV ′ Σy , where U ′ Σx U = V ′ Σy V = Ir . (2) Here U = [u1 , ..., ur ] and V = [v1 , ..., vr ] collect the canonical coefficient vectors and Λ = diag(λ1 , . . . , λr ) with 1 > λ1 ≥ · · · ≥ λr > 0 are the ordered canonical correlations. Let Su = supp(U ) and Sv = supp(V ) be the indices of nonzero rows of U and V . One way to impose sparsity on the canonical coefficient vectors is to require the sizes of Su and Sv to be small, namely |Su | ≤ su and |Sv | ≤ sv for some su ≤ p and sv ≤ m. Under this model, Gao et al. [18] showed that the minimax rate for estimating U and V under the joint loss function b Vb ′ − U V ′ k2 is kU F ep em  1  r(s + s ) + s log + s log . (3) u v u v nλ2r su sv However, to achieve the rate, Gao et al. [18] used a computationally infeasible and nonadaptive procedure, which requires exhaustive search of all possible subsets with the given cardinality and the knowledge of su and sv . Moreover, it is unclear from (3) whether the estimation error of U depends on the sparsity and the ambient dimension of V and vice versa. The goal of the present paper is to study three fundamental questions in sparse CCA: (1) What are the minimax rates for estimating the canonical coefficient vectors on the two sets of variables separately? (2) Is there a computationally efficient and sparsity-adaptive method that achieves the optimal rates? (3) What is the price one has to pay to achieve the optimal rates in a computationally efficient way? 1.1 Main contributions We now introduce the main contributions of the present paper from three different viewpoints as suggested by the three questions we have raised. b Vb ′ − U V ′ k2 studied by [18] characterizes the Separate minimax rates The joint loss kU F joint estimation error of both U and V . In this paper, we provide a finer analysis by studying individual estimation errors of U and V under a natural loss function that can be interpreted as prediction error of canonical variates. The exact definition of the loss functions is given in Section 2. Separate minimax rates are obtained for U and V . In particular, we show that the minimax rate of convergence in estimating U depends only on n, r, λr , p and su , but not on 2 either m or sv . Consequently, if U is sparser than V , then convergence rate for estimating U can be faster than that for estimating V . Such a difference is not reflected by the joint loss, since its minimax rate (3) is determined by the slower of the rates of estimating U and V . Adaptive estimation As pointed out in [14] and [18], sparse CCA is a more difficult problem than the well-studied sparse PCA. A naive application of sparse PCA algorithm to sparse CCA can lead to inconsistent results [14]. The additional difficulty in sparse CCA mainly comes from the presence of the nuisance parameters Σx and Σy , which cannot be estimated consistently in a high-dimensional regime in general. Therefore, our goal is to design an estimator that is adaptive to both the nuisance parameters and the sparsity levels. Under the canonical pair model, we propose a computationally efficient algorithm. The algorithm has two stages. In the first stage, we propose a convex program for sparse CCA based on a tight convex relaxation of a combinatorial program in [18] by considering the smallest convex set containing all matrices of the form AB ′ with both A and B being rankr orthogonal matrices. The convex program can be efficiently solved by the Alternating Direction Method with Multipliers (ADMM) [16, 11]. Based on the output of the first stage, we formulate a sparse linear regression problem in the second stage to improve estimation b and Vb can be obtained via a group-Lasso algorithm [45]. accuracy, and the final estimator U Under the sample size condition that n ≥ Csu sv log(p + m)/λ2r (4) b and Vb recover the true canonical for some sufficiently large constant C > 0, we show U coefficient matrices U and V within optimal error rates adaptively with high probability. Computational lower bound We require the sample size condition (4) for the adaptive procedure to achieve optimal rates of convergence. Assuming hardness of certain instances of the Planted Clique detection problem, we provide a computational lower bound to show that a condition of this kind is unavoidable for any computationally feasible estimation procedure to achieve consistency. Up to an asymptotically equivalent discretization which is necessary for computational complexity to be well-defined, our computational lower bound is established directly for the Gaussian canonical pair model used throughout the paper. An analogous sample size condition has been imposed in the sparse PCA literature [24, 29, 13, 37], namely n ≥ Cs2 log p/λ2 where s is the sparsity of the leading eigenvector and λ the gap between the leading eigenvalue and the rest of the spectrum. Berthet and Rigollet [7] showed that if there existed a polynomial-time algorithm for a generalized sparse PCA detection problem while such a condition is violated, the algorithm could be made (in randomized polynomial-time) into a detection method for the Planted Clique problem in a regime where it is believed to be computationally intractable. However, both the null and the alternative hypotheses in the sparse PCA detection problem were generalized in [7] to include all multivariate distributions whose quadratic forms satisfy certain uniform tail probability bounds and so the distributions need not be Gaussian or having a spiked covariance structure [24]. The same remark also applies to the subsequent work on sparse PCA estimation [39]. Hence, the computational lower bound in sparse PCA was only established for such enlarged parameter spaces. As a byproduct of our analysis, we establish the desired computational lower bound for sparse PCA in the Gaussian single spiked covariance model. 3 1.2 Organization After an introduction to notation, the rest of the paper is organized as follows. In Section 2, we formulate the sparse CCA problem by defining its parameter space and loss function. Section 3 presents separate minimax rates for estimating U and V . Section 4 proposes a two-stage adaptive estimator that is shown to be minimax rate optimal under an additional sample size condition. Section 5 shows a condition of this kind is necessary for all randomized polynomial-time estimator to achieve consistency by establishing new computational lower bounds for sparse PCA and sparse CCA. Section 6 presents proofs of theoretical results in Section 4. Implementation details of the adaptive procedure, numerical studies, additional proofs and technical details are deferred to the supplement [19]. 1.3 Notation For any t ∈ Z+ , [t] denotes the set {1, 2, ..., t}. For any set S, |S| denotes its cardinality and S c its complement. For any event E, 1{E} denotes its indicator function. For any a, b ∈ R, ⌈a⌉ denotes the smallest integer no smaller than a, ⌊a⌋ the largest integer qP no 2 larger than a, a ∨ b = max(a, b) and a ∧ b = min(a, b). For a vector u, ||u|| = i ui , P P ||u||0 = i 1{ui 6=0} , and ||u||1 = i |ui |. For any matrix A = (aij ) ∈ Rp×k , Ai· denotes its i-th row and supp(A) = {i ∈ [p] : kAi· k > 0}, the index set of nonzero rows, is called its support. For any subset J ⊂ [p] × [k], AJ = (aij 1{(i,j)∈J} ) ∈ Rp×k is obtained by keeping all entries in J and replacing all entries in J c with zeros. We write AJ1 J2 for AJ1 ×J2 and A(J1 J2 )c for A(J1 ×J2 )c . Note that AJ1 ∗ = AJ1 ×[k] ∈ Rp×k while AJ1 · stands for the corresponding nonzero submatrix of size |J1 | × k. In addition, PA ∈ Rp×p stands for the projection matrix onto the column space of A, O(p, k) denotes the set of all p × k orthogonal matrices and O(k) = O(k, k). Furthermore, σi (A) stands for the i-th largest singular value of A and σmax (A) = σq 1 (A), σmin (A) = σp∧k (A). The Frobenius norm and the operator norm of A P a2 and kAkop = σ1 (A), respectively. The l1 norm and the nuclear norm are kAkF = P i,j ij P are ||A||1 =P ij |aij | and kAk∗ = i σi (A), respectively. If A is a square matrix, its trace is Tr(A) = i aii . For two square matrices A and B, we write A  B if B − A is positive semidefinite. For any positive semi-definite matrix A, A1/2 denotes its principal square root that is positive semi-definite and satisfies A1/2 A1/2 = A. The trace inner product of two matrices A, B ∈ Rp×k is hA, Bi = Tr(A′ B). Given a random element X, L(X) denotes its probability distribution. The symbol C and its variants C1 , C ′ , etc. are generic constants and may vary from line to line, unless otherwise specified. The symbols P and E stand for generic probability and expectation when the distribution is clear from the context. 2 2.1 Problem Formulation Parameter space Consider a canonical pair model where the observed pairs of measurement vectors (Xi′ , Yi′ )′ , i = 1, . . . , n are i.i.d. from a multivariate Gaussian distribution Np+m (0, Σ) where   Σx Σxy , Σ= Σyx Σy 4 with the cross-covariance matrix Σxy satisfying (2). We are interested in the situation where the leading canonical coefficient vectors are sparse. One way to quantify the level of sparsity is to bound how many nonzero rows there are in the U and V matrices. This notion of sparsity has been used previously in both sparse PCA [13, 37] and sparse CCA [18] problems when one seeks multiple sparse vectors simultaneously. Recall that for any matrix A, supp(A) collects the indices of nonzero rows in A. Adopting the above notion of sparsity, we define F(su , sv , p, m, r, λ; M ) to be the collection of all covariance matrices Σ with the structure (2) satisfying 1. U ∈ Rp×r and V ∈ Rm×r with |supp(U )| ≤ su and |supp(V )| ≤ sv ; 2. σmin (Σx ) ∧ σmin (Σy ) ≥ M −1 and σmax (Σx ) ∨ σmax (Σy ) ≤ M ; (5) M −1 . 3. λr ≥ λ and λ1 ≤ 1 − The probability space we consider is  P(n, su , sv , p, m, r, λ; M ) = L((X1′ , Y1′ )′ , ..., (Xn′ , Yn′ )′ ) : iid (Xi′ , Yi′ )′ ∼ Np+m (0, Σ) with Σ ∈ F(su , sv , p, m, r, λ; M ) , (6) where n is the sample size. We shall allow su , sv , p, m, r, λ to vary with n, while M > 1 is restricted to be an absolute constant. 2.2 Prediction loss From now on, the presentation of definitions and results will focus on U only since those b = [b for V can be obtained via symmetry. Given an estimator U u1 , . . . , u br ] of the leading canonical coefficient vectors for X, a natural way of assessing its quality is to see how well it predicts the values of the canonical variables U ′ X ⋆ ∈ Rr for a new observation X ⋆ which b . This is independent of and identically distributed as the training sample used to obtain U leads us to consider the following loss function b, U) = L(U b ′ X ⋆ − U ′ X ⋆ k2 , E⋆ kW ′ U (7) b W − U )′ Σx (U b W − U )]. Tr[(U (8) inf W ∈O(r) b , U ) is still a random quantity where E⋆ means taking expectation only over X ⋆ and so L(U b . Since L(U b , U ) is the expected squared error for predicting the due to the randomness of U ′ ⋆ ′ ⋆ b canonical variables U X via U X , we refer to it as prediction loss from now on. It is worth noting that the introduction of an r × r orthogonal matrix W is unavoidable. To see this, we can simply consider the case where λ1 = · · · = λr = λ in (2), then we can replace the pair (U, V ) in (2) by (U W, V W ) for any W ∈ O(r). In other words, the canonical coefficient vectors are only determined up to a joint orthogonal transform. If we work out the E⋆ part in the definition (7), then the loss function can be equivalently defined as b, U) = L(U inf W ∈O(r) b , X ⋆ and Σx in (7) and (8) By symmetry, we can define L(Vb , V ) by simply replacing U , U with V , Vb , Y ⋆ and Σy . 5 A related loss function is kPUb −PU k2F measuring the difference between two subspaces. By b , U ) is a stronger Proposition 9.2 in the supplementary material [19], the prediction loss L(U 2 b loss function. That is, kPUb − PU kF ≤ CL(U , U ) for some constant C > 0 only depending on b , U ) is strictly stronger. To see this, let Σx = Ip , U ∈ O(p, r) and U b = 2U . M . Actually, L(U 2 b , U ) = inf W ∈O(r) kΣ1/2 b Then, kPUb − PU k2F = 0, while L(U x (U W − U )kF = inf W ∈O(r) (5r − b , U ), and provide brief Tr(W )) = r > 0. In this paper, we will focus on the stronger loss L(U 2 remarks on results for kPUb − PU kF . 3 Minimax Rates We first provide a minimax upper bound using a combinatorial optimization procedure, and then show that the resulting rate is optimal by further providing a matching minimax lower bound. Let (Xi′ , Yi′ )′ ∈ Rp+m, i = 1, . . . , n, be i.i.d. observations following Np+m (0, Σ) for some Σ ∈ F(su , sv , p, m, r, λ; M ). For notational convenience, we assume the sample size is divisible by three, i.e., n = 3n0 for some n0 ∈ N. Procedure To obtain minimax upper bound, we propose a two-stage combinatorial op0 timization procedure. We split the data into three equal size batches D0 = {(Xi′ , Yi′ )′ }ni=1 , 2n0 ′ ′ ′ ′ ′ ′ n D1 = {(Xi , Yi ) }i=n0 +1 and D2 = {(Xi , Yi ) }i=2n0 +1 , and denote the sample covariance mab (j) b (j) b (j) trices computed on each batch by Σ x , Σy and Σxy for j ∈ {0, 1, 2}. b (0) , Vb (0) ) which solves the following program: In the first stage, we find (U max L∈Rp×r ,R∈Rm×r b (0) R), Tr(L′ Σ xy b (0) L = R′ Σ b (0) R = Ir , and subject to L′ Σ x y (9) |supp(L)| ≤ su , |supp(R)| ≤ sv . b (1) solving In the second stage, we further refine the estimator for U by finding U min L∈Rp×r ′ b (1) b (0) b (1) Tr(L′ Σ ) x L) − 2 Tr(L Σxy V (10) subject to |supp(L)| ≤ su . b (1) , defined as The final estimator is a normalized version of U b =U b (1) ((U b (1) )′ Σ b (2) b (1) )−1/2 . U x U (11) The purpose of sample splitting employed in the above procedure is to facilitate the proof. Theory and discussion The program (9) was first proposed in [18] as a sparsity constrained version of the classical CCA formulation. However, the resulting estimator will have a convergence rate that involves the sparsity level sv and the ambient dimension m of the V 6 matrix [18, Theorem 1], which is sub-optimal. The second stage in the procedure is thus proposed to further pursue the optimal estimation rates. First, if we were given the knowledge of V , then the least square solution of regressing V ′ Y ∈ Rr on X ∈ Rp is U Λ = argmin EkY ′ V − X ′ Lk2 L∈Rp×r = argmin Tr(L′ Σx L) − 2 Tr(L′ Σxy V ) + Tr(V ′ Σy V ) L∈Rp×r (12) = argmin Tr(L′ Σx L) − 2 Tr(L′ Σxy V ), L∈Rp×r where the expectation is with respect to the distribution (X ′ , Y ′ )′ ∼ Np+m (0, Σ). The second equality results from taking expectation over each of the three terms in the expansion of the square Euclidean norm, and the last equality holds since Tr(V ′ Σy V ) does not involve the argument to be optimized over. In fact, from the canonical pair model, one can easily derive a regression interpretation of CCA, V ′ Y = ΛU ′ X + E, where E ∼ N (0, Ir − Λ2 ). Then, (10) is a least square formulation of the regression interpretation. However, CCA is different from regression because the response V ′ Y depends on an unknown V . Comparing (10) with (12), it is clear that (10) is a sparsity constrained version of (12) where the knowledge of V and the covariance matrix Σ are replaced by the initial estimator Vb (0) and sample covariance matrix b (1) can be viewed as an estimator of U Λ. Hence, from an independent sample. Therefore, U a final normalization step is taken in (11) to transform it to an estimator of U . We now state a bound for the final estimator (11). Theorem 3.1. Assume ep em  1 r(su + sv ) + su log + sv log ≤c n su sv (13) for some sufficiently small constant c > 0. Then there exist constants C, C ′ > 0 only depending on c such that   b , U ) ≤ C su r + log ep , L(U (14) nλ2 su with P-probability at least 1 − exp (−C ′ (su + log(ep/su ))) − exp (−C ′ (sv + log(em/sv ))) uniformly over P ∈ P(n, su , sv , p, m, r, λ; M ). Remark 3.1. The paper assumes that M is a constant. However, it is worth noting that the minimax upper bound of Theorem 3.1 does not depend on M even if M is allowed to grow with n. To be specific, assume the eigenvalues of Σx are bounded in the interval [M1 , M2 ].
b , U ) would still be 1 2 su r + log ep , because the dependence on The convergence rate of L(U su nλ M1 , M2 has been implicitly built into the prediction loss. On the other hand, a convergence
1 ep
M2 2 rate for the loss kPUb − PU kF would be M1 nλ2 su r + log su , with an extra factor of the condition number of Σx . Under assumption (13), Theorem 3.1 achieves a convergence rate for the prediction loss in U that does not depend on any parameter related to V . Note that the probability tail still ′ involves m and sv . However, it can be shown that exp (−C ′ (sv + log(em/sv ))) ≤ m−C /2 , and so the corresponding term in the tail probability goes to 0 as long as m → ∞. The optimality of this upper bound can be justified by the following minimax lower bound. 7 v Theorem 3.2. Assume that r ≤ su ∧s 2 . Then there exists some constant C > 0 only depending on M and an absolute constant c0 > 0, such that    C ep  b inf sup P L(U , U ) ≥ c0 ∧ su r + log ≥ 0.8, b P∈P nλ2 su U where P = P(n, su , sv , p, m, r, λ; M ). By Theorem 3.1 and Theorem 3.2, the rate in (14), whenever it is upper bounded by a constant, is the minimax rate of the problem. 4 Adaptive and Computationally Efficient Estimation Section 3 determines the minimax rates for estimating U under the prediction loss. However, there are two drawbacks of the procedure (9) – (11). One is that it requires the knowledge of the sparsity levels su and sv . It is thus not adaptive. The other is that in both stages one needs to conduct exhaustive search over all subsets of given sizes in the optimization problems (9) and (10), and hence the computation cost is formidable. In this section, we overcome both drawbacks by proposing a two-stage convex program approach towards sparse CCA. The procedure is named CoLaR, standing for Convex program with group-Lasso Refinement. It is not only computationally feasible but also achieves the minimax estimation error rates adaptively over a large collection of parameter spaces under an additional sample size condition. The issues related to this additional sample size condition will be discussed in more detail in the subsequent Section 5. 4.1 Estimation scheme The basic principle underlying the computationally feasible estimation scheme is to seek tight convex relaxations of the combinatorial programs (9) – (10). In what follows, we introduce convex relaxations for the two stages in order. As in Section 3, we assume that the data is b (j) b (j) b (j) split into three batches D0 , D1 and D2 of equal sizes and for j = 0, 1, 2, let Σ x , Σy and Σxy be defined as before. First stage By the definition of trace inner product, the objective function in (9) can be b xy R) = hΣ b xy , LR′ i. Since it is linear in F = LR′ , this suggests treating rewritten as Tr(L′ Σ ′ LR as a single argument rather than optimizing over L and R separately. Next, the support size constraints |supp(L)| ≤ su , |supp(R)| ≤ sv imply that the vector ℓ0 norm kLR′ k0 ≤ su sv . Applying the convex relaxation of ℓ0 norm by ℓ1 norm and including it as a Lagrangian term, we are led to consider a new objective function b (0) , F i − ρ||F ||1 , max hΣ xy F ∈Rp×m (15) P where F serves as a surrogate for LR′ , kF k1 = i∈[p],j∈[m] |Fij | denotes the vector ℓ1 norm of the matrix argument, and ρ is a penalty parameter controlling sparsity. Note that (15) is the maximization problem of a concave function, which becomes a convex program if the 8 constraint set is convex. Under the identity F = LR′ , the normalization constraint in (9) reduces to 1/2 1/2 b (0) b (0) (Σ F (Σ ∈ Or = {AB ′ : A ∈ O(p, r), B ∈ O(m, r)}. x ) y ) (16) Cr = {G ∈ Rp×m : kGk∗ ≤ r, kGkop ≤ 1} = conv(Or ) (17) 1/2 F (Σ 1/2 ∈ C where b (0) b (0) Naturally, we relax it to (Σ x ) y ) r is the smallest convex set containing Or . The relation (17) is stated in the proof of Theorem 3 of [40]. Combining (15) – (17), we use the following convex program for the first stage in our adaptive estimation scheme: max F ∈Rp×m b (0) , F i − ρ||F ||1 hΣ xy b (0) )1/2 F (Σ b (0) )1/2 k∗ ≤ r, k(Σ b (0) )1/2 F (Σ b (0) )1/2 kop ≤ 1. subject to k(Σ x y x y (18) Implementation of (18) is discussed in Section 10 in the supplement [19]. Remark 4.1. A related but different convex relaxation was proposed in [37] for the sparse PCA problem, where the set of all rank r projection matrices (which are symmetric) is relaxed to its convex hull – the Fantope {P : Tr(P ) = r, 0  P  Ip }. Such an idea is not directly applicable in the current setting due to the asymmetric nature of the matrices included in the set Or in (16). Remark 4.2. The risk of the solution to (18) for estimating U V ′ is sub-optimal compared to the optimal rates determined in [18]. See Theorem 4.1 below. Nonetheless, it leads to a reasonable estimator for the subspaces spanned by the first r left and right canonical coefficient vectors under a sample size condition, which is sufficient for achieving the optimal estimation rates for U and V in a further refinement stage to be introduced below. Although it is possible that some other penalty function rather than the ℓ1 penalty in (18) could also achieve this goal, ℓ1 is appealing due to its simplicity. Second stage Now we turn to the convex relaxation to (10) in the second stage. By the discussion following Theorem 3.1, if we view the rows of L as groups, then (10) becomes a least square problem with a constrained number of active groups. A well-known convex relaxation for such problems is the group-Lasso [45], where the number of active groups constraint is relaxed by bounding the sum of ℓ2 norms of the coefficient vector of each group. b (0) (resp. Vb (0) ) be the matrix consisting of its first r left b be the solution to (18) and U Let A (resp. right) singular vectors. Thus, in the second stage of the adaptive estimation scheme, we propose to solve the following group-Lasso problem: b (1) L) − 2 Tr(L′ Σ b (1) Vb (0) ) + ρu min Tr(L′ Σ x xy L∈Rp×m p X j=1 kLj· k, (19) P where pj=1 kLj· k is the group sparsity penalty, defined as the sum of the ℓ2 norms of all the row vectors in L, and ρu is a penalty parameter controlling sparsity. Note that the group sparsity penalty is crucial, since if one uses an ℓ1 penalty instead, only a sub-optimal rate can 9 b (1) , then our final estimator in the adaptive be achieved. Suppose the solution to (19) is U estimation scheme is its normalized version b =U b (1) ((U b (1) )′ Σ b (2) b (1) )−1/2 . U x U (20) As before, sample splitting is only used for technical arguments in the proof. Simulation results in Section 11 in the supplement [19] show that using the whole dataset repeatedly in (18) – (20) yields satisfactory performance and the improvement by the second stage is considerable. 4.2 Theoretical guarantees b to the convex program (18). We first state the upper bound for the solution A Theorem 4.1. Assume that su sv log(p + m) , (21) λ2 for some sufficiently large constant C1 > 0. Then there p exist positive constants γ1 , γ2 and ′ C, C only depending on M and C1 , such that when ρ = γ log(p + m)/n for γ ∈ [γ1 , γ2 ], n ≥ C1 b − U V ′ k2 ≤ C kA F su sv log(p + m) , nλ2 with P-probability at least 1 − exp(−C ′ (su + log(ep/su ))) − exp(−C ′ (sv + log(em/sv ))) for any P ∈ P(n, su , sv , p, m, r, λ; M ). Note that the error bound in Theorem 4.1 can be much larger than the optimal rate for joint estimation of U V ′ established in [18]. Nonetheless, under the sample size condition (21), b is close to U V ′ in Frobenius norm distance. This fact, together with it still ensures that A the proposed refinement scheme (19) – (20), guarantees the optimal rates of convergence for the estimator (20) as stated in the following theorem. Theorem 4.2. Assume (21) holds for some sufficiently large C1 ≥ 0. p Then there exist ′ [log(p + m)]/n constants γ and γ only depending on C and M such that if we set ρ = γ u 1 p and ρu = γu′ (r + log p)/n for any γ ′ ∈ [γ, C2 γ] and γu′ ∈ [γu , C2 γu ] for some absolute constant C2 > 0, there exist a constants C, C ′ > 0 only depending on C1 , C2 and M , such that b, U) ≤ C L(U su (r + log p) , nλ2 with P-probability at least 1 − exp(−C ′ (su + log(ep/su ))) − exp(−C ′ (sv + log(em/sv ))) − exp(−C ′ (r + log(p ∧ m))) uniformly over P ∈ P(n, su , sv , p, m, r, λ; M ). Remark 4.3. The result of Theorem 4.2 assumes a constant M . Explicit dependence on the eigenvalues of the marginal covariance can be tracked even when M is diverging. Assuming b, U) the eigenvalues of Σx all lie in the interval [M1 , M2 ], then the convergence rate of L(U

2 3 su (r+log p) su (r+log p) M2 M2 and a convergence rate of kPUb − PU k2F would be M . would be M nλ2 nλ2 1 1
M2 2 Compared with Remark 3.1, there is an extra factor M1 , which is also present for the Lasso error bounds [8, 32]. Evidence has been given in the literature that such an extra factor can be intrinsic to all polynomial-time algorithms [46]. 10 Although both Theorem 4.1 and Theorem 4.2 assume Gaussian distributions, a scrutiny of the proofs shows that the same results hold if the Gaussian assumption is weakened to subgaussian. By Theorem 3.2, the rate in Theorem 4.2 is optimal. By Theorem 4.1 and Theorem 4.2, the choices of the penalty parameters ρ and ρu in (18) and (19) do not depend on su or sv . Therefore, the proposed estimation scheme (18) – (20) achieves the optimal rate adaptively over sparsity levels. A full treatment of adaptation to M is beyond the scope of the current paper, though it seems possible in view of the recent proposals in [6, 35, 12]. A careful examination of the proofs shows that the dependence of ρ and ρu on M is through 1/2 1/2 kΣx kop kΣy kop and kΣx kop , respectively. When p and m are bounded from above by a constant multiple of n, we can upper bound the operator norms by the sample counterparts to remove the dependence of these penalty parameters on M . We conclude this section with two more remarks. Remark 4.4. The group sparsity penalty used in the second stage (19) plays an important role in achieving the optimal rate su (r + log p)/(nλ2 ). Except for the extra λ−2 term, this convergence rate is a typical one for group Lasso [28]. If we simply use an ℓ1 penalty, then we will obtain the rate rsu log p/(nλ2 ), which is clearly sub-optimal. Remark 4.5. Comparing Theorem 3.1 with Theorem 4.2, the adaptive estimation scheme achieves the optimal rates of convergence for a smaller collection of parameter spaces of interest due to the more restrictive sample size condition (21). We examine the necessity of this condition in more details in Section 5 below. 5 Computational Lower Bounds In this section, we provide evidence that the sample size condition (21) imposed on the adaptive estimation scheme in Theorems 4.1 and 4.2 is probably unavoidable for any computationally feasible estimator to be consistent. To be specific, we show that for a sequence of parameter spaces in (5) – (6), if the condition is violated, then any computationally efficient consistent estimator of sparse canonical coefficients leads to a computationally efficient and statistically powerful test for the Planted Clique detection problem in a regime where it is believed to be computationally intractable. Planted Clique Let N be a positive integer and k ∈ [N ]. We denote by G(N, 1/2) the Erdős-Rényi graph on N vertices where each edge is drawn independently with probability 1/2, and by G(N, 1/2, k) the random graph generated by first sampling from G(N, 1/2) and then selecting k vertices uniformly at random and forming a clique of size k on these vertices. For an adjacency matrix A ∈ {0, 1}N ×N of an instance from either G(N, 1/2) or G(N, 1/2, k), the Planted Clique detection problem of parameter (N, k) refers to testing the following hypotheses H0G : A ∼ G(N, 1/2) v.s. H1G : A ∼ G(N, 1/2, k). (22) It is widely believed that when k = O(N 1/2−δ ), the problem (22) cannot be solved by any randomized polynomial-time algorithm. In the rest of the paper, we formalize the conjectured hardness of Planted Clique problem into the following hypothesis. 11 Hypothesis A. For any sequence k = k(N ) such that lim supN →∞ randomized polynomial-time test ψ,   2 lim inf PH G ψ + PH G (1 − ψ) ≥ . 0 1 N →∞ 3 log k log N < 1 2 and any Evidence supporting this hypothesis has been provided in [34, 17]. Computational lower bounds in several statistical problems have been established by assuming the above hypothesis and its close variants, including sparse PCA detection [7] and estimation [39] in classes defined by a restricted covariance concentration condition, submatrix detection [30] and community detection [21]. Necessity of the sample size condition (21) Under Hypothesis A, the necessity of condition (21) is supported by the following theorem. Theorem 5.1. Suppose that Hypothesis A holds and that as n → ∞, p = m satisfying 2n ≤ p ≤ na for some constant a > 1, su = sv , n(log n)5 ≤ cs4u for some sufficiently small su sv c > 0, and λ = 7290n(log(12n)) 2 . If for some δ ∈ (0, 1), lim inf n→∞ (su sv )1−δ log(p + m) > 0, nλ2 then for any randomized polynomial-time estimator u b, n 1 o 1 > . lim inf sup P L(b u, u) > n→∞ P∈P(n,s ,s ,p,m,1,λ;3) 300 4 u v (23) (24) Comparing (21) with (23), we see that subject to a sub-polynomial factor, the condition (21) is necessary to achieve consistent sparse CCA estimation within polynomial time complexity. Remark 5.1. The statement in Theorem 5.1 is rigorous only if we assume the computational complexities of basic arithmetic operations on real numbers and sampling from univariate continuous distributions with analytic density functions are all Θ(1) [10]. To be rigorous under the probabilistic Turing machine model [2], we need to introduce appropriate discretization of the problem and be more careful with the complexity of random number generation. To convey the key ideas in our computational lower bound construction, we focus on the continuous case throughout this section and defer the formal discretization arguments to Section 8 in the supplement [19]. In what follows, we divide the reduction argument leading to Theorem 5.1 into two parts. In the first part, we show Hypothesis A implies the computational hardness of the sparse PCA problem under the Gaussian spiked covariance model. In the second part, we show computational hardness of sparse PCA implies that of sparse CCA as stated in Theorem 5.1. 5.1 Hardness of sparse PCA under Gaussian spiked covariance model Gaussian single spiked model [24] refers to the distribution Np (0, Σ) where Σ = τ θθ ′ + Ip . Here, θ is the eigenvector of unit length and τ > 0 is the eigenvalue. Define the following 12 Gaussian single spiked model parameter space for sparse PCA iid Q(n, s, p, λ) = {L(W1 , . . . , Wn ) :Wi ∼ Np (0, τ θθ ′ + Ip ), kθk0 ≤ s, τ ∈ [λ, 3λ]}. (25) ep The minimax estimation rate for θ under the loss kPθb − Pθ k2F is λ+1 nλ2 s log s . See, for instance, [13]. However, to achieve the above minimax rate via computationally efficient methods such as those proposed in [9, 29, 13], researchers have required the sample size to satisfy n ≥ 2 p C s λlog for some sufficiently large constant C > 0. Moreover, no computationally efficient 2 estimator is known to achieve consistency when the sample size condition is violated. As a first step toward the establishment of Theorem 5.1, we show that Hypothesis A implies hardness 2−δ p >0 of sparse PCA under Gaussian spiked covariance model (25) when lim inf n→∞ s nλlog 2 for some δ > 0. We note that previous computational lower bounds for sparse PCA in [7, 39] cannot be used here directly because they are only valid for parameter spaces defined via the restricted covariance concentration (RCC) condition. As pointed out in [39], such parameter spaces include (but are not limited to) all subgaussian distributions with sparse leading eigenvectors and the covariance matrices need not be of the spiked form Σ = τ θθ ′ + Ip . Therefore, the Gaussian single spiked model parameter space defined in (25) only constitutes a small subset of such RCC parameter spaces. The goal of the present subsection is to establish the computational lower bound for the Gaussian single spiked model directly. b 1 , . . . , Wn ) of the leading sparse eigenvector, we Suppose we have an estimator θb = θ(W propose the following reduction scheme to transform it into a test for (22). To this end, we first introduce some additional notation. Consider integers k and N . Define δN = k , N ηN = k . 45N (log N )2 (26) For any µ ∈ R, let φµ denote the density function of the N (µ, 1) distribution, and let φ̄µ = 1 (φµ + φ−µ ) 2 (27) e0 denote the density function of the Gaussian mixture 12 N (µ, 1) +√12 N (−µ,√1). Next, let Φ be the √ restriction of the N (0, 1) distribution on the interval [−3 log N , 3 log N ]. For any |µ| ≤ 3 ηN log N , define two probability distributions Fµ,0 and Fµ,1 with densities
−1 [φ̄µ (x) − φ0 (x)] 1{|x|≤3√log N } , fµ,0 (x) = M0 φ0 (x) − δN (28)
−1 (29) [φ̄µ (x) − φ0 (x)] 1{|x|≤3√log N } , fµ,1 (x) = M1 φ0 (x) + δN R where the Mi ’s are normalizing constants such that R fµ,i = 1 for i = 0, 1. √It can be verified that fµ,i are properly defined probability density function when |µ| ≤ 3 ηN log N . For details, see Lemma 7.4 in the supplement [19]. With the foregoing definition, the proposed reduction scheme can be summarized as Algorithm 1. Here, the starting point is the adjacency matrix A of the random graph, and the reduction is well defined for all instances of N ≥ 12n and p ≥ 2n. 13 Algorithm 1: Reduction from Planted Clique to Sparse PCA (in Gaussian Single Spiked Model) Input: 1. Graph adjacency matrix A ∈ {0, 1}N ×N ; 2. Estimator θb for the leading eigenvector θ. Output: A solution to the hypothesis testing problem (22). e 0 . Set 1 Initialization. Generate i.i.d. random variables ξ1 , . . . , ξ2n ∼ Φ 1/2 µi = ηN ξi , 2 i = 1, . . . , 2n. (30) Gaussianization. Generate two matrices B0 , B1 ∈ R2n×2n where conditioning on the µi ’s, all the entries are mutually independent satisfying L((B0 )ij |µi ) = Fµi ,0 and L((B1 )ij |µi ) = Fµi ,1 . (31) Let A0 ∈ {0, 1}2n×2n be the lower–left 2n × 2n submatrix of the matrix A. Generate a ′ ]′ ∈ R2n×p where for each i ∈ [2n], if j ∈ [2n], we set matrix W = [W1′ , . . . , W2n Wij = (B0 )ij (1 − (A0 )ij ) + (B1 )ij (A0 )ij . 3 (32) If 2n < j ≤ p, we let Wij be an independent draw from N (0, 1). b 1 , . . . , Wn ) be the estimator of the leading Test Construction. Let θb = θ(W n eigenvector by treating {Wi }i=1 as data. It is normalized to be a unit vector. We reject H0G if 1 θb′ ( n 2n X i=n+1 1 Wi Wi′ )θb ≥ 1 + k ηN . 4 (33) We now explain how the reduction achieves its goal. For simplicity, focus on the case where p = 2n. Let ǫ = (ǫ1 , ..., ǫ2n ) ∈ {0, 1}2n where ǫi is the indicator of whether the i-th row of A0 (defined in Step 2 of Algorithm 1) belongs to the planted clique or not, and γ = (γ1 , ..., γ2n ) the indicators of the columns of A0 . In what follows, we discuss the distributions of W when A ∼ H0G and H1G , respectively. When A ∼ H0G , the ǫi ’s and γj ’s are all zeros. In this case, we can verify that the entries of W are mutually independent and for each (i, j) the marginal distribution of Wij is close to the N (0, 1) distribution (c.f., Lemma 7.1 in the supplement [19]). Hence, the rows of W b 1 , . . . , Wn ) is are close to i.i.d. random vectors from the Np (0, Ip ) distribution. Since θb = θ(W 2n 2 independent of {Wi }i=n+1 , the LHS of (40) is close in distribution to a χn random variable scaled by p n which concentrates around its expected value one. Indeed, it is upper bounded by 1 + O( log(n)/n) with high probability. If A ∼ H1G , then the (i, j)-th entry of A0 is an edge in the planted clique if and only if ǫi = γj = 1. Moreover, the joint distribution of {ǫ1 , . . . , ǫ2n , γ1 , . . . , γ2n } is close to that of 4n i.i.d. Bernoulli random variables {e ǫ1 , . . . , e ǫ2n , γ e1 , . . . , γ e2n } with success probability δN = k/N . For simplicity, suppose that these indicators are indeed i.i.d. Bernoulli(δN ) variables 14 {e ǫ1 , . . . , e ǫ2n , γ e1 , . . . , γe2n }. Then, one can show that conditioning on γ ej = 0, for any i ∈ [2n], the conditional distribution of (Wij |e γj = 0), after integrating over the conditional distribution of e ǫi , µi and (A0 )ij , is approximately N (0, 1). In contrast, conditioning on γ ej = 1, for any i ∈ [2n], the conditional distribution of (Wij |e γj = 1) is approximately N (0, 1 + ηN ). Therefore, conditioning on γ e the distribution of the Wi ’s is close to that of 2n i.i.d. random vectors sampled from Np (0, τ θθ ′ + Ip ), where θ = e γ /ke γ k and τ = ηN ke γ k2 , i.e., a Gaussian spiked covariance P model in (25). Here, the leading eigenvector θ has sparsity level |supp(θ)| = |supp(e γ )| = j γ ej , which concentrates around its mean value nδN ≍ k if b N ≍ n. Thus, if θ estimates θ well, then the LHS of (40) approximately follows a non-central p χn distribution scaled by n, which should exceed 1 + O( log(n)/n) with high probability under the alternative hypothesis. Hence, Algorithm 1 is expected to yield a test with small error for the Planted Clique problem (22) when θb is a good estimator. The materialization of the foregoing discussion leads to the following result which demonstrates quantitatively that a decent estimator of the leading sparse eigenvector results in a good test (by applying the reduction (30) – (33)) for the Planted Clique detection problem (22). 5 N) ≤ c, cN ≤ n ≤ Theorem 5.2. For some sufficiently small constant c > 0, assume N (log k4 b N/12 and p ≥ 2n. Then, for any θ such that n 1o ≤ β, (34) sup Q kPθb − Pθ k2F > 3 Q∈Q(n,3k/2,p,kηN /2) the test ψ defined by (30) – (33) satisfies PH G ψ + PH G (1 − ψ) < β + 0 1 4n ′ + C(n−1 + N −1 + e−C k ), N for sufficiently large n with some constants C, C ′ > 0. If the estimator θb is uniformly consistent over Q(n, 3k/2, p, kηN /2), then β is close to zero. Hence the conclusion of Theorem 5.2 implies that for appropriate growing sequences of n, N and k, the testing error for (22) can be made smaller than any fixed nonzero probability. Further invoking Hypothesis A, we obtain the following computational lower bounds for sparse PCA. Theorem 5.3. Suppose that Hypothesis A holds and that as n → ∞, 2n ≤ p ≤ na for some s2 constant a > 1, n(log n)5 ≤ cs4 for some sufficiently small c > 0, and λ = 2430n(log(12n)) 2 . If for some δ ∈ (0, 2), s2−δ log p lim inf > 0, (35) n→∞ nλ2 b then for any randomized polynomial-time estimator θ, lim inf n→∞ n 1o 1 > . Q kPθb − Pθ k2F > 3 4 Q∈Q(n,s,p,λ) sup 15 (36) In addition to estimation, we can also consider the following sparse PCA detection problem: Let Q denote the joint distribution of W1 , . . . , Wn , and we want to test H0 : Q ∈ Q(n, s, p, 0), v.s. H1 : Q ∈ Q(n, s, p, λ). (37) iid Note that Q(n, s, p, 0) contains only one distribution Q0 where Wi ∼ Np (0, Ip ). Given any testing procedure φ = φ(W1 , . . . , Wn ), we can obtain a solution to (22) by replacing the third step in Algorithm 1 with the direct testing result of φ(W1 , . . . , Wn ). Following the lines of the proof of Theorem 5.3, we have the following theorem. Theorem 5.4. Under the same condition of Theorem 5.3, for any randomized polynomialtime test φ for testing (37),   1 (38) sup Q(1 − φ) ≥ . lim inf Q0 φ + n→∞ 4 Q∈Q(n,s,p,λ) Remark 5.2. Theorems 5.3–5.4 are the first computational lower bounds for sparse PCA that are valid in the setting of Gaussian single spiked covariance models (25). 5.2 Hardness of sparse CCA In the second step, we show that computational hardness of sparse PCA under Gaussian spiked covariance model implies the desired result in Theorem 5.1. To this end, we propose the following reduction. Algorithm 2: Reduction from Sparse PCA to Sparse CCA Input: 1. Observations W1 , . . . , Wn ∈ Rp ; 2. Estimator u b of the first leading canonical correlation coefficient u. Output: An estimator θb of the leading eigenvector of L(W1 ). 1 Generate i.i.d. random vectors Z1 , . . . , Zn ∼ Np (0, Ip ). Set 1 Xi = √ (Wi + Zi ), 2 2 1 Yi = √ (Wi − Zi ), 2 i = 1, . . . , n. Compute u b=u b(X1 , Y1 , . . . , Xn , Yn ). Set b 1 , . . . , Wn ) = u θb = θ(W b/kb uk. (39) (40) iid To see why Algorithm 2 is effective, one can verify that if Wi ∼ Np (0, τ θθ ′ + Ip ), then iid (Xi′ , Yi′ )′ ∼ Np+m (0, Σ) where Σx = Σy = with u = v = √ θ , τ /2+1 λ= τ /2 τ /2+1 . τ ′ θθ + Ip , Σxy = Σx (λuv ′ )Σy 2 (41) This is a special case of the Gaussian canonical pair model (2). Thus, the leading eigenvector of Wi aligns with the leading canonical coefficient vectors of (Xi , Yi ). Exploiting this connection, we obtain the following theorem. 16 Theorem 5.5. Consider p = m, su = sv and λ ≤ 1. Then for any u b such that n 1 o ≤ β, P L(b u, u) > 300 P∈P(n,su ,sv ,p,m,1,λ/3;3) sup (42) the estimator θb defined by Algorithm 2 satisfies n 1o ≤ β. Q kPθb − Pθ k2F > 3 Q∈Q(n,s,p,λ) sup If we start with an estimator u b of the leading canonical coefficient vector, then we can construct the reduction from Planted Clique to sparse CCA directly by essentially following the steps in Algorithm 1 while using Algorithm 2 to construct θb from u b in the third step. Finally, the desired Theorem 5.1 is a direct consequence of Theorems 5.3 and 5.5. 6 Proofs This section presents proofs of Theorems 4.1 and 4.2. The proofs of the other theoretical results are given in the supplement [19]. 6.1 Proof of Theorem 4.1 Before presenting the proof, we state some technical lemmas. The proofs of all the lemmas are given in Section 9.3 in the supplement [19]. First, note that the estimator is normalized b (0) b (0) with respect to Σ x and Σy , while the truth U and V is normalized with respect to Σx and b (0) b (0) Σy . To address this issue, we normalize the truth with respect to Σ x and Σy to obtain −1/2 and V −1/2 . Also define Λ 1/2 Λ(V ′ Σ 1/2 . e = U (U ′ Σ b (0) e = V (V ′ Σ b (0) e = (U ′ Σ b (0) b (0) U x U) y V) x U) y V) For notational convenience, define r  r  ep  em  1 1 su + log sv + log , ǫn,v = . (43) ǫn,u = n su n sv The following lemma bounds the normalization effect. Lemma 6.1. Assume ǫ2n,u + ǫ2n,v ≤ c for some sufficiently small constant c ∈ (0, 1). Then there exist some constants C, C ′ > 0 only depending on c such that 1/2 e e e kΣ1/2 x (U − U )kop ≤ Cǫn,u , kΣy (V − V )kop ≤ Cǫn,v , kΛ − Λkop ≤ C(ǫn,u + ǫn,v ), with probability at least 1 − exp (−C ′ (su + log(ep/su ))) − exp (−C ′ (sv + log(em/sv ))). e and Ve , let us state the following lemma, which asserts that the Using the definitions of U ′ e e e matrix A = U V is feasible to the optimization problem (18). e=U e Ve ′ . When A e exists, we have Lemma 6.2. Define A 1/2 e b (0) 1/2 b (0) k(Σ A(Σy ) k∗ = r x ) and 17 1/2 e b (0) 1/2 b (0) k(Σ A(Σy ) kop = 1. x ) As was argued in Section 4.1, the set Cr is the convex hull of Or . The following curvature lemma shows that the relaxation Cr preserves the restricted strong convexity of the objective function. Lemma 6.3. Let F ∈ O(p, r), G ∈ O(m, r), K ∈ Rr×r and D = diag(d1 , ..., dr ) with d1 ≥ ... ≥ dr > 0. If E satisfies kEkop ≤ 1 and kEk∗ ≤ r, then hF KG′ , F G′ − Ei ≥ Define dr kF G′ − Ek2F − kK − DkF kF G′ − EkF . 2 ′ b (0) e xy = Σ b (0) Σ x U ΛV Σy . (44) (45) Lemma 6.4 is instrumental in determining the proper value of the tuning parameter required in the program (18). p Lemma 6.4. Assume r [log(p + m)]/n ≤ c for some sufficiently small constant c ∈ (0, 1). b (0) Then there p exist some constants C, C ′ > 0 only depending on M and c such that ||Σ xy − ′ −C e Σxy ||∞ ≤ C [log(p + m)]/n, with probability at least 1 − (p + m) . We also need a lemma on restricted eigenvalue. For any p.s.d. matrix B, define φB max (k) = u′ Bu , ||u||0 ≤k,u6=0 u′ u φB min (k) = max u′ Bu . ||u||0 ≤k,u6=0 u′ u min The following lemma is adapted from Lemma 12 in [18], and its proof is omitted.
Lemma 6.5. Assume n1 (ku ∧ p) log(ep/(ku ∧ p)) + (kv ∧ m) log(em/(kv ∧ m)) ≤ c for some sufficiently small constant c > 0. Then C ′ > 0 only depending q there exist some constants C,q on M and c such that for δu (ku ) = we have (ku ∧p) log(ep/(ku ∧p)) n b (j) b (j) b (j) Σ b (j) Σ and δv (kv ) = (kv ∧m) log(em/(kv ∧m)) , n Σx x M −1 − Cδu (ku ) ≤ φΣ min (ku ) ≤ φmax (ku ) ≤ M + Cδu (ku ), y y (kv ) ≤ φmax (kv ) ≤ M + Cδv (kv ), M −1 − Cδv (kv ) ≤ φmin
with
probability at least 1−exp −C ′ (ku ∧p) log(ep/(ku ∧p)) −exp −C ′ (kv ∧m) log(em/(kv ∧ m)) , for j = 0, 1, 2. Finally, we need a result on subspace distance. Recall that for a matrix F , PF denotes the projection matrix onto its column subspace. Lemma 6.6. For any matrix F ∈ O(d, r) and any matrix G ∈ Rd×r , we have inf kF − GW k2F = W 1 kPF − PG k2F . 2 If further G ∈ O(d, r), then inf W ∈O(r,r) kF − GW k2F = 21 kPF − PG k2F . Proofs of Lemma 6.1-6.6 are given in Section 9.3.1 of the supplement [19]. 18 b (0) b (0) b (0) b b Proof of Theorem 4.1. In the rest of this proof, we denote Σ x , Σy and Σxy by Σx , Σy and b xy for notational convenience. We also let ∆ = A b − A. e The proof consists of two steps. Σ b x1/2 ∆Σ b y1/2 kF . In the second In the first step, we are going to derive an upper bound for kΣ b 1/2 b 1/2 step, we derive a generalized cone condition and use it to lower bound kΣ x ∆Σy kF by a b 1/2 b 1/2 constant multiple of k∆kF and hence the upper bound on kΣ x ∆Σy kF leads to an upper bound on k∆kF . e and Ve are well-defined with high probability. Thus, A e is Step 1. By Lemma 6.1, U well-defined with high probability, and we have ′ 1/2 e kΣ1/2 x (A − U V )Σy kop ≤ C(ǫn,u + ǫn,v ). (46) with probability at least 1 − exp (−C ′ (su + log(ep/su ))) − exp (−C ′ (sv + log(em/sv ))). Ace is feasible. Then, by the definition of A, b we have cording to Lemma 6.2, A b xy , Ai b − ρ||A|| b 1 ≥ hΣ b xy , Ai e − ρ||A|| e 1. hΣ After rearrangement, we have
b xy − Σ e xy , ∆i, e xy , ∆i ≤ ρ ||A|| e 1 − ||A e + ∆||1 + hΣ − hΣ (47) e xy is defined in (45). For the first term on the right hand side of (47), we have where Σ e 1 − ||A e + ∆||1 = ||A eSu Sv ||1 − ||A eSu Sv + ∆Su Sv ||1 − ||∆(S S )c ||1 ||A|| u v ≤ ||∆Su Sv ||1 − ||∆(Su Sv )c ||1 . b xy − Σ e xy , ∆i ≤ ||Σ b xy − For the second term on the right hand side of (47), we have hΣ e xy ||∞ ||∆||1 . Thus when Σ b xy − Σ e xy ||∞ , ρ ≥ 2||Σ (48) we have e xy , ∆i ≤ 3ρ ||∆Su Sv ||1 − ρ ||∆(S S )c ||1 . − hΣ u v 2 2 Using Lemma 6.3, we can lower bound the left hand side of (49) as e xy , ∆i = hΣ b x1/2 U ΛV ′ Σ b 1/2 b 1/2 e b b 1/2 − hΣ y , Σx (A − A)Σy i b x1/2 U eΛ e Ve ′ Σ b 1/2 b 1/2 e b b 1/2 = hΣ y , Σx (A − A)Σy i ≥ 1 b 1/2 e b b 1/2 2 b 1/2 e b b 1/2 λr kΣ x (A − A)Σy kF − δkΣx (A − A)Σy kF , 2 e − ΛkF . Combining (49) and (50), we have where δ = kΛ (49) (50) b 1/2 ∆Σ b 1/2 k2 ≤ 3ρ||∆Su Sv ||1 − ρ||∆(S S )c ||1 + 2δkΣ b 1/2 ∆Σ b 1/2 kF λr kΣ x y F x y u v 1/2 b 1/2 b ≤ 3ρ||∆Su Sv ||1 + 2δkΣx ∆Σy kF . (51) 2 2 b 1/2 b 1/2 2 kΣ x ∆Σy kF ≤ 6ρ||∆Su Sv ||1 /λr + 4δ /λr . (53) (52) Solving the quadratic equation (52) by Lemma 2 of [13], we have 19 Combining (51) and (53), we have b 1/2 b 1/2 2 0 ≤ 3ρ||∆Su Sv ||1 − ρ||∆(Su Sv )c ||1 + δ2 /λr + λr kΣ x ∆Σy kF ≤ 9ρ||∆Su Sv ||1 − ρ||∆(Su Sv )c ||1 + 5δ2 /λr , (54) which gives rise to the generalized cone condition that we are going to use in Step 2. Finally, √ by the bound ||∆Su Sv ||1 ≤ su sv ρk∆Su Sv kF and (53), we have √ 2 2 b 1/2 b 1/2 2 kΣ x ∆Σy kF ≤ 6 su sv ρk∆Su Sv kF /λr + 4δ /λr , (55) which completes the first step. Step 2. By (54), we have obtained the following condition ||∆(Su Sv )c ||1 ≤ 9||∆Su Sv ||1 + 5δ2 /(ρλr ). (56) Due to the existence of the extra term 5δ2 /(ρλr ) on the RHS, we call it a generalized cone b x1/2 ∆Σ b 1/2 condition. In this step, we are going to lower bound kΣ y kF by k∆kF on the generalized cone. Motivated by the argument in [8], let the index set J1 = {(ik , jk )}tk=1 in (Su × Sv )c correspond to the entries with the largest absolute values in ∆, and we define the set Je = (Su × Sv ) ∪ J1 . Now we partition Jec into disjoint subsets J2 , ..., JK of size t (with |JK | ≤ t), such that Jk is the set of (double) indices corresponding to the entries of t largest absolute Sk−1 values in ∆ outside Je ∪ j=2 Jj . By triangle inequality, b 1/2 b 1/2 b 1/2 b 1/2 kΣ x ∆Σy kF ≥ kΣx ∆JeΣy kF − ≥ q bx φΣ min (su + by Σ (sv t)φmin K X k=2 + t)k∆JekF − b 1/2 b 1/2 kΣ x ∆Jk Σy kF q K X by bx Σ φΣ (t)φ (t) k∆Jk kF . max max k=2 By the construction of Jk , we have K X k=2 k∆Jk kF ≤ K K X √ X ||∆Jk−1 ||1 ≤ t−1/2 ||∆(Su Sv )c ||1 ||∆Jk ||∞ ≤ t−1/2 t k=2 k=2 r   su sv 5δ2 5δ2 −1/2 √ , ≤t 9||∆Su Sv ||1 + k∆JekF + ≤9 ρλr t ρλr t (57) where we have used the generalized cone condition (56). Hence, we have the lower bound √ 2 b 1/2 b 1/2 kΣ x ∆Σy kF ≥ κ1 k∆JekF − κ2 δ /(ρλr t), with κ1 = q bx φΣ min (su + by Σ (sv t)φmin q by bx Σ κ2 = 5 φΣ max (t)φmax (t). r q by su sv bx Σ + t) − 9 φΣ max (t)φmax (t), t 20 (58) Taking t = c1 su sv for some sufficiently large constant c1 > 1, with high probability, κ1 can be lower bounded by a positive constant κ0 only depending p that by p on M . To see this, note −1 Lemma 6.5, (58) can be lower bounded by the difference of M − Cδu (2c1 su sv ) M −1 − Cδv (2c1 su sv ) p −1/2 p and 9c1 M + Cδu (c1 su sv ) M + Cδv (c1 su sv ), where δu and δv are defined as in Lemma 6.5. It is sufficient to show that δu (2c1 su sv ), δv (2c1 su sv ), δu (c1 su sv ) and δv (c1 su sv ) are sufficiently small to get a positive absolute constant κ0 . For the first term, when 2c1 su sv ≤ p, it is bounded by 2c1 su svnlog(ep) and is sufficiently small under the assumption (13). When 2c1 su sv > p, it is bounded by 2c1 nsu sv and is also sufficiently small under (13). The same argument also holds for the other terms. Similarly, κ2 can be upper bounded by some constant. Together with (55), this brings the inequality √
√ k∆Jek2F ≤ C1 ( su sv ρ/λr )k∆JekF + C2 δ2 /λ2r + (δ2 /(ρλr t))2 . Solving this quadratic equation, we have k∆Jek2F ≤ C  s s ρ2 δ2  δ2 2  u v √ . + + λ2r λ2r ρλr t (59) By (57), we have r 5δ2 su sv √ . k∆JekF + k∆Jk kF ≤ 9 k∆Jec kF ≤ t ρλ t r k=2 K X (60) Summing (59) p and (60), we obtain a bound for k∆kF . According to Lemma 6.4, we may choose ρ = γ [log(p p + m)]/n for some large γ, so that √ (48) holds with high probability. By ′ Lemma 6.1, δ ≤ C r(su + sv + log(p + m))/n ≤ C ρ t with high probability. Hence, √ k∆kF ≤ C su sv ρ/λr , (61) with high probability. This completes the second step. Finally, the triangle inequality leads b − U V ′ kF ≤ k∆kF + kA e − U V ′ kF . By (46) and (61), the proof is complete. to kA 6.2 Proof of Theorem 4.2 b (1) − U ∗ . Define U ∗ = U ΛV ′ Σy Vb (0) and ∆ = U r+log p ≤ c for some sufficiently small constant c ∈ n ′ C, C > 0 only depending on M and c such that max Lemma 6.7. Assume (0, 1). Then there b (1) b (0) − exist some constants 1≤j≤p ||[Σxy V p
∗ ′ b (1) Σ x U ]j· || ≤ C (r + log p)/n, with probability at least 1 − exp − C (r + log p) . The proof of Lemma 6.7 is given in Section 9.3.1 of the supplement [19]. b (1) b (1) b (1) b b Proof of Theorem 4.2. In the rest of this proof, we denote Σ x , Σy and Σxy by Σx , Σy b xy for simplicity of notation. Note that they depends on D1 , while the estimator Vb (0) and Σ depends on D0 . Hence, Vb (0) is independent of the sample covariance matrices occurring in this proof. The proof consists of three steps. In the first step, we derive a bound for b x ∆). In the second step, we derive a cone condition and use it to obtain a bound for Tr(∆′ Σ 21 b x ∆) upper bounds k∆kF . In the last step, we derive the desired k∆kF by arguing that Tr(∆′ Σ b , U ). bound for L(U b (1) , we have Tr((U b (1) )′ Σ b xU b (1) )−2 Tr((U b (1) )′ Σ b xy Vb (0) )+ρu Pp ||U b (1) || ≤ Step 1. By definition of U j=1 j· b x U ∗ ) − 2 Tr((U ∗ )′ Σ b xy Vb (0) ) + ρu Pp ||U ∗ ||. After rearrangement, we have Tr((U ∗ )′ Σ j· j=1 ′b Tr(∆ Σx ∆) ≤ ρu p h X j=1 i h i b xy Vb (0) − Σ b xU ∗ ) . ||Uj·∗ || − ||Uj·∗ + ∆j· || + 2 Tr ∆′ (Σ (62) For the first term on the right hand side of (62), we have p X j=1 X X X
||Uj·∗ + ∆j· || − ||∆j· || ||Uj·∗ || − ||Uj·∗ || − ||Uj·∗ + ∆j· || = ≤ X j∈Su j∈Suc j∈Su j∈Su ||∆j· || − X j∈Suc ||∆j· ||. For the second term on the right hand side of (62), we have p    X ′ b (0) ∗ b xy Vb (0) − Σ b x U ∗ ]j· ||, b b ||∆j· || max ||[Σ Tr ∆ (Σxy V − Σx U ) ≤ 1≤j≤p j=1 where [·]j· means the j-th row of the corresponding matrix. When b xy Vb (0) − Σ b x U ∗ ]j· ||, ρu ≥ 4 max ||[Σ 1≤j≤p we have Since P j∈Su ||∆j· || ≤ b x ∆) ≤ Tr(∆′ Σ ρu X 3ρu X ||∆j· || − ||∆j· ||. 2 2 c j∈Su (63) (64) j∈Su √ qP 2 su j∈Su ||∆j· || , (64) can be upper bounded by √ s 3 s u ρu X Tr(∆ Σx ∆) ≤ ||∆j· ||2 . 2 ′b (65) j∈Su This completes the first step. Step 2. The inequality (64) implies the cone condition X X ||∆j· ||. ||∆j· || ≤ 3 j∈Suc (66) j∈Su Let the index set J1 = {j1 , ..., jt } in Suc correspond to the rows with the largest ℓ2 norm in ∆, and we define the extended support Seu = Su ∪ J1 . Now we partition Seuc into disjoint subsets J2 , ..., JK of size t (with |JK | ≤ t), such that Jk is the set of indices corresponding to the t 22 Sk−1 b x ∆) = kn−1/2 X∆k2 , rows with largest ℓ2 norm in ∆ outside Seu ∪ j=2 Jj . Note that Tr(∆′ Σ F where X = [X1 , ..., Xn ]′ ∈ Rn×p denotes the data matrix. By triangle inequality, we have X kn−1/2 X∆kF ≥ kn−1/2 X∆Seu ∗ kF − kn−1/2 X∆Jk ∗ kF ≥ q k≥2 b x φΣ min (su + t)k∆Seu ∗ kF − q b x φΣ max (t) X k≥2 k∆Jk ∗ kF , where for a subset B ⊂ [p], ∆B∗ = (∆ij 1{i∈B,j∈[r]} ), and X k≥2 √ X √ X1 X t max ||∆j· || ≤ t ||∆j· || j∈Jk t k≥2 k≥2 j∈Jk−1 X X ||∆j· || ≤ t−1/2 ||∆j· || ≤ 3t−1/2 k∆Jk ∗ kF ≤ j∈Suc (67) j∈Su r s r su X su 2 ≤ 3 ||∆j· || ≤ 3 k∆Seu ∗ kF . t t (68) j∈Su In the above derivation, we have used the construction of Jk and the qcone condition (66). q p bx b Σ x Hence, kn−1/2 X∆kF ≥ κk∆Seu ∗ kF with κ = φmin (su + t) − 3 stu φΣ max (t). In view of Lemma 6.5, taking t = c1 su for some sufficiently large constant c1 , with high probability, κ can be lower bounded by a positive constant κ0 only depending on M . Combining with (65), we have √ k∆Seu ∗ kF ≤ C su ρu /(2κ20 ). (69) By (67)-(68), we have k∆(Seu )c ∗ kF ≤ X p −1/2 k∆Jk ∗ kF ≤ 3 su /tk∆Seu ∗ kF ≤ 3c1 k∆Seu ∗ kF . (70) k≥2 √ and (70), we have k∆k ≤ C su ρ. By Lemma 6.7, we may choose ρu ≥ Summing (69) F q p for some large γu so that (63) holds with high probability. Hence, k∆kF ≤ γu r+log p n C su (r + log p)/n with high probability. This completes the second step. Step 3. Using the same argument in Step 2 of the proof of Theorem 3.1 (see supplementary b , U ). The proof is complete. material [19]), we obtain the desired bound for L(U References [1] T. W. Anderson. An Introduction to Multivariate Statistical Analysis. Wiley, 1958. [2] S. Arora and B. Barak. Computational complexity: a modern approach. Cambridge University Press, 2009. [3] B. B. Avants, P. A. 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Lower bounds on the performance of polynomial-time algorithms for sparse linear regression. arXiv preprint arXiv:1402.1918, 2014. 26 Supplement to “Sparse CCA: Adaptive Estimation and Computational Barriers” 7 Proofs of Results in Section 5 In this section, we present the proofs of Theorems 5.1–5.5. Here we do not consider the issue of discretization. The main purpose is to help the readers get the intuition behind the problem without worrying about rigor at the theoretical computer science level. A rigorous treatment of the computational lower bounds is deferred to Section 8 where the asymptotic equivalent discretization and the statement of rigorous results for the discretized models will be presented. 7.1 Proof of Theorem 5.2 The reason for introducing the two distributions (28)–(29) is to match specific mixtures of them to φ0 and φ̄µ respectively as summarized in the following lemma. For two probability distributions P and Q, the total variation distance is defined as TV(P, Q) = supB |P(B) − Q(B)|. We also write TV(p, q) if p and q are the densities of P and Q, respectively. Lemma 7.1. There exists √ an absolute constant C > 0, such that for all integers N ≥ 12, k ≤ N/12 and all |µ| ≤ 3 ηN log N , TV(hµ,0 , φ0 ) ≤ CN −3 and TV(hµ,1 , φ̄µ ) ≤ CN −3 , where hµ,0 = 12 (fµ,0 + fµ,1 ) and hµ,1 = δN fµ,1 + (1 − δN ) 21 (fµ,0 + fµ,1 ). The proof of the above lemma will be given in Section 7.4. To facilitate the proof of Theorem 5.2, we need to state and prove another two lemmas which characterize the distributions of the Wi ’s under H0G and H1G respectively. Let L({Wi }2n i=1 ) denote the joint distribution of 2n ′ {Wi }i=1 . In addition, denote Np (0, τ θθ + Ip ) by Qθ,τ . When τ = 0, Np (0, Ip ) is denoted by Q0 . The first lemma shows that under H0G , the joint distribution of {Wi }2n i=1 is close in total variation to that of a random sample of size 2n from Q0 . Lemma 7.2. Suppose A ∼ G(N, 1/2). There exists an absolute constant C > 0 such that 2n −1 TV(L({Wi }2n . i=1 ), Q0 ) ≤ CN Proof. Recall ηN defined in (26) and hµ,0 in Lemma√7.1. Let ν be √ N (0, ηN ), and ν̄ be the distribution obtained by restricting ν on the set [−3 ηN log N , 3 ηN log N ]. Then the µi ’s in (30) are i.i.d. r.v.’s following the distribution ν̄. For each i ∈ [2n] and each j ∈ [2n], define i.i.d. random variables W ij ∼ N (0, 1). For each i ∈ [2n] and 2n < j ≤ p, define W ij = Wij . Let W i = (W i1 , ..., W ip )′ . It is straightforward 2n to verify that L({W i }2n i=1 ) = Q0 . By the data-processing inequality, we have 2n n n TV(L({Wi }i=1 ), Q2n 0 ) ≤ TV(L({Wi }i=1 ), L({W i }i=1 )). 27 Hence, it is sufficient to bound TV(L({Wi }ni=1 ), L({W i }ni=1 )). Conditioning on µi , Wij follows hµi ,0 when A ∼ G(N, k). Therefore, Z TV(Wij , W ij ) = TV( hµi ,0 dν̄(µi ), φ0 ) ≤ sup √ |µi |≤3 ηN log N TV(hµi ,0 , φ0 ) ≤ CN −3 . Here the last inequality is due to Lemma 7.1. Applying Lemma 7 of [30], we obtain TV(L({Wi }ni=1 ), L({W i }ni=1 )) P n Pn −1 . This completes the proof. i=1 j=1 TV(Wij , W ij ) ≤ CN The next lemma shows that the joint distribution {Wi }2n i=1 is close in total variation to a mixture of the joint distribution of a random sample of size 2n from Qθ,τ . Here, the mixture is defined by a prior distribution π on the (θ, τ ) pair, which is supported on a region where θ is sparse and τ is bounded away from zero. For notational convenience, for any distribution R Pβ indexed by parameter β ∈ B and any probability R measure π on B, we let Pβ dν(β) denote the probability measure P defined by P(E) = Pβ (E)dν(β) for any event E. When β ∼ ν Ris a random variable and Pβ = L(W |β) is the conditional distribution of W |β, we also write L(W |β)dν(β) to represent the marginal distribution of W after integrating out β. Lemma 7.3. Suppose A ∼ G(N, 1/2, k). There exists a distribution π supported on the set  (θ, τ ) : θ ∈ S p−1 , |supp(θ)| ≤ 3k/2, τ ∈ [kηN /2, 3kηN /2] , (71) such that for some absolute constants C1 , C2 > 0,   Z 4n 1 −C2 k 2n 2n . + + TV(L({Wi }i=1 ), Qθ,τ dπ(θ, τ )) ≤ C1 e N N Proof. Recall ηN defined in (26) and hµ,0 and hµ,1 defined in Lemma 7.1. As in the proof of Lemma √ 7.2, let ν√ be N (0, ηN ), and ν̄ the distribution obtained by restricting ν on the set [−3 ηN logRN , 3 ηN log N ]. Then the µi ’s in (30) are i.i.d. r.v.’s followingRν̄. Simple calculus shows that φ0 (x)dν(µ) = φ0 (x) is the density function of N (0, 1), and φ̄µ (x)dν(µ) gives the density function of N (0, 1 + ηN ). We first focus on the case p = 2n. The case of p ≥ 2n will be treated at the end of the proof. Recall that (ǫ1 , ..., ǫ2n ) are the indicators of the rows of A0 whether the corresponding vertices belong to the planted clique, and (γ1 , ..., γp ) are the corresponding indicators of the columns of A0 . Let (e ǫ1 , ..., e ǫ2n ) and (e γ1 , ..., γep ) be i.i.d. Bernoulli random variables with e e0 )ij = 1 if e mean δN = k/N . Define a matrix A0 , where an entry (A ǫi = γ ej = 1 and is an f independent instantiation of the Bernoulli(1/2) distribution otherwise. Then, we define W with entries fij = (B0 )ij (1 − (A e0 )ij ) + (B1 )ij (A e0 )ij . W Then, by Theorem 4 of [15] and the data-processing inequality, we have f ), L(W )) ≤ TV(L(e TV(L(W ǫ, γ e), L(ǫ, γ)) ≤ 4n . N (72) f , conditioning on µi and Recall hµ,0 and hµ,1 defined in Lemma 7.1. By the definition of W fij ∼ hµ ,0 , while conditioning on µi and γ fij ∼ hµ ,1 . γ ej = 0, W ej = 1, W i i 28 Further define W ij by setting W ij |(e γj = 0, µi ) ∼ φ0 , W ij |(e γj = 1, µi ) ∼ φ̄µi , where φ̄µi is √ defined according to (27). By Lemma 7.1 and Lemma 7 of [30], uniformly over maxi |µi | ≤ 3 ηN log N, we have p 2n X    X  f |e fij |e TV L(W γ , µ), L(W |e TV L(W γj , µi ), L(W ij |e γ , µ) ≤ γj , µi ) ≤ CN −1 i=1 j=1 for some constant C > 0. Next, we integrate the above bound over µ. To this end, first note that Z Z dν(µ) = φ0 (x)dx ≤ CN −4 . TV(ν, ν̄) = √ √ |x|>3 log N |µ|>3 ηN log N R R f |e f |e With slight abuse of notation, let L(W γ , µ)dν̄(µ) (resp. L(W γ , µ)dν(µ)) denote the f |e conditional distribution of W γ if the coordinates of µ = (µ , . . . , µ 1 2n ) were i.i.d. following R R ν̄ (resp. ν), and let L(W |e γ , µ)dν̄(µ) and L(W |e γ , µ)dν(µ) be analogously defined. Then, conditioning on γ e, we obtain Z Z f TV( L(W |e γ , µ)dν̄(µ), L(W |e γ , µ)dν(µ)) Z Z f |e ≤ TV( L(W γ , µ)dν̄(µ), L(W |e γ , µ)dν̄(µ)) Z Z + TV( L(W |e γ , µ)dν̄(µ), L(W |e γ , µ)dν(µ)) ≤ sup √ maxi |µi |≤3 ηN log N f |e TV(L(W γ , µ), L(W |e γ , µ)) + CnTV(ν̄, ν) ≤ CN −1 . Here, the first inequality comes from the triangle inequality, the second Pnfrom thePdefinition of total variation distance. For each given γ e = (e γ1 , ..., γen ), define s = j=1 γ ej = nj=1 γ ej2 = ke γj k2 , θ = s−1/2 γ e and τ = sηN . Note that both θ and τ are functions of γ e. Then observe R R f |e f |e , which implies for L( W γ ) = L( W γ , µ)dν̄(µ), γ , µ)dν(µ) = Q2n that L(W |e θ,τ   −1 f |e . TV L(W γ ), Q2n θ,τ ≤ CN Define the event Q = {e γ : |s − k| ≤ k/2}. Then, by Bernstein’s inequality, P(Qc ) ≤ e−Ck . Let π e be the joint distribution of (θ, τ ), and π be the distribution obtained from renormalizing the restriction of π e on {(θ(e γ ), τ (e γ )) : γ e ∈ Q} which is exactly the set in (71). Then we have c −Ck f |e f |θ, τ ) since there TV(π, π e) ≤ CP(Q ) ≤ Ce . In addition, we note that L(W γ ) = L(W exists one-to-one identification between the pair (θ, τ ) and e γ . Therefore, we have Z Z f ), L(W f |θ, τ )dπ(θ, τ )) f ), Q2n dπ(θ, τ )) ≤ TV(L(W TV(L(W θ,τ Z Z f |θ, τ )dπ(θ, τ ), Q2n dπ(θ, τ )) +TV( L(W θ,τ f |θ, τ ), Q2n ) ≤ TV(e π , π) + sup TV(L(W θ,τ  θ,τ −Ck ≤ C e 29 +N −1  . R f ) = L(W f |θ, τ )de Here, the Rsecond inequality holds since L(W π(θ, τ ). Hence, by (72),
4n −Ck −1 2n +N + N . Note that on the support of π, the TV(L(W ), Qθ,τ dπ(θ, τ )) ≤ C e parameter (θ, τ ) belongs to the set (71). An application of data-processing inequality leads to the conclusion. When p ≥ 2n, we may first analyze the distribution of the first 2n coordinates using the above arguments. The remaining 2n − p coordinates are exact, and the total variation bound is zero. P ′ b Proof of Theorem 5.2. Abbreviate n1 2n i=n+1 Wi Wi by Σ. We can rewrite the testing function ψ as n o b θb ≥ 1 + kηN /4 . ψ(W ) = ψ(A, µ, B0 , B1 ) = 1 θb′ Σ Here, µ = (µ1 , . . . , µ2n ) collects the random variables in (30). Thus, it is clear that ψ is a randomized test for the Planted Clique detection problem (22). Note that for any (θ, τ ) in the support of π, we have  3k kηN  . (73) Qnθ,τ ∈ Q n, , p, 2 2 We now bound the testing errors. For Type-I error, Lemma 7.2 implies PH G ψ ≤ Qn0 ψ + CN −1 . 0 b are independent. Conditioning on θb and using Bernstein’s Note that under Qn0 , θb and Σ inequality, we have 2n   X b 2 > 1 + kηN , b θb = 1 + 1 |θb′ Wi |2 − ||θ|| θb′ Σ n 4 i=n+1   4 b with probability at most exp − N 2Cnk (log N )4 . Integrating over θ, we have  PH G ψ ≤ exp − 0 Cnk 4  + CN −1 ≤ C(n−1 + N −1 ), N 2 (log N )4 where the last inequality holds under the assumptions some sufficiently small constant c > 0. Turn to the Type-II error. Lemma 7.3 implies N (log N )5 k4 (74) ≤ c and cN ≤ n ≤ N/12 for  4n  , (75) PH G (1 − ψ) ≤ Qπ (1 − ψ) + C e−Ck + N −1 + 1 N R where we have used the notation Qπ = Qnθ,τ dπ. For each Qnθ,τ in the support of π, Wn+i has √ representation Wn+i = τ gi θ + ǫi , where the gi ’s and the ǫi ’s are independently distributed according to N (0, 1) and Np (0, Ip ), and are independent across i = 1, ..., n, and τ ≥ kηN /2. Therefore, √ n n n  1X 1 X 2 τ b′ X ′ b 2 ′ 2 ′ ′ 2 b θb = τ |θb θ| gi ǫi θ. gi + |θb ǫi | + θθ θb Σ n n n i=1 i=1 30 i=1 After rearrangement, we have n b θb − (1 + τ ) θb′ Σ ≤ 1X 2 (gi − 1) + τ min{|(θb − θ)′ θ|2 , |(θb + θ)′ θ|2 } n + i=1 n X 1 n i=1 n (|θb′ ǫi |2 − 1) + 2X b , gi (ǫ′i θ) n i=1  where min{|(θb − θ)′ θ|2 , |(θb + θ)′ θ|2 } is bounded by min{|(θb − θ)′ θ|2 , |(θb + θ)′ θ|2 } ≤ min ||θb − θ||2 , ||θb + θ||2 ≤ kPθb − Pθ k2F . Together with (34), the above bound implies that for each (θ, τ ) pair in the support of π, n 1o ≤ β. Qnθ,τ min{|(θb − θ)′ θ|2 , |(θb + θ)′ θ|2 } > 3 (76) By Bernstein’s inequality, we have Qnθ,τ r n n n n 1X 1 X b′ 2 2X ′ log n o 2 ′b ≤ n−C . (gi − 1) + (|θ ǫi | − 1) + gi (ǫi θ) > C n n n n i=1 i=1 i=1 Combining the above analysis and using the assumptions that N/12, we have N (log N )5 k4 ≤ c and cN ≤ n ≤ ′ Qnθ,τ (1 − ψ) ≤ β + n−C . (77) Integrating over (θ, τ ) according to the prior π and applying (75), we obtain   4n ′ . PH G (1 − ψ) ≤ β + n−C + C e−Ck + N −1 + 1 N Summing up the Type-I and Type-II errors, we have PH G ψ + PH G (1 − ψ) ≤ β + 0 1 4n ′ + C(n−1 + N −1 + e−C k ). N (78) Thus, the proof is complete. 7.2 Proofs of Theorems 5.3, 5.4 and 5.1 Consider a Planted Clique detection problem with N = 12n and  k = ⌊2s/3⌋. Then the assumptions of Theorem 5.2 are satisfied. Thus, supQ∈Q(n,s,p,λ) Q kPθb − Pθ k2F > 13 ≤ 41 ′ implies a testing error bounded by 41 + 31 + C(n−1 + N −1 + e−C k ), which is smaller than 2/3 as n → ∞. This contradicts the condition (35) that implies the condition of Hypothesis A. Hence, we must have (36) and the proof of Theorem 5.3 is complete. To prove Theorem 5.4, note that according to the proof of Theorem 5.2, a testing error bound 1/4 for the sparse PCA ′ problem implies a testing error bound 14 + 31 + C(n−1 + N −1 + e−C k ) for the Planted Clique detection problem. The same argument we have just used leads to the desired conclusion. Finally, Theorem 5.1 can be derived from Theorem 5.5 and Theorem 5.2 by applying a similar argument. 31 7.3 Proof of Theorem 5.5 Let Wi ∼ Np (0, τ θθ ′ + Ip ), then (Xi′ , Yi′ )′ ∼ Np+m (0, Σ) with Σ given in (41). We complete the proof by noting o 16 min kb n u − uk2 , kb u + uk2 kPθb − Pθ k2F ≤ 4 min kθb − θk2 , kθb + θk2 ≤ kuk2   3 2 σ 2 (Σx ) L(b u, u) ≤ 16 1 + ≤ 16 max L(b u, u). 2 2 σmin (Σx ) 7.4 Proof of Lemma 7.1 √ We first verify that (28)–(29) are proper density functions when |µ| ≤ 3 ηN log N , which is a corollary of the following lemma. √ √ Lemma 7.4. If k ≤ N/12, |µ| ≤ 3 ηN log N and |x| ≤ 3 log N , then 4 −1 φ̄µ (x) − φ0 (x) ≤ φ0 (x). δN 5 Proof. By definition,   µ2  µ2  −1 φ̄µ (x) − φ0 (x) = (2δN )−1 φ0 (x) exp µx − δN + exp − µx − −2 . 2 2 Under the conditions of the lemma, we have |µx| + µ2 2 ≤ 12 , and so   µ2  4 µ4 µ2  + exp − µx − − 2 ≤ µ2 + |µx|2 + ≤ 8µ2 log N. exp µx − 2 2 3 3 We complete the proof by combining the last two displays. The following lemma controls the rescaling constants in (28) and (29). Lemma 7.5. There exists an absolute constant C > 0 such that for any |µ| ≤ 1, |Mi − 1| ≤ CN −4 for i = 0, 1. Proof. Note that Z Z 1 = fµ,0 (x)dx = M0 −M0 Z
−1 (φ̄µ (x) − φ0 (x)) dx φ0 (x) − δN √ |x|>3 log N = M0 − M0 Z
−1 (φ̄µ (x) − φ0 (x)) dx φ0 (x) − δN √ |x|>3 log N The integral on the RHS is upper bounded by Z Z
−1 −1 φ (x)dx + δ 1 + δN 0 N √
−1 (φ̄µ (x) − φ0 (x)) dx. φ0 (x) − δN √ |x|>3 log N |x|>3 log N φ̄µ (x)dx ≤ CN −4 , where the last inequality comes from standard Gaussian tail bounds. This readily implies |M0 − 1| ≤ CN −4 The desired bound on M1 follows from similar arguments. 32 Proof of Lemma 7.1. Define −1 (φ̄µ (x) − φ0 (x)), gi (x) = φ0 (x) − (−1)i δN for i = 0, 1. Then we have for i = 0 and 1, fµ,i (x) = gi (x) − (1 − Mi 1{|x|≤3√log N } )gi (x), By Lemma 7.4 and Lemma 7.5, Z Z (1 − Mi 1{|x|≤3√log N } )gi (x) dx |fµ,i (x) − gi (x)|dx ≤ Z Z ≤ |1 − Mi | |gi (x)|dx + Mi √ |x|>3 log N |gi (x)|dx ≤ CN −3 . (79) Therefore, we have   Z 1 1 1 TV (fµ,0 + fµ,1 ), φ0 = φ0 (x) − (fµ,0 (x) + fµ,1 (x)) dx 2 2 2 Z Z 1 1 1 X ≤ |fµ,i (x) − gi (x)|dx ≤ CN −3 , φ0 (x) − (g0 (x) + g1 (x)) dx + 2 2 4 i=0,1 where the last inequality is due to the identity φ0 = 12 (g0 + g1 ) and (79). In addition, we have   1 TV δN fµ,1 + (1 − δN ) (fµ,0 + fµ,1 ) , φ̄µ 2 Z 1 − δN 1 (fµ,0 (x) + fµ,1 (x)) − φ̄µ (x) dx δN fµ,1 (x) + = 2 2 Z 1 1 − δN ≤ (g0 (x) + g1 (x)) − φ̄µ (x) dx δN g1 (x) + 2 2 X 1 − (−1)i δN Z + |fµ,i (x) − gi (x)|dx 4 ≤ CN i=0,1 −3 . Here, the last inequality is due to the identity δN g1 + completes the proof. 8 1−δN 2 (g0 + g1 ) = φ̄µ and (79). This Discretization and Computational Lower bounds To formally address the computational complexity issue in a continuous statistical model, we adopt the framework in [30]. After introducing the asymptotically equivalent discretized models, we state the computational lower bounds for sparse PCA and sparse CCA under the discretized models in Section 8.1. The necessary modifications to Algorithms 1 and 2 are spelled out in Section 8.2 to ensure that they are truly of randomized polynomial time complexity. 33 For any t ∈ N, define the function [·]t : R → 2−t Z by   [x]t = 2−t 2t x . (80) For any matrix, the function is defined component-wise. Let (p,n) EM  iid = L(X1 , . . . , Xn ) : Xi ∼ Np (µ, Σ), 1/M ≤ σmin (Σ) ≤ σmax (Σ) ≤ M be the class of joint distributions of n i.i.d. samples from all multivariate Gaussian distributions with spectrum contained in [1/M, M ], and (p,n,t) EM (p,n) = {L([X1 ]t , . . . , [Xn ]t ) : L(X1 , . . . , Xn ) ∈ EM } be its discretized counterpart. The following lemma bounds the Le Cam distance [26] between the two classes of distributions. Its proof is given below in Section 8.3. (p,n) Lemma 8.1. When 2t t−1/2 ≥ 2(pM )3/2 , the Le Cam distance between EM (p,n) (p,n,t) ) ≤ n(pM )3/2 t1/2 2−t . satisfies ∆(EM , EM (p,n,t) and EM For any t ∈ N, we define discretized sparse PCA parameter space as Qt (n, s, p, λ) = {L([W ]t ) : L(W ) ∈ Q(n, s, p, λ)} , and discretized sparse CCA probability space as P t (n, su , sv , p, m, r, λ; M ) = {L([X]t , [Y ]t ) : L(X, Y ) ∈ P(n, su , sv , p, m, r, λ; M )} . In view of Theorem 5.1, we are primarily interested in P(n, su , sv , p, m, 1, λ; 3) and its discretized counterpart. For the sparse PCA parameter spaces, with the choice of n, s, p and λ in Theorem (p,n) (p,n,t) 5.3, under condition (35), Q(n, s, p, λ) ⊂ E4 and Qt (n, s, p, λ) ⊂ E4 . Thus, if we set the discretization level at t = ⌈4 log2 (p + n)⌉, then Q(n, s, p, λ) and Qt (n, s, p, λ) are asymptotically equivalent. Similarly, with the choice of n, su , sv , p, m, λ in Theorem 5.1, un(p+m,n) and der condition (23), when t = ⌈4 log2 (p + m + n)⌉, P(n, su , sv , p, m, 1, λ; 3) ⊂ E5 (p+m,n,t) t P (n, su , sv , p, m, 1, λ; 3) ⊂ E5 are also asymptotically equivalent. Therefore, with the foregoing discretization levels, the statistical difficulties of the original sparse PCA and CCA problems are asymptotically equivalent to those of the discretized problems. In particular, the conditions for any procedure to be consistent are the same for the original and the discretized parameter spaces. 8.1 Computational lower bounds for discretized models We now state computational lower bounds for the discretized sparse PCA and sparse CCA problems. The meaning of “randomized polynomial-time estimators” is now based on the probabilistic Turing machine computation model rather than the computation model mentioned in Remark 5.1. 34 Theorem 8.1. Let t = ⌈4 log2 (p + n)⌉. Under the condition of Theorem 5.3, for any ranb domized polynomial-time estimator θ, n 1o 1 > . (81) lim inf sup Q kPθb − Pθ k2F > n→∞ Q∈Qt (n,s,p,λ) 3 4 Theorem 8.2. Let t = ⌈4 log2 (p + m + n)⌉. Under the condition of Theorem 5.1, for any randomized polynomial-time estimator u b, n 1 o 1 lim inf sup P L(b u, u) > > . (82) n→∞ P∈P t (n,s ,s ,p,m,1,λ;3) 300 4 u v To prove these theorems, we need to modify Algorithms 1 and 2 which are not compatible with the Turing machine computation model. The details are spelled out in the next subsection. After these modifications, the proofs can be obtained by essentially following the lines of the proofs of their continuous counterparts while controlling some additional negligible terms in total variation bounds due to additional truncation. The details are omitted. 8.2 Randomized polynomial-time reduction for discretized models We first introduce a way to approximately sample with polynomial time complexity from a distribution obtained from discretizing a continuous distribution with density [30, Section 4.2]. The modifications to Algorithms 1 and 2 then follow. For any w, K ∈ N and K + w + 1 < b ∈ N, define the discrete distribution Aw,b,K [F] with probability mass function Aw,b,K [f ] as   R K −w −2 +i2 f (x)dx i − 1
−2K +(i−1)2−w  , (83) Aw,b,K [f ] − 2K + w =  R 2K 2 K f (x)dx −2 b for i ∈ [2K+w+1 − 1], and let K −w Aw,b,K [f ](2 − 2 ) =1− 2K+w+1 X −1 i=1 Aw,b,K [f ](−2K + (i − 1)2−w ). (84) In (83), [·]b is the quantization defined previously in (80), and (84) ensures that Aw,b,K [F] is a proper probability distribution. By the definition of total variation distance, it is straightforward to verify that the approximation error in total variation distance by Aw,b,K [F] to the distribution of [U 1{U ∈[−2K ,2K ]} ]w with U ∼ F is upper bounded by 2K+w+1−b . As discussed in Section 4.2 of [30], regardless of the original distribution F, the computational complexity of drawing a random number from Aw,b,K (F) is O(b2K+w ). This fact is crucial in ensuring the modified reduction below is of randomized polynomial-time. Randomized polynomial-time reduction For any t ∈ N, we let p w = t + ⌈4 log2 p⌉, K = ⌈log2 (3 log(N + p))⌉, b = w + K + 1 + ⌈4 log2 p⌉. (85) The reduction in Algorithm 1 is modified to Algorithm 3 and the reduction in Algorithm 2 is modified to Algorithm 4. As in the continuous case, a direct reduction from Planted Clique 35 Algorithm 3: Reduction from Planted Clique to Sparse PCA (in Discretized Gaussian Single Spiked Model) Input: 1. Graph adjacency matrix A ∈ {0, 1}N ×N ; 2. Estimator θb for the leading eigenvector θ. Output: A solution to the hypothesis testing problem (22). e 0 ]. Set 1 Initialization. Generate i.i.d. random variables ξ1 , . . . , ξ2n ∼ Aw,b,K [Φ 1/2 µi = [ηN ]w ξi , 2 i = 1, . . . , 2n. Gaussianization. Generate two matrices B0 , B1 ∈ R2n×2n where conditioning on the µi ’s, all the entries are mutually independent satisfying L((B0 )ij |µi ) = Aw,b,K [Fµi ,0 ] and L((B1 )ij |µi ) = Aw,b,K [Fµi ,1 ]. Let A0 ∈ {0, 1}2n×2n be the lower–left 2n × 2n submatrix of the matrix A. Generate a ′ ]′ ∈ R2n×p where for each i ∈ [2n], if j ∈ [2n], we set matrix W = [W1′ , . . . , W2n Wij = (B0 )ij (1 − (A0 )ij ) + (B1 )ij (A0 )ij . 3 If 2n < j ≤ p, we let Wij be an independent draw from Aw,b,K [N (0, 1)]. b 1 ]t , . . . , [Wn ]t ) be the estimator of the leading Test Construction. Let θb = θ([W eigenvector by treating {[Wi ]t }ni=1 as data. It is normalized to be a unit vector. We reject H0G if θb′ 2n  1 X n [Wi ]t [Wi ]′t i=n+1   1 θb ≥ 1 + [ k ηN ]w . w 4 to discretized sparse CCA can be obtained by constructing the estimator θb in the third step of Algorithm 3 from Algorithm 4. By (85) and the discussion following (84), the complexity for sampling any random variable in the above reduction is O(p8 (log p)3/2 ), and in total, we need to generate no more than O(n(p + n)) random variables. Hence, the total complexity for random number generation is O(p10 (log p)3/2 ) in view of the condition p ≥ 2n. On the other hand, it is straightforward b have complexity to verify that all the other computations (except for the estimator u b or θ) 10 3/2 no more than O(p (log p) ). Since the conditions of Theorems 8.2 and 8.1 ensure that for some constant a > 1, 2n ≤ p ≤ na and n ≤ N/12, we obtain that the additional computational complexity induced by the proposed reductions is O(N 10a (log N )3/2 ). Therefore, they are of randomized polynomial-time complexity. 8.3 Proof of Lemma 8.1 We need the following lemma for the proof. 36 Lemma 8.2. For X ∼ Np (µ, Σ) with M −1 ≤ σmin (Σ) ≤ σmax (Σ) ≤ M and U = (U1 , . . . , Up )′ iid where Ui ∼ Unif[0, 1], we have for any t−1/2 2t ≥ 2(pM )3/2 , TV(X, [X]t + 2−t U ) ≤ (pM )3/2 t1/2 2−t . (p,n,t) comes from discretizing a corresponding distribution in Since each distribution in EM (p,n,t) (p,n) (p,n) ) = 0. On the other hand, EM on a grid with equal spacing 2−t , we have δ(EM , EM (p,n) (p,n,t) , EM ) ≤ n(pM )3/2 t1/2 2−t . This completes Lemma 8.2 and Lemma 7 of [30] lead to δ(EM the proof. −t Proof of Lemma 8.2. Let f and g denote the density functions of X and [X] Qpt + 2−t U , respectively. Then g is a piecewise constant function. For any (x1 , ..., xp ) ∈ B = i=1 [2 ij , 2−t (ij + 1)), where ij ∈ Z, we have Z 1 f (x1 , ..., xp )dx1 ...dxp , g(x1 , ..., xp ) = ν(B) B where ν is the Lebesgue measure. Hence, sup ||x−µ||∞ ≤K ≤ ≤ g(x) −1 ≤ f (x) ′ e|(x−µ) Σ sup sup ||x−µ||∞ ≤K ||x−y||∞ ≤2−t f (x) −1 f (y) −1 (x−µ)−(y−µ)′ Σ−1 (y−µ)|/2 ||x−µ||∞ ≤K ||x−y||∞ ≤2−t ekΣ sup −1 −1 k k(x−µ)(x−µ)′ −(y−µ)(y−µ)′ k /2 F F ||x−µ||∞ ≤K ||x−y||∞ ≤2−t −1   ≤ exp p3/2 M K2−t − 1 ≤ (87) 3 3/2 p M K2−t , 2 Algorithm 4: Reduction from Discretized Sparse PCA to Discretized Sparse CCA Input: 1. Observations W1 , . . . , Wn ∈ (2−t Z)p ; 2. Estimator u b of the first leading canonical correlation coefficient u. Output: An estimator θb of the leading eigenvector of L(W1 ). 1 Generate i.i.d. random vectors Zi = (Zi1 , . . . , Zip )′ for i ∈ [n] with iid Zij ∼ Aw,b,K [N (0, 1)]. Set h 1 i (Wi + Zi ), Xi = √ 2 w 2 (86) h 1 i Yi = √ (Wi − Zi ), 2 w Compute u b=u b([X1 ]t , [Y1 ]t , . . . , [Xn ]t , [Yn ]t ). Set b 1 , . . . , Wn ) = [b θb = θ(W u/kb uk]w . 37 i = 1, . . . , n. whenever p3/2 M K2−t ≤ 21 . The inequality (86) holds since |(x − µ)′ Σ−1 (x − µ) − (y − µ)′ Σ−1 (y − µ)|
= Tr Σ−1 [(x − µ)(x − µ)′ − (y − µ)(y − µ)′ ] ≤ kΣ−1 kF k(x − µ)(x − µ)′ − (y − µ)(y − µ)′ kF √ by Cauchy-Schwarz inequality. The inequality (87) holds because kΣ−1 kF ≤ pkΣ−1 kop ≤ √ pM and k(x − µ)(x − µ)′ − (y − µ)(y − µ)′kF ≤ p||x − y||∞ (||x − µ||∞ + ||y − µ||∞) ≤ 2pK2−t . Note that Z Z Z g(x) − 1 dx. f (x) |f (x) − g(x)|dx + |f − g| ≤ f (x) ||x−µ||∞ ≤K ||x−µ||∞ >K According to Gaussian tail probability, the first term can be bounded by 2p q √ 2 2 M − (K−1) 2M . π K−1 e The second term is bounded by 23 p3/2 M K2−t according to our previous analysis. Choosing √ K = 2M t log 2 + 1, we obtain the bound 2(pM )3/2 t1/2 2−t for Rall t−1/2 2t ≥ 2(pM )3/2 . The conclusion follows the simple fact that TV(X, [X]t + 2−t U ) = 21 |f − g|. 9 9.1 Additional Proofs Proof of Theorem 3.1 We first present a bound for the estimator defined by (9) under the joint loss. Theorem 9.1. Assume (13) for some sufficiently small c > 0. Then there exist constants C, C ′ > 0 only depending on c such that     b (0) (Vb (0) )′ − U V ′ Σ1/2 k2 ≤ C r(su + sv ) + su log ep + sv log em , kΣ1/2 U y F x nλ2 su sv with P-probability at least 1 − exp (−C ′ (su + log(ep/su ))) − exp (−C ′ (sv + log(em/sv ))) uniformly over P ∈ P(n, su , sv , p, m, r, λ; M ). Theorem 9.1 is similar to Theorem 1 of [18], except that the loss function depends on the marginal covariances so that the error bound is independent of M . Its proof is omitted given the similarity with that of Theorem 1 of [18]. ∗ b (1) b (1) b (1) From now on, we omit the superscript in Σ x , Σy and Σxy for simplicity. Let U = b (1) − U ∗ . The following lemmas are needed in the proof of Theorem U ΛV ′ Σy Vb (0) and ∆ = U 3.1. Their proofs are given in Section 9.3.2. u) Lemma 9.1. Assume su log(ep/s ≤ c for some sufficiently small constant c ∈ (0, 1). Then, n there exist some constants C, C ′ > 0 only depending on c, such that with probability at least 1 − exp(−C ′ su log(ep/su )), r r s log(ep/s ) su log(ep/su ) u u b (0) kop ≤ 1 + C , k(Vb (0) )′ Σy Vb (0) − Ikop ≤ C . kΣ1/2 y V n n 38 u) Lemma 9.2. Assume su log(ep/s ≤ c for some sufficiently small constant c > 0. Then, there n ′ exist some constants C, C > 0 only depending on c, such that ′ 2 ′ 1/2 2 b 1/2 2 (1 − δC )kΣ1/2 x ∆kF ≤ kΣx ∆kF ≤ (1 + δC )kΣx ∆kF , q u) ′ ′ . with probability at least 1 − exp(−C su log(ep/su )), with δC = C su log(ep/s n Lemma 9.3. Assume n1 (sv log(em/sv ) + su log(ep/su ) + rsu ) ≤ c for some sufficiently small constant c > 0. Then, there exist some constants C, C ′ > 0 only depending on c, such that r   rsu + su log(ep/su ) 1/2 ′ b (0) kΣx ∆kF , Tr ∆ (Σxy − Σxy )Vb ≤C n with probability at least 1 − exp (−C ′ (su log(ep/su ) + rsu )) − exp(−C ′ sv log(em/sv )). Lemma 9.4. Assume n1 (sv log(em/sv ) + su log(ep/su ) + rsu ) ≤ c for some sufficiently small constant c > 0. Then, there exist some constants C, C ′ > 0 only depending on c, such that r   b x − Σx )U ∗ ≤ C rsu + su log(ep/su ) kΣ1/2 ∆kF , Tr ∆′ (Σ x n with probability at least 1 − exp (−C ′ (su log(ep/su ) + rsu )) − exp(−C ′ sv log(em/sv )). 1/2 Proof. The proof consists of two steps. First, we derive a bound for kΣx ∆kF . Next, we b , U ). derive the desired bound for L(U Step 1. By the definition of the estimator, we have b (1) )′ Σ b xU b (1) ) − 2 Tr((U b (1) )′ Σ b xy Vb (0) ) ≤ Tr((U ∗ )′ Σ b x U ∗ ) − 2 Tr((U ∗ )′ Σ b xy Vb (0) ). Tr((U After rearrangement, we have
b x ∆) ≤ 2 Tr ∆′ (Σ b xy Vb (0) − Σ b xU ∗ ) Tr(∆′ Σ

b xy − Σxy )Vb (0) + 2 Tr ∆′ (Σ b x − Σx )U ∗ . ≤ 2 Tr ∆′ (Σ Using Lemma 9.2, Lemma 9.3 and Lemma 9.4, we have r 1 1/2 2 rsu + su log(ep/su ) 1/2 kΣx ∆kF ≤ 4C kΣx ∆kF , 2 n 1/2 with high probability, which immediately implies a bound for kΣx ∆k2F . This completes Step 1. Step 2. We claim that
−1 ′ b (0) ≤ C , σmin Σ1/2 x U ΛV Σy V λr b − Σ1/2 U b (1) ((U b (1) )′ Σx U b (1) )−1/2 kF ≤ C rsu + su log(ep/su ) , kΣx1/2 U x n 39 (88) (89) with high probability. The two claims (88) and (89) will be proved in the end. We bound b , U ) by L(U q b , U ) = inf kΣ1/2 b L(U x (U W − U )kF W ∈O(r,r) 1/2 b (1) b (1) ′ b b (1) )−1/2 kF ≤ kΣ1/2 ((U ) Σx U x U − Σx U b (1) ((U b (1) )′ Σx U b (1) )−1/2 W − Σ1/2 U kF . + inf kΣ1/2 U W ∈O(r,r) x x With high probability, we could further bound the rightmost side by r 1 rsu + su log(ep/su ) C + √ kPΣ1/2 Ub − PΣ1/2 U kF x x n 2 r   rsu + su log(ep/su ) ′ −1 b (0) kΣ1/2 ≤ C Σ1/2 + Cσmin x ∆kF x U ΛV Σy V n r rsu + su log(ep/su ) ≤ C . nλ2 (90) (91) The bound (90) is due to the claim (89), Lemma 6.6 and the fact that PΣ1/2 Ub = PΣ1/2 Ub (1) . The inequality (91) is derived from the sin-theta theorem [41]. Thus, we have obtained b , U ). To finish the proof, we need to prove (88) and (89). Since the desired bound for L(U 1/2 Σx U ∈ O(p, r), we have
−1 ′ b (0) ≤ λ−1 k(V ′ Σy Vb (0) )−1 kop . σmin Σ1/2 x U ΛV Σy V Thus, it is sufficient to bound k(V ′ Σy Vb (0) )−1 kop . By Theorem 9.1 and sin-theta theorem [41], kPΣ1/2 Vb (0) − PΣ1/2 V kF is sufficiently small. In view of Lemma 6.6, there exists W ∈ O(r, r), y y such that kΣy1/2 Vb (0) (Vb (0) Σy Vb (0) )−1/2 − Σ1/2 y V W kF is sufficiently small. Therefore, together with Lemma 9.1, kV ′ Σy Vb (0) − W kop ≤ kV ′ Σy Vb (0) − V ′ Σy V W (Vb (0) Σy Vb (0) )1/2 kop + kW kop k(Vb (0) Σy Vb (0) )1/2 − Ikop b (0) Σy Vb (0) )1/2 kop + k(Vb (0) Σy Vb (0) )1/2 − Ikop ≤ kΣy1/2 Vb (0) (Vb (0) Σy Vb (0) )−1/2 − Σ1/2 y V W kF k(V is also sufficiently small. By Weyl’s inequality [20, p.449], |σmin (V ′ Σy Vb (0) )−1| ≤ kV ′ Σy Vb (0) − W kop is sufficiently small. Hence, k(V ′ Σy Vb (0) )−1 kop ≤ 2 with high probability, which implies the desired bound in (88). Finally, we need to prove (89). We have b − Σ1/2 b (1) ((U b (1) )′ Σx U b (1) )−1/2 kF kΣx1/2 U x U b (1) kF k((U b (1) )′ Σ b (2) U b (1) )−1/2 − ((U b (1) )′ Σx U b (1) )−1/2 kop ≤ kΣx1/2 U x   b (1) )′ (Σ b (2) b (1) kop . ≤ C kΣx1/2 U ΛV ′ Σy Vb (0) kF + kΣ1/2 x − Σx )U x ∆kF k(U 1/2 1/2 We have already shown that kΣx ∆kF is sufficiently small. The term kΣx U ΛV ′ Σy Vb (0) kF √ √ √ is bounded by rkV ′ Σy Vb (0) kop ≤ r(1 + kV ′ Σy Vb (0) − W kop ) ≤ C r by using the bound 40 b (1) )′ (Σ b (2) b (1) kop , note that Σ b (2) derived for kV ′ Σy Vb (0) − W kop . To bound k(U x − Σx )U x only (1) b depends on D2 and is independent of U . Using union bound and an ǫ-net argument (see, 1/2 for example, [36]) and the fact q that r ≤ su (which is implied by Σx U ∈ O(p, r)), we have b (1) )′ (Σ b (2) b (1) kop ≤ C rsu +su log(ep/su ) with high probability. Hence, the proof is k(U x − Σx )U n complete. 9.2 Proof of Theorem 3.2 For any probability measures P, Q, define the Kullback-Leibler(KL) divergence by D(P||Q) =  R dP log dQ dP. The following result is Lemma 14 in [18]. It gives explicit formula for the KL divergence between distributions generated by a special kind of covariance matrices. " # ′ Ip λU(i) V(i) Lemma 9.5. For i = 1, 2, let Σ(i) = with λ ∈ (0, 1), U(i) ∈ O(p, r) ′ λV(i) U(i) Im and V(i) ∈ O(m, r). Let P(i) denote the distribution of a random i.i.d. sample of size n from the Np+m (0, Σ(i) ) distribution. Then D(P(1) ||P(2) ) = nλ2 ′ kU V ′ − U(2) V(2) k2F . 2(1 − λ2 ) (1) (1) The main tool for our proof is the following Fano’s lemma [44, Lemma 3]. Proposition 9.1. Let (Θ, ρ) be a metric space and {Pθ : θ ∈ Θ} a collection of probability measures. For any totally bounded T ⊂ Θ, denote by M(T, ρ, ǫ) the ǫ-packing number of T with respect to ρ, i.e., the maximal number of points in T whose pairwise minimum distance in ρ is at least ǫ. Define the KL diameter of T by dKL (T ) , supθ,θ′ ∈T D(Pθ ||Pθ′ ). Then  
ǫ2 dKL (T ) + log 2 2 b . inf sup Pθ ρ θ(X), θ ≥ ≥1− 4 log M(T, ρ, ǫ) b θ θ∈Θ (92) Finally, we lower bound the prediction loss by the squared subspace distance. Its proof is given in Section 9.3.2. Proposition 9.2. Suppose the eigenvalues of Σx lie in the interval [M1 , M2 ]. Then, we have kPUb − PU k2F ≤ A similar inequality holds for L(Vb , V ). √ M2 b , U ). 2 L(U M1 Proof of Theorem 3.2. Let us first give an outline of the proof. By Proposition 9.2, we have
 b , U ) ≥ Cǫ2 ≥ inf sup P kP b − PU k2F ≥ C1 ǫ2 , inf sup P L(U U b P∈P U b P∈P U for any rate ǫ2 . Therefore, it is sufficient to derive a lower bound for the loss kPUb − PU k2F . Without loss of generality, we assume su /3 is an integer and su ≤ 3p/4. The case su > 41 3p/4 is harder and thus it shares the same lower bound. The subset of covariance class F(p, m, su , sv , r, λ; M ) we consider is      e 0 Ip λU V0′ U e ∈ B, :U = T = Σ= ,U λV0 U ′ Im 0 ur  p−2su /3 ur ∈ R , ||ur || = 1, |supp(ur )| ≤ su /3 ,  ′ where V0 = Ir 0′ ∈ O(m, r) and B is a subset of O(2su /3, r − 1) to be specified later. e and the vector ur . As U e and ur vary, we From the construction, U depends on the matrix U ∗ always have U ∈ O(p, r). We use T (ur ) to denote a subset of T where ur = u∗r is fixed, and e ∗ ) to denote a subset of T where U e =U e ∗ is fixed. use T (U rsu ∗ The proof has three steps. In the first step, we derive the part nλ 2 using the subset T (ur ) s log(ep/s ) u u for some particular u∗r . In the second step, we derive the other part using the nλ2 ∗ ∗ e e subset T (U ) for some fixed U . Finally, we combine the two results in the third step.   e0 = Ir−1 0′ ′ ∈ Step 1. Let u∗r = (1, 0, ..., 0)′ , and we consider the subset T (u∗r ). Let U √ O(2su /3, r − 1) and ǫ0 ∈ (0, r] to be specified later. Define By Lemma 9.5, e ∈ O(2su /3, r − 1) : kU e −U e0 kF ≤ ǫ0 }. B = B(ǫ0 ) = {U dKL (T (u∗r )) = 2 2 nλ2 e(1) − U e(2) k2 ≤ 2nλ ǫ0 . k U F 2(1 − λ2 ) 1 − λ2 ∈B(ǫ0 ) sup e(i) U (93) Here, the equality is due to the definition of V0 and the inequality due to the definition of B(ǫ0 ). We now establish a lower bound for the packing number of T (u∗r ). For some α ∈ (0, 1) e(1) , . . . , U e(N ) } ⊂ O(2su /3, r − 1) be a maximal set such that for to be specified later, let {U any i 6= j ∈ [N ], √ e(i) U e′ − U e0 U e0′ kF ≤ ǫ0 , e(i) U e′ − U e(j) U e ′ kF ≥ 2αǫ0 . (94) kU kU (i) (i) (j) Then by [13, Lemma 1], for some absolute constant C > 1, N≥  1 Cα (r−1)(2su /3−r+1) . e(i) U e′ − It is easy to see that the loss function kPU(i) − PU(j) k2F on the subset T (u∗r ) equals kU (i) √ e(j) U e ′ k2 . Thus, for ǫ = 2αǫ0 with sufficiently small α, log M(T (u∗ ), ρ, ǫ) ≥ (r−1)(2su /3− U r (j) F 1 1 1 1 r+1) log Cα ≥ (r−1)( 16 su −1) log Cα ≥ 12 rsu log Cα when r is sufficiently large and r ≤ su /2. rsu 2 Taking ǫ0 = c1 nλ2 for sufficiently small c1 , we have   ǫ2 2 inf sup P kPUb − PU kF ≥ ≥1− 4 b T (u∗ ) U r 2c1 rsu 1−λ2 + log 2 1 . 1 12 rsu log Cα (95) Since λ is bounded away from 1, we may choose sufficiently small c1 and α, so that the right hand side of (95) can be lower bounded by 0.9. This completes the first step. 42 u) can be obtained from the rank-one argument spelled out in Step 2. The part su log(ep/s nλ2   e ∗ ) with U e ∗ = Ir−1 0′ ′ ∈ O(2su /3, r − 1). [14]. To be rigorous, consider the subset T (U e ∗ ), the loss function is Restricting on the set T (U kPU(i) − PU(j) k2F = kur,(i) u′r,(i) − ur,(j) u′r,(j) k2F . Let X = [X1 X2 ] with X1 ∈ Rn×(r−1) and X2 ∈ Rn×(p−r+1) , and Y = [Y1 Y2 ] with Y1 ∈ Rn×(r−1) and Y2 ∈ Rn×(m−r+1) . Then it is further equivalent to estimating u1 under projection loss based on the observation (X2 , Y2 ), because (X2 , Y2 ) is a sufficient statistic for ur . Applying the argument in [14, Appendix G] and choosing the appropriate constant, we have   su log(ep/su ) ∧ c inf sup P kPUb − PU k2F ≥ C (96) 0 ≥ 0.9, nλ2 b U e ∗) T (U for some constant C > 0. This completes the second step. Step 3. For any P ∈ P, by union bound, we have  P kPUb − PU k2F ≥ ǫ21 ∨ ǫ22   ≥ 1 − P kPUb − PU k2F < ǫ21 − P kPUb − PU k2F < ǫ22   = P kPUb − PU k2F ≥ ǫ21 + P kPUb − PU k2F ≥ ǫ22 − 1. rsu Taking supT (u∗ )∪T (Ue ∗ ) on both sides of the inequality, and letting ǫ21 = C1 nλ 2 in (95) and r u) ǫ22 = C2 su log(ep/s ∧ c0 in (96), we have nλ2  sup P kPUb − PU k2F ≥ ǫ21 ∨ ǫ22 ≥ 0.9 + 0.9 − 1 = 0.8, P∈P e where we have used the identity supU∈T e e ∗ ) (f (ur ) + g(U )) = supur ∈T (U e ∗ ) f (ur ) + (u∗r ),ur ∈T (U e supUe ∈T (u∗ ) g(U ). Careful readers may notice that we have assume sufficiently large r in Step r 1. For r which is not sufficiently large, a similar rank-one argument as in Step 2 gives the desired lower bound. Thus, the proof is complete. 9.3 Proofs of technical lemmas This section gathers the proofs of all technical results used in the above sections. The proofs are organized according to the order of their first appearance. To simplify notation, we denote b x(j) , Σ b (j) b (j) b b b Σ y and Σxy by Σx , Σy and Σxy for j ∈ {0, 1, 2} whenever there is no confusion from the context. 9.3.1 Proofs of lemmas in Section 6 In order to prove Lemma 6.1, we need an auxiliary result. 43 Lemma 9.6. Assume n1 (su + sv + log(ep/su ) + log(em/sv )) ≤ c for some sufficiently small constant c ∈ (0, 1). Then there exist some constants C, C ′ > 0 only depending on c such that s   ep 1 ′b ′b 1/2 kU Σx U − Ikop ∨ k(U Σx U ) − Ikop ≤ C , su + log n su s   em 1 ′b ′b 1/2 , kV Σy V − Ikop ∨ k(V Σy V ) − Ikop ≤ C sv + log n sv with probability at least 1 − exp(−C ′ (su + log(ep/su ))) − exp(C ′ (sv + log(em/sv ))). Proof. Using the definition of operator norm and the sparsity of U , we have b x U − Ir kop = kU ′ (Σ b x − Σx )U kop = k(USu ∗ )′ (Σ b xSu Su − ΣxSu Su )USu ∗ kop kU ′ Σ −1/2 b xSu Su − ΣxSu Su )(USu ∗ v) ≤ kΣ1/2 USu ∗ k2 kΣ−1/2 Σ b = sup (USu ∗ v)′ (Σ op xSu Su xSu Su ΣxSu Su − Ikop , xSu Su ||v||=1 1/2 −1/2 −1/2 b xSu Su Σ where kΣxSu Su USu ∗ k2op ≤ 1 and kΣxSu Su Σ xSu Su − Ikop is bounded by the desired rate b x U )1/2 − with high probability according to Lemma 16 in [18]. Lemma 15 in [18] implies k(U ′ Σ b x U − Ikop , and thus k(U ′ Σ b x U )1/2 − Ikop also shares same upper bound. The Ikop ≤ CkU ′ Σ ′ ′ b y V − Ikop ∨ k(V Σ b y V )1/2 − Ikop can be derived by the same argument. upper bound for kV Σ Hence, the proof is complete. Proof of Lemma 6.1. According to the definition, we have 1/2 ′b 1/2 e b x U )−1/2 kop , kΣ1/2 − Ikop k(U ′ Σ x (U − U )kop ≤ kΣx U kop k(U Σx U ) 1/2 ′b 1/2 e b y V )−1/2 kop , kΣ1/2 − Ikop k(V ′ Σ y (V − V )kop ≤ kΣy V kop k(V Σy V ) e − Λkop ≤ k(U ′ Σ b x U )1/2 − Ikop kΛ(V ′ Σ b y V )1/2 kop kΛ b y V )1/2 − Ikop . +kΛkop k(V ′ Σ Applying Lemma 9.6, the proof is complete. e , we have U e ′Σ b xU e = I, and thus Σ b 1/2 e Proof of Lemma 6.2. By the definition of U x U ∈ O(p, r). b 1/2 e Similarly Σ y V ∈ O(m, r). Thus, b 1/2 e b 1/2 b 1/2 e b 1/2 e kΣ x AΣy kop ≤ kΣx U kop kΣy V kop ≤ 1. (97) b y V )−1 (V ′ Σ b y V )) = r. Tr(Q′ Q) = Tr((V ′ Σ (98) ′ b 1/2 e b 1/2 e Now let us use the notation Q = Σ x AΣy . Then, by the definition of A, we have Q Q = ′b −1 ′ b 1/2 b 1/2 Σ y V (V Σy V ) V Σy , and Combining (97) and (98), it is easy to see that all eigenvalues of Q′ Q are 1. Thus, we have kQk∗ = r and kQkop = 1. The proof is complete. 44 Proof of Lemma 6.3. Denote F = [f1 , ..., fr ], G = [g1 , ..., gr ] and cj = fj′ Egj . By kEkop ≤ 1, we have |cj | ≤ 1. The left hand side of (44) is lower bounded by hF KG′ , F G′ − Ei ≥ hF DG′ , F G′ − Ei − P kK − DkF kF G′ − EkF , where hF DG′ , F G′ − Ei = hD, I − F ′ EGi = P r r l=1 dl (1 − cl ) ≥ dr l=1 (1 − cl ). The first term on the right hand side of (44) is  dr  dr kF G′ k2F + kEk2F − 2 Tr(F ′ EG) kF G′ − Ek2F = 2 2 r  X dr  ≤ cj Tr(F ′ F G′ G) + kEkop kEk∗ − 2 2 j=1 ≤ dr r X j=1 (1 − cj ). This completes the proof. b xy − Σ e xy ||∞ can be upper bounded by the Proof of Lemma 6.4. Using triangle inequality, ||Σ following sum, b xy − Σxy ||∞ + ||(Σ b x − Σx )U ΛV ′ Σy ||∞ ||Σ b y − Σy )||∞ + ||(Σ b x − Σx )U ΛV ′ (Σ b y − Σy )||∞ . +||Σx U ΛV ′ (Σ The first term can be bounded by the desired rate by union bound and Bernstein’s inequality [36, Prop. 5.16]. For the second term, it can be written as n 1X max (Xij [Xi′ U ΛV ′ Σy ]k − EXij [Xi′ U ΛV ′ Σy ]k ) , j,k n i=1 where Xij is the j-th element of Xi and the notation [·]k means the k-th element of the referred vector. Thus, it is a maximum over average of centered sub-exponential random variables. Then, by Bernstein’s inequality and union bound, it is also bounded by the desired rate. we can bound the third term. For the last term, it can be bounded by Pr Similarly, ′ b ′ b b b λ ||( Σ − Σ x x )ul vl (Σy − Σy )||∞ , where for each l, ||(Σx − Σx )ul vl (Σy − Σy )||∞ can be l=1 l written as n n 1 X  1 X  max (Xij Xi′ ul − EXij Xi′ ul ) (Yik Yi′ vl − EYik Yi′ vl ) . j,k n n i=1 i=1 with the desired probability using union bound and It can be bounded by the rate log(p+m) n . Under the Bernstein’s inequality. Hence, the last term can be bounded by λ1 r log(p+m) n q log(p+m) assumption that r is bounded by a constant, it can further be bounded by the n q rate log(p+m) with high probability. Combining the bounds of the four terms, the proof is n complete. Proof of Lemma 6.6. By the property of least squares, we have inf kF − GW k2F = kF − G(G′ G)− G′ F k2F = kF − PG F k2F = r − Tr(PF PG ). W Since kPF − PG k2F = 2r − 2 Tr(PF PG ), the proof is complete. 45 Proof of Lemma 6.7. By the definition of U ∗ , we have Σxy Vb (0) = Σx U ∗ . Thus, b xy Vb (0) − Σ b x U ∗ ]j· || ≤ max ||[(Σ b xy − Σxy )Vb (0) ]j· || + max ||[(Σ b x − Σx )U ∗ ]j· ||. max ||[Σ 1≤j≤p 1≤j≤p 1≤j≤p b x − Σx )U ∗ ]j· ||. Note that the sample covariance can be Let us first bound max1≤j≤p ||[(Σ written as n   X 1/2 1 b Zi Zi′ Σ1/2 Σx = Σx x , n i=1 where 1/2 {Zi }ni=1 are i.i.d. Gaussian vectors distributed as N (0, Ip ). Let Tj′ be the j-th row of Σx , and then we have n b x − Σx )U ∗ ]j· = [(Σ 1X ′ ∗ ′ 1/2 ∗ (Tj Zi Zi′ Σ1/2 x U − Tj Σx U ). n i=1 For each i and j, define vector (j) Wi 1/2 # Tj′ Zi . = 1/2 (U ∗ )′ Σx Zi " (j) (j) b x −Σx )U ∗ ]j· || ≤ k 1 Since Tj′ Zi Zi′ Σx U ∗ is a submatrix of Wi (Wi )′ , we have ||[(Σ n (j) (j) EWi (Wi )′ )kop . Hence, for any t > 0, we have n o b x − Σx )U ∗ ]j· || > t P max ||[(Σ Pn (j) (j) ′ i=1 (Wi (Wi ) − 1≤j≤p p n o n 1X X (j) (j) (j) (j) (Wi (Wi )′ − EWi (Wi )′ )kop > t P k ≤ n ≤ (j) j=1 p X j=1 i=1  n exp C1 r − C2 n min t t2 , kW (j) kop kW (j) k2op o , (99) (j) where W (j) = EWi (Wi )′ , and we have used concentration inequality for sample covariance [36, Thm. 5.39]. Since kW (j) kop ≤ C3 for some constant C3 only depending on M , (99) can be bounded by   exp C1′ (r + log p) − C2′ n(t ∧ t2 ) . p Take t2 = C4 r+log for some sufficiently large C4 , and under the assumption n−1 (r + log p) ≤ n q b x −Σx )U ∗ ]j· || ≤ C r+log p with probability at least 1−exp(−C ′ (r+log p)). C1 , max1≤j≤p ||[(Σ n b xy − Σxy )Vb (0) ]j· ||. Let us sketch the Similar arguments lead to the bound of max1≤j≤p ||[(Σ proof. Note that we may write n
1X ′ (0) b b [(Σxy − Σxy )V ]j = Tj Zi Yi′ Vb (0) − E(Tj′ Zi Yi′ Vb (0) ) . n i=1 46 (j) Then, define Hi =   Tj′ Zi , and we have (Vb (0) )′ Yi b xy − Σxy )Vb (0) ]j || ≤ max k 1 max ||[(Σ 1≤j≤p n 1≤j≤p n X i=1 (j) Using the same argument, we can bound this term by C 1 − exp(−C ′ (r + log p)). Thus, the proof is complete. 9.3.2 (j) (j) (j) (Hi (Hi )′ − EHi (Hi )′ )kop . q r+log p n with probability at least Proofs of lemmas in Section 9 1/2 (0) Proof of Lemma 9.1. Let Tv = Sbv ∪Sv , where Sbv = supp(Vb (0) ). First, let us bound kΣyTv Tv VbTv ∗ kop . b y Vb (0) = Ir , we have Since (Vb (0) )′ Σ 1/2 (0) 1/2 b −1/2 b 1/2 Vb (0) kop ≤ kΣ1/2 Σ b −1/2 kop kΣ kΣyTv Tv VbTv ∗ kop ≤ kΣyTv Tv Σ yTv Tv yTv Tv kop yTv Tv yTv Tv −1/2 −1/2 1/2 b b −1 Σ1/2 k1/2 = σmin (Σ−1/2 Σ = kΣyTv Tv Σ yTv Tv yTv Tv ΣyTv Tv ) yTv Tv yTv Tv op r −1/2  su log(ep/su ) −1/2 b −1/2 ≤ 1 − kΣyTv Tv ΣyTv Tv ΣyTv Tv − Ikop ≤ 1+C , n with probability at least 1 − exp(−C ′ su log(ep/su )), where the last inequality is by Lemma 12 of [18]. Hence, b yTv Tv )Vb (0) kop b y )Vb (0) kop = k(Vb (0) )′ (ΣyTv Tv − Σ k(Vb (0) )′ Σy Vb (0) − Ikop = k(Vb (0) )′ (Σy − Σ Tv ∗ Tv ∗ r su log(ep/su ) −1/2 b −1/2 (0) 1/2 , ≤ kΣyTv Tv VbTv ∗ k2op kΣyTv Tv Σ yTv Tv ΣyTv Tv − Ikop ≤ 4C n with probability at least 1 − exp(−C ′ su log(ep/su )). The proof is completed by realizing 1/2 (0) 1/2 kΣyTv Tv VbTv ∗ kop = kΣy Vb (0) kop . b ). Using the definition of FrobeProof of Lemma 9.2. Let Tu = Sbu ∪ Su , where Sbu = supp(U nius norm, we have 2 ′ b ′ b b 1/2 2 kΣ1/2 x ∆kF − kΣx ∆kF = Tr(∆ (Σx − Σx )∆) = Tr((∆Tu ∗ ) (ΣxTu Tu − ΣxTu Tu )∆Tu ∗ ) r su log(ep/su ) 1/2 2 −1/2 b −1/2 ≤ kΣxTu Tu ∆Tu ∗ k2F kΣxTu Tu Σ kΣx ∆kF , xTu Tu ΣxTu Tu − Ikop ≤ C n 1/2 with high probability, where we have used kΣxTu Tu ∆Tu ∗ k2F = kΣx ∆k2F and Lemma 12 in [18] in the last inequality. After rearrangement, the proof is complete. b x is constructed from D0 and Σ b y is constructed from D1 . Proof of Lemma 9.3. In this proof, Σ b ) and Sbv = supp(Vb (0) ). We use the notation Tu = Su ∪Sbu and Tv = Sv ∪Sbv , where Sbu = supp(U Note that Tu depends on D1 and Tv depends on D0 . We first condition on D0 , and then we have
b xy − Σxy )Vb (0) = hΣ b xyTu Tv − ΣxyTu Tv , ∆Tu ∗ (Vb (0) )′ i Tr ∆′ (Σ Tv ∗ 1/2 (0) 1/2 −1/2 −1/2 b xyTu Tv − ΣxyTu Tv )Σ ≤ kΣyTv Tv VbTv ∗ kop kΣxTu Tu ∆Tu ∗ kF hΣxTu Tu (Σ yTv Tv , KTu i 1/2 (0) 1/2 −1/2 b −1/2 ≤ kΣyTv Tv VbTv ∗ kop kΣxTu Tu ∆Tu ∗ kF sup hΣxT T (Σ xyT Tv − ΣxyT Tv )ΣyTv Tv , KT i T 47 where T ranges over all subsets with cardinality bounded by 2su , and for each such T , 1/2 1/2 (0) 1/2 b (0) ′ 1/2 KT = kΣxT T ∆T ∗ (VbTv ∗ )′ ΣyTv Tv k−1 F ΣxT T ∆T ∗ (VTv ∗ ) ΣyTv Tv satisfying kKT kF = 1. We do not put Tv in the subscript of K because conditioning on D0 , Tv is fixed. For each T , we can −1/2 b −1/2 use Lemma 7 in [18] to bound hΣxT T (Σ xyT Tv − ΣxyT Tv )ΣyTv Tv , KT i . A direct union bound argument leads to r rsu + su log(ep/su ) −1/2 b −1/2 sup hΣxT T (ΣxyT Tv − ΣxyT Tv )ΣyTv Tv , KT i ≤ C , n T (0) 1/2 with probability at least 1−exp (−C ′ (su log(ep/su ) + rsu )). By Lemma 9.1, we have kΣyTv Tv VbTv ∗ kop = 1/2 1/2 1/2 kΣy Vb (0) kop ≤ 2 with high probability. Finally, observing that kΣxTu Tu ∆Tu ∗ kF = kΣx ∆kF , we have completed the proof. Proof of Lemma 9.4. It is omitted due to similarity to that of Lemma 9.3. Proof of Proposition 9.2. Let the singular value decomposition of U be U = ΘRH ′ . Then we have HRΘ′ Σx ΘRH ′ = U ′ Σx U = I, from which we derive Θ′ Σx Θ = R−2 . Using Lemma 6.6, we have √ b W − ΘkF kPUb − PU kF = 2 inf kU W √ √ b W HR−1 − ΘRH ′ HR−1 kF ≤ 2 inf kU b W − U kF kR−1 kop ≤ 2 inf kU W W √ −1/2 ′ 1/2 b inf kΣ1/2 ≤ 2M1 x (U W − U )kF kΘ Σx Θkop W √ b ≤ 2(M2 /M1 )1/2 inf kΣ1/2 x (U W − U )kF W √ b ≤ 2(M2 /M1 )1/2 inf kΣ1/2 x (U W − U )kF . W ∈O(r,r) 1/2 b W − U )k2 = Tr((U b W − U )′ Σx (U b W − U )), the proof is complete. Finally, by kΣx (U F 10 Implementation of (18) To implement the convex programming (18), we turn to the Alternating Direction Method b x and Σ b y for Σ b (0) of Multipliers (ADMM) [16, 11]. In the rest of this section, we write Σ x and (0) b Σy for notational convenience. First, note that (18) can be rewritten as minimize subject to where f (F ) + g(G), b 1/2 F Σ b 1/2 − G = 0, Σ x b xy , F i + ρkF k1 , f (F ) = −hΣ g(G) = ∞1{kGk∗ >r} + ∞1{kGkop >1} . 48 (100) y (101) (102) Thus, the augmented Lagrangian form of the problem is b 1/2 F Σ b 1/2 − Gi + η kΣ b 1/2 F Σ b 1/2 − Gk2 . Lη (F, G, H) = f (F ) + g(G) + hH, Σ x y y F 2 x (103) Following the generic algorithm spelled out in Section 3 of [11], suppose after the kth iteration, the matrices are (F k , Gk , H k ), then we update the matrices in the (k + 1)-th iteration as follows: F k+1 = argmin Lη (F, Gk , H k ), (104) Gk+1 = argmin Lη (F k+1 , G, H k ), (105) k+1 b 1/2 b 1/2 H k+1 = H + η(Σ Σy − Gk+1 ). x F (106) F G k The algorithm iterates over (104) – (106) till some convergence criterion is met. It is clear that the update (106) for the dual variable H is easy to calculate. Moreover the updates (104) and (105) can be solved easily and have explicit meaning in giving solution to sparse CCA. We are going to show that (104) can be viewed as a Lasso problem. Thus, this step targets at the sparsity of the matrix U V ′ . The update (105) turns out to be equivalent to a singular value capped soft thresholding problem, and it targets at the low-rankness of the 1/2 1/2 matrix Σx U V ′ Σy . In what follows, we study in more details the updates for F and G. First, we note that (104) is equivalent to b 1/2 F Σ b 1/2 − Gk k2 b 1/2 F Σ b 1/2 i + η kΣ F k+1 = argmin f (F ) + hH k , Σ x y F x y 2 F 1 k 1 b −1/2 b b −1/2 2 η b 1/2 b 1/2 k Σxy Σy )kF + ρkF k1 . = argmin kΣ x F Σy − (G − H + Σx 2 η η F (107) Thus, it is clear that the update of F in (104) can be viewed as a Lasso problem as summarized in the following proposition. Here and after, for any positive semi-definite matrix A, A−1/2 denotes the principal square root of its pseudo-inverse. Proposition 10.1. Let vec be the vectorization operation of any matrix and ⊗ the Kronecker product. Then vec(F k+1 ) is the solution to the following standard Lasso problem min kΓx − bk2F + x 2ρ kxk1 η b −1/2 Σ b xy Σ b −1/2 b y1/2 ⊗ Σ b x1/2 and b = vec(Gk − 1 H k + 1 Σ ). where Γ = Σ y η η x Remark 10.1. It is worth mentioning that the vectorized formulation in Proposition 10.1 is for illustration only. In practice, we solve the problem in (107) directly, since the vectorized version, especially the Kronecker product, would great increase the computation cost. The solver to (107) can be easily implemented in standard software packages for convex programming, such as TFOCS [5]. Since each update of F is the solution of some Lasso problem, it should be sparse in the sense that its vector ℓ1 norm is well controlled. 49 Turning to the update for G, we note that (105) is equivalent to η b 1/2 k+1 b 1/2 Gk+1 = argmin g(G) − hH k , Gi + kΣ Σy − Gk2F x F 2 G 1 η k+1 b 1/2 2 b 1/2 Σy )kF = argmin kG − ( H k + Σ x F 2 η G + ∞1{kGk∗ >r} + ∞1{kGkop >1} 1 b 1/2 F k+1 Σ b 1/2 )k2 = argmin kG − ( H k + Σ x y F η G + ∞1{kGk∗ >r} + ∞1{kGkop >1} . (108) The solution to the last display has a closed form according to the following result. Proposition 10.2. Let G∗ be the solution to the optimization problem: minimize kG − W kF subject to kGk∗ ≤ r, kGkop ≤ 1. P ′ Let the SVDPof W be W = m i=1 ωi ai bi with ω1 ≥ · · · ≥ ωm ≥ 0 the ordered singular values. m Then G∗ = i=1 gi ai b′i where for any i, gi = 1 ∧ (ωi − γ ∗ )+ for some γ which is the solution to m X 1 ∧ (ωi − γ)+ ≤ r. minimize γ, subject to γ > 0, i=1 Proof. The proof essentially follows that of Lemma 4.1 in [37]. In addition to the fact that the current problem deals P with asymmetric matrix, the only difference that we now have an inequality constraint i gi ≤ r rather than an equality constraint as in [37]. The asymmetry of the current problem does not matter since it is orthogonally invariant. Here and after, we call the operation in Proposition 10.2 singular value capped soft thresholding (SVCST) and write G∗ = SVCST(W ). Thus, any update for G results from the SVCST operation of some matrix, and so it has well controlled singular values. In summary, the convex program (18) is implemented as Algorithm 5. 11 Numerical Studies This section presents numerical results demonstrating the competitive finite sample performance of the proposed adaptive estimation procedure CoLaR on simulated datasets. Simulation settings We consider three simulation settings. In all these settings, we set p = m, Σx = Σy = Σ, and r = 2 with λ1 = 0.9 and λ2 = 0.8. Moreover, the nonzero rows of both U and V are set at {1, 6, 11, 16, 21}. The values at the nonzero coordinates are obtained from normalizing (with respect to Σ) random numbers drawn from the uniform distribution on the finite set {−2, 1, 0, 1, 2}. The choices of Σ in the three settings are as follows: 1. Identity: Σ = Ip . 50 Algorithm 5: An ADMM algorithm for SCCA Input: b x, Σ b y and Σ b xy , 1. Sample covariance matrices Σ 2. Penalty parameter ρ, 3. Rank r, 4. ADMM parameter η and tolerance level ǫ. 1 2 3 4 5 6 b Output: Estimated sparse canonical correlation signal A. b xy ), G0 = 0, H 0 = 0. Initialize: k = 0, F 0 = SVCST(Σ repeat Update F k+1 as in (104) (Lasso) ; k+1 −1 k+1 Σ b 1/2 b 1/2 Update G ← SVCST(η H k + Σ (SVCST) ; x F y ) 1/2 k+1 b 1/2 k+1 k k+1 b Update H ← H + η(Σx F Σy − G ) ; k ←k+1 ; until max{kF k+1 − F k kF , ρkGk+1 − Gk kF } ≤ ǫ; b = F k. Return A 2. Toeplitz: Σ = (σij ) where σij = 0.3|i−j| for all i, j ∈ [p]. In other words, Σx and Σy are Toeplitz matrices. q 0 / σ 0 σ 0 ). We set Σ0 = (σ 0 ) = Ω−1 where Ω = (ω ) with 3. SparseInv: Σ = (σij ij ij ii jj ωij = 1{i=j} + 0.5 × 1{|i−j|=1} + 0.4 × 1{|i−j|=2} , i, j ∈ [p]. In other words, Σx and Σy have sparse inverse matrices. In all three settings, we normalize the variance of each coordinate to be one. Implementation details The proposed CoLaR estimator in Section 4.1 has two stages. The convex program (18) in the first stage can be solved via an ADMM algorithm [11]. The details of the ADMM approach are presented in Section 10. The optimization problem (19) in the second stage can be solved by a standard group-Lasso algorithm [45]. In p all numerical results reported in this section, we used the same penalty level ρ = 0.55 log(p + m)/n in (18) and we used η = 2 in (106). Inp(19), we used five-fold cross validation to select a common penalty parameter ρu = ρv = b (r + log p)/n. In particular, test , Y test ) and the other four for l = 1, . . . , 5, we use one fold of the data as the test set (X(l) (l) train , Y train ). For any choice of b, we solved (19) on (X train , Y train ) folds as the training set (X(l) (l) (l) (l) b(l) , Vb(l) ). Then we computed the sum of canonical correlations between to obtain estimates (U b(l) ∈ Rn×r and Y test Vb(l) ∈ Rn×r to obtain CVl (b). Finally, CV(b) = P5 CVl (b). X test U (l) l=1 (l) Among all the candidate penalty parameters, we select the b value such that CV(b) is maximized. The candidate b values used in the simulation below are {0.5, 1, 1.5, 2}. Throughout the simulation, we used all the sample {(Xi , Yi )}ni=1 to form the sample covariance matrices used in (18) – (20). 51 In addition to the performance of CoLaR, we also report that of the method proposed in [43] (denoted by PMA here and on). The PMA seeks the solution to the optimization problem b xy v, subject to ||u|| ≤ 1, ||v|| ≤ 1, ||u||1 ≤ c1 , ||v||1 ≤ c2 . max u′ Σ u,v The solution is used to estimate the first canonical pair (b u1 , vb1 ). Then the same procedure is ′ ′ b b b repeated after Σxy is replaced by Σxy − (b u1 Σxy vb1 )b u1 vb1 , and the solution gives the estimator of the second canonical pair (b u2 , vb2 ). This process is repeated until u br , vbr is obtained. Note that the normalization constraint kuk ≤ 1 and kvk ≤ 1 implicitly assumes that the marginal covariance matrices Σx and Σy are identity matrices. We used the R implementation of the method (function CCA in the PMA package in R) by the authors of [43]. To remove undesired amplification of error caused by normalization, we renormalized each individual u bj with b x and each individual vbj with respect to Σ b y before calculating the error under respect to Σ the loss (7). For each simulated dataset, we set the sparsity penalty parameters penaltyx and penaltyz of the function CCA at each of the eleven different values {0.6l : l = 0, 1, . . . , 10} and only the smallest estimation error out of all eleven trials was used to compute the error reported in the tables below. Results Tables 1 – 3 report, in each of the three settings, the medians of the prediction errors of CoLaR and PMA out of 100 repetitions for four different configurations of (p, m, n) values. In each table, the columns U -PMA and V -PMA report the medians of the smallest estimation errors out of the eleven trials on each simulated dataset. The columns U -init and V -init report the median estimation errors of the renormalized r left singular vectors and right singular vectors of the solutions to the initialization step (18), where the renormalization is the same as in (20) and in both (18) and renormalization we used all the n pairs of observations. Last but not least, the columns U -CoLaR and V -ColaR report the median estimation errors of the CoLaR estimators where both stages were carried out. In all simulation settings, both the renormalized initial estimators and the CoLaR estimators consistently outperform PMA. Comparing the last four columns within each table, we also find that the CoLaR estimators with both stages carried out significantly improve over the renormalized initial estimators, which is in accordance with our theoretical results in Section 4. In summary, the proposed method delivers consistent and competitive performance in all three covariance settings across all dimension and sample size configurations, and its behavior agrees well with the theory. (p, m, n) (300, 300, 200) (600, 600, 200) (300, 300, 500) (600, 600, 500) U -PMA 2.1316 3.4154 0.2683 2.0335 V -PMA 2.1297 3.3584 0.2701 2.0368 U -init 0.2653 0.3167 0.1207 0.1448 V -init 0.1712 0.2087 0.0665 0.0817 U -CoLaR 0.0498 0.0671 0.0135 0.0166 V -CoLaR 0.0646 0.0776 0.0159 0.0203 Table 1: Prediction errors (Identity): Median in 100 repetitions. 52 (p, m, n) (300, 300, 200) (600, 600, 200) (300, 300, 500) (600, 600, 500) U -PMA 2.1853 3.4247 0.2358 2.1214 V -PMA 2.1840 3.4852 0.2191 2.0889 U -init 0.2885 0.3236 0.1202 0.1408 V -init 0.1706 0.2004 0.0664 0.0811 U -CoLaR 0.0511 0.0638 0.0135 0.0176 V -CoLaR 0.0601 0.0764 0.0166 0.0209 Table 2: Prediction errors (Toeplitz): Median in 100 repetitions. (p, m, n) (300, 300, 200) (600, 600, 200) (300, 300, 500) (600, 600, 500) U -PMA 2.9697 4.6908 2.3967 2.8707 V -PMA 2.9619 4.3339 2.0620 2.8609 U -init 0.5552 0.5596 0.2695 0.3068 V -init 0.5718 0.6133 0.1917 0.2368 U -CoLaR 0.1568 0.2123 0.0242 0.0338 V -CoLaR 0.1194 0.1572 0.0219 0.0271 Table 3: Prediction errors (SparseInv): Median in 100 repetitions. Model misspecification We now examine the performance of our estimator when the model is misspecified. To this end, we consider the case where there are three pairs of nontrivial canonical correlations present in the data but we set r = 2 in our algorithm. As before, we consider three different types of marginal covariance matrices: Identity, Toeplitz and SparseInv. In addition, we generate the first two pairs of canonical correlation vectors in the same way as before. For the generation of the third pair of canonical directions, we consider two different scenarios. In the first scenario, the support of the third pair of canonical direction vectors are set at {1, 6, 11, 16, 21} and so they are the same as those of the first two pairs. In the second scenario, we put no constraint on the support of these vectors. For both scenarios, we set λ3 = 0.3. Table 4 reports the prediction errors of the first two pairs of canonical correlations in both scenarios when (p, m, n) = (300, 300, 500). The implementation details are exactly the same as before. The first two columns contain results in the first scenario, and the third and the fourth columns the second scenario. Comparing these results with their counterparts in correctly specified models (the last two cells in the second last rows of Tables 1–3), we have found that the performance of our estimator was robust to model misspecification in both scenarios. Covariance Identity Toeplitz SparseInv Scenario 1 U -CoLaR V -CoLaR 0.0190 0.0195 0.0197 0.0197 0.0348 0.0263 Scenario 2 U -CoLaR V -CoLaR 0.0144 0.0160 0.0138 0.0147 0.0221 0.0328 Table 4: Prediction errors with misspecified models: Median in 100 repetitions. (p, m, n) = (300, 300, 500). 53 

EQUIVARIANT MODELS OF SPHERICAL VARIETIES MIKHAIL BOROVOI WITH AN APPENDIX BY arXiv:1710.02471v3 [math.AG] 17 Feb 2018 GIULIANO GAGLIARDI Abstract. Let G be a connected semisimple group over an algebraically closed field k of characteristic 0. Let Y = G/H be a spherical homogeneous space of G, and let Y ′ be a spherical embedding of Y . Let k0 be a subfield of k. Let G0 be a k0 -model (k0 -form) of G. We show that if G0 is an inner form of a split group and if the subgroup H of G is spherically closed, then Y admits a G0 -equivariant k0 -model. If we replace the assumption that H is spherically closed by the stronger assumption that H coincides with its normalizer in G, then both Y and Y ′ admit G0 -equivariant k0 -models, and these models are unique. Contents 0. Introduction 2 1. Semi-morphisms of k-schemes 4 2. Semi-morphisms of G-varieties 9 3. Quotients 11 4. Semi-morphisms of homogeneous spaces 12 5. k-automorphisms of homogeneous spaces 14 6. Equivariant models of G-varieties 15 7. Spherical homogeneous spaces and their combinatorial invariants 17 8. Action of an automorphism of the base field on the combinatorial invariants of a spherical homogeneous space 21 9. Equivariant models of automorphism-free spherical homogeneous spaces 24 10. Equivariant models of spherically closed spherical homogeneous spaces 25 11. Equivariant models of spherical embeddings of automorphism-free spherical homogeneous spaces 31 Appendix A. Algebraically closed descent for spherical homogeneous spaces Appendix B. The action of the automorphism group on the colors of a spherical homogeneous space References 32 34 36 Date: February 20, 2018. 2010 Mathematics Subject Classification. 14M27, 14M17, 14G27, 20G15. Key words and phrases. Spherical variety, spherical homogeneous space, spherical embedding, color, model, form, semi-automorphism. This research was partially supported by the Hermann Minkowski Center for Geometry and by the Israel Science Foundation (grant No. 870/16). 1 2 MIKHAIL BOROVOI WITH AN APPENDIX BY GIULIANO GAGLIARDI 0. Introduction Let G be a connected semisimple group over an algebraically closed field k of characteristic 0. Let Y be a G-variety, that is, an (irreducible) algebraic variety over k together with a morphism θ : G ×k Y → Y defining an action of G on Y . We say that (Y, θ) is a G-k-variety or just that Y is a G-k-variety. Let k0 ⊂ k be a subfield. Let G0 be a k0 -model (k0 -form) of G, that is, an algebraic group over k0 together with an isomorphism of algebraic k-groups ∼ κG : G0 ×k0 k → G. By a G0 -equivariant k0 -model of the G-k-variety (Y, θ) we mean a G0 -k0 -variety (Y0 , θ0 ) ∼ together with an isomorphism κY : Y0 ×k0 k → Y such that the diagram (26) commutes, see Section 6 below. From now on till the end of the Introduction we assume that Y is a spherical homogeneous space of G. This means that Y = G/H (with the natural action of G) for some algebraic subgroup H ⊂ G and that a Borel subgroup B of G has an open orbit in Y . Let Y ֒→ Y ′ be a spherical embedding of Y = G/H. This means that Y ′ is a G-kvariety, that Y ′ is a normal variety, and that Y ′ contains Y as an open dense G-orbit. Then B has an open dense orbit in Y ′ . Inspired by the works of Akhiezer and Cupit-Foutou [ACF14], [Akh15], [CF15], for a given k0 -model G0 of G we ask whether there exist a G0 -equivariant k0 -model Y0 of Y and a G0 -equivariant k0 -model Y0′ of Y ′ . Since char k = 0, by a result of Alexeev and Brion [AB05, Theorem 3.1], see Knop’s MathOverflow answer [Kn17b] and Appendix A below, the spherical subgroup H of G is conjugate to some (spherical) subgroup defined over the algebraic closure of k0 in k. Therefore, from now on we assume that k is an algebraic closure of k0 . We set Γ = Gal(k/k0 ) (the Galois group of k over k0 ). Let T be a maximal torus of G contained in a Borel subgroup B. We consider the Dynkin diagram Dyn(G) = Dyn(G, T, B). The k0 -model G0 of G defines the so-called ∗-action of Γ = Gal(k/k0 ) on the Dynkin diagram Dyn(G), see Tits [Tits66, Section 2.3, p. 39]. In other words, we obtain a homomorphism ε : Γ → Aut Dyn(G). The k0 -group G0 is called an inner form (of a split group) if the ∗-action is trivial, that is, if εγ = id for all γ ∈ Γ. For example, if G is a simple group of any of the types A1 , Bn , Cn , E7 , E8 , F4 , G2 , then any k0 -model G0 of G is an inner form, because in these cases Dyn(G) has no nontrivial automorphisms. If G is a split k-group, then of course G is an inner form. Let D(Y ) denote the set of colors of Y = G/H, that is, the (finite) set of the closures of B-orbits of codimension one in Y . A spherical subgroup H ⊂ G is called spherically closed if the automorphism group AutG (Y ) = NG (H)/H acts on D = D(Y ) faithfully, that is, if the homomorphism AutG (Y ) → Aut(D) is injective. Here NG (H) denotes the normalizer of H in G. Example 0.1. Let k = C, G = PGL2,C , H = T (a maximal torus), Y = G/T . Then |NG (T )/T | = 2, and the spherical homogeneous space Y of G has exactly two colors, which are swapped by the non-unit element of NG (T )/T . We see that the subgroup H = T of G is spherically closed. EQUIVARIANT MODELS OF SPHERICAL VARIETIES 3 Theorem 0.2. Let G be a connected semisimple group over an algebraically closed field k of characteristic 0. Let Y = G/H be a spherical homogeneous space of G. Let k0 be a subfield of k such that k is an algebraic closure of k0 . Let G0 be a k0 -model of G. Assume that: (i) G0 is an inner form, (ii) H is spherically closed. Then Y admits a G0 -equivariant k0 -model Y0 . Theorem 0.2 is a special case of the more general Theorem 10.2 below, where instead of assuming that G0 is an inner form, we assume only that for all γ ∈ Γ the automorphism εγ of Dyn(G) preserves the combinatorial invariants (Luna-Losev invariants) of the spherical homogeneous space Y . Note that we have to assume that char k = 0 because in the proof we use Losev’s uniqueness theorem [Lo09, Theorem 1], which has been proved only in characteristic 0. Note also that Y = G/H might have no G0 -equivariant k0 -models if H is not spherically closed, see Example 10.11 below. Theorem 0.2 was inspired by Theorem 1.1 of Akhiezer [Akh15] and by Corollary 1 of Cupit-Foutou [CF15, Section 2.5]. Note that the G0 -equivariant k0 -model Y0 in Theorem 0.2 is in general not unique. The following theorem is a special case of the more general theorem 10.13 below. Theorem 0.3. In Theorem 0.2 the set of isomorphism classes of G0 -equivariant k0 -models of Y = G/H is naturally a principal homogeneous space of the abelian group H 1 (Γ, AutG (Y )) ≃ (Hom(Γ, S2 ))Ω (2) . Here S2 is the symmetric group on two symbols (isomorphic to Z/2Z), Ω(2) = Ω(2) (Y ) is (2) the finite set defined in Section 7 below (before Definition 7.3), and ( . )Ω denotes the group of maps from the set Ω(2) to the group in the parentheses. In particular, for k0 = R we have Hom(Γ, S2 ) = S2 , and therefore, the number of these isomorphism classes is 2s , where s = |Ω(2) |. For G and Y as in Example 0.1 we have s = 1, hence for each of the two R-models of G there are exactly two non-isomorphic equivariant R-models of Y , see Example 10.15 below. Corollary 0.4 (Akhiezer’s theorem). In Theorem 0.2 instead of (ii) assume that (ii′ ) H is self-normalizing, that is, NG (H) = H. Then Y = G/H admits a G0 -equivariant k0 -model Y0 , and this model is unique up to a unique isomorphism. Indeed, since H is self-normalizing, it is spherically closed. By Theorem 0.2 Y admits a G0 -equivariant k0 -model. The uniqueness assertion is obvious because AutG (Y ) = {1}. Corollary 0.4 generalizes Theorem 1.1 of Akhiezer [Akh15], where the case k0 = R was considered. Theorem 0.5. Under the assumptions of Corollary 0.4, any spherical embedding Y ′ of Y = G/H admits a G0 -equivariant k0 -model Y0′ . This k0 -model Y0′ is compatible with the unique G0 -equivariant k0 -model Y0 of Y from Corollary 0.4, and hence is unique up to a unique isomorphism. Theorem 0.5 generalizes Theorem 1.2 of Akhiezer [Akh15], who proved in the case k0 = R that the wonderful embedding of Y admits a unique G0 -equivariant R-model. Our proof of Theorem 0.5 uses results of Huruguen [Hu11]. Note that in Theorem 0.5 we do not assume that Y ′ is quasi-projective. 4 MIKHAIL BOROVOI GIULIANO GAGLIARDI WITH AN APPENDIX BY Theorems 0.2, 0.3, and 0.5 seem to be new even in the case k0 = R. The plan of the rest of the paper is as follows. In Sections 1–6 we consider semilinear morphisms and models for general G-varieties and homogeneous spaces of G, not necessarily spherical. In Sections 7–8 we consider combinatorial invariants of spherical homogeneous spaces. Following ideas of Akhiezer [Akh15, Theorem 1.1] and Cupit-Foutou [CF15, Theorem 3(1), Section 2.2], for γ ∈ Γ = Gal(k/k0 ) we give a criterion of isomorphism of a spherical homogeneous space Y = G/H and the “conjugate” variety γ∗ Y = G/γ(H) in terms of the action of γ on the combinatorial invariants of G/H. In Sections 9–11 we prove Corollary 0.4, Theorem 0.2, Theorem 0.3, and Theorem 0.5. In Appendix A for a connected reductive group G0 defined over an algebraically closed field k0 of characteristic 0 and for an algebraically closed extension k ⊃ k0 , it is proved that any spherical subgroup H of the base change G = G0 ×k0 k is conjugate to a (spherical) subgroup defined over k0 . In Appendix B, following Friedrich Knop’s MathOverflow answer [Kn17a] to the author’s question, Giuliano Gagliardi gives a proof of an unpublished theorem of Ivan Losev that describes the image of AutG (G/H) = NG (H)/H in the group of permutations of D(G/H). Our proofs of Theorems 0.2, 0.3, and 0.5 use this result of Losev. Acknowledgements. The author is very grateful to Friedrich Knop for answering the author’s numerous MathOverflow questions, especially for the answer [Kn17a], to Giuliano Gagliardi for writing Appendix B, and to Roman Avdeev for suggesting Example 10.9 and proving Proposition 10.10. It is a pleasure to thank Michel Brion for very helpful e-mail correspondence. The author thanks Dmitri Akhiezer, Stéphanie Cupit-Foutou, Cristian D. González-Avilés, David Harari, Boris Kunyavskiı̆, and Stephan Snigerov for helpful discussions. This preprint was written during the author’s visits to the University of La Serena (Chile) and to the Paris-Sud University, and he is grateful to the departments of mathematics of these universities for support and excellent working conditions. Notation and assumptions. k is a field. In Section 2 and everywhere starting Section 4, k is algebraically closed. k0 is a subfield of the algebraically closed field k such that k is a Galois extension of k0 (except for Appendix A), hence k0 is perfect. A k-variety is a reduced separated scheme of finite type over k, not necessarily irreducible. An algebraic k-group is a smooth k-group scheme of finite type over k, not necessarily connected. All algebraic k-subgroups are assumed to be smooth. 1. Semi-morphisms of k-schemes Let k be a field and let Spec k denote the spectrum of k. By a k-scheme we mean a pair (Y, pY ), where Y is a scheme and pY : Y → Spec k is a morphism of schemes. Let (Y, pY ) and (Z, pZ ) be two k-schemes. By a k-morphism λ : (Y, pY ) → (Z, pZ ) we mean a morphism of schemes λ : Y → Z such that the following diagram commutes: Y λ /Z pZ pY  Spec k id  / Spec k Let γ : k → k be an automorphism of k (we write γ ∈ Aut(k)). Let γ ∗ := Spec γ : Spec k → Spec k denote the induced automorphism of Spec k, then (γγ ′ )∗ = (γ ′ )∗ ◦ γ ∗ . EQUIVARIANT MODELS OF SPHERICAL VARIETIES 5 Let (Y, pY ) be a k-scheme. By abuse of notation we write just that Y is a k-scheme. We define the γ-conjugated k-scheme γ∗ (Y, pY ) = (γ∗ Y, γ∗ pY ) to be the base change of (Y, pY ) from Spec k to Spec k via γ ∗ . By abuse of notation we write just γ∗ Y for γ∗ (Y, pY ). Lemma 1.1. Let (Y, pY ) be a k-scheme, and let γ ∈ Aut(k). Then the γ-conjugated k-scheme γ∗ (Y, pY ) is canonically isomorphic to (Y, (γ ∗ )−1 ◦ pY ) Proof. Write (X, pX ) = γ∗ (Y, pY ), then X comes with a canonical morphism λ : X → Y such that the following diagram commutes: λ /Y γ∗  X pY pX  Spec k / Spec k Since (γ −1 )∗ (γ∗ (Y, pY )) is canonically isomorphic to (Y, pY ), one can easily see that λ is an isomorphism of schemes. From the above diagram we obtain a commutative diagram λ X /Y pY  Spec k pX (γ ∗ )−1  Spec k id  / Spec k ∼ which gives a canonical isomorphism of k-schemes (X, pX ) → (Y, (γ ∗ )−1 ◦ pY ).  We define an action of γ : k → k on k-points. Let y be a k-point of Y , that is, a morphism y : Spec k → Y such that pY ◦ y = idSpec k . We denote (1) γ! (y) = y ◦ γ ∗ : Spec k → Spec k → Y, then an easy calculation shows that γ! (y) a k-point of γ∗ Y . Thus we obtain a bijection (2) γ! : Y (k) → (γ∗ Y )(k), y 7→ γ! (y). Let G be a k-group scheme. Following Flicker, Scheiderer, and Sujatha [FSS98, (1.2)], we define the k-group scheme γ∗ G to be the base change of G from Spec k to Spec k via γ ∗ . Then the map (2) γ! : G(k) → (γ∗ G)(k) is an isomorphism of groups (because for any field extension λ : k ֒→ k′ the corresponding map on rational points λ! : G(k) → (G ×k k′ )(k′ ) is a homomorphism). If H ⊂ G is a k-group subscheme, then γ∗ H is naturally a k-group subscheme of γ∗ G (because a base change of a group subscheme is a group subscheme). From the commutative diagram H(k)  G(k) γ! γ! / (γ∗ H)(k)  / (γ∗ G)(k) we see that (3) (γ∗ H)(k) = γ! (H(k)) ⊂ (γ∗ G)(k). 6 MIKHAIL BOROVOI GIULIANO GAGLIARDI WITH AN APPENDIX BY Let (Y, θ) be a G-k-scheme (a G-scheme over k), where θ : G ×k Y → Y, is an action of G on Y . By abuse of notation we write just that Y is a G-k-scheme. Again we define the γ∗ G-k-scheme γ∗ (Y, θ) = (γ∗ Y, γ∗ θ) to be the base change of (Y, θ) from Spec k to Spec k via γ ∗ . Definition 1.2. Let (Y, pY ) and (Z, pZ ) be two k-schemes. A semilinear morphism (γ, ν) : (Y, pY ) → (Z, pZ ) is a pair (γ, ν) where γ : k → k is an automorphism of k, and ν : Y → Z is a morphism of schemes such that the following diagram commutes: Y ν /Z  (γ ∗ )−1  pZ pY Spec k / Spec k We shorten “semilinear morphism” to “semi-morphism”. We write “ν : (Y, pY ) → (Z, pZ ) is a γ-semi-morphism” if (γ, ν) : (Y, pY ) → (Z, pZ ) is a semi-morphism. Then by abuse of notation we write just that ν : Y → Z is a γ-semi-morphism. Note that if we take γ = idk , then a idk -semi-morphism (Y, pY ) → (Z, pZ ) is just a morphism of k-schemes. Lemma 1.3. If (γ, ν) : (Y, pY ) → (Z, pZ ) is a semi-morphism of nonempty k-schemes, then the morphism of schemes ν : Y → Z uniquely determines γ. Proof. We may and shall assume that Y and Z are affine, Y = Spec RY , Z = Spec RZ . Then we have a commutative diagram (4) RO Y o ν∗ RO Z γ −1 ko k Since k is a field, the vertical arrows are injective, and therefore, the homomorphism of rings ν ∗ uniquely determines the automorphism γ −1 .  We define an action of a semi-morphism (γ, ν) : (Y, pY ) → (Z, pZ ) on k-points. If y : Spec k → Y is a k-point of (Y, pY ), we set (5) (γ, ν)(y) = ν ◦ y ◦ γ ∗ : Spec k → Z, which is a k-point of (Z, pZ ). This formula is compatible with the usual formula for the action of a k-morphism on k-points. By abuse of notation we write ν(y) instead of (γ, ν)(y). Definition 1.4. By a γ-semi-isomorphism ν : (Y, pY ) → (Z, pZ ) we mean a γ-semimorphism ν : (Y, pY ) → (Z, pZ ) for which the morphisms of schemes ν : Y → Z is an isomorphism. By a γ-semi-automorphism of a k-scheme (Y, pY ) we mean a γ-semiisomorphism µ : (Y, pY ) → (Y, pY ). Let us fix γ ∈ Aut(k). The commutative diagram (6) ν /Z Y ▲▲ ▲▲▲ ▲▲γ▲∗ pY pZ pY ▲▲ ▲▲▲  & ∗ −1  (γ ) / Spec k Spec k EQUIVARIANT MODELS OF SPHERICAL VARIETIES 7 shows that (7) (γ, ν) : (Y, pY ) → (Z, pZ ) is a semi-morphism (that is, ν : (Y, pY ) → (Z, pZ ) is a γ-semi-morphism) if and only if (8) (idk , ν) : γ∗ (Y, pY ) → (Z, pZ ) is a semi-morphism (that is, ν : γ∗ (Y, pY ) → (Z, pZ ) is a k-morphism). For brevity we write ν ♮ : γ∗ Y → Z (9) for the k-morphism (8), then the k-morphism ν♮ acts on k-points as follows: (10) (y ′ : Spec k → γ∗ Y ) 7−→ (ν ◦ y ′ : Spec k → Z). Example 1.5. Let (Y, pY ) be a k-scheme. Recall that γ∗ (Y, pY ) = (Y, (γ ∗ )−1 ◦ pY ). The commutative diagram idY Y /Y pY  pY Spec k (γ ∗ )−1  Spec k (γ ∗ )−1  / Spec k shows that (γ, idY ) : Y → γ∗ Y is a γ-semi-isomorphism. We denote this γ-semi-isomorphism by γ! : Y → γ∗ Y. Comparing formulas (1) and (5), we see that the γ-semi-isomorphism γ! : Y → γ∗ Y acts on k-points as the bijective map γ! : Y (k) → (γ∗ Y )(k) defined by formula (1). Now let ν : Y → Z be a γ-semi-morphism. The commutative diagram (6) shows that (11) γ ν♮ ν = ν♮ ◦ γ! = (idk , ν) ◦ ((γ ∗ )−1 , idY ) : Y −−!→ γ∗ Y −−→ Z, where γ! is a γ-semi-isomorphism and ν♮ is a k-morphism (an idk -semi-morphism). It follows that (12) ν(y) = ν♮ (γ! (y)) for y ∈ Y (k) (this follows also from comparing formulas (1), (5), and (10)). Example 1.6. Let Y0 be a k0 -scheme, where k0 is a subfield of k. Let γ ∈ Aut(k/k0 ), that is, γ is an automorphism of k that fixes all elements of k0 . Consider Y := Y0 ×k0 k = Y0 ×Spec k0 Spec k and µγ = idY0 × (γ ∗ )−1 : Y → Y. It follows from the construction of µγ that the following diagram commutes: Y µγ /Y pY pY  Spec k (γ ∗ )−1  / Spec k We see that µγ is a γ-semi-automorphism of Y . Let Y be an affine k-variety, Y = Spec RY , then RY is the ring of regular functions on Y . If f ∈ RY , then for any y ∈ Y (k) the value f (y) ∈ k is defined. 8 MIKHAIL BOROVOI WITH AN APPENDIX BY GIULIANO GAGLIARDI Lemma 1.7. Let ν : (Y, pY ) → (Z, pZ ) be a γ-semi-isomorphism of affine k-varieties, where γ : k → k is an automorphism of k. Let Y = Spec RY , Z = Spec RZ , and let ν ∗ : RZ → RY denote the morphism of rings corresponding to ν. Let fZ ∈ RZ . Then fZ (ν(y)) = γ((ν ∗fZ )(y)) (13) for all y ∈ Y (k). Proof. The assumption that ν : Y → Z is a γ-semi-morphism means that the diagram (4) commutes. A k-point y ∈ Y (k) corresponds to a homomorphism of k-algebras ϕy : RY → k, and the following diagram commutes: RY o ν∗ RZ γ −1  ϕν(y) ϕy  ko hence ϕν(y) = γ ◦ ϕy that ◦ ν ∗. We set fY = ν ∗f Z k ∈ RY , then fY (y) = ϕy (fY ), and (13) means (γ ◦ ϕy ◦ ν ∗ )(fZ ) = γ(ϕy (ν ∗ fZ )), which is obvious.  Now let ν : (Y, pY ) → (Z, pZ ) be a γ-semi-isomorphism of irreducible k-varieties, where γ : k → k is an automorphism of k. Then the isomorphism of schemes ν : Y → Z induces an isomorphism of the fields of rational functions ν∗ : K(Y ) → K(Z), f 7→ ν∗ f. For f ∈ K(Y ) and y ∈ Y (k), the value f (y) ∈ k ∪ {∞} of f at y is defined, where we write f (y) = ∞ if f is not regular at y. Corollary 1.8. With the above notation and assumptions we have (ν∗ fY )(z) = γ(fY (ν −1 (z))) for all fY ∈ K(Y ), z ∈ Z(k). Proof. We consider the isomorphism ν ∗ = ν∗−1 : K(Z) → K(Y ), fZ 7→ ν ∗fZ , where fZ ∈ K(Z). Set fZ = ν∗ fY , then fY = ν ∗ fZ . We must prove that (13) holds. We may and shall assume that Y and Z are affine varieties, Y = Spec RY , Z = Spec RZ , fZ ∈ RZ , and that the morphism ν corresponds to a homomorphism of rings ν ∗ : RZ → RY . Now the corollary follows from Lemma 1.7.  Remark 1.9. (Classical language.) In this remark we describe the variety γ∗ Y and the map γ! : Y (k) → (γ∗ Y )(k) in the language of the classical algebraic geometry. First, consider the affine space Ank , then Ank (k) = kn . Let k0 be the prime subfield of k, that is, the subfield generated by 1, then Ank = Ank0 ×k0 k. Let γ ∈ Aut(k) = Aut(k/k0 ), then γ induces a γ-semi-automorphism µγ : Ank → Ank , see Example 1.6. For i = 1, . . . , n, let fi denote the i-th coordinate function on Ank , which is a regular function. Since fi comes from a regular function on Ank0 , we have (µγ )∗ fi = fi , and by Lemma 1.7 we have fi (µγ (x)) = γ(fi (x)) for x ∈ Ank (k) = kn . If we write x = (xi )ni=1 ∈ kn , where xi = fi (x) ∈ k, then µγ (x) = γ(xi )ni=1 . Now let Y ⊂ Ank be an affine variety. Let ι : Y ֒→ Ank denote the inclusion morphism, then γ induces a k-morphism γ∗ ι : γ∗ Y → γ∗ Ank . We have a k-isomorphism (µγ )♮ : γ∗ Ank → Ank , EQUIVARIANT MODELS OF SPHERICAL VARIETIES 9 and we obtain a k-morphism (γ∗ ι)′ = (µγ )♮ ◦ γ∗ ι : γ∗ Y → Ank . From the commutative diagram Y (k) γ! / (γ∗ Y )(k) γ∗ ι ι  Ank (k) γ!  / (γ∗ An )(k) k we see that (γ∗ ι)(γ! (y)) = γ! (ι(y)) for y ∈ Y (k), hence (γ∗ ι)′ (y) = (µγ )♮ (γ! (ι(y))) = µγ (ι(y)). Now we assume that k is algebraically closed. As usual in the classical algebraic geometry, we identify Y with the algebraic set Y (k) ⊂ Ank (k) = kn . Furthermore, we identify γ∗ Y with the algebraic set (γ∗ ι)′ (Y (k)) = µγ (Y (k)) ⊂ kn . We see that γ∗ Y = µγ (Y ) = {γ(yi )ni=1 | (yi )ni=1 ∈ Y }, and that the map γ! : Y (k) → (γ∗ Y )(k) sends a point y with coordinates (yi )ni=1 to the point with coordinates γ(yi )ni=1 . If Y ⊂ kn is defined by a family of polynomials (Pα )α∈A , then γ∗ Y ⊂ kn is defined by the family (γ(Pα ))α∈A , where γ(Pα ) is the polynomial obtained from Pα by acting by γ on the coefficients. 2. Semi-morphisms of G-varieties 2.1. In this section k is an algebraically closed field, and Y is a k-variety, that is, a reduced separated scheme of finite type over k. Let G be an algebraic group over k (we write also “an algebraic k-group”), that is, a smooth group scheme of finite type over k. Let (Y, θ) be a G-k-variety, that is, a k-variety Y together with an action θ : G ×k Y → Y of G on Y . If g ∈ G(k) and y ∈ Y (k), we write just g ∗ y or g · y for θ(g, y) ∈ Y (k). Y Definition 2.2 (cf. [FSS98, (1.2)]). Let γ ∈ Aut(k). A γ-semi-automorphism of an algebraic k-group G is a γ-semi-automorphism of k-schemes τ : G → G such that the corresponding isomorphism of k-varieties τ♮ : γ∗ G → G, see (9), is an isomorphism of algebraic k-groups. This condition is the same as to require that certain diagrams containing τ commute, see [Brv93, 1.2]. Let H ⊂ G be an algebraic k-subgroup. By Definition 2.2 we have τ (H) = τ♮ (γ∗ H)), hence τ (H) is a k-subgroup of G. We have (τ (H))(k) = τ (H(k)). We shall always assume that G = G0 ×k0 k, where G0 is an algebraic group defined over a subfield k0 of k, and that γ ∈ Aut(k/k0 ), that is, γ is an automorphism of k fixing all elements of k0 . Then we have a γ-semi-automorphism σγ = idG0 × (γ ∗ )−1 : G → G, 10 MIKHAIL BOROVOI WITH AN APPENDIX BY GIULIANO GAGLIARDI compare Example 1.6. If α is any k-automorphism of G, then τ := α ◦ σγ : G → G is a γ-semi-automorphism of G, and all γ-semi-automorphisms of G (for given γ) can be obtained in this way. Definition 2.3. Let G be an algebraic k-group, and let (Y, θY ) and (Z, θZ ) be two Gk-varieties. Let γ ∈ Aut(k), and let τ : G → G be a γ-semi-automorphism of G. A τ -equivariant γ-semi-morphism ν : (Y, θY ) → (Z, θZ ) is a γ-semi-morphism ν : Y → Z such that the following diagram commutes: G ×k Y (14) θY /Y ν τ ×ν  G ×k Z θZ  /Z where we write τ × ν for the product of τ and ν over the automorphism (γ ∗ )−1 of Spec k. Since k is algebraically closed, G is smooth (reduced), and Y and Z are reduced, we see that the diagram (14) commutes if and only if ν(g · y) = τ (g) · ν(y) for all g ∈ G(k), y ∈ Y (k). Construction 2.4. Let G be an algebraic k-group, and let (Y, θY ) be a G-k-variety. The group γ∗ G naturally acts on γ∗ Y : the action θ : G ×k Y → Y gives an action γ∗ θ : γ∗ G ×k γ∗ Y → γ∗ Y. (15) By definition, a γ-semi-automorphism τ of G defines an isomorphism of algebraic k-groups τ♮ : γ∗ G → G. We identify G and γ∗ G via τ♮ and obtain from (15) an action τ ∗ γ∗ θ : G ×k γ∗ Y → γ∗ Y, (g, y ′ ) 7→ (γ∗ θ)(τ♮−1 (g), y ′ ) for g ∈ G(k), y ′ ∈ (γ∗ Y )(k). By abuse of notation we write τ ∗ θ for τ ∗ γ∗ θ and we write γ∗ Y for the G-k-variety (γ∗ Y, τ ∗ γ∗ θ). We write g ∗τ y ′ for (τ ∗ θ)(g, y ′ ), where g ∈ G(k), y ′ ∈ (γ∗ Y )(k). By formula (12) we have τ (g) = τ♮ (γ! (g)), hence (16) g ∗τ y ′ = (γ∗ θ)(τ♮−1 (g), y ′ ) = γ! (θ(γ!−1 (τ♮−1 (g)), γ!−1 (y ′ ))) = γ! (θ(τ −1 (g), γ!−1 (y ′ )) = γ! (τ −1 (g) ∗ γ!−1 (y ′ )). Y Lemma 2.5. Let G be an algebraic k-group, and let (Y, θ) be a G-k-variety. Let γ ∈ Aut(k), and let τ : G → G be a γ-semi-automorphism of G. Let y (0) ∈ Y (k) be a k-point, and write H = StabG (y (0) ). Consider the action τ ∗ θ : G ×k γ∗ Y → γ∗ Y. Then the stabilizer in G(k) of the point γ! (y (0) ) ∈ (γ∗ Y )(k) under the action τ ∗ θ is τ (H(k)) = (τ (H))(k). EQUIVARIANT MODELS OF SPHERICAL VARIETIES 11 Proof. By formula (16) we have g ∗τ γ! (y (0) ) = γ! (τ −1 (g) · y (0) ). Since the stabilizer in G(k) of y (0) ∈ Y (k) is H(k), the lemma follows.  Note that ν in Definition 2.3 defines a k-morphism ν♮ : γ∗ Y → Z, see (9). Lemma 2.6. Let γ ∈ Aut(k) and let τ : G → G be a γ-semi-automorphism of G. Let (Y, θY ) and (Z, θZ ) be two G-k-varieties. A morphism of schemes ν : Y → Z is a τ equivariant γ-semi-morphism if and only if ν♮ : γ∗ Y → Z is a G-equivariant morphism of k-varieties. Proof. By (6) the morphism of schemes ν is a γ-semi-morphism Y → Z if and only if it is a k-morphism γ∗ Y → Z. Let g ∈ G(k), y ′ ∈ (γ∗ Y )(k). Using formula (16) we obtain ν♮ (g ∗τ y ′ ) = ν(γ!−1 (g ∗τ y ′ )) = ν(τ −1 (g) ∗ γ!−1 (y ′ )). Y We have also g ∗ ν(γ!−1 (y ′ )) = g ∗ ν♮ (y ′ ). Z Z If ν is τ -equivariant, then ν(τ −1 (g) ∗ γ!−1 (y ′ )) = g ∗ ν(γ!−1 (y ′ )), Z Y and we obtain that ν♮ (g ∗τ y ′ ) = g ∗ ν♮ (y ′ ) for all g ∈ G(k), y ′ ∈ (γ∗ Y )(k), Z hence ν♮ is G-equivariant. Conversely, if ν♮ is G-equivariant, we obtain from the above calculations that ν(τ −1 (g) ∗ γ!−1 (y ′ )) = ν♮ (g ∗τ y ′ ) = g ∗ ν♮ (y ′ ) = g ∗ ν(γ!−1 (y ′ )). Z Y Set y = γ!−1 (y ′ ), Z g′ = τ −1 (g), then we obtain that ν(g ′ ∗ y) = τ (g′ ) ∗ ν(y) for all g′ ∈ G(k), y ∈ Y (k). Y Z Thus ν is τ -equivariant.  Corollary 2.7. Let γ be an automorphism of k, and let τ : G → G be a γ-semi-automorphism of G. Let (Y, θ) be a G-k-variety. There exists a τ -equivariant γ-semi-automorphism µ : Y → Y if and only if the G-k-variety (γ∗ Y, τ ∗ θ) is isomorphic to (Y, θ). Proof. We take Z = Y in Lemma 2.6.  3. Quotients Let k be a field (not necessarily algebraically closed). By an algebraic scheme over k we mean a scheme of finite type over k. By an algebraic group scheme over k we mean a group scheme over k whose underlying scheme is of finite type over k. Let H be an algebraic group subscheme of an algebraic group k-scheme G. A quotient of G by H is an algebraic scheme Y over k equipped with an an action θ : G ×k Y → Y and a point y (0) ∈ Y (k) fixed by H satisfying certain properties (a) and (b), see Milne [Mi18, Definition 5.20]. 12 MIKHAIL BOROVOI WITH AN APPENDIX BY GIULIANO GAGLIARDI By [Mi18, Theorem 5.28] there exists a quotient of G by H. By [Mi18, Proposition 5.22] this quotient (Y, θ, y (0) ) has the following universal property: (U) Let Z be a k-scheme on which G acts, and let z (0) ∈ Z(k) be a point fixed by H. Then there exists a unique G-equivariant map Y → Z making the following diagram commute: g7→g·y (0) /Y ◆◆◆ ◆◆◆ ◆◆ g7→g·z (0) ◆◆◆  & G ◆◆ Z Clearly the universal property (U) uniquely determines the quotient up to a unique isomorphism, so we may take (U) as a definition of the quotient. We return to our settings: k is an algebraically closed field, G is a linear algebraic kgroup (a smooth affine group k-scheme) and H is a smooth algebraic k-subgroup of G. Since G is smooth, so is the quotient Y , see [Mi18, Corollary 5.26]. Since G is smooth and affine, the quotient Y is a separated algebraic scheme, see [Mi18, Theorem 7.18]. Thus Y is a k-variety, and therefore, in the universal property U defining Y we may assume that Z is a k-variety. Since k is algebraically closed and H is smooth, the condition “fixed by H” is equivalent to “fixed by H(k)”. Thus we arrive to the following definition of Springer: Definition 3.1 (cf. Springer [Sp98, Section 5.5]). Let k be an algebraically closed field, and let G be a linear algebraic k-group. Let H ⊂ G be a smooth k-subgroup. A quotient of G by H is a pointed G-k-variety (Y, θ : G ×k Y → Y, y (0) ∈ Y (k) ) such that H(k) fixes y0 , with the following universal property: (U′ ) For any pointed G-k-variety (Z, θZ , z (0) ) such that the k-point z (0) ∈ Z(k) is fixed by H(k), there exists a unique morphism of pointed G-k-varieties (Y, θ, y (0) ) → (Z, θZ , z (0) ). For G and H as in Definition 3.1, let (Y, θ, y (0) ) be a quotient of G by H. The action of G on Y induces a G-k-morphism (17) G → Y, g 7→ g · y (0) , where G acts on itself by left translations. As usual, we write G/H for Y and g · H or gH for g · y (0) , where g ∈ G(k). In particular, we write 1 · H for y (0) . The G-equivariant morphism G/H = Y → Z of (U′ ) sends 1·H ∈ (G/H)(k) to z (0) , hence for any g ∈ G(k) it sends the k-point gH ∈ (G/H)(k) to g · z (0) ∈ Z(k). Thus the quotient G/H has the following universal property: (U′′ ) For any pointed G-k-variety (Z, θZ , z (0) ) such that the k-point z (0) is fixed by H(k), there exists a unique G-k-morphism G/H → Z sending gH to g · z (0) for any g ∈ G(k). By [Mi18, Definition 5.20(a)] the morphism (17) induces an injective map G(k)/H(k) → (G/H)(k) sending g · H(k) to gH. By [Mi18, Proposition 5.25] the morphism (17) is faithfully flat, and therefore, since k is algebraically closed, we see that the induced map G(k)/H(k) → (G/H)(k) is surjective. We conclude that this map is bijective. Thus any k-point of G/H is of the form gH, where g ∈ G(k). 4. Semi-morphisms of homogeneous spaces Let k be an algebraically closed field. Lemma 4.1 (well-known). Let G be a linear algebraic k-group over an algebraically closed field k, and let H1 , H2 be two k-subgroups. Then Y1 = G/H1 and Y2 = G/H2 are isomorphic as G-k-varieties if and only if the subgroups H1 and H2 are conjugate. To be more precise, for a ∈ G(k) the following two assertions are equivalent: EQUIVARIANT MODELS OF SPHERICAL VARIETIES 13 (i) There exists an isomorphism of G-k-varieties φa : G/H1 → G/H2 taking g · H1 to ga−1 · H2 for g ∈ G(k); (ii) H1 = a−1 H2 a. Proof. (i)⇒(ii). Clearly StabG(k) (1 · H1 ) = H1 (k) and StabG(k) (a−1 · H2 ) = a−1 · H2 (k) · a. Since φa (1 · H1 ) = a−1 · H2 , these stabilizers coincide, whence (ii). (0) (0) (ii)⇒(i). Set Y2 = G/H2 , y2 = 1 · H2 ∈ Y2 (k), y ′ = a−1 · y2 ∈ Y2 (k), then StabG(k) (y ′ ) = a−1 H2 (k)a = H1 (k), so by the property (U′′ ) of the quotient G/H1 there exists a unique morphism of G-varieties φa : G/H1 → G/H2 such that φa (g · H1 ) = g · a−1 · H2 for g ∈ G(k). Similarly, since the stabilizer in G(k) of a · H1 ∈ (G/H1 )(k) unique morphism of G-varieties ψa : G/H2 → G/H1 such that ψa (g · H2 ) = g · a · H1 is H2 (k), there exists a for g ∈ G(k). Clearly these two morphisms are mutually inverse, hence both φa and ψa are isomorphisms.  4.2. Let k be an algebraically closed field. Let G be a linear algebraic group over k. Let γ ∈ Aut(k). Let τ : G → G be a γ-semi-automorphism of G. Let H ⊂ G be a smooth k-subgroup. Set Y = G/H, then we have a morphism θ : G ×k Y → Y defining the action of G on Y . Furthermore, the variety Y has a k-point y (0) = 1 · H such that StabG(k) (y (0) ) = H(k), and the group of k-points G(k) acts on Y (k) transitively. Consider the variety γ∗ Y , the action γ∗ θ : γ∗ G ×k γ∗ Y → γ∗ Y of γ∗ G on γ∗ Y , and the k-point γ! (y (0) ) ∈ (γ∗ Y )(k). As in Construction 2.4 we obtain an action τ ∗ θ : G ×k γ∗ Y → γ∗ Y. Lemma 4.3. Let k, G, γ, τ , H be as in Subsection 4.2. Set Y = G/H and let θ : G×k Y → Y denote the canonical action. Consider the map on k-points (18) (G/H)(k) → (G/τ (H))(k), g · H 7→ τ (g) · τ (H) for g ∈ G(k). Then the following assertions hold: (i) The pointed G-k-variety (γ∗ Y, τ ∗ θ, γ! (y (0) )) is isomorphic to G/τ (H); (ii) the map (18) is induced by some γ-semi-isomorphism ν : G/H → G/τ (H). Proof. Let (Z, θZ , z (0) ) be a pointed G-k-variety, and assume that τ (H(k)) fixes z (0) . Consider the pointed G-k-variety ((γ −1 )∗ Z, (τ −1 )∗ θZ , γ!−1 (z (0) )), where the action (19) (τ −1 )∗ θZ : G ×k (γ −1 )∗ Z → (γ −1 )∗ Z is defined as in Construction 2.4, but for the pair (γ −1 , τ −1 ) instead of (γ, τ ). By Lemma 2.5, H(k) fixes γ!−1 (z (0) ) ∈ (γ −1 )∗ Z. For any morphism of pointed G-k-varieties (20) κ : (Y, θ, y (0) ) → ((γ −1 )∗ Z, (τ −1 )∗ θZ , γ!−1 (z (0) )) we obtain a morphism of pointed G-k-varieties (21) γ∗ κ : (γ∗ Y, τ ∗ θ, γ! (y (0) )) → (Z, θZ , z (0) ). We see that the map κ 7→ γ∗ κ is a bijection between the set of morphisms as in (20) and the set of morphisms as in (21). Since Y = G/H and H(k) fixes γ!−1 (z (0) ) under the action (19), we conclude by the universal property (U′′ ) for the quotient Y = G/H that 14 MIKHAIL BOROVOI WITH AN APPENDIX BY GIULIANO GAGLIARDI the former set contains exactly one element. It follows that the latter set contains exactly one element, that is, the triple (γ∗ Y, τ ∗ θ, γ! (y (0) )) has the universal property (U′′ ). This means that (γ∗ Y, τ ∗ θ, γ! (y (0) )) is a quotient of G by τ (H), which proves (i). It follows that there exists an isomorphism of G-k-varieties λ : γ∗ Y → G/τ (H) such that g ∗τ γ! (y (0) ) 7−→ g · τ (H). (22) We set γ λ ν = λ ◦ γ! : G/H = Y −−!→ γ∗ Y −−→ G/τ (H), (23) where γ! : Y → γ∗ Y is the γ-semi-morphism of Example 1.5. Then we have ν♮ = λ. Since ν♮ is an isomorphism of G-k-varieties, by Lemma 2.6 ν is a τ -equivariant γ-semi-isomorphism. Since ν(1 · H) = ν(y (0) ) = λ(γ! (y (0) )) = 1 · τ (H) by (22), we have ν(g · H) = ν(g · (1 · H)) = τ (g) · ν(1 · H) = τ (g) · ν(H), which proves (ii).  Corollary 4.4. Let G be a linear algebraic k-group and H ⊂ G be an algebraic k-subgroup. Set Y = G/H. Let γ ∈ Aut(k) and let τ : G → G be a γ-semi-automorphism of G. The following three conditions are equivalent: (i) There exists a τ -equivariant γ-semi-automorphism µ : Y → Y ; (ii) The G-k-variety G/τ (H) is isomorphic to G/H; (iii) The algebraic subgroup τ (H) ⊂ G is conjugate to H. Proof. By Corollary 2.7 there exists µ : Y → Y as in (i) if and only if the G-k-variety (γ∗ Y, τ ∗ θ) is isomorphic to (Y, θ). By construction (Y, θ) = G/H, and by Lemma 4.3 (γ∗ Y, τ ∗ θ) ∼  = G/τ (H). Thus (i)⇔(ii). By Lemma 4.1 (ii)⇔(iii). 4.5. With the assumptions of Subsection 4.2, assume also that G is connected, then the homogeneous spaces Y1 = G/H and Y2 = G/τ (H) of G are irreducible k-varieties. We consider the fields of rational functions K(Y1 ) and K(Y2 ). The γ-semi-isomorphism ν : Y1 → Y2 of (23) induces an isomorphism of fields ∼ ν∗ : K(Y1 ) → K(Y2 ), f1 7→ f2 = ν∗ f1 . and by Lemma 1.8 we have f2 (y2 ) = γ! (f1 (ν −1 (y2 ))) (24) for y2 ∈ Y2 (k). 5. k-automorphisms of homogeneous spaces Let G be a linear algebraic group over an algebraically closed field k. Let Y be a G-kvariety. We denote by AutG (Y ) the group of G-equivariant k-automorphisms of Y , that is, of k-automorphisms ψ : Y → Y such that ψ(g · y) = g · ψ(y) for g ∈ G, y ∈ Y. We assume that Y is a homogeneous space of G, that is, Y = G/H, where H is a k-subgroup of G. Set N = NG (H), the normalizer of H in G. Lemma 5.1 (well-known). For n ∈ N (k) we define a map on k-points n∗ : G/H → G/H, gH 7→ gn−1 H for g ∈ G(k). Then (i) The map n∗ is induced by some automorphism φn ∈ AutG (G/H); EQUIVARIANT MODELS OF SPHERICAL VARIETIES 15 (ii) The map φ : N (k) → AutG (G/H), (25) n 7→ φn is a homomorphism inducing an isomorphism ∼ N (k)/H(k) → AutG (G/H). Proof. By assumption n−1 Hn = H, and by Lemma 4.1 there exists an isomorphism φn : G/H → G/H such that φn (g · H) = gn−1 · H, which proves (i). Clearly the map φ of (25) is a homomorphism with kernel H(k). To prove (ii) it remains to show that φ is surjective. Let ψ ∈ AutG (G/H). Write ψ(1 · H) = a · H with a ∈ G(k). Since ψ is an isomorphism of G-k-varieties, we have StabG(k) (a · H) = StabG(k) (1 · H) = H(k). On the other hand, StabG(k) (a · H) = a · H(k) · a−1 . Thus a · H · a−1 = H, hence a ∈ N (k). We have ψ(g · H) = ga · H. Write a = n−1 , then n ∈ N (k) and ψ = φn . Thus the homomorphism φ is surjective, as required.  Corollary 5.2. If NG (H) = H, then AutG (G/H) = {1}.  6. Equivariant models of G-varieties Let k be an algebraically closed field, and k0 ⊂ k be a subfield such that k is a Galois extension of k0 , that is, k0 is a perfect field and k is an algebraic closure of k0 . We write Γ = Gal(k/k0 ) := Aut(k/k0 ). Let Y be a k-variety. Let γ, γ ′ ∈ Γ. If µ is a γ-semi-automorphism of Y , and µ′ is a of Y , then µ ◦ µ′ is a γ ◦ γ ′ -semi-automorphism of Y and µ−1 is a −1 γ -semi-automorphism of Y . γ ′ -semi-automorphism We denote by SAutk/k0 (Y ) or just by SAut(Y ) the group of all γ-semi-automorphisms µ of Y where γ runs over Γ = Gal(k/k0 ). A k0 -model of Y is a k0 -variety Y0 together with an isomorphism of k-varieties ∼ κY : Y0 ×k0 k → Y. Note that γ ∈ Γ defines a γ-semi-automorphism of Y0 ×k0 k idY0 × (γ ∗ )−1 : Y0 ×Spec k0 Spec k → Y0 ×Spec k0 Spec k and thus, via κY , a γ-semi-automorphism µγ of Y . We obtain a homomorphism Γ → SAut(Y ), γ 7→ µγ . Conversely: Lemma 6.1 (Borel and Serre [BS64, Lemma 2.12]). Let k, k0 , Γ, Y be as above. Assume that for any γ ∈ Γ we have a γ-semi-automorphism µγ of Y such that (i) the map Γ → SAutk/k0 (Y ), γ 7→ µγ , is a homomorphism, (ii) the restriction of this map to Gal(k/k1 ) for some finite Galois extension k1 /k0 in k comes from a k1 -model Y1 of Y , (iii) Y is quasi-projective. Then there exists a k0 -model Y0 of Y that defines this homomorphism γ 7→ µγ .  6.2. Let G be a linear algebraic group over k. We assume that we are given a k0 -model of G, that is, a linear algebraic group G0 over k0 together with an isomorphism of algebraic ∼ k-groups over κG : G0 ×k0 k → G. For γ ∈ Γ the automorphism (γ ∗ )−1 of Spec k induces a γ-semi-automorphism idG0 × (γ ∗ )−1 of G0 ×Spec k0 Spec k. We identify G with G0 ×Spec k0 16 MIKHAIL BOROVOI WITH AN APPENDIX BY GIULIANO GAGLIARDI Spec k via κG , then for any γ ∈ Γ we obtain a γ-semi-automorphism σγ : G → G. The map Γ → SAut(G), γ 7→ σγ is a homomorphism. We identify γ∗ G with G using (σγ )♮ : γ∗ G → G. Let (Y, θ) be a G-k-variety. By a G0 -equivariant k0 -model of the G-k-variety Y we mean ∼ a G0 -k0 -variety (Y0 , θ0 ) together with an isomorphism κY : Y0 ×k0 k → Y such that the following diagram commutes: (26) G0,k ×k Y0,k θ0,k / Y0,k κY κG ×κY  G ×k Y  θ /Y where G0,k := G0 ×k0 k and Y0,k := Y0 ×k0 k. For a given k0 -model G0 of G we ask whether there exists a G0 -equivariant k0 -model Y0 of Y . Let (Y, θ) be a G-k-variety. We write g · y for θ(g, y). Recall (Definition 2.3) that a γ-semi-automorphism µ of Y is σγ -equivariant if the following diagram commutes: G ×k Y θ /Y θ  µ σγ ×µ  G ×k Y /Y Since k is algebraically closed, G is smooth (reduced) and Y is reduced, this is the same as to require that µ(g · y) = σγ (g) · µ(y) for all g ∈ G(k), y ∈ Y (k). We ask whether there exists such µ. A G0 -equivariant k0 -model of Y defines a homomorphism Γ → SAutk/k0 (Y ), γ 7→ µγ , where for any γ ∈ Γ, the γ-semi-automorphism µγ of Y is σγ -equivariant. Conversely: Lemma 6.3. Let k, k0 , Γ = Gal(k/k0 ), G, (Y, θ) be as above and let G0 be a k0 -model of G. Assume that for any γ ∈ Γ we have a γ-semi-automorphism µγ of Y such that the following conditions are satisfied: (i) the map Γ → SAutk/k0 (Y ), γ 7→ µγ is a homomorphism, (ii) the restriction of this map to Gal(k/k1 ) for some finite Galois extension k1 /k0 in k comes from a G1 -equivariant k1 -model Y1 of Y , where G1 = G0 ×k0 k1 , (iii) Y is quasi-projective, (iv) for any γ ∈ Γ, the γ-semi-automorphism µγ is σγ -equivariant. Then there exists a G0 -equivariant k0 -model Y0 of Y that defines this homomorphism γ 7→ µγ . Proof. By Lemma 6.1 the homomorphism Γ → SAut(Y ), γ 7→ µγ defines a k0 -model Y0 of Y . Using Galois descent for morphisms (see e.g. Jahnel [Ja00, Proposition 2.8]) we obtain from condition (iv) that θ comes from some morphism θ0 : G0 ×k0 Y0 → Y0 , and the k0 -model (Y0 , θ0 ) of (Y, θ) is G0 -equivariant.  EQUIVARIANT MODELS OF SPHERICAL VARIETIES 17 Remark 6.4. If in Lemma 6.3 we do not assume that Y is quasi-projective, then we obtain a k0 -model Y0 in the category of algebraic k0 -spaces (see Wedhorn [We15, Proposition 8.1]), but not necessarily in the category of k0 -schemes (even when k = C and k0 = R, see Huruguen [Hu11, Theorem 2.35]). Lemma 6.5. Let k, k0 , Γ = Gal(k/k0 ), G, (Y, θ) be as above and let G0 be a k0 -model of G. Assume that AutG (Y ) = {1}. Assume that for any γ ∈ Γ there exists a γ-semiautomorphism µγ of Y satisfying condition (iv) of Lemma 6.3. Then such µγ is unique, and the map γ 7→ µγ satisfies conditions (i) and (ii) of Lemma 6.3. G ′ Proof. If µ′γ another such γ-semi-automorphism, then µ−1 γ µγ ∈ Aut (Y ) = {1}, hence G µ′γ = µγ . If γ, δ ∈ Γ, then µ−1 γδ µγ µδ ∈ Aut (Y ) = {1}, hence µγδ = µγ µδ , hence the map γ 7→ µγ is a homomorphism, that is, condition (i) holds. Since G and Y are of finite type over k, there exists a finite Galois extension k1 /k in k and a G1 -equivariant k1 -model (Y1 , θ1 ) of (Y, θ), where G1 := G0 ×k0 k1 . This k1 -model defines a homomorphism (27) γ 7→ µ′γ : Gal(k/k1 ) → SAut(Y ) such that µ′γ is σγ -equivariant for all γ ∈ Gal(k/k1 ). Since a σγ -equivariant γ-semiautomorphism is unique, we see that for all γ ∈ Gal(k/k1 ) we have µγ = µ′γ , and hence, the restriction of the map (28) γ 7→ µγ : Γ → SAut(Y ) to Gal(k/k1 ) comes from the k1 -model (Y1 , θ1 ) of (Y, θ), that is, condition (ii) of Lemma 6.3 is satisfied.  7. Spherical homogeneous spaces and their combinatorial invariants Let G be a connected reductive group over an algebraically closed field k. We describe combinatorial invariants (invariants of Luna and Losev) of a spherical homogeneous space Y = G/H of G. We start with combinatorial invariants of G. We fix T ⊂ B ⊂ G, where B is a Borel subgroup and T is a maximal torus. Let BRD(G) = BRD(G, T, B) denote the based root datum of G. We have BRD(G, T, B) = (X, X ∨ , R, R∨ , S, S ∨ ) where X = X∗ (T ) := Hom(T, Gm,k ) is the character group of T ; X ∨ = X∗ (T ) := Hom(Gm,k , T ) is the cocharacter group of T ; R = R(G, T ) ⊂ X is the root system; R∨ ⊂ X ∨ is the coroot system; S = S(G, T, B) ⊂ R is the system of simple roots (the basis of R) defined by B; S ∨ ⊂ R∨ is the system of simple coroots. There is a canonical pairing X × X ∨ → Z, (χ, x) 7→ hχ, xi, and a canonical bijection α 7→ α∨ : R → R∨ such that S ∨ = {α∨ | α ∈ S}. See Springer [Sp79, Sections 1 and 2] for details. We consider also the Dynkin diagram Dyn(G) = Dyn(G, T, B), which is a graph with the set of vertices S. The edge between two simple roots α, β ∈ S is described in terms of the integers hα, β ∨ i and hβ, α∨ i. We call a pair (T, B) as above a Borel pair. If (T ′ , B ′ ) is another Borel pair, then by Theorem 11.1 and Theorem 10.6(4) in Borel’s book [B91], there exists g ∈ G(k) such that (29) g · T · g −1 = T ′ , g · B · g −1 = B ′ . 18 MIKHAIL BOROVOI WITH AN APPENDIX BY GIULIANO GAGLIARDI This element g induces an isomorphism ∼ g∗ : BRD(G, T ′ , B ′ ) → BRD(G, T, B). If g′ ∈ G(k) another element as in (29), then g = gt for some t ∈ T (k), and therefore, the isomorphism ∼ (g′ )∗ : BRD(G, T ′ , B ′ ) → BRD(G, T, B) coincides with g∗ . Thus we can canonically identify BRD(G, T ′ , B ′ ) with BRD(G, T, B) and write BRD(G) for BRD(G, T, B). We say that BRD(G) is the canonical root datum of G. We see that the based root datum BRD(G) is an invariant of G. In particular, the character lattice X = X∗ (T ) with the subset S ⊂ X is an invariant, and the Dynkin diagram Dyn(G) is an invariant. Now we describe combinatorial invariants of a homogeneous spherical G-variety Y = G/H. Let K(Y ) denote the field of rational functions of Y . The group G(k) acts on K(Y ) by (g · f )(y) = f (g−1 · y) for f ∈ K(Y ), g ∈ G(k), and y ∈ Y (k). (B) For χ ∈ X∗ (B) let K(Y )χ denote the space of χ-eigenfunctions in K(Y ), that is, the k-space of rational functions f ∈ K(X) such that b · f = χf (b) · f for all b ∈ B(k). (B) Since B has an open dense orbit in Y , the k-dimension of K(Y )χ is ≤ 1. Let X = (B) X (Y ) ⊂ X∗ (B) denote the set of characters χ of B such that K(Y )χ 6= 0, which is a subgroup of X∗ (B) called the weight lattice of Y . We set V = V (Y ) = HomZ (X , Q). Let Val(K(Y )) denote the set of Q-valued valuations of the field K(Y ) that are trivial on k. The group G(k) naturally acts on K(Y ) and on Val(K(Y )). We will consider the set ValB (K(Y )) of B(k)-invariant valuations, and the set ValG (K(Y )) of G(k)-invariant valuations. We have a canonical map ρ : ValB (K(Y )) → V, v 7→ (χ 7→ v(fχ )), (B) where v ∈ ValB (K(Y )), χ ∈ X , fχ ∈ K(Y )χ , fχ 6= 0. It is known, see Knop [Kn89, Corollary 1.8], that the restriction of ρ to ValG (K(Y )) is injective. We denote by V = V(Y ) := ρ(ValG (K(Y ))) ⊂ V the image of ValG (K(Y )) in V . It is a cone in V called the valuation cone of Y . Let D = D(Y ) denote the set of colors of Y , that is, the set of closures of B-orbits of codimension 1 in Y ; it is finite. Each D ∈ D is a B-invariant divisor, which defines a B-invariant valuation of K(Y ) that we denote by val(D). Thus we obtain a map val : D → ValB (K(Y )). By abuse of notation we denote ρ(val(D)) ∈ V by ρ(D). Thus we obtain a map ρ : D → V, which in general is not injective (for example, it is not injective for G and Y as in Example 0.1). For D ∈ D, let StabG (D) denote the stabilizer of D ⊂ Y in G. Clearly StabG (D) ⊃ B, hence StabG (D) is a parabolic subgroup of G. For α ∈ S, let Pα ⊃ B denote the corresponding minimal parabolic subgroup of G containing B. Let ς(D) denote the set of α ∈ S for which Pα is not contained in StabG (D). We obtain a map ς : D → P(S), where P(S) denotes the set of all subsets of S. EQUIVARIANT MODELS OF SPHERICAL VARIETIES 19 Lemma 7.1. Any fiber of the map ς has ≤ 2 elements. Proof. Let D ∈ D. Since Y is a homogeneous G-variety, clearly StabG (D) 6= G, hence ς(D) 6= ∅. We see that there exists α ∈ ς(D). Consider the set D(α) consisting of those D ∈ D for which α ∈ ς(D). By Proposition B.2 in Appendix B below we have |D(α)| ≤ 2, and the lemma follows.  Consider the map ρ × ς : D → V × P(S). Corollary 7.2. Any fiber of the map ρ × ς has ≤ 2 elements.  Consider the subset Ω := im(ρ × ς) ⊂ V × P(S). Let Ω(2) (resp. Ω(1) ) denote the subset of Ω consisting of the elements with two preimages (resp. with one preimage) in D. We obtain two subsets Ω(1) , Ω(2) ⊂ V × P(S), and by Corollary 7.2 we have Ω = Ω(1) ∪ Ω(2) (disjoint union). Definition 7.3. Let G be a connected reductive group over an algebraically closed field k. Let Y = G/H be a spherical homogeneous space of G. By the combinatorial invariants of Y we mean X ⊂ X∗ (B), V ⊂ V := HomZ (X , Q), and Ω(1) , Ω(2) ⊂ V × P(S). 7.4. Let G be a connected reductive k-group. Let H1 ⊂ G be a spherical subgroup, then we set Y1 = G/H1 . We consider the set of colors D(Y1 ) and the canonical maps ρ1 : D(Y1 ) → V (Y1 ), ς1 : D(Y1 ) → P(S). If H2 ⊂ G is another spherical subgroup, then we set Y2 = G/H2 and consider the set of colors D(Y2 ) and the canonical maps ρ2 : D(Y2 ) → V (Y2 ), ς2 : D(Y2 ) → P(S). Now assume that there exists a ∈ G(k) such that H2 = aH1 a−1 . Then we have an isomorphism of G-varieties of Lemma 4.1 φa : Y1 → Y2 , g · H1 7→ ga−1 · H2 . It follows that X (Y1 ) = X (Y2 ), V (Y1 ) = V (Y2 ), V(Y1 ) = V(Y2 ). Moreover, the Gequivariant map φa : Y1 → Y2 induces a bijection (φa )∗ : D(Y1 ) → D(Y2 ) satisfying (30) ρ2 ◦ (φa )∗ = ρ1 , ς2 ◦ (φa )∗ = ς1 . It follows that Ω(1) (Y1 ) = Ω(1) (Y2 ) and Ω(2) (Y1 ) = Ω(2) (Y2 ). Conversely: Proposition 7.5 (Losev’s Uniqueness Theorem [Lo09, Theorem 1]). Let G be a connected reductive group over an algebraically closed field k of characteristic 0. Let H1 , H2 ⊂ G be two spherical subgroups, and let Y1 = G/H1 and Y2 = G/H2 be the corresponding spherical homogeneous spaces. If X (Y1 ) = X (Y2 ), V(Y1 ) = V(Y2 ), Ω(1) (Y1 ) = Ω(1) (Y2 ), and Ω(2) (Y1 ) = Ω(2) (Y2 ), then there exists a ∈ G(k) such that H2 = aH1 a−1 . 7.6. Consider the group AutG (Y ) = NG (H)/H, this group acts on D. We consider the surjective map ζ = ρ × ς : D → Ω. By Corollary 7.2 each of the fibers of ζ has either one or two elements. We denote by AutΩ (D) the group of permutations π : D → D such that ζ ◦ π = ζ. It is clear that 20 MIKHAIL BOROVOI WITH AN APPENDIX BY GIULIANO GAGLIARDI the group AutG (Y ), when acting on D, acts on the fibers of the map ζ, so we obtain a homomorphism AutG (Y ) → AutΩ (D). Theorem 7.7 (Losev, unpublished). Let G be a connected reductive group over an algebraically closed field k of characteristic 0. Let Y = G/H be a spherical homogeneous space of G. Then, with the above notation, the homomorphism AutG (Y ) → AutΩ (D). (31) is surjective. This theorem will be proved in Appendix B, see Theorem B.4. Corollary 7.8 (Strong version of Losev’s Uniqueness Theorem). Let G, H1 , H2 , Y1 = G/H1 , Y2 = G/H2 be as in Proposition 7.5, in particular, X (Y1 ) = X (Y2 ), V(Y1 ) = V(Y2 ), Ω(1) (Y1 ) = Ω(1) (Y2 ), and Ω(2) (Y1 ) = Ω(2) (Y2 ). Let ϕ : D(Y1 ) → D(Y2 ) be any bijection satisfying ρ2 ◦ ϕ = ρ1 , ς2 ◦ ϕ = ς1 (such a bijection exists because Ω(1) (Y1 ) = Ω(1) (Y2 ) and Ω(2) (Y1 ) = Ω(2) (Y2 )). Then there exists a′ ∈ G(k) such that H2 = a′ H1 (a′ )−1 and (φa′ )∗ = ϕ : D(Y1 ) → D(Y2 ). Proof. By Proposition 7.5 there exists a ∈ G(k) such that H2 = aH1 a−1 . This element a defines a map (φa )∗ : D(Y1 ) → D(Y2 ) satisfying (30). Set ψ = (φa )−1 ∗ ◦ ϕ : D(Y1 ) → D(Y1 ), then ψ satisfies ρ1 ◦ ψ = ρ1 , ς1 ◦ ψ = ς1 , hence ψ ∈ AutΩ D(Y1 ). By Theorem 7.7 there exists n ∈ NG (H1 ) such that (φn )∗ = ψ : D(Y1 ) → D(Y1 ). We set a′ = an, then a′ H1 (a′ )−1 = H2 , φa′ = φa ◦ φn , and (φa′ )∗ = (φa )∗ ◦ (φn )∗ = (φa )∗ ◦ ψ = ϕ.  Corollary 7.9. If in Theorem 7.7 H is spherically closed, then the homomorphism (31) is an isomorphism. Proof. Indeed, since H is spherically closed, the homomorphism (31) is injective, and by Theorem 7.7 it is surjective, hence it is bijective, as required.  Note that AutΩ (D) = Y Aut(ζ −1 (ω)), ω∈Ω Aut(ζ −1 (ω)) where is the group of permutations of the set ζ −1 (ω). It is clear that for any ω ∈ Ω, the restriction homomorphism (32) AutΩ (D) → Aut(ζ −1 (ω)) is surjective. Corollary 7.10. If in Theorem 7.7 NG (H) = H, then the surjective map ζ is bijective, hence D injects into V × P(S). EQUIVARIANT MODELS OF SPHERICAL VARIETIES 21 Proof. It follows from Theorem 7.7 and the surjectivity of the homomorphism (32), that the group AutG (Y ) = NG (H)/H acts transitively on the fiber ζ −1 (ω) for any ω ∈ Ω. Since by assumption NG (H)/H = {1}, we conclude that each fiber of ζ has exactly one element, hence ζ is bijective, as required.  8. Action of an automorphism of the base field on the combinatorial invariants of a spherical homogeneous space 8.1. Let k is an algebraically closed field, G be a connected reductive group over k, H1 ⊂ G be a spherical subgroup, and Y1 = G/H1 be the corresponding spherical homogeneous space. Let k0 ⊂ k be a subfield and let γ ∈ Aut(k/k0 ). We assume that “G is defined over k0 ”, that is, we are given a k0 -model G0 of G. Then we have a γ-semi-automorphism σγ of G, see Subsection 6.2. Set H2 = σγ (H1 ) ⊂ G and denote by Y2 := G/H2 the corresponding spherical homogeneous space. We wish to know whether the spherical homogeneous spaces Y1 and Y2 are isomorphic as G-varieties. For this end we compare their combinatorial invariants. We fix a Borel pair (T, B), then T ⊂ B ⊂ G. Consider σγ (T ) ⊂ σγ (B) ⊂ G. Then (σγ (T ), σγ (B)) is again a Borel pair, hence there exists gγ ∈ G(k) such that gγ · σγ (T ) · gγ−1 = T, gγ · σγ (B) · gγ−1 = B. Set τ = inn(gγ ) ◦ σγ : G → G, then τ is a γ-semi-automorphism of G, and (33) Set H2′ and Y2′ τ (B) = B, Y2′ τ (T ) = T. G/H2′ . = τ (H1 ) ⊂ G and = We have H2′ = gγ · H2 · gγ−1 , so by Lemma 4.1 Y2 are isomorphic, and we wish to know whether Y1 and Y2′ are isomorphic. By (33), τ acts on the characters of T and B; we denote the corresponding automorphism by εγ . By definition (34) εγ (χ)(b) = γ(χ(τ −1 (b))) for χ ∈ X∗ (B), b ∈ B(k), and the same for the characters of T (recall that X∗ (B) = X∗ (T )). Since τ (B) = B, we see that εγ , when acting on X∗ (T ), preserves S = S(G, T, B) ⊂ X∗ (T ). It is well known (see e.g. [BKLR14, 3.2 and Proposition 3.1(a)]) that the automorphism εγ does not depend on the choice of gγ and that the map ε : Aut(k/k0 ) → Aut(X∗ (T ), S), is a homomorphism. Since εγ acts on εγ (Ω(1) (Y1 )), εγ (Ω(2) (Y1 )). X∗ (B) γ 7→ εγ and on S, one can define εγ (X (Y1 )), εγ (V(Y1 )), Following Akhiezer [Akh15], we compute the combinatorial invariants of the spherical homogeneous space Y2′ . We define a map (35) Y1 (k) → Y2′ (k), g · H1 7→ τ (g) · H2′ , where g ∈ G(k). By Lemma 4.3 the map (35) is induced by some τ -equivariant γ-semi-isomorphism ν : Y1 → Y2′ , and so we obtain an isomorphism of the function fields ν∗ : K(Y1 ) → K(Y2′ ). (B) (B) Lemma 8.2. Let χ1 ∈ X∗ (B) and assume that f1 ∈ K(Y1 )χ1 . Then ν∗ f1 ∈ K(Y2′ )χ2 , where χ2 = εγ (χ1 ). 22 MIKHAIL BOROVOI WITH AN APPENDIX BY GIULIANO GAGLIARDI Proof. By assumption f1 (b−1 y1 ) = χ1 (b) · f1 (y1 ) for all y1 ∈ Y1 (k), b ∈ B(k). We write f2′ = ν∗ f1 ∈ K(Y2′ ). Since ν : Y1 → Y2′ is a γ-semi-isomorphism, by Corollary 1.8 we have f2′ (y2′ ) = γ(f1 (ν −1 (y2′ ))) for y2′ ∈ Y2′ (k). Note that τ −1 : G → G is a γ −1 -semi-automorphism of G, and ν −1 : Y2′ → Y1 is a τ −1 equivariant γ −1 -semi-isomorphism. Moreover, τ −1 (T ) = T and τ −1 (B) = B. We compute: f2′ (b−1 · y2′ ) = γ(f1 (ν −1 (b−1 · y2′ ))) = γ(f1 ((τ −1 (b))−1 · ν −1 (y2′ ))) = γ(χ1 (τ −1 (b))) · γ(f1 (ν −1 (y2′ ))) = εγ (χ1 )(b) · f2′ (y2′ ). (B) Thus f2′ ∈ K(Y2′ )χ2 , where χ2 = εγ (χ1 ), as required.  Corollary 8.3. X (Y2′ ) = εγ (X (Y1 )). Proof. By Lemma 8.2 we have εγ (X (Y1 )) ⊂ X (Y2′ ). Applying Lemma 8.2 to the triple (γ −1 , τ −1 , ν −1 ), we obtain that εγ −1 (X (Y2′ )) ⊂ X (Y1 ), hence X (Y2′ ) ⊂ εγ (X (Y1 )). Thus X (Y2′ ) = εγ (X (Y1 )), as required.  Let v1 ∈ ValB (K(Y1 )). We define ν∗ v1 ∈ ValB (K(Y2′ )) by (ν∗ v1 )(f2′ ) = v1 (ν∗−1 (f2′ )) for f2′ ∈ K(Y2′ ). We consider the maps ρ1 : ValB (K(Y1 )) → V (Y1 ) and ρ′2 : ValB (K(Y2′ )) → V (Y2′ ). Lemma 8.4. For any v1 ∈ ValB (K(Y1 )) we have ρ′2 (ν∗ v1 ) = εγ (ρ1 (v1 )). Proof. See Huruguen [Hu11, Proposition 2.18].  Corollary 8.5. V(Y2′ ) = εγ (V(Y1 )).  Let D1 ∈ D(Y1 ) be a color, that is, D1 is the closure of a B-orbit of codimension one in Y1 . We set D2′ := ν(D1 ) ⊂ Y2′ , then D2′ ∈ D(Y2′ ). We also write ν∗ D1 for ν(D1 ). Let D1 ∈ D(Y1 ) and D2′ ∈ D(Y2′ ), then we denote by val1 (D1 ) ∈ ValB (K(Y1 )) and val′2 (D2′ ) ∈ ValB (K(Y2′ )) the corresponding B-invariant valuations. Lemma 8.6. For any D1 ∈ D(Y1 ) we have val′2 (ν∗ D1 ) = ν∗ (val1 (D1 )). Proof. See Huruguen [Hu11, Proposition 2.19].  Remark 8.7. Propositions 2.18 and 2.19 of Huruguen [Hu11, Section 2.2] are proved in his paper under certain additional assumptions. Namely, Huruguen assumes that that k/k0 is a Galois extension, that the triple (G, Y, θ) has a k0 -model (G0 , Y0 , θ0 ), and that Y0 has a k0 -point y (0) . Those assumptions are not used in his proof. By abuse of notation, if D1 ∈ D(Y1 ) and D2′ ∈ D(Y2′ ), we write ρ1 (D1 ) for ρ1 (val1 (D1 )) ∈ V (Y1 ) and ρ′2 (D2′ ) for ρ′2 (val′2 (D2′ )) ∈ V (Y2′ ). Corollary 8.8 (from Lemma 8.4 and Lemma 8.6). For any D1 ∈ D(Y1 ) we have ρ′2 (ν∗ D1 ) = εγ (ρ1 (D1 )).  Lemma 8.9. For any D1 ∈ D(Y1 ) we have: EQUIVARIANT MODELS OF SPHERICAL VARIETIES 23 (i) StabG (ν∗ D1 ) = τ (StabG (D1 )); (ii) ς(ν∗ D1 ) = εγ (ς(D1 )). Proof. (i) follows from the fact that the map ν : Y1 (k) → Y2′ (k) is τ -equivariant, and (ii) follows from (i).  Corollary 8.10 (from Corollary 8.8 and Lemma 8.9). Ω(1) (Y2′ ) = εγ (Ω(1) (Y1 )) and Ω(2) (Y2′ ) = εγ (Ω(2) (Y1 )).  Proposition 8.11. X (Y2 ) = εγ (X (Y1 )), V(Y2 ) = εγ (V(Y1 )), Ω(1) (Y2 ) = εγ (Ω(1) (Y1 )), Ω(2) (Y2 ) = εγ (Ω(2) (Y1 )). Proof. Since H2′ and H2 are conjugate, by Lemma 4.1 the G-varieties Y2′ and Y2 are isomorphic, hence they have the same combinatorial invariants, and the proposition follows from Corollaries 8.3, 8.5, and 8.10.  Note that Proposition 8.11 generalizes Propositions 5.2, 5.3, and 5.4 of Akhiezer [Akh15]. Namely, in the case when γ 2 = 1, our Proposition 8.11 is equivalent to those results of Akhiezer. Our proofs are similar to his. Proposition 8.12. With the notation and assumptions of Subsection 8.1, if the subgroups H1 and H2 = σγ (H1 ) are conjugate, then εγ preserves the combinatorial invariants of Y1 , that is (36) εγ (X (Y1 )) = X (Y1 ), εγ (V(Y1 )) = V(Y1 ), εγ (Ω(1) (Y1 )) = Ω(1) (Y1 ), εγ (Ω(2) (Y1 )) = Ω(2) (Y1 ). Conversely, if equalities (36) hold and char k = 0, then H1 and H2 are conjugate. Proposition 8.12 generalizes Theorem 3(1) of Cupit-Foutou [CF15], where the case k0 = R was considered. Proof. If H1 and H2 are conjugate, then by Lemma 4.1 the homogeneous spaces Y1 = G/H1 and Y2 = G/H2 are isomorphic as G-varieties, hence then they have the same combinatorial invariants, that is, (37) X (Y2 ) = X (Y1 ), V(Y2 ) = V(Y1 ), Ω(1) (Y2 ) = Ω(1) (Y1 ), Ω(2) (Y2 ) = Ω(2) (Y1 ), and (36) follows from Proposition 8.11. Conversely, if equalities (36) hold and char k = 0, then by Proposition 8.11 the equalities (37) hold, and by Proposition 7.5 (Losev’s Uniqueness Theorem) the subgroups H1 and H2 are conjugate.  Corollary 8.13. With the notation and assumptions of Subsection 8.1, if there exists a σγ -equivariant γ-semi-automorphism µ : Y1 → Y1 , then εγ preserves the combinatorial invariants of Y1 , that is, equalities (36) hold. Conversely, if equalities (36) hold and char k = 0, then there exists a σγ -equivariant γ-semi-automorphism µ : Y1 → Y1 . Proof. By Corollary 4.4 there exists a σγ -equivariant γ-semi-automorphism µ : Y1 → Y1 if and only if the subgroup σγ (H1 ) of G is conjugate to H1 . Now the corollary follows from Proposition 8.12.  24 MIKHAIL BOROVOI WITH AN APPENDIX BY GIULIANO GAGLIARDI 9. Equivariant models of automorphism-free spherical homogeneous spaces 9.1. Let k0 be a perfect field and let k be a fixed algebraic closure of k0 with Galois group Γ = Gal(k/k0 ). Let G be a connected reductive group over k. Let T ⊂ B ⊂ G be as in Section 7. We consider the based root datum BRD(G) = BRD(G, T, B). Let G0 be a k0 -model of G. For any γ ∈ Γ, this model defines a γ-semi-automorphism σγ : G → G, which induces an automorphism εγ ∈ AutBRD(G), see Section 8. We obtain a homomorphism ε : Γ → AutBRD(G), γ 7→ εγ . Let Y = G/H be a spherical homogeneous space of G. We consider the combinatorial invariants of Y : X = X (Y ) ⊂ X∗ (T ), V = V(Y ) ⊂ HomZ (X , Q), Ω(1) = Ω(1) (Y ), Ω(2) = Ω(2) (Y ) ⊂ HomZ (X , Q) × P(S), see Section 7. Since εγ acts on BRD(G), we can define εγ (X ), εγ (V), εγ (Ω(1) ), εγ (Ω(2) ). Recall that Y = G/H. By Lemma 4.3(i) we have γ∗ Y ∼ = G/σγ (H). Proposition 9.2. If Y = G/H admits a G0 -equivariant k0 -model Y0 , then for all γ ∈ Γ, εγ preserves the combinatorial invariants of Y , that is (38) εγ (X ) = X , εγ (V) = V, εγ (Ω(1) ) = Ω(1) , εγ (Ω(2) ) = Ω(2) . Proposition 9.2 follows from formulas of Huruguen [Hu11, Section 2.2]. For the reader’s convenience we prove it here. Proof. A G0 -equivariant k0 -model Y0 of Y defines, for any γ ∈ Γ, a σγ -equivariant γ-semiautomorphism µγ of Y , hence an isomorphism of G-k-varieties (µγ )♮ : G/σγ (H) = γ∗ Y → Y . We see that the G-varieties G/H and G/σγ (H) are isomorphic, hence they have the same combinatorial invariants. By Proposition 8.11 the combinatorial invariants of the spherical homogeneous space G/σγ (H) are εγ (X ), εγ (V), εγ (Ω(1) ), εγ (Ω(2) ), and (38) follows.  The next theorem is a partial converse of Proposition 9.2. Theorem 9.3. Let k, k0 , Γ, G, H, G0 be as in 9.1. Assume that: (i) For all γ ∈ Γ, εγ preserves the combinatorial invariants of Y , that is, equalities (38) hold; (ii) NG (H) = H; (iii) char k = 0. Then Y = G/H admits a G0 -equivariant k0 -model Y0 . This k0 -model is unique up to a unique isomorphism. Proof. Let γ ∈ Γ. Since char k = 0 and εγ preserves the combinatorial invariants of Y , by Corollary 8.13 there exists a σγ -equivariant γ-semi-automorphism µγ : Y → Y. EQUIVARIANT MODELS OF SPHERICAL VARIETIES 25 Thus condition (i) of Lemma 6.3 is satisfied. Since NG (H) = H, by Corollary 5.2 AutG (Y ) = {1}, and by Lemma 6.5 conditions (ii) and (iii) of Lemma 6.3 are satisfied. The variety Y = G/H is quasi-projective, hence condition (iv) of Lemma 6.3 is satisfied. By Lemma 6.3 there exists a G0 -equivariant k0 -model Y0 of Y . Since AutG (Y ) = {1}, for any given γ ∈ Γ the γ-semi-automorphism µγ is unique, hence the model Y0 is unique up to a unique isomorphism.  Recall that a k0 -model G0 of a connected reductive k-group G is called an inner form (of a split group) if εγ = 1 for all γ ∈ Γ = Gal(k/k0 ). Lemma 9.4. Let k, k0 , Γ, G, H, G0 be as in 9.1. Then each of the conditions below imply condition (i) of Theorem 9.3. (i) G0 is an inner form; (ii) G0 is absolutely simple (that is, G is simple) of one of the types A1 , Bn , Cn , E7 , E8 , F4 , G2 ; (iii) G0 is split. Proof. (i) If G0 is an inner form, then εγ = 1 for any γ ∈ Γ, hence condition (i) of Theorem 9.3 is clearly satisfied. (ii) In these cases Dyn(G) has no nontrivial automorphisms, hence Γ acts trivially on Dyn(G). We see that (ii) implies (i). (iii) In this case clearly εγ = 1 for all γ ∈ Γ.  Corollary 9.5. If char k = 0, NG (H) = H, and at least one of the conditions (i–iii) of Lemma 9.4 is satisfied, then Y admits a G0 -equivariant k0 -model, and this k0 -model is unique.  Remark 9.6. Assume that k = R and NG (H) = H. The assertion that if condition (iii) of Lemma 9.4 is satisfied, then Y has a unique G0 -equivariant R-model Y0 , is Theorem 4.12 of Akhiezer and Cupit-Foutou [ACF14]. The similar assertion when only condition (i) of Lemma 9.4 is satisfied, is Theorem 1.1 of Akhiezer [Akh15]. Our paper is inspired by this result of Akhiezer, and our proof of Theorem 9.3 is similar to his proof. 10. Equivariant models of spherically closed spherical homogeneous spaces In this section we do not assume that NG (H) = H. Let k, G, H, Y = G/H, T ⊂ B ⊂ G be as in Section 7, in particular, k is algebraically closed, and we assume that char k = 0. The group AutG (Y ) = NG (H)/H acts on Y and on the set D of colors of Y . Definition 10.1. A spherical homogeneous space Y = G/H is called spherically closed if NG (H)/H acts on D faithfully, that is, if the homomorphism AutG (Y ) → Aut(D) is injective. (Here Aut(D) denotes the group of permutations of the finite set D.) Let k0 ⊂ k be a subfield such that k is an algebraic closure of k0 . Let G0 be a k0 -model of G, and for γ ∈ Γ := Gal(k/k0 ) let σγ : G → G be the γ-semi-automorphism defined by G0 . Let εγ : X∗ (T ) → X∗ (T ) be as in (34). Theorem 10.2. Let k be an algebraically closed field. Let G, H, Y = G/H be as in Section 7. Let G0 be a k0 -model of G, where k0 ⊂ k is a subfield such that k is an algebraic closure of k0 . Assume that 26 MIKHAIL BOROVOI WITH AN APPENDIX BY GIULIANO GAGLIARDI (i) εγ preserves the combinatorial invariants of Y for all γ ∈ Γ, (ii) Y is spherically closed, (iii) char k = 0. Then Y admits a G0 -equivariant k0 -model. Theorem 10.2 generalizes the existence assertion of Theorem 9.3. It was inspired by Corollary 1 of Cupit-Foutou [CF15, Section 2.5], where the case k0 = R was considered. In order to prove the theorem we need a few lemmas. Lemma 10.3. Let ζ : D → Ω be a mapping of nonempty finite sets. Let Γ be a group acting on Ω by a homomorphism s : Γ → Aut(Ω), γ 7→ sγ . Assume that for any γ ∈ Γ there exists a permutation mγ : D → D covering sγ , that is, such that the following diagram commutes: D mγ /D ζ ζ  Ω sγ  /Ω Then there exists a homomorphism m′ : Γ → Aut(D) such that: (i) for any γ ∈ Γ the permutation m′γ covers sγ ; (ii) for any γ ∈ Γ we have m′γ = aγ ◦ mγ , where aγ ∈ AutΩ (D); (iii) m′γ = idD for all γ ∈ ker s. Proof. We may and shall assume that Γ acts transitively on Ω. Let ω, ω ′ ∈ Ω, then there exists γ ∈ Γ such that sγ (ω) = ω ′ . By hypotheses there exists mγ ∈ Aut(D) covering sγ , then mγ induces a bijection ζ −1 (ω) → ζ −1 (ω ′ ), hence the cardinalities of ζ −1 (ω) and ζ −1 (ω ′ ) are equal. We see that ω 7→ |ζ −1 (ω)| is a constant function on Ω; we denote its value by n. For each ω ∈ Ω we fix some bijection between ζ −1 (ω) and the set {1, . . . , n}; (i) we denote the element of ζ −1 (ω) ⊂ D corresponding to i ∈ {1, . . . , n} by dω . Then we define m′γ ∈ Aut(D) by (i) m′γ (d(i) ω ) = dsγ (ω) . Since s : γ 7→ sγ is a homomorphism, we see that m′ : γ 7→ m′γ is a homomorphism, and clearly m′γ covers sγ , which proves (i). Set aγ = m′γ ◦ m−1 γ , then clearly (ii) holds, and the assertion (iii) holds by construction.  10.4. Write ζ = ρ × ς : D → V × P(S), Ω = im ζ, sγ = εγ |Ω : Ω → Ω, then the map Γ → Aut(Ω), γ 7→ sγ is a homomorphism. Assume that for all γ ∈ Γ there exists a σγ -equivariant γ-semiautomorphism µ : Y → Y , that is, a γ-semi-automorphism satisfying (39) µ(g · y) = σγ (g) · µ(y) for all g ∈ G(k), y ∈ Y (k). The following lemma is obvious: Lemma 10.5. If γ, δ ∈ Aut(k/k0 ), µ is a σγ -equivariant γ-semi-automorphism, and ν is a σδ -equivariant δ-semi-automorphism, then µν is a σγδ -equivariant γδ-semi-automorphism and µ−1 is a σγ −1 -equivariant γ −1 -semi-automorphism.  EQUIVARIANT MODELS OF SPHERICAL VARIETIES 27 10.6. Consider σγ (T ) ⊂ σγ (B) ⊂ G. There exists gγ ∈ G(k) such that if we set σγ′ = inn(gσ ) ◦ σγ , then σγ′ (T ) = T (40) and σγ′ (B) = B. For µ : Y → Y as in (39), we define a γ-semi-automorphism µ′ = gγ ◦ µ : Y → Y, y 7→ gγ · µ(y) for y ∈ Y (k). Then for g ∈ G(k), y ∈ Y (k) we have (41) µ′ (g · y) = gγ · µ(g · y) = gγ · σγ (g) · µ(y) = (gγ · σγ (g) · gγ−1 ) · (gγ · µ(y)) = σγ′ (g) · µ′ (y). Let D ∈ D = D(Y ) be a color, this means that D is the closure of a codimension one B-orbit in Y . Since σγ′ (B) = B, it follows from (41) that the divisor µ′ (D) in Y is the closure of a codimension one B-orbit, that is, a color. We obtain a permutation mµ : D → D, (42) D 7→ µ′ (D), covering sγ . Since gγ for which (40) holds is defined uniquely up to multiplication on the left by an element t ∈ T (k) ⊂ B(k), we see that mµ depends only on µ and does not depend on the choice of gγ . Lemma 10.7. The map µ 7→ mµ is a homomorphism: for γ, µ, δ, ν as in Lemma 10.5, we have mµ◦ν = mµ ◦ mν . Proof. Let (γ, µ) and (δ, ν) be as in Lemma 10.5. Choose gγ , gδ ∈ G(k) such that (43) gγ · σγ (T, B) · gγ−1 = (T, B), gδ · σδ (T, B) · gδ−1 = (T, B). Set µ′ = gγ ◦ µ : Y → Y, ν ′ = gδ ◦ ν : Y → Y. Then for y ∈ Y (k) we have (44) (µ′ ν ′ )(y) = µ′ (ν ′ (y)) = gγ · µ(gδ · ν(y)) = gγ · σγ (gδ ) · (µν)(y). On the other hand, from (43) we obtain gγ · σγ (gδ · σδ (T, B) · gδ−1 ) · gγ−1 = (T, B), hence gγ σγ (gδ ) · σγδ (T, B) · (gγ σγ (gδ ))−1 = (T, B). Thus we may set (µν)′ = (gγ σγ (gδ )) ◦ µν , that is, (µν)′ (y) = gγ σγ (gδ ) · (µν)(y) for y ∈ Y (k). Comparing with (44), we see that with this (µν)′ we have (µν)′ = µ′ ◦ ν ′ . hence mµν = mµ ◦ mν , as required.  28 MIKHAIL BOROVOI WITH AN APPENDIX BY GIULIANO GAGLIARDI Proof of Theorem 10.2. Let γ ∈ Γ. Since char k = 0 and εγ preserves the combinatorial invariants of Y = G/H, by Corollary 8.13 there exists a σγ -equivariant γ-semi-automorphism µγ : Y → Y . Set mγ = mµγ ∈ Aut(D), see Subsection 10.6, then mγ covers sγ , where sγ ∈ Aut(Ω) is the restriction of εγ to Ω. By Lemma 10.3 there exists a homomorphism m′ : Γ → Aut(D), γ 7→ m′γ such that for any γ ∈ Γ the permutation m′γ ∈ Aut(D) covers sγ (property (i)) and we have m′γ = aγ ◦ mγ , where aγ ∈ AutΩ (D) (property (ii)). By Theorem 7.7 there exists an automorphism e aγ ∈ AutG (Y ) inducing aγ on D. We set µ′γ = e aγ ◦ µγ , then ′ ′ µγ is a σγ -equivariant γ-semi-automorphism of Y , and by Lemma 10.7, µγ acts on D by aγ ◦ mγ = m′γ . Let γ, δ ∈ Γ, then µ′γ µ′δ (µ′γδ )−1 ∈ AutG (Y ) and by Lemma 10.7 it acts on D by m′γ m′δ (m′γδ )−1 = idD . Since Y is spherically closed, we conclude that µ′γ µ′δ (µ′γδ )−1 = idY , hence µγδ = µγ ◦ µδ . Thus the map γ 7→ µ′γ is a homomorphism. It is easy to see that Y admits a Gk2 -equivariant k2 -model Y2 over some finite Galois extension k2 /k0 in k. Let Γ2 = Gal(k/k2 ), and for γ ∈ Γ2 let µ′′γ denote the γ-semiautomorphism of Y defined by the k2 -model Y2 . After passing to a finite extension, we may assume that for γ ∈ Γ2 we have sγ = idΩ , and by property (iii) of Lemma 10.3 we have m′γ = idD . Moreover, we may assume that for γ ∈ Γ2 the semi-automorphism µ′′γ acts trivially on D. It follows that (µ′′γ )−1 µ′γ acts trivially on D, and clearly (µ′′γ )−1 µ′γ ∈ AutG (Y ). Since Y is spherically closed, we conclude that (µ′′γ )−1 µ′γ = idY , and hence, µ′γ = µ′′γ for γ ∈ Γ2 . We see that the homomorphism γ 7→ µ′γ satisfies condition (iii) of Lemma 6.3. Note that Y = G/H is quasi-projective, that is, condition (iv) of Lemma 6.3 is satisfied as well. By Lemma 6.3 the homomorphism γ 7→ µ′γ defines a G0 -equivariant k0 -model Y0 of Y , which completes the proof of the theorem.  10.8. In Example 0.1 we considered a spherically closed spherical variety Y = G/H, where G = SL2,k and H = T , a maximal torus in G. In this case it is obvious that for any k0 -model G0 of G there exists a G0 -equivariant k0 -model Y0 of Y = G/H. Indeed, there exists a maximal torus T0 ⊂ G0 defined over k0 , and it is clear that Y0 := G0 /T0 is a G0 -equivariant k0 -model of Y = G/T . In the following example we consider a spherically closed spherical subgroup that is not conjugate to a subgroup defined over k0 . Example 10.9. Let k = C, k0 = R. Following a suggestion of Roman Avdeev, we take G = SO2n+1,C , where n ≥ 2, and we take for H a Borel subgroup of SO2n,C , where SO2n,C ⊂ SO2n+1,C = G. By Proposition 10.10 below, H is a spherically closed spherical subgroup of G and NG (H) 6= H. Take G0 = SO2n+1,R , then G0 is an anisotropic (compact) R-model of G. Since the Dynkin diagram of G has no nontrivial automorphisms, G0 is an inner form. We wish to show that Y = G/H admits a G0 -equivariant R-model. Clearly H is not conjugate to any subgroup of G0 defined over R because H is not reductive, and therefore, we cannot argue as in Subsection 10.8. Since NG (H) 6= H, we cannot apply Theorem 9.3 either. However, since H is spherically closed, by Theorem 10.2 the homogeneous variety Y = G/H does admit a G0 -equivariant R-model Y0 . Proposition 10.10 (Roman Avdeev, private communication). Let G = SO2n+1,C , where n ≥ 2. Let H be a Borel subgroup of SO2n,C , where SO2n,C ⊂ SO2n+1,C = G. Then H is a spherically closed spherical subgroup of G, and NG (H) 6= H. Proof. Set g = Lie(G). Choose a Borel subgroup B ⊂ G and a maximal torus T ⊂ B. Let X = X∗ (T ) denote the character lattice of T and let R = R(G, T ) ⊂ X be the root system. The Borel subgroup B defines a set of positive roots R+ ⊂ R and the corresponding set of EQUIVARIANT MODELS OF SPHERICAL VARIETIES 29 simple roots S ⊂ R+ ⊂ R. Let U denote the unipotent radical of B and put u = Lie(U ). We have M M g = Lie(T ) ⊕ gβ , u= gβ , β∈R β∈R+ where gβ is the root subspace corresponding to a root β. Let Rl ⊂ R denote the root subsystem consisting of the long roots. Observe that R is a root system of type Bn , and Rl is a root system of type Dn . Set Rl+ = R+ ∩ Rl . We set M M gl = Lie(T ) ⊕ gβ , ul = gβ . β∈R+ l β∈Rl Let Gl (resp., Ul ) be the connected algebraic subgroup of G with Lie algebra gl (resp., ul ). Set H = T Ul . Then Gl ≃ SO2n,C and H is a Borel subgroup of Gl . It is well known that H is a spherical subgroup of G. For example, this fact follows from Theorem 1 of Avdeev [Av11] (to apply this theorem one needs to check that all the short positive roots in R are linearly independent). By [Av15, Proposition 5.25] H is spherically closed. We consider the Weyl group W = W (G, T ) = W (R). Let r ∈ W = NG (T )/T denote the reflection with respect to the short simple root, and let ρ be a representative of r in NG (T ). Since r preserves Rl+ , we see that ρ ∈ NG (H). Since ρ ∈ NG (T ) r T and NG (T ) ∩ B = T , we see that ρ ∈ / B. By construction H ⊂ B, and we conclude that ρ ∈ / H, hence NG (H) 6= H. In fact, NG (H) = H ∪ ρH by [Av13, Theorem 3].  The following example shows that G/H might have no G0 -equivariant k0 -model when H is not spherically closed. Example 10.11. Let k = C, k0 = R. Let n ≥ 1, G = Sp2n,C ×C Sp2n,C , Y = Sp2n,C , the group G acts on Y by (g1 , g2 ) ∗ y = g1 y g2−1 , g1 , g2 , y ∈ Sp2n (C). Let H denote the stabilizer in G of 1 ∈ Sp2n (C) = Y (C), then H = Sp2n,C embedded diagonally in G. We have Y = G/H, and Y is a spherical homogeneous space of G. We have NG (H) = Z(G)·H, where Z(G) denotes the center of G. It follows that NG (H)/H ≃ {±1} = 6 {1}. Clearly NG (H)/H acts trivially on D(G/H), so H is not spherically closed. Consider the following real model of G: G0 = Sp2n,R ×R Sp(n), where Sp(n) is the compact real form of Sp2n . We show that Y cannot have a G0 equivariant real model, although G0 is an inner form of a split group. Indeed, assume for the sake of contradiction that such a real model Y0 of Y exists. We have Y = Sp2n,C , and Y0 is simultaneously a principal homogeneous space of Sp2n,R and of Sp(n). Since H 1 (R, Sp2n,R ) = 1, we see that Y0 (R) is not empty. It follows that the topological space Y0 (R) is simultaneously a principal homogeneous space of Sp(2n, R) and of Sp(n). Thus Y0 (R) is simultaneously homeomorphic to the noncompact Lie group Sp(2n, R) and to the compact Lie group Sp(n), which is clearly impossible. Thus, there is no G0 -equivariant real model Y0 of Y . 10.12. Let k, k0 , Γ, G, H, G0 be as in Subsection 9.1, in particular, Γ = Gal(k/k0 ). We do not assume that char k = 0. We assume that H is spherically closed and that Y = G/H admits a G0 -equivariant k0 -model Y0 . Then by Corollary 8.13 εγ preserves the combinatorial invariants of Y for all γ ∈ Γ, in particular, Γ acts on the finite set Ω(2) = Ω(2) (Y ). Let U1 , U2 , . . . , Ur be the orbits of Γ in Ω(2) . For each i = 1, 2, . . . , r, let us choose a point ui ∈ Ui . Set Γi = StabΓ (ui ). 30 MIKHAIL BOROVOI WITH AN APPENDIX BY GIULIANO GAGLIARDI Theorem 10.13. With the notation and assumptions of 10.12 we have: (i) The set of isomorphism classes of G0 -equivariant k0 -models of Y is canonically a 1 principal homogeneous Qr space of the abelian group H (Γ, AutΩ (D)); 1 (ii) H (Γ, AutΩ (D)) ≃ i=1 Hom(Γi , S2 ), where S2 is the symmetric group on two symbols. Note that S2 ∼ = {±1} ∼ = Z/2Z. Corollary 10.14. In Theorem 10.13 assume that |Γ| = 2. Then the number of isomorphism classes of G0 -equivariant k0 -models of Y = G/H is 2s , where s is the number of fixed points of Γ in Ω(2) . Proof. Let Ui be an orbit of Γ in Ω(2) . If |Ui | = 2, then Γi = {1}, hence |Hom(Γi , S2 )| = 1. If |Ui | = 1, then Γi = Γ, and hence, |Hom(Γi , S2 )| = 2. Now the corollary follows from the theorem.  Proof of Theorem 10.13. Since Y is quasi-projective, the set of the isomorphism classes in the theorem is in a canonical bijection with the pointed set H 1 (Γ, AutG (Y )); see Serre [Se97], Proposition 5 in Section III.1.3. By Theorem 2 of Losev [Lo09] (see also Theorem B.1 below), the group AutG (Y ) is abelian, hence H 1 (Γ, AutG (Y )) is an abelian group, and the set of isomorphism classes in the theorem is canonically a principal homogeneous space of this abelian group. By Corollary 7.9 there is a canonical isomorphism of abelian ∼ groups AutG (Y ) → AutΩ (D), and (i) follows. We compute H 1 (Γ, AutΩ (D)). Recall that we have a surjective map ζ : D → Ω. Set D (2) = ζ −1 (Ω(2) ), then clearly   r Y Y Y  AutΩ (D) = AutΩ(2) (D (2) ) = S2 = S2  , ω∈Ω(2) hence 1 H (Γ, AutΩ (D)) = r Y H 1 (Γ, i=1 i=1 Y ω∈Ui S2 ). ω∈Ui Since Γ acts on Ui transitively, by the lemma of Faddeev and Shapiro, see Serre [Se97, I.2.5, Proposition 10], we have Y H 1 (Γ, S2 ) ≃ H 1 (Γi , S2 ) = Hom(Γi , S2 ). ω∈Ui Thus H 1 (Γ, AutΩ (D)) ≃ r Y Hom(Γi , S2 ), i=1 which proves (ii).  Example 10.15. Let G = SO3,C ≃ PGL2,C . Let T ⊂ G be a maximal torus. We take H = T and consider Y = G/H = G/T , which is a spherical homogeneous space of G. We have AutG (G/T ) = NG (T )/T ≃ {±1}. Let G0 be an R-form of G, and let T0 be a maximal torus of G0 (defined over R), then clearly G0 /T0 is a G0 -equivariant R-model of Y = G/T (so we do not have to refer to Theorem 10.2 in order to see that Y admits a G0 -equivariant real model). Since H 1 (Γ, AutG (G/T )) = Hom(Γ, {±1}) ≃ {±1}, EQUIVARIANT MODELS OF SPHERICAL VARIETIES 31 Y has exactly two G0 -equivariant R-models. We describe such models for each R-model of G = SO3,C . Consider the indefinite real quadratic form in three variables F2,1 (x1 , x2 , x3 ) = x21 + x22 − x23 , xi ∈ R. Set G0 = SO(F2,1 ) = SO2,1 , which is a noncompact (split) R-model of G. We consider + the affine quadric Y2,1 ⊂ A3R given by the equation F2,1 (x) = +1, and the affine quadric − + − 3 Y2,1 ⊂ AR given by the equation F2,1 (x) = −1. Then Y2,1 and Y2,1 are SO2,1 -equivariant + R-models of Y = G/T . It is well known that Y2,1 (R) is a hyperboloid of one sheet, hence − it is connected, while Y2,1 (R) is a hyperboloid of two sheets, hence it is not connected. + − It follows that the R-varieties Y2,1 and Y2,1 are two non-isomorphic SO2,1 -equivariant R-models of Y = G/T . Now consider the positive definite real quadratic form in three variables F3 (x1 , x2 , x3 ) = x21 + x22 + x23 , xi ∈ R. Set G0 = SO(F3 ) = SO3,R , which is a compact (anisotropic) R-model of G. We consider the affine quadric Y3+ ⊂ A3R given by the equation F3 (x) = +1, and the affine quadric Y3− ⊂ A3R given by the equation F3 (x) = −1. Then Y3+ and Y3− are SO3,R -equivariant R-models of Y = G/T . Clearly, Y3+ (R) is the unit sphere in R3 , hence it is nonempty, while Y3− (R) is empty. It follows that the R-varieties Y3+ and Y3− are two non-isomorphic SO3,R -equivariant R-models of Y = G/T . 11. Equivariant models of spherical embeddings of automorphism-free spherical homogeneous spaces In this section we assume that NG (H) = H. Theorem 11.1. Let k, G, H, Y = G/H, k0 , Γ, G0 be as in Section 9.1 . We assume that (i) G0 is an inner form of a split group, (ii) NG (H) = H, (iii) char k = 0. Let Y ֒→ Y ′ be an arbitrary spherical embedding of Y . Then Y ′ admits a G0 -equivariant k0 -model Y0′ . This model is compatible with the k0 -model Y0 of Y from Theorem 9.3 and is unique up to a canonical isomorphism. This theorem generalizes Theorem 1.2 of Akhiezer [Akh15], who considered the case k0 = R. Note that Akhiezer considered only the wonderful embedding of Y , while we consider an arbitrary spherical embedding, so our result is new even in the case k0 = R. Proof. We show that a k0 -model of Y ′ , if exists, is unique. Indeed, let Y0′ be such a k0 -model. For γ ∈ Γ := Gal(k/k0 ), let µ′γ : Y ′ → Y ′ be the corresponding γ-semiautomorphism of Y ′ . Since Y is the only open G-orbit in Y ′ , it is stable under µ′γ for all γ ∈ Γ. Since k0 is a perfect field, this defines a G0 -equivariant k0 -model Y0 of Y , which is unique because NG (H) = H and hence AutG (Y ) = {1}. Since Y is Zariski-dense in Y ′ , we conclude that the model Y0′ of Y ′ is unique. (This argument does not use the assumption that char k = 0.) We prove the existence. By Theorem 9.3 Y admits a unique G0 -equivariant k0 -model Y0 . The model Y0 defines an action of Γ on the finite set D, see e.g. Huruguen [Hu11, 2.2.5]. Namely, for any γ ∈ Γ we have a σγ -equivariant γ-semi-automorphism µγ , which induces an automorphism mγ : D → D covering sγ : Ω → Ω, see (42). We show that this 32 MIKHAIL BOROVOI WITH AN APPENDIX BY GIULIANO GAGLIARDI action of Γ on D is trivial. Indeed, since NG (H) = H, by Corollary 7.10 the surjective map ζ: D → Ω is bijective. Since by assumption G0 is an inner form, for all γ ∈ Γ we have εγ = 1, hence sγ = 1. Thus Γ acts trivially on Ω and on D. Let CF(Y ′ ) denote the colored fan of Y ′ (see Knop [Kn89] or Perrin [Pe14, Definition 3.1.9]) which is a set of colored cones (C, F) ∈ CF(Y ′ ), where C ⊂ V and F ⊂ D. We know that Γ acts trivially on V = HomZ (X , Q) and on D. It follows that for any γ ∈ Γ and for any colored cone (C, F) ∈ CF(Y ′ ), we have (45) γ∗ (C) = C, γ∗ (F) = F. It follows that the colored fan CF(Y ′ ) is Γ-stable. Moreover, it follows from (45) that the hypothesis of Theorem 2.26 of Huruguen [Hu11] is satisfied, that is, Y ′ has a covering by G-stable and Γ-stable open quasi-projective subvarieties. By this theorem Y ′ admits a G0 -equivariant k0 -model compatible with Y0 .  Remark 11.2. Huruguen [Hu11] assumes that Y0 has a k0 -point, but he does not use that assumption. Remark 11.3. In Theorem 11.1 we do not assume that Y ′ is quasi-projective. Appendix A. Algebraically closed descent for spherical homogeneous spaces The proofs in this appendix were communicated to the author by experts. Theorem A.1. Let G0 be a connected reductive group defined over an algebraically closed field k0 of characteristic 0. Let k ⊃ k0 be a larger algebraically closed field. We set G = G0 ×k0 k, the base change of G0 from k0 to k. Let H ⊂ G be a spherical subgroup of G (defined over k). Then H is conjugate to a (spherical) subgroup defined over k0 . Proof. The theorem will be proved in five steps. 1) Let X0 be a variety equipped with an action of G0 . Then X0 is the disjoint union of locally closed G0 -stable subvarieties X0m consisting of all orbits of a fixed dimension m. The orbits of maximal (resp. minimal) dimension form an open (resp. closed) subvariety. To see this, consider the product G0 × X0 and the subvariety Y0 consisting of the pairs (g, x) such that gx = x, equipped with the projection to X0 . By a theorem of Chevalley, see EGA [Gr66, 13.1.3], the dimension of fibers of this projection is an upper semicontinuous function on Y0 . Restricting this function to X0 (viewed as the closed subvariety of Y0 on which g = e), it follows that the dimension of the G0 -orbit is an upper semicontinuous function on X0 . 2) Take for X0 the variety of Lie subalgebras of g0 = Lie G0 of a fixed codimension, say r, and let X be the k-variety obtained from X0 by scalar extension. Then X is the variety of Lie subalgebras of codimension r in g = Lie G. Moreover, the stabilizer of a k-point h in X is the normalizer of h (viewed as a Lie algebra) in G. So the dimension of the orbit of h is dim(G) − dim NG (h) = dim(G) − dim(h) − (dim NG (h) − dim(h)) = r − dim ng (h)/h, where ng (h) denotes the normalizer of h in g. Thus, if there exists h such that h = ng (h), then the Lie subalgebras h satisfying this property are the k-points of the open subset of orbits of maximal dimension. Note that ng (h) is the Lie algebra of NG (h). So if h = ng (h), then h is an algebraic Lie algebra. EQUIVARIANT MODELS OF SPHERICAL VARIETIES 33 3) Let H be a spherical subgroup of G with Lie algebra h such that ng (h) = h. We claim that the orbit G · h in X is open. To prove this, recall that G · h has maximal dimension among G-orbits in X. Since h is a spherical Lie subalgebra, all Lie subalgebras h′ in an open neighborhood U of h in X are spherical and their orbits are of the same (maximal) dimension; thus, NG (h′ ) is a spherical subgroup of G, with Lie algebra h′ by Step 2. By Theorem 3.1 of Alexeev and Brion [AB05], only finitely many conjugacy classes of spherical subgroups of the form NG (h′ ) for h′ ∈ U are obtained in this way; let H1 , . . . , Hr be representatives of the conjugacy classes. We write hi = Lie Hi , then we see that any the spherical Lie algebra h′ ∈ U is conjugate to one of h1 , . . . , hr ; in particular, U intersects nontrivially with finitely many G-orbits in X, and all these orbits are of the same dimension. It follows that all these orbits are open, in particular, the orbit G · h in X is open. 4) By Step 3, the Lie algebras h of spherical subgroups H of G such that ng (h) = h form finitely many G-orbits, and the closures of these orbits are irreducible components of the variety X, which is defined over k0 . It follows that every such orbit is defined over k0 . Since k0 is algebraically closed, every such G-orbit has a k0 -point, which proves the theorem for spherical subgroups such that ng (h) = h. Also NG (h) = NG (H 0 ) = NG (H), where the latter equality follows from Corollary 5.2 of Brion and Pauer [BP87]. Thus the condition that ng (h) = h is equivalent to the condition that NG (H)/H is finite. 5) To handle the case of an arbitrary spherical k-subgroup H of G, consider the spherical closure of H, that is, the algebraic subgroup H of NG (H) containing H such that H/H = ker [NG (H)/H → Aut D(G/H)] . By Corollary A.3 below, the spherical closure H is spherically closed, that is, NG (H)/H acts faithfully on the finite set of colors of G/H, hence the group NG (H)/H is finite, and therefore, ng (Lie H) = Lie H. By Step 4 we may assume that H is defined over k0 . Now H is an intersection of kernels of characters of H (since the quotient H/H is diagonalizable) and every such character is defined over k0 . Thus H is defined over k0 , as required. An alternative proof, also based on Theorem 3.1 of Alexeev and Brion [AB05], is sketched in Knop’s MathOverflow answer [Kn17b].  From now on till the end of this appendix we follow Avdeev [Av15]. Let G be a connected reductive group over an algebraically closed field k of characteristic 0. Fix a e → G such that G e is a direct product of a torus with a simply finite covering group G e connected semisimple group. For every simple G-module V , the corresponding projective space P(V ) has the natural structure of a G-variety. Every G-variety arising in this way is said to be a simple projective G-space. Proposition A.2 (Bravi and Luna [BL11, Lemma 2.4.2], see also Avdeev [Av15, Corollary 3.24]). For any spherical subgroup H of a connected reductive group G over an algebraically closed field k of characteristic 0, the spherical closure H of H is the common stabilizer in G of all H-fixed points in all simple projective G-spaces. Corollary A.3 (well known). Let H be a spherical subgroup of a connected reductive group G defined over an algebraically closed field k of characteristic 0. Let H ′ denote the spherical closure of H, and let H ′′ denote the spherical closure of H ′ . Then H ′′ = H ′ , that is, H ′ is spherically closed. This result was stated without proof in Section 6.1 of Luna [Lu01] (see also Avdeev [Av15, Corollary 3.25]). 34 MIKHAIL BOROVOI WITH AN APPENDIX BY GIULIANO GAGLIARDI Deduction of Corollary A.3 from Proposition A.2. Let P(V ) be a simple projective G-space. Let P(V )H denote the set of fixed points of H in P(V ). Since H ⊂ H ′ , we have ′ ′ P(V )H ⊂ P (V )H . By Proposition A.2 applied to H, we have P(V )H ⊃ P (V )H . Thus ′ P(V )H = P(V )H . By Proposition A.2 applied to H ′ , the group H ′′ (k) is the set of g ∈ G(k) that fix ′ P(V )H for all simple projective G-spaces P(V ). By Proposition A.2 applied to H, the group H ′ (k) is the set of g ∈ G(k) that fix P(V )H for all simple projective G-spaces P(V ). ′  Since P(V )H = P(V )H , we have H ′′ = H ′ , as required. Appendix B. The action of the automorphism group on the colors of a spherical homogeneous space By Giuliano Gagliardi In this appendix we prove Theorem 7.7, which we restate below as Theorem B.4. Our proof is based on Friedrich Knop’s MathOverflow answer [Kn17a] to Borovoi’s question. Knop writes that Theorem B.4 was communicated to him by Ivan Losev. Let G be a connected reductive group over an algebraically closed field k of characteristic 0. Let Y = G/H be a spherical homogeneous space. We present results of Knop [Kn96] and Losev [Lo09] describing AutG (Y ), using the notation of Section 7. Consider the uniquely determined set Σ ⊂ X of linearly independent primitive elements in the lattice X such that \ {v ∈ V : hγ, vi ≤ 0}. V= γ∈Σ The elements of Σ are called the spherical roots of Y . For the description of AutG (Y ) a related subset Σ ⊂ X is used, see Losev [Lo09, Introduction, before Theorem 2]. Losev has shown how Σ can be computed from Σ and D. Theorem B.1 (Knop [Kn96, Theorem 5.5]). For every φ ∈ AutG (Y ) and every χ ∈ X there exists aφ,χ ∈ k× such that φ|K(Y )(B) = aφ,χ id. χ The resulting homomorphism AutG (Y ) → Hom(X , k× ) is injective and its image is {ψ ∈ Hom(X , k× ) : ψ(Σ) = {1}}. For α ∈ S, let D(α) denote the set of colors D ∈ D such that the parabolic subgroup Pα moves D, that is, α ∈ ς(D). We need the following results of Luna [Lu97], [Lu01]: Proposition B.2. Let α ∈ S. (1) We have |D(α)| ≤ 2. Moreover, |D(α)| = 2 if and only if α ∈ Σ ∩ S. (2) Assume |D(α)| = 2 and write D(α) = {Dα+ , Dα− }. If ρ(Dα+ ) = ρ(Dα− ), then: (i) hρ(Dα+ ), χi = hρ(Dα− ), χi = 21 hα∨ , χi for all χ ∈ X , where α∨ ∈ X∗ (T ) is the corresponding simple coroot; (ii) we have ς(Dα+ ) = ς(Dα− ) = {α}. Proof. For (1), see Luna [Lu97, Sections 2.6 and 2.7] or Timashev [Tim11, 30.10]. For (2), we use that [Lu01, Theorem 2] or [Tim11, Theorem 30.22] implies that the invariants of a spherical homogeneous space satisfy the axioms of a homogeneous spherical datum. These axioms are stated in [Lu01, Sections 2.1 and 2.2] and [Tim11, Definition 30.21]. In particular, we have ρ(Dα+ ) + ρ(Dα− ) = α∨ |X and for every β ∈ ς(Dα± ) we have β ∈ X and hρ(Dα± ), βi = 1. With the assumption ρ(Dα+ ) = ρ(Dα− ), we obtain (i) and then (ii).  EQUIVARIANT MODELS OF SPHERICAL VARIETIES 35 We need the following result of Losev: Proposition B.3. The set Σ has the following properties: (1) The set Σ can be obtained from Σ by replacing some elements γ ∈ Σ by 2γ and leaving the other elements γ ∈ Σ unchanged. (2) If α ∈ Σ ∩ S and hρ(Dα+ ), χi = hρ(Dα− ), χi = 21 hα∨ , χi for all χ ∈ X , then 2α ∈ Σ (hence α ∈ / Σ). Proof. See Losev [Lo09, Theorem 2 and Definition 4.1.1(1)].  The following theorem is the main result of this appendix: Theorem B.4 (Losev, unpublished). The homomorphism (46) AutG (Y ) → AutΩ (D) is surjective. Proof of Theorem B.4. Let A denote the set of simple roots α ∈ S such that |D(α)| = 2 and ρ(Dα+ ) = ρ(Dα− ). By Proposition B.2, for every α ∈ A we have ς(Dα+ ) = ς(Dα− ) = {α}, hence the map α 7→ {Dα+ , Dα− } is a bijection between A and the set of unordered pairs {Dα+ , Dα− } such that (ρ × ς)(Dα+ ) = (ρ × ς)(Dα− ). Note that there is a canonical bijection A → Ω(2) , α 7→ (ρ × ς)(Dα+ ). By Proposition B.2(1), for every α ∈ A we have α ∈ S ∩ Σ ⊂ X , hence there exists (B) fα ∈ K(Y )α , fα 6= 0. Moreover, from Propositions B.2 and B.3 we obtain 2α ∈ Σ (and α∈ / Σ). We want to show that for any α ∈ A there exists φα ∈ AutG (Y ) such that φα swaps and Dα− , but fixes all Dβ+ and Dβ− for β ∈ A, β 6= α. Dα+ Let α ∈ A. Since the field k is algebraically closed and the set Σ ⊂ X is linearly independent, we can construct a homomorphism ψα : X → k× with ψα (α) = −1 and ψα (γ) = 1 for every γ ∈ Σ r {α}, and such that ψα is of finite order in the group Hom(X , k× ). Then we have ψα (Σ) = {1}. By Theorem B.1 there exists an automorphism of finite order φα ∈ AutG (Y ) with ( −fβ for β = α, φα (fβ ) = (47) fβ for β ∈ A r {α}. e ⊂ NG (H) denote the subgroup containing H such that Let H e H/H = hφα i ⊂ NG (H)/H = AutG (Y ), e We use the where hφα i denotes the finite subgroup generated by φα . We set Ye = G/H. same notation for the combinatorial objects associated to the spherical homogeneous space Ye as for Y , but with a tilde above the respective symbol. The morphism of G-varieties Y → Ye induces an embedding K(Ye ) ֒→ K(Y ), and K(Ye ) is the fixed subfield of φα . Since K(Ye ) is a G-invariant subfield of K(Y ), we have K(Ye )(B) ⊂ K(Y )(B) and Xe ⊂ X . By (47) we have φα (fα ) = −fα 6= fα . We see that fα ∈ / K(Ye ), hence α ∈ X r Xe, in e e particular α ∈ / Σ. By Proposition B.2(1) we have |D(α)| ≤ 1, hence the two colors in D(α) are mapped to one color by the map Y → Ye , that is, φα swaps Dα+ and Dα− . On the other hand, for any β ∈ Ar{α}, by (47) we have φα (fβ ) = fβ , hence fβ ∈ K(Ye ) and β ∈ Xe. Since β is a primitive element of X , it is a primitive element of Xe ⊂ X . The e (see Knop natural map V → Ve induced by Y 7→ Ye is bijective and identifies V and V e = −V. [Kn89, Section 4]). Since β ∈ Σ is dual to a wall of −V, it is dual to a wall of −V 36 MIKHAIL BOROVOI WITH AN APPENDIX BY GIULIANO GAGLIARDI e hence |D(β)| e It follows that β ∈ S ∩ Σ, = 2, and the two colors in D(β) are mapped to e  distinct colors under Y → Y , that is, φα fixes Dβ+ and Dβ− . References [Akh15] Dmitri Akhiezer, Satake diagrams and real structures on spherical varieties, Internat. J. Math. 26 (2015), no. 12, 1550103, 13 pp. [ACF14] Dmitri Akhiezer and Stéphanie Cupit-Foutou, On the canonical real structure on wonderful varieties, J. reine angew. Math., 693 (2014), 231–244. [AB05] Valery Alexeev and Michel Brion, Moduli of affine schemes with reductive group action, J. Algebraic Geom. 14 (2005), 83–117. [Av11] Roman Avdeev, On solvable spherical subgroups of semisimple algebraic groups, Trans. Moscow Math. Soc. 2011, 1–44. [Av13] Roman Avdeev, Normalizers of solvable spherical subgroups, Math. Notes 94 (2013), 20–31. 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Simulation studies on the design of optimum PID controllers to suppress chaotic oscillations in a family of Lorenz-like multi-wing attractors Saptarshi Dasa,b, Anish Acharyac,d, and Indranil Pana a) Department of Power Engineering, Jadavpur University, Salt Lake Campus, LB-8, Sector 3, Kolkata-700098, India. b) Communications, Signal Processing and Control (CSPC) Group, School of Electronics and Computer Science, University of Southampton, Southampton SO17 1BJ, United Kingdom. c) Department of Instrumentation and Electronics Engineering, Jadavpur University, Salt Lake Campus, LB-8, Sector 3, Kolkata-700098, India. d) Department of Electrical and Computer Science Engineering, University of California Irvine, CA 92697-2620, USA. Authors’ Emails: [email protected], [email protected] (S. Das) [email protected] (A. Acharya) [email protected], [email protected] (I. Pan) Abstract: Multi-wing chaotic attractors are highly complex nonlinear dynamical systems with higher number of index-2 equilibrium points. Due to the presence of several equilibrium points, randomness and hence the complexity of the state time series for these multi-wing chaotic systems is much higher than that of the conventional double-wing chaotic attractors. A real-coded Genetic Algorithm (GA) based global optimization framework has been adopted in this paper as a common template for designing optimum Proportional-Integral-Derivative (PID) controllers in order to control the state trajectories of four different multi-wing chaotic systems among the Lorenz family viz. Lu system, Chen system, Rucklidge (or Shimizu Morioka) system and Sprott-1 system. Robustness of the control scheme for different initial conditions of the multi-wing chaotic systems has also been shown. Keywords - chaos control; chaotic nonlinear dynamical systems; Lorenz family; multiwing attractor; optimum PID controller 1. Introduction Chaos is a field in mathematical physics which has found wide applications, ranging from weather prediction to geology, mathematics, biology, computer science, economics, philosophy, sociology, population dynamics, psychology, robotics etc. [29]. Chaos theory studies the behavior of dynamical systems which are nonlinear, highly sensitive to changes in initial conditions, and have deterministic (rather than probabilistic) underlying rules which dictates the evolution of the future states of a 1 system. Such systems exhibit aperiodic oscillations in the time series of the state variables and long term prediction of such systems is not possible due to the sensitive dependence on initial conditions. Theoretical investigations of chaotic time series have percolated into real world applications like those in computer networks, data encryption, information processing systems, pattern recognition, economic forecasting, stock market prediction etc. [29]. Historically, chaos was first observed in natural convection [16], weather prediction and subsequently in several other fluid mechanics related problems [2]. Survey of a wide variety of physical systems exhibiting chaos and pattern formation in nature ranging from convection, mixing of fluids, chemical reaction, electrohydrodynamics to solidification and nonlinear optics have been summarized in [10]. An interesting ramification of chaos theory is the investigation of synchronization and control of such systems [19]. Even with their unpredictable dynamics, proper control laws can be designed to suppress the undesirable excursions of the state variables or make the state variables follow a definite trajectory [35]. These chaos control schemes have a variety of engineering applications and hence control of chaotic systems has received increased attention in last few years [5]. In chaos control, the prime objective is to suppress the chaotic oscillations completely or reduce them to regular oscillations. Recent chaos control techniques include open loop control methods, adaptive control methods, traditional linear and nonlinear control methods, fuzzy control techniques etc. among many others [5]. Most chaos control techniques exploit the fact that any chaotic attractor containing infinite number of unstable periodic orbits can be modified using an external control action to produce a stable periodic orbit. The chaotic system’s states never remains in any of these unstable orbits for a long time but rather it continuously switches from one orbit to the other which gives rise to this unpredictable, random wandering of the state variables over a longer period of time. The control scheme basically tries to achieve stabilization, by means of small system perturbations, of one of these unstable periodic orbits. The result is to render an otherwise chaotic motion more stable and predictable. The perturbation must be tiny, to avoid significant modification of the system’s natural dynamics. Several techniques have been used to control chaos, but most of them are developments of two basic approaches: the OGY (Ott, Grebogi and Yorke) method [20], and Pyragas continuous control method [22]. Both methods require a prior determination of the unstable periodic orbits of the chaotic system before the control algorithm can be designed. The basic difference between the OGY and Pyragas methods of chaos control is that the former relies on the linearization of the Poincare map and the latter is based on time delay feedback. PID controller based designs are popular in the control engineering community for several decades, due to its design simplicity, ease of use, and implementation feasibility. PID type controller design can be found in recent literatures for synchronization of chaotic systems with different initial guess [1, 3, 7-9, 14, 15]. However, except few attempts like [4], optimum PID controller design for chaos control is not well investigated yet, especially for the control of highly complex chaotic systems like multi-wing attractors, as attempted in this paper. The rationale behind particularly choosing PID controllers is due to its simplicity, ease of implementation and tuning methods compared to unavailability of Lypunov based stabilization and sliding mode techniques [19, 35] for multi-wing chaotic attractors. The classical approach of dealing with chaos control and synchronization is to formally eliminate the nonlinear terms for 2 designing a suitable nonlinear control scheme, in order to make the error dynamical system linear and then find a suitable Lypunov based stabilization scheme [19]. The present approach needs no additional requirement of canceling complex multi-segment nonlinear terms by a nonlinear control scheme for each chaotic system. The success of the present approach lies in the fact that a generalized linear PID control framework can stabilize a family of complex multi-wing chaotic system when only the second state variable needs to be sensed and manipulated using an external PID control action. The controller gains are optimized to damp the oscillations as early as possible and the control scheme is also tested for required robustness against system’s initial condition variation. Previously, design of PID type controllers equipped with fractional order integrodifferential operators and fuzzy logic have been researched for chaos synchronization [11] and chaos control [21], within a common global optimization framework without paying much attention to the specific nonlinear structure and complexity of the chaotic system. This idea has been extended here for the control of highly complex nonlinear dynamical systems i.e. multi-wing chaotic attractors in the Lorenz family. Many researches have focused on theoretical and experimental studies on the synchronization and control of classical double-wing chaotic and hyper-chaotic systems from the Lorenz family like the Lorenz, Chen and Lu attractors [19, 35, 5, 12]. However, there is almost no significant result for relatively complex chaotic systems like the multiscroll and multi-wing attractors which is addressed in the present paper. Multi-scroll attractors first emerged as an extension of the Chua’s circuit [31] and it has been found to be more suitable than the classical double-wing chaotic attractors in applications like secure communication and data encryption. The multi-scroll attractors are produced by suitably modifying the nonlinear terms of the governing differential equations of the chaotic system which has been successfully applied to produce 1D, 2D and 3D scroll grid attractors in [17]. The reason behind obtaining highly complex time-series out of multiscroll and multi-wing attractors is that they have more number of equilibrium points than the classical double-wing attractors. Recent research has shown that such highly complex time-series and phase space behavior can also be obtained from typical chaotic and hyper-chaotic systems with no equilibrium point [27] and infinite number of equilibrium points [36]. Theoretical studies on the complexity analysis of such multi-wing attractors have been reported in He et al. [13] using spectral entropy and statistical complexity measure. Among the two major families of complex chaotic systems, the control and synchronization of multi-scroll attractors are proposed in [30] but there is almost no result for the multi-wing attractors. The purpose of the present study is to develop a common template to suppress highly complex chaotic oscillations in the Lorenz family of multi-wing attractors using a simple controller structure like the PID and a global optimization based gain tuning mechanism. An objective function for the controller design problem is framed in terms of the state variables of the chaotic system and the desired state trajectories of the system. The real coded GA based optimization is then employed to find out the values of the PID controller gains. 2. Basics of the Lorenz family of multi-wing chaotic systems Four classical examples of symmetric double-wing chaotic attractors are chosen as the base cases here among the Lorenz family to study the control of multi-wing attractors [32]. In spite of having several literatures on the generation of multi-scroll attractors as an extension of Chua’s circuit, the first extension of the Lorenz family of 3 attractors from double-wing to multi-wing was proposed by Yu et al. [32]. It is also observed that in the Lorenz family of chaotic systems, similar attractors could be generated by replacing the cross-products with quadratic terms. The common characteristics of the source of nonlinearity in the state equations of the original doublewing Lorenz family of chaotic systems are due to having either a square and/or crossterms of the state variables. These particular terms can be replaced by a multi-segment parameter adjustable quadratic function (1) to generate multi-wing attractor with additional flexibility of modifying the numbers and locations of index-2 equilibrium points. As reported in the pioneering works of multi-wing chaos [32]-[33], the segment characteristics like the slope and width can be adjusted using the parameters { F0 , Fi , Ei } of equation (1). This typical function increases the number of index-2 equilibrium points along a particular axis for Lorenz family of chaotic systems from two to ( 2 N + 2 ) , thereby increasing the randomness or complexity of the state trajectories of the nominal (double-wing) chaotic system which is difficult to control by analytical methods. N f ( x ) = F0 x 2 − ∑ Fi 1 + 0.5sgn ( x − Ei ) − 0.5sgn ( x + Ei )  (1)  1 for x > 0  sgn ( x ) =  0 for x = 0 −1 for x < 0  (2) i =1 where, and N being a positive integer responsible for the number of wings of the chaotic attractor. In this paper, two multi-wing chaotic systems among the Lorenz family are obtained by replacing the cross-product terms (in Lu and Chen system) by the above mentioned multi-segment function. In the other two systems among the Lorenz family, the quadratic terms (in Rucklidge and Sprott-1 system) are replaced by the multi-segment function (1). 2.1. Chaotic multi-wing Lu system The double-wing chaotic Lu system [18] is represented by (3). xɺ = − ax + ay yɺ = cy − xz (3) zɺ = xy − bz The typical parameter settings for chaotic double-wing Lu attractor is given as a = 36, b = 3, c = 20 . The equilibrium points of the Lu system are located at ( 0, 0, 0 ) ; ( ± ) bc , ± bc , c . The state equations of the multi-wing chaotic Lu attractor whose states are to be controlled using a PID controller are given by (4). xɺ = − ax + ay yɺ = cy − (1 P ) xz + u zɺ = f ( x ) − bz 4 (4) Here, P reduces the dynamic range of the attractors so as to facilitate hardware realization. The suggested parameters for N = 4 are given in (5) [32] for which the chaotic Lu system exhibits multi-wing attractors in the phase portraits. P = 0.05, F0 = 100, F1 = 10, F2 = 12, F3 = 16.67, F4 = 18.18, (5) E1 = 0.3, E2 = 0.45, E3 = 0.6, E4 = 0.75 Here, the original cross product ( xy ) is replaced by the multi-segment quadratic function f ( x ) . Fig. 1 shows the projections of the 3-dimensional (x-y-z) phase space dynamics (Fig. 1d) of the multi-wing Lu system on the x-y (Fig. 1a), y-z (Fig. 1b), x-z (Fig. 1c) planes. Also, in (4) the PID control action is added to the second state variable to suppress the chaotic oscillations and the control action is given by (6). de u = K p e + K i ∫ e.dt + K d , e = (r − y) (6) dt Here, { K p , Ki , K d } are the PID controller gains which are to be found out by a suitable optimization technique for the reference signal ( r ) as a unit step. Fig. 1. Uncontrolled phase plane portraits for multi-wing Lu system [32]. 2.2. Chaotic multi-wing Chen system The double-wing Chen system [6] is represented by (7). xɺ = − ax + ay yɺ = ( c − a ) x + cy − xz (7) zɺ = xy − bz The typical parameter settings for chaotic double-wing Chen attractor is given as a = 35, b = 3, c = 28 . The equilibrium points of the Chen system are located at ( 0, 0, 0 ) ; ( ± b ( 2c − a ) , ± b ( 2c − a ) , ( 2c − a ) ) . The state equations of the multi-wing chaotic Chen attractor whose states are to be controlled are given by (8). 5 xɺ = −ax + ay yɺ = ( c − a ) x + cy − (1 P ) xz + u (8) zɺ = f ( x ) − bz The suggested parameters for N = 4 are given in (9) for which the chaotic Chen system exhibits multi-wing attractors in phase portraits. P = 0.05, F0 = 100, F1 = 12, F2 = 12, F3 = 16.67, F4 = 18.75, (9) E1 = 0.3, E2 = 0.45, E3 = 0.6, E4 = 0.75 The multi-wing Chen system in Fig. 2 shows similar phase space behavior to that of the multi-wing Lu system in Fig. 1, except the fact that more number of trajectories are observed away from the equilibrium points. Similar to the previous case of multi-wing Lu system, the nonlinear cross-product ( xy ) in the third state variable is replaced by f ( x ) . Fig. 2. Uncontrolled phase plane portraits for multi-wing Chen system [32]. 2.3. Chaotic Multi-wing Rucklidge system The double-wing Shimizu-Morioka system [24] is given by (10). xɺ = −ax + by − yz yɺ = x (10) zɺ = y 2 − z The typical parameter settings for chaotic double-wing Shimizu-Morioka attractor is given as a = 2, b = 7.7 . The equilibrium points of the Shimizu-Morioka system are located ( ) at ( 0, 0, 0 ) ; 0, ± b , b . The state equations of the multi-wing chaotic Shimizu-Morioka attractor whose states are to be controlled are given by (11). xɺ = − ax + by − (1 P ) yz yɺ = x + u (11) zɺ = f ( y ) − z 6 The above mentioned multi-wing chaotic Shimizu-Morioka system [34] is also known as modified Rucklidge system [23]. The suggested parameters for N = 3 are given in (12) [32] for which the chaotic Rucklidge system exhibits multi-wing attractors in the phase portraits (Fig. 3). (12) P = 0.5, F0 = 4, F1 = 9.23, F2 = 12, F3 = 18.18, E1 = 1.5, E2 = 2.25, E3 = 3.0 Here in the case of Rucklidge system, the quadratic term ( y 2 ) (unlike the cross-product of states in previous cases of Lu and Chen system) is replaced by the multi-segment function f ( y ) in the third state equation. Fig. 3. Uncontrolled phase plane portraits for multi-wing Rucklidge (Shimizu-Morioka) system [32]. 2.4. Chaotic Multi-wing Sprott-1 system The double-wing Sprott-1 system [25]-[26] is given by (13). xɺ = yz yɺ = x − y (13) zɺ = 1 − x 2 The equilibrium points of the Sprott-1 system are located at ( ±1, ±1, 0 ) . State equations of the multi-wing chaotic Sprott-1 attractor whose states are to be controlled are given by (14). xɺ = yz (14) yɺ = x − y + u zɺ = 1 − f ( x ) The suggested parameters for N = 4 are given by (15) [32] for which the chaotic Sprott-1 system exhibits multi-wing attractors in phase portraits (Fig. 4). (15) F0 = 1, F1 = 5, F2 = 5, F3 = 6.67, F4 = 8.89, E1 = 2, E2 = 3, E3 = 4, E4 = 5 7 Here, the quadratic term ( x 2 ) is replaced by the multi-segment function f ( x ) similar to the previous case of Rucklidge system. It is evident from the phase portraits of all the four multi-wing chaotic systems that the wandering of the states in phase space are highly complex, thus indicating the corresponding state time series being highly jittery which is hard to regularize using analytically derived external control action. Here, we have shown that a simple PID type linear controller structure which is widely used in industrial process control is capable of suppressing chaotic oscillations in such complex dynamical systems. Fig. 4. Uncontrolled phase plane portraits for multi-wing Sprott-1 system [32]. 3. Optimum PID control to suppress chaotic oscillations in multi-wing attractors Each of the above four classes of multi-wing chaotic systems are to be controlled using a PID controller (6) which will enforce the second state variable ( y ) to track a unit reference step input ( r ). Instead of simple error minimization criteria for PID controller tuning, the well-known Integral of Time multiplied Absolute Error (ITAE) criterion has been used as the performance index J (16) in order to ensure fast tracking of the second state variable with lesser oscillations. T J = ∫ t ⋅ e ( t ) dt , e ( t ) = r ( t ) − y ( t ) (16) 0 For time domain simulations, the upper time limit of the above integral ( T ) is restricted to realistic values depending on the speed of the chaotic time series to ensure that all oscillations in the state variables have died down due to introduction of the PID control action in the second state. It has also been shown through simulation examples that controlling the second state variable with PID automatically damps chaotic oscillations in the other two state variables for the Lorenz family of multi-wing attractors. Tuning of the PID controller gains have been done in this study using the widely used population based global optimizer known as the real coded genetic algorithm [28]. The GA is a stochastic optimization process which can be used to minimize a chosen 8 objective function. A solution vector is initially randomly chosen from the search space and undergoes reproduction, crossover and mutation, in each generation, to give rise to a better population of solution vectors in the next generation. Reproduction implies that solution vectors with higher fitness values can produce more copies of themselves in the next generation. Crossover refers to information exchange based on probabilistic decisions between solution vectors. In mutation a small randomly selected part of a solution vector is occasionally altered, with a very small probability. This way the solution is refined iteratively until the objective function is minimized below a certain tolerance level or the maximum number of generations are exceeded. In this study, the population size in GA is chosen to be 20. The crossover and mutation fraction are chosen to be 0.8 and 0.2 respectively for the present minimization problem using (16). Due to the randomness of the chaotic time series of the multi-wing attractors, the error signal with respect to step command input also becomes highly jittery and will contain several minima which justifies the application of GA in such controller tuning problems as compared to other gradient based optimization algorithms. GA being a global optimization algorithm is able to get out of the local minima, whereas the other gradient based optimization algorithms often get stuck in the local minima and cannot give good solutions. For the control and synchronization of chaotic systems, an evolutionary and swarm based PID controller design with other time domain performance index optimization have been used previously as reported in [11], [21]. But for the sake of simplicity, we have restricted the study for ITAE based PID design only to handle multi-wing attractors in chaotic nonlinear dynamical systems. The GA based optimization results for the PID controller parameters (gains) are given in Table 1 for the four respective multi-wing attractors among the Lorenz family. Also, in order to ensure that the best possible solution is found in the global optimization process, the algorithm has been run several times and only the controller gains, resulting in the fastest tracking performance (and hence the lowest value of Jmin) for respective systems are reported here. Table 1: GA Based Optimum PID Controller Settings for Chaos Suppression in Multiwing Attractors Multi-wing Chaotic systems Jmin Kp Ki Kd Lu system 244.986 3.156 27.562 1.449 Chen system 307.709 3.305 21.274 1.591 Rucklidge system 1.161 19.435 30.097 0.237 Sprott-1 system 1468.193 0.272 0.433 0.393 In all cases except the Sprott-1 system, the integral gains ( K i ) take high value signifying that the controller gives extra effort to reduce the steady state error using the integral action. Also, due the sluggish nature of the uncontrolled dynamics of Rucklidge system, the proportional gain ( K p ) also takes high value to make the overall closed loop system faster. It is obvious that such controller gains justifies the final objective of ITAE (16) that puts extra penalty for sluggish controlled response and pushes the system to reduce oscillations as soon as possible. A simpler Integral of Absolute Error (IAE) error criterion without the temporal effect in the objective function, unlike ITAE, would have yielded different controller parameters since it only enforces tracking of the set-point but not the fastest possible set-point tracking. 9 3.1. PID Control of Multi-wing Lu System Fig. 5. Uncontrolled and PID controlled response of the state variable for multi-wing Lu system (a) state-x (b) state-y (c) state-z. Simulation results for the uncontrolled and PID controlled state variables of the multi-wing Lu system (4) has been shown in Fig. 5 with the corresponding control signal and error of the second state depicted in Fig. 6. In Fig. 5(a)-(c), it can be observed that the chaotic evolution of the state variables in the uncontrolled state disappears when the PID controller is applied to the system. All the controlled states evolve to a steady state value. It is evident that the second state variable of the multi-wing Lu system not only tracks the unit step reference but the randomness in the other two state variables is also stabilized. As can be observed from Fig. 5, the controller achieves its design objective of suppressing the oscillations in the second state and making it track the desired trajectory of a unit step. Although the oscillations in the other states x and z are suppressed, the final value at which they settle is dependent on the controller parameters and cannot be obtained a-priori. A more customized control action could have been incorporated like that in [11] utilizing the sum of error for all the three state variables with the intention of getting set-point tracking in all the three state variables. But it would unnecessarily increase the control action which will create difficulty in implementing the control scheme in an analog circuit. A closer look on the set-point tracking error in Fig. 6 shows that the error asymptotically goes to zero and correspondingly the PID control action stabilizes to its final value. This essentially implies that in the steady state, the controller has to continuously give a constant signal to the second state of the chaotic system so that the error remains zero and the oscillations are suppressed. 10 Fig. 6. Control signal and the second state error for the PID controlled multi-wing Lu system. Also, the time series of the controlled states in Fig. 5(a)-(c) show initial oscillations which get damped quite fast even with a simple GA based optimum PID controller having a linear structure, though better performance can be expected at the cost of implementation of complex nonlinear controller structures and associated computational complexity like that used in [21]. 3.2. PID control of multi-wing Chen system Fig. 8. Uncontrolled and PID controlled response of the state variables for multi-wing Chen system (a) state-x (b) state-y (c) state-z. 11 Fig. 8 shows the controlled three state trajectories of the multi-wing Chen system (8). The associated error signal and control signal (Fig. 9) are quite similar to that of the Lu system. Damping of the chaotic oscillations in the PID controlled phase space is shown in Fig. 10 for this particular system among the Lorenz family. Similar to the case of Lu system, control of Chen system also shows initial jerk in the state trajectories which finally settles down asymptotically. A comparison can be made between the control efforts of Fig. 6 and Fig. 9. It can be observed that the area under the curve is less for Fig. 9 (a) than Fig. 6 (a). This implies that the overall control effort required for controlling the multi-wing Chen system is less than that required for controlling the multi-wing Lu system. Fig. 9. Control signal and the second state error for the PID controlled multi-wing Chen system. 3.3. PID control of multi-wing Rucklidge system Similar nature of chaos control can be obtained in the multi-wing Rucklidge system (11) also with the GA based optimum PID controller which enforces fast tracking of the second state variable. Also the irregular oscillations of this system are found to be more sluggish compared to the Lu system which is controlled by the PID to track a reference input using ITAE criterion. Also, controlled state trajectories are smoother at initial stages unlike that for the Lu and Chen system. Here, Fig. 11 shows the respective uncontrolled and controlled three state trajectories. The control and error signals are shown in Fig. 12. A comparison can be made between Fig. 11 and Figs. 5 and 8. It can be seen that the oscillations in the multi-wing Rucklidge system is suppressed within a few seconds of applying the control signal, unlike that in the two former cases. A comparison of the control and error signals in this case (Fig. 12) as compared to the previous cases (Figs. 6 and 9) show that the steady state control signal required in this case is much smaller than the former two. The error goes to zero within the first 10 seconds while it takes almost 200 seconds in the former two cases. Therefore it can be concluded that the 12 Rucklidge system is comparatively easier to control using a simple linear PID controller structure than the previously studied chaotic systems i.e. Lu and Chen system. Fig. 11. Uncontrolled and PID controlled response of the state variables for multi-wing Rucklidge system (a) state-x (b) state-y (c) state-z. Fig. 12. Control signal and the second state error for the PID controlled multi-wing Rucklidge system. 3.4. PID control of multi-wing Sprott-1 system For the multi-wing Sprott-1 system (14), the ITAE based GA tuned PID enforces fast reference tracking and simultaneously damps chaotic oscillations (Fig. 14) in an efficient way. It can be noted that in this case, although the original objective of making 13 the second state (y) follow a step function is achieved, there are small oscillations in the x and the z states. This is unlike the previous three cases where controlling one of the states resulted in controlling all the other states automatically. If the oscillations in the other two states are desired to be minimized then they have to be explicitly taken into account in the objective function itself. It is also evident from the control and error signals in Fig. 15 (which shows the settling of the error signal to zero and control action approaching to its final value) that this system is easier to control than the multi-wing Chen and Lu system. But it is more difficult to control than the multi-wing Rucklidge system, in the sense that it takes more control effort than the former and also takes much more time to settle than the former. Fig. 14. Uncontrolled and PID controlled response of the state variables for multi-wing Sprott-1 system (a) state-x (b) state-y (c) state-z. It is well known that chaotic systems are highly sensitive to the initial conditions of the states. Since in the presented approach only a single initial condition of the state variables are assumed to tuned the GA based PID controllers with minimum ITAE for the second state, hence under different initial conditions, effective damping of chaotic oscillations need to be investigated. Therefore, study of the robustness of the present PID control scheme for effective control is shown in next section for each of the four classes of multi-wing chaotic systems. 14 Fig. 15. Control signal and the second state error for the PID controlled multi-wing Sprott-1 system. 4. Test of robustness for the PID control scheme with different initial conditions of state variables Fig. 17. Robustness of the PID controller for chaos suppression in multi-wing Lu system for different initial conditions. In this section, the proposed GA based PID control scheme has also been tested for its robustness with variation in the initial conditions of multi-wing chaotic systems. Question may arise whether the optimum PID control scheme would be able to stabilize the system for other initial values of the state variables, since the controller is tuned with 15 any one initial guess of the states. This is particularly important to investigate since the proposed approach does not rely on classical Lyapunov criterion based analytical stabilization which has been extensively studied for double wing attractors [19], [35], [5]. Fig. 18. Robustness of the PID controller for chaos suppression in multi-wing Chen system for different initial conditions. Fig. 17-20 shows that in the phase portraits the chaotic oscillations get suppressed along different trajectories for the four complex nonlinear dynamical systems, even if the initial conditions for the first two states are gradually decreased from unity to zero. In spite of high sensitivity to the initial conditions for the states in all chaotic systems, the proposed PID control scheme is capable of suppressing the wandering of the states in phase space for the Lu and Chen system which are quite similar and shown in Fig. 17-18. For the Rucklidge system as shown in Fig. 19, the controlled phase space trajectories converges towards a particular direction even with variation in initial condition of the system’s states. The controlled phase portraits for different initial conditions are much more complex for the Sprott-1 system as portrayed in Fig. 20 but finally converge to a stable equilibrium point, similar to the other multi-wing systems. A comparison of Figs 17 and 18 can be made with that of Fig. 19. It can be seen that in case of the Rucklidge system in Fig. 19, the state trajectories from the different initial conditions to the final equilibrium solution are much smaller than those of the Lu or the Chen systems (in Figs. 17 and 18 respectively). This also provides an insight into the amount of controller effort and settling time required to suppress the chaotic oscillations in these systems. As discussed previously, the Lu and the Chen systems are more difficult to control and this can also be understood from the phase portraits of the trajectories in Figs. 17 and 18. In these figures the trajectories evolve through multiple small scrolls and hence take more time than that of the Rucklidge system in Fig. 19. The trajectories of the controlled systems in Fig. 20 show that they take larger excursions than the Rucklidge system but do not get trapped in small scrolls, therefore making them 16 easier to control than the Lu and the Chen systems, but more difficult than the Rucklidge system. Fig. 19. Robustness of the PID controller for chaos suppression in multi-wing Rucklidge system for different initial conditions. Fig. 20. Robustness of the PID controller for chaos suppression in multi-wing Sprott-1 system for different initial conditions. 5. Discussion: It is well known that it is difficult to find a quadratic Lyapunov function in such complicated nonlinear dynamical systems like multi-wing chaotic attractors except few attempts like that in [30]. The present approach has been shown to work reliably well, even though analytical stabilization has not yet been addressed for such systems due to 17 high system complexity. The proposed optimum PID control scheme has another advantage over the conventional Lyapunov function based stabilization approach i.e. the optimum tracking of the desired state variable which is difficult to achieve with the classical Lyapunov approach. With slight modification of the objective function different states or even more than one state variable can easily be enforced to track different reference inputs [11], unlike the Lyapunov based approach. The Lyapunov and sliding mode approaches enforce guaranteed stabilization but not optimum set-point tracking and also the nonlinear control law needs to be changed every time, depending on the structure of the chaotic system which restricts the designer to have a common controller structure (like a simple PID here) to control all the multi-wing systems in the Lorenz family. Thus to the best of authors’ knowledge, this is the first approach to suppress chaotic oscillations in the family of multi-wing Lorenz family of chaotic systems with a simple control scheme augmented with a global optimization framework. Another important part of the present control scheme is the course of control action to derive the control law i.e. the effect of the upper limit of the integral objective function (16). Also, the trajectory of control for different chaotic systems plays an important role here with respect to the quality of the control. Depending on the number of equilibrium points and speed of oscillations for a particular chaotic system, the control action and the associated controller gains would vary widely within the same common PID structure. Also, the upper limit of the time domain performance index needs to be chosen suitably keeping in mind the number of oscillations within a same window. For example, the multi-wing Rucklidge system has got less number of irregular oscillations with a chosen 100 sec of simulation time window thus it gets settled down very quickly. Whereas both the multi-wing Lu and multi-wing Chen system produces more number irregular oscillation (in the uncontrolled mode) within that chosen window which takes much higher time get damped in the controlled mode. The multi-wing Sprott-1 system has the slowest dynamics compared to the other three chaotic systems thus it also settles down fast. The particular improvements of the present paper over state-of-the-art techniques are highlighted below. • • • Due to the structure specific Lyapunov stabilization scheme for a particular chaotic system, the proposed PID control scheme is more suited where the control scheme does not need to be changed every time except the PID controller gains. Deriving an analytical control action to suppress chaotic oscillation is based on the choice of the quadratic Lyapunov candidate which guarantees stabilization but not the fastest possible stabilization. The optimum PID controller not only suppresses the oscillations but enforces optimum tracking of one or multiple state variables (with suitable choice of the objective function as in [11]). Most of the popular analytical control schemes for a particular chaotic system rely on manipulating all the state variables [19], [35], [5]. The present PID control scheme only senses the second state variables and manipulates it to perform the same stabilization task. This is practical from the hardware implementation point of view as less number of sensors is required. Also not all the states of the chaotic system may be measurable. In such cases, this scheme scores over the others. 18 • The control scheme is particularly useful even in the presence of noise in the measured variable as previously studied in [21] which is difficult tackle with using analytical methods of nonlinear state feedback law design [19], [35], [5]. The PID control scheme, due to not being dependent on the system structure and directly working on the sensed state time series instead of the system model, is capable of stabilization of such nonlinear dynamical systems [12]. This is difficult to achieve with the classical state feedback controller. In spite of having several advantages, it is well known that stabilization of chaotic systems cannot be guaranteed by simulation due to their sensitive dependence on initial conditions. Although the simulations presented in section 4 in order to prove the robustness of PID control for multi-wing chaotic systems show promising results, still after testing with thousands of different initial conditions, the stabilization is not theoretically guaranteed unlike the Lyapunov based approach. Except the unavailability of guaranteed stabilization for all possible initial condition of the states, the proposed control scheme enjoys more flexibility of the controller design in a common template and can easily be tweaked by modifying the objective function (16) and the global optimizer used in this context. 6. Conclusion A real coded Genetic Algorithm based optimum PID controller design has been proposed in this paper to suppress chaotic oscillations in a family of highly complex multi-wing Lorenz family of chaotic systems. The PID controller enforces fast tracking of the second state due to presence of the ITAE criterion in the controller design phase which also removes chaotic oscillations in other state variables. The present approach is based on a heuristic optimization framework which can be easily modified to enforce optimum tracking performance in the other state variables and also with a combination of them. Using credible numerical examples, the optimum PID control scheme has been shown to be robust enough with different initial conditions, even for such highly complex nonlinear dynamical systems known as the multi-wing Lorenz family of chaotic systems. 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1 Pseudometrically Constrained Centroidal Voronoi Tessellations: Generating uniform antipodally symmetric points on the unit sphere with a novel acceleration strategy and its applications to Diffusion and 3D radial MRI Cheng Guan Koay Department of Medical Physics University of Wisconsin School of Medicine and Public Health Madison, WI 53705 Corresponding author: Cheng Guan Koay, PhD Department of Medical Physics University of Wisconsin School of Medicine and Public Health 1161 Wisconsin Institutes for Medical Research (WIMR) 1111 Highland Avenue Madison, WI 53705 E-mail: [email protected] Word Counts: 4953 Keywords: antipodal symmetry, uniform point set, Centroidal Voronoi Tessellations, diffusion MRI, 3D radial MRI, uniform points on the sphere. 2 ABSTRACT Purpose: The purpose of this work is to investigate the hypothesis that uniform sampling measurements that are endowed with antipodal symmetry play an important role when the raw data and image data are related through the Fourier relationship as in q-space diffusion MRI and 3D radial MRI. Currently, it is extremely challenging to generate large uniform antipodally symmetric point sets suitable for 3D radial MRI. A novel approach is proposed to solve this important and longstanding problem. Methods: The proposed method is based upon constrained centroidal Voronoi tessellations of the upper hemisphere with a novel pseudometric. Geometrically intuitive approach to tessellating the upper hemisphere is also proposed. Results: The average time complexity of the proposed centroidal tessellations was shown to be effectively linear. For small sample size, the proposed method was comparable to the state-ofthe-art iterative method in terms of the uniformity. For large sample size, in which the state-ofthe-art method is infeasible, the reconstructed images from the proposed method has less streak and ringing artifact as compared to those of the commonly used methods. Conclusion: This work solved a long-standing problem on generating uniform sampling points for 3D radial MRI. 3 INTRODUCTION A centroidal Voronoi tessellation (2) is a Voronoi tessellation (3) in which the center of mass (or centroid) of the Voronoi region is also its generator and it has been found useful in many applications, from analysis of cellular pattern (4), neuronal density (5) and territorial behavior of animals(6) in biological sciences to optimal data acquisition, data quantization, compression and clustering (7) in the engineering and physical sciences. However, it remains a challenge to generate these tessellations via Lloyd's algorithm even though Lloyd's algorithm has been found to be very robust (8). The main computational bottleneck in the computation of the centroidal Voronoi tessellations has been the reconstruction step in which the Voronoi tessellations are reconstructed anew from the newly computed centroids at each iteration. From the conventional point of view, if the time complexity of the Voronoi tessellations of a specific manifold, e.g., plane, sphere or other esoteric surfaces, is O(n log n), then the time complexity of the centroidal Voronoi tessellations on this manifold must be O(m n log n), i.e., invoking the O(n log n) algorithm m number times. Note that m is the number of iterations needed to reach convergence and n is the number of generators. We have discovered strong heuristic strategies to reduce the average time complexity of the problem from O(m n log n) to O(m n) by using less optimal, i.e., O(n2), but simpler algorithm innovatively and will present these strategies in the context of generating uniform antipodally symmetric points on the sphere. The approach used in this work for accelerating the centroidal Voronoi tessellations is completely novel, representing a clear departure from the current paradigms. 4 The problem of generating uniform points on the sphere was proposed by J.J. Thomson (9) more than a century ago and has spurred many theoretical and computational investigations, and applications (1,10-15). A variant of the Thomson problem is that of generating uniform antipodally symmetric points on the sphere (16); the resultant point set plays a particular important role in MRI, which will be discussed next. It is interesting to note that the problem of generating nearly uniform antipodally symmetric points on the sphere via a deterministic method was solved only recently (13,14). Conceptually, diffusion MRI (17) and 3D radial MRI (18) share many similar features. The most notable feature relevant to this work is the data redundancy associated with symmetries inherent in each of these imaging techniques. Specifically, antipodal symmetry of the diffusion propagator (19) and Hermitian symmetry due to the real-valuedness of the object in the image domain influence how data are acquired in diffusion MRI and 3D radial MRI, respectively. Hermitian symmetry and antipodal symmetry are closely related to each other and both symmetries are defined through parity transformation or spatial inversion with Hermitian symmetry having an additional operation, which is complex conjugation. Specifically, a realvalued function f possesses antipodal symmetry if f ( x)  f (  x) and a complex-valued ______ function g possesses Hermitian symmetry if it is a Hermitian function, i.e., g (x)  g ( x) where complex conjugation is denoted by the overhead bar. Due to these symmetries, it is sufficient to sample only half of the space, e.g., q-space in diffusion MRI or k-space in MRI. However, certain constraints on the imaging gradients (20) force the sampling trajectory in 3D radial MRI to take the usual diametrical line or a curve consists of two straight radial lines adjoined with a slight non-smooth bend at the center of k-space (21); the former sampling strategy cannot take advantage of the data redundancy because sampling along a diametrical line through k-space only is equivalent to acquiring repeated measurements and not new measurements and the 5 latter sampling strategy is more desirable with the potential of acquiring new and different points in k-space. The use of uniformity and antipodal symmetry in sampling scheme was advocated by Jones (16) in diffusion tensor imaging on the basis that the diffusion tensor is symmetric. Here, we argue that uniformity and antipodal symmetry should be incorporated into sampling design in diffusion MRI (q-space formalism) and 3D radial MRI (k-space formalism) because both the qspace and k-space possess Hermitian symmetry. Uniform sampling measurements endowed with antipodal symmetry can maximize sampling coverage and at the same time take advantage of the inherent symmetry of the imaging systems. It is to the best of our knowledge that none of the works in 3D radial MRI (18,21-28) used antipodally symmetric point sets to acquire k-space data. In fact, many studies (22,23,25,27) used the method of Wong and Roos (15) or the generalized spiral scheme (10,11). The point sets generated from any of the methods, i.e., generalized spiral scheme, the method of Wong and Roos and the method of Bauer (12), have the spiral pattern although each of these methods was formulated and derived differently. Antipodally symmetric point sets have not been used in any 3D radial MRI study may be due to the following reasons: the lack of a robust iterative method or a deterministic method capable of generating large number of antipodally symmetric points on the sphere, and the availability of other deterministic methods (1,11,12,15,29) for generating non-antipodally symmetric point sets. Ad hoc strategy of collecting half of the non-antipodally symmetric points on one of the hemispheres is usually adopted to come up with an antipodally symmetric point set. We should note that the diffusion MRI sampling method based on vertices of a multifold-tessellated icosahedral hemisphere was used by Tuch et al. (30) but this method is inflexible and limited in that it cannot generate point set of any size except certain "magic" numbers, e.g., 6, 26, 46, 91, 126, 196, 246, 341, 406, 526 and so on. Note that fivefold-tessellated icosahedral hemisphere 6 produces 126 points on the upper hemisphere, see Tuch (30). Due to the above mentioned limitation, this sampling method used by Tuch et al. may not be very appealing to the practitioners of 3D radial MRI. Even with the advent of the deterministic method capable of generating nearly uniform antipodally symmetric points on the sphere (13,14), it is still extremely challenging to generate highly uniform antipodally symmetric point set for use in 3D radial MRI through iterative function minimization even when the initial solution is taken from the deterministic methods. The number of points (on the sphere) required in 3D radial MRI and diffusion MRI is in the thousands (4000 to 20000 or more) and in the hundreds (6 to 500), respectively. Therefore, the number of parameters in spherical coordinate representation to be optimized are twice as many (8000 to 40000) and optimization methods that rely on approximate or exact Hessian or its inverse are completely infeasible. While some of the iteratively optimized antipodally symmetric point sets (with sample size in the hundreds) have been tabulated and available for public use (31), the lack of highly uniform and antipodally symmetric point sets applicable to 3D radial MRI motivates the present investigation. Here, we propose a constrained centroidal Voronoi tessellation endowed with a pseudometric to robustly and efficiently generating uniform antipodally symmetric point sets on the unit sphere. The uniformity of the point sets generated from the proposed method is shown to be comparable to the state-of-the-art iterative method (31) for small sample size relevant to diffusion MRI but not to 3D radial MRI. For large point set with sample size in the range of thousands, which is relevant to 3D radial MRI and other applications (32), the state-of-the-art method is completely infeasible but the proposed approach remains feasible. Further, we used our recently developed three-dimensional analytical MRI phantom in the Fourier domain (33), which was based upon the three-dimensional version of the famous Shepp-Logan phantom 7 (34,35) in the image domain, to compare and contrast the qualitative features of the reconstructed images obtained through the proposed method, the method of Wong and Roos, the generalized scheme and the tessellated-icosahedral of the upper hemisphere (the method used by Tuch et al.). We found that the reconstructed image from the proposed method has less streak and ringing artifacts as compared to those of other methods. 8 METHODS A centroidal Voronoi tessellation (2) is a Voronoi tessellation in which the center of mass (or centroid) of the Voronoi region is also its generator and the Voronoi regions can be prescribed with a density function. To the best of our knowledge, a centroidal Voronoi tessellation capable of generating uniform antipodally symmetric points on the unit sphere has never been proposed. Instead of dealing with a density function that is constant and invariant with respect to spatial inversion or antipodally symmetry, it is more convenient and efficient to introduce a novel pseudometric in the centroidal Voronoi tessellation so as to make generating uniform antipodally symmetric points on the unit sphere possible. The spherical Voronoi regions, { Vi }iN1 , on the upper hemisphere are characterized by the following properties:  No two distinct regions share the same point. That is, the intersection between any two distinct regions is empty, Vi  V j   with i  j . Points on the boundary between any two regions belong to the closure of these regions, which is denoted by an overhead bar, e.g., Vi .  The union of all the closures of the spherical Voronoi regions covers the upper hemisphere, denoted by S 2 .  Vi  {x  S2 | d (x, gi )  d (x, g j ) for j  1,..., N and j  i} [1] Each of the unit vectors, { g i }iN1 , on S 2 is called the generator of its respective Voronoi region. The pseudometric, d , will be defined later. The center of mass of each Voronoi region does not necessarily coincide with the generator of that region. An iterative method such as the Lloyd algorithm (8,36) may be used to 9 make the generators from each successive iterations closer and closer to the centers of mass of their respective regions. The resultant tessellations are known as the centroidal Voronoi tessellations (2). The center of mass of a spherical Voronoi region, ĝi   vd i  vd i Vi , can be expressed as: . [2] Note that  i is the spherical surface of Vi , v represents unit vector normal to the spherical surface element d i and . denotes the Euclidean norm, which is needed to ensure ĝ i is unit length. In practice, the computation of ĝ i is based on the discretized version of Eq.[2]. Specifically, it is obtained through the sum of the products between the area of the spherical triangle formed by the generator and each pair of consecutive vertices at the boundary surrounding the generator in counterclockwise order. We further note that the density function, which usually appears as a factor in the integrand of Eq.[2], is taken to be a unit constant function in order to ensure uniformity of the generators. The distance measure or metric, d (,) , used in this work is a novel extension of the modified electrostatic potential energy term studied in Ref.(37). For completeness, we will introduce the notion of real and virtual points for manipulating the antipodally symmetric point set suggested in Ref.(37). Due to the constraint of antipodal symmetry, we classify points as real and their corresponding antipodal points as virtual. If we have N real points (on the upper hemisphere), denoted by unit vectors ri with i  1,  , N , then the total electrostatic energy for the complete configuration (37) of 2 N points of both real and virtual points on the whole sphere is given by: 10   N 2   1  2   r  i  1 j  i  1  ij  N 1 N    4  rij2   1 [3] with rij  ri  r j . Note that Eq.[3] is expressed solely in terms of real points. If we define S (ri , r j )  1 1 then S (ri , r j ) may be thought of as a reciprocal metric (or reciprocal of  2 rij 4  rij the distance measure) between two real points.). Here, we exploit this reciprocal metric fully by defining our pseudometric, d (,) , as d (ri , r j )  1 / S (ri , r j ) . [4] Note that d (r j , r j )  0  d (r j ,r j ) , hence the term pseudometric. Please refer to Appendix B for the proof that d (,) is indeed a pseudometric. We should note that implementation of spherical Voronoi tessellations is not a trivial computational task (38,39) and our approach is different from the existing methods, Refs.(38,39). To compare and contrast our proposed approach, we will first present an outline of the Lloyd's algorithm below: Pseudometrically Constrained Centroidal Voronoi Tessellation: Let n be the number of desired points on the upper hemisphere. Step 1. Deterministically generate 2 n highly uniform points on the unit sphere via Analytically Exact Spiral Scheme(1) and select those points that are on the upper hemisphere as the generators. Step 2. Construct the spherical Voronoi regions of the upper hemisphere with the chosen generators. Step 3. Compute the normalized centroids using the discretized version of Eq.[2]. Step 4. Adopt the normalized centroids as the generators. Step 5. Iterate Steps 2, 3, 4 until convergence it reached. 11 We would like to point out that time complexity of Step 2 above is O(n log n), see Ref.(39). Therefore, the time complexity of the centroidal Voronoi tessellation is O(m n log n) where m is the number of iterations needed to reach convergence. Our approach to generating the Voronoi tessellations of the upper hemisphere is based on the following steps and observations: 1. The Voronoi region of a generator can be found by first identifying surrounding generators (approximately neighbors of neighbors) within a spherical cap with a prescribed radius, which depends on the total number of generators, away from the generator of interest, see Figure 1A. This step can always be done because the initial point set obtained from the Analytically Exact Spiral Scheme(1) is already reasonably uniform. When the total number of generators is large, the number of surrounding generators remain constant and is around 20. The time complexity of this step for all the generators is O(n2). 2. The generator and its surrounding generators are rotated in such a way that the generator is situated along the z-axis, see Figure 1A. The time complexity of this step for all the generators is O(n). 3. To compute the vertices of the Voronoi region, the smallest convex region formed by the surrounding generators and around the generator is needed, see Figure 1B. Finding this smallest convex region is equivalent to finding the boundary points of the convex hull of the surrounding generators after these generators have been stereographically projected (40) onto the x-y plane, see Figure 1C. The validity of this step hinges on the angle preserving (or conformal) property of steoreographic projection (41). The boundary points of the convex hull of a set of points on the plane is then computed through the Graham's scan, (42). The time 12 complexity of Graham's scan is O(k log k) and k is the number of points in the convex hull, i.e., the number of surrounding generators, which is about 20. Therefore, the time complexity of this step for all the generators is O(n) because k is a constant. 4. The vertices of the Voronoi region, the area of spherical triangles formed by the Voronoi vertices and the generator can be computed in O(n) for all the generators. The steps proposed above are intuitive and geometrically motivated but we should note that Step 1 in our proposed approach is O(n2), which is the main bottleneck of our approach if this step were to be invoked at every iteration. In what follows, we will describe a simple way to avoid tessellating at every iteration and provide empirical evidence that our approach is effectively O(n) for generating the centroidal Voronoi tessellations. We can compute the distance between a generator at the current iteration and the same generator from the previous iteration. This information has already been used to determine convergence, i.e., Ref.(8). The novelty of our proposal is in using this information to compute the cumulative sum of Euclidean distances made by each generators. If the maximum value of these cumulative sums is less than some prescribed value, see Appendix A, then Step 1 will not be invoked. Otherwise, Step 1 will be invoked and the cumulative sum is reset to zero. When Step 1 is not invoked, the connectivity network between a generator and its surrounding generators will not be altered but the coordinates of the generator and its surrounding generators will likely be different from one iteration to the next. Here, we discuss other important implementation details. Note that some of the vertices around the equator may be located at the lower hemisphere and the resultant centroids may also be on the lower hemisphere. Therefore, the centroids or generators should be reoriented 13 onto S 2 at each iteration. Finally, we note that the convergence of the proposed algorithm is based on that of local deviation as described in Ref.(8). 14 RESULTS We implemented the proposed pseudometrically constrained centroidal Voronoi tessellation in Java. We conducted four tests to illustrate the proposed method. The first two tests were run on a machine with an Intel® Core™ i7 CPU at 1.73 GHz and with 8 GB of RAM and the last two tests were run on a different machine, Four X 8-Core 2.3GHz AMD Opteron Processors (6134) with 128GB of RAM. The first test is to show that the performance of the point sets generated by the proposed method is of comparable quality in terms of modified electrostatic energy to that of the state-ofthe-art iterative method and to show that the proposed centroidal Voronoi algorithm is O(m n) through empirical analysis of the execution time per iteration as a function of number of generators (or points on the upper hemisphere), which shows a linear trend, see Figure 2. Figure 2A shows the percent of relative error in terms of the modified electrostatic energy of initial deterministic generators, which were obtained our recent analytically exact spiral scheme(1) and of the final generators obtained by the proposed method with respect to the state-of-the-art method. At the level of percent of relative error achieved by the proposed method, the uniformity of the points between the proposed method and the state-of-the-art method is indistinguishable visually or quantitatively. However, we observed that the point sets generated centroidal Voronoi tessellations has consistently shown to be higher in modified electrostatic energy than those obtained through iterative scheme. We believe this observation has to do with the intrinsic property of centroidal Voronoi tessellations, that is the fundamental centroidal constraint itself. The centroidal constraint forces generators to be in geometrically frustrated positions even though the effect of the centroidal constraint on geometric frustration is very slight. 15 Figure 2B shows the performance of the proposed method in terms of execution time (in seconds) per iteration of the proposed method as a function of the number of points on the upper hemisphere. It is clear that the execution time per iteration has a linear trend with respect to N. Figure 2C shows the frequency of invocation of Step 1, which is an O(N2) algorithm, as a function of N, number of points on the upper hemisphere and Figure 2D shows the total number of iterations as a function of N and the instances of invocation of Step1 that occurred within these iterations are shown in red. Note that the highest peak is at (N=324, 4161 iterations). The second test is to give an example of a point set with sample size large enough that the state-of-the-art method is currently not feasible. In this example, we used sample size of 888. Figure 3A shows the initial generators and their Voronoi regions as obtained the analytically exact spiral scheme(1). Figures 3B and 3C shows the centroids and centroidal Voronoi regions on the upper and the lower hemispheres, respectively. Figures 3D shows the centroids on the whole sphere by combining Figures 3B and 3C. The third test is intended to show the feasibility of the proposed method in generating point set of sample size large enough for 3D MRI applications and beyond. Figure 4 shows only the Voronoi regions on the sphere with 16000 generators on the upper hemisphere. It took 10.38 minutes to generate the centroidal Voronoi tessellations (at the tolerance level of 10-12) on the machine with Four X 8-Core 2.3GHz AMD Opteron Processors (6134). The final test is intended to show the qualitative features of the reconstructed images of the three-dimensional analytical MRI phantom (33) obtained from the proposed method, the method of Wong and Roos, the generalized spiral scheme and the tessellated icosahedral of the upper hemisphere. Note that the reconstruction algorithm is based upon the following nonuniform discrete Fourier transform for the image domain signal at r : 16 m I (r )   S (k i ) | k i |2 exp(2 i k i  r ) , i 1 where S (k i ) is the signal value of the three-dimensional Shepp-Logan phantom evaluated analytically at k i , see (33). The term, | k i |2 , came about from the spherical coordinate transformation in the Fourier expansion. Two target matrix sizes of the reconstructed imaging volume were chosen and they were 128  128  128 (standard resolution) and 256  256  256 (high resolution). We have m  526  64  33664 , kmax  32 (arbitrary units) and k  0.5 (arbitrary units) at the standard resolution and m  526  128  67328 , kmax  64 (arbitrary units) and k  0.5 (arbitrary units) at the high resolution. Figure 5 shows the reconstructed images generated from the different schemes studied in this work at the standard resolution (the first two rows) and at the high resolution (the last two rows). At the standard resolution of 128x128x128 with z = -0.181, the images generated by the method of Wong and Roos and the generalized spiral scheme have more noticeable ringing artifacts around the bright region than that of the proposed method. The image generated by the tessellated icosahedral scheme has more streak artifacts in the dark regions than that of the proposed method. At the same resolution with z = 0.228, the images generated from these methods have more ringing artifacts than that of the proposed method. Similar patterns of artifacts showed up more noticeably at a higher resolution of 256x256x256. At this resolution, the images generated from the proposed method have less ringing artifacts than those from other methods. 17 DISCUSSION Even though our recent works (13,14) were the first to solve the problem of deterministically distributing antipodally symmetric points on the unit sphere, the generated points are structurally very regular and are prone to causing degeneracy in triangulation (43), i.e., when four points rather than three sit on the same circumcircle, and hence, point sets from these works could not be used as the initial generators for the proposed method. The point sets generated from the analytically exact spiral scheme did not have the above mentioned defect and contribute immensely to the feasibility of the present approach by providing highly efficient technique for generating nearly uniform points on the sphere and half of which are then used as the initial generators in the present approach. The novelty of this work lies in the realization that the reciprocal of the reciprocal metric can be treated as a pseudometric in the antipodally symmetric space, in the extension of the centroidal Voronoi tessellations to the antipodally symmetric space, in the heuristic strategies suggested above for accelerating centroidal Voronoi tessellations without tessellating at every iteration, and in the utilization of the proposed constrained centroidal Voronoi tessellations to generate uniform antipodally symmetric points on the sphere. It should be clear that the pseudometric used in this work can also be used in any K-mean clustering algorithm of discrete data that are endowed with antipodal symmetry. The results showed that antipodally symmetric uniform sampling strategy does play a positive role in image quality and such a strategy should be incorporated into sampling design considerations of any 3D radial MRI study. One of the most exciting results of this work is the heuristic strategies used to accelerate centroidal Voronoi tessellations. This result highlights the emergence of strong heuristics in which traditional optimal algorithm that is invoked iteratively may not outperform less optimal algorithm that is invoked sparingly in an innovative way to accomplish the same computational 18 task. For small sample size, the point sets generated from the proposed method are comparable to those generated from the state-of-the-art method in all the cases tested. One of the important highlights of this work is that the proposed method is capable of generating large uniform point sets relevant to 3D radial MRI applications; the state-of-the-art iterative method is completely infeasible in this case particularly when the desired number of points is in the thousands. We must note that the reconstruction algorithm adopted in this work is for testing purposes as it is computational expensive but it serves the present purpose very well because whatever the differences in the reconstructed images these differences have to come from uniformity or lack thereof of each of the methods studied in this work. It is not fortuitous that the reconstructed image from the proposed method has less streak and ringing artifacts as compared to those of other methods studied in this work. Uniform antipodally symmetric point sets have continued to play a major role in MRI acquisition design (14,16,23,37,44-49) and are beginning to make an impact in other fields (32). We hope this work contributes to further advances to these scientific endeavors beyond diffusion and 3D radial MRI. 19 Acknowledgment The author dedicates this work to Lean Choo Khoo. He would like to thank Drs. M. Elizabeth Meyerand, Charles A. Mistretta and Peter J. Basser for the encouragement. Thanks go to Dr. Orhan Unal, Director of Medical Physics Computing, for making available the computing resources needed for this work and for providing the specifications of the Opteron Processors. The author would like to thank Drs. Steven R. Kecskemeti, Kevin M Johnson, and Michael A. Speidel for helpful discussion on image reconstruction and image quality assessment. This work was supported in part by the National Institutes of Health IRCMH090912. The software related to this work will be made freely http://sites.google.com/site/hispeedpackets/ . Appendix A available for research use at URL: 20 In this appendix, we present a method, which is an special case of our previous work(50), to compute the radius of a spherical cap so as to cover sufficient number of surrounding generators for the purposes of computing Voronoi region. We also mention the criterion adopted to decide whether to invoke Step 1 or otherwise. If the center of a circle of radius r on a plane that is tangent to a unit sphere at point (0,0,1) or on the z-axis, then its inverse Gnomonic projection is a spherical cap, see Figure 2A of Ref.(50). Since the area of the upper hemisphere of unit radius is 2 , the area associated with a generator can be approximated by 2 / N where N is the total number of generators on the upper hemisphere. This approximate works well for large N , i.e., 15 and above. The unnormalized areal measure, Eq.(33) of Ref.(50), of a spherical cap with area equals to 2 / N is related to r by the following expression: 2 / N  2 (1  Therefore, r  1 ). 1 r 2 NN1 2  1 . The angle subtended by an longest arc passing through the interior of the spherical cap is given by:   2 tan 1( r ) . This angle is also the approximate spherical distance between two generators. Therefore, any generators that are within 5 / 2 in spherical distance away for the generator of interest will be classified as surrounding generators. Previously, we mentioned that if the maximum value of the cumulative sums of distances made by generators in successive iterations is less than some prescribed threshold value, 21 which is a small fraction of the prescribed radius in Step 1, then Step 1 will not be invoked. The prescribed value is based on Euclidean distance for simplicity as the local convergence criterion is also based on Euclidean distance. The prescribed threshold value is given C 2(1  cos(tan 1( r ))) where C is a positive real number less than unity. The specific value of C adopted in this study was C  0.15 . Note that if C assumes too high a value, one may run into the problem in which the initial surrounding generators may not be the current surrounding generators, which may cause the error in Voronoi tessellations. If C assumes too low a value, Step1 may be invoked many more times than necessary. Appendix B: The proposed pseudometric for the unit sphere 22 The distance measure, d (,) , proposed in this work is a novel extension of the modified electrostatic potential energy term studied in (37). For the sake of completeness, the notion of real and virtual points for manipulating the antipodally symmetric point set as suggested in (37) will be reintroduced here. Due to the constraint of antipodal symmetry, we classify points as real and their corresponding antipodal points as virtual. If we have N real points (say on the upper hemisphere), denoted by unit vectors ri with i  1, , N , then the total electrostatic energy for the whole configuration (37) of 2 N points of both real and virtual points on the whole sphere is given by:  N 2  1 1 2     4  rij2 i 1 j i 1  rij  N 1 N   ,   [B1] with  ri  r j rij    ri  r j if if arccos(ri  r j )   / 2 . arccos(ri  r j )   / 2 [B2] We should note that we could have defined rij to be simply ri  r j , which is in fact the actual definition used in the implementation of the proposed algorithm, and the expression in Eq.[B1] would still be valid but the definition of rij in Eq.[B2] would bring much clarity to the proof later. Note that Eq.[B1] is expressed solely in terms of real points. The interaction term in Eq.[B1], 1 I (ri , r j )  1  rij 4  rij2 [B3] was first postulated in (37) as a reciprocal metric (or reciprocal of the distance measure) between two real points.). Here, we prove that m(,) given by 23 m(ri , r j )  1 / I (ri , r j )  1 rij  1 [B4] 1 4  rij2 is indeed a pseudometric. Before we begin the proof to show that m (,) is a pseudometric on the unit sphere, we first list all the necessary information about (1) the pseudometric conditions on the unit sphere, (2) concave functions, (3) sub-additive functions and (4) a well known property about the vertices of any spherical triangle on the unit sphere that is endowed with antipodal symmetry. Definition 2.1. Let S 2 be the three-dimensional unit sphere. A pseudometric on S 2 is a mapping, d : S 2  S 2  [0,2] , such that (a) d (ri , r j )  0 , (b) d (ri , r j )  d (r j , ri ) , (c) ri  r j implies d (ri , r j )  0 , and (d) d (ri , r j )  d (ri , rk )  d (r j , rk ) . Note that ri , r j , and rk are unit vectors. Note further that the d (,) is considered a metric if it satisfies the above conditions and also satisfies the converse of (c), i.e., d (ri , r j )  0 implies ri  r j . In brief, a metric is automatically a pseudometric but the converse is not generally true. It is not hard to see that m (,) satisfies the first three conditions trivially by virtue of Eq.[B2]. The proof will focus on showing that m (,) also satisfies the last condition, i.e., the triangle inequality. 24 Definition 2.2. A function g is concave on an interval [ a , b] if, for any x and y in [ a , b] and for any   [0,1] , we have g ( x  (1   ) y )  g ( x)  (1   ) g ( y ) . [B5] Lemma 2.1. If a function g is concave on an interval [0, b] for some b  0 and g (0)  0 , then g is subadditive, i.e., g ( x  y )  g ( x )  g ( y ) for any x, y  0 . Proof. By setting y  0 and invoking the fact that g is concave and g (0)  0 , Eq.[B5] reduces to g ( x )  g ( x ) . [B6] Therefore, y y g ( x  y)  x g ( x  y)  g ( x  y )  g ( x ( x  y ))  g ( ( x  y ))  g ( x)  g ( y ). x y x y x y x y The following lemma is well known and it is stated for completeness. Lemma 2.2. Any three points, ri , r j , and rk , on S 2 that are endowed with antipodal symmetry, two of the eight triplets, ( ri ,r j ,rk ) , have vertices bounded by a spherical octant. ri , ~ rj , and ~ rk , satisfies the following Consequently, any one of these two triplets, say, ~ properties: arccos((~ rk  ~ ri )  (~ rj  ~ ri ))   / 2 , arccos((~ rk  ~ rj )  (~ ri  ~ r j ))   / 2 , and arccos((~ rj  ~ rk )  (~ ri  ~ rk ))   / 2 . 25 If you define a new function based on Eq.[B4] as f (r )  1  r 1 1 , [B7] 4r 2 it should be clear that f is concave on [0,2] because f is continuous and twice-differentiable, and d2 f dr 2  0 for any r  (0,2) . Further, f (0)  0 . Therefore, f is sub-additive. We should note that f is an increasing function on [0, 2 ] and a decreasing function on [ 2 ,2] . ri , ~ rj , To prove the triangle inequality, we invoke Lemma 2.2 to focus on one of the triplets, i.e., ~ rk , and write and ~ ~ ~ ~ ~ ~ ~ ~ ~ ( r  r )( r  r ) ( r  r )( r  r ) ~ ri  ~ r j  a  b  2 , with a  i ~j ~ k j  0 , and b  j ~i ~ k i  0 . r r r r i  j i j  By Lemma 2.1, f ~ ri  ~ r j  f (a  b)  f (a )  f (b) . By Lemma 2.2 again, we have ri  ~ r j )( ~ rk  ~ rj) rj  ~ ri )( ~ rk  ~ ri ) (~ (~ , and  1  1 , and therefore, ~ ~ ri  ~ rj ~ rk  ~ rj ri  ~ rj ~ rk  ~ ri a ~ rk  ~ r j  2 and b  ~ rk  ~ ri  2 . Note that   f (a )  f (b)  f ~ rk  ~ rj  f~ rk  ~ ri  because f is an increasing function on [0, 2 ] . Therefore,     f ~ ri  ~ rj  f ~ rk  ~ rj  f~ rk  ~ ri  . We conclude that m (,) satisfies the triangular inequality and therefore, it is a pseudometric. 26 References 1. 2. 3. Koay CG. 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A signal transformational framework for breaking the noise floor and its applications in MRI. Journal of Magnetic Resonance 2009;197(4):108-119. Deriche R, Calder J, Descoteaux M. Optimal real-time Q-ball imaging using regularized Kalman filtering with incremental orientation sets. Medical Image Analysis 2009;13(4):564-579. Nett EJ, Johnson KM, Frydrychowicz A, Del Rio AM, Schrauben E, Francois CJ, Wieben O. Four-dimensional phase contrast MRI with accelerated dual velocity encoding. Journal of Magnetic Resonance Imaging 2012;35(6):1462-1471. Koay CG, Nevo U, Chang LC, Pierpaoli C, Basser PJ. The elliptical cone of uncertainty and its normalized measures in diffusion tensor imaging. IEEE Transactions on Medical Imaging 2008;27(6):834-846. FIGURE CAPTIONS 29 Figure 1. (A) A generator (in blue) and its surrounding generators (in red) within a spherical cap (blue closed curve) of predetermined radius. (B) The smallest convex region formed by the surrounding generators that contains the generator. (C) Finding the smallest convex region that contains the generator is equivalent to finding the convex hull of the surrounding generators in the stereographic projection. 30 Figure 2. (A) Percent of relative error of initial generators obtained from the analytically exact spiral scheme and of the final generators or centroids obtained from the proposed method compared to the state-of-the-art method in terms of modified electrostatic energies of N antipodally symmetric point sets with N ranges from 16 to 356. (B) The performance of the proposed method in terms of execution time (in seconds) per iteration of the proposed method as a function of the number of points on the upper hemisphere has a linear trend with respect to N. (C) The frequency of invocation of Step 1, which is an O(N2) algorithm, as a function of N, number of points on the upper hemisphere. (D) Total number of iterations as a function of N with instances of invocation of Step1 throughout the iteration are shown in red. The highest peak is at (N=324, 4161 iterations). 31 Figure 3. (A) 888 deterministic generators taken from the analytically exact spiral scheme and their Voronoi regions on the upper hemisphere. 888 final (B) real and (C) virtual generators from the proposed method. (D) 1776 antipodally symmetric points on the sphere obtained from (B) and (C). 32 Figure 4. Pseudometrically constrained Voronoi tessellations of the sphere with 16000 generators (not shown in the figure) on the upper hemisphere. 33 Figure 5. Reconstructed images of the analytical phantom generated from different sampling schemes at various slice locations. At the standard resolution of 128x128x128 with z = -0.181, the images generated by the method of Wong and Roos and the generalized spiral scheme have more noticeable ringing artifacts around the bright region than that of the proposed method. The image generated by the tessellated icosahedral scheme has more streak artifacts in the dark regions than that of the proposed method. At the same resolution with z = 0.228, the images generated from these methods have more ringing artifacts than that of the proposed method. Similar patterns of artifacts showed up more noticeably at the high resolution of 256x256x256. At this resolution, the images generated from the proposed method have less ringing artifacts than those from other methods. 

NON-ASYMPTOTIC RESULTS FOR CORNISH-FISHER EXPANSIONS arXiv:1604.00539v1 [] 2 Apr 2016 V.V. ULYANOV, M. AOSHIMA, AND Y. FUJIKOSHI Abstract. We get the computable error bounds for generalized Cornish-Fisher expansions for quantiles of statistics provided that the computable error bounds for Edgeworth-Chebyshev type expansions for distributions of these statistics are known. The results are illustrated by examples. 1. Introduction and main results In statistical inference it is of fundamental importance to obtain the sampling distribution of statistics. However, we often encounter situations, where the exact distribution cannot be obtained in closed form, or even if it is obtained, it might be of little use because of its complexity. One practical way of getting around the problem is to provide reasonable approximations of the distribution function and its quantiles, along with extra information on their possible errors. It can be made with help of Edgeworth–Chebyshev and Cornish–Fisher type expansions. Recently the interest for Cornish–Fisher type expansions stirred up because of intensive study of VaR (Value at Risk) models in financial mathematics and financial risk management (see, e.g. [14] and [15]). Mainly, it is studied the asymptotic behavior of the expansions mentioned above. It means that accuracy of approximation for distribution of statistics or its quantiles is given as O(·), that is in the form of order with respect to some parameter(s) (usually, w.r.t. n as a number of observations and/or p as dimension of observations). In this paper we construct non-asymptotic error bounds, in other words – computable error bounds, for Cornish–Fisher type expansions, that is for an error of approximation we prove upper bounds with closed-form dependence on n and/or p and, perhaps, on some moment characteristics of observations. We get these bounds under condition that similar nonasymptotic results are already known for accuracy of approximation of distributions of statistics by Edgeworth–Chebyshev type expansions. Let X be a univariate random variable with a continuous distribution function F . For α : 0 < α < 1, there exists x such that F (x) = α, Key words and phrases. computable bounds, non-asymptotic results, CornishFisher expansions. This work was supported by RSCF grant No. 141100364. 1 2 V.V. ULYANOV, M. AOSHIMA, AND Y. FUJIKOSHI which is called the (lower) 100α% point of F . If F is strictly increasing, the inverse function F −1 (·) is well defined and the 100α% point is uniquely determined. We also speak of “quantiles” without reference to particular values of α meaning the values given by F −1 (·). Even in the general case, when F (x) is not necessarily continuous nor is it strictly increasing, we can define its inverse function by formula F −1 (u) = inf{x; F (x) > u}. This is a right-continuous nondecreasing function defined on the interval (0, 1) and F (x0 ) ≥ u0 if x0 = F −1 (u0 ). Let Fn (x) be a sequence of distribution functions and let each Fn admit the Edgeworth-Chebyshev type expansion (ECE) in the powers of ǫ = n−1/2 or n−1 : (1) Fn (x) = Gk,n (x) + Rk (x) with Rk (x) = O(ǫk ) and  Gk,n (x) = G(x) + ǫa1 (x) + · · · + ǫk−1 ak−1 (x) g(x), where g(x) is a density function of the limiting distribution function G(x). An important approach to the problem of approximating the quantiles of Fn is to use their asymptotic relation to those of G’s. Let x and u be the corresponding quantiles of Fn and G, respectively. Then we have (2) Fn (x) = G(u). Write x(u) and u(x) to denote the solutions of (2) for x in terms of u and u in terms of x, respectively [i.e. u(x) = G−1 (Fn (x)) and x(u) = Fn−1 (G(u))]. Then we can use the ECE (1) to obtain formal solutions x(u) and u(x) in the form (3) and x(u) = u + ǫb1 (u) + ǫ2 b2 (u) + · · · (4) u(x) = x + ǫc1 (x) + ǫ2 c2 (x) + · · · . Cornish and Fisher in [3] and [6] obtained the first few terms of these expansions when G is the standard normal distribution function (i.e., G = Φ). Both (3) and (4) are called the Cornish–Fisher expansions, (CFE). Concerning CFE for random variables obeying limit laws from the family of Pearson distributions see, e.g., [1]. Hill and Davis in [13] gave a general algorithm for obtaining each term of CFE when G is an analytic function. Usually the CFE are applied in the following form with k = 1, 2 or 3: (5) xk (u) = u + k−1 X j=1 ǫj bj (u) + R̂k (u) with R̂k (u) = O(ǫk ). NON-ASYMPTOTIC RESULTS FOR CORNISH-FISHER EXPANSIONS 3 It is known (see, e.g., [15]) how to find the explicit expressions for b1 (u) and b2 (u) as soon as we have (1). By Taylor’s expansions for G, g, and a1 , we obtain b1 = −a1 (u), (6) 1 b2 = {g ′(u)/g(u)}a21(u) − a2 (u) + a′1 (u)a1 (u), 2 provided that g and a1 are smooth enough functions. In the following Theorems we show how xk (u) from (5) could be expressed in terms of u. Moreover, we show what kind of bounds we can get for R̂k (x) as soon as we have some bounds for Rk (x) from (1). Theorem 1. Suppose that for the distribution function of a statistic U we have (7) F (x) ≡ Pr{U ≤ x} = G(x) + R1 (x), where for remainder term R1 (x) there exists a constant c1 such that |R1 (x)| ≤ d1 ≡ c1 ǫ. Let xα and uα be the upper 100α% points of F and G respectively, that is (8) Pr{U ≤ xα } = G(uα ) = 1 − α. Then for any α such that 1 − c1 ǫ > α > c1 ǫ > 0 we have (i): uα+d1 ≤ xα ≤ uα−d1 . (ii): |xα − uα | ≤ c1 ǫ/g(u(1) ), where g is the density function of the limiting distribution G and g(u(1) ) = min g(u). u∈[uα+d1 ,uα−d1 ] Theorem 2. In the notation of Theorem 1 we assume that F (x) ≡ Pr{U ≤ x} = G(x) + ǫg(x)a(x) + R2 (x), where for remainder term R2 (x) there exists a constant c2 such that |R2 (x)| ≤ d2 ≡ c2 ǫ2 . Let T = T (u) be a monotone increasing transform such that Pr{T (U) ≤ x} = G(x) + R̃2 (x) with |R̃2 (x)| ≤ d˜2 ≡ c̃2 ǫ2 . Let x̃α and uα be the upper 100α% points of Pr{T (U) ≤ x} and G, respectively. Then for any α such that we have (9) where 1 − c̃2 ǫ2 > α > c̃2 ǫ2 > 0, |x̃α − uα | ≤ c̃2 ǫ2 /g(u(2) ), g(u(2) ) = min u∈[uα+d̃ ,uα−d̃ ] 2 2 g(u). 4 V.V. ULYANOV, M. AOSHIMA, AND Y. FUJIKOSHI Theorem 3. We use the notation of Theorem 2. Let b(x) be a function inverse to T , i.e. b(T (x)) = x. Then xα = b(x̃α ) and for α such that 1 − c̃2 ǫ2 > α > c̃2 ǫ2 we have |b′ (u∗ )| 2 (10) |xα − b(uα )| ≤ c̃2 ǫ, g(u(2) ) where |b′ (u∗ )| = max u∈[uα+d̃ ,uα−d̃ ] 2 2 |b′ (u)|. Moreover, (11) b(x) = x − ǫa(x) + O(ǫ2 ). Remark 1. The main assumption of the Theorems is that for distributions of statistics and for distributions of transformed statistics we have some approximations with computable error bounds. There are not many papers with this kind of non-asymptotic results because it requires technique which is different from the asymptotic results methods (cf., e.g., [10] and [20]). In series of papers [7], [8], [10], [11], [2], [16], [18], [19] we got non-asymptotic results for wide class of statistics including multivariate scale mixtures and MANOVA tests. We considered as well the case of high dimensions, that is the case when the dimension of observations and sample size are comparable. The results were included in the book [9]. See also [5]. Remark 2. The results of Theorems 1–3 could not be extended to the whole range of α ∈ (0, 1). It follows from the fact that the Cornish-Fisher expansion does not converge uniformly in 0 < α < 1. See corresponding example in Section 2.5 of [12]. Remark 3. In Theorem 2 we required the existence of a monotone increasing transform T (z) such that distribution of transformed statistic T (U) is approximated by some limit distribution G(x) in better way than the distribution of original statistic U. We call this transformation T (z) the Bartlett type correction. See corresponding examples in Section 3. Remark 4. According to (10) and (11) the function b(uα ) in Theorem 3 could be considered as an ”asymptotic expansion” for xα up to order O(ǫ2 ). 2. Proofs of main results Proof of Theorem 1. By the mean value theorem, |G(xα ) − G(uα )| ≥ |xα − uα | min g(uα + θ(xα − uα )). 0<θ<1 From (7) and the definition of xα and uα in (8), we get |G(xα ) − G(uα )| = |G(xα ) − Pr{U ≤ xα }| = |R1 (xα )| ≤ d1 . NON-ASYMPTOTIC RESULTS FOR CORNISH-FISHER EXPANSIONS 5 Therefore, (12) |xα − uα | ≤ d1 . min0<θ<1 g(uα + θ(xα − uα ) On the other hand, it follows from (7) that G(xα ) = 1 − α − R1 (α) ≤ 1 − (α − d1 ) = G(uα−d1 ). This implies that xα ≤ uα−d1 . Similarly, we have uα+d1 ≤ xα . Therefore, we proved Theorem 1 (i). It follows from Theorem 1 (i) that min u∈[uα+d1 ,uα−d1 ] g(u) ≤ min g(uα + θ(xα − uα )). 0<θ<1 Thus, using (12) we get statement of Theorem 1 (ii). Proof of Theorem 2. It is easy to see that it is sufficient to apply Theorem 1 (ii) to the transformed statistic T (U). Proof of Theorem 3. we obtain (13) Using now (9) and the mean value theorem
x̃α − uα = b−1 (xα ) − b−1 (b(uα )) = (b−1 )′ (x∗ ) xα − b(uα ) ,
where x∗ is a point on the interval min{xα , b(uα )} , max{xα , b(uα )} . By Theorem 1 (i) we have uα+d˜2 ≤ x̃α ≤ uα−d˜2 . Therefore, for xα = b(x̃α ) we get (14)

min{b−1 (xα ), uα } , max{b−1 (xα ), uα } ⊆ uα+d˜2 , uα−d˜2 . Since by properties of derivatives of inverse functions (b−1 )′ (z) = 1/ b′ (b−1 (z)) = 1/b′ (y) for z = b(y), the relations (13) and (14) imply (10). Representation (11) for b(x) follows from (6) and (10). 3. Examples In [17] we gave sufficient conditions for transformation T (x) to be the Bartlett type correction (see Remark 3 above) for wide class of statistics U allowing the following represantion (15) Pr{U ≤ x} = Gq (x) + k 1 X aj Gq+2j (x) + R2k , n j=0 where R2k = O(n−2 ) and Gq (x) is the distribution function of chisquared distribution with q degrees of freedom and coefficients aj ’s P satisfy the relation kj=0 aj = 0. Some examples of the statistic U are 6 V.V. ULYANOV, M. AOSHIMA, AND Y. FUJIKOSHI as follows: for k = 1, the likelihood ratio test statistic; for k = 2, the Lawley-Hotelling trace criterion and the Bartlett-Nanda-Pillai trace criterion, which are test statistics for multivariate linear hypothesis under normality; for k = 3, the score test statistic and Hotelling’s T 2 statistic under nonnormality. The results of [17] were extended in [4] and [5]. In [10] we were interested in the null distribution of Hotelling’s generalized T02 statistic defined by T02 = n trSh Se−1 , where Sh and Se are independently distributed as Wishart distributions Wp (q, Ip ) and Wp (n, Ip ) with identity operator Ip in Rp , respectively. In Theorem 4.1 (ii) in [10] we proved (15) for all n ≥ p with k = 3 and computable error bound: r |Pr(T02 ≤ x) − Gr (x) − {(q − p − 1)Gr (x) 4n −2qGr+2 (x) + (q + p + 1)Gr+4 (x)}| cp,q ≤ 2, n where r = pq and for constant cp,q we gave expicit formula with dependence on p and q. Therefore, according to [17] we can take in this case the Bartlett type correction T (z) as s 2 z a−1 a−1 + T (z) = + , 2b 2b b where a= 1 p(q − p − 1), 2n 1 p(q + p + 1)(q + 2) −1. 2n It is clear that T (z) is invertable and we can apply Theorem 3. Other examples and numerical calculations and comparisons of approximation accuracy see in [4] and [5]. One more example is connected with sample correlation coefficient. ~ = (X1 , ..., Xn )T , and Y~ = (Y1 , ..., Yn )T be two vectors from an Let X n-dimensional normal distribution N(0, In ) with zero mean, identity covariance matrix In and the sample correlation coefficient Pn Xk Y k ~ ~ . R = R(X, Y ) = pPn k=1 2 Pn 2 k=1 Yk k=1 Xk b= In [2] it was proved for n ≥ 7 and N = n − 2.5: √  Bn x3 ϕ(x) supx Pr ≤ 2, N R ≤ x − Φ(x) − 4N N NON-ASYMPTOTIC RESULTS FOR CORNISH-FISHER EXPANSIONS 7 with Bn ≤ 2.2. It is easy to see that we can take T (z) as the Bartlett type correction in the form T (z) = z + z 3 /(4N). Then the inverse function b(z) = T −1 (z) is defined by formula 1/3  p 2 3 b(z) = 2 N z + (2Nz) + (4N/3) 1/3  p − − 2 N z + (2Nz)2 + (4N/3)3 z3 3 z5 + + O(N −3 ). 4N 16 N 2 Now we can apply Theorem 3. = z− References [1] L.N. Bol’shev, ”Asymptotically Pearson transformations”, Theor. Probab. Appl., 8, 121–146 (1963). [2] G. Christoph, V.V. Ulyanov, and Y. Fujikoshi, ”Accurate approximation of correlation coefficients by short Edgeworth-Chebyshev expansion and its statistical applications”, Springer Proceedings in Mathematics and Statistics, 33, 239–260 (2013). [3] E.A. Cornish and R.A. Fisher, ”Moments and cumulants in the specification of distributions”, Rev. Inst. Internat. Statist., 4, 307–320 (1937). [4] H. Enoki and M. Aoshima, ”Transformations with improved chi-squared approximations”, Proc. Symp., Res. Inst. Math. Sci., Kyoto Univ., 1380, 160– 181 (2004). [5] H. Enoki and M. Aoshima, ”Transformations with improved asymptotic approximations and their accuracy”, SUT Journal of Mathematics, 42, no.1, 97–122 (2006). [6] R.A. Fisher and E.A. Cornish, ”The percentile points of distributions having known cumulants”, J. Amer. Statist. Assoc., 80, 915–922 (1946). [7] Y. Fujikoshi, and V.V. Ulyanov, ”Error bounds for asymptotic expansions of Wilks’ lambda distribution”, Journal of Multivariate Analysis, 97, no.9, 1941– 1957 (2006). [8] Y. Fujikoshi, and V.V. Ulyanov, ”On accuracy of approximations for location and scale mixtures”, Journal of Mathematical Sciences, 138, no.1, 5390–5395 (2006). [9] Y. Fujikoshi, V.V. Ulyanov, and R. Shimizu, Multivariate Statistics : HighDimensional and Large-Sample Approximations, Wiley Series in Probability and Statistics, John Wiley & Sons, Hoboken, N.J., (2010). [10] Y. Fujikoshi, V.V. Ulyanov, and R. Shimizu, ”L1 -norm error bounds for asymptotic expansions of multivariate scale mixtures and their applications to Hotelling’s generalized T02 ”, Journal of Multivariate Analysis, 96, no.1, 1–19 (2005). [11] Y. Fujikoshi, V.V. Ulyanov, and R. Shimizu, ”Error bounds for asymptotic expansions of the distribution of multivariate scale mixture”, Hiroshima Mathematical Journal, 35, no.3, 453–469 (2005). [12] P. Hall, The Bootstrap and Edgeworth Expansion, Springer-Verlag, New York (1992). [13] G.W. Hill and A.W. Davis, ”Generalized asymptotic expansions of CornishFisher type”, Ann. Math. Statist., 39, 1264–1273 (1968). [14] S. Jaschke, ”The Cornish-Fisher-expansion in the context of delta-gammanormal approximations”, J. Risk, 4, no.4, 33–52 (2002). 8 V.V. ULYANOV, M. AOSHIMA, AND Y. FUJIKOSHI [15] V.V. Ulyanov, ”Cornish-Fisher Expansions”. In International Encyclopedia of Statistical Science, Ed. M.Lovric, Springer-Verlag, Berlin–Heidelberg, 312–315 (2011). [16] V.V. Ulyanov, G.Christoph, and Y. Fujikoshi, ”On approximations of transformed chi-squared distributions in statistical applications”, Siberian Mathematical Journal, 47, no.6, 1154–1166 (2006). [17] V.V. Ulyanov and Y. Fujikoshi, ”On accuracy of improved χ2 -approximations”, Georgian Mathematical Journal, 8, no.2, 401–414 (2001). [18] V.V. Ulyanov, Y. Fujikoshi, and R. Shimizu, ”Nonuniform error bounds in asymptotic expansions for scale mixtures under mild moment conditions”, Journal of Mathematical Sciences, 93, no.4, 600–608 (1999). [19] V.V. Ulyanov, H. Wakaki, and Y. Fujikoshi, ”Berry-Esseen bound for high dimensional asymptotic approximation of Wilks’ lambda distribution”, Statistics and Probability Letters, 76, no.12, 1191–1200 (2006). [20] H. Wakaki, Y. Fujikoshi, and V.V. Ulyanov ”Asymptotic Expansions of the Distributions of MANOVA Test Statistics when the Dimension is Large”, Hiroshima Mathematical Journal, 44, no.3, 247–259 (2014). V.V. Ulyanov, Faculty of Computational Mathematics and Cybernetics, Moscow State University, Moscow, 119991, Russia, and National Research University Higher School of Economics (HSE), Moscow, 101000, Russia E-mail address: [email protected] M. Aoshima, Institute of Mathematics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan E-mail address: [email protected] Y. Fujikoshi, Department of Mathematics, Hiroshima University, Higashi-Hiroshima, 739-8526, Japan E-mail address: fujikoshi [email protected] 

1 Millimeter Wave MIMO Prototype: Measurements and Experimental Results arXiv:1710.09449v1 [] 25 Oct 2017 Vasanthan Raghavan, Andrzej Partyka, Ashwin Sampath, Sundar Subramanian, Ozge Hizir Koymen, Kobi Ravid, Juergen Cezanne, Kiran Mukkavilli, Junyi Li Abstract Millimeter-wave multi-input multi-output (mm-Wave MIMO) systems are one of the candidate schemes for 5G wireless standardization efforts. In this context, the main contributions of this article are threefold. 1) We describe parallel sets of measurements at identical transmit-receive location pairs with 2.9, 29 and 61 GHz carrier frequencies in indoor office, shopping mall, and outdoor settings. These measurements provide insights on propagation, blockage and material penetration losses, and the key elements necessary in system design to make mm-Wave systems viable in practice. 2) One of these elements is hybrid beamforming necessary for better link margins by reaping the array gain with large antenna dimensions. From the class of fully-flexible hybrid beamformers, we describe a robust class of directional beamformers towards meeting the high data-rate requirements of mm-Wave systems. 3) Leveraging these design insights, we then describe an experimental prototype system at 28 GHz that realizes high data-rates on both the downlink and uplink and robustly maintains these rates in outdoor and indoor mobility scenarios. In addition to maintaining large signal constellation sizes in spite of radio frequency challenges, this prototype leverages the directional nature of the mm-Wave channel to perform seamless beam switching and handover across mm-Wave base-stations thereby overcoming the path losses in non-line-of-sight links and blockages encountered at mm-Wave frequencies. Index Terms Millimeter-wave, experimental prototype, MIMO, channel measurements, beamforming, handover, RF. I. I NTRODUCTION Millimeter-wave multi-input multi-output (mm-Wave MIMO) systems are one of the candidates for the physical layer (PHY) in the currently ongoing standardization efforts for the Fifth The authors are with Qualcomm Corporate R&D, Bridgewater, NJ 08807, USA and Qualcomm Corporate R&D, San Diego, CA 92121, USA. 2 Generation (5G) air link specifications. Over the last few years, there has been an exploding interest on mm-Wave systems (see, e.g., [1]–[3] and references therein). Most of these works either present expectations from mm-Wave systems, or use-case analysis, or channel measurements, or performance studies assuming certain PHY abstractions. With this backdrop, the focus of this paper is on understanding the implementation of mm-Wave systems in practice and to present a complete picture starting from channel measurements to system design implications to prototype performance. Towards this goal, we first study electromagnetic propagation in the mm-Wave regime using a number of parallel measurements at the same transmit-receive location pairs in different usecases (indoor office, shopping mall and outdoor) at 2.9, 29 and 61 GHz. Such a parallel set of measurements minimizes the number of confounding factors and allows a direct comparison of propagation across different carrier frequencies. A limited number of such measurement studies at the same location pairs are available in the literature. While our studies show that losses at mm-Wave frequencies are typically higher than with sub-6 GHz systems, these losses are not substantially worse. Nevertheless, additional losses due to hand/human blockages and material penetration can be significantly detrimental to the link margins and are expected to play the role of a serious differentiator for a mm-Wave chipset solution. The above observations motivate the use of beamforming to overcome these losses. We then briefly describe the radio frequency (RF) component challenges that impact the design of a practical mm-Wave chipset. The use of near-optimal beamforming structures needed to improve the link margin in both single- and multi-user contexts in a practical mm-Wave chipset is not viable due to cost and complexity challenges of radio frequency (RF) components. Thus, secondly, to overcome these constraints, we are motivated to use a certain subset of directional beamformers for MIMO transmissions. In addition to low complexity, the proposed approaches also enjoy advantages such as robustness to phase changes across paths (an issue of immense importance for small wavelength systems such as mm-Wave) and a simpler system design for initial user equipment (UE) discovery and subsequent beam refinement. With this background, thirdly, we describe our prototype system operating at 28 GHz that realizes a robust directional beamforming solution leading to high data-rates on both the downlink and uplink. We describe various experiments performed with the prototype in both outdoor and indoor scenarios and provide unique insights into the operations of a practical mm-Wave system. 3 The key elements tested by these experiments include robustness of mm-Wave links in nonline-of-sight (NLOS) settings via beam switching in response to mobility and blockage, interbase-station handover, and interference management. Comparable prototypes in the literature such as [4] mostly emphasize peak throughputs in line-of-sight (LOS) settings and do not provide lessons applicable for practical deployments. Other prototypes of importance in practical deployments include the CAP-MIMO architecture [5] and [6]–[8] that apply lens array techniques for steering multiple beams from the base-station. II. M ILLIMETER -WAVE C HANNEL M EASUREMENTS AND S YSTEM I MPLICATIONS The focus of this section is on reporting mm-Wave channel measurements at 2.9, 29 and 61 GHz, which are representative of the three most-likely commercial offerings in the 2018-20 time-frame and likely to be compared against each other in terms of performance: sub-6 GHz 5G-NR, mm-Wave MIMO 5G-NR, and 802.11ad/ax/ay. While a number of mm-Wave channel measurement campaigns have been reported in the literature, the novelty of this work is on channel propagation comparisons across these three carrier frequencies at identical transmitreceive location pairs in different use-cases. Such studies are important as they eliminate most confounding factors that prevent a direct comparison across frequencies. Towards this goal, channel sounding is performed with both omni-directional antennas as well as directional horn antennas. For directional measurements, an azimuthal scan (or a 360o view) with a 10 dBi gain horn antenna and producing 39 directional slices, and a spherical scan (360o azimuth view and −30o to 90o view in elevation) with a 20 dBi gain horn antenna and producing 331 directional slices are generated. The time-resolution of the channel sounder is approximately 5 ns. An Agilent E8267D signal generator is used to generate a pseudo-noise (PN) sequence at a chip rate of 100 Mc/s, which is then used to sound the channel. At the receiver, an Agilent N9030A signal analyzer is used for acquisition and the PN chip sequence is despread using a sampler at 200 MHz and with 16 bit resolution. These sounding measurements are obtained for the indoor office setting (two floors of the Qualcomm building in Bridgewater, NJ), indoor shopping mall setting (Bridgewater Commons Mall, Bridgewater, NJ), outdoor settings (open areas outside the Qualcomm building), including the suburban setting (residential location in Bedminster, NJ) and the Urban Micro setting (New Brunswick, NJ), etc., as well as emulation of stadium deployments. These measurement scenarios 4 are representative of typical applications considered for future deployment efforts. While macroscopic channel properties such as path loss are studied with omni-directional scans, other properties such as delay spread, path diversity, etc., are studied with both omnidirectional and directional scans. Processing of these measurements lead to the following observations and implications on system design for mm-Wave channels. More technical details on these studies can be found in [9]. Path Loss: Measurements in different deployments are used to study macroscopic properties of LOS and NLOS links. We use a frequency-dependent path loss model with a close-in free space reference distance of d0 = 1 m where the path loss (in dB) at a distance of d m is modeled as PL(d) = PL(d0 ) + α · 10 log10 (d/d0 ) + X, X ∼ N (0, σX2 ). (1) The path loss exponents (PLEs), denoted as α, and the shadowing factors (σX ) for different types of links (LOS/NLOS) in different use-cases are learned with a least-squares fitting of the model in (1) to the measured data. These parameters are listed in Table I and they show that PLEs and shadowing factors for NLOS links generally increase with frequency. For LOS links, PLEs are generally smaller than those for NLOS links and in indoor settings can be smaller than the freespace PLE of 2. A plausible explanation for this observation is waveguide effect where long enclosures such as walkways/corridors, dropped/false ceilings, etc., tend to propagate electromagnetic energy via alternate modes/more reflective paths decreasing the PLE. Shadowing factors show inconsistent behavior with frequency. From Table I, we conclude that while mmWave systems experience higher path losses than sub-6 GHz systems, the differential impact of the PLEs and shadowing factors on link margin at higher carrier frequencies is not dramatic. Delay Spread: The delay spread of the channel is an important metric to understand the system overhead (in terms of the cyclic prefix length for a multi-carrier design). In this context, frequency-dependent delay spreads are observed in NLOS settings both with omni-directional and directional antennas. While omni-directional delay spreads are small in most scenarios (for example, the essential spread is on the order of 30-50 ns in indoor office, 50-90 ns in indoor shopping mall and 150-300 ns in outdoor street canyon settings), there are also scenarios where a significantly large delay spread is seen (e.g., even up to 800 ns in outdoor open square settings). These extreme scenarios can be explained with the radar cross-section effect, where seemingly small objects that do not participate in electromagnetic propagation at lower frequencies show 5 TABLE I PATH L OSS M ODEL PARAMETERS IN D IFFERENT U SE - CASES Parameter ↓ fc (in GHz) → LOS 2.9 29 NLOS 61 2.9 LOS 29 61 2.9 Indoor office 29 NLOS 61 2.9 29 61 Indoor shopping mall PLE (α) 1.62 1.46 1.59 3.08 3.46 4.17 1.93 1.98 2.05 2.61 2.76 2.98 σX (in dB) 5.49 4.25 4.81 6.60 8.31 13.83 5.32 3.56 4.29 9.08 9.47 12.86 Urban Micro street canyon Outdoor open areas PLE (α) 2.18 2.19 2.22 2.95 3.07 3.27 2.41 2.73 2.83 3.01 3.39 3.42 σX (in dB) 4.41 4.37 4.84 7.82 8.16 10.70 4.60 5.73 6.78 4.00 8.03 1.97 up at higher frequencies. Such behavior happens as the wavelength approaches the roughness of surfaces (e.g., walls, light poles, etc.). Supporting these extremes without incurring a high fixed system overhead is important. In most indoor scenarios, sparse scattering implies that the beamformed delay spread is comparable with the omni-directional delay spread. While the same trend holds for most scenarios in the outdoor setting, the beamformed delay spread can be significantly smaller than the omni-directional delay spread for the tail values. Blockage: An important feature that makes mm-Wave propagation significantly different from propagation at sub-6 GHz frequencies is that a large area of the UE can be easily covered and blocked by parts of the human body, other humans, vehicles, foliage, etc. Additional link impairments due to these blockers (not seen at sub-6 GHz) are observed at mm-Wave frequencies and the practical viability of mm-Wave systems are more dependent on blockage than the path losses reported in Table I. For example, a typical UE design with multiple linear subarray units of four antenna elements (on the Top and Long edges of the device) is presented for the Landscape mode in Fig. 1(a). Corresponding to this UE antenna design, Figs. 1(b)-(d) show the received gain in azimuth and elevation in Freespace, and with hand blocking in the Landscape and Portrait modes, respectively. While almost the entire sphere is covered around the UE in the Freespace mode, the presence of hand leads to an angular blockage region of 160o × 75o (blue areas) in the Landscape mode. The region in blue stretches from behind the palm to the thumb. In this setting, the Top edge subarray is not useful and the Long edge subarray allows good signal reception. 6 Fig. 1. (a) (c) (b) (d) (a) A typical UE design with multiple subarrays (Top and Long edges) in Landscape mode. Received gain as a function of azimuth and elevation angles for the UE design at 28 GHz in (b) Freespace mode, and with hand blocking in (c) Landscape and (d) Portrait modes. Furthermore, in the Portrait mode, a blockage region of 120o × 80o (blue areas) is seen and the Long edge subarray does not play an important role in signal reception as it is blocked with the fingers resulting in significantly deteriorated antenna efficiencies. However, the Top edge subarray is not affected much with the presence of the hand. These observations suggest that subarray diversity in UE design is critical in overcoming near-field obstructions as well as to ensure coverage at the UE side over the entire sphere. Penetration Loss: For the outdoor-to-indoor coverage scenario, material measurements show that penetration loss generally increases with frequency. Further, periodic notches that are several GHz wide and often with more than 30 dB in loss are seen. These losses are attributed to changing material properties with frequency due to which signals constructively/destructively interfere from different surfaces that make the material. While a similar trend is observed across these experiments for both polarizations and different choices of incidence angles, the precise loss at a frequency and the depth of the notches depend on the material, incidence angle and 7 polarization. This observation motivates the need for designs that support both frequency and spatial diversity. Path Diversity: Low pre-beamforming signal-to-noise ratios (SNRs) are typically the norm when mm-Wave path losses and additional blockage/penetration losses are incorporated with typical equivalent isotropically radiated power (EIRP) constraints. Thus, a viable system design has to overcome these huge losses with beamforming array gains from the packing of a large number of antennas within the same array aperture [3]–[5], [10]–[13]. In this context, a small number of (at most 4-6) well-spread out (in direction) clusters/paths rendering multi-mode/multi-layer signaling viable are typically observed. The viability of multiple modes suggests the use of both single-user MIMO strategies for increasing the peak rate as well as multi-user MIMO strategies for increasing the sum-rate [11], [14]. Such modes also offer robustness against blockages via intra-base-station beam switching. In addition to the likelihood of multiple viable paths to a certain base-station, there are also viable paths to multiple base-stations. These observations suggest the criticality of a dense deployment of base-stations for robust mm-Wave operation and inter-base-station handover to leverage these paths. Integrated access and backhaul operation is highly desirable for small cell deployment, which also leads to the need to study inter-base-station interference management issues more carefully. III. E XPERIMENTAL P ROTOTYPE D ESCRIPTION Motivated by the system design intuition developed in the previous section, we now describe our experimental prototype system operating in a time-division duplexing framework at 28 GHz. In this setup, baseband analog in-phase and quadrature (IQ) signals are routed to/from the modem to an IQ modulator/demodulator at 2.75 GHz center frequency. The 2.75 GHz intermediate frequency signal is translated to 28 GHz using a 25.25 GHz tunable local oscillator (with a 100 MHz step size). The bandwidth supported is 240 MHz at a sampling rate of 240 Msps. ADCs with an effective number of bits (ENOB) resolution of 8 bits are used at both ends. At the basestation end, the 28 GHz signal is routed to an 16 × 8 element planar array (a waveguide design) and analog beamforming is applied using tunable four bit phase shifters and gain controllers. The prototype uses a transmit power dynamic range of 19 dB with a maximum EIRP of 55 dBm. As motivated earlier, to overcome blockage, the UE end is made of four selectable subarrays, each a four element phased array of either dipoles or patches as in Fig. 1(a). 8 With beamforming being a central component in meeting the mm-Wave link budget, RF component and architecture-driven challenges (e.g., cost, power, complexity, form factor, regulatory constraints, etc.) play a principal role in determining practically viable hybrid beamforming solutions. In this context, the sparse and directional channel structure suggests the use of a certain subset of directional beamforming strategies along the dominant clusters/paths at both ends [5], [12]–[14] relative to optimal beamforming along the dominant eigen-modes/singular vectors of the channel matrix. Directional beamforming structures offer robustness to small perturbations in the channel matrix and also allow a tradeoff between peak beamforming gain and initial UE discovery latency (with minimal loss relative to the optimal schemes) via the construction of a hierarchy of directional codebooks [13]. The directional channel structure can be leveraged for scheduling and can also be generalized to multi-user beamforming design with the following solutions: i) beam steering to each UE (with complete agnosticism of the interference caused to other users), ii) zeroforcing (where each user’s beamforming vector steers a beam null to the other users), and iii) generalized eigenvector precoding (that performs a weighted combination of beam steering and beam nulling). These solutions can result in substantial performance improvement over single-user solutions. The readers are referred to [14] for technical details on these constructions as well as performance studies in outdoor and indoor deployments. Motivated by the robustness of directional beamformers, this solution is implemented in the prototype by leveraging the beam broadening principles described in [13, Sec. IVB] to construct static analog beam codebooks to be used at both the base-station and UE ends. The experimental system implements mm-Wave beamforming by initially determining the best beam direction to be used at either end. After this, the system continuously evaluates all possible beam directions from all available transmitters and switches to the best beam and the best transmitter (handover) with little to no performance degradation. In addition, the system adjusts its parameters to optimize for the link type (LOS/NLOS) by SNR control allowing upto 64-QAM operation. The seamless beam switching and capability to maintain high SNR enable the experimental system to realize high rates (on both the downlink and uplink) and robustly maintain these rates despite channel variations. That said, the main focus behind the experimental system is not the optimization of data-rates, but to study the various fundamental difficulties in realizing mm-Wave systems in practice, especially with NLOS links. In the next two sections, we describe some experimental results illustrating the versatility of the prototype. 9 (a) Fig. 2. (b) (a) Elevated gNB and UE inside the testing vehicle used in outdoor testing. (b) Aerial layout of the testing range including gNB locations, achieved rates and important features in rates as the UE is driven over the trajectory. IV. O UTDOOR M OBILITY S TUDIES An outdoor mobility testing experiment (see Fig. 2(a)) is conducted in the parking spaces adjacent to the Qualcomm building (see aerial layout in Fig. 2(b)). One base-station is mounted on a mast elevated 14 feet in a testing vehicle and located in the parking lot (marked gNB1 ), and another base-station is mounted in the sixth floor of the Qualcomm building facing the window and elevated 5 feet from the ground (marked gNB2 ). gNB1 and gNB2 have a 90o and 110o downtilt, respectively. The UE is mounted on the dashboard of another testing vehicle and testing is done by driving through the parking spaces at 10-15 mph speeds. Fig. 2(b) plots the achieved throughput (in Mbps) as a function of the driving trajectory. From this plot, we note that a high throughput close to 600 Mbps is realized (Scenario 1) when gNB2 has an unobstructed LOS path to the UE. As the UE moves over the trajectory (Scenario 2), seamless handover is realized between gNB2 and gNB1 . Further, as the UE is driven on this trajectory, the LOS path from gNB1 is obstructed and communication is realized through a reflected path first (Scenario 3) and other paths subsequently (Scenario 4) leading to a drop in throughput (gradings in the heat map). This experiment illustrates the prototype’s capability to maintain mm-Wave links robustly in outdoor scenarios with intra- and inter-base-station beam switching and handover. 10 Fig. 3. Left: Outdoor aerial layout of experiment including key geographical features. Right: Achieved rates as a function of outdoor trajectory and time as well as beam indices at gNB and subarray indices at UE. In a second study illustrated in Fig. 3, an outdoor mobility experiment around the Qualcomm building is conducted. This environment is mostly a tree-lined open square-type setting with some street canyon-type features. Specific points-of-interest include parking lots and structures with bordering buildings having glass window panes, foliage (a mix of pine and spruce trees), a large shopping mall in close vicinity (Bridgewater Commons Mall), highways (US Rt. 202), etc. For the specific experiment reported here, a testing vehicle is driven for a period of ≈ 55 seconds through the exit lane of US Rt. 202 at a speed of 20-30 mph, onto the ramp and into a side street enveloping the Qualcomm building (see trajectory in red in Fig. 3, Left side). In terms of notable observations from this experiment, a base-station mounted on a raised platform at 24 feet with a 90o downtilt (marked gNB1 ) offers a LOS path to the UE as it starts exiting from Rt. 202 (throughput of 375 Mbps). However, as the UE traverses the exit lane, a small hillock-like feature blocks the LOS path leading to a coverage hole that cannot bridged with any reasonable NLOS path from this base-station and a significant deterioration in rate. This observation points at the necessity of sufficient base-station density to enhance mm-Wave coverage under blockages. For example, a base-station on the opposite side of the highway (Rt. 202) could have provided coverage to the UE over this coverage hole. As the UE crosses this feature, a beam recovery process recovers the LOS path albeit with a different subarray offering complementary coverage 11 in the LOS direction leading to an improved throughput of 375 Mbps. Further, as the testing vehicle enters the ramp, blockage loss due to foliage results in a throughput drop (125-250 Mbps) and two subarrays turn out to be useful over this period. As the UE exits the ramp onto the adjoining street, the LOS path is recovered leading to a throughput of over 375 Mbps. The distance between the gNB and UE varies from 50-100 m over the whole experiment. (a) (b) (c) Fig. 4. (a) Typical UE testing with pedestrian mobility in an indoor scenario. (b) Indoor layout and building plan along with two gNB locations and coverage areas. (c) Achieved rate along with key features in the rate trajectory over a certain indoor segment. V. I NDOOR M OBILITY S TUDIES Complementary to the above discussion, an indoor mobility study (see Fig. 4(a)) in the third floor of the Qualcomm building (see building layout in Fig. 4(b)) is now described. The floor plan is mostly comprised of cubicles along the edge with walled offices and conference rooms towards the center. Two base-stations (marked gNB1 and gNB2 ) are placed at the far corners of the floor plan and the UE is moved at pedestrian speeds through the layout. From our studies, these two base-stations are sufficient to guarantee adequate coverage with at least 1 bps/Hz 12 spectral efficiency as the UE is traversed through the floor plan (coverage areas with each gNB marked in red and blue of Fig. 4(b), respectively). Nevertheless, the coverage area corresponding to each gNB does not lead to a well-defined cell boundary and is clearly dependent on the environment, material properties, etc. This observation points to necessity of further system coverage studies with irregular cell boundaries and base-station density to overcome coverage holes in such scenarios. As a particular illustration of this study, the throughput achieved with an ≈ 90 second trajectory is illustrated in Fig. 4(c) which illustrates both link drops due to penetration loss through obstructions (concrete, elevator area, wall, metallic material, etc.) and link variability due to changing material properties. Such link drops can be mitigated with enhanced beamforming, fast subarray switching, network densification, etc. (a) Fig. 5. (b) (a) A successful handover from eNB1 to eNB2 in an indoor setup. (b) Layout of another indoor coverage experiment with downlink/uplink rates using the proposed beamforming solutions. More indoor mobility experiments can be seen in the video demonstration at [15]. In one experiment, illustrated in Fig. 5(a) and seen in [15], a LOS link is initially established between the transmitter (labeled eNB1 ) and receiver (labeled UE) by beam scanning at both ends. With a high SNR from the LOS link, a high rate is established on both the downlink and uplink. As the UE is moved across the long edge of the hallway (see relative positions of eNB1 , eNB2 and UE in the bottom left inset of Fig. 5(a)), the link connecting eNB1 with the UE becomes NLOS with increasing path loss as the distance between eNB1 and UE increases. As the UE turns the corner 13 at the short edge, the NLOS1 link between eNB2 and UE becomes better than the NLOS link between eNB1 and UE and a successful handover (illustrating robustness to blockage) happens from eNB1 to eNB2 , as shown in Fig. 5(a). Figure 5(b) illustrates the layout of yet another indoor experiment where the UE is moved from an initial position (marked “1” in a white circle) towards its final destination (marked “1” in an orange square) via the dashed white-line trajectory. As the UE is moved over the trajectory, the achieved downlink and uplink rates show many disruptions as the connected path is blocked by the pillars in the layout (marked as gray squares). For example, when the pillar blocks the LOS path, connection is established to the dominant NLOS path (again through reflections) leading to the first disruption in rate(s). Connection is re-established to the LOS path as the UE moves past the pillar until the next pillar is reached where the second disruption happens. The third disruption corresponds to the switch from the re-established LOS path to the dominant NLOS path as the UE turns the corner. Thus, these examples illustrate the robustness of our proposed beamforming solutions to blockages in real deployments. VI. C ONCLUDING R EMARKS A. Summary This article provides a brief overview of mm-Wave channel measurements and what the implications of these measurements are for system design. An immediate consequence of the blockage and penetration losses inherent and specific to mm-Wave systems are the poor link margins. These motivate the necessity to reap spatial array gains via the use of near-optimal beamforming solutions over large antenna arrays. Prominent challenges in this goal include the limited range and performance of mm-Wave components, as well as the robustness of the beamforming solution to spatio-temporal channel variations and its impact on overall system design. Further, cost considerations may allow only the use of a small number of RF chains at either end and thus the beamforming solutions should be adaptive to changes in the RF architecture(s). Towards this goal, directional beamforming approaches can be used as robust, low-complexity, near-optimal solutions that help overcome the high propagation losses at mmWave frequencies. Such solutions are demonstrated with our experimental prototype, illustrating 1 Note that the LOS link between eNB2 and UE is blocked by a pillar in the layout, all marked in white squares. 14 the viability of mm-Wave systems for high data-rate requirements. In particular, the prototype system demonstrates: i) beamforming and beam scanning, ii) outdoor coverage and mobility, iii) resilience to blockage of paths, iv) inter-base-station handover, v) indoor mobility, and vi) interference management in both outdoor and indoor settings. B. Future Research Directions Important issues that require further study include: i) a more exhaustive study on realistic channel modeling for mm-Wave propagation, ii) models for spatio-temporal channel variations, iii) models for impairments such as hand/body/human blockages, phase noise, etc., iv) advanced MIMO techniques for both single- and multi-user multi-carrier transmissions, v) impact of mm-Wave channel properties on mm-Wave system/network design issues such as coverage and network latency tradeoffs, mm-Wave handover, interworking with sub-6 GHz bands and applications, integrated access-bachkaul solutions, vi) advanced MIMO RF architectures such as [5]–[8] for prototype studies and real deployments, vii) RF tradeoffs in form factor UE design, etc. R EFERENCES [1] F. Boccardi et al., “Five disruptive technology directions for 5G,” IEEE Commun. Magaz., vol. 52, no. 2, Feb. 2014, pp. 74–80. [2] T. S. Rappaport et al., “Millimeter wave mobile communications for 5G cellular: It will work!,” IEEE Access, vol. 1, 2013, pp. 335–349. [3] R. W. Heath, Jr. et al., “An overview of signal processing techniques for millimeter wave MIMO systems,” IEEE Journ. Sel. Topics in Sig. Proc., vol. 10, no. 3, Apr. 2016, pp. 436–453. [4] W. Roh et al., “Millimeter-wave beamforming as an enabling technology for 5G cellular communications: Theoretical feasibility and prototype results,” IEEE Commun. Magaz., vol. 52, no. 2, Feb. 2014, pp. 106–113. [5] J. Brady, N. Behdad, and A. M. Sayeed, “Beamspace MIMO for millimeter-wave communications: System architecture, modeling, analysis and measurements,” IEEE Trans. Ant. Propagat., vol. 61, no. 7, July 2013, pp. 3814–3827. [6] Y. Zeng and R. Zhang, “Millimeter wave MIMO with lens antenna array: A new path division multiplexing paradigm,” IEEE Trans. Commun., vol. 64, no. 4, Apr. 2016, pp. 1557–1571. [7] J. A. Laurinaho et al., “2-D beam-steerable integrated lens antenna system for 5G E-band access and backhaul,” IEEE Trans. on Microwave Theory and Tech., vol. 64, no. 7, July 2016, pp. 2244–2255. [8] T. Kwon et al., “RF lens-embedded massive MIMO systems: Fabrication issues and codebook design,” IEEE Trans. on Microwave Theory and Tech., vol. 64, no. 7, July 2016, pp. 2256–2271. [9] V. Raghavan et al., “Millimeter wave channel measurements and implications for PHY layer design,” IEEE Trans. Ant. Propagat., vol. 65, no. 12, Dec. 2017. 15 [10] S. Hur et al., “Millimeter wave beamforming for wireless backhaul and access in small cell networks,” IEEE Trans. Commun., vol. 61, no. 10, Oct. 2014, pp. 4391–4403. [11] S. Sun et al., “MIMO for millimeter wave wireless communications: Beamforming, spatial multiplexing, or both?,” IEEE Commun. Magaz., vol. 52, no. 12, Dec. 2014, pp. 110–121. [12] O. El Ayach et al., “Spatially sparse precoding in millimeter wave MIMO systems,” IEEE Trans. Wireless Commun., vol. 13, no. 3, Mar. 2014, pp. 1499–1513. [13] V. Raghavan et al., “Beamforming tradeoffs for initial UE discovery in millimeter-wave MIMO systems,” IEEE Journ. Sel. Topics in Sig. Proc., vol. 10, no. 3, Apr. 2016, pp. 543–559. [14] V. Raghavan et al., “Single-user vs. multi-user precoding for millimeter wave MIMO systems,” IEEE Journ. Sel. Areas in Commun., vol. 35, no. 6, June 2017, pp. 1387–1401. [15] Qualcomm, “5G mm-Wave demonstration,” Accessed on Sept. 29, 2017. https://www.qualcomm.com/videos/5g-mmwave-demonstration, 

Preprint submitted to Energy and Buildings, November 2013 Pseudo Dynamic Transitional Modeling of Building Heating Energy Demand Using Artificial Neural Network Subodh Paudel a,b,c b c , Mohamed Elmtiri , Wil L. Kling , Olivier Le Corre a* , Bruno Lacarrière a a Department of Energy System and Environment, Ecole des Mines, Nantes, GEPEA, CNRS, UMR 6144, France b Environnement Recherche et Innovation, Veolia, France c Department of Electrical Engineering, Technische Universiteit Eindhoven, Netherlands *Corresponding author. Tel.: +33 2 51 85 82 57 E-mail: [email protected] Abstract This paper presents the building heating demand prediction model with occupancy profile and operational heating power level characteristics in short time horizon (a couple of days) using artificial neural network. In addition, novel pseudo dynamic transitional model is introduced, which consider time dependent attributes of operational power level characteristics and its effect in the overall model performance is outlined. Pseudo dynamic model is applied to a case study of French Institution building and compared its results with static and other pseudo dynamic neural network models. The results show the coefficients of correlation in static and pseudo dynamic neural network model of 0.82 and 0.89 (with energy consumption error of 0.02%) during the learning phase, and 0.61 and 0.85 during the prediction phase respectively. Further, orthogonal array design is applied to the pseudo dynamic model to check the schedule of occupancy profile and operational heating power level characteristics. The results show the new schedule and provide the robust design for pseudo dynamic model. Due to prediction in short time horizon, it finds application for Energy Services Company (ESCOs) to manage the heating load for dynamic control of heat production system. Keywords: Building Energy Prediction; Short term building energy forecasting; Operational Heating Characteristics; Occupancy Profile; Artificial Neural Network; Orthogonal Arrays 1. Introduction The global concerns of climate change and regulation in energy emissions have drawn more attention towards researchers and industries for the design and implementation of energy systems for low energy buildings. According to IEA statistics [1], total energy use globally accounts for around 7200 Mtoe (Mega Tonnes Oil Equivalents). Residential and commercial buildings consume 40% of final energy use in the world and European countries consume 76% of energy towards thermal comfort in buildings. The small deviations in design parameters of buildings could bring large adverse effect in the energy efficiency and which, additionally, results in huge emissions from the buildings. It is estimated that improvement in energy efficiency of the buildings in European Union by 20% will result in saving at least 60 billion Euro annually [2]. So, research is very active in driving towards the sustainable/low energy buildings. In order to accomplish this and to ensure thermal comfort, it is essential to know energy flows and energy demand of the buildings for the control of heating and cooling energy production from plant systems. The energy demand of the building system, thus, depends on physical and geometrical parameters of buildings, operational characteristics of heating and cooling energy plant systems, weather conditions, appliances characteristics and internal gains. There  are  various  approaches  to  predict   building energy demand based on physical methods and data-driven methods (statistical and regression methods and artificial intelligence methods) as mentioned by Zhao et al. [3]. Physical methods are based on physical engineering methods and uses thermodynamics and heat transfer characteristics to determine the energy demand of the building. There are numerous physical simulation tools developed as EnergyPlus [4], ESP-r [5], IBPT [6], SIMBAD [7], TRNSYS [8], CARNOT [9] etc… to compute the building energy demand. A simplified physical model based on physical, geometrical, climatic and occupant model was presented by Duanmu et al. [10] to bridge the complexities of collecting more physical data required in simulation tools. Other possible approaches for building energy prediction are semi-physical models like response factor method, transfer function method, frequency analysis method and lumped method [11]. Though methodologies adapted to estimate energy demand of buildings are different in physical and semi-physical models, both are highly parameterized. In addition, physical parameters of buildings are not always known or even sometimes data are missing. And also, these models are computationally expensive for Energy Services Company (ESCOs) to manage heating and cooling loads for control applications. Other approaches to predict building energy demand with limited physical parameters are data-driven methods, which strongly dependent on the measurements of historical data. Statistical and regression methods seem more feasible to predict building energy demand with limited physical parameters. The statistical approaches have been widely used by Girardin et al. [12] to determine the best model parameters by fitting actual data. Different approaches (physical and behaviour characteristics based on statistical data) were presented by Yao et al. [13] to bridge the gap between semi-physical and statistical methods. In their work, statistical daily load profile was grounded on energy consumption per capita and human behaviour factor, and semi-physical method was based on thermal resistance capacitance network. Nevertheless, these statistical models used linear characteristics of input and output variables to evaluate the building parameters and are not adapted to non-linear energy demand behavior. Regression models [14-15] have also been used to predict the energy demand, but, they are not accurate enough to represent short term horizon (couple of days) with hourly (or couple of minutes) sampling time energy demand prediction. In order to find the best fitting from the actual data, this kind of models requires significant effort and time. In recent years, there is a growth in research work in the field of artificial intelligence (AI) like artificial neural network [3, 16] and support vector machines [3, 17-18]. These methods are known for solving the complex non-linear function of energy demand models with limited physical parameters. Neural network method has shown better performances than physical, statistical and regression methods. Authors [19-20] used static neural network to predict energy demand of the building and compared results with physical models. For instance, Kalogirou et al. [19] used climate variables (mean and maximum of solar radiation, wind speed, and other parameters as wall and roof type) coupled with artificial neural network (ANN) to predict daily heating and cooling load of the buildings. In their work, results obtained using ANN are similar to those given by the physical modelling tool TRNSYS. Neto et al. [20] presented a comparison of neural network approach with physical simulation tool EnergyPlus. In this work, authors used climate variables as external dry temperature, relative humidity and solar radiation as input variables to predict daily consumption of the building. Results showed that neural network is slightly more accurate than EnergyPlus when comparing with real data. Static neural network model proposed by Shilin et al. [21] consider climate variables as dry bulb temperature and information regarding schedule of holiday’s to predict cooling power of residential buildings. Dong et al. [17] used support vector machine (SVM) to predict the monthly building energy consumption using dry bulb temperature, relative humidity and global solar radiation. Performance of SVM and neural network model wee compared and results show that SVM was better than neural network in prediction. Various authors [22-26] performed hourly building energy prediction using ANN. Mihalakakou et al. [22] performed hourly prediction of residential buildings with solar radiation and multiple delays of air temperature predictions as input variables. Ekici et al. [23] used building parameters (window’s transmittivity, building’s orientation, and insulation thickness) and Dombayci [24] used time series information of hour, day and month, and energy consumption of the previous hour to predict the hourly heating energy consumptions. Gonzalez et al. [25] used time series information hour and day, current energy consumption and predicted values of temperature as input variables to predict hourly energy consumption of building system. Popescu et al. [26] used climate variables as solar radiation, wind speed, outside temperature of previous 24 hours, and other variables as mass flow rate of hot water of previous 24 hours and hot water temperature exit from plant system to predict the space hourly heat consumptions of buildings. Li et al. [18] used SVM to predict hourly cooling load of office building using climate variables as solar radiation, humidity and outdoor temperature. In their work, SVM was compared with static neural network and result showed SVM better than static neural network in terms of model performance. Dynamic neural network method which includes time dependence was presented by Kato et al. [27] to predict heating load of district heating and cooling system based on maximum and minimum air temperature. Kalogirou et al. [28] used Jordan Elman recurrent dynamic network to predict energy consumption of a passive solar building system based on seasonal information, masonry thickness and thermal insulation. For many authors [29-31] occupancy profile has a significant impact on building energy consumption. Sun et al. [29] mentioned that occupancy profile period has a significant impact on initial temperature requirement in the building during morning. In their work, reference day (the targeted day prediction which depends on previous day and beginning of following day based on occupancy and non-occupancy profile period) was calculated based on occupancy profile period. In addition to this value, correlated weather data and prediction errors of previous 2 hours were used as input variables to predict hourly cooling load. Yun et al. [30] used ARX (autoregressive with exogeneous i.e., external, inputs) time and temperature indexed model with occupancy profile to predict hourly heating and cooling load of building system and compared this with results given by neural network. Results showed that occupancy profile has a significant contribution in determination of auto regressive terms during different intervals of time and further showed a variation of it in the building heating and cooling energy consumption. The proposed ARX model showed similar performance with neural network. Sensitivity analysis for heating, cooling, hot water, equipment and lighting energy consumption based on occupancy profile was performed by Azar et al. [31] for different sizes of office buildings. In their work, they found that heating energy consumption has the highest sensitivity compared to cooling, hot water, equipment and lighting energy consumption for small size buildings. Also, results showed that heating energy consumption is highly influenced by occupancy profile for medium and small buildings during the occupancy period. Moreover, few literatures focused on operational power level characteristics (schedule of heating and cooling energy to manage energy production from plant system). For example, Leung et al. [32] used climate variables and operational characteristics of electrical power demand (power information of lighting, air-conditioning and office equipment which implicitly depends on occupancy schedule of electrical power demand) to predict hourly and daily building cooling load using neural network. In conclusion, it can be reiterated that physical and semi-physical models [4-11], though give precise prediction of building energy, they are highly parameterized and are computationally expensive to manage the energy for control applications for ESCOs. Data-driven methods which depend on measurement historical data are not effective during the early stage of building operation and construction since measurement data are not available at these stages. When building energy data are available, data-driven methods can be considered if measurement data are accurate and reliable as this kind of models can be sensitive on the quality of measured data. Sensitivity of the accuracy of data driven models, thus, depends on the measurement data. Data-driven models based on statistical and regression methods [12-15, 26] cannot precisely represent short time horizon (couple of days) with hourly (or couple of minutes) sampling time prediction, though they perform prediction of energy consumptions of buildings with limited physical parameters. They also require significant efforts and time to compute the best fitting of the actual data. Static neural network models [19-21] are used for daily prediction and [22-25] are used for hourly prediction of the buildings energy consumptions. Though dynamic neural network model [27-28] gives better precision in compared to static neural network, they do not consider occupancy profile and operational power level characteristics of the plant system and therefore not adapted for the ESCOs to manage energy production for control applications. The important features like transition and time dependent attributes of operational power level characteristics of the plant system are still missing, though, authors [29-30] consider occupancy profile and author [32] considers operational characteristics of electrical power demand. The detailed variables and application of models developed in the literature reviews are summarized in Table 1. Table  1  :  Summary of variables and application models in the literature Input Variable of Model Climate Variables Author and Year Type of Model Outside Tempeature Inner Temperature Ambient Dry Bulb Wet Bulb Girardin et al. (2009) Yao et al. (2005) Catalina et al. (2008) Wan et al. (2012) Statistical Thermal and Statistical Regression Horizon of Forecast Annually √ (2*) Daily √ Regression √ √ Static NN Mihalakakou et al. (2002) Ekici et al. (2009) Dombayci (2010) Gonzalez et al. (2005) Popescu et al. (2009) Kato et al. (2008) Kalogirou et al. (2000) Li et al. (2010) Static NN Static NN Static NN Static NN Static NN Dynamic NN Dynamic NN SVM √(4*) Regression √ Autoregressive with exogeneous √ √ √ √ √ √ √ √ √ (3*) √ √ √ √ Monthly Monthly & Yearly Monthly Daily Daily Daily √ √ (5*) √ (6*) √ (7*) √ (8*) √ √ √ √ √(9*) √ (10*) √(11*) Static NN Physical Occupancy Operational Other Profile Characteristics Parameters √ (1*) Shilin et al. (2010) Leung et al. (2012) Duanmu et al. (2013) Relative Humidity √ SVM Static NN Static NN Yun et al. (2012) Wind Speed √ Dong et al. (2005) Kalogirou et al. (2001) Neto et al. (2008) Sun et al. (2013) Global Solar Radiation √(11*) √ √ √ √ √ √ √ √ √ √ √ √ √ √ (12*) √ √ √ (14*) √ √ (13*) Hourly Hourly Hourly Hourly Hourly Hourly Hourly Hourly Hourly Hourly √ √ (15*) √ (16*) Hourly & Daily Hourly Type of Applications for Buildings 80 Residential (heating and cooling) Residential (space heating) Residential Office (heating and cooling) 4 Buildings (total energy consumptions) 9 Buildings (heating and cooling) Office (3000 m2) Residential (cooling power) Residential (200 m2) Heating Energy of Buildings Residential (heating energy) Electrical load 8 Buildings District (heating energy) Passive solar buildings Office building and library Cooling load for high rise buildings (440,000 m2) Small building for heating load (464 m2) Office (space electrical power demand) Cooling load of buildings Remarks: 1*: Nominal Temperature of heating, cooling and hot water system; Threshold heating and cooling temperature 2*: Appliances Model 3*: Climate Index based on principal component 4*: Multiple lag output predictions of ambient air temperature 5*: Transmittivity, orientation and insulation thickness 6*: Heating degree hour method 7*: Predict value of temperature, present electricity load, hour and day 8*: Outside temperature and mass flow rate in previous 24 hour, hot water temperature 9*: Highest and Lowest open air temperature 10*:Season, insulation, wall thickness, heat transfer coefficient 11*: Multiple lag of dry bulb temperature and solar radiation 12*: Reference day of each day based on occupancy schedule 13*: Correlated weather data based on reference day and accuracy of calibrated prediction error of previous 2 hours 14*: Occupancy profile represented by space electrical power demand 15*: Clearness of sky, rainfall, cloudiness conditions 16*: Physical and geometrical parameters, hourly cooling load factor None of these studies has evaluated the transition and time dependent effects of operational power level characteristics of heating plant system and has predicted building heating energy demand in short time horizon (a couple of days). This short term prediction is important to ESCOs for dynamic control of heat plant system. This paper bridges the gap between static and dynamic neural network methods with occupancy profile and operational power level characteristics of heating plant system. It introduces novel pseudo dynamic model, which incorporates time dependent attributes of operational power level characteristics. Their effects on neural network model performances are compared to static neural network for building heating demand. Orthogonal arrays are applied to the proposed pseudo dynamic model for robust design and confirmed the new schedule of occupancy profile and operational heating power level characteristics obtained from ESCOs. The proposed method allows short term horizon prediction (around 4 days with sampling interval of 15 minutes) to make decision (e.g. management of wood power plant) for the ESCOs. The next section describes methodology including scope of study, design of transitional and pseudo dynamic characteristics, neural network model and orthogonal arrays. Finally, a case study is presented and results and discussion are drawn to analyze the performance of different static and pseudo dynamic models along with robustness of proposed pseudo dynamic model for heating demand prediction of the building. 2. Methodology Collection of climate data Collection of Building Heating Energy data Operational heating power level characteristics Settling and Steady State time of the control system Transitional and Pseudo Dynamic Model Occupancy Profiles Artificial Neural Network Static and Pseudo dynamic model for heating demand prediction The development and implementation of models proposed in this work are based on collection of real building heating demand, operational heating power level characteristics, climate variables and approximated occupancy profile data (see Appendix A for selection of relevant input variables). An outline of the methodology presented in this paper is shown in figure (1). The input of this methodology is in form of time-series climate and building heating energy data. The other inputs data are occupancy profile and operational heating power level characteristics for working and off-days for 24 hours. Dynamics of building heating demand is also an input to the methodology which includes settling and steady state time and is estimated from real building data. Based on operational heating power level and dynamics of building characteristics, transitional and pseudo dynamic models are designed. Finally, neural networks for static and pseudo dynamic models are designed to predict heating demand in short time horizon (couple of days). For the robustness of pseudo dynamic model, occupancy profile and operational power level characteristics are analyzed for different time intervals to confirm occupancy schedule profile and operation of plant system from the orthogonal arrays. The pseudo dynamic model after optimum orthogonal arrays design is used for final prediction of the building heating demand. Scope of this study, details of transitional and pseudo dynamic model, neural network model and orthogonal arrays are described in section 2.1 - 2.4. Figure 1: Outline of the proposed methodology on heating demand prediction 2.1 Scope of Study The scope of this paper is heating demand prediction in short time horizon for the large building. The overall objective is to make an energy services decisions (e.g. management of wood power plant) for ESCOs. The assumptions carried for this study are highlighted as: 1. Winter period is studied. 2. Existing building is considered and space heating demand of this building is fed up from a heat network to a central substation. Domestic hot water (DHW) is out of the scope. 3. The heating demand data was recorded in data acquisition system database and thermal comfort inside the building was performed in this database. Thus, the effects of ventilation and air-conditioning on heating are already included in this database. 4. Simple occupancy profile of building is anticipated approximately to assist the ESCOs to schedule their heat production system. In such a system, individual occupant’s behavior or precise occupancy profile is not considered. Thus, the modeling constraints are closer to the operational condition of ESCOs to estimate the heat demand. 5. The wind speed and direction are not taken into consideration. This is due to the fact that present weather variables data are taken from data acquisition system but future weather variables values are coming from an atmospheric modeling system which mesh size can be 15 km (as ARPEGE, see [33]), 10 km (as ALADIN, see [34]) or 2.5 km (as AROME, see [35]). In such a case, wind impact on heating demand prediction of a specific building located inside the mesh is very difficult or even impossible to consider for precise effect. Further, heating energy demand is highly dependent on outside temperature and other climate variables have less significant impact on heat energy [36]. 2.2 Transitional and Pseudo Dynamic Model The operational heating power level characteristics gives operational features of the plant system, however, they do not give abstract information about transition attributes of operational heating power level which is illustrated through an example in figure (2). The y-axis represents set up power level from the production system and x-axis represents operation schedule. 100 3' State 1 3 90 4 4' 7' 7 α 43 State 3 8' 8 α 87 40 Transition 3 Transition 2 Transition 1 Transition 0 Power Level (%) 75 State 2 5' 6' 5 α 65 6 25 1 10 1' State 0 State 4 2' 2 α 21 0 9 6 12 Hour 14 20 9' α 109 10' 10 24 Figure 2: Operational heating power level characteristics of the plant system (for a day) In figure (2), operational power levels are identified by different states and transition levels and each level has its own significant effects on the operational power level characteristics. State means consistency in the power level from one operation schedule to another and transition means change in power level from one operation schedule to another in heat production system. The transition level 0, 1, 2 and 3 have similar feature of transitional power level characteristics on the overall operational performance, however, power level required for transition from point 2 to 3, point 4 to 5, point 6 to 7 and point 8 to 9 is different for each level. If the power level of state 0, 1, 2, 3 and 4 in operational heating power level characteristics is represented by the from point α uv , then the power required for transition v to point u can be represented as β uv in the transitional characteristics as shown in figure (3). Thus, the power level transition in transitional characteristics corresponding to operational characteristics can be written as: β uv = β (u −2 )(v−2 ) + 2Δβ α uv − α (u −2 )(v−2 ) , ∀ u = 4,6,8....., v = 3,5,7... where, (1) , v = 1, u = 2 β0 β 0 , Δβ and represents initial power level, step size of transition power level and absolute values respectively. Each level ( β 21 , β 43 , β 65 , β 87 and β 109 ) represents transitional level and depends on the power level of operational characteristics. 9 β109 Transition Level 7 Transition Level β87 β87 β109 10 8 5 β65 6 3 β43 β43 4 PDL 1 β21 0 β21 2 6 12 14 Hour 20 24 Figure 3: Transitional and Pseudo dynamic characteristics (for a day) The transitional characteristics explicate the power transition level of operational characteristics, however, dynamic information of power level attributes is still lacking. It means that power content in operational characteristics of figure (2) of point 1-1’ is not equal to 2-2’; point 3-3’ is not equal to the 4-4’; 5-5’ is not equal to 6-6’; 7-7’ is not equal to 8-8’ and 9-9’ is not equal to 10-10’. Dynamic transition information, thus, is necessary in the model which considers dynamic characteristics of the building. The simple first order dynamics of building characteristic is shown in figure (4), where τ represents time constant. Amplitude of Heating Power (%) 100 80 60 Tsteady 40 Ts τ 20 delay 0 -20 0 τ Time Constant 5τ 6τ Figure 4: Dynamics of building characteristics In figure (4), delay represents time it takes from plant system to reach the building for heating operation and after this, power is sufficient to provide heating demand. The τ represents the 63% of power transferred to the building heating system from plant system. Other dynamics to incorporate is settling time ( Ts ), which is the time elapsed for heating power to reach and remain within the specified error band and equal to [2 τ , 5 τ ] and have almost similar behavior like steady state time. The steady state time corresponds to [3 τ , 6 τ ]. Thus, τ , settling time ( Ts ) and steady state time ( Tsteady ) gives information about dynamic characteristics of heating demand. This dynamic information of building, thus, depends on the transitional attributes of power level and this information is not totally dynamic but pertaining to the appearance of dynamic behavior, so pseudo dynamic name is chosen. Thus, pseudo dynamic is just a lag of transitional attribute information and further depends on time constant τ or range between settling and steady state of the dynamic building heating characteristics. The simplified pseudo dynamic lag (PDL) is calculated from equation (2), where, ts represents the sampling time of building data and Tu represents the new unknown time which lies between settling and steady state time. The concise value of Tu depends on dynamics of the heating demand and pseudo dynamic characteristics can be seen from figure (3), where PDL is pseudo dynamic lag. Ts ≤ Tu ≤ Tsteady , where Ts ∈ [2τ ,5τ ]; Tsteady ∈ [3τ ,6τ ] PDL ∈ τ ts [3,6] (2) 2.3 Neural Network Model The neural network consists of neurons to interconnect the inputs, model parameters and activation function. Each interconnection between the neurons represents model parameters. Input- output mapping in neural network is based on the linear and non-linear activation function. From input and targeted data, model parameters are adjusted to minimize the error i.e. difference between actual values and predicted values produced by the network. Learning/training of data are repeated until there is no significant change in the model parameters and only stops the training. This type of learning approach is called supervised learning since predicted value of the model is guided by actual values. There are numerous ANN model like Feed-forward Multilayer Perceptron (MLP), Radial Basis Function (RBF) Network, Recurrent Network and Self-Organizing Maps (SOM) [37]. All of these networks have their own learning algorithm to learn and generalize the network. In this paper, MLP is taken as a neural network model since pseudo dynamic model is not fully dynamic (in time behavior). There are two ways of learning mechanism in the neural network: sequential learning and batch learning. In sequential learning, cost function is computed and model parameters are adjusted after each input is applied to the network. In batch learning, all the inputs are fed to the network before model parameters are updated. In batch learning, model parameter adjustment is done at the end of epoch (one complete representation of the learning process) and for this paper, batch learning is carried out. MLP network consists of three layers: input layer, hidden layer and output layer and there can exist more than one hidden layer. However, according to the Kolmogorov’s theorem [38], single hidden layer is sufficient to map the function provided suitable hidden neurons and for this paper, single hidden layer is used as shown in figure (5). The hidden layer assists to solve non-linear separable problems. Figure 5: Neural Network Architecture In figure (5), xi , wk and neuron which varies from lag of 1 and z -M y represents input neuron which varies from i = 0 to i = q , hidden k = 0 to k = p and output neuron respectively. The z-1 signifies transition signifies transition lag corresponding to PDL, where maximum value of M ( M max ) equals to PDL i.e. M max = {1,2,.....PDL}. The MLP uses logistic function or hyperbolic tangent as a threshold function in the hidden layer. It has been identified empirically [39] that network using logistic functions tends to converge slower than hyperbolic tangent activation function in the hidden layer during the learning phase. Hyperbolic tangent activation functions is chosen in the hidden layer and pure linear activation function is chosen in the output layer for this paper and hyperbolic tangent T function is shown in equation (3), where θ represents model parameter with transpose of matrix. Division of input and output data into learning, validation and testing gives more generalization of model. Learning data sets are used to learn the behavior of input data and to adjust the model parameters. Validation data is used to minimize the overfitting. It is not used to adjust the model parameter but it is used to verify if any increase in accuracy over learning dataset actually yields an increase in accuracy over dataset that has not learned to the network before. Testing data sets are used to confirm the actual prediction from neural network model which is unknown to neural network before. For this paper, data is divided into learning, validation and testing sets. Normalization of input data is also important for faster convergence to achieve desire performance goal. If input data are poorly scaled during learning process, there is a risk of inaccuracy and slower convergence. It is, thus, essential to standardize the input data before applying to neural network. There are various methods for normalization of input and output variable, and for this paper, normalization with zero mean and i unit standard deviation is done as shown in equation (4). In equation (4), x , X and m represents mean of input variable, overall vector of input variable and number of datasets respectively and thus, applies similarly for output variable. ( ) T hθ ,x = eθ T x − e −θ T x eθ T x + e −θ T x (3) xi − x Xi = (4) 1 ( xi − x ) ∑ m −1 i The cost function of MLP network is computed in equation (5): 1 m (l ) (l ) J (θ ) = y − ya ∑ 2m l =1 [ ] 2 (5) y , y a , l and J (θ ) represents predicted values produced from the network, actual values of given datasets, individual data from m number of datasets and cost function of the neural network model respectively. Further, y of the network is computed as: where p ⎛ q ⎞ y = ∑θ k h⎜⎜ ∑θ ki xi ⎟⎟ k =1 ⎝ i =0 ⎠ (6) In order to update the model parameters for a higher degree approximation on unknown nonlinear function for learning process, there are different methods as – gradient descent, Newton’s method and so on [37]. Gradient descent is too slow for the convergence, and it takes more time to compute the hessian matrix in Newton’s method as well. Levenberg-Marquardt algorithm is used for this paper which takes approximation of hessian matrix in the form of Newton’s method and model parameter update equation [ θ t +1 is given as: ] θ t +1 = θ t − [LT L + µI ] LT J (θ ) −1 (7) T In equation (7), hessian matrix is approximated as [ L L] LT J (θ ), parameter, µ is and gradient is computed as L is Jacobian matrix, J (θ ) is vector of cost function, θ t   is initial model suitable chosen scalar and I is identity matrix. Update model parameter, thus, depends on the cost where, function and scalar value µ . 2.3.1 Stopping Criteria There are different criteria for stopping the neural network model. For this paper, the stopping criteria depend on number of epochs to learn the network, performance goal, maximum range of µ and maximum failures in the validation. The performance goal (PG) is given as: m PG = 0.01 ∑ y a (l ) (8) l =1 The maximum failures in validation or accuracy over validation datasets is defined to stop the learning process if the accuracy of learning datasets increase and validation accuracy stays same or decrease. 2.3.2 Model Performance Performances of models are characterized by mean square error (MSE) and coefficient of 2 2 correlation (R ). The MSE and R can be calculated as: m MSE = ∑ [y − ya (l ) l =1 ] (9) m m R2 = 2 (l ) ∑ [y ( ) − y (l ) l a l =1 m ] 2 (10) ∑ (y ) (l ) 2 a l =1 2.3.3 Degree of Freedom Adjustment One of the issues of neural network model is over learning of the network. With increase of hidden neurons, model performance can be increased, but, it will lead neural network to over learning. Validation accuracy and degree of freedom (DOF) adjustments are done in this paper to avoid over fitting. Number of learning equations that model could deliver are given by equation (11), where learning equations of the network and Le is L y is length of vector output neurons ( y ), and in this case equal to 1 since there is only heating demand load. Le = m * L y (11) The number of model parameters for a single hidden layer MLP neural network are given by the equation (12), where Lθ , L x and Lw represents number of model parameters, vector length of input neurons ( xi ) and vector length of hidden neurons ( wk ) respectively. Lθ = (Lx + 1) * Lw + (Lw + 1) * L y (12) DOF of neural network model is the difference between number of learning equations and number of model parameters in the network. It should be always >>1 and depends on the optimum size of hidden neurons. DOF and maximum hidden neurons are given by equation (13) and (14), where, δ represents the scalar constant value and depends on DOF required for design and Wmax is the maximum hidden neurons. DOF = Le − Lθ Wmax ≅ 1 (L (13) − Ly ) θ (14) δ (L x + L y + 1) Modified performance goal according to degree of freedom adjustment is given as: m PG = 0.01 DOF ∑ y a (l ) l =1 (15) Le Model performance is also further modified based on degree of freedom adjustment. The 2 modified MSE and R can be calculated as: m MSE mod ified = l =1 (l ) ] 2 (16) DOF * m m R 2 mod ified = [ Le ∑ y (l ) − y a [ Le ∑ y (l ) − y a l =1 m ] 2 (17) ( ) DOF ∑ y a l =1 (l ) (l ) 2 For each hidden neurons, optimal MSE mod ified and maximum R 2 mod ified for learning and validation are calculated from the different initialized random parameters. For different number of hidden neurons, R 2 mod ified and MSE mod ified for each model is performed for learning and validation, and based on it, optimal configuration of model is identified for the final prediction. 2.4 Orthogonal Arrays It is essential to know whether schedule of occupancy profile and operational characteristics obtained from ESCOs is reliable for the robust design of pseudo dynamic model. Occupancy profile and operational characteristics transition period, thus, plays an important role in the model performance and if all these transition period are consider for finding the best robust model, it takes long time to compute. Orthogonal arrays (OA) identify the main effects with minimum number of trials to find the best design. These are applied in various fields: mechanical and aerospace engineering [40], electromagnetic propagation [41] and signal processing [42] for the robust design model. The orthogonal array allows the effect of several parameters to find best design with given different levels of parameters. It can be defined as matrix with column representing number of parameters with different settings to be studied and rows representing number of experiments. In orthogonal arrays, parameters are called factors and parameter settings are called levels. In general, OA(N , k , s, t ) is used to represent the orthogonal arrays, where N , k , s and t represents number of experiments, number of design parameters, number of levels and strength. There are different methods as Latin square [43]; Juxtaposition [44]; Finite geometries [45] etc... to create orthogonal arrays with different strength and levels. Orthogonal arrays with different number of design parameter, level, and strength are available from OA databases or libraries. The orthogonal arrays used for this paper is taken from OA library [46]. 3. Case Study The methodology is applied for case study at Ecole des Mines de Nantes, French Institution. 2 The building has floor area of 25,000 m . It has 600 students and 200 employees. The building consists of 120 research and administration rooms, 30 class rooms, 3 laboratories, and 8 seminar halls. Class rooms have different sizes and can accommodate to 18 to 28 students. The 2 big seminar halls can be occupied by 250 students and 6 small seminar halls can be occupied by 80 students. 2 Each floor area of the laboratory is 600 m . The data is taken from data acquisition system and consists of day/month/time, solar radiation, outside air temperature and heating demand from mid of January to February 2013 with sampling interval of 15 minutes. The 70% of data (outside temperature, solar radiation and heating demand as shown in figure 5) are used for learning phase i.e. in mathematical equation in neural network, see section 2.3, equivalent to 19 days with 15 minute sampling time, and each 15% of data (4 days with 15 minute sampling time) is used for validation and testing phase. Outside temperature taken for this 0 0 0 study has minimum, average and maximum value of 1.2 C, 8.95 C and 15.3 C respectively. Global 2 2 solar radiation has an average and maximum value of 7 W/m and 438 W/m respectively. The simplified/theoretical occupancy profile and operational heating power level characteristics for working and off-days for 24 hours is shown in figure (6) and (7). 1000 Working Day Off Day 900 800 Number of Occupants 700 600 500 400 300 200 100 0 0 8 12 13:30 Hour 17:45 Figure 6: Occupancy profiles for working and off-day 24 Working Day Off Day 100 90 Power Level (%) 75 40 25 10 0 6 12 Hour 14 20 24 Figure 7: Operational heating power level characteristics for working and off-day Power demand and occupancy profile during working day is depicted from figure (8). From figure (8), occupancy profile almost gives information about power demand characteristics, however, from 18 hour onwards, power demand characteristics is not accordance with occupancy profile. Thus, it further shows that simplified occupancy profile is not enough to characterize the heating demand. Working Day Off Day 100 90 Power Level (%) 75 40 25 10 0 6 12 14 Hour 20 24 Figure 8: Heating power demand and occupancy profile during working days Different neural network models are designed based on climate variables (outside temperature and solar radiation), work/off day information, occupancy profile and operational characteristics as shown in figure (5). For this case study, 10 represent working day and 5 represent off day information (work/off day) to the input of neural network model. Static neural network model 1 consists of operational characteristics and occupancy profile, external temperature and solar radiation as input variables and heating power demand as an output variable, and thus, vector length of input neurons ( L x ) in equation (12) equals to 5. Model 2 comprises additional transitional characteristics in model 1 and vector length of input neurons ( L x ) in equation (12) equal to 6. For this case study the sampling time ( ts ) of real building data is 15 minutes, settling time ( Ts ) is estimated approximately 45 minutes and steady state time ( Tsteady ) is approximately 1 hour. The PDL, thus, is calculated from equation (2), where PDL corresponds to settling and steady state time is nearly equal to 3 and 4 respectively. Since pseudo dynamic model depends on transition lag of operational heating power level and building dynamic characteristics, PDL is varied from 3-4, and to understand the phenomena of pseudo dynamic lag, PDL is varied from 1-4. Model 3 comprises model 2 with additional parameters of one PDL i.e. i.e. L x equals to 7; model 4 consists model 2 with additional parameters of two PDL i.e. model 5 includes model 2 with additional parameters of three PDL i.e. L x equals L x equals to 8; to 9 and model 6 comprises model 2 with additional parameters of four PDL in the transitional characteristics i.e. L x equals to 10. Transitional and pseudo dynamic characteristic with four lags during working day is shown in figure (9). Transition level in figure (9) is calculated from equation (1) and for this case study, 25 is chosen for each β0 and Δβ . In figure (9), lag 0 means static model which contains transition attributes, lag 1 means pseudo dynamic model with transition lag 1 (PDL=1), lag 2 means pseudo dynamic model with transition lag 2 (PDL=2) and so on. Further, effects of transitional and pseudo dynamic effects on the heating demand can be understood from figure (10). It is clear that the information hidden in heating demand which climate variables could not answer can be justify from transitional and pseudo dynamic attributes of operational characteristics. The summary of models is shown in table (2). No Lag Lag 1 Lag 2 Lag 3 Lag 4 Transition Level 825 575 425 275 25 0 6 7 12 13 14 15 Hour 20 21 24 Figure 9: Transitional and pseudo dynamic characteristics during working day Heating Demand No Lag Lag 1 Lag 2 Lag 3 Lag 4 1000 1000 Pseudo Dynamic transitional effects 800 800 Heating Demand (kW) Transition effects 600 PDL 600 400 400 200 200 0 0 24 48 Hour 72 82 Figure 10: Pseudo dynamic transitional effects on heating demand Table 2: Summary of models Model No. Type of Model Input Variables Remarks Model 1 Static Climates, occupancy profile and operational characteristics No Lag Model 2 Static Model 1 with transitional characteristics No Lag Model 3 Pseudo Dynamic Model 2 with pseudo dynamic transition in dead band Lag 1 Model 4 Pseudo Dynamic Model 2 with pseudo dynamic transition in t Lag 2 Model 5 Pseudo Dynamic Model 2 with pseudo dynamic transition in settling time Lag 3 Model 6 Pseudo Dynamic Model 2 with pseudo dynamic transition in steady state time Lag 4 For each model, cost function J (θ ) in equation (5) is computed iteratively up to 1000 for each of the minimum and maximum number of hidden neurons. The maximum number of hidden neurons is calculated from equation (14), where δ is chosen 8 as it gives the flexibility in the degree of model parameters. Thus, three minimum hidden neurons are chosen as 3 for this case study. Hidden neurons length ( Lw ), thus, is varied from 3 to Wmax . Performance of model at each iteration (number of epochs) is computed from equation (16) and (17) and model parameters are updated based on equation (7), where initial value of µ is chosen as 0.01 and its value is increased with a factor of 10 and decreased with a factor of 0.1. The maximum value of µ is chosen as 1e10. Neural network model in this study will be stopped if the number of epochs reached to 1000 and performance goal reached the value given by equation (15). Under the scope of study (see subsection 2.1), the accuracy on the number of occupants are not relevant, however, it is essential to know inside the sampling time, when the staff and students come and leaves the buildings. It is necessary to check occupancy and operational power level characteristics provided by ESCOs are right or not for robust design model. And, the main controlling factors for robust design model are the transition schedule of occupancy and operational characteristics. From figure (6), it is clear that there is no transition of occupancy during off-day, but there is transition of occupancy during the interval at 8 hour, 12 hour, 13:30 hour and 17:45 hour and these are represented by t1, t2, t3 and t4 factors respectively. Similarly, there is a transition of operational characteristics for working and off day as shown in figure (7) and these transition factors are represented by t5, t6, t7 and t8 for working day for 6 hour, 12 hour, 14 hour and 20 hour; t9 and t10 for off day for 6 hour and 20 hour. Since the sampling interval taken for this case study is 15 minutes, three levels are used for orthogonal arrays so that the model will represent the 15 minutes ahead and before from occupancy and operational characteristics schedule period. The summary of control factors and their levels are shown in table (3), where OSW represents occupancy schedule at work day, OCSW represents operational characteristics schedule at work day and OCSO represent operational characteristics schedule at off day. Table 3: Summary of control factors and their levels Levels Factors 1 2 3 OSW at 8 hour (f1) t1-15 min t1 t1+15 min OSW at 12 hour (f2) t2-15 min t2 t2+15 min OSW at 13:30 hour (f3) t3-15 min t3 t3+15 min OSW at 17:45 hour (f4) t4-15 min t4 t4+15 min OCSW at 6 hour (f5) t5-15 min t5 t5+15 min OCSW at 12 hour (f6) t6-15 min t6 t6+15 min OCSW at 14 hour (f7) t7-15 min t7 t7+15 min OCSW at 20 hour (f8) t8-15 min t8 t8+15 min OCSO at 6 hour (f9) t9-15 min t9 t9+15 min OCSO at 20 hour (f10) t10-15 min t10 t10+15 min Thus, there are 10 factors and 3 levels that govern the robustness of the model and if the full 10 factorials are used to generalize the model, it takes 3 = 59049 experiments. The orthogonal arrays reduce the number of experiments to 729 with 5 strengths. OA (729,10,3,5) is applied to the proposed pseudo dynamic model in this case study. 4. Result and Discussion Optimal configuration of the model is based on maximum R 2 mod ified and minimum MSE mod ified   from different random initialized parameters. For each hidden neurons in the model, five random initialized parameters is assigned for learning phase and based on it, the neurons with minimum MSE mod ified and maximum R 2 mod ified for learning and validation are chosen from random initialized parameters. Optimal configuration of each model is chosen from maximum minimum R 2 mod ified and MSE mod ified   model performance from learning and validation datasets for different hidden neurons. Figure (11) and (12) shows R 2 mod ified and MSE mod ified performance for learning, validation and testing for different hidden neurons sizes of model 5 and from this optimal configuration is chosen from the best performance model. It is clear from figure (11) and (12) that the maximum and minimum R 2 mod ified   MSE mod ified performance is achieved in hidden neuron size 13 and which is the optimal 2 configuration of the model. It can also be noticed that although R testing performance increases for 2 hidden neuron size 15, R for validation and learning does not increase optimally. The model 5 is just an example and similarly, the process is repeated for each model to find the optimal configuration of the neural network model. The optimal configurations of the different neural network model are summarized in table (4). Learning Validation Testing 0.95 Best Performance Coefficient of Correlation 0.9 0.85 0.8 0.75 0.7 0.65 2 4 6 8 10 12 14 Hidden Neurons 16 18 Figure 11: Coefficient of correlation performance (Model 5) 20 22 Learning Validation Testing 0.55 0.5 0.45 Mean Square Error 0.4 0.35 0.3 Best Performance 0.25 0.2 0.15 0.1 2 4 6 8 10 12 14 Hidden Neurons 16 18 20 22 Figure 12: Mean Square Error performance (Model 5) Table 4: Optimal configuration of models Model Hidden Coefficient of Correlation Neurons Learning Validation Testing Mean Square Error Learning Validation Testing Model 1 10 0.82 0.81 0.61 0.18 0.18 0.40 Model 2 19 0.87 0.85 0.80 0.13 0.15 0.21 Model 3 7 0.88 0.86 0.75 0.12 0.14 0.25 Model 4 9 0.89 0.87 0.82 0.12 0.13 0.18 Model 5 13 0.89 0.87 0.83 0.11 0.13 0.18 Model 6 9 0.89 0.87 0.85 0.11 0.13 0.15 Table (4) shows that with static neural network model 1, best R 2 mod ified for learning and validation can be obtained up to 0.82 and 0.81. From this, it is clear that occupancy profile and operational characteristics are not enough to determine and generalize the unknown function of the building heating demand. As transitional attributes of operational characteristic is introduced in model 2, R 2 mod ified model performance increases significantly from 0.82 to 0.87 for learning phase and from MSE mod ified decreases in contrast to model 1. Pseudo dynamic transitional attributes in model 3 and time constant τ in model 4 leads increase in 0.81 to 0.85 for validation phase and correspondingly model performance. Further, dynamics of settling time and steady state plays an important role in characterizing the neural network model. It is seen that R 2 mod ified performance increases from 0.87 to 0.89 for learning and 0.85 to 0.87 for validation in model 5 compare to model 2 although transition attributes is introduce in model 2. In addition, hidden neuron size is also reduces from 19 to 13. Moreover, it is distinguish that learning and validation performances remained the same in the model 6 compared to model 5. The optimal choice of the model, thus, lies in between settling and steady state time. It can be further view that model 5 and model 6 show reasonable and consistent model performances. However, minimum hidden neuron size and maximum learning criteria is essential for the overall network generalization. Since the hidden neurons size decreases from 13 to 9 and model performance R 2 mod ified remained the same (0.89) in model 6 comparing to model 5, model 6 is chosen as the best configuration of the overall models. The optimal choice of the model 5 and model 6 can be delineated by the error in percentage of energy consumption (kWh) in actual and prediction for the learning and validation phase. Heating energy consumption error in actual and prediction in learning phase in Model 6 is 0.02% compare to 0.32% in Model 5. For validation phase, heating energy consumption error is 2.39% in Model 6 compare to 2.57% in Model 5. From this energy consumption error, it is clear that there is a small heating energy consumption error in Model 6 compare to Model 5 during the learning and validation phase. So, one can conclude that Model 6 can be chosen as optimal configuration of the overall model. The model 6, thus, bridges the gap between static and dynamic neural network model in the sense that it is better than static model and increases the performance comparable to dynamic neural network model. For the robustness of pseudo dynamic model, orthogonal arrays are applied to determine the highest coefficient of correlation for learning and validation for the optimum 9 hidden neuron size of model 6. Table (5) shows OA(729,10,3,5) and coefficient of correlation for learning and validation phase. It is clear from table (5) that the schedule taken from the ESCOs is from experiment 1 and from the orthogonal arrays, the optimal schedule that fits the best for model 6 is experiment 398. The orthogonal arrays, thus, ensures that there is transition in occupancy in 7:45 hour, 12 hour, 13:45 hour and 18 hour instead of 8 hour, 12 hour, 13:30 hour and 17:45 hour period in the existing case respectively. There is also a transition in 5:45 hour, 11:45 hour, 14 hour and 17:45 hour instead of 6 hour, 12 hour, 14 hour and 17:45 hour for working day; 5:45 hour and 20 hour instead of 6 hour and 20 hour in off days for operational characteristics. The coefficient of correlation after the orthogonal array design is 0.90 for learning, 0.88 for validation and 0.86 for training phase. Nevertheless, other issue of overall model is that it is difficult to increase the coefficient of correlation beyond 0.90 and this is due to the sampling time of 15 minutes. With short sampling time, it is very difficult to learn the datasets which changes in 15 minutes sample, nonetheless, for good generalization of the model, R 2 mod ified value of 0.90 during the learning phase is always acceptable. Coefficient of correlation of linear regression obtained from neural network model in the actual and prediction of heating demand for learning, validation and testing phase of Model 6 after optimum orthogonal array design are 0.95, 0.95 and 0.93 respectively. The prediction of heating demand for model 6 after optimum orthogonal array design during validation phase is shown in figure (13). Prediction gives the power heating demand and the area under the curve gives the heating energy demand. From figure (13), it is clear that heating demand tremendously increases approximately 990 kW during third and fourth day and pseudo dynamic model is able to predict and learn the behavior. However, there is a fluctuation in the power demand in the morning for each consecutive 4 days and it is difficult to learn datasets which transits rapidly in actual power demand. The prediction of heating demand for model 6 during testing phase after optimum orthogonal array design is shown in figure (14). It is vivid that pseudo dynamic model is able to predict heating demand, however during the third day, the pseudo dynamic model is not able to meet 1.1 MW of heating demand. This is due to the fact that neural network does not learn this threshold maximum heating demand in the learning phase as this kind of information is not available in the database. This data, thus, needs to be improved in the learning phase through feature extraction techniques. Nonetheless, pseudo dynamic model (model 6) prediction is in accordance to the actual target except for some rapid transits in the actual target. To sum up, pseudo dynamic transition attributes in model 6 after orthogonal array design leads best prediction of heating demand. Table 5: OA(729,10,3,5) and coefficient of correlation for learning and validation for model 6 Element f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 1 2 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 1 3 3 2 2 2 2 2 2 4 2 1 2 2 2 2 3 5 1 1 2 2 2 2 6 3 1 2 2 2 7 2 3 2 2 8 1 3 2 2 9 3 3 2 10 2 2 11 1 12 3 …. Experiment Coefficient of Correlation Learning Validation Testing 2 0.89 0.87 0.85 1 1 0.89 0.88 0.81 3 3 3 0.90 0.86 0.76 2 1 3 0.89 0.86 0.79 3 1 3 2 0.89 0.87 0.78 2 3 3 2 1 0.89 0.88 0.79 2 2 1 2 3 1 0.89 0.87 0.83 2 2 1 1 2 3 0.89 0.87 0.84 2 2 2 1 3 1 2 0.90 0.86 0.85 1 2 2 2 3 1 2 1 0.89 0.87 0.76 2 1 2 2 2 3 3 1 3 0.89 0.87 0.67 2 1 2 2 2 3 2 3 2 0.89 0.87 0.67 . . . . . . . . . . . . . … . . . . . . . . . . . . . 394 2 3 1 3 1 1 3 3 3 3 0.89 0.87 0.80 395 1 3 1 3 1 1 3 2 2 2 0.90 0.87 0.76 396 3 3 1 3 1 1 3 1 1 1 0.90 0.87 0.76 397 2 2 3 3 1 1 2 2 2 3 0.89 0.88 0.81 398 1 2 3 3 1 1 2 1 1 2 0.90 0.88 0.86 399 3 2 3 3 1 1 2 3 3 1 0.90 0.87 0.70 400 2 1 3 3 1 1 3 2 1 1 0.90 0.87 0.77 401 1 1 3 3 1 1 3 1 3 3 0.89 0.88 0.84 402 3 1 3 3 1 1 3 3 2 2 0.87 0.88 0.74 …. . . . . . . . . . . . . . … . . . . . . . . . . . . . 725 1 1 3 3 3 3 2 1 2 3 0.89 0.87 0.80 726 3 1 3 3 3 3 2 3 1 2 0.90 0.87 0.84 727 2 3 3 3 3 3 3 2 2 2 0.90 0.87 0.80 728 1 3 3 3 3 3 3 1 1 1 0.89 0.88 0.78 729 3 3 3 3 3 3 3 3 3 3 0.89 0.88 0.61 Validation Phase 1000 Actual Predict 900 Heating Demand (kW) 800 700 600 500 400 300 0 24 48 Hour 72 96 Figure 13: Prediction of heating demand in model 6 during validation phase (after optimum orthogonal array design) Test Phase 1100 Actual Predict 1000 Heating Demand (kW) 900 800 700 600 500 400 300 0 24 48 Hour 72 96 Figure 14: Prediction of heating demand in model 6 during testing phase (after optimum orthogonal array design) 4. Conclusion This paper introduces pseudo dynamic transitional model for the building heating demand prediction in a short time horizon using artificial neural network. Occupancy profile and operational heating power level characteristics are included in the model. Dynamic characteristic of the building is included in the model for the determination of pseudo dynamic transition lag. Settling time and steady state time of the heating demand give an increment in precision of the model, however, choice of model depends on their actual time between settling and steady state. The results were based on case study where occupancy profile is already known and results may vary for more fluctuating occupancy buildings. Coefficient of correlation increases from 0.82 to 0.89 for learning, 0.81 to 0.87 for validation and 0.61 to 0.85 for testing in pseudo dynamic comparing to static neural network model. Also, the size of hidden neuron is further reduced, which reduces complexities and increases generalization of the model. Moreover, minimum energy consumption error is achieved in pseudo dynamic model as 0.02% for learning and 2.57% for validation phase. Further, orthogonal array is applied to optimal pseudo dynamic model to confirm the schedule of occupancy profile and operational level characteristics, and robustness of the model. The orthogonal array design leads to the increases in coefficient of correlation in pseudo dynamic model and confirmed the new schedule of the occupancy profile and operational level characteristics. The major contribution of this paper, thus, is the introduction of transition and novel time dependent attributes of operational heating power level characteristics, which is the dominant factor for building heating demand. Also, orthogonal array design in the model makes flexibility in cross checking the schedule of occupancy profile and operational heating power level characteristics obtained from ESCOs to design the robust model. The prediction is in short time horizon (4 days) with sampling interval of 15 minutes and thus useful for dynamic control of building heating demand. Further, research will be focused towards the feature extraction of data before learning phase of the neural network so that abnormalities in the data can be corrected in the learning phase. Also adaptive and real time learning criteria with seasonal behaviour will be studied. Acknowledgement This research has been done in collaboration with Ecole des Mines, Nantes, Technische Universiteit Eindhoven and VEOLIA Environnement Recherche et Innovation, funded through Erasmus Mundus Joint Doctoral Programme SELECT+, the support of which is gratefully acknowledged. References 1. J. 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Suen, Construction of mixed orthogonal arrays by juxtaposition, Statistics and Proabability Letters, 65 (2003) 161-163. 45. C.Y. Suen, A. Dey, Construction of asymmetric orthogonal arrays through finite geometries, Journal of Statistical Planning and Inference, 115 (2003) 623-635. 46. N.J.A. Sloane, A library of orthogonal arrays (Online). http://www2.research.att.com/~njas/oadir/ (Access on : 28/04/2013)       Appendix A The influence of input variables on the model output is evaluated based on the correlation analysis. Correlation measures the strength and weakness of linear relationship between two variables. There are several coefficients that measure the correlation degree and Pearson’s correlation coefficient is used to determine the input variables relevance for this paper. Pearson’s correlation coefficient is calculated by dividing covariance of two variables by product of their standard deviation as shown in equation (A.1 – A.2), where r represents Pearson’s correlation coefficient. In equations (A.1-A.2), cov(xy)   is covariance which represents strength of linear relationship between two variables x y ; x and y are mean values of variables x and y ; s x and s y are standard deviations of variables x and y ; and n is the number of data. and r= cov(xy ) sx s y cov(xy ) = (A.1) 1 n ∑ xi − x y i − y n − 1 i =1 ( )( ) (A.2) The correlation coefficients can range from -1 to +1: r =1 : perfect positive linear correlation r = -1 : perfect negative linear correlation 0.1< r <0.25 : small positive linear correlation 0.25< r <0.6 0.6< r <1 -1< r <0 : medium positive linear correlation : strong positive linear correlation : negative linear correlation Climatic conditions (outside temperature and solar radiation), operational power level characteristics and approximate occupancy profile are used to evaluate the relevance variables that affect building heat demand based on case study data. Other variables pseudo dynamic transitional attributes, which signifies the dynamics of building characteristics is not consider for relevance variable determination since it only signifies time and phase interval of heating power transition. Results show the linear coefficient of correlation of outside air temperature, solar radiations, occupancy profile and operational power level characteristics with the heat load are -0.84, -0.40, 0.32 and 0.35 respectively. Results, thus, signifies that climatic conditions (outside temperature and solar radiations) are relevant input variables to predict the heat load. Also, it is clearer that occupancy profile and operational power level characteristics has medium positive correlation with heat load and shows relevance to characterize the heat demand behaviour.   

Experimental Biological Protocols with Formal Semantics Alessandro Abate2 , Luca Cardelli1,2 , Marta Kwiatkowska2 , Luca Laurenti2 , and Boyan Yordanov1 arXiv:1710.08016v1 [] 22 Oct 2017 1 2 Microsoft Research Cambridge Department of Computer Science, University of Oxford Abstract. Both experimental and computational biology is becoming increasingly automated. Laboratory experiments are now performed automatically on high-throughput machinery, while computational models are synthesized or inferred automatically from data. However, integration between automated tasks in the process of biological discovery is still lacking, largely due to incompatible or missing formal representations. While theories are expressed formally as computational models, existing languages for encoding and automating experimental protocols often lack formal semantics. This makes it challenging to extract novel understanding by identifying when theory and experimental evidence disagree due to errors in the models or the protocols used to validate them. To address this, we formalize the semantics of a core protocol language as a stochastic hybrid process, which provides a unified description for the models of biochemical systems being experimented on, together with the discrete events representing the liquid-handling steps of biological protocols. Such a representation captures uncertainties in equipment tolerances, making it a suitable tool for both experimental and computational biologists. We illustrate how the proposed protocol language can be used for automated verification and synthesis of laboratory experiments on case studies from the fields of chemistry and molecular programming. 1 Introduction The classical cycle of observation, hypothesis formulation, experimentation, and falsification, which has driven scientific and technical progress since the scientific revolution, is lately becoming automated in all its separate components. Data gathering is conducted by high-throughput machinery. Models are automatically synthesized, at least in part, from data [6,11,4]. Experiments are selected to maximize knowledge acquisition. Laboratory protocols are run under reproducible and auditable software control. However, integration between these automated components is lacking. Theories are not placed in the same formal context as the (coded) protocols that are supposed to test them. Theories talk about changing in physical quantities, while protocols talk about steps carried out by machines: neither knows about the other, although they both try to describe the same process. The consequence is that often it is hard to tell what happened when experiments and models do not match: was it an error in the model, or an error in the protocol? Often both the model and the protocol have unknown parameters: do we use the experimental data to fit the model or to fit the protocol? When most activities are automated, we need a way to answer those questions that is equally automated. In this paper, we present a core language to model experimental biological protocols that gives an integrated description of the protocol and of the underlying molecular process. A basic example of experimental biological protocol is shown in Figure 1. From this integrated representation both the model of a phenomenon (for possibly automated mathematical analysis), and the steps carried out to test it (for automated execution by lab equipment) can be separately extracted. This is essential to perform automated model synthesis and falsification by taking into account uncertainties in the both model structure and in equipment tolerances. We map our language into a Piecewise Deterministic Markov Process (PDMP). That is, a class of Markov stochastic hybrid processes where the continuous variables evolve according to ordinary differential equations (ODEs) and the Fig. 1: Graphical representation of an discrete variables evolve by means of acid-base titration protocol. The protorandom jumps [12]. The discrete dy- col is initialized with samples A (connamics are used to map the discrete taining H + and Cl− ) and B (containoperations of a lab protocol, while ing N a+ and OH − ). Some fraction of continuous dynamics model the evo- each sample (p1 and p2 ) is mixed tolution of the physical variables. In our gether and the resulting sample is let to language physical variables are de- equilibrate for t seconds. scribed with Chemical Reaction Networks (CRNs), a widely used formalism to model molecular interactions [16]. Our goal is to define a simple core language and focusing on formalizing its semantics. We then show how our language can easily be extended to collect observations of the process and to model complicate protocols. Giving to the language a formal semantics in terms of a PDMP allows us to include in our semantics the uncertainties intrinsic in the discrete operations of an experimental protocol. Uncertainties for experimental protocols have also been standardized (standards ISO 17025 and 8655). On examples from chemistry and molecular programming, we demonstrate how our integrated representation allows one to perform analysis and synthesis of both the discrete steps of the protocol and of the underlying physical system. Related Work Several factors contribute to the growing need for a formalization of experimental protocols in biology. First, better record-keeping of experimental operations is recognized as a step towards tackling the reproducibility crisis in biology [17]. Second, the emergence of cloud labs [18] creates a need for precise, machine-readable descriptions of the experimental steps to be executed. To address these needs, frameworks allowing protocols to be recorded, shared, and reproduced locally or in a remote lab have been proposed. These frameworks introduce different programming languages for experimental protocols including BioCoder [3], Autoprotocol, and Antha [24]. These languages provide expressive, high-level protocol descriptions but consider each experimental sample as a labelled black-box. This makes it challenging to study a protocol together with the biochemical systems it manipulates in a common framework. In contrast, we consider a simpler set of protocol operations but capture the details of experimental samples, enabling us to track properties of chemical species (e.g. amounts, concentrations, etc) as they react during the execution of a protocol. This allows us to formalize and verify requirements for the correct execution of a protocol or to optimize various protocol or system parameters to satisfy these specifications. 2 Background We first introduce the formalism of PDMP, which we use to model experimental protocols. Then, we introduce Chemical Reaction Networks (CRN), which are used to model the underlying physical process. 2.1 Piecewise Deterministic Markov Process The syntax of a PDMP is given as follows. Definition 1 A Piecewise Deterministic Markov Process (PDMP) H is a tuple H = (Q, d, G, F, Λ, R), where – Q = {q1 , ..., q|Q| } is the set of discrete modes – d : Q → N is a map such that Rd(q) , is the state space of the continuous dynamics for state q. The hybrid state space is defined as D = ∪q∈Q {q} × Rd(q) – G : Q × Rd(q) → {0, 1} is a set of guards – F : Q × Rd(q) → Rd(q) is a family of vector fields – Λ : S × Q → R≥0 is an intensity function, where for (qi , x) ∈ S, qj ∈ Q, we P define Λ((qi , x), qj ) = λi,j (x) and qj 6=qi λi,j (x) = λqi (x) – R : B(S) × S → [0.1] is the reset function, which assigns to each (q, x) ∈ S a measure R(·, q, x) on (S, B(S)). Let B(S) be the smallest σ−algebra on S containing all the sets of the form ∪q∈Q {q}×Aq , where Aq is a Borel subeset of Rd(q) . For t ∈ R≥0 , q ∈ Q, x ∈ Rd(q) , we call Φ(q, t, x), the solution of the following differential equation dΦ(q, t, x) = F (q, Φ(q, t, x)), Φ(q, 0, x) = x. dt The solution of a PDMP is a stochastic process Y = (α, X), whose semantics is classically defined according to the notion of execution (see Definition 2 below) [13]. In order to introduce such a notion, we define the exit time t∗ (q, x, G) as t∗ (q, x, G) = inf{t ∈ R≥0 | G(q, Φ(q, t, x)) = 1} (1) ( Rt e− 0 λq (Φ(q,τ,x))dτ if t < t∗ (q, x, G) . and the survival function f (q, t, x) = 0 otherwise. Here t∗ (q, x, G) represents the first time instant, starting from state (q, x), when the guard set is reached by a solution of the process; further f (q, t, x) denotes the probability that the system remains within q, starting from x, at time t [12], which depends on random arrivals induced by the intensity function Λ. The semantics of a PDMP is provided next. Definition 2 (Execution of PDMP H) Set t := 0 Set (α(0), X(0)) := q0 , x0 While t < ∞ Extract R≥0 -valued random variable T such that P rob(T > t̄) = f (α(t), t̄, X(t)) ∀τ ∈ [t, t + T ) Set (α(τ ), X(τ )) := (α(t), Φ(α(t), τ − t, X(t))) If t + T < ∞ Extract (α(t + T ), X(t + T )) according to R(·, (α(t), Φ(α(t), T, X(t))) End If Set t := t + T End While Let T < t∗ (qi , x̄, G) be the dwelling time in state qi ∈ Q, with x̄ such that Φ(qi , T, x̄) = x, (qj , A) ∈ B(S). Assume that x is such that G(x, qi ) = 0. Then, the reset has the following form, which results from a transition that is not due to the crossing of a guard: R((qj , A), (qi , x)) = Rc (A|qj , (qi , x)) P ΛTqi ,qj (Φ, x) qk 6=qi where ΛTqi ,qj (Φ, x) = RT 0 ΛTqi ,qk (Φ, x) , (2) λi,j (Φ(s, qi , x))ds, and Rc (A|qj , (qi , x)) = P rob(X(T ) ∈ A|α(T ) = qj , (α(0) = qi , X(0) = x)) is the conditional reset of the continuous dynamics. 2.2 Chemical Reaction Networks A CRN C = (A, R) is a pair of finite sets, where A denotes a set of chemical species, |A| is its cardinality, and R denotes a set of reactions. A reaction τ ∈ R is a triple τ = (rτ , pτ , kτ ), where rτ ∈ N|Λ| is the source complex, pτ ∈ N|Λ| is the product complex and kτ ∈ R>0 is the coefficient associated with the rate of the reaction.The quantities rτ and pτ represent the stoichiometry of reactants and products. Given a reaction τ1 = ([1, 0, 1], [0, 2, 0], k1 ) we often refer to it visually as τ1 : λ1 + λ3 →k1 2λ2 . The net change associated to τ is defined by υτ = pτ − rτ . Many models have been introduced to study CRNs [9,7,15,8]. Here we consider the reaction rate equations [15], which describe the time evolution of the concentration of the species in C, in a sample of temperature T and volume V , as follows: X dΦ(t) = F (t) = υτ · γS (Φ(t), kτ , V, T ), (3) dt τ ∈R where γS (Φ(t), kτ , V, T )) is the propensity rate, and in case of mass action kinetics we have Y r γS (Φ(t), kτ , V, T )) = kτ (T ) ΦSS,τ (t), S∈Λ where ΦS and rS,τ are the components of vectors Φ and rτ relative to species S, and where in kτ (T ) we make explicit the dependence from temperature T . Definition 3 (Chemical Reaction System) A chemical reaction system (CRS) C = (A, R, x0 ) is defined as a tuple, where (A, R) is a CRN and x0 ∈ N|Λ| represents its initial condition. Example 1. Consider the CRS C = (A, R, x0 ), evolving in a volume V and at temperature T , where A = {H2 O, N a+ , OH − , Cl− , H + } and R is composed of the following reactions: N a+ + OH − + H + + Cl− →k H2 O + N a+ + Cl− where k = 2.81e−10 is the rates at temperature T = 298 Kelvin. Then, According to Equation (3), we have that the state of H + is given by the solution of the following ordinary differential equation: dH + (t) = − kN a+ (t)OH − (t)H + (t)Cl− (t), dt with H + (0) = 3 x0,H + V , where x0,H + is the component of x0 relative to H + . A Language for Experimental Biological Protocols In this section we introduce the syntax of a language for modelling experimental protocols. A formal semantics of the language, based on denotational semantics [25], is then discussed. We model the physical process underlying a biological experimental protocol as a CRS. As a consequence, in order to introduce formal semantics for experimental protocols, we first need to define semantics for a CRS, which has been only introduced informally in the previous section. Let S = (R|A| × R≥0 × R≥0 ) be a sample. We define the semantics for a CRS as follows. |A| Definition 4 (CRS Semantics) Let C = (A, R) be a CRN, x0 ∈ R≥0 , V, T ∈ R≥0 be the initial concentration (moles), volume (liters) and temperature (degrees Kelvin). Call F (V, T ) : R|A| → R|A| the drift at volume V and temperature T for C. Then, the semantics of the CRS (A, R, x0 ) at volume V , temperature T and time t, for a time horizon H ∈ R≥0 ∪ {∞}, [[·]] : (CRS × R≥0 × R≥0 ) → R≥0 ∪ {∞} → R≥0 → S is defined as [[((A,R, x0 ), V, T )]](H)(t) = let G : [0...H) → R |A| 0 Z be the solution of G(t ) = x0 + t0 F (V, T )(G(s))ds 0 (G(t), V, T ) where the above operation reads as follows: ’first line’ = ’third line’, where G is defined as in ’second line’. If for such an H, G is not unique, then we say that [[((A, R, x0 ), V, T )]] (H)(t) is ill posed. In Definition 4 we have explicitly introduced a dependence on a time horizon H, because it may happen that the solution of the rate equations is defined only for a finite time horizon [15]. 3.1 A Language for Experimental Protocols Our goal is to build a simple core language that gives an integrated representation of a discrete protocol within the physical process being implemented on. We consider the following language modelling the basic operation of a lab protocol. Definition 5 (Syntax of a Protocol) Given a set of variables V ar, the syntax of a protocol P for a given fixed CRN C = (A, R) is P = x (sample variable) (x0 , V, T ) (initial condition) M ix(P1 , P2 ) let x = P1 in P2 let x, y = Dispense(P1 , p) in P2 Equilibrate(P, t) Dispose(P ) (mix samples) (define variable) (dispense samples) (let time pass) (discard P) where T, V, t ∈ R≥0 , x, y ∈ V ar, p ∈ R(0,1) . Moreover, let-bound variables must occur exactly once ( that is, be free) in P2 . A protocol P yields a sample, which is the result of operations of Equilibrate, Mix, Dispose and Dispense, over a CRS. This syntax allows one to create and manipulate new samples using Mix (put together different samples), Dispense (separate samples) and Dispose (discard samples) operations. Note that the CRN is common for all samples. However, different samples may have different initial conditions. The single-occurrence (linearity) restriction implies that a sample cannot be duplicated or eliminated from the pool. Example 2. We use let x, = Dispense(P1 , p) in P2 as a short-hand for let x, y = Dispense(P1 , p) in M ix(Dispose(y), x). The protocol (call it P ro1 ) represented graphically in Figure 1 is defined formally as P ro1 =let A = ([(H + , 0.1M ); (Cl− , 0.1M )], 1.0mL, 25.0o C) in let B = ([(N a+ , 0.1M ); (OH − , 0.1M )], 1.0mL, 25.0o C) in let a, = Dispense(A, p1 ) in let b, = Dispense(B, p2 ) in Equilibrate(M ix(a, b), t). In order to define the semantics of a protocol we introduce the following definitions. Definition 6 (Free Variables) The set of Free Variables (FV) of a protocol P is defined inductively as follows: F V (x) = {x} F V ((x0 , V, T )) = {} F V (M ix(P1 , P2 )) = F V (P1 ) ∪ F V (P2 ) F V (let x = P1 in P2 ) = F V (P1 ) ∪ (F V (P2 ) − {x}) F V (let x, y = Dispense(P1 , p) in P2 ) = F V (P1 ) ∪ (F V (P2 ) − {x, y}) F V (Equilibrate(P, t)) = F V (P ) F V (Dispose(P )) = F V (P ). We define the operation of substitution of a protocol into a variable as follows. Definition 7 (Substitution) P1 {x ← P2 } is defined inductively as follows: x{x ← P } = P y{x ← P } = y, for x 6= y M ix(P1 , P2 ){x ← P3 } = M ix(P1 {x ← P3 }, P2 {x ← P3 }) (let x = P1 in P2 ){x ← P3 } = (letx = P1 {x ← P3 } in P2 ) (let y = P1 in P2 ){x ← P3 } = (let y = P1 {x ← P3 }inP2 {x ← P3 }), for x 6= y and y 6∈ F V (P3 ) Equilibrate(P, t){x ← P1 } = Equilibrate(P {x ← P1 , t}) Dispose(P ){x ← P1 } = Dispose(P {x ← P1 }). The following equivalences can be shown structurally, based on the definitions above. Proposition 1. (Equivalence Relationships) let x = P1 in P2 = P2 {x ← P1 } let x = P1 in P2 = let y = P1 in (P2 {x ← y}) for y 6∈ F V (P2 ). 3.2 Deterministic Semantics of a Protocol We introduce the deterministic semantics for a protocol. Then, in the next Section, we extend such a semantics in order to take into account errors and inaccuracies within the protocol, which in practice may be quite relevant. The deterministic semantics of a protocol P for a CRN C = (A, R), under a given environment ρ : V ar → S, is a function [[P ]]ρ : (V ar → S) → S that is defined inductively as follows. Definition 8 (Deterministic Semantics of a Protocol) Let S = (R|A| × R≥0 × R≥0 ), then the deterministic semantics of a protocol P for CRN C = (A, R), under environment ρ : V ar → S is defined inductively as follows [[x]]ρ = ρ(x) [[x0 , V, T ]]ρ = (x0 , V, T ) [[M ix(P1 , P2 )]]ρ = let (x10 , V1 , T1 ) = [[P1 ]]ρ let (x20 , V2 , T2 ) = [[P2 ]]ρ T1 V1 + T2 V2 x10 V1 + x20 V2 , V1 + V2 , ) V1 + V2 V1 + V2 [[let x = P1 in P2 ]]ρ = ( let (x0 , V, T ) = [[P1 ]]ρ let ρ1 = ρ{x ← (x0 , V, T )} [[P2 ]]ρ1 [[let x, y = Dispense(P1 , p) in P2 ]]ρ = let (x0 , V, T ) = [[P1 ]]ρ let ρ1 = ρ{x ← (x0 , V · p, T ), y ← (x0 , V · (1 − p), T )} [[P2 ]]ρ1 [[Equilibrate(P, t)]]ρ = let (x0 , V, T ) = [[P ]]ρ [[(A, R, x0 ), V, T )]](H)(t) [[Dispose(P )]]ρ = (0|Λ| , 0, 0), where H ∈ R≥0 is such that for any Equilibrate(P, t), [[(A, R), x0 , V, T )]](H)(t) is well posed. If such an H does not exist, we say that P is ill posed. The above semantics identifies a protocol which outputs the concentration of the species, the volume, and the temperature of the sample at final time. 3.3 Deterministic Semantics of a Protocol as a PDMP Given a protocol P and an environment ρ, [[P ]]ρ induces semantics that correspond to the solution of a PDMP H = (Q, d, G, F, Λ, R) as per Definitions 1 and 2. In the corresponding PDMP model H, Q represents the set of discrete operations, so that d(q) = |A| + 1 denotes the continuous dimension (the number of continuous variables). The vector field F is given by Definition 4, with an additional clock variable time representing time as dtime = 1. For each dt Equilibrate(P, t) step, there is a guard set defined as time ≥ t: when a trajectory enters this guard, an associated reset is modeled with the identity function. The resulting process is a PDMP without random jumps (that is, where Λ(q, x) = 0 for any q ∈ Q, x ∈ R|A|+1 ) and with non-probabilistic resets R. As such, the resulting PDMP is also a (non-probabilistic) hybrid model [2]. We elucidate this with the next example. Example 3. Consider the protocol P ro1 introduced in Example 2. The CRN of the system comprises the reactions given in the CRN in Example 1. According to Definition 11, the state of variable H + in time is given by the solution of the following equation: Z t + + H (t) = H (0) − kN a+ (s)OH − (s)H + (s)Cl− (s)ds, 0 −7.4 2 10 . Then, given an environment ρ, [[P ro1 ]]ρ is the where H + (0) = p1 0.1+p p1 +p2 solution of the PDMP H = (Q, d, G, F, Λ, R), where Q = {q}; d(q) = R8 ; G(q, x) = 1 iff xtime ≥ t, where xtime is the component of x relative to variable time; and F is given by Definition 11. Notice again that Λ(q, x) = 0 for any x and R(q, x) = δ(x), where δ(x) is the Dirac delta function centered at x. 3.4 Stochastic Semantics of a Protocol, and Interpretation as a PDMP The semantics of Definition 11 are fully deterministic, and indeed are shown to map into a fully non-probabilistic PDMP model. However, it is often the case that operations of Dispense and Equilibrate are stochastic in nature, due to the fact that they are performed by humans and in view of experimental inaccuracies related to lab equipment. In what follows, we encompass this features by extending the previously defined semantics with stochasticity. More precisely, – in the Equilibrate(P, t) step, time is sampled from a distribution; – the resulting volume after a Dispense step is sampled from a distribution. The first characteristic models the fact that in real experiments the system is not equilibrated for exactly t seconds, as it may start or be stopped at different time instants, and accounts for the fact that after a mix of samples well mixed conditions are not reached instantaneously, whereas the second takes into account the error of pipetting devices, whose ranges and parameters have been standardized t0 (standard ISO 8655). For the first feature, consider the function T (t0 , t) = e− t , defined for two values t0 , t ∈ R≥0 . This function corresponds to the density function of an exponential random variable, modelling random arrivals. For the m second function, let B(Rm ≥0 ) be the Borel sigma-algebra over R≥0 , m > 0. Then, we consider the following function D : B(R(0,1) ) × R≥0 × R[0,1] → [0, 1], which assigns to D(·, V, p) a probability measure in B(R[0,1] ). Function D is used to reset the volume randomly, after a discrete operation. (As an anticipation of results, notice that both functions T and D can be mapped to elements in the syntax in a PDMP model.) We define the Stochastic Semantics of a protocol as an extension of the deterministic ones in Definition 11. For the sake of compactness, we explicitly write only the operators that differ from the earlier ones. Definition 9 (Stochastic Semantics of a Protocol) Let S = (R|A| × R≥0 × R≥0 ), then the semantics of a protocol P for CRN C = (A, R), under environment ρ : V ar → S and functions T , D, as defined above, is defined inductively as follows [[let x, y = Dispense(P1 , p) in P2 ]]ρ = let (x0 , V, T ) = [[P1 ]]ρ let p0 being sampled f rom D(·, V, p) let ρ1 = ρ{x ← (x0 , V · p0 , T ), y ← (x0 , V · (1 − p0 ), T )} [[P2 ]]ρ1 [[Equilibrate(P, t)]]ρ = let (x0 , V, T ) = [[P ]]ρ let I be a R≥0 − valued random variable such that for s ∈ R≥0 P rob(I > s) = T (s, t) [[(A, R, x0 ), V, T )]](H)(I), where H ∈ R≥0 is such that for any Equilibrate(P, t), and any I random variable such that P rob(I > s) = T (s, t), [[(A, R), x0 , V, T )]](H)(I) is well posed with probability 1. If such an H does not exist, we say that [[P ]]ρ is ill posed. D is a transition kernel that depends only on the current state of the system. T is the cumulative probability distribution of a random variable with exponential distribution. As a consequence, according to Definition 2, [[P ]]ρ induces semantics that are again solution of a PDMP. However, here T determines the probability of changing discrete state and D acts as a probabilistic reset, there are no guards, and the continuous dynamics evolve according to the ODE in Definition 4. Next, we leverage results from the analysis of PDMP models and export them over the protocol language. The following assumptions guarantee that the solution of the PDMP induced by [[P ]]ρ exists, establish that it is a strong Markov process, and allow to exclude pathological Zeno behaviours [12,19]. Assumption 1 – Let A0 , A1 ⊂ B(R[0,1] ) be the smallest sets in B(R[0,1] ) containing respectively 0 and 1. Then, D(A0 , V, p) = D(A1 , V, p) = 0 for any p ∈ (0, 1), V 6= 0. That is, the Volume after a dispense is zero with probability zero. – Let F : R|A| → R|A| be the drift term of the rate equations (Eqn (4)). Then, F is a globally Lipschitz function. – For any Equilibrate(·, t) we have that t > 0. Let us interpret these assumptions over the protocol languages. The first assumption guarantees that the volume of a non-empty sample cannot be 0 almostsurely. The second assumption guarantees that the solution of (4) exists and does not hit infinity in finite time. This excludes non-physical reactions like X + X → X + X + X. The third assumption guarantees that we have a finite number of jumps over a finite time, thus excluding Zeno behaviours [12,13]. Example 4. Consider the protocol introduced in Example 2. For σ1 > 0, A ⊂ − R[0.1,0.8] . Assume that D(A, p, V̄ ) = R A e R 0.8 0.1 x−p 2 2σ1 − e dx x−p 2 2σ1 . That is, D(·, p, V ) is a trun- dx cated Gaussian measure centered at p and independent of the volume. Then, according to Definition 9, we have the following Stochastic Semantics: Z I H + (I) =H + (0) − kN a+ (s)OH − (s)H + (s)Cl− (s)ds, 0 −7.4 2 10 . Here I is a random variable with an exponential with HCl(0) = V1 0.1+V V1 +V2 distribution with rate T1 , V1 is a random variable sampled from D(·, p1 , 1), and V2 is a random variable sampled from D(·, p2 , 1). 4 Extending the Protocol Language with Observations The language introduced in Section 3 can be extended in a number of directions, according to specific envisioned scenarios for the protocols. A common task is to take observations of the state of the protocol. That is, often it is useful to store the state of the system at different times or when a particular event happens. As some of the events may be stochastic, in general, it is not possible to know before the simulation starts when a particular event happens. Consequently, observations need to be included in the language. Definition 10 (Extended Syntax). Given a set of variables V ar, the syntax of a protocol for a given fixed CRN C = (A, R) and idn ∈ N is P = x (sample variable) (x0 , V, T ) (initial condition) M ix(P1 , P2 ) let x = P1 in P2 let x, y = Dispense(P1 , p) in P2 Equilibrate(P, t) Dispose(P ) Observe(P, idn) (mix samples) (define variable) (dispense samples) (let time pass) (discard P) (observe sample) where T, V, t ∈ R≥0 , x, y ∈ V ar, p ∈ R[0,1] . Moreover, let-bound variables must occur exactly once ( that is, be free) in P2 . Observe(P, idn) makes an observation of protocol P after its execution, and identifies such an observation with identifier idn. In order to include observations we extend the semantics as detailed next, where we consider in detail just the deterministic semantics, focusing on a few key operators. The other operators and the Stochastic Semantics follow similarly. Definition 11 (Extended Deterministic Semantics) For CRN C = (A, R) let S = R|A| × R≥0 × R≥0 , Obs = R≥0 × N × R|A| , Obs∗ , an eventually empty set of Obs and M = S × Obs∗ × R≥0 . The semantics of a protocol P , under environment ρ : V ar → M, is a function [[P ]] : (V ar → M) × R≥0 → M defined inductively as follows [[M ix(P1 , P2 )]]ρt = let ((x10 , V1 , T1 ), Obs1 , t1 ) = [[P1 ]]ρt let ((x20 , V2 , T2 ), Obs2 , t2 ) = [[P2 ]]ρt T1 V1 + T2 V2 x10 V1 + x20 V2 , V1 + V2 , ), Obs1 :: Obs2 , max(t1 , t2 )) V1 + V2 V1 + V2 [[Observe(P, idn)]]ρt = (( let ((x0 , V, T ), Obs, t1 ) = [[P ]]ρt let O = (x0 , idn, t1 ) ((x0 , V, T ), Obs ∪ O, t1 ) [[Equilibrate(P, t)]]ρt0 = let ((x0 , V, T ), Obs, t1 ) = [[P ]]ρt0 ([[(A, R), x0 , V, T )]](H)(t), Obs, t1 + t), where H ∈ R≥0 is such that for any Equilibrate(P, t), [[(A, R), x0 , V, T )]](H)(t) is well posed. If such H does not exist, we say that P is ill posed. Note that the above syntax does not prevent the programmer to assign the same identifier to two distinct observations. We further stress that often observations of the state of an experiment are not exact, but corrupted by sensing noise. For instance, this is what happens with noisy fluorescence measurements. This noise can be easily taken into account at a semantical level by sampling an observation from a distribution with added noise, where the noise level depends on the particular measure technique or instrumentation. Finally, we can as well extend the sample semantic to take into account noise in Dispense operations. 5 Case Study As a case study we consider an experimental protocol for DNA strand displacement. DNA strand displacement (DSD) is a design paradigm for DNA nano-devices [10]. In such a paradigm, single-stranded DNA acts as signals and double-stranded (or more complex) DNA structures act as gates. The interactions between signals and gates allow one to generate computational mechanisms that can operate autonomously at the molecular level [26]. The DSD programming language has been developed as a means of formally programming and analyzing such devices [20,10]. Here, we consider an AN D circuit implemented in DSD, which can be represented with the reactions in Figure 2b. Strands Input1 =< 1∗ 2 > and Input2 =< 3 4∗ > represent the two inputs, while strand Output =< 2 3 > is the output. Strand Gate = {1∗ }[2 3]{4∗ } is an auxiliary strand. The Output strand is released only if both the inputs react with the Gate gate. We consider the protocol in Figure 2a, which can be written formally as follows. We use let x,= Dispense(P 1, p) in P 2 as a short-hand for let x, y = Dispense(P 1, p) in M ix(Dispose(y), x) P1 =let In1 = ((Input1, 100.0nM ), 0.1mL, 25.0o C) in let In2 = ((Input2, 100.0nM ), 0.1mL, 25.0o C) in let GA = ((Output, 100.0nM ), 0.1mL, 25.0o C) in let GB = ((GateB , 100.0nM ), 0.1mL, 25.0o C) in let sGA,= Dispense(GA, p1 ) in let sGB,= Dispense(GB, p2 ) in let sIn1,= Dispense(In1, p3 ) in let sIn2,= Dispense(In1, p4 ) in Observe(Equilibrate(M ix(M ix(Equilibrate(M ix(sGA, sGB), t1 ), sIn1), sIn2), t2 ), idn). The protocol proceeds as follow: Output and GateB strands are dispensed from the original samples. Then, they are let evolve for t1 seconds to create Gate strands. Then, the two inputs are dispensed from their samples. The resulting samples are mixed and the resulting solution evolves for t2 seconds. Finally, we collect the final sample, observe the results, and associate to the observation the identifier ’idn’. According to the standard ISO 8655 for a volume of 1mL, Fig. 2: (A) Graphical representation of the protocol. (B) Graphical representation of the reactions between the different DNA strands in the considered solution. For example, in the second reaction, stand {1∗ }[2 3]{4∗ } reacts with < 1∗ 2 > at a rate 0.0003, and there exists an inverse reaction with rate 0.1126. the maximum standard deviation of a particular pipetting device is 0.3µL per single operation. To incorporate such an error in our model, we make use of the Stochastic Semantics. Thus, the concentration of the Output strand at the end of the protocol is a random variable. It is common that the reaction rates of the physical system are not known exactly and they may be affected by extrinsic noise [23]. This leads to another source of uncertanity in the output of the protocol. To estimate the distribution of the output under the effect of both these sources of noise we extend our semantics to sample the rate of a reaction from a normal distribution with variance equals to half of its mean (sub-Poisson noise). In Figure 3a we plot 4500 executions resulting from the protocol. From the figure it is easy to realize how the difference source of noise may have a distinctive effect on the final outcome of the experiments. In many experimental protocols, one of the key challenges is to synthesize their optimal discrete parameters, to optimize the probability of obtaining desired behaviours. Here, we assume perfect knowledge of the reaction rates of the physical system and p1 = p2 = 0.4. Our goal is to see how the concentration of the Output changes varying (p3 , p4 ) ∈ [0.45, 0.65]×[0.45, 0.65]. We are interested in the following property PSaf e ([3.0·10−4 , 3.5·10−4 ]) = P rob(Output(t0 ) ∈ [3.0·10−4 , 3.5·10−4 ]|t0 = tf inal ), where tf inal is the final time of the protocol. The following probability is estimated using Statistical Model Checking [22] in Figure 3b, which in this context reduces to Monte-Carlo sampling. From Figure 3b it is easy to infer that the Fig. 3: (A): (red) 1500 execution of the protocol assuming the physical model is fully known, and the only source of noise is the discrete parameters of the protocols (p1 , p2 , p3 , p4 ). (yellow) 1500 executions of the protocol when the rates of the physical system are sampled from a sub-Poisson distribution, and discrete operations are exact. (blue) 1500 simulations of the protocol when both sources of noise are active.(B): PSaf e ([3.0 · 10−4 , 3.5 · 10−4 ]) as a function of p3 and p4 . Each cell is estimated from 20000 executions of the protocol. optimal value for such property is not unique (it is attained at values over the yellow band) and obtained, for instance, at (p3 , p4 ) = (0.5, 0.54). 6 Discussion We presented a language to model experimental biological protocols, and provided semantics to this protocol in terms of PDMPs. That is, to each experimental biological protocol is associated a particular instance of a stochastic hybrid process. Our language provides a unified description of the model of the system being experimented on, together with the discrete events representing the parts of biological protocols dealing with handling samples. Moreover, we allow the modeller to take into account uncertainties in both the model structure and the equipment tolerances. This makes our language a suitable tool for both experimental and computational biologists. Our objective has been that of providing a basic language, yielding an integrated representation of an experimental biological protocol. To this end, we have kept the language as simple as possible, showing how different extensions are easy to be integrated. For instance, in our denotational semantics the dynamics of a physical process is given by a set of ODEs. This is accurate when the number of molecules involved is big enough, as in the discussed example of DNA strand displacement (DSD). However, in other scenarios, such as localized computation or gene expression, this might be unsatisfactory as stochasticity becomes important [5,14]: nevertheless, the semantics presented here can be easily extended to incorporate such stochasticity, which can be done for example by considering more general classes of stochastic hybrid processes, such as switching diffusions [21,27]. Another relatively simple extension is to include finite loops or operations based on concentrations. 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arXiv:1604.01946v1 [cs.LG] 7 Apr 2016 Optimizing Performance of Recurrent Neural Networks on GPUs Jeremy Appleyard⋆ Tomáš Kočiský†‡ Phil Blunsom†‡ ⋆ NVIDIA † University of Oxford ‡ Google DeepMind [email protected] {tomas.kocisky,phil.blunsom}@cs.ox.ac.uk Abstract As recurrent neural networks become larger and deeper, training times for single networks are rising into weeks or even months. As such there is a significant incentive to improve the performance and scalability of these networks. While GPUs have become the hardware of choice for training and deploying recurrent models, the implementations employed often make use of only basic optimizations for these architectures. In this article we demonstrate that by exposing parallelism between operations within the network, an order of magnitude speedup across a range of network sizes can be achieved over a naive implementation. We describe three stages of optimization that have been incorporated into the fifth release of NVIDIA’s cuDNN: firstly optimizing a single cell, secondly a single layer, and thirdly the entire network. 1 Introduction Recurrent neural networks have become a standard tool for modelling sequential dependencies in discrete time series and have underpinned many recent advances in deep learning, from generative models of images [1, 2] to natural language processing [3, 4, 5, 6, 7, 8, 9]. A key factor in these recent successes has been the availability of powerful Graphics Processing Units (GPUs) which are particularly effective for accelerating the large matrix products at the heart of recurrent networks. However as recurrent networks become deeper [10] and their core units more structured [11, 12], it has become increasingly difficult to maximally utilise the computational capacity of the latest generation of GPUs. There have been several studies optimizing the implementation of neural networks for GPUs, particularly in the case of convolutional neural networks [13, 14]. While GPUs are already widely used to compute RNNs [15, 16, 17, 18], there has been less work on the optimization of RNN runtime. In this article we present a number of options for going beyond straightforward RNN GPU implementations that allow us to achieve close to maximum computational throughput for common network architectures. These enhancements are implemented in the fifth version of NVIDIA’s cuDNN library for Simple RNN, GRU, and LSTM architectures. 2 Implementation In this section we will consider the performance of a forward and backward propagation passes through an LSTM network[11]. This is a standard four-gate LSTM network without peephole connections. The equations to compute the output at timestep t in the forward pass of this LSTM are given below: 1 for l in layers for j in iterations i(l,j)’ = A_i(l) * i(l,j) h(l,j)’ = A_h(l) * h(l,j) for pointwiseOp in pointwiseOps do pointwiseOp Listing 1: Pseudocode demonstrating the starting point for optimization of the forward pass. ii = σ(Wi xi + Ri ht−1 + bi ) ft = σ(Wf xt + Rf ht−1 + bf ) ot = σ(Wo xt + Ro ht−1 + bo ) c′t = tanh(Wc xt + Rc ht−1 + bc ) ct = ft ◦ c′t−1 + it ◦ c′t ht = ot ◦ tanh(ct ) Most of the same strategies found to be beneficial to LSTM performance are easily transferable to other types of RNN. 2.1 Naive implementation There are many ways to naively implement a single propagation step of a recurrent neural network. As a starting point we will consider an implementation where each individual kernel (ie. matrix multiplication, sigmoid, point-wise addition, etc.) is implemented as a separate kernel. While the GPU executes the operations within each kernel in parallel, the kernels are executed back-to-back sequentially. The forward pass performance of this implementation is poor, achieving approximately 0.4 TFLOPS on a test case with hidden state size 512 and minibatch 64, less than 10% of the peak performance of the hardware (approximately 5.8 TFLOPS when running at base clocks).1 A widely used optimization is to combine matrix operations sharing the same input into a single larger matrix operation. In the forward pass the standard formulation of an LSTM leads to eight matrix matrix multiplications: four operating on the recurrent input (R∗ ht−1 ), four operating on the input from the previous layer (W∗ xt ). In these groups of four the input is shared, although the weights are not. As such, it is possible to reformulate a group of four matrix multiplications into a single matrix multiplication of four times the size. As larger matrix operations are more parallel (and hence more efficient), this roughly doubles forward pass throughput to 0.8 TFLOPs in the test case described above. This reformulation is very easy to implement in most deep learning frameworks leading to its wide use. A similar optimization is possible for GRU units, with two groups of three matrices able to be grouped. Single gate RNNs cannot benefit from this optimization as they have only one matrix multiplication at each input. The backward pass also benefits from this optimization as four inputs are transformed into one output. Pseudocode for this implementation is given in Listing 1. While some of the optimizations that follow have been implemented before, those that have are by no means universal nor standard practice. 2.2 Single Cell 2.2.1 Streamed matrix operations The matrix multiplications performed by RNNs often have insufficient parallelism for optimal performance on the GPU. Current state-of-the-art GEMM kernels are implemented with each CUDA 1 All runtime and FLOP measurements reported are based off the mean of 100 executions on an NVIDIA M40 GPU (https://images.nvidia.com/content/tesla/pdf/ nvidia-teslam40-datasheet.pdf), at default application clocks with auto-boost disabled. The host CPU is an Intel Xeon CPU E5-2690 v2 @ 3.00GHz. 2 for l in layers for j in iterations set stream 0 i(l,j)’ = A_i(l) * i(l,j) set stream 1 h(l,j)’ = A_h(l) * h(l,j) wait for stream 0 do pointwise ops Listing 2: Pseudocode demonstrating forward pass single cell optimizations. block computing a rectangular tile of the output. The dimensions of these tiles is typically in the range 32 to 128. Partitioning the matrix multiplications required for the forward pass of a hidden state size 512, minibatch 64 LSTM with 128x64 tiles gives a total of 16 CUDA blocks. As blocks reside on a single GPU streaming multiprocessor (SM), and modern top-of-the-range GPUs (eg. our M40) currently have 24 streaming multiprocessors, this matrix multiplication will use at most two thirds of the available GPU performance. As it is desirable to have multiple blocks per SM to maximise latency hiding it is clear that to achieve better performance, it is required that we increase parallelism. One easy way to increase the parallelism of a single RNN cell is to execute both matrix multiplications on the GPU concurrently. By using CUDA streams we can inform hardware that the matrix multiplications are independent. This doubles the amount of parallelism available to the GPU, increasing performance by up to 2x for small matrix multiplications. For larger matrix multiplications, streaming is still useful as it helps to minimize the so called “tail effect”. If the number of blocks launched to the GPU are only sufficient to fill its SMs a few times they can be thought of as passing through the GPU in waves. All of the blocks in the first wave finishes at approximately the same time, and as they finish the second wave begins. This continues until there is no more work. If the number of waves is small, the last wave will often have less work to do than the others, creating a “tail” of low performance. By increasing parallelism this tail can be overlapped with another operation, reducing the performance penalty. 2.2.2 Fusion of point-wise operations Although parallelism comes naturally to point-wise operations, it was found that they were being executed inefficiently. This is for two reasons: firstly because there is a cost associated with launching a kernel to the GPU; secondly because it is inefficient to move the output of one point-wise operation all the way out to GPU main memory before reading it in again moments later for the next. By their nature point-wise operations are independent, and as such, it is possible to fuse all of the point-wise kernels into one larger kernel. 2.3 Single Layer A single recurrent layer comprises many cells, the recurrent input of each depending on the output of the previous. The input from the previous layer may not have such a dependency and it is often possible to concatenate the inputs for multiple time steps producing a larger, more efficient, matrix multiplication. Selecting the amount of steps to concatenate over is not trivial: more steps leads to a more efficient matrix multiplication, but fewer steps reduces the time a recurrent operation may potentially be waiting for. The exact amount of steps will depend not only on the hyper-parameters, but also on the target hardware. Another operation that is possible, when considering a layer in its entirety, is re-ordering the layout of the weight matrices. As the same weight matrices are used repeatedly over the course of a layer the cost of reordering is typically small compared to the cost of operating on the matrices. In our tests it was found that pre-transposing the weight matrix lead to noticeable performance improvements. Note that because the transpose of the weight matrix is used in the backward pass this pre-transposition must be performed every pass through the network. 3 for l in layers A_it(l) = transpose(A_i(l)) A_ht(l) = transpose(A_h(l)) for j in iterations, step s set stream 0 i(l,j:j+s)’ = A_it(l) * i(l,j:j+s) for k in 1,s set stream 1 h(l,j+k)’ = A_ht(l) * h(l,j+k) wait for stream 0 to complete operation on j+k do pointwise ops Listing 3: Pseudocode demonstrating optimizations across the forward pass of a layer. for l in layers A_it(l) = transpose(A_i(l)) A_ht(l) = transpose(A_h(l)) while not Complete l,it = get next task for j in it->it+s set stream 0+2*l i(l,j:j+s)’ = A_it(l) * i(l,j:j+s) for k in 1,s set stream 1+2*l h(l,j+k)’ = A_ht(l) * h(l,j+k) wait for stream 0+2*l to complete operation on j+k do pointwise ops Listing 4: Pseudocode demonstrating the final optimized forward pass. 2.4 Multiple Layers It is becoming increasingly common for RNNs to feature multiple recurrent layers “stacked” such that each recurrent cell feeds its output directly into a recurrent cell in the next layer. In this situation, it is possible to exploit the parallelism between recurrent layers: the completion of a recurrent cell not only resolves the dependency on the next iteration of the current layer, but also on the current iteration of the next layer. This allows multiple layers to be computed in parallel, greatly increasing the amount of work the GPU has at any given time. 2.4.1 Scheduling As launching work to the GPU takes a small, but not insignificant, amount of time it is important to consider the order in which the kernels are launched to the GPU. For example, if GPU resources are available it is almost always preferable to launch a kernel with all of its dependencies resolved, rather than a kernel which may be waiting some time for its dependencies to be cleared. In this way as much parallelism as possible can be exposed. In order to do this we chose a simple scheduling rule whereby the next work to be scheduled is that with the fewest edges to traverse before reaching the “first” recurrent cell. If one considers a recurrent network as a 2D grid of cells, this leads to a diagonal “wave” of launches propagating from the first cell. 2.5 Performance The impact of each of the optimizations described on the forward pass of a 1000 step, four layer LSTM network with hidden state size 512 and an input minibatch of 64 is shown in Table 1. For this network we achieve an ~11x speedup over a completely naive implementation, and a ~6x speedup over an implementation with the standard GEMM grouping optimization. 4 Optimization Naive #1 Grouped GEMMs #2 Streamed GEMMs #3 Fused point-wise #4 Pre-transpose #5 Batching inputs (2-way) #6 Overlapping layers Time per cell (us) 777 400 280 146 125 119 70 Speedup (1.0x) 1.9x 2.8x 5.3x 6.2x 6.5x 11.1x Table 1: LSTM forward pass performance. Each optimization was applied on top of the previous. These measurements were made on a 100 iteration, four layer LSTM with a hidden state size of 512 and a minibatch of 64 using cuBLAS 7.5. Given sufficient recurrent steps there are three variables that are expected to significantly influence performance of an RNN implementation. These are: hidden state size, minibatch size and number of layers. Fixing the number of layers to four, Figure 1 shows the impact of each of these optimizations across a wide range of hidden state sizes and minibatch sizes. In some cases increasing the number of layers from one to four doubles throughput (ie. 4x the work in only 2x the time). This performance improvement is particularly high in low parallelism cases, where the minibatch is small, the hidden state size is small, or both are small. One feature of note is the reduction of performance in the minibatch 32 case at high hidden state size. This is attributable to cuBLAS choosing a different path of execution for matrix multiplications due to an internal heuristic. As cuBLAS is unaware of the algorithm it is being used for, this is arguably reasonable behaviour, and performance for a single layer continues to climb as the problem size increases. The most significant speedup is seen when the minibatch is 64. As the minibatch size increases so does the amount of parallelism already available to the GPU, so optimization strategies focused around increasing parallelism are less effective. Speedup is also lower with larger hidden layer sizes, for the same reason. Despite this, even for the largest problem benchmarked (minibatch 256, hidden state size 4096) the increase in parallelism due to layer overlapping still brings better performance. At larger minibatches it becomes less clear that some of the individual optimizations bring improvement. Batching inputs is actually found to be detrimental in many cases for minibatch sizes other than 32, likely due to the trade-off discussed in Section 2.3. In other cases, changes can bring a significant improvement for particular problem sizes, while causing a slowdown for others. This makes it hard to say for sure that the combination of all optimizations will give the fastest runtime for a given set of hyperparameters, however it is the case that, excepting batched inputs, each optimization helps in most cases. 2.6 Weight Update The above optimizations only apply to propagation steps. By completing the gradient propagation before starting the weight update, the weight update becomes very efficient. A single large matrix multiplication can be used to update each matrix with no dependencies and this will usually achieve close to peak performance. Updating the bias weights is very cheap in comparison to updating the matrices. 3 cuDNN The optimizations described in Section 2 have been implemented in the fifth version of NVIDIA’s cuDNN library for single gate RNNs, GRUs and LSTMs. The performance of this implementation is shown in Figure 2. For this implementation it was possible to interact at a lower level with cuBLAS than is available via the current interface, and to tune the heuristics used to determine the mode of operation of cuBLAS to this use-case. In particular, cuBLAS will often pick a more parallel but less resource efficient route if it detects that the GPU is likely to be underused by a call. As we know the expected amount of parallelism at a higher level than cuBLAS, overriding this behaviour to favour 5 Minibatch 32 7 6 TFLOPS 5 Minibatch 64 7 Naive Opt #1 Opt #2 Opt #3 Opt #4 1 Layer 4 Layers 6 5 4 4 3 3 2 2 1 1 0 128 256 512 1024 2048 4096 0 128 Naive Opt #1 Opt #2 Opt #3 Opt #4 1 Layer 4 Layers 256 Hidden State Size Minibatch 128 7 6 TFLOPS 5 Naive Opt #1 Opt #2 Opt #3 Opt #4 1 Layer 4 Layers 6 5 4 3 3 2 2 1 1 256 512 1024 1024 2048 4096 2048 4096 Minibatch 256 7 4 0 128 512 Hidden State Size 2048 4096 Hidden State Size 0 128 Naive Opt #1 Opt #2 Opt #3 Opt #4 1 Layer 4 Layers 256 512 1024 Hidden State Size Figure 1: Impact of optimizations on a the forward pass of a four layer LSTM network. The peak performance of the M40 GPU used at fixed base clocks is approximately 5.8 TFLOPS. 6 RNN 6 TFLOPS 5 GRU 6 4 Layers Forward 4 Layers Backward 1 Layer Forward 1 Layer Backward 5 4 4 3 3 2 2 1 1 0 512 1024 2048 Hidden State Size 4096 0 256 4 Layers Forward 4 Layers Backward 1 Layer Forward 1 Layer Backward 512 1024 Hidden State Size LSTM 6 5 4 Layers Forward 4 Layers Backward 1 Layer Forward 1 Layer Backward TFLOPS 4 3 2 1 0 256 512 1024 Hidden State Size 2048 Figure 2: Forward and backward performance of different networks types using cuDNN v5 RC. The peak performance of the M40 GPU used at fixed base clocks is approximately 5.8 TFLOPS. more resource efficient paths in cases of high streamed parallelism sometimes resulted in an overall speedup. It is hoped that an interface to allow this sort of manual tuning will be incorporated into a future release of cuBLAS. 4 Conclusions We have presented a method by which recurrent neural networks can be executed on GPUs at high efficiently. While previous implementations exist achieving good acceleration on GPUs [19, 20], to our knowledge none achieve the levels of performance we achieve using the methods discussed above. The primary strategy used was to expose as much parallelism to the GPU as possible so as to maximize the usage of hardware resources. The methods are particularly efficient when working on smaller deeper recurrent networks where individual layers have less inherent parallelism. 7 2048 One feature of the problem that we do not exploit for performance benefit is the reuse of the parameters between recurrent iterations. It is conceivable that these parameters could be stored in a lower level of GPU memory and reused from iteration to iteration. In bandwidth bound regimes this could potentially greatly improve performance. There are several drawbacks to this method; firstly the amount of storage for parameters is limited, and hence there would be an upper limit on the number of parameters. Secondly, any implementation would have to make assumptions which are invalid in the CUDA programming model, and hence would be prone to unexpected failure. Source code able to reproduce the forward pass timings for each optimization step is available at https://github.com/parallel-forall/code-samples/blob/master/ posts/rnn/LSTM.cu. This code closely mirrors the code used to write the core RNN functionality from version 5 of NVIDIA’s cuDNN library, which is available for download at https:// developer.nvidia.com/cudnn. References [1] Karol Gregor, Ivo Danihelka, Alex Graves, Danilo Jimenez Rezende, and Daan Wierstra. DRAW: A recurrent neural network for image generation. In Proceedings of the 32nd International Conference on Machine Learning, ICML 2015, Lille, France, 6-11 July 2015, pages 1462–1471, 2015. [2] Ivo Danihelka Nal Kalchbrenner, Alex Graves. Grid long short-term memory. In Proceedings of the International Conference on Learning Representations, ICLR, Puerto Rico, 2016. [3] Nal Kalchbrenner and Phil Blunsom. Recurrent continuous translation models. In Proceedings of the 2013 Conference on Empirical Methods in Natural Language Processing, pages 1700– 1709, Seattle, Washington, USA, October 2013. Association for Computational Linguistics. [4] Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio. Neural machine translation by jointly learning to align and translate. In Proceedings of the International Conference on Learning Representations, ICLR, San Diego, 2015. [5] Ilya Sutskever, Oriol Vinyals, and Quoc V. V Le. Sequence to sequence learning with neural networks. In Advances in Neural Information Processing Systems 27. 2014. [6] Oriol Vinyals, Alexander Toshev, Samy Bengio, and Dumitru Erhan. Show and tell: A neural image caption generator. In IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2015, Boston, MA, USA, June 7-12, 2015, pages 3156–3164, 2015. [7] Kelvin Xu, Jimmy Ba, Ryan Kiros, Kyunghyun Cho, Aaron C. Courville, Ruslan Salakhutdinov, Richard S. Zemel, and Yoshua Bengio. Show, attend and tell: Neural image caption generation with visual attention. In Proceedings of the 32nd International Conference on Machine Learning, ICML 2015, Lille, France, 6-11 July 2015, pages 2048–2057, 2015. [8] Tomas Mikolov, Martin Karafiát, Lukás Burget, Jan Cernocký, and Sanjeev Khudanpur. Recurrent neural network based language model. In INTERSPEECH 2010, 11th Annual Conference of the International Speech Communication Association, Makuhari, Chiba, Japan, September 26-30, 2010, pages 1045–1048, 2010. [9] Chris Dyer, Miguel Ballesteros, Wang Ling, Austin Matthews, and Noah A. Smith. Transitionbased dependency parsing with stack long short-term memory. In Proceedings of the 53rd Annual Meeting of the Association for Computational Linguistics and the 7th International Joint Conference on Natural Language Processing, pages 334–343, Beijing, China, July 2015. Association for Computational Linguistics. [10] Alex Graves. Supervised Sequence Labelling with Recurrent Neural Networks, volume 385 of Studies in Computational Intelligence. Springer, 2012. [11] Sepp Hochreiter and Jürgen Schmidhuber. Long short-term memory. Neural Computation, 9(8):1735–1780, November 1997. [12] Kyunghyun Cho, Bart van Merrienboer, Dzmitry Bahdanau, and Yoshua Bengio. On the properties of neural machine translation: Encoder–decoder approaches. In Proceedings of SSST-8, Eighth Workshop on Syntax, Semantics and Structure in Statistical Translation, pages 103– 111, Doha, Qatar, October 2014. Association for Computational Linguistics. 8 [13] Andrew Lavin. Fast algorithms for convolutional neural networks. CoRR, abs/1509.09308, 2015. [14] Nicolas Vasilache, Jeff Johnson, Michaël Mathieu, Soumith Chintala, Serkan Piantino, and Yann LeCun. Fast convolutional nets with fbfft: A GPU performance evaluation. CoRR, abs/1412.7580, 2014. [15] Dario Amodei, Rishita Anubhai, Eric Battenberg, Carl Case, Jared Casper, Bryan C. Catanzaro, Jingdong Chen, Mike Chrzanowski, Adam Coates, Greg Diamos, Erich Elsen, Jesse Engel, Linxi Fan, Christopher Fougner, Tony Han, Awni Y. Hannun, Billy Jun, Patrick LeGresley, Libby Lin, Sharan Narang, Andrew Y. Ng, Sherjil Ozair, Ryan Prenger, Jonathan Raiman, Sanjeev Satheesh, David Seetapun, Shubho Sengupta, Yi Wang, Zhiqian Wang, Chong Wang, Bo Xiao, Dani Yogatama, Jun Zhan, and Zhenyao Zhu. Deep speech 2: End-to-end speech recognition in English and Mandarin. CoRR, abs/1512.02595, 2015. [16] Rafal Józefowicz, Oriol Vinyals, Mike Schuster, Noam Shazeer, and Yonghui Wu. Exploring the limits of language modeling. CoRR, abs/1602.02410, 2016. [17] Nicholas Léonard, Sagar Waghmare, Yang Wang, and Jin-Hwa Kim. RNN : Recurrent library for torch. CoRR, abs/1511.07889, 2015. [18] Felix Weninger. Introducing currennt: The Munich open-source CUDA recurrent neural network toolkit. Journal of Machine Learning Research, 16:547–551, 2015. [19] Tomas Kocisky. oxnn. https://github.com/tkocisky/oxnn, 2015. [20] Justin Johnson. Torch-rnn. https://github.com/jcjohnson/torch-rnn, 2016. 9 

Convergence Results for Neural Networks via Electrodynamics arXiv:1702.00458v4 [] 21 Nov 2017 Rina Panigrahy Google Inc. Mountain View, CA [email protected] Ali Rahimi Google Inc. Mountain View, CA [email protected] Sushant Sachdeva∗ University of Toronto Toronto, Canada [email protected] Qiuyi Zhang† University of California Berkeley, Berkeley, CA [email protected] November 22, 2017 Abstract We study whether a depth two neural network can learn another depth two network using gradient descent. Assuming a linear output node, we show that the question of whether gradient descent converges to the target function is equivalent to the following question in electrodynamics: Given k fixed protons in Rd , and k electrons, each moving due to the attractive force from the protons and repulsive force from the remaining electrons, whether at equilibrium all the electrons will be matched up with the protons, up to a permutation. Under the standard electrical force, this follows from the classic Earnshaw’s theorem. In our setting, the force is determined by the activation function and the input distribution. Building on this equivalence, we prove the existence of an activation function such that gradient descent learns at least one of the hidden nodes in the target network. Iterating, we show that gradient descent can be used to learn the entire network one node at a time. 1 Introduction Deep learning has resulted in major strides in machine learning applications including speech recognition, image classification, and ad-matching. The simple idea of using multiple layers of nodes with a non-linear activation function at each node allows one to express any function. To learn a certain target function we just use (stochastic) gradient descent to minimize the loss; this approach has resulted in significantly lower error rates for several real world functions, such as those in the above applications. Naturally the question remains: how close are we to the optimal values of the network weight parameters? Are we stuck in some bad local minima? While there are several recent works [CHM+ 15, DPG+ 14, Kaw16] that have tried to study the presence of local minima, the picture is far from clear. There has been some work on studying how well can neural networks learn some synthetic function classes (e.g. polynomials [APVZ14], decision trees). In this work we study how well can neural networks learn neural networks with gradient descent? Our focus here, via the framework of ∗ † This work was done when the author was a Research Scientist at Google, Mountain View, CA. Part of this work was done when the author was an intern at Google, Mountain View, CA. 1 proper learning, is to understand if a neural network can learn a function from the same class (and hence achieve vanishing error). Specifically, if the target function is a neural network with randomly initialized weights, and we attempt to learn it using a network with the same architecture, then, will gradient descent converge to the target function? Experimental simulations (see Figure 1 and Section 5 for further details) show that for depth 2 networks of different widths, with random network weights, stochastic gradient descent of a hypothesis network with the same architecture converges to a squared `2 error that is a small percentage of a random network, indicating that SGD can learn these shallow networks with random weights. Because our activations are sigmoidal from -1 to 1, the training error starts from a value of about 1 (random guessing) and diminishes quickly to under 0.002. This seems to hold even when the width, the number of hidden nodes, is substantially increased (even up to 125 nodes), but depth is held constant at 2. In this paper, we attempt to understand this phenomenon theoretically. We prove that, under some assumptions, depth-2 neural networks can learn functions from the same class with vanishingly small error using gradient descent. Figure 1: Test Error of Depth 2 Networks of Varying Width. 1.1 Results and Contributions. We theoretically investigate the question of convergence for networks of depth two. Our main conceptual contribution is that for depth 2 networks where the top node is a sum node, the question of whether gradient descent converges to the desired target function is equivalent to the following question in electrodynamics: Given k fixed protons in Rd , and k moving electrons, with all the electrons moving under the influence of the electrical force of attraction from the protons and repulsion from the remaining electrons, at equilibrium, are all the electrons matched up with all the fixed protons, up to a permutation? 2 In the above, k is the number of hidden units, d is the number of inputs, the positions of each fixed charge is the input weight vector of a hidden unit in the target network, and the initial positions of the moving charges are the initial values of the weight vectors for the hidden units in the learning network. The motion of the charges essentially tracks the change in the network during gradient descent. The force between a pair of charges is not given by the standard electrical force of 1/r2 (where r is the distance between the charges), but by a function determined by the activation and the input distribution. Thus the question of convergence in these simplified depth two networks can be resolved by studying the equivalent electrodynamics question with the corresponding force function. Theorem 1.1 (informal statement of Theorem 2.3). Applying gradient descent for learning the output of a depth two network with k hidden units with activation σ, and a linear output node, under squared loss, using a network of the same architecture, is equivalent to the motion of k charges in the presence of k fixed charges where the force between each pair of charges is given by a potential function that depends on σ and the input distribution. Based on this correspondence we prove the existence of an activation function such that the corresponding gradient descent dynamics under standard Gaussian inputs result in learning at least one of the hidden nodes in the target network. We then show that this allows us to learn the complete target network one node at a time. For more realistic activation functions, we only obtain partial results. We assume the sample complexity is close to its infinite limit. Theorem 1.2 (informal statement of Theorem 4.1). There is an activation function such that running gradient descent for minimizing the squared loss along with `2 regularization for standard Gaussian inputs, at convergence, we learn at least one of the hidden weights of the target neural network. We prove that the above result can be iterated to learn the entire network node-by-node using gradient descent (Theorem 4.6). Our algorithm learns a network with the same architecture and number of hidden nodes as the target network, in contrast with several existing improper learning results. In the appendix, we show some weak results for more practical activations. For the sign activation, we show that for the loss with respect to a single node, the only local minima are at the hidden target nodes with high probability if the target network has a randomly picked top layer. For the polynomial activation, we derive a similar result under the assumption that the hidden nodes are orthonormal. Name of Activation Almost λ-harmonic Sign Polynomial Potential (Φ(θ, w)) Complicated (see Lem 4.2) 1 − π2 cos−1 (θT w) (θT w)m Convergence? Yes, Thm 4.6 Yes for d = 2, Lem G.2 Yes, for orthonormal wi . Lem G.3 Table 1: Activation, Potentials, and Convergence Results Summary 1.2 Intuition and Techniques. Note that for the standard electric potential function given by Φ = 1/r where r is the distance between the charges, it is known from Earnshaw’s theorem that an electrodynamic system with some 3 fixed protons and some moving electrons is at equilibrium only when the moving electrons coincide with the fixed protons. Given our translation above between electrodynamic systems and depth 2 networks (Section 2), this would imply learnability of depth 2 networks under gradient descent under `2 loss, if the activation function corresponds to the electrostatic potential. However, there exists no activation function σ corresponding to this Φ. The proof of Earnshaw’s theorem is based on the fact that the electrostatic potential is harmonic, i.e, its Laplacian (trace of its Hessian) is identically zero. This ensures that at every critical point, there is direction of potential reduction (unless the hessian is identically zero). We generalize these ideas to potential functions that are eigenfunctions of the Laplacians, λ-harmonic potentials (Section 3). However, these potentials are unbounded. Subsequently, we construct a non-explicit activation function such that the corresponding potential is bounded and is almost λ-harmonic, i.e., it is λ-harmonic outside a small sphere (Section 4). For this activation function, we show at a stable critical point, we must learn at least one of the hidden nodes. Gradient descent (possibly with some noise, as in the work of Ge et al. [GHJY15]) is believed to converge to stable critical points. However, for simplicity, we descend along directions of negative curvature to escape saddle points. Our activation lacks some regularity conditions required in [GHJY15]. We believe the results in [JGN+ 17] can be adapted to our setting to prove that perturbed gradient descent converges to stable critical points. There is still a large gap between theory and practice. However, we believe our work can offer some theoretical explanations and guidelines for the design of better activation functions for gradientbased training algorithms. For example, better accuracy and training speed were reported when using the newly discovered exponential linear unit (ELU) activation function in [CUH15, SKS+ 16]. We hope for more theory-backed answers to these and many other questions in deep learning. 1.3 Related Work. If the activation functions are linear or if some independence assumptions are made, Kawaguchi shows that the only local minima are the global minima [Kaw16]. Under the spin-glass and other physical models, some have shown that the loss landscape admits well-behaving local minima that occur usually when the overall error is small [CHM+ 15, DPG+ 14]. When only training error is considered, some have shown that a global minima can be achieved if the neural network contains sufficiently many hidden nodes [SC16]. Recently, Daniely has shown that SGD learns the conjugate kernel class [Dan17]. Under simplifying assumptions, some results for learning ReLU’s with gradient descent are given in [Tia17, BG17]. Our research is inspired by [APVZ14], where the authors show that for polynomial target functions, gradient descent on neural networks with one hidden layer converges to low error, given a large number of hidden nodes, and under complex perturbations, there are no robust local minima. Even more recently, similar results about the convergence of SGD for two-layer neural networks have been established for a polynomial activation function under a more complex loss function [GLM17]. And in [LY17], they study the same problem as ours with the RELU activation and where lower layer of the network is close to identity and the upper layer has weights all one. This corresponds to the case where each electron is close to a distinct proton – under these assumptions they show that SGD learns the true network. Under worst case assumptions, there has been hardness results for even simple networks. A neural network with one hidden unit and sigmoidal activation can admit exponentially many local minima [AHW96]. Backprogration has been proven to fail in a simple network due to the abundance of bad local minima [BRS89]. Training a 3-node neural network with one hidden layer is NP-complete [BR88]. But, these and many similar worst-case hardness results are based on worst case training data assumptions. However, by using a result in [KS06] that learning a neural network with threshold 4 activation functions is equivalent to learning intersection of halfspaces, several authors showed that under certain cryptographic assumptions, depth-two neural networks are not efficiently learnable with smooth activation functions [LSSS14, ZLWJ15, ZLJ16]. Due to the difficulty of analysis of the non convex gradient descent in deep learning, many have turned to improper learning and the study of non-gradient methods to train neural networks. Janzamin et. al use tensor decomposition methods to learn the shallow neural network weights, provided access to the score function of the training data distribution [JSA15]. Eigenvector and tensor methods are also used to train shallow neural networks with quadratic activation functions in [LSSS14]. Combinatorial methods that exploit layerwise correlations in sparse networks have also been analyzed provably in [ABGM14]. Kernel methods, ridge regression, and even boosting were explored for regularized neural networks with smooth activation functions in [SSSS11, ZLWJ15, ZLJ16]. Non-smooth activation functions, such as the ReLU, can be approximated by polynomials and are also amenable to kernel methods[GKKT16]. These methods however are very different from the simple popular SGD. 2 2.1 Deep Learning, Potentials, and Electron-Proton Dynamics Preliminaries. We will work in the space M = Rd . We denote the gradient and Hessian as ∇Rd f and ∇2Rd f respectively. The Laplacian is defined as ∆Rd f = Tr(∇2Rd f ). If f is multivariate with variable xi , then let fxi be a restriction of f onto the variable xi with all other variables fixed. Let ∇xi f, ∆xi f to be the gradient and Laplacian, respectively, of fxi with respect to xi . Lastly, we say x is a critical point of f if ∇f does not exist or ∇f = 0. We focus on learning depth two networks with a linear activation on the output node. If the network takes inputs x ∈ Rd (say from some distribution D), then the network output, denoted f (x) is d d d a sum over k = poly(d) hidden units with weight vectors Pkwi ∈ R , activation σ(x, w) : R × R → R, and output weights bi ∈ R. Thus, we can write f (x) = i=1 bi σ(x, wi ). We denote this concept class Cσ,k . Our hypothesis concept class is also Cσ,k . P Let a = (a1 , ..., ak ) and θ = (θ1 , ..., θk ); similarly for b, w and our guess is fˆ(x) = ki=1 ai σ(x, θi ). We define Φ, the potential function corresponding to the activation σ, as Φ(θ, w) = E [σ(X, θ)σ(X, w)]. X∼D We work directly with the true squared loss error L(a, θ) = Ex∼D [(f − fˆ)2 ]. To simplify L, we re-parametrize a by −a and expand.  !2  k k X X L(a, θ) = E  ai σ(X, θi ) + bi σ(X, wi )  X∼D = k X k X i=1 i=1 ai aj Φ(θi , θj ) + 2ai bj Φ(θi , wj ) + bi bj Φ(wi , wj ), (1) i=1 j=1 Given D, the activation function σ, and the loss L, we attempt to show that we can use some variant of gradient descent to learn, with high probability, an -approximation of wj for some (or all) j. Note that our loss is jointly convex, though it is quadratic in a. In this paper, we restrict our attention to translationally invariant activations and potentials. Specifically, we may write Φ = h(θ − w) for some function h(x). Furthermore, a translationally invariant function Φ(r) is radial if it is a function of r = kx − yk. 5 Remark: Translationally symmetric potentials satisfy Φ(θ, θ) is a positive constant. We normalize Φ(θ, θ) = 1 for the rest of the paper. We assume that our input distribution D = N (0, Id×d ) is fixed as the standard Gaussian in Rd . This assumption is not critical and a simpler distribution might lead to better bounds. However, for arbitrary distributions, there are hardness results for PAC-learning halfspaces [KS06]. We call a potential function realizable if it corresponds to some activation σ. The following theorem characterizes realizable translationally invariant potentials under standard Gaussian inputs. Proofs and a similar characterization for rotationally invariant potentials can be found in Appendix B . Theorem 2.1. Let M = Rd and Φ is square-integrable and F(Φ) is integrable. Then, Φ is realizable under standardpGaussian inputs if F(Φ)(ω) ≥ 0 and the corresponding activation is T σ(x) = (2π)d/4 ex x/4 F−1 ( F(Φ))(x), where F is the generalized Fourier transform in Rd . 2.2 Electron-Proton Dynamics By interpreting the pairwise potentials as electrostatic attraction potentials, we notice that our dynamics is similar to electron-proton type dynamics under potential Φ, where wi are fixed point charges in Rd and θi are moving point charges in Rd that are trying to find wi . The total force on each charge is the sum of the pairwise forces, determined by the gradient of Φ. We note that standard dynamics interprets the force between particles as an acceleration vector. In gradient descent, it is interpreted as a velocity vector. Definition 2.2. Given a potential Φ and particle locations θ1 , ..., θk ∈ Rd along with their respective charges a1 , ..., ak ∈ R. We define Electron-Proton Dynamics under Φ with some subset S ⊆ [k] of fixed particles to be the solution (θ1 (t), ..., θk (t)) to the following system of differential equations: For each pair (θi , θj ), there is a force from θj exerted on θi that is given by Fi (θj ) = ai aj ∇θi Φ(θi , θj ) and dθi X − = Fi (θj ) dt j6=i for all i 6∈ S, with θi (0) = θi . For i ∈ S, θi (t) = θi . For the following theorem, we assume that θ is fixed. Theorem 2.3. Let Φ be a symmetric potential and L be as in (1). Running continuous gradient descent on 12 L with respect to θ, initialized at (θ1 , ..., θk ) produces the same dynamics as ElectronProton Dynamics under 2Φ with fixed particles at w1 , ..., wk with respective charges b1 , .., bk and moving particles at θ1 , ..., θk with respective charges a1 , ..., ak . 3 Earnshaw’s Theorem and Harmonic Potentials When running gradient descent on a non-convex loss, we often can and do get stuck at a local minima. In this section, we use second-order information to deduce that for certain classes of potentials, there are no spurious local minima. The potentials In this section are often unbounded and un-realizable. However, in the next section, we apply insights developed here to derive similar convergence results for approximations of these potentials. Earnshaw’s theorem in electrodynamics shows that there is no stable local minima for electronproton dynamics. This hinges on the property that the electric potential Φ(θ, w) = kθ − wk2−d , d 6= 2 is harmonic, with d = 3 in natural setting. If d = 2, we instead have Φ(θ, w) = − ln(kθ − wk). First, 6 we notice that this is a symmetric loss, and our usual loss in (1) has constant terms that can be dropped to further simplify. L(a, θ) = 2 k X X ai aj Φ(θi , θj ) + 2 i=1 i<j k X k X ai bj Φ(θi , wj ) (2) i=1 j=1 Definition 3.1. Φ(θ, w) is a harmonic potential on Ω if ∆θ Φ(θ, w) = 0 for all θ ∈ Ω, except possibly at θ = w. Definition 3.2. Let Ω ⊆ Rd and consider a function f : Ω → R. A critical point x∗ ∈ Ω is a local minimum if there exists  > 0 such that f (x∗ + v) ≥ f (x∗ ) for all kvk ≤ . It is a strict local minimum if the inequality is strict for all kvk ≤ . Fact 3.3. Let x∗ be a critical point of a function f : Ω → R such that f is twice differentiable at x∗ . Then, if x∗ is a local minimum then λmin (∇2 f (x∗ )) ≥ 0. Moreover, if λmin (∇2 f (x∗ )) > 0, then x∗ is a strict local minimum. Note that if λmin (∇2 f (x∗ )) < 0 then moving along the direction of the corresponding eigenvector decreases f locally. If Φ is harmonic then it can be shown the trace of its Hessian is 0 so if there is any non zero eigenvalue then at least one eigenvalue is negative. This idea results in the following known theorem (see full proof in supplementary material) that is applicable to the electric potential function 1/r in 3-dimensions since is harmonic. It implies that a configuration of n electrons and n protons cannot be in a strict local minimum even if one of the mobile charges is isolated (however note that this potential function goes to ∞ at r = 0 and may not be realizable). Theorem 3.4. (Earnshaw’s Theorem. See [AKN85]) Let M = Rd and let Φ be harmonic and L be as in (2). Then, L admits no differentiable strict local minima. Note that the Hessian of a harmonic potential can be identically zero. To avoid this possibility we generalize harmonic potentials. 3.1 λ-Harmonic Potentials In order to relate our loss function with its Laplacian, we consider potentials that are non-negative eigenfunctions of the Laplacian operator. Since the zero eigenvalue case simply gives rise to harmonic potentials, we restrict our attention to positive eigenfunctions. Definition 3.5. A potential Φ is λ-harmonic on Ω if there exists λ > 0 such that for every θ ∈ Ω, ∆θ Φ(θ, w) = λΦ(θ, w), except possibly at θ = w. Note that there are realizable versions of these potentials; for example Φ(a, b) = e−ka−bk1 in R1 . In the next section, we construct realizable potentials that are λ-harmonic almost everywhere except when θ and w are very close. Theorem 3.6. Let Φ be λ-harmonic and L be as in (1). Then, L admits no local minima (a, θ), except when L(a, θ) = L(0, θ) or θi = wj for some i, j. Proof. Let (a, θ) be a critical point of L. On the contrary, we assume that θi 6= wj for all i, j. WLOG, we can partition [k] into S1 , ..., Sr such that for all u ∈ Si , v ∈ Sj , we have θu = θv iff i = j. Let S1 = {θ1 , . . . , θl }. We consider changing all θ1 , . . . , θl by the same v and define H(a, v) = L(a, θ1 + v, ..., θl + v, θl+1 . . . , θk ). 7 P P ∂L The optimality conditions on a are 0 = ∂a = 2 j aj Φ(θi , θj ) + 2 kj=1 bj Φ(θi , wj ). Thus, by the i definition of λ-harmonic potentials, we may differentiate as θi 6= wj and compute the Laplacian as   l k k X X X ∆v H = λ ai 2 bj Φ(θi , wj ) + 2 aj Φ(θi , θj ) i=1 =λ l X i=1 j=1  ai −2 l X j=l+1  aj Φ(θi , θj ) = −2λ j=1 l X i=1  ai  l X  aj  = −2λ j=1 l X !2 ai i=1 Pl If i=1 ai 6= 0, then we conclude that the Laplacian is strictly P negative, so we are not at a local minimum. Similarly, we can conclude that for each Si , u∈Si au = 0. In this case, since Pk i=1 ai σ(θi , x) = 0, L(a, θ) = L(0, θ). 4 Realizable Potentials with Convergence Guarantees In this section, we derive convergence guarantees for realizable potentials that are almost λ-harmonic, specifically, they are λ-harmonic outside of a small neighborhood around the origin. First, we prove the existence of activation functions such that the corresponding potentials are almost λ-harmonic. Then, we reason about the Laplacian of our loss, as in the previous section, to derive our guarantees. We show that at a stable minima, each of the θi is close to some wj in the target network. We may end up with a many to one mapping of the learned hidden weights to the true hidden weights, instead of a bijection. To make sure that kak remains controlled throughout the optimization process, we add a quadratic regularization term to L and instead optimize G = L + kak2 . Our optimization procedure is a slightly altered version of gradient descent, where we incorportate a second-order method (which we call Hessian descent as in Algorithm 1) that is used when the gradient is small and progress is slow. The descent algorithm (Algorithm 2) allows us to converge to points with small gradient and small negative curvature. Namely, for smooth functions, in poly(1/) iterations, we reach a point in MG, , where n o MG, = x ∈ M k∇G(x)k ≤  and λmin (∇2 G(x)) ≥ − We show that if (a, θ) is in MG, for  small, then θi is close to wj for some j. Finally, we show how to initialize (a(0) , θ (0) ) and run second-order GD to converge to MG, , proving our main theorem. Algorithm 1 x = HD(L, x0 , T, α) Input: L : M → R; x0 ∈ M; T ∈ N; α ∈ R Initialize x ← x0 for i = 1 to T do Find unit eigenvector vmin corresponding to λmin (∇2 f (x)) β ← −αλmin (∇2 f (x))sign(∇f (x)T vmin ) x ← x + βvmin Theorem 4.1. Let M = Rd for d ≡ 3 mod 4 and k = poly(d). For all  ∈ (0, 1), we can construct an activation σ such that if w1 , ..., wk ∈ Rd with wi randomly chosen from wi ∼ N (0, O(d log d)Id×d ) and b1 , ..., bk be randomly chosen at uniform from [−1, 1], then with high probability, we can choose an initial point (a(0) , θ (0) ) such that after running SecondGD (Algorithm 2) on the regularized objective G(a, θ) for at most (d/)O(d) iterations, there exists an i, j such that kθi − wj k < . 8 We start by stating a lemma concerning the construction of an almost λ-harmonic function on The construction is given in Appendix B and uses a linear combination of realizable potentials that correspond to an activation function of the indicator function of a n-sphere. By using Fourier analysis and Theorem 2.1, we can finish the construction of our almost λ-harmonic potential. Rd . Lemma 4.2. Let M = Rd for d ≡ 3 mod 4. Then, for any  ∈ (0, 1), we can construct a radial activation σ (r) such that the corresponding radial potential Φ (r) is λ-harmonic for r ≥ . Furthermore, we have Φ (d−1) (r) ≥ 0 for all r > 0, Φ (k) (r) ≥ 0, and Φ (k+1) (r) ≤ 0 for all r > 0 and d − 3 ≥ k ≥ 0 even. (k) When λ = 1, |Φ (r)| ≤ O((d/)2d ) for all 0 ≤ k ≤ d−1. And when r ≥ , Ω(e−r r2−d (d/)−2d ) ≤ Φ (r) ≤ O((1 + r)d e1−r (r)2−d ) and Ω(e−r r1−d (d/)−2d ) ≤ |Φ0 (r)| ≤ O((d + r)(1 + r)d e1−r r1−d ) Our next lemma use the almost λ-harmonic properties to show that at an almost stationary point of G, we must have converged close to some wj as long as our charges ai are not too small. The proof is similar to Theorem 3.6. Then, the following lemma relates the magnitude of the charges ai to the progress made in the objective function. Lemma 4.3. Let M = Rd for d ≡ 3 mod 4 and let G be the regularized loss corresponding to the activation σ given by Lemma 4.2 with λ = 1. For any  ∈ (0, 1) and δ ∈ (0, 1), if (a, θ) ∈ MG,δ , then for all i, either 1) there exists j such that kθi − wj k < k or 2) a2i < 2kdδ. p p Lemma 4.4. Assume the conditions of Lemma 4.3. If G(a, θ) ≤ G(0, 0) − δ and (a, θ) ∈ MG,δ2 /(2k3 d) , then there exists some i, j such that kθi − wj k < k. Finally, we guarantee that our initialization substantially decreases our objective function. Together with our previous lemmas, it will imply that we must be close to some wj upon convergence. This is the overview of the proof of Theorem 4.1, presented below. of Theorem 4.1 and Lemma 4.3. With high probability, we can Lemma 4.5. Assume the conditions q p (0) (0) initialize (a , θ ) such that G(a(0) , θ (0) ) ≤ G(0, 0) − δ with δ = (d/)−O(d) . Proof of Theorem 4.1. Let our potential Φ/k be the one as constructed in Lemma 4.2 that is 1harmonic for all r ≥ q /k and as always, k = poly(d). First, by Lemma 4.5, we can initialize p (a(0) , θ (0) ) such that G(a(0) , θ (0) ) ≤ G(0, 0) − δ for δ = (d/)−O(d) . If we set α = (d/)−O(d) and η = γ = δ 2 /(2k 3 d), then running Algorithm 2 will terminate and return some (a, θ) in at most (d/)O(d) iterations. This is because our algorithm ensures that our objective function decreases by at least min(αη 2 /2, α2 γ 3 /2) at each iteration, G(0, 0) is bounded by O(k), and G ≥ 0 is non-negative. Let θ = (θ1 , ...θk ). If there exists θi , wj such that kθi − wj k < , then we are done. Otherwise, we claim that (a, θ) ∈ MG,δ2 /(2k3 d) . For the sake of contradiction, assume otherwise. By our algorithm termination conditions, then it must be that after one step of gradient or Hessian descent from (a, θ), we reach some (a0 , θ 0 ) and G(a0 , θ 0 ) > G(a, θ) − min(αη 2 /2, α2 γ 3 /2). Algorithm 2 x = SecondGD(L, x0 , T, α, η, γ) Input: L : M → R; x0 ∈ M; T ∈ N; α, η, γ ∈ R for i = 1 to T do if k∇L(xi−1 )k ≥ η then xi ← xi−1 − α∇L(xi−1 ) else xi ← HD(L, xi−1 , 1, α) if L(xi ) ≥ L(xi−1 ) − min(αη 2 /2, α2 γ 3 /2) then return xi−1 9 Algorithm 3 Node-wise Descent Algorithm Input: (a, θ) = (a1 , ..., ak , θ1 , ..., θk ), ai ∈ R, θi ∈ M; T ∈ N; L; α, η, γ ∈ R; for i = 1 to k do Initialize (ai , θi ) (ai , θi ) = SecondGD (Lai ,θi , (ai , θi ), T, α, η, γ) return a = (a1 , ..., ak ), θ = (θ1 , ..., θk ) Now, Lemma 4.2 ensures all first three derivatives of Φ/k are bounded by O((dk/)2d ), except at w1 , ..., wk . Furthermore, since there do not exist θi , wj such that kθi − wj k < , G is three-times continuously differentiable within a α(dk/)2d = (d/)−O(d) neighborhood of θ. Therefore, by Lemma D.1 and D.2 in the appendix , we must have G(a0 , θ 0 ) ≤ G(a, θ) − min(αη 2 /2, α2 γ 3 /2), a contradiction. p pLastly, since our algorithm maintains that our objective function is decreasing, so G(a, θ) ≤ G(0, 0) − δ. Finally, we conclude by Lemma 4.4. 4.1 Node-by-Node Analysis We cannot easily analyze the convergence of gradient descent to the global minima when all θi are simultaneously moving since the pairwise interaction terms between the θi present complications, even with added regularization. Instead, we run a greedy node-wise descent (Algorithm 3) to learn the hidden weights, i.e. we run a descent algorithm with respect to (ai , θi ) sequentially. The main idea is that after running SGD with respect to θ1 , θ1 should be close to some wj for some j. Then, we can carefully induct and show that θ2 must be some other wk for k 6= j and so on. Let L1 (a1 , θ1 ) be the objective L restricted to a1 , θ1 being variable, and a2 , ..., ak = 0 are fixed. The tighter control on the movements of θ1 allows us to remove our regularization. While our previous guarantees before allow us to reach a -neighborhood of wj when running SGD on L1 , we will strengthen our guarantees to reach a (d/)−O(d) -neighborhood of wj , by reasoning about the first derivatives of our potential in an -neighborhood of wj . By similar argumentation as before, we will be able to derive the following convergence guarantees for node-wise training. Theorem 4.6. Let M = Rd and d ≡ 3 mod 4 and let L be as in 1 and k = poly(d). For all  ∈ (0, 1), we can construct an activation σ such that if w1 , ..., wk ∈ Rd with wi randomly chosen from wi ∼ N (0, O(d log d)Id×d ) and b1 , ..., bk be randomly chosen at uniform from [−1, 1], then with high probability, after running nodewise descent (Algorithm 3) on the objective L for at most (d/)O(d) iterations, (a, θ) is in a (d/)−O(d) neighborhood of the global minima. 5 Experiments For our experiments, our training data is given by (xi , f (xi )), where xi are randomly chosen from a standard Gaussian in Rd and f is a randomly generated neural network with weights chosen from a standard Gaussian. We run gradient descent (Algorithm 4) on the empirical loss, with stepsize around α = 10−5 , for T = 106 iterations. The nonlinearity used at each node is sigmoid from -1 to 1, including the output node, unlike the assumptions in the theoretical analysis. A random guess for the network will result in a mean squared error of around 1. Our experiments (see Fig 1) show that for depth-2 neural networks, even with non-linear outputs, the training error diminishes quickly to under 0.002. This seems to hold even when the width, the number of hidden nodes, is substantially increased (even up to 125 nodes), but depth is held constant; although as the number of nodes 10 Depth Depth Depth Depth Depth Width 5 0.0015 0.0033 0.0036 0.0085 0.0845 2 3 5 9 17 Width 10 0.0017 0.0264 0.0579 0.1662 0.3862 Width 20 0.0018 0.1503 0.2400 0.4171 0.4934 Width 40 0.0019 0.2362 0.4397 0.6071 0.5777 Table 2: Test Error of Learning Neural Networks of Various Depth and Width increases, the rate of decrease is slower. This substantiates our claim that depth-2 neural networks are learnable. However, it seems that for depth greater than 2, the test error becomes significant when width is high (see Fig 2). Even for depth 3 networks, the increase in depth impedes the learnability of the neural network and the training error does not get close enough to 0. It seems that for neural networks with greater depth, positive convergence results in practice are elusive. We note that we are using training error as a measure of success, so it’s possible that the true underlying parameters are not learned. Figure 2: Test Error of Varying-Depth Networks vs. Width References [ABGM14] Sanjeev Arora, Aditya Bhaskara, Rong Ge, and Tengyu Ma. Provable bounds for learning some deep representations. In ICML, pages 584–592, 2014. [AHW96] Peter Auer, Mark Herbster, and Manfred K Warmuth. Exponentially many local minima for single neurons. pages 316–322, 1996. 11 [AKN85] Vladimir I Arnold, Valery V Kozlov, and Anatoly I Neishtadt. Mathematical aspects of classical and celestial mechanics. Encyclopaedia Math. Sci, 3:1–291, 1985. [APVZ14] Alexandr Andoni, Rina Panigrahy, Gregory Valiant, and Li Zhang. 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The initial values are the same. Notice that continuous gradient descent on L(a, θ) with i (t) respect to θ produces dynamics given by dθdt = −∇θi L(a, θ). Therefore, k X X dθi (t) = −2 ai aj ∇θi Φ(θi , θj ) − 2 ai bj ∇θi Φ(θi , wj ) dt j=1 j6=i And gradient descent does not move wi . By definition, the dynamics corresponds to Electron-Proton Dynamics as claimed. B B.1 Realizable Potentials Activation-Potential Calculations First define the dual of a function f : R → R is defined to be fb(ρ) = [f (X)f (Y )], E X,Y ∼N (ρ) where N (ρ) is the bivariate normal distribution with X, Y unit variance and ρ covariance. This is as in [DFS16]. b is the corresponding potential Lemma B.1. Let M = S d−1 and σ be our activation function, then σ function. Proof. If u, v have norm 1 and if X is a standard Gaussian in Rd , then note that X1 = uT X and X2 = v T X are both standard Gaussian variables in R1 and the covariance is E[X1 X2 ] = uT v. Therefore, the dual function of the activation gives us the potential function. E[σ(uT X)σ(v T X)] = X E X,Y ∼N (uT v) [σ(X)σ(Y )] =σ b(uT v). By Lemma B.1, the calculations of the activation-potential for the sign, ReLU, Hermite, exponential functions are given in [DFS16]. For the Gaussian and Bessel activation functions, we can calculate directly. In both case, we notice that we may write the integral as a product of integrals in each dimension. Therefore, it suffices to check the following 1-dimensional identities. Z ∞√ 1 2 2√ 2 2 2 2ex /4 e−(x−θ) 2ex /4 e−(x−w) √ e−x /2 dx 2π −∞ r Z ∞ 2 2 2 2 = e−(x−θ) e−(x−w) dx = e−(θ−w) /2 π −∞ 14 Z ∞ 2 1 2 2 ( )3/2 ex /2 K0 (|x − θ|)K0 (|x − w|) √ e−x /2 dx 2π −∞ π Z ∞ 2 = K (|x − θ|)K0 (|x − w|) dx = e−|θ−w| 2 0 −∞ π The last equality follows uniqueness and taking the Fourier transform of both sides, p by Fourier 2 −1 which are both equality 2/π(ω + 1) . B.2 Characterization Theorems Theorem 2.1. Let M = Rd and Φ is square-integrable and F(Φ) is integrable. Then, Φ is realizable under standardpGaussian inputs if F(Φ)(ω) ≥ 0 and the corresponding activation is T σ(x) = (2π)d/4 ex x/4 F−1 ( F(Φ))(x), where F is the generalized Fourier transform in Rd . p Proof. Since Φ is square-integrable, its Fourier transform exists. Let h(x) = F−1 ( F(Φ))(x) and this is well-defined since the Fourier transform was non-negative everywhere and the Fourier inverse p 2 exists since F(Φ)(x) is square-integrable. Now, let σ(x, w) = (2π)1/4 ex /4 h(x − w). Realizability follows by the Fourier inversion theorem: Z E [σ(X, w)σ(X, θ)] = h(x − w)h(x − θ) dx X∼N n R Z = h(x)h(x − (θ − w)) dx Rn −1 =F (F(h ∗ h)(θ − w)) = F−1 (F(h)2 (θ − w)) = F−1 (F(Φ)(θ − w)) = Φ(θ − w) Note that ∗ denotes function convolution. When our relevant space is M = S d−1 , we let ΠM be the projection operator on M. The simplest way to define the gradident on S d−1 is ∇S d−1 f (x) = ∇Rd f (x/kxk), where k · k denotes the l2 norm and x ∈ S d−1 . The Hessian and Laplacian are analogously defined and the subscripts are usually dropped where clear from context. We say that a potential Φ on M = S d−1 is rotationally invariant if for all θ, w ∈ S d−1 , we have Φ = h(θT w). Theorem B.2. Let M = S d−1 and Φ(θ, w) = f (θT w). Then, Φ is P realizable if f has non-negative √ Taylor coefficients, ci ≥ 0 , and the corresponding activation σ(x) = ∞ i=1 ci hi (x) converges almost everywhere, where hi (x) is the i-th Hermite polynomial. P Proof. By B.1 and due to the orthogonality of hermite polynomials, if f = i ai hi , where hi (x) is the i-th Hermite polynomial, then X fb(ρ) = a2i ρi i Therefore, any function with non-negative taylor coefficients is a valid potential function, with the corresponding activation function determined by the sum of hermite polynomials, and the sum is bounded almost everywhere by assumption. 15 B.3 Further Characterizations To apply Theorem 2.1, we need to check that the Fourier transform of our function is non-negative. Not only is this is not straightforward to check, many of our desired potentials do not satisfy this criterion. In this section, we would like to have a stronger characterization of realizable potentials, allowing us to construct realizable potentials that approximates our desired potential. Definition B.3. Let Φ be a positive semidefinite function if for all x1 , ..., xn , the matrix Aij = Φ(xi − xj ) is positive semidefinite. Lemma B.4. Let M = Rd and Φ(θ, w) = f (θ − w) is is realizable, then it is positive semidefinite. Proof. If Φ is realizable, then there exists σ such that Φ(θ, w) = EX∼N [σ(X, w)σ(X, θ)]. For x1 , ..., xn , we note that the quadratic form:  !2  X X X Φ(xi , xj )vi vj = E [σ(X, xi )σ(X, xj )]vi vj = E  vi σ(X, xi )  ≥ 0 i,j i,j X∼N X∼N i Since Φ is translationally symmetric, we conclude that Φ is positive semidefinite. Definition B.5. A potential Φ is F-integrable if it is square-integrable and F(Φ(ω)) is integrable, where F is the standard Fourier transform. Rb Lemma B.6. Let w(x) ≥ 0 be a positive weighting function such that a w(x) dx is bounded. If Rb Φx is a parametrized family of F-integrable realizable potentials, then, a w(x)Φx is F-integrable realizable. Rb Rb Proof. Let Φ = a w(x)Φx . From linearity of the Fourier transform and a w(x) dx is bounded, we know that Φ is F-integrable. Since Φx are realizable, they are positive definite by Lemma B.4 and by Bochner’s theorem, their Fourier transforms are non-negative. And since w(x) ≥ 0, we conclude by linearity and continuity of the Fourier transform that F(Φ) ≥ 0. By Theorem 2.1, we conclude that Φ is realizable. Lemma B.7. Let M = Rd for d ≡ 3 mod 4. Then, for any , t > 0, there exists a F-integrable realizable Φ such that for t ≥ r > , Φ(d−1) (r) = t − r and for r ≤ , Φ(d−1) (r) = t−  r. Furthermore, Φ(k) (r) = 0 for r > t for all 0 ≤ k ≤ d. Proof. Our construction is based on the radial activation function ht (x, θ) = 1kθ−xk≤t/2 , which is the indicator in the disk of radius t/2. This function, when re-weighted correctly as σt (x, θ) = 2 (2π)1/4 ex /4 ht (x, θ) gives rise to a radial potential function that is simply the convolution of ht with itself, measuring the volume of the intersection of two spheres of radius t centered at θ and w. ( R t/2 C kθ−wk/2 ((t/2)2 − x2 )(d−1)/2 dx kθ − wk ≤ t Φt (θ, w) = E[σt (X, θ)σt (X, w)] = X 0 otherwise. R t/2 2 2 (d−1)/2 dx r/2 ((t/2) − x ) and Φ0t (r) = −C 0 ((t/2)2 − (r/2)2 )(d−1)/2 . Since d ≡ 3 mod 4, we notice that Φ0t has a positive coefficient in the leading rd−1 term and since it is a function of r2 , it has a zero rd−2 term. Therefore, Therefore, as a function of r = kθ − wk, we see that when r ≤ t, Φt (r) = C we can scale Φt such that ( r r≤t (d−1) Φt (r) = 0 otherwise. 16 Φt is clearly realizable and now we claim that it is F-integrable. First, Φt is bounded on a compact set so it is square-integrable. Now, since Φt = ht ∗ ht can be written as a convolution, F(Φt ) = F(ht )2 . Since ht is square integrable, then by Parseval’s, F(ht ) is square integrable, allowing us to conclude that Φt is F-integrable. Now, for any  > 0, let us construct our desired Φ by taking a positive sum of Φt and then appealing to Lemma B.6. Consider Z t 1 Φ(r) = Φ (r) dx 2 x  x Rt First, note that the total weight  x12 is bounded. Then, when r ≥ t, since Φx (r) = 0 for x ≤ t, we conclude that Φ(k) (r) = 0 for any k. Otherwise, for  < r < t, we can apply dominated convergence theorem to get Z t Z t Z r 1 (d−1) r 1 (d−1) (d−1) Φx (r) dx + Φx (r) dx = 0 + dx = 1 − r/t Φ (r) = 2 2 2 r x r x  x Scaling by t gives our desired claim. For r ≤ , we integrate similarly and scale by t to conclude. Lemma B.8. Let M = Rd for d ≡ 3 mod 4 and let Φ(r) be a radial potential. Also, Φ(k) (r) ≥ 0 and Φ(k+1) (r) ≤ 0 for all r > 0 and k ≥ 0 even, and limr→∞ Φ(k) (r) = 0 for all 0 ≤ k ≤ d. (k) Then, for any  > 0, there exists a F-integrable realizable potential Φ such that Φ (r) = Φ(k) (r) (d−1) (k) for all 0 ≤ k ≤ d − 1 and r ≥ . Furthermore, we have Φ (r) ≥ 0 for all r > 0 and Φ (r) ≥ 0 (k+1) and Φ (r) ≤ 0 for all r > 0 and d − 3 ≥ k ≥ 0 even. k−j+1 P (d−1−k) (d−j) ()| Lastly, for r <  and 0 ≤ k ≤ d − 1, |Φ (r)| ≤ |Φ(d−1−k) ()| + kj=1 (−r) (k−j+1)! |Φ Proof. By Lemma B.7, we can find Φt such that  t−    r (d−1) Φt = t−r   0 0≤r≤ <r≤t r>t (k) Furthermore, Φt (r) = 0 for r > t for all 0 ≤ k ≤ d. Therefore, we consider Z ∞ Φ(r) = Φ(d+1) (x)Φx (r) dx  R∞ Note that this is a positive sum with  Φ(d+1) (x) dx = −Φ(d) () < ∞. By the non-negativity of our summands, we can apply dominated convergence theorem and Fubini’s theorem to get Z ∞ (d−1) Φ (r) = Φ(d+1) (x)(Φ(d−1) (r)) dx x  Z ∞ = Φ(d+1) (x)(Φx(d−1) (r)) dx Zr ∞ Z x (d+1) = Φ (x) 1 dy dx r Zr ∞ Z ∞ Z ∞ (d+1) = Φ (x) dx dy = −Φ(d) (y) dy r y (d−1) =Φ r (r) 17 (d−1) (k) Now, since Φ (r) = Φ(d−1) (r) for r ≥  and limr→∞ Φ (r) = limr→∞ Φ(k) (r) = 0 for 0 ≤ k ≤ d − 1, repeated integration gives us our claim. Finally, for the second claim, notice that for r ≤ , we get Z ∞ Z ∞ x− (d−1) (d+1) (d−1) Φ(d+1) (x) Φ (x)(Φx (r)) dx = r Φ (r) = dx = Cr    Note that our constant C ≥ 0 since the summands are non-negative. Therefore, we conclude (d−1) (k) that Φ (r) ≥ 0 for all r > 0. Repeated integration and noting that limr→∞ Φ (r) = 0 for 0 ≤ k ≤ d − 1 gives us our claim. Lastly, we prove the last claim of the theorem with induction on k. This holds trivially for k = 0 (d−1) (d−1) since Φ (r) ≤ Φ () = Φ(d−1) () for r ≤ . Then, assume we have the inequality for k < d − 1. By integration, we have Z  (d−k−2) (d−k−2) (d−1−k) |Φ |Φ (r)| ≤ |Φ ()| + (y)| dy r Z  (d−k−2) (d−1−k) |Φ ()| + ()| dy ≤ |Φ r Z X k ( − y)k−j+1 (d−j) + |Φ ()| dy (k − j + 1)! r j=1 ≤ |Φ (d−k−2) ()| + k+1 X ( − y)k−j+2 j=1 (k − j + 2)! |Φ(d−j) ()| Therefore, we conclude with induction. Lemma 4.2. Let M = Rd for d ≡ 3 mod 4. Then, for any  ∈ (0, 1), we can construct a radial activation σ (r) such that the corresponding radial potential Φ (r) is λ-harmonic for r ≥ . Furthermore, we have Φ (d−1) (r) ≥ 0 for all r > 0, Φ (k) (r) ≥ 0, and Φ (k+1) (r) ≤ 0 for all r > 0 and d − 3 ≥ k ≥ 0 even. (k) When λ = 1, |Φ (r)| ≤ O((d/)2d ) for all 0 ≤ k ≤ d−1. And when r ≥ , Ω(e−r r2−d (d/)−2d ) ≤ d Φ (r) ≤ O((1 + r) e1−r (r)2−d ) and Ω(e−r r1−d (d/)−2d ) ≤ |Φ0 (r)| ≤ O((d + r)(1 + r)d e1−r r1−d ) Proof. This is a special case of the following lemma. Lemma B.9. Let M = Rd for d ≡ 3 mod 4. Then, for any 1 >  > 0, we can construct a radial activation σ (r) with corresponding normalized radial potential Φ (r) that is λ-harmonic when r ≥ . Furthermore, we have Φ (d−1) (r) ≥ 0 for all r > 0 and Φ (k) (r) ≥ 0 and Φ (k+1) (r) ≤ 0 for all r > 0 and d − 3 ≥ k ≥ 0 even. √ √ √ (k) λ for all 0 ≤ k ≤ d − 1. And for r ≥ , e− λr r 2−d (2d + Also, |Φ (r)| ≤ 3(2d + λ)2d −2d e √ √ −2d 2d √ d √λ(1−r) 2−d √ λ)  /3 ≤ Φ (r) ≤ (1 + r √ λ) e (r) . Also for r ≥ , e− λr r1−d (2d + λ)−2d 2d /3 ≤ √ √ |Φ0 (r)| ≤ (d + λr)(1 + r λ)d e λ(1−r) r1−d √ Proof. Consider a potential of the form Φ(r) = p(r)e− λr /rd−2 . We claim that there exists a polynomial p of degree k = (d − 3)/2 with non-negative coefficients and p(0) √ = 1 such that Φ is λ-harmonic. Furthermore, we will also show along√the way that p(r) ≤ (1 + λr)d . When d = 3, it is easy to check that Φ(r) = e(− λ)r /r is our desired potential. Otherwise, by our formula for the radial Laplacian in d dimensions, we want to solve the following differential equation: ∆Φ = 1 rd−1 ∂ d−1 ∂Φ (r ) = λΦ ∂r ∂r 18 Solving this gives us the following second-order differential equation on p √ √ rp00 − (d − 3 + 2 λr)p0 + λ(d − 3)p = 0 P Let us write p(r) = ki=0 ai ri . Then, substituting into our differential equation gives us the following equations by setting each √ coefficient of ri to zero: i r : ai+1 (i + 1)(i − (d − 3)) = ai λ(2i − (d − 3)) rk : (−2k + d − 3)ak = 0 The last equation explains why we chose k = (d − 3)/2, so that it is automatically zero. Thus, setting a0 = 1 and running the recurrence gives us our desired polynomial. Note that the recurrence is valid and produces positive coefficients since i√< k = (d √ − 3)/2. Our claim follows and √ Φ is λ-harmonic. And furthermore, notice that ai+1 ≤ λai ≤ ( λ)i+1 . Therefore, p(r) ≤ (1 + r λ)d . Lastly, we assert that Φ(j) (r) is non-negative for j even and non-positive for √ j odd. To prove our − assertion, we note that it suffices to show that if Φ is of the form Φ(r) √ = p(r)e λr /rl for some p of degree k < l and p has non-negative coefficients, then Φ0 (r) = −q(r)e− λr /rl+1 for some q of degree k + 1 with non-negative coefficients. Differentiating Φ gives: √ e−r Φ0 = l+1 (rp0 (r) − (l + λr)p(r)) r √ It is clear that if p has degree k, then q(r) = (l + λr)p(r) − rp0 (r) has degree k + 1, so it suffices to show that it has non-negative coefficients. Let p0 , ..., pk be the non-negative coefficients of p. Then, by our formula, we see that q0 = lp0 √ √ qi = lpi √ − ipi + λpi−1 = (l − i)pi + λpi−1 qk+1 = λpk Since i ≤ k < l, we conclude that q has non-negative coefficients. Finally, our assertion follows with induction since Φ(0) (r) is non-negative and has our desired form with k = (d − 3)/2 < d − 2. By Lemma B.8, our primary theorem follows, we can construct a realizable radial potential Φ (r) that is λ-harmonic when r ≥  and has alternating-signed derivatives. (k) (k) Lastly, we prove the following preliminary bound on Φ (r) when k ≤ d: |Φ (r)| ≤ 3(2d + √ 2d −2d (k)  λ)  for all 0 ≤ k ≤ d − 1. First, notice that by the results of Lemma B.8, Φ (r) is monotone (k) (k) and limr→∞ Φ (r) = 0. So, it follows that we just have to bound |Φ (0)|. From our construction, √ (k) Φ () = pk ()e− λ 2−d−k , for some √ polynomial pk . Furthermore, from our construction, we have the recurrence pk () = (d − 2 + k + λ)pk−1 () − p0k−1 (). Therefore, we conclude that for k ≤ d, √ √ √ √ pk () ≤ (2d + λ)k p0 () ≤ (2d + λ)k (1 + λ)d ≤ (2d + λ)2d . √ (k) Therefore, we can bound |Φ ()| ≤ (2d + λ)2d −2d . Finally, by Lemma B.8, |Φ(d−1−k) (0)| ≤ |Φ(d−1−k) ()| +  k X ()k−j+1 |Φ(d−j) ()| (k − j + 1)! j=1 √ k X k−j+1 ) (k − j + 1)! j=1 √ 2d −2d  √ ≤ (2d + λ)  e ≤ 3(2d + λ)2d −2d ≤ (2d + λ)2d −2d (1 + √ √ √ − λr And for r ≥ , we see that |Φ (r)| = |Φ(r)| ≤ |p(r)| erd−2 = (1 + r λ)d e− λr r2−d . And √ √ √ √ e− λr 0 0 |Φ (r)| = |Φ (r)| ≤ |p1 (r)| rd−1 ≤ (d + λr)(1 + r λ)d e− λr r1−d . 19 e  = Φ /Φ (0). Note that since Φ is monotonically Finally, we consider the normalized potential: Φ √ decreasing, we can lower bound Φ (0) ≥√Φ () ≥ e− λ . Therefore, we can derive √the following upper √ 2d −2d λ √ e (k) e  (r)| ≤ (1 + r λ)d e λ(1−r) r2−d and its bounds: |Φ e and for r ≥ , |√Φ  (r)| ≤ 3(2d + λ)  √ √ e 0 (r)| ≤ (d + λr)(1 + r λ)d e λ(1−r) r1−d . derivative is bounded by |Φ  e  and the first derivative when r ≥ , by using the And lastly, we derive some lower bounds on √ √ Φ−2d √ e  (r) ≥ Φ (r)(2d + λ) 2d /3 ≥ e− λr r2−d (2d + λ)−2d 2d /3. For the upper bound on Φ (0): Φ √ √ e 0 (r)| ≥ e− λr r1−d (2d + λ)−2d 2d /3. derivative, we get |Φ Lemma B.10. The λ-harmonic radial potential Φ(r) = e−r /r in 3-dimensions is realizable by the activation σ(r) = K1 (r)/r. Proof. The activation is obtained from the potential function by first taking its Fourier transform, then taking its square root, and then taking the inverse fourier transform. Since the functions in consideration are radially symmetric F (y) of f (x) (and inverse) are obtained by R ∞ the Fourier transform √ the Hankel Transfom yF (y) = 0 xf (x)J1/2 (xy) xydx. Plugging f (x) = e−x /x, from the Hankel 2 2 tranform tables we get yF we wish to find the inverse p(y) = cy/(1 + y ) giving F (y) = cy/(1 + y ). SoR ∞ √ 2 Fourier transform for 1/ 1 + y . The inverse f (x) is given by xf (x) = 0 yF (y)J1/2 (xy) xydy = cK1 (x). So σ(r) = K1 (r)/r. C Earnshaw’s Theorem Theorem 3.4. (Earnshaw’s Theorem. See [AKN85]) Let M = Rd and let Φ be harmonic and L be as in (2). Then, L admits no differentiable strict local minima. Proof. If (a, θ) is a differentiable strict local minima, then for any i, we must have ∇θi L = 0, and Tr(∇2θi L) > 0. Since Φ is harmonic, we also have Tr(∇2θi L(θ1 , ..., θn )) = ∆θi L = 2 X ai aj ∆θi Φ(θi , θj ) + 2 k X ai bj ∆θi Φ(θi , wj ) = 0, j=1 j6=i which is a contradiction. In the first line, there is a factor of 2 by symmetry. D Descent Lemmas and Iteration Bounds Algorithm 4 x = GD(L, x0 , T, α) Input: L : M → R; x0 ∈ M; T ∈ N; α ∈ R Initialize x = x0 for i = 1 to T do x = x − α∇L(x) x = ΠM x Lemma D.1. Let f : Ω → R be a thrice differentiable function such that |f (y)| ≤ B0 , k∇f (y)k ≤ B1 , k∇2 f (y)k ≤ B2 , k∇2 f (z) − ∇2 L(y)k ≤ B3 kz − yk for all y, z in a (αB1 )-neighborhood of x. If k∇f (x)k ≥ η and x0 is reached after one iteration of gradient descent (Algorithm 4) with stepsize α ≤ B12 , then kx0 − xk ≤ αB1 and f (x0 ) ≤ f (x) − αη 2 /2. 20 Proof. The gradient descent step is given by x0 = x − α∇f (x). The bound on kx0 − xk is clear since k∇f (x)k ≤ B1 . f (x0 ) ≤ f (x) − α∇f (x)T ∇f (x)T + α2 ≤ f (x) − (α − α2 For 0 ≤ α ≤ 1 B2 , B2 k∇f (x)k2 2 B2 2 )η 2 we have α − α2 B2 /2 ≥ α/2, and our lemma follows. Lemma D.2. Let f : Ω → R be a thrice differentiable function such that |f (y)| ≤ B0 , k∇f (y)k ≤ B1 , k∇2 f (y)k ≤ B2 , k∇2 f (z) − ∇2 L(y)k ≤ B3 kz − yk for all y, z in a (αB2 )-neighborhood of x. If λmin (∇2 f (x)) ≤ −γ and x0 is reached after one iteration of Hessian descent (Algorithm 1) with stepsize α ≤ B13 , then kx0 − xk ≤ αB2 and f (x0 ) ≤ f (x) − α2 γ 3 /2. Proof. The gradient descent step is given by x0 = x + βvmin , where vmin is the unit eigenvector corresponding to λmin (∇2 f (x)) and β = −αλmin (∇2 f (x))sgn(∇f (x)T vmin ). Our bound on kx0 − xk is clear since |λmin (∇2 f (x))| ≤ B2 . T f (x0 ) ≤ f (x) + β∇f (x)T vmin + β 2 vmin ∇2 f (x)vmin + B3 3 |β| kvmin k3 6 B3 3 |β| 6 ≤ f (x) − |β|2 γ + The last inequality holds since the sign of β is chosen so that β∇f (x)T vmin ≤ 0. Now, since |β| = αγ ≤ Bγ3 , −|β|2 γ + B63 |β|3 ≤ −α2 γ 3 /2. E Convergence of Almost λ-Harmonic Potentials Lemma 4.3. Let M = Rd for d ≡ 3 mod 4 and let G be the regularized loss corresponding to the activation σ given by Lemma 4.2 with λ = 1. For any  ∈ (0, 1) and δ ∈ (0, 1), if (a, θ) ∈ MG,δ , then for all i, either 1) there exists j such that kθi − wj k < k or 2) a2i < 2kdδ. Proof. The proof is similar to Theorem 3.6. Let Φ be the realizable potential in 4.2 such that Φ (r) is λ-harmonic when r ≥  with λ = 1. Note that Φ (0) = 1 is normalized. And let (a, θ) ∈ MG,δ . WLOG, consider θ1 and a initial set S0 = {θ1 } containing it. For a finite set of points S and a point x, define d(x, S) = miny∈S kx − yk. Then, we consider the following set growing process. If there exists θi , wi 6∈ Sj such that d(θi , Sj ) <  or d(wi , Sj ) < , add θi , wi to Sj to form Sj+1 . Otherwise, we stop the process. We grow S0 to until the process terminates and we have the grown set S. If there is some wj ∈ S, then it must be the case that there exists j1 , · · · jq such that kθ1 −θj1 k <  and kθji − θji+1 k < , and kθjq − wj k <  for some wj . So, there exists j, such that kθ1 − wj k < k. Otherwise, notice that for each θi ∈ S, kwj − θi k ≥  for all j, and kθi − θj k ≥  for all θj 6∈ S. WLOG, let S = {θ1 , . . . , θl }. We consider changing all θ1 , . . . , θl by the same v and define H(a, v) = G(a, θ1 + v, ..., θl + v, θl+1 . . . , θk ). The optimality conditions on a are k X X ∂H = |4ai + 2 aj Φ (θi , θj ) + 2 bj Φ (θi , wj )| ≤ δ ∂ai j=1 j6=i 21 Next, since Φ (r) is λ-harmonic for r ≥ , we may calculate the Laplacian of H as   l k k X X X ∆v H = λ 2 ai bj Φ (θi , wj ) + 2 ai aj Φ (θi , θj ) i=1 ≤ l X j=1 j=l+1  λ −4a2i − 2 i=1 l X  ai aj Φ (θi , θj ) + δ = −2λ E  l X !2  ai σ(θi , X) λ|ai | i=1 j=1,j6=i  l X  − 2λ i=1 l X i=1 a2i + δλ l X |ai | i=1 The second line follows from our optimality conditions and the third Pl line2 follows from completing the square. Since (a, θ)√∈√ MG,δ , we have ∆v H ≥ −2kdδ. Let S = i=1 ai . Then, by √ Cauchy-Schwarz, √ we have −2λS + δλ k S ≥ −2kdδ. When S ≥ δ 2 k, we see that −λS ≥ −2λS + δλ k S ≥ −2kdδ. Therefore, S ≤ 2kdδ/λ. We conclude that S ≤ max(δ 2 k, 2kdδ/λ) ≤ 2kdδ/λ since δ ≤ 1 ≤ 2d/λ and λ = 1. Therefore, 2 ai ≤ 2kdδ. p p Lemma 4.4. Assume the conditions of Lemma 4.3. If G(a, θ) ≤ G(0, 0) − δ and (a, θ) ∈ MG,δ2 /(2k3 d) , then there exists some i, j such that kθi − wj k < k. Proof. If there does not exists i, j such that kθi − wj k < k, p then by Lemma 4.3, this implies 2 for all i. Now, for a integrable function f (x), kf k = a2i < δ 2 /kP EX [f (X)2 ] is a norm. Therefore, X if f (x) = i bi σ(wi , x) be our true target function, we conclude that by triangle inequality k X p G(a, θ) ≥ ai σ(θi , x) − f (x) i=1 ≥ kf (x)kX − k X kai σ(θi , x)kX ≥ p G(0, 0) − δ i=1 X This gives a contradiction, so we conclude that there must exist i, j such that θi is in a k neighborhood of wj . Lemma 4.5. Assume the conditions of Theorem 4.1 and Lemma 4.3. With high probability, we can q p (0) (0) initialize (a , θ ) such that G(a(0) , θ (0) ) ≤ G(0, 0) − δ with δ = (d/)−O(d) . Proof. Consider choosing θ1 = 0 and then optimizing a1 . Given θ1 , the loss decrease is: G(a1 , 0) − G(0, 0) = min 2a21 + 2 a1 k X j=1  2 k X 1 a1 bj Φ (0, wj ) = −  bj Φ (0, wj ) 2 j=1 Because wj are random Gaussians with variance O(d log d), we have kwj k ≤ O(d log d) with high probability for all j. By Lemma 4.2, our potential satisfies Φ (0, wj ) ≥ (d/)−O(d) . And since bj are uniformly chosen in [−1, 1], we conclude that with high probability over the choices of bj , 2 P k b Φ(θ , w ) ≥ (d/)−O(d) by appealing to Chebyshev’s inequality on the squared term. − 21 1 j j=1 j Therefore, we conclude that with high probability, G(a1 , 0) ≤ G(0, 0) − 12 (d/)−O(d) . Let p p G(a1 , 0) = G(0, 0) − ∆ ≥ 0. Squaring and rearranging gives ∆ ≥ √ 1 (d/)−O(d) . Since 4 G(0, 0) ≤ O(k) = O(poly(d)), we are done. 22 G(0,0) E.1 Node by Node Analysis The first few lemmas are similar to the ones proven before in the simultaneous case. The proof are presented for completeness because the regularization terms are removed. Note that our loss function is quadratic in a. Therefore, let a∗1 (θ1 ) denote the optimal value of a1 to minimize our loss. Lemma E.1. Let M = Rd for d ≡ 3 mod 4 and let L1 be the loss restricted to (a1 , θ1 ) corresponding to the activation function σ given by Lemma 4.2 with λ = 1. For any  ∈ (0, 1) and δ ∈ (0, 1), we can construct σ such that if (a1 , θ1 ) ∈ ML1 ,δ , then for all i, either 1) there exists j such that kθ1 − wj k <  or 2) a21 < 2dδ. Proof. The proof is similar to Lemma 4.3. Let Φ be the realizable potential in 4.2 such that Φ (r) is λ-harmonic when r ≥ . Note that Φ (0) = 1 is normalized. And let (a1 , θ1 ) ∈ ML1 ,δ . Assume that there does exist wj such that kθ1 − wj k < . The optimality condition on a1 is k X ∂L = |2a1 + 2 bj Φ (θ1 , wj )| ≤ δ ∂a1 j=1 Next, since Φ (r) is λ-harmonic for r ≥ , we may calculate the Laplacian of L as   k X ∆θ1 L = λ 2 a1 bj Φ (θ1 , wj ) ≤ −2λa21 + δλ|a1 | j=1 The inequality follows from our optimality conditions. Since (a1 , θ1 ) ∈ ML,δ , we have ∆θ1 L ≥ −2dδ. When a21 ≥ δ 2 , we see that −λa21 ≥ −2λa21 + δλ|a1 | ≥ −2dδ. Therefore, a21 ≤ 2dδ/λ. We conclude that a21 ≤ max(δ 2 , 2dδ/λ) ≤ 2dδ/λ for δ ≤ 2d ≤ 2d/λ since λ = 1. Therefore, a21 ≤ 2dδ. p p Lemma E.2. Assume the conditions of Lemma E.1. If L1 (a1 , θ1 ) ≤ L1 (0, 0) − δ and (a1 , θ1 ) ∈ MG,δ2 /(2d) , then there exists some j such that kθ1 − wj k < . Proof. The proof follows similarly from Lemma 4.4. Now, our main observation is below, showing that in a neighborhood around wj , descending along the gradient direction will move θ1 closer to wj . Our tighter control of the gradient of Φ around wj will eventually allow us to show that θ1 converges to a small neighborhood around wj . Lemma E.3. Assume the conditions of Theorem E.5 and Lemma E.1. If kθ1 − wj k ≤ d and |bj | ≥ 1/poly(d) and |a1 − a∗1 (θ1 )| ≤ (d/)−O(d) is almost optimal and for i, kwi − wj k ≥ Ω(d log d), w −θ 1 (d/)−8d and ξ ≤ (d/)−O(d) . then −∇θ1 L1 = ζ kθ1j−w1j k + ξ with ζ ≥ poly(d) Proof. Through the proof, we assume k = poly(d). Now, our gradient with respect to θ1 is ∇θ1 L1 = 2a1 bj Φ0 (kθ1 − wj k) X θ 1 − wj θ 1 − wi +2 a1 bi Φ0 (kθ1 − wi k) kθ1 − wj k kθ1 − wi k i6=j √ √ Since kθ1 − wj k ≤ d, we may lower bound |Φ0 (kθ1 − wj k)| ≥ e− λd d1−d (2d + λ)−2d 2d /3 ≥ Ω((d/)−4d ). Similarly, Φ (kθ1 − wj k) ≥ Ω((d/)−4d ). On the other hand since kwi − wj k ≥ Ω(d log d) for all i = 6 j, we may upper bound |Φ (kθ1 − wi k)| ≤ (d/)−O(d) and |Φ0 (kθ1 − wi k)| ≤ (d/)−O(d) . θ −w Together, we conclude that ∇θ1 L1 = 2a1 bj Φ0 (kθ1 − wj k) kθ11 −wjj k + 2a1 ξ, where kξk ≤ (d/)−O(d) . 23 By assumption, |a1 − a∗1 (θ1 )| ≤ (d/)−O(d) , so X ∂L1 = |2a1 + 2bj Φ (kθ1 − wj k) + 2 bi Φ (kθ1 − wi k)| ≤ (d/)−O(d) ∂a1 i6=j By a similar argument as on the derivative, we see that a1 = −bj Φ (kθ1 − wj k) + (d/)−O(d) . Therefore, the direction of −∇θ1 L1 is moving θ1 closer to wj since −∇θ1 L1 = b2j Φ (kθ1 − wj k)Φ0 (kθ1 − wj k) θ 1 − wj + (d/)−O(d) kθ1 − wj k and we know Φ > 0 and Φ0 < 0, thereby −b2j Φ (kθ1 − wj k)Φ0 (kθ1 − wj k) ≥ 1/poly(d)(d/)−8d . Lemma E.4 (Node-wise Initialization). Assume the conditions q of Theorem E.5pand Lemma E.1. (0) (0) (0) (0) With high probability, we can initialize (a1 , θ1 ) such that L(a1 , θ1 ) ≤ L(0, 0) − δ with 1 δ = poly(d) (d/)−18d in time log(d)O(d) . Proof. By our conditions, there must exist some |bj | such that |bj | ≥ 1/poly(d) and for all i, kwi − wj k ≥ Ω(d log d). Note that if we randomly sample points in a ball of radius O(d log d), we will land in a d-neighborhood of wj with probability log(d)−O(d) since kwj k ≤ O(d log d). Let θ1 be such that kθ1 − wj k ≤ d and then we can solve for a1 = a∗1 (θ1 ) since we are simply minimizing a quadratic in one variable. Then, by Lemma E.3, we see that k∇θ1 L1 k ≥ 1/poly(d)(d/)−8d . Finally, by Lemma 4.2, we know that the Hessian is bounded by poly(d)(d/)2d . 1 So, by Lemma D.1, we conclude by taking a stepsize of α = poly(d) (d/)−2d to reach (a01 , θ10 ), we can guarantee that L1 (a01 , θ10 ) ≤ L1 (a∗1 (θ1 ), θ1 ) − 1 −18d . poly(d) (d/) conclude that L1 (a01 , θ10 ) 1 But since L1 (a∗1 (θ1 ), θ1 ) ≤ L1 (0, 0), we ≤ L1 (0, 0) − poly(d) (d/)−18d . Let p p 1 −18d . L1 (a01 , θ10 ) = L1 (0, 0) − ∆ ≥ 0. Squaring and rearranging gives ∆ ≥ √ 1 poly(d) (d/) 4 L1 (0,0) Since L1 (0, 0) ≤ O(k) = O(poly(d)), we are done. Lemma E.5. Assume the conditions of Lemma E.1. Also, assume b1 , ..., bk are any numbers in [−1, 1] and w1 , ..., wk ∈ Rd satisfy kwi k ≤ O(d log d) for all i and there exists some |bj | ≥ 1/poly(d) with kwi − wj k ≥ Ω(d log d) for all i. (0) (0) Then with high probability, we can choose an initial point (a1 , θ1 ) such that after running SecondGD (Algorithm 2) on the restricted regularized objective L1 (a1 , θ1 ) for at most (d/)O(d) iterations, there exists some wj such that kθ1 − wj k < . Furthermore, if |bj | ≥ 1/poly(d) and kwi − wj k ≥ Ω(d log d) for all i, then kθ1 − wj k < (d/)−O(d) and |a + bj | < (d/)−O(d) . q p (0) (0) (0) (0) Proof. First, by Lemma E.4, we can initialize (a1 , θ1 ) such that L1 (a1 , θ1 ) ≤ L1 (0, 0) − δ 1 for δ = poly(d) (d/)−18d . If we set α = (d/)−O(d) and η = γ = λδ 2 /(2d), then running Algorithm 2 will terminate and return some (a1 , θ1 ) in at most (d/)O(d) iterations. This is because our algorithm ensures that our objective function decreases by at least min(αη 2 /2, α2 γ 3 /2) at each iteration and G(0, 0) is bounded by O(k) and G ≥ 0 is non-negative. Assume there does not exist wj such that kθ1 − wj k < (d/)−O(d) . Then, we claim that (a1 , θ1 ) ∈ ML,λδ2 /(2d) . For the sake of contradiction, assume otherwise. By our algorithm termination conditions, then it must be that after one step of gradient or Hessian descent from (a1 , θ1 ), we reach 24 some (a0 , θ0 ) and L1 (a0 , θ0 ) > L1 (a1 , θ1 ) − min(αη 2 /2, α2 γ 3 /2). Now, Lemma 4.2 ensures all first three derivatives of Φ are bounded by (d/)2d , except at w1 , ..., wk . Furthermore, since there does not exists wj such that kθ1 − wj k < (d/)−O(d) , L1 is three-times continuously differentiable within a α(d/)2d = (d/)−O(d) neighborhood of θ1 . Therefore, by Lemma D.1 and D.2, we know that L(a0 , θ0 ) ≤ L1 (a0 , θ0 ) ≤ L1 (a1 , θ1 ) − min(αη 2 /2, α2 γ 3 /2), a contradiction. So, it must be (a1 , θ1 ) ∈ ML,λδ2 /(2d) . Since our algorithm maintains that our objective function p p is decreasing, so L1 (a1 , θ1 ) ≤ L1 (0, 0) − δ. So, by Lemma E.2, there must be some wj such that kθ − wj k ≤ . Now, if |bj | ≥ 1/poly(d) and kwi − wj k ≥ Ω(d log d) for all i, then since (a, θ) ∈ ML,λδ2 /(2d) and kθ − wj k ≤ , by Lemma E.3, we have k∇θ1 L1 k ≥ 1/poly(d)(d/)−8d > δ 2 /(2d), a contradiction. Therefore, we must conclude that our original assumption was false and kθ − wj k < (d/)−O(d) for some wj . Finally, we see that the charges also converge since a = −2bj Φ (kθ − wj k) + O(d/)−O(d) and kθ − wj k = (d/)−O(d) . By noting that Φ (0) = 1 and Φ is O((d/)2d )-Lipschitz, we conclude. Finally, we have our final theorem. Theorem 4.6. Let M = Rd and d ≡ 3 mod 4 and let L be as in 1 and k = poly(d). For all  ∈ (0, 1), we can construct an activation σ such that if w1 , ..., wk ∈ Rd with wi randomly chosen from wi ∼ N (0, O(d log d)Id×d ) and b1 , ..., bk be randomly chosen at uniform from [−1, 1], then with high probability, after running nodewise descent (Algorithm 3) on the objective L for at most (d/)O(d) iterations, (a, θ) is in a (d/)−O(d) neighborhood of the global minima. Proof. Let our potential Φ be the one as constructed in Lemma 4.2 that is λ-harmonic for all r ≥  with λ = 1. Let (ai , θi ) be the i-th node that is initialized and applied second order gradient descent onto. We want to show that the nodes (ai , θi ) will converge, in a node-wise fashion, to some permutation of {(b1 , w1 ), ..., (bk , wk )}. First, with high probability we know that 1 − 1/poly(d) ≥ |bj | ≥ 1/poly(d) and kwi k ≤ O(d log d) and kwi − wj k ≥ Ω(d log d) for all i, j. By Lemma E.5, we know that with high probability (a1 , θ1 ) will converge to some (d/)−O(d) neighborhood of (bπ(1) , wπ(1) ) for some function π : [k] → [k]. Now, we treat a1 , θ1 as one of the fixed charges and note that |a1 | ≤ 1 and kθ1 k ≤ O(d log d) and as long as k > 1 (if k = 1, we are done), then there exists |bj | ≥ 1/poly(d) with kwi − wj k ≥ Ω(d log d) for all i and kθ1 − wj k ≥ Ω(d log d). q p (0) (0) (0) (0) Then, by Lemma E.4, we can initialize (a2 , θ2 ) such that L2 (a2 , θ2 ) ≤ L2 (0, 0) − δ, with δ = 1/poly(d)(d/)−18d . Then, by Lemma E.5, we know that (a2 , θ2 ) will converge to some wπ(2) such that kθ2 − wπ(2) k <  (or kθ2 − θ1 k <  but θ2 is still -close to wπ(1) ). We claim that π(1) 6= π(2). By optimality conditions on a2 , we see that X a∗2 (θ2 ) = a1 Φ (kθ2 − θ1 k) + bj Φ (kθ1 − wj k) + bi Φ (kθ1 − wi k) i6=j P If wπ(1) = wπ(2) , then note that kθ1 − wi k ≥ Ω(d log d) for all i 6= π(1). Therefore, 2 i6=j bi Φ (kθ1 − wi k) = (d/)−O(d) . And by our convergence guarantees and the (d/)2d -Lipschitzness of Φ , a1 Φ (kθ2 − θ1 k) + bj Φ (kθ1 − wj k) ≤ (d/)−O(d) . Therefore, a∗2 (θ2 ) ≤ (d/)−O(d) . However, we see that L2 (a2 , θ2 ) ≥ L2 (a∗2 (θ2 ), θ2 ) = L2 (0, 0) − 12 a∗2 (θ2 )2 ≥ L2 (0, 0) − (d/)−O(d) . But since L2 is non-increasing, this contradicts our initialization and therefore π(1) 6= π(2). Therefore, our claim is done and by Lemma E.5, we see that since |bπ(2) | ≥ 1/poly(d) and for all i, kwi −wπ(2) k ≥ 25 Ω(d log d) and kθ1 − wπ(2) k ≥ Ω(d log d), we conclude that (a2 , θ2 ) is in a (d/)−O(d) neighborhood of (bπ(2) , θπ(2) ). Finally, we induct and by similar reasoning, π is a permutation. Now, our theorem follows. F Convergence of Almost Strictly Subharmonic Potentials Definition F.1. Φ(θ, w) is a strictly subharmonic potential on Ω if it is differentiable and ∆θ Φ(θ, w) > 0 for all θ ∈ Ω, except possibly at θ = w. An example of such a potential is Φ(θ, w) = kθ − wk2−d− for any  > 0. Although this potential is unbounded at θ = w for most d, we remark that it is bounded when d = 1. Furthermore, the sign of the output weights ai , bi matter in determining the sign of the Laplacian of our loss function. Therefore, we need to make suitable assumptions in this framework. Under Assumption 1, we are working with an even simpler loss function: L(θ) = 2 k X X Φ(θi , θj ) − 2 i=1 i<j k X k X Φ(θi , wj ) (3) i=1 j=1 Theorem F.2. Let Φ be a symmetric strictly subharmonic potential on M with Φ(θ, θ) = ∞. Let Assumption 1 hold and let L be as in (3). Then, L admits no local minima, except when θi = wj for some i, j. Proof. First, let Φ be translationally invariant and M = Rd . Let θ be a critical point. Assume, for sake of contradiction, that for all i, j, θi 6= wj . If θi are not distinct, separating them shows that we are not at a local minima since Φ(θi , θj ) = ∞ and finite elsewhere. The main technical detail is to remove interaction terms between pairwise θi by considering a correlated movement, where each θi are moved along the same direction v. In this case, notice that our objective, as a function of v, is simply H(v) = L(θ1 + v, θ2 + v, ..., θk + v) =2 k X X Φ(θi + v, θj + v) − 2 i=1 i<j k X k X Φ(θi + v, wj ) i=1 j=1 Note that the first term is constant as a function of v, by translational invariance. Therefore, ∇2v H = −2 k k X X ∇2 Φ(θi , wj ) i=1 j=1 P P By the subharmonic condition, Tr(∇2v H) = −2 ki=1 kj=1 ∆θi Φ(θi , wj ) < 0. Therefore, we conclude that θ is not a local minima of H and L. We conclude that θi = wj for some i, j. The above technique generalizes to Φ being rotationally invariant case by working in spherical coordinates and correlated translations are simply rotations. Note that we can change to spherical coordinates (without the radius parameter) and let θe1 , ..., θek be the standard spherical representation of θ1 , ..., θk . We will consider a correlated translation in the spherical coordinate space, which are simply rotations on the sphere. Let v be a vector in Rd−1 and our objective is simply H(v) = L(θe1 + v, ..., θek + v) 26 Then, we apply the same proof since Φ(θei + v, θej + v) is constant as a function of v by rotationally invariance. Corollary F.3. Assume the conditions of Theorem F.2 and Φ(θ, θ) < ∞. Then, L admits no local minima, except at the global minima. Proof. From the same proof from theorem F.2, we conclude that there must exists i, j such that θi = wj . Then, since Φ(θ, θ) < ∞, notice that θi , wj cancels each other out and by drop θi , wj from the loss function, we have a new loss function L with k − 1 variables. Then, using induction, we see that θi = wπ(i) at the local minima for some permutation π. For concreteness, we will focus on a specific potential function with this property: the Gaussian kernel Φ(θ, w) = exp(−kθ − wk2 /2). In Rd , the Laplacian is ∆Φ = (kθ − wk2 − d) exp(−kθ − wk2 /2), which becomes positive when kθ − wk2 ≥ d. Thus, Φ√is strictly subharmonic outside a ball of radius √ d. This informally implies that θ1 converges to a d-ball around some wj . For concreteness, we will focus on a specific potential function with this property: the Gaussian kernel Φ(θ, w) = exp(−ckθ − wk2 /2), which corresponds to a Gaussian activation. In Rd , the Laplacian is ∆Φ = (ckθ − wk2 − d) exp(−ckθ − wk2 /2),p which becomes positive when kθ − wk2 ≥ d/c. Thus, Φ is strictly subharmonic outside a ball of radius d/c. Note that Gaussian potential restricted to S d−1 gives rise to the exponential activation function, so we can show convergence similarly. 2 Theorem F.4. Let M = Rd and Φ(θ, w) = e−ckθ−wk /2 and Assumption 1 holds. Let L be as in (3) and kwk ≤ poly(d). If c = O(d/) and (a, θ) ∈ Me−poly(d,1/) , then there exists i, j such that kθi − wj k2 ≤ . Proof. Consider again a correlated movement, where each θi are moved along the same direction v. As before, this drops the pairwise θi terms. If for all i, j kθi − wj k2 ≤ , then we see that ∆θi Φ = (ckθ − wk2 − d) exp(−ckθ − wk2 /2) > e−poly(d,1/) . 2 Tr(∇ L) = −2 k X k X ∆θi Φ(θi , wj ) < −e−poly(d,1/) i=1 j=1 Therefore, ∇2 L must admit a strictly negative eigenvalue that is less than e−c3 d , which implies our claim (we drop the poly(d, k) terms). G Common Activations First, we consider the sign activation function. Under restrictions on the size of the input dimension or the number of hidden units, we can prove convergence results under the sign activation function, as it gives rise to a harmonic potential. Assumption 1. All output weights bi = 1 and therefore the output weights ai = −bi = −1 are fixed throughout the learning algorithm. Lemma G.1. Let M = S 1 and let Assumption 1 hold. Let L be as in (2) and σ is the sign activation function. Then L admits no strict local minima, except at the global minima. 27 We cannot simply analyze the convergence of GD on all θi simultaneously since as before, the pairwise interaction terms between the θi present complications. Therefore, we now only consider the convergence guarantee of gradient descent on the first node, θ1 , to some wj , while the other nodes are inactive (i.e. a2 , ..., ak = 0). In essence, we are working with the following simplified loss function. k X L(a1 , θ1 ) = a21 Φ(θ1 , θ1 ) + 2 a1 bj Φ(θ1 , wj ) (4) j=1 Lemma G.2. Let M = S 1 and L be as in (4) and σ is the sign activation function. Then, almost surely over random choices of b1 , ..., bk , all local minima of L are at ±wj . For the polynomial activation and potential functions, we also can show convergence under orthogonality assumptions on wj . Note that the realizability of polynomial potentials is guaranteed in Section B. Theorem G.3. Let M = S d−1 . Let w1 , ..., wk be orthonormal vectors in Rd and Φ is of the form Φ(θ, w) = (θT w)l for some fixed integer l ≥ 3. Let L be as in (4). Then, all critical points of L are not local minima, except when θ1 = wj for some j. G.1 Convergence of Sign Activation Lemma G.1. Let M = S 1 and let Assumption 1 hold. Let L be as in (2) and σ is the sign activation function. Then L admits no strict local minima, except at the global minima. Proof. We will first argue that unless all the electrons and protons have matched up as a permutation it cannot be a strict local minimum and then argue that the global minimum is a strict local minimum. First note that if some electron and proton have merged, we can remove such pairs and argue about the remaining configuration of charges. So WLOG we assume there are no such overlapping electron and proton. First consider the case when there is an isolated electron e and there is no charge diagonally opposite to it. In this case look at the two semicircles on the left and the right half of the circle around the isolated electron – let q1 and q2 be the net charges in the left and the right semi-circles. Note that q1 = 6 q2 since they are integers and q1 + q2 = +1 which is odd. So by moving the electron slightly to the side with the larger charge you decrease the potential. If there is a proton opposite the isolated electron the argument becomes simpler as the proton benefits the motion of the electron in either the left or right direction. So the only way the electron does not benefit by moving in either direction is that q1 = −1 and q2 = −1 which is impossible. If there is an electron opposite the isolated electron then the combination of these two diagonally opposing electrons have a zero effect on every other charge. So it is possible rotate this pair jointly keeping them opposed in any way and not change the potential. So this is not a strict local minimum. Next if there is a clump of isolated electrons with no charge on the diagonally opposite point then again as before if q1 6= q2 we are done. If q1 = q2 then the the electrons in the clump locally are unaffected by the remaining charges. So now by splitting the clump into two groups and moving them apart infinitesimally we will decrease the potential. Now if there is only protons in the diagonally opposite position an isolated electron again we are done as in the case when there is one electron diagonally opposite one proton. Finally if there is only electrons diagonally opposite a clump of electrons again we are done as we have found at least one pair of opposing electrons that can be jointly rotated in any way. 28 Next we will argue that a permutation matching up is a strict local minumum. For this we will assume that no two protons are diagonally opposite each other (as they can be removed without affecting the function). Now given a perfect matching up of electrons and protons, if we perturb the electrons in any way infinitesimally, then any isolated clump of electrons can be moved slightly to the left or right to improve the potential. Lemma G.2. Let M = S 1 and L be as in (4) and σ is the sign activation function. Then, almost surely over random choices of b1 , ..., bk , all local minima of L are at ±wj . Proof. In S 1 , notice that the pairwise potential function is Φ(θ, w) = 1 − 2 cos−1 (θT w)/π = 1 − 2α/π, where α is the angle between θ, w. So, let us parameterize in polar coordinates, calling our true parameters as w f1 , ..., w fk ∈ [0, 2π] and rewriting our loss as a function of θe ∈ [0, 2π]. Since Φ is a linear function of the angle between θ, wj , each wj exerts a constant gradient on θe towards w fj , with discontinuities at w fj , π + w fj . Almost surely over b1 , .., bk , the gradient is non-zero almost everywhere, except at the discontinuities, which are at w fj , π + w fj for some j. G.2 Convergence of Polynomial Potentials Theorem G.3. Let M = S d−1 . Let w1 , ..., wk be orthonormal vectors in Rd and Φ is of the form Φ(θ, w) = (θT w)l for some fixed integer l ≥ 3. Let L be as in (4). Then, all critical points of L are not local minima, except when θ1 = wj for some j. Proof. WLOG, we can consider w1 , ..., wd to be the basis vectors e1 , ..., ed . Note that this is a manifold optimization problem, so our optimality conditions are given by introducing a Lagrange multiplier λ, as in [GHJY15]. d X ∂L =2 abi (θi )l + 2a = 0 ∂a i=1 (∇θ L)i = 2abi l(θi )l−1 − 2λθi = 0 where λ is chosen that minimizes λ = arg min λ X (abi l(θi )l−1 − λθi )2 = X abi l(θi )l i Therefore, either θi = 0 or bi (θi )l−2 = λ/(al). From [GHJY15], we consider the constrained Hessian, which is a diagonal matrix with diagonal entry: (∇2 L)ii = 2abi l(l − 1)(θi )l−2 − 2λ Assume that there exists θi , θj 6= 0, then we claim that θ is not a local minima. First, our optimality conditions imply bi (θi )l−2 = bj (θj )l−2 = λ/(al). So, (∇2 L)ii = (∇2 L)jj = 2abi l(l − 1)(θi )l−2 − 2λ = 2(l − 2)λ = −2(l − 2)la2 Now, there must exist a vector v ∈ S d−1 such that vk = 0 for k 6= i, j and v T θ = 0, so v is in the tangent space at θ. Finally, v T (∇2 L)v = −2(l − 2)la2 < 0, implying θ is not a local minima when a 6= 0. Note that a = 0 occurs with probability 0 since our objective function is non-increasing throughout the gradient descent algorithm and is almost surely initialized to be negative with a optimized upon initialization, as by observed before. 29 Under a node-wise descent algorithm, we can show polynomial-time convergence to global minima under orthogonality assumptions on wj for these polynomial activations/potentials. We will not include the proof but it follows from similar techniques presented for nodewise convergence in Section E. H Proof of Sign Uniqueness For the sign activation function, we can show a related result. Theorem H.1. Let M = S d−1 and σ be the sign activation function and b2 , ..., √ bk = 0. If the loss (1) at (a, θ) is less than O(1), then there must exist θi such that w1T θi > Ω(1/ k). Proof. WLOG let w1 = e1 . Notice that our loss can be bounded below by Jensen’s:  !2  k X E ai σ(θiT X) − σ(X1 )  X i=1  ≥ E X1 " k X E X2 ...Xd !2  ai σ(θiT X) − σ(X1 )  , # i=1 where X is a standard Gaussian in Rd .   " k # k X X X EX2 ,..,Xd ai σ(θiT X) = ai EX2 ,...Xd σ(θi1 X1 + θij Xj ) i=1 i=1 j>1   k q X 2 = EY σ(θi1 X1 + 1 − θi1 Y ) = i=1 k X  ai EY σ( √ θi1  2 1−θi1 i=1 X1 + Y ) , where Y is an independent standard Gaussian and for any small δ, if p(y) is the standard Gaussian density, Z δ EY [σ(δ + Y )] = p(y) dy = 2p(0)δ + O(δ 2 ) −δ If w1T θi = θi1 <  for all i, then notice that with high probability on X1 (say condition on |X1 | ≤ 1),   θ i1 E σ( √ 2 X1 + Y ) = 2p(0) √ θi1 2 X1 + O(2 X12 ) 1−θi1 Y 1−θi1 √ Therefore, since  < O(1/ k), " k # k X X T E ai σ(θi X) = X1 2p(0)ai √ θi1 X2 ,..,Xd i=1 i=1 = cX1 + O(1) 30 2 1−θi1 + O(k2 X12 ) Finally, our error bound is now  E X1 " E X2 ...Xd ≥ k X # ai σ(θiT X) − σ(X1 ) !2   i=1 E [(cX1 + O(1) − σ(X1 ))2 ] |X1 |≤1 And the final expression is always larger than some constant, regardless of c. 31 

An Optimal Polarization Tracking Algorithm for Lithium-Niobate-based Polarization Controllers Joaquim D. Garcia and Gustavo C. Amaral arXiv:1603.06751v1 [] 12 Mar 2016 February 27, 2018 Abstract We present an optimal algorithm for the three-stage arbitrary polarization tracking using LithiumNiobate-based Polarization Controllers: device calibration, polarization state rotation, and stabilization. The theoretical model representing the lithium-niobate-based polarization controller is derived and the methodology is successfully applied. Results are numerically simulated in the MATLAB environment. 1 Introduction Keeping the polarization state stable in long-haul fiber optical communication links is a difficult task due to the local changes in the silica structure along the fiber which induces bi-refringence and, therefore, variation of the polarization state [1, 2]. Although until the early 1990’s most optical communication links did not account for it, Polarization Mode Dispersion (PMD) has been a more serious threat to modern optical links as the bit rate grows [3]. In this context, polarization control has gained much attention, specially with the development PMD-compensation techniques [4] and of the so-called Polarization Shift Keying [5]. Lithium-niobate (LiN bO3 ) is a material capable of altering its refractive index upon application of a difference of potential between its terminals [6]. This device represented a huge step in the polarization stabilization and control technology since it allowed extremely fast polarization controlling and tracking devices to be developed, once no mechanical structures were necessary [7]. In this work, we present a complete algorithm for polarization control and stabilization that relies on the use of the aforementioned LiN bO3 structures, more specifically, the EOSpace Polarization Controller Module [8]. The polarization control itself is composed of two main steps: firstly, an analytical rotation along the Poincaré Sphere relying on basic analytic geometry [9] and quaternion arithmetic [10]; secondly, stabilization method based on adaptive filtering to achieve fine adjustment [11]. A state estimator that is fundamental for the good functioning of the rotation algorithm is also presented. 2 Mathematical representation of polarization The state of polarization of light has been represented mathematically as Jones Vectors and Stokes Vectors [12]. We shall stick to the Stokes representation since visualization in the Poincaré Sphere is direct. Neverthelees, the conversion between these two representations require straightforward computations. Stokes vectors are 4-dimensional vectors that carry information about the State of Polarization (SOP) of light. Since the first component (S0 ) is associated to the total light intensity, it is common to normalize the Stokes Vector by dividing it by S0 . In the case of coherent light, the 3-dimensional Stokes Vector formed of the remaining three normalized components of the former 4-dimensional vector, has norm 1. Since we have 3-dimensional normalized vectors representing the SOP, it is usual to represent it graphically in a 3-sphere known as the Poincaré sphere. Since all SOPs are mapped bijectively in the 3-sphere, we shall treat, from now on, an SOP as a point in the 3-sphere. SOP changes that do not affect the light intensity can be represented by rotations in 3-space. These rotations are a class of unitary transformations and can be represented by orthonormal matrices [13]. Even though, polarisers do affect the light intensity and, as such, cannot be represented by orthonormal 1 matrices, they can be represented by projection matrices. The rotation matrix in 3-space has a very interesting characterization via the Spectral Theorem: they always have 3 eigenvalues (since they are normal); all of the eigenvalues have norm 1; one of the eigenvalues is always equal to 1 and its eigenvector is the rotation axis, e; the remaining eigenvalues are complex conjugate numbers whose real part correspond to the cosine of the rotation angle, θ, and whose imaginary part correspond to the sine of the rotation angle. The quaternions are a number system that extend the complex numbers. Since the unitary quaternions are homeomorphically mapped to the 3-space rotation matrices [14], it is possible to use them to perform 3-space rotations for which they were shown to be way more stable [15]. For the aforementioned reasons, our algorithm will rely on quaternions representation instead of on the matrix representation. 3 Lithium-Niobate-based Polarization Controller Characteristics Literature around polarization control is very rich and diverse techniques were proposed and verified along the last years: [16] shows that three elements, two quarter wave plates and one half wave plate, are sufficient to reach any SOP from any other SOP; other methods appear in [17, 18, 19]. As mentioned before our methodology focuses on the electro-optic LiN bO3 EOSpace Polarization Controller Module (PCM). Such device is available commercially as a multi-stage component but, for simplicity, we are going to develop an algorithm that uses a single stage. The algorithm is easily extended to account for the multi-stage version to reduce the input voltage that may vary within a ±70 Volts range. A single stage of the PCM has 3 electrodes [8] and realizes an arbitrary Linear Retarder : linear wave plates that induce relative phase differences between the TE and TM modes of the propagating electromagnetic field due to the variation of the refractive index in both axes as a function of the applied difference of potential which causes birefringence and, thus, altering the polarization state. Linear Retarders have a main polarization axis, also known as eigen-mode, e ∈ {v ∈ R3 |z = 0}, and a characteristic phase delay, θ ∈ [0, 2π). It is possible to show that, by changing the eigen-mode and the phase delay of a linear retarder, one can shift from one SOP to any other SOP. The proof of this is actually constructive and our algorithm provides such construction. In order to set the eigen-mode to e = (cos(α/2), sin(α/2), 0) and the phase delay to θ = 2πδ the electrodes voltages must be set to: Va = 2V0 δsin(α) − Vπ δcos(α) + Vab (1) Vb = 0 (2) Vc = 2V0 δsin(α) − Vπ δcos(α) + Vcb (3) where Vπ is the voltage required to induce a 180o phase shift between the TE and TM modes for a single stage, V0 is the voltage required to rotate all power from the TE to the TM mode, or vice versa, for a single stage, and Vab and Vcb are the bias voltages required on electrodes A and C, respectively, in order to achieve zero birefringence between the TE and TM modes [8]. Even though the data-sheet of the device provides the range within which V0 , Vπ ,Vab and Vcb should be, their actual values for an arbitrary stage must be determined via a calibration procedure. The calibration procedure adopted in our methodology is presented in [20]. 4 Analytical Rotation Algorithm The feasible eigen-modes of Linear retarders are linear SOPs, hence their name, so the rotation axis must lie in the s1 -s2 plane, i.e., along the equator of the Poincaré Sphere. Therefore, given two polarization states represented by sin and starget , we must find a rotation axis lying on the s1 -s2 plane and a rotation angle such that the corresponding linear transformation converts sin into starget . Since the possibility of measuring the output SOP after a transformation is of paramount importance of our adaptive algorithm, we assume, throughout the development, that a polarimeter such as the one described in [21] is included in the control loop. We start the rotation step by defining the following vectors: v1 , orthogonal to the rotation axis; v2 , the normalized cross-product between v1 and s3 (supposing v1 is not parallel to s3 ); v3 , the centre of the 2 rotation, i.e., the point in the rotation plane that intersects the line parallel to the rotation axis. In the case that v1 is parallel to s3 , we use the first of these that is not parallel to v1 : sin , starget or s1 . Now, we set: v4 = sin − v3 ; and v5 = starget − v3 where one can imagine v4 and v5 as two clock arrows where the rotation angle, θ, is the oriented angle between them. The cosine of θ is given by: cos(θ) = hv4 , v5 i ||v4 ||2 ||v5 ||2 (4) Note that the angle −θ has the same cosine as θ. To determine which rotation direction is the right one, we take the sign of the cross product between v4 and v5 with the following implication: if its negative, then θ = −θ and v3 = −v3 ; if its positive, then do nothing. After all those computations, θ is the rotation angle and v3 is the rotation axis. Fig. 1 presents the graphical interpretation on the 3-space Poincaré Sphere of the determination of both rotation axis and rotation angle given two arbitrary SOPs for sin and starget . Figure 1: Rotation example: input SOP in green, target SOP in black, output SOP in red, rotation axis in black, rotation circle in pink. The rotation arc is the smaller arc (least angular distance) defined by the points in the pink circle. 5 Stabilization algorithm Generally, non-linear optimization problems such as the one presented suffer from an intrinsic issue: finding a rotation that changes from one polarization state to another is extremely useful when the SOPs are distant but become highly unstable when sin and starget are close to each other. Taking the Levenberg-Marquadt procedure as inspiration [], we devise a two-step control algorithm that alternates between rotation and stabilization depending on the distance between SOPs. The stabilization step also helps eliminating problems involving measurement errors and numerical approximations that may influence the stability. For this step, we resort to an old optimization algorithm known as the Gradient Descent, which has been successfully applied to adaptive filtering [22] and machine learning [23]. 3 Given a cost function J(x) to be minimized, where x is a n-dimensional vector, the Gradient Descent algorithm updates x in the direction of the negative (descending) gradient, in search for the minimum of the functional, with the rule x = x − α∇J(x), where α is a step size parameter that must be calibrated [24]. Since the cost function depends non-linearly on the input and target SOP’s, there is no simple analytical form for the gradient and it must be estimated by measurements with intrinsic error. This measurement error is the reason why we employ a variant of the algorithm known as the Stochastic Gradient Descent (SGD) [25]. It is possible to estimate all the components of the gradient ∇J(x) by using the well-known secant method. Given ei , the ith element of the canonical basis of Rn , we can perturb the current value x by a small value , and accounting for measurement imprecisions by evaluating the cost function m times for ˆ a better estimate, the ith component of ∇J(x) is approximated by: m X J k (x + ei ) − J k (x − ei ) ˆ i≈ 1 ∇J(x) m  (5) k=1 When the gradient is estimated instead of being computed analytically, the descending path is way less smoother, and it is usual to see some roaming around the local minimum []. For our problem, x = [Va , Vc ]> and J(x) = ||Sout (x) − Sin ||22 since the 3-Space output Stokes Vector is a function of the linear retarder input voltages. The algorithm was simulated numerically via MATLAB with the step α empirically set as directly proportional to the error between starget and sin . 6 State estimation For a given calibration, we have estimates of the values V0 , Vπ ,Vab and Vcb . Thus, at any time, if we know the voltages Va and Vc applied to the PCM stage’s electrodes, we can obtain α and δ by solving a 2 × 2 non-linear system which has an analytical solution. With the pair (α, δ), we can easily obtain the pair (e, θ) and figure out the rotation implemented by the stage and, by taking its inverse and applying on Sout , we can obtain an estimate of Sin . Let A = 2V0 , B = −Vπ , C = Va − Vab , and D = Vc − Vcb . Then: q 2 2 δ = ((C + D) / (2A)) + ((C − D) / (2B)) (6) 2α = f ((C + D) / (2Aδ) , (C − D) / (2Bδ)) (7) where f (·, ·) is the function that receives cos(α) and sin(α) and returns α. It is worth emphasizing that an analogous procedure can be performed if the calibration resorts on LUTs [26]. The complete two-stage methodology to control and to stabilize polarization making use of the rotation algorithm , the stochastic gradient descent algorithm and, the state estimator is presented in Fig.2 in the form of a workflow chart. It contains two main loops that compute the voltages to be applied to the Polarization Controller stage’s electrodes. The choice for which pair of voltages to be used depends on the distance ε between the output SOP and the target SOP: when Sout and Starget are distant, the SGD algorithm may take a long time to reach stability so the rotation algorithm is employed; on the other hand, if they are close enough, the rotation algorithm can be unstable so the SGD is employed. 7 Simulation Results In our simulation procedure, two issues were taken into account: the polarization drift of sin , which is inherent to fiber optical communication systems; and the necessity of shift between two or more starget ’s. To clearly depict the algorithm’s performance through the simulation results, we present, in Fig. 3, the Poincaré Sphere and, in it, the green dots represent the wandering state of polarization at the input of the controlling apparatus and the red dots represent the output state of polarization. The algorithm is capable of maintaining the polarization stable in the vicinity of the set of starget defined by the three main linear states of polarization states: horizontal; vertical; diagonal; and anti-diagonal. In Fig. 4, we present the values of each component of sout and starget , as well as the associated error between starget and sout , as a function of time. We observe that the algorithm uses the rotation 4 Figure 2: Workflow chart diagram of the proposed three-stage control algorithm. step only after the shifts in starget , while the SGD is responsible for stabilizing the polarization around its value while sin drifts. This result is in correspondence to the expected behaviour of the algorithm, and confirms its good performance and applicability since the associated error is very small, i.e., even though the vicinity into which the algorithm keeps the output state of polarization in relation to the target polarization state seems large when observing Fig. 3, we see that, in the majority of time, inside a smaller vicinity define as the algorithm’s error tolerance. Figure 4: Simulation of real-time polarization control considering both the inherent polarization drift in the output of the controller and the target polarization shift. All graphs are in the same horizontal scale displayed at the bottom of the figure. The first three graphs display the values of the Stokes parameters that compose the SOP vector. The bottom two graphs display the associated error between sout and starget , where the second one is a re-scaled version of the first to clarify that the error does not exceed the threshold between SGD and the rotation step except in the cases of target polarization shift. The time scale was determined based on the time responses of off-the-shelf Analog-to-Digital and Digital-to-Analog Converters (ADC and DAC, respectively), of General Photonic’s Polarimeter module, and of the EOSpace Polarization Controller Module [27, 28, 29, 8]. It yielded an overall control loop 5 Figure 3: Simulation of real-time polarization control considering both the inherent polarization drift in the output of the controller and the target polarization shift. The green dots represent the wandering state of polarization at the input of the controlling apparatus and the red dots represent the output state of polarization. The algorithm is capable of maintaining the polarization stable in the vicinity of starget . iteration of approximately 1 µs. The polarization shift was set to match a Polarization Shift Keying system working at 50 × 103 symbols per second. The drift in polarization was set at 6 krad/s, a value well that attempts to mimic the operation of an optical communication system under severe conditions. 8 Conclusion We presented a three-step methodology to calibrate a Lithium-Niobate-based polarization controller module, to control the polarization and to stabilize the output SOP. Calibration, rotation algorithm, the Stochastic Gradient Descent and the state estimation algorithm were successfully tested in MATLAB simulations. The algorithms presented take simple forms and are readily embeddable in an FPGA or micro-controlled unit so we leave it as a future point of investigation. Acknowledgment The authors would like to thank brazilian agency CNPq for financial support. The authors are indebted to F. Calliari and V. Lima for help with the polarimeter unit and with the electronic circuitry, specially the analog-to-digital and digital-to-analog converters. Supplemental Material The authors provide digital supplemental material to accompany the article: a video file depicting the simulation run of the algorithm can be accessed in [30]; and the Matlab source files necessary for running the algorithm and the simulation are available at [31]. References [1] G. P. Agrawal, Fiber-Optic Communication Systems, 2nd ed. John Wiley & Sons, 2002. [2] D. Derickson, Fiber Optics: Tests and Measurements, 1st ed. Prentice Hall, 1998. [3] N. Gisin, J. P. von der Weid, J.-P. Pellaux et al., “Polarization mode dispersion of short and long single-mode fibers,” Lightwave Technology, Journal of, vol. 9, no. 7, pp. 821–827, 1991. 6 [4] J. Gordon and H. Kogelnik, “Pmd fundamentals: Polarization mode dispersion in optical fibers,” Proceedings of the National Academy of Sciences, vol. 97, no. 9, pp. 4541–4550, 2000. [5] S. Benedetto and P. Poggiolini, “Theory of polarization shift keying modulation,” Communications, IEEE Transactions on, vol. 40, no. 4, pp. 708–721, 1992. [6] A. J. Van Haasteren, J. J. Van der Tol, M. O. Van Deventer, and H. J. Frankena, “Modeling and characterization of an electrooptic polarization controller on linbo 3,” Lightwave Technology, Journal of, vol. 11, no. 7, pp. 1151–1157, 1993. [7] B. Koch, A. Hidayat, H. Zhang, V. Mirvoda, M. Lichtinger, D. Sandel, and R. Noé, “Optical endless polarization stabilization at 9 krad/s with fpga-based controller,” Photonics Technology Letters, IEEE, vol. 20, no. 12, pp. 961–963, 2008. [8] EOSpace, “Lithium niobate polarization controller,” http://www.hanamuraoptics.com/device/ EOSPACE/PC030123 EO.pdf, accessed: 2016-01-22. [9] S. K. Stein, “Calculus and analytic geometry,” AMC, vol. 10, p. 12, 1977. [10] J. B. Kuipers, Quaternions and rotation sequences. vol. 66. Princeton university press Princeton, 1999, [11] S. Haykin, “Adaptive filters,” Signal Processing Magazine, vol. 6, 1999. [12] B. E. Saleh, M. C. Teich, and B. E. Saleh, Fundamentals of photonics. vol. 22. [13] G. Strang, Linear Algebra and Its Applications. [14] M. D. Crossley, Essential topology. Wiley New York, 1991, Wellesley, MA: Wellesley-Cambridge Press, 2009. Springer Science & Business Media, 2006. [15] M. Karlsson and M. Petersson, “Quaternion approach to pmd and pdl phenomena in optical fiber systems,” Lightwave Technology, Journal of, vol. 22, no. 4, pp. 1137–1146, 2004. [16] F. Heismann, “Analysis of a reset-free polarization controller for fast automatic polarization stabilization in fiber-optic transmission systems,” Lightwave Technology, Journal of, vol. 12, no. 4, pp. 690–699, 1994. [17] T. Imai, K. Nosu, and H. Yamaguchi, “Optical polarisation control utilising an optical heterodyne detection scheme,” Electronics Letters, vol. 21, no. 2, pp. 52–53, 1985. [18] R. Noé, H. Heidrich, and D. Hoffmann, “Automatic endless polarization control with integratedoptical ti: Linbo 3 polarization transformers,” Optics letters, vol. 13, no. 6, pp. 527–529, 1988. [19] F. Heismann, “Integrated-optic polarization transformer for reset-free endless polarization control,” Quantum Electronics, IEEE Journal of, vol. 25, no. 8, pp. 1898–1906, 1989. [20] L. Xi, X. Zhang, X. Tang, X. Weng, and F. Tian, “A novel method to calibrate linbo3-based polarization controllers,” Chinese Optics Letters, vol. 8, no. 8, pp. 804–806, 2010. [21] F. Calliari, Electrical Engineering, PUC-Rio 2014 Monography, “Desenvolvimento de interface gráfica para análise do estado de polarização da luz através de plataforma fpga,” http://www.maxwell.vrac. puc-rio.br/23790/23790.PDF, accessed: 2016-01-22. [22] B. Widrow, J. M. McCool, M. G. Larimore, and C. R. Johnson Jr, “Stationary and nonstationary learning characteristics of the lms adaptive filter,” Proceedings of the IEEE, vol. 64, no. 8, pp. 1151–1162, 1976. [23] C. Burges, T. Shaked, E. Renshaw, A. Lazier, M. Deeds, N. Hamilton, and G. Hullender, “Learning to rank using gradient descent,” in Proceedings of the 22nd international conference on Machine learning. ACM, 2005, pp. 89–96. 7 [24] S. Haykin, Adaptive Filter Theory, ser. Prentice-Hall information and system sciences series. Prentice Hall, 1996. [Online]. Available: https://books.google.com.br/books?id=l78QAQAAMAAJ [25] J. J. Shynk, Probability, random variables, and random processes: theory and signal processing applications. John Wiley & Sons, 2012. [26] A. Hidayat, B. Koch, H. Zhang, V. Mirvoda, M. Lichtinger, D. Sandel, and R. Noé, “High-speed endless optical polarization stabilization using calibrated waveplates and field-programmable gate array-based digital controller,” Optics express, vol. 16, no. 23, pp. 18 984–18 991, 2008. [27] Texas Instruments, “Adc324x dual-channel, 14-bit, 25-msps to 125-msps, analog-to-digital converters,” http://www.ti.com/lit/ds/symlink/adc3242.pdf, accessed: 2016-01-22. [28] General Photonics, “High-speed polarimeter – poladetect: Pod-201,” http://www.generalphotonics. com/wp-content/uploads/2015/05/POD-201-5-8-15.pdf, accessed: 2016-01-22. [29] Texas Instruments, “Dac3484 quad-channel, 16-bit, 1.25 gsps digital-to-analog converter,” http: //www.ti.com/lit/ds/symlink/dac3484.pdf, accessed: 2016-01-22. [30] Optoelectronics Laboratory – Gustavo C. Amaral and Joaquim D. Garcia, “Polarization tracking algorithm – simulation run in youtube,” https://www.youtube.com/channel/ UCcfPvWdcMmyhIS7lmXIhm-w, accessed: 2016-02-01. [31] Optoelectronics Laboratory – Joaquim D. 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Biological and Shortest-Path Routing Procedures for Transportation Network Design arXiv:1803.03528v1 [physics.soc-ph] 7 Mar 2018 François Queyroi∗ CNRS, UMR8504 Géographie-cités (CNRS/Université, Paris-1, Panthéon-Sorbonne/Université Paris Diderot) Abstract Background. The design of efficient transportation networks is an important challenge in many research areas. Among the most promising recent methods, biological routing mimic local rules found in nature. However comparisons with other methods are rare. Methods. In this paper we define a common framework to compare network design method. We use it to compare biological and a shortest-path routing approaches. Results. We find that biological routing explore a more efficient set of solution when looking to design a network for uniformly distributed transfers. However, the difference between the two approaches is not as important for a skewed distribution of transfers. 1 Introduction Transportation networks. The transportation network design corresponds to the problem of selecting a set of possible links between locations (for example cities) in order for transfers (for example of goods, people, etc.) to be made possible [8]. The automated design of transportation network has a range of applications going from solving transshipment problems [9] to the computation of space trajectories [17]. In the social sciences, researchers want to compare efficient simulated networks with the real ones (railroads, railways, etc.) in order to assess the existence and nature of suboptimal choices [18]. Transportation networks are also important for the simulation of the development of a city system [23]. The design of transportation networks as a computational problem is part of a large domain of inquiry which is known as network flow problems [1]. Transportation network design can be here described as a multi-objective variant of the multi-commodity flow problem [10] with unlimited edge capacities. While most analytical research focus on cost minimisation, we are looking for transportation networks that are efficient with respect to multiple criteria. The choice therefore often involves a cost/benefits analysis. The most common criteria are the time performance (how quickly can we travel using the network), the cost (the size or total length of the network) and the tolerance to fault (how travel is affected by random perturbations). Obviously, finding a good balance between these ∗ [email protected] criteria make the problem hardly solvable analytically and solutions are often explored using heuristics. Networks computation models. Several computation models can be used to explore potential solutions. The first approach is what we call the biological approach. The reason for this name is that those methods are derived from actual natural phenomena and the most cited one is probably the behaviour of the Physarum polycephalum organism [27]. This slime mould development is indeed capable to solving mazes or discover shortest-paths in a difficult terrain [25]. Experiments involving this organism caught a lot of attention in the scientific press. One important achievement was the simulation of the Greater Tokyo Area transportation network [26] where the authors introduce an algorithm replicating the behaviour of the organism. It should be noted however that similar behaviours can be found in other natural phenomena such as ant colonies [7] or current in an electrical network [16]. The second possible approach to the problem of network design is what we call the shortest-path routing method. We should stress out the fact that this category has known less extensions and led to less applications than the previous one. To the best of our knowledge, Levinson and Yerra [14] are the first to study this model of computation. Their objective was to show that a hierarchy of routes can emerge in the transportation network from a uniform distribution of transfers [28]. This method has been rediscovered by others in the domain of Information Visualisation [13]. Others methods such as greedy algorithms could be used. The idea is here to incrementally build a network by adding at each iteration the links that contribute the most to the performance of the network [5, 22]. A common variant is to start from a minimum cost spanning tree covering all transfers and then iteratively adding the best alternative routes. The Physarium simulation was actually compared to this approach in [26]. It is also possible to start with a more complete network and prune the least used paths. This last method actually mimics the development of neural networks [21]. Routing and Reinforcement. Both the biological and shortest-path approaches actually rely on two common mechanisms. First, the routing of goods (food in the case of the slime mould) is done by assigning them to paths depending of their “attractiveness” (the diameter of the tubes for the slime mould). Then, paths attractiveness is updated according to the amount of goods using these paths (the slime mould’s tubes expand or shrink due to the pressure). We call this second phase reinforcement. By repeating those steps we can mimic a continuous process where the network gradually appears from a starting grid as least used paths in the grid gradually disappear while others gather more and more transfers. Biological routing uses local rules of dispersion. Transfers will be assumed to behave like a liquid flowing through pipes of various size to reach a sink. On the other hand, the shortest-path method routes the transfers along the shortest-path between their source and destination. Biological routing therefore explores different paths even if there are longer while the shortest-path routing only selects the best paths. Contribution. The aim of this study is to analyse the differences between biological and shortestpath approaches. We introduce a common framework for the two algorithms. Most of the previous studies (in particular with the biological models) focus on uniform transfers between locations (either there is an exchange of commodities or not). However this setting may not be appropriate for different applications (in urban planning for example) since locations may differ in term of attraction potential. 2 We therefore expand previous definitions of the algorithms in order to take into account arbitrary transfers distributions between several sources and destinations. This common framework allows us to compare biological and shortest-path routing only focusing on the way transfers are routed without the interference of other minor differences. Moreover, we use as experiments a random generated set of grid-embedded locations and transfers while previous studies use a few small toy examples or real world configurations. Biological network design such as the one simulating the Physarum polycephalum organism has been shown to produce efficient network but was never, to the best of our knowledge, compared to the shortest-path approach. Our hypothesis is that the shortest-path method while not being as popular as biologically based approaches in the literature may be worthwhile to pursue if the transfers between location are modelled according to distributions found in human mobility (such as the gravity model). 2 Reinforced Routing Procedure Overview Notations. Throughout this paper, we call the support graph G = (V, E, l) a graph (or network) with nodes (or vertices) set V , edge set E and edge length l : E → R+ . Let n = |V | the number of nodes and m = |E| the number of edges. We call F : V × V → R+ the transfer matrix between nodes in G and we call Q : E → R+ a flow distributionPon the network (i.e. the way transfers in F are distributed along the edges of G). We have F = s,t F (s, t) the total amount of transfers between the nodes of G. A transportation network on the support graph G is simply a subgraph N = (V, E 0 , l) of G with E 0 ⊆ E. For edge defined functions such as Q or l and an edge e = (u, v), we may write Q(u, v) or l(u, v) to refer to Q(e) or l(e). Networks flows are mostly defined for transfers between a source node and one or several destination nodes. Here Q corresponds to the distribution of the total F units among E i.e. if all units transferred travel through the network G to reach their destination according to F then Q corresponds to the number of units that went through each edge. Notice that, for a given G and F , there are multiple possible values of Q. Although we use the same terminology of flow, we do not expect Q to respect to classic networks flow rules [1] and we consider G to be undirected without a loss of generality. Main procedure backbone. We design the procedure so that the differences between biological and shortest-path approaches only rest on the way transfers are routed. As previously explained, the procedure can be divided into two parts: the routing of the transfers in F (which gives Q) and the adaptation of edge length according to the flow Q. The routing depends on the length of the edges of the network l. Algorithm 1 details this generic procedure. Here, the reinforcement modifies the length of network edges by modifying σ which can be interpreted as edges’ “diameter” of tubes in the slime mould organism, the “speed” in a railway network or the “resistance” in an electrical circuit. Notice that Algorithm 1 does not directly produce a transportation network. Rather, the σ values indicate whether an edge is likely to be part of an efficient transportation network. For the experiments, we create a network by selecting edges with a σ value higher than . Flow routing. The function FlowRouting (line 5) will depend on the chosen method (Biological or Shortest-Paths routing). We assume that, in both cases, the result Q is a normalized flow i.e. such 3 Algorithm 1: Reinforced Routing of flow F along network G Input: G = (V, E, l), F : V × V → R+ , α > 0, µ > 1, δ ∈]0, 1],  > 0 Output: σ : E → [0, 1] 1 σ 0 (e) ← 1, e ∈ E; 2 n ← 0; 3 do   l 0 4 G ← G V, E, n ; σ 5 Q ← FlowRouting(G0 , F ); 6 σ n+1 ← δf (Q; µ, α) + (1 − δ)σ n ; 7 n ← n + 1; 8 while max |σ n+1 − σ n | > ; 9 return σ n ; that Q ∈ [0, 1]m . To do this we simply divide the number of units transferred going through a given edge by F. The two following sections will describe each routing procedure in details. Reinforcement. First, notice Q is an aggregation of transfers with different sources and destinations. Previous works [26] used to adapt network edges after the routing of a single line of F (the transfers connecting a single node s) or a single exchange in F (the transfers between s and t). This obviously requires to take lines or elements of F in a random order. This approach may speed up the convergence of the procedure but we believe this addition of randomness is of little interest in the context of our study. In Algorithm 1, support network’s edges’ resistance σ is modified according to a reinforcement function f and an update rate δ (line 6). The latter is a parameter that influences the convergence rate of the algorithm. In previous studies, this parameter was often implicit: the reinforcement is defined as a change in resistance over time in a continuous fashion so the convergence speed is given by the way time is split into steps. The reinforcement function (sometimes called “value function”) f we use is a logit-like function frequently used in the literature [25]. f (Q; µ, α) = (1 + α)Qµ 1 + αQµ (1) Function f is a response function where parameter µ > 1 controls the slope of the curve while parameter α > 0 controls the inflection point of the function. Indeed the higher α is the greater is the response value of small signals (see examples of curves with various values of α in Fig. 1). In the biology analogy, this function models the responses to the flow going through the Physarium tubes. We initialize σ 0 (e) = 1, ∀e ∈ E (line 1) and, since f is dimensionless with f (0) = 0 and f (1) = 1, we have σ n ∈ [0, 1]m at each iteration of the algorithm. The length edges of the modified support graph G0 will increase (line 4) which will affect the next routing phase. Running time. The stopping condition of the algorithm (line 8) depends of the difference in the 4 Figure 1: f (q; µ, α) for µ = 1.8 and different values of α distribution of σ values between successive steps. The parameter  is used to set the precision of the algorithm (in this study we used  = 5.10−4 ). Previous definition of the Physarium algorithm uses a arbitrary number of iterations instead. The convergence of resistance σ to a stable solution has been studied in the case of the Physarium solver [19, 11]. In our case, Algorithm 1 seems to always converge to a solution whether we use biological or shortest-path routing although the number of steps required for the shortest path routing procedure is lower. The running time of function FlowRouting may also be affected by the ordering of the vertices. This is true whether we adopt the biological or the shortest-path routing procedure (described below). In both cases, we only need to route the transfers connected to the minimal vertex cover of the graph formed with the transfers in F higher than 0. The reason is that the flow we compute is an aggregation of different “commodities” (we can not exchange a person travelling from a to b with one going from c to d). Both routing procedure will route one “commodity” (the people travelling from a to b or from a to c) at a time. We can find a small enough vertex cover by taking vertices in decreasing order of their number of strictly positive transfers. An implementation of the two procedures is available at https://github.com/fqueyroi/tulip_ plugins/tree/master/TransportationNetworks as plugins for the Tulip software [2] http://tulip. labri.fr. 2.1 Shortest-path routing Using the shortest-path model, the flow going through support network’s edges is given by what we call the flow betweenness. Informally, flow betweenness corresponds to the total number of travellers going through a given edge if the travellers choose the fastest (shortest) route. In the general case, we 5 should therefore have: X Q(e) = (s,t)∈V ×V πe (s, t) F (s, t) π(s, t) (2) where πe (s, t) is the number of shortest-paths from node s to t going through edge e and π(s, t) is the total number of shortest paths from node s to t. In practice, if we use continuous edge length on a connected network, we can assume that πe (s, t) ∈ {0, 1} and π(s, t) = 1 for all pairs (s, t). This means that there is only one shortest-path between each pair of nodes in G. On way to compute Eq. 2 is therefore to compute all pairwise shortest-paths between the locations evolved in the transfers using a shortest-path algorithm such as Dijkstra’s algorithm [6]. However, notice that if F (s, t) = 1 for all (s, t) ∈ V × V then Eq. 2 corresponds to the betweenness centrality measure [3]. This measure is well-known in network analysis as it allows to identify vital elements of the network (hubs). It is possible to generalize the algorithm proposed in [3] in order to take account of transfer values different than 1. The computation of FlowRouting therefore runs in O(nm + n log m). In practice it involves computing as many shortest-path tree as the size of the minimal vertex cover of transfers higher than 0 (which is at most n). 2.2 Biological routing As previously explained, “biological” routing is designed to mimic several known physical phenomena that can be found in biological organisms such as the slime mould Physarum Polycephalum. As such, the routing of flow in biological models corresponds to a flow Q that respects classic flow constraints and minimizes the total energy of the model: ξσ (Q) = X l(e) Q(e) σ(e) (3) e∈E This biological routing of flow is also commonly introduced as a solution of a linear equation system: Q(u, v) = l(u, v) (p(u) − p(v)) σ(u, v) (4) l(u, v) is the conductance of edge (u, v). σ(u, v) This routing is therefore also close to other network analysis measures such as Eigenvector centrality or PageRank [4]. The potential p(u) of a node u to attract flows depends on u’s neighbours’ potential w. r. t. the conductance (speed or diameter) of adjacent edges. However, in our case, we are not interested in the potential value of nodes but only by the flow going through each edge. Eq. 4 corresponds to Ohm’s law for electrical circuit where The computation of the flow Q in the biological model corresponds to a solution of a linear equation system. This computation can be cumbersome due to the number of variables. We adopt the approximation algorithm described in [12]. It starts with a suboptimal solution where the flows are routed on a low-stretch spanning tree (distance preserving tree). Then, we randomly select a “short-cut” edge (a, b) that does not belong to that tree and send a portion of the flow going from 6 a to b in the tree through that short-cut. Those modifications are called “cycle updates”. The number of cycle updates performed is set so that the distribution of flow is a -approximation of the solution of the linear equation system. This approach leads to a near-linear time computation of O(m log2 n log log n log(−1 )). Note however that a solution Q to Eq. 3 can only be found if all transfers in F have for source a single unique vertex (in order to respect flow constraints). Again, we need to find as many solutions the size of the minimal vertex cover of non-null transfers (which is at most n). The running time of the biological routing therefore highly depends on the density of transfers. 3 Experiment details We present in this section the choices made for comparing the two transportation network construction models described above. Fitness Measures. We introduce here the fitness measures used. We use concepts found previously in the literature [26] (the three dimensions: performance, fault-tolerance and cost). The differences with previous definitions of performance or fault-tolerance come from the fact that we consider arbitrary transfers distributions. Even though Algorithm 1 outputs a real vector of speed/diameter σ, we take as transportation net¯ work N = (V, {e ∈ E : σ(e) > }, Pl) for simplicity sake. We call dN the diameter of N (length of the longest shortest-path) and F = s,t F (s, t) the total amount of transfers on the network. We define the following indicators: 1. Performance P corresponds to the total time taken for units to go from their source to their destination in N . 1 X P (N, F ) = 1 − ¯ F (s, t)dN (s, t) (5) F dN s,t where dN (s, t) is the distance between s and t in N (i.e. the sum of the length of the edges along the shortest-path from s to t). Notice we have P (N ) ∈ [0, 1] and we say the network N has good performance when P (N ) is close to 1. The main difference with previous definitions of performance is that the amount of transfers is taken into account. Indeed, the measure P corresponds to the mean time taken by travellers to reach there destination while previous definition (using uniform transfers) correspond to the mean time of travel for each pair of locations. 2. Fault Tolerance F T corresponds to the proportion of transfers still able to reach destination after the removal of a random edge in N . X 1 X 1 F T (N, F ) = 1 − F (s, t)1PN \e (s,t)6=∅ (6) |E(N )| F e∈E(N ) (s,t)∈V ×V where 1PN \e 6=∅ (s, t) is equal to 1 if there is still a path between s et t in N after the removal of edge e or 0 otherwise. Using this normalisation, we have F T (N ) ∈ [0, 1] and we say that N is highly tolerant to fault when F T (N ) is close to 1. Notice that F T is a generalisation of the classic metric that just focused on whether the network is disconnected or not (a measure used in [26]). If an area is loosely connected to the network, it may not affect F T much if the amount of transfers with this area is relatively low. 7 3. Cost C corresponds to the normalized sum of the length of edges in N P e∈E(N ) l(e) C(N ) = P e∈E(G) l(e) (7) Using this normalisation, we have C(N ) ∈ [0, 1] and we say that N is a costly network when C(N ) is close to 1. Obviously, a costly network is more likely to have a higher tolerance to fault and performance. Therefore it is interesting to look at the ratio P/C and F T /C for comparison purpose. Samples. In order to compare the two algorithms, we first use synthetic data generated randomly. We sample a set of 150 points in the [0, 1]2 plain and connect the points using a standard Delaunay triangulation. This random grid corresponds to the possible adjacencies of the future network (the support graph G). We then select a subset of 8 points that will correspond to the sources and destinations of transfers. The matrix F is then generated using two different models: 1. Uniform distribution: set F (s, t) = 1 for all (s, t). It is the same as in [26]. P (s)P (t) where P : V → R is the population of the nodes and dE (s, t)γ dE (s, t) is the euclidean distance between s and t. Here we set γ = 1.2. This model is often used in urban geography to model human mobility [24]. The population P is generated using an Zipf exponential model i.e. the population exponentially decreases with the rank of the points by a factor of 1.5 (the ranks of the 8 locations being chosen randomly). This model produces few important centres of attractions and isolated areas. 2. Gravity model: set F (s, t) = The way transfers are distributed corresponds to two experimental settings. In addition, we develop a third and a forth using as locations cities of the French region Pays-de-la-Loire (West of France). The resulting grid can be seen in Fig. 4 (red nodes represent important cities in the region). We also analyse the difference between a uniform and a gravity-like distribution of transfers. For the latter, we use as locations weight the actual population of the cities according to the 1999 national census. Parameters choices. Algorithm 1 has many different parameters. The most influential however is the parameter α. For the experiments, we take for α values powers of 2 (see the different behaviour of f in Fig. 1). In [26], the authors choose to modify the total amount of transfers (which here corresponds to F) since their reinforcement function does not include a parameter similar to α. The effect is however similar. Indeed, the higher α is, the more the route that are less used are given a high weight. In [26], the more transfers amount there is, the more those routes are likely to be used. The α value can be used to influence the final cost of the network. Small α values are likely to result in a tree-like organisation of the transportation network while higher α values are more likely to produce to a grid-like organisation [15]. Regarding the other parameters we use an update rate γ = 0.5, a slope of reinforcement µ = 1.8 (similar to the one in Fig. 1) and a precision of  = 5.10−4 (i.e. the algorithm will stop when the greatest difference in σ values is smaller than 5.10−4 ). Hypothesis on the difference between the two approaches. Previous studies show that the biological routing is efficient when compared for example to greedy approaches [26]. The efficiency here 8 involves finding a good compromise between performance and fault tolerance. These results where made using uniformly distributed transfers between locations. Using the same distribution, we expect shortest-path routing to perform worst than biological routing. The reason is that shortest-paths explore fewer solutions and the procedure could quickly fall in a local minima. However, we could expect the results to be different when modifying the distribution of transfers. A skewed distribution of transfers should favour the shortest-path routing approach since the direct routing of the most important transfers along the fastest route will have the most impact. 4 Results In this section, we discuss the results of the various experiments using the indicators of performance (P ), fault tolerance (F T ) and cost (C). We first look at the statistical results for the synthetic experiments then we discuss some qualitative aspect of the results in the case of the geographical grid. For clarity purpose, we use SP and BIO to refer to the shortest-path routing and biological routing respectively. The two types of transfers distribution are referred as U M for the uniformly distributed transfers and GM for the gravity model. We generated 200 instances of random grid and applied the transfers distribution models described above. We shall first stress out the fact that BIO computation is very slow. It requires several minutes when SP only takes a second. The distributions of the indicators according to α can be found in Fig. 2. Note that we do not have to compare each algorithm using the same α value. Therefore, the evolution of the ratio P/C and F T /C can be found in Fig. 3. For the geographical grid, we report the statistics computed for some values of α for U M (Table 1) and GM (Table 2). A representation of the computed networks can be seen in Figures 4 and 5. Here are the conclusions that can be drawn from these results: 1. BIO seems to achieve better results than SP if we look at the ratio. In the U M experiments and for α ∈ [1, 4], the ratio P/C and F T /C are both greater for BIO than for SP with any α values. The same can be said in GM with α < 4. However, the variation of the indicators is important (height of the boxplots in Fig. 2). It means there are configurations when SP routing may find a better network. It is actually the case for the geographical grid with both U M and GM . 2. BIO explores a broader set of results. A trade-off between performance and fault-tolerance is clearly visible in U M with both BIO and SP . This phenomenon was already observed for BIO in [26] using different fitness indicators. One important difference between the two procedures is that the set of networks that can be found using BIO corresponds to a wide range of cost. This is more limited for SP as the networks found for various value of α may not differ a lot. 3. There is important difference in the results between the two transfers distribution models. The trade-off between perf6ormance and fault-tolerance is not clearly apparent in the GM case. We can observe that the behaviour of the ratio P/C and F T /C is similar for BIO and SP . It can be explained by the fact that the simulated transfers create a single 9 Figure 2: Distribution of the various indicators according to α values (x-axis) for the four experimental settings (see Legend). Bottom-right plot: similarity between the results of SP and BIO routing procedures (proportion of edges found in both networks). 10 Table 1: Results for the geographical grid (with a uniform distribution of transfers) Routing SP BIO α 2 16 64 2 16 64 Perf .469 .479 .489 .471 .487 .548 FT .793 .797 .822 .782 .956 .998 Cost .034 .035 .035 .035 .042 .054 Perf/Cost 13.794 13.686 13.971 13.457 11.595 10.148 FT/Cost 23.324 22.771 23.486 22.343 22.762 18.481 important centre of attraction in the grid (the most “populated” location). In this context, achieving good performance and fault-tolerance is not hard: we just need to connect this centre to the periphery. Accordingly, we observe that the networks found with BIO and SP are more similar in this model (see Fig. 2). Moreover, the value of the ratio P/C and F T /C is higher for both algorithms. Still, further expansions of the network (using higher α values) are less and less cost-effective since they connect locations whose transfers between them are exponentially smaller. 4. The differences in performance or fault tolerance seems to correspond to different behaviours when looking at the geographical example (Figs. 4 and 5). In the U M case, networks found using SP have a tree-like structure for most values of α. BIO finds similar networks using small α value. However, it can also find more costly networks with higher α value such as a grid-like network (see Fig. 4b). Table 1 shows this trade-off between P/C and F T /C. Notice however that BIO produces a lot of alternative paths that are actually parallel routes going between the same locations. This type of behaviour can explain the important increase of cost for higher α values. In the GM case, the result are very different. Small α values still correspond to tree-like structure for both BIO and SP . For higher α values however, SP produces alternative paths. This is also the case for BIO but again with redundant routes. This apparently strange behaviour from BIO could be traced back to the “double edges” often found in experimental settings using a real Physarum polycephalum organism [20]. In our case, it seems that the redundant edges do not add much in term of fault-tolerance since alternatives (but longer) routes already exist. 5 Conclusion and Related Questions In this paper we compared two different approaches of transportation network design. We provide a common analysis framework and an implementation of the algorithms. The methods were compared based on synthetic random grids using statistical indicators. From a quantitative point-of-view, we can conclude that the biological inspired approach is overall better than the routing of flows based on shortest-paths. However, the difference between the two is not always significant. The biologicallyinspired model can be used to explore a wider range of solution. Contrary to our hypothesis, those observations are still valid when using a more realistic non-uniform model of transfers even though the difference in terms of performance and fault-tolerance is even smaller than in the uniform case. 11 Figure 3: Evolution of the median value of the ratio Perf/Cost and FT/Cost (middle of the corresponding boxplots in Fig. 2) according α values (points labels). Table 2: Results for the geographical network (with a gravity model distribution of transfers) Routing SP BIO α 0 128 256 512 1024 0 128 256 512 1024 Perf .644 .661 .661 .677 .689 .578 .578 .627 .637 .689 FT .851 .928 .928 .981 .987 .831 .831 .938 .967 .987 12 Cost .031 .034 .034 .04 .053 .033 .033 .037 .045 .053 Perf/Cost 20.799 19.369 19.161 16.734 12.968 17.343 17.343 16.743 14.15 10.759 FT/Cost 27.483 27.201 26.913 24.242 18.559 24.914 24.914 25.05 21.469 15.398 (a) Bio routing – α = 4 (b) Bio routing – α = 64 (c) SP routing – α = 4 (d) SP routing – α = 64 Figure 4: Comparison of the result on the geographical example with uniform transfers. Red nodes correspond to major cities in the regions (préfecture and sous-préfecture). Our experiments still have important limitations since there are numerous dimensions still unexplored. One could access, for example, the influence of the size of the grid. First results tend to reveal a similar behaviour but the discrepancy between the two methods may be amplified. The influence of others parameters such as the decay rate γ or the reinforcement curve gradient µ should also be investigated but we expect they will have less of an impact. Our study is also limited by the fact that our evaluation relies on objectives functions related to performance, fault-tolerance and cost. However, geographers, biologists or urban planners might want to know whether or not simulated networks are close to real-world networks. In this context, the objective functions are useful but it is not possible to infer the closeness between two networks based on the proximity in these functions values. What we learn from this study is that biologically-inspired routing may be better suited for the researcher since it allows an exploration of a broader set of solutions. 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Finite element method for thermal analysis of concentrating solar receivers Stanko Shtrakov and Anton Stoilov South-West University, Blagoevgrad, Bulgaria Application of finite element method and heat conductivity transfer model for calculation of temperature distribution in receiver for dish-Stirling concentrating solar system is described. The method yields discretized equations that are entirely local to the elements and provides complete geometric flexibility. A computer program solving the finite element method problem is created and great number of numerical experiments is carried out. Illustrative numerical results are given for an array of triangular elements in receiver for dish-Stirling system. 1. Introduction. Cavity receivers for solar concentrating systems absorb a concentrated solar energy, convert it to heat, and transfer heat to the working gas in power unit (Stirling engine, steam turbine). Heat flux in such elements is great and material tension is very high. Different shapes and materials were used to optimize the performance of cavity receivers. Besides the natural experimental tests, there are possibilities to use serious theoretical studies for the thermal behavior of the cavity receivers. It would be very useful, because the processes in the cavity are complicated by high energy intensity and a real measure of temperature distribution is difficult and expensive. Because of the special form of the solar receivers, it is not suitable to use ordinary mathematical technique, such as analytic methods or numeric solutions with finite difference approximation. Finite elements method is now one of the most popular ways to solve complicated mathematical problems, especially with irregular definition area. There are many works treating the heat conductivity problems by using finite elements methods and there is practical experience in this field. In this paper is presented a mathematical model for heat conductivity processes in cavity receiver for dish-Stirling system with appliance of finite elements method for solving differential equations. The studies and numerical tests are made for a typical cylindrical cavity receiver, but algorithm and computer program created for calculation of temperature distribution in receiver are suited to solving a different forms of receivers. The simple construction scheme of typical Absorber tube type cavity receiver for dish-Stirling system is shown in Fig. 1. It comprises absorber plate (tube system for gas heating), cavity space, walls, Cavity aperture and insulation on the external surfaces of walls. The absorbing surface is usually placed Walls behind the focal point of the concentrator so that the flux density on the absorbing surface is reduced. The size of the absorber and cavity Aperture walls is typically kept to a minimum to reduce Insulation Fig. 1. Cavity receiver for dish-Stirling system heat loss and receiver cost. Concentrated radiation entering the receiver aperture diffuses inside the cavity. Most of the energy is directly absorbed by the absorber, and most of the remainder is reflected or reradiated within the cavity and is eventually absorbed by absorber and cavity walls. Converted by walls heat is conducted to the absorber plate and is transferred to the working gas. The major advantage of cavity receiver is that the size of the absorber may be different from the size of aperture. With a cavity receiver, the concentrator’s focus is placed at the cavity aperture and the highly concentrated flux spread inside the cavity before encountering the larger absorbing surface area. This spreading reduces the flux incident on the absorber surface. When incident flux on the absorbing surface is high, it is difficult to transfer heat through surface without thermally overstressing materials. The bottom of the receiver is a heat exchanger for the Stirling engine. Part of solar energy strikes directly the bottom wall of the receiver and by conduction it delivers heat to exchange pipes. Cylindrical walls absorb the other part of solar energy and conduct it to the bottom of the receiver. The external cylindrical surfaces are well insulated for protecting heat losses. Natural convection in cavity produces air circulation and convective losses. Heat losses are further caused by insensitive radiation from cavity aperture to the ambient. These losses are not big, because of the small measurements of the receiver. Nevertheless, all heat losses can be taking into account because they influence the temperature distribution in the receiver. 2. Theoretical Model Mathematical model of processes in cavity receiver can be derived on the base of theoretical treating of heat conductivity in receiver walls, heat transfer to the working gas by absorber plate and heat losses to the ambient. Because of symmetry, the problem domain can be simply presented as in fig.2. The main mathematical equation in this problem is the heat conductivity equation for receiver construction written in cylindrical coordinate system: ∂ ⎛ ∂T ⎞ ∂ ⎛ ∂T ⎞ 1 ∂T = ∆2T = 0 ⎟+ λ ⎜λ ⎟ + ⎜λ ∂r ⎝ ∂r ⎠ ∂z ⎝ ∂z ⎠ r ∂r (1) where T is the temperature in the walls, r and z – space variables in radial and high direction (fig.2) and λ – heat conductivity coefficient [W/m K]. ∇ 2 is Poason’s operator for vector form description of equation. z E C D The boundary conditions, needed for the solution of eqn (1), can be obtained from consideration of energy balance for different surfaces of the cavity receiver. External surfaces of the receiver (D) are insulated and the heat transfer to the ambient can be presented by the next equation: λ A ∂Τ = K D (Τ − Τa ) ∂r (2) Here Ta is the ambient temperature, T – temperature of external surface and Kd – overall heat transfer coefficient from the receiver external surface to the ambient air (including insulation). r The inner surfaces of receiver (A and C) B exchange heat energy with air in cavity and absorb solar Fig.2 Theoretical scheme of receiver radiation entering the space in receiver. Boundary condition for these surfaces can be written as: λ ( ) ∂Τ = α B Τ − Τf + q ∂r (3) where αB is a convective transfer coefficient in receiver cavity [W/m2 K], Tf – air temperature in cavity, T – temperature of inner receiver surface and q – solar energy flux [W/m2]. Surface E can be modeled as a heat exchanger for heating the working gas in Stirling motor. Boundary condition in this case is: λ ∂Τ = K W (Τ − Τwg ) ∂r (4) Heat transfer coefficient Kw depends on parameters of convective heat transfer to the working fluid. Temperature of working fluid Twg is defined from the thermodynamic treating of Stirling process. In general, boundary conditions can be written in common form as follow: λ ∂T = h eT + C e , ∂n (5) where he refers to the convection coefficient and Ce describes the convection coefficient with the external temperature and solar flux. Superscription e refers to the surface index (e). Derivative ∂T/∂n is taken to the outward normal direction n for every boundary surface. Differential problem (1) – (4) can be solved by numerical methods. The special form of problem domain (fig.2) leads to creating an unstructured grid for descretization of calculation area. This is the reason for using the finite element method for solving the problem. 3. Finite Element Method In the finite element method (FEM), the problem domain is discretized and represented by an assembly of finite elements. The method yields discretized equations that are entirely local to the element. As a result, the discrete equations are developed in isolation and are independent of the mesh configuration. In this way, finite elements can readily accommodate unstructured and complex grids, unlike other numerical methods (finite difference method) requiring structured (rows and columns) format. Finite element method provides complete geometric flexibility. Various approaches to the finite element method formulation are used, but the most prevalent is the Galerkin method [1,2]. It provides the correct number of basis function and the same resulting equations as methods based on other (more complicated) types of formulations. Galerkin’s method selects the weight functions equal to the basis functions (shape functions) of the approximate solution. It will be demonstrated on the example of solving the two-dimensional thermal conductivity problem in cavity receiver for dish-Stirling system – equation (1) with boundary condition (2) – (4). Commonly encountered elements in two-dimensional configuration include a linear triangle, bilinear rectangle, and bilinear quadrilateral. Triangular elements are well suited to irregular boundaries, and spatial and scalar interpretation can be accommodated in terms of a linear polynomial. Nodes are assigned to location in the element at local node 1 which unknown function, such as temperature, is to be (r1,z1) determined. Nodes are often placed only at the corners of the elements (Fig.3), but additional nodes can be placed internally or along the element boundaries. The main problem in finite elements method is to local node 3 choose shape or integration function to provide (r3,z3) approximate variation of the depended variable within local node 2 each element between the values of the nodes. The (r2,z2) simplest shape function is linear. Each shape function is a local interpretation function that is defined only Fig. 3. Linear triangle within elements containing a particular node. Linear triangle element (Fig.3) refers to linear interpolation along a side or within the element. For an interpolation involving a scalar shape form function, N(r,z) within a linear triangle can be described as: N(r,z)=a+br+cz. (6) where the unknown coefficients, a, b, c, can be determined on the basis of the substitution of nodal values. For example, at node 1, the position is (r1, z1) and the scalar shape function is N1. By using the shape functions, an approximate solution Tˆ ( r , z ) for T(r,z) is assumed in the form N Tˆ (r , z ) = ∑ T j N j (r , z ) (7) j =1 where Tj are the temperatures at the nodes desired for the solution, and Nj are the shape functions that are equal to 1 at each node. When this approximation is substituted into the energy equation (1), there is a residual that depends on r and z: ∂ ⎛ ∂Tˆ ⎞ ∂ ⎛ ∂Tˆ ⎞ 1 ∂Tˆ ⎟+ λ ⎟ + ⎜λ ⎜λ = ∆2Tˆ = Re s (r , z ) ⎟ ⎜ ⎟ ⎜ ∂r ⎝ ∂r ⎠ ∂z ⎝ ∂z ⎠ r ∂r (8) It is desired to be obtained a solution that, in average sense over the entire volume, is as close as possible to the exact solution each time. Variational principles are applied to minimize the residual. A set Wi(r,z) of independent weighting function is applied, and the residual is made orthogonal with respect to each of the weighting functions. This provides the following integral, which is evaluated for each of the sets of independent weighting functions, ∫∫ Re s(r , z )W (r , z )dS (9) i S The integration is over the whole volume in which the solution is being obtained. In the Galerkin method the weighting functions are chosen to be the same function set as the shape functions. Equation (9) provides the Galerkin form of energy equation for each of the weighting functions, ⎡ N ⎤ T j ∇ 2 N j (r , z )⎥ ⋅ N i (r , z ) ⋅ dS ∫∫S ⎢⎣λ ⋅ ∑ j =1 ⎦ (10) Boundary conditions can be incorporated through the boundary integral term in above equation. Considering the general form of the boundary conditions (5) and using integration by parts, the following form of equation (10) can be received [1]: ⎡ N ∂N i ∂N j ∂N i ∂N j 1 ∂N j ⎤ + + ( N i )T j ⎥ ⋅ dS = ∫ [ N i (r , z )(h eTi + C e )dn ∫∫S ⎢⎣λ ∑ ∂r ∂r ∂z ∂z r ∂r j =1 n ⎦ (11) Evaluating equation (11) for each i provides N simultaneous algebraic equations for the Tˆ ( r , z ) . In the matrix form, this is: [Kij] [Tj] = [Fj] (12) Since each shape function Nj(r,z) (and hence each weighting function in Galerkin method) is zero except within an element containing Tj, the resulting matrix 7 8 9 10 11 12 [Kij] for solving the simultaneous 6 4 10 2 8 equations for Tj is banded and sparse. It is 3 1 9 5 7 usually banded along the diagonal (banded 3 4 5 6 1 2 matrix). The finite element method yields Fig. 4. Discretization of domain discretized equations that are entirely local to the element and hence the global matrix [Kij] is a simple combination (sum) from local matrixes of each node. Solving the system of algebraic equations (12), the unknown temperatures Tj can be received as a result. Because of the special form of [Kij] matrix, different numerical techniques have been developed [1,2] for solving the system (12). 4. Numerical solution techniques The mentioned above technique is demonstrated on the example of cavity receiver (fig.1). Using triangle elements the problem domain can be discretized as it is shown in fig.4. Each element is numbered and nodes coordinates (ri,zi) are specified. Interpolation of the temperature in triangle element can be written in terms of the shape functions, Ni, Nj, Nk, as follow: T (e ) (r , z ) = Ti N i(e ) (r , z ) + T j N (je ) (r , z ) + Tk N k(e ) (r , z ) where function N ie = a ie + bie r + cie z ( i =1, 2, 3 ) is determined by considering the next conditions: N ie (ri , z i ) = 1 N ie (r j , z j ) = N ie (rk , z k ) = 0 . Superscription e refers to the element number (e). Coefficients a, b, and c in shape form functions have the following form: ⎧ai = (r j z k − rk z j ) / 2 S ⎪ ⎨ bi = (z j − z k ) / 2 S ⎪ c = (r − r ) / 2 S k j ⎩ i ⎧a j = (rk z i − rk z i ) / 2 S ⎪ ⎨b j = ( z k − z i ) / 2S ⎪ ⎩c j = (ri − z k ) / 2S ⎧a k = (ri z j − r j z i ) / 2S ⎪ (13) ⎨bk = (z i − z j ) / 2S ⎪ ⎩c k = (r j − ri ) / 2S where S is the area of triangle element. Finite element method is well-established numerical technique for heat conduction problem in orthogonal coordinate system. Heat conductive problem in cylindrical coordinate system (equation (1)) differs from the orthogonal coordinate problem with next integral in equation (11): λ ∫∫ S 1 ∂N j N i drd r ∂r (14) The shape function derivatives in equation (11) are constants: ∂Ni/∂r = bi and ∂Ni/∂z = bi. This gives a simple form of the local matrix for element n (part of global matrix [Kij]) in orthogonal coordinate system (without contribution of integral (14)). With regarding the above values of derivatives and not considering the boundary conditions the matrix can be written in the following form [1]: k (n) ⎡ (bi n ) 2 + (ci n ) 2 ⎢ n n n n = λ ⎢bi b j + ci c j ⎢b n b n + c n c n i k ⎣ i k bi b j + ci c j n n (b j ) 2 + (c j ) 2 n n n n b j bk + c j c k n n n n n n n n bi bk + ci c k ⎤ n n n n⎥ b j bk + c j c k ⎥ n n (bk ) 2 + (c k ) 2 ⎥⎦ (15) Integral (14) is more complicated because there is 1/r member and Ni is in form (6) with coordinates r and z. The integral (14) can be divided in three parts: I = λ ⋅ b j ∫∫ Ni drdz = λ ⋅ b j [ r ai ∫∫ r drdz + ∫∫ b drdz + ∫∫ c i i z drdz ] = λ ⋅ b j [ I1 + I 2 + I 3 ] r ntegral I2 is easy to be calculated because bi is constant and: I2 = bi S, where S is the area of triangle element. Other integrals can be calculated by integrating over triangle area (fig. 5). By using the triangle elements with particular disposition as it is shown in fig.4 and dimensions presented on the fig.5 the integrals I1 and I3 can be presented as: I1 = α i r2 [1 +ν (lnν − 1)] µ 2 r 3 2 r I 3 = [ r1 ln ν ( z 2 + 1 ) − z 2 ∇ r ( r1r2 − r1 − 1 )] µ 4 4 2µ ci (17) I µ= where r ∇r и ν = 2 r1 ∇z z 3 3 3 2 r2= r3 z 1 2 r 1 r2 r1= r3 r 2 r1 (б ) (a) r2 1 r1= r3 (в ) Fig.5 Triangle elements These values must be added to the corresponding members in matrix [Kij]. The cylindrical heat conductivity problem can be solved by approximate methods. It is necessary in case of common disposition and form of triangle elements. One of these methods is to make integration in (14) assuming coordinates r and z as constants and equal to the coordinates of the mass center of the element - rm and zm. In this case approximate solution can be received, and it will be so close to the real solution as much as small are elements. Integral (16) can be presented as: ⎡a z ⎤ I = b j S e ⎢ i + bi + ci m ⎥ rm ⎦ ⎣ rm (18) Other approximate method is determined by rearranging the original equation (1) in form: ∂ ⎛ ∂T ⎞ ∂ ⎛ ∂T ⎞ ⎜ λr ⎟ + ⎜ λr ⎟=0 ∂r ⎝ ∂r ⎠ ∂z ⎝ ∂z ⎠ (19) This form resembles to the form of conductivity equation written in Decart’s coordinates if product λr is interpreted as modified conductivity coefficient. It will vary for different elements and will depend on radius r. For integration in (11) it can be assumed approximate approach, for example to use constant radius rm of the mass center of element. In this case the matrix (15) can be used, but the coefficient λ will be different for each element of the matrix. Boundary conditions (5) contribute the matrix elements by adding values for triangle sides that touch the boundaries of the receiver: surfaces A, B, C, D, E (fig.2). Contribution is determined by integrating the last members in (11). It differs for elements with equal indexes and with different indexes: gii = gjj = gkk = he∆r/3 ; gij = gjk = gik = he∆r/6 ; (20) Values in right column of matrix equation are determined for triangle sides that contact the ambient or exchange elements of receiver. These are formed by integrating the members not containing temperature in last part of (11): Fi = 1 e 1 ⋅ C ⋅ ∇r or Fi = ⋅ C e ⋅ ∇z 2 2 (21) 5. Programming and result A computer program solving the finite element problem has been created and great number of numerical experiments has been carried out. The program comprises two main parts. The first part includes program modules for automatic discretization and numbering the elements and nodes in mesh configuration. These modules are specific for different problems and configuration of domain. The 85 4 85 0 80 0 73 5 80 0 85 0 73 5 73 2 2 73 4 85 main result of performance of program modules is forming the matrix with numbers and coordinates of triangle element nodes. The second part of the program modules uses created in the first module data to form the basic matrix Kij – equation (12). Special procedure for solving the band matrix is used to calculate the requested parameters (temperatures, heat flux etc.). This part is standard and can be used for solving different tasks. The program and mathematical treating of finite element method were verified by using the exact analytical solution of thermal conductivity problem. It is known, that for simple configuration of domain, the heat conductivity problem has analytical solution. Such configuration is a cylindrical domain. For such a simple configuration the finite element method gives results, which a very close to the analytical solution (difference is below 0.1%). Fig. 7 presents the 73 0 0 73 temperature distribution in receiver for dish-Stirling 85 8 8 85 system, calculated by computer program. Some of the used data for boundary conditions are assessed 710 710 approximately. This means that the temperature distribution in receiver must be interpreted only in rough figures. When the conditions for receiver performance are 700 700 defined more precise, the results for temperature distribution will be more accurately. Fig.7 Temperature distribution in Receiver 6. Conclusion Finite element method has been applied for heat transfer problem in receiver for dish-Stirling system. It uses unstructured and complex grids, which are suited to the special forms of absorber element. Finite element method provides complete geometric flexibility. Presented mathematical model and algorithm for program is universal and can be used for different tasks. In this paper are discussed only principle aspects of using the finite element method for heat transfer processes in receiver for solar receivers in concentrating solar systems. Numerical experiments carried out in this research show that this technique gives very good approaches to the real thermal processes. It can be used successfully to investigate different conditions and forms of receivers for concentrating solar systems and other thermal equipments. References: 1. Zienkiewich O.C., K. Morgan, Finite Elements and Approximation, John Wiley & Sons, New York, 1983. 2. Zienkiewich O.C., The Finite Element Method in Engineering Science, McGraw-Hill, London, 1971. 3. Naterer G.F, Heat Transfer in Single and Multiphase Systems, Taylor & Francis, New York, 1997 4. Siegal R, J.R.Howell, Thermal Radiation Heat Transfer, Taylor & Francis, New York, 1998 5. Strang G, G.J. Fox, An Analysis of the Finite Element Method, Prentice-Hall,Inc, Engeleword Cliffs, 1973. 6. Norrie D.H., G de Vires, An Introduction to Finite Element Analysis, Calgary, Alberta, Canada. 7. Kreith F., W.B. Black, Basic Heat Transfer, Harper and Row, Publishers, New York, 1980. 

Lisco: A Continuous Approach in LiDAR Point-cloud Clustering arXiv:1711.01853v1 [] 6 Nov 2017 Hannaneh Najdataei, Yiannis Nikolakopoulos, Vincenzo Gulisano, Marina Papatriantafilou Chalmers University of Technology Gothenburg, Sweden {hannajd,ioaniko,vinmas,ptrianta}@chalmers.se Abstract—The light detection and ranging (LiDAR) technology allows to sense surrounding objects with fine-grained resolution in a large areas. Their data (aka point clouds), generated continuously at very high rates, can provide information to support automated functionality in cyberphysical systems. Clustering of point clouds is a key problem to extract this type of information. Methods for solving the problem in a continuous fashion can facilitate improved processing in e.g. fog architectures, allowing continuous, streaming processing of data close to the sources. We propose Lisco, a singlepass continuous Euclidean-distance-based clustering of LiDAR point clouds, that maximizes the granularity of the data processing pipeline. Besides its algorithmic analysis, we provide a thorough experimental evaluation and highlight its up to 3x improvements and its scalability benefits compared to the baseline, using both real-world datasets as well as synthetic ones to fully explore the worst-cases. Keywords-streaming, clustering, pointcloud, LiDAR I. I NTRODUCTION Active sensors that are able to measure properties of the surrounding environment with very fine-grained time resolution are utilized more and more in cyber-physical systems, such as autonomous vehicles, digitalized automated industrial environments and more. These sensors can produce large streams of readings, with the LiDAR (light detection and ranging) sensor being a prominent example. A LiDAR sensor commonly mounts several lasers on a rotating column; at each rotation step, these lasers shoot light rays and, based on the time the reflected rays take to reach back the sensor, they produce a stream of distance readings at high rates, in the realm of million of readings per second. As common in big data applications, one of the challenges in leveraging the information carried by such large streams is the need for efficient methods that can rapidly distill the valuable information from the raw measurements [5], [8], [17], [28]. A common problem in the analysis of the LiDAR sensor data is clustering of the raw distance measurements, in order to detect objects surrounding the sensor [22]. This can, for instance, enable the detection of surrounding obstacles and prevent accidents (e.g. avoiding pedestrians in autonomous driving) or study the motion feasibility of objects in factories’ production paths [1]. Challenges and contributions The processing time incurred by clustering of raw measurements (aka point clouds) represents one of the main challenges in this context because of the high rates and the need for the clustering outcome to be available in a timely manner for it to be useful. Furthermore, the accuracy of the clustering is challenging as well, since readings from objects that are at different distances from the sensor can vary a lot in density. A key performance enabler for high-rate data streaming analysis is the pipelining of its composing tasks. Nevertheless, common state-of-the art approaches for clustering of LiDAR data (cf [27], [21], [24] and more detail in § VII elaborating on related work) first organize the points to be clustered (e.g. sorting them so that points close in space are also close in the data structure maintaining them) and only then perform the clustering by querying the organized data (e.g. by running neighbor-query as discussed in § II). By doing this, they introduce a batch based processing that affects the clustering performance. To overcome this, we target a single-pass analysis that will enable fine-grained pipelining in processing the data. We propose a new method that achieves LiDAR data-point clustering, called Lisco, that allows to boost processing throughput by maximizing the internal pipelining of the analysis steps, through a key idea that can exploit the inner ordering of the data generated by a LiDAR sensor. In more detail, we make the following contributions: 1) We introduce Lisco, a new algorithm for Euclideandistance-based clustering of LiDAR point clouds, that maximizes the granularity of the data processing pipeline, without the need for supporting sorting data structures. 2) We provide a fully implemented prototype and we discuss Lisco’s complexity in connection to the stateof-the-art Euclidean-distance-based clustering method in the Point Cloud Library (PCL), which we adopt as baseline due to its known accuracy, efficiency and wide use-base [22].1 3) We perform a thorough comparative evaluation, using both real-world datasets as well as synthetic ones to fully explore the worst-cases and the spectrum of trade-offs. We achieve a significant improvement, up to 3 times faster than the baseline and we also show significant scalability benefits. 1 Available through: http://pointclouds.org The rest of the paper is organized as follows. § II overviews the LiDAR sensor, the data it produces and clustering-related techniques that exist for such data. § III presents the main idea, the outline and argues about the properties of the proposed Lisco method, while the algorithmic implementation is given in § IV. We evaluate the proposed method in § VI. Finally, we discuss related work and conclude in § VII and § VIII, respectively. II. P RELIMINARIES In this section, we give details about the key properties of LiDAR sensors and the data they generate. We also provide a detailed problem description and evaluation criteria of solutions. Finally, we briefly describe the Euclidean-distance based clustering method in PCL, which we use as baseline as explained in the introduction. A. LiDAR - sensor and data The light detection and ranging (LiDAR) technology allows sensing surrounding objects with fine-grained resolution in a large areas. The LiDAR sensor mounts L lasers in a column, each measuring the distance from the target by means of the time difference from emitted and reflected light pulses. The column of lasers performs R rotations per second, each consisting of S steps, producing a set of n points, also called point cloud. The number of points reported by the LiDAR sensor for each rotation can be lower than L×S, since some of the emitted pulses might not hit any obstacle. We refer to the angle in the x-y plane between two consecutive steps2 as ∆α (e.g. measured in radians) and to the elevation angle from the x-y plane between two consecutive lasers as ∆θ. Each point p is described through attributes hd, l, si where d, l, s are the measured distance, the laser index and the step index. The measured distance, d, is a value greater than or equal to 0. Value 0 for d shows no reflection in the point. In the following, we use the notation px to refer to attribute x of point p (i.e., pd refers to the distance reported for point p). Let us consider in the following examples that L = 8 and S = 8. We visually present the points by unfolding them as a 2D matrix, where each column contains L rows and each rotation step is a column. We assume that new data is delivered from the physical sensor with the granularity of one rotation step (Figure 1). B. Problem formulation: from point clouds to clusters Given a set of points corresponding to LiDAR measurements, we want to identify disjoint groups of them that can be potential objects in the environment surrounding the sensor. A natural criterion, commonly used in the literature and 2 If the horizontal angle between steps is not constant and lasers are not perfectly aligned in the x-y plane, then ∆α refers to the minimum such angle. Similarly, if the vertical angle between lasers is not constant, ∆θ refers to the minimum such angle. applications, is the distance between points. In particular, adopting the problem definition in [20] (Chapter 4), which we paraphrase here for ease of reference: Definition 1. [Euclidean-distance based clustering] Given n points in 3D space, we seek to partition them into some (unknown) number of clusters C using the Euclidean-distance metric, such that every cluster contains at least a predefined number of points (minP ts), that is ∀j, |Cj | ≥ minP ts, and all clusters are disjoint, that is Ci ∩ Cj = ∅, ∀i 6= j. Two points pi and pj should be clustered together if their Euclidean distance ||pi − pj ||2 is at most , with  being a predefined threshold. To facilitate the presentation of the baseline algorithm and our proposed one, we introduce the -neighborhood of a point p: the set of points of the input whose Euclidean distance from p is at most . A set of points closed under the union of their -neighborhoods is characterized as noise if its cardinality is less than minP ts. It should be noticed that, when clustering data from scenarios like the vehicular one, a pre-processing task is usually defined to filter out points that refer to the ground, since many objects laying on it would be otherwise clustered together. Since ground removal can be implemented as a non expensive and continuous filtering operation [10] (e.g. by removing points below a certain threshold, as we do in § VI) we do not further discuss it in the remainder. C. Euclidean-distance based clustering in PCL PCL [22] provides a set of tools based on a collection of state-of-the-art algorithms to process 3D data, including filtering, clustering, surface reconstruction and more. In this section we review its method for cluster extraction, which is an Euclidean-distance based clustering and we use it as a baseline. For brevity we call this algorithm PCL E Cluster in the rest of this document. PCL E Cluster works on batches of data points. It first builds a kd-tree to facilitate finding the nearest neighbors of points. Subsequently, it proceeds as described by algorithm 1 to extract the clusters. Algorithm 1 Main loop of PCL E Cluster 1: clusters = ∅ 2: for p ∈ P do 3: Q=∅ 4: if p.status 6= processed then 5: Q.add(p) 6: for q ∈ Q do 7: q.status = ’processed’ 8: N = GetNeighbors(q, ) 9: Q.addAll(N ) 10: if size(Q) ≥ minPts then clusters.add(Q) Starting from any arbitrary unprocessed point p, the algorithm adds it in an empty list, Q (line 5). Then, S steps S steps pd Δα p A. Top view Object L lasers Object L lasers ct Obje pd p Δθ B. Side view C. 2D view Figure 1: Top and side views for the LiDAR’s emitted light pulses showing steps and lasers, together with the resulting 2D view. In the 2D view, non-reflected pulses are white while reflected ones are coloured. PCL E Cluster adds all the points of the -neighborhood of each member of Q to it. After processing all members of the list Q, if its size is greater than or equal to minPts, the list is returned a cluster. The algorithm continues with the next unprocessed point, to explore another cluster. The procedure is terminated when all input points have been processed. We discuss the computational complexity of algorithm 1 in § V. III. Lisco We present in this section Lisco. We first discuss the intuition behind the algorithm, i.e. a continuous, singlepass approach in clustering in contrast to existing batch based methods as discussed in § II. Subsequently, we focus on the challenges and trade-offs that continuous clustering introduces. A. Towards continuous clustering Based on the clustering requirements as introduced in Definition 1, each point p reported by the LiDAR sensor is temporarily clustered with each neighbor point p0 within distance  from p. A cluster of points is eventually delivered if it contains at least minPts points, otherwise its points are characterized as noise. As discussed in § II-C, implementations such as the one provided by PCL limit the pipelining of the analysis because of processing stages that cannot be executed concurrently. More concretely, they first traverse all the points to populate a supporting data structure that facilitates finding the points in the -neighborhood of each point p (a kd-tree in the case of PCL) and subsequently they traverse a second time all the points to cluster them. As shown in Figure 2, the intuition behind Lisco is that the search space for the -neighborhood of point p can be translated into a set of readings given by certain steps (Figure 2.A) and lasers (Figure 2.B) around p, and within a certain distance range from the LiDAR’s emitting sensors. It should be noted that these constraints describe a region of space that may also contain points whose distance from p is greater than . Nevertheless, if points within distance  from p exist, they will be returned by one of these steps and lasers and will fall within the given distance range, as we further explain in § V. Thus, in order to discover the -neighborhood of point p, by leveraging the sorted delivering of tuples from the LiDAR sensor step after step, it is enough to explore a neighbor mask centered in p, as shown in Figure 2.C. In this way, we eliminate the need for a search-optimized data structure like kd-trees, and we allow the algorithm to process data as they are received from the sensor. B. Coping with the challenges of continuous clustering The continuous one-pass analysis of Lisco introduces several challenges that are not found in batch based approaches. Here we discuss and explain how they are addressed in Lisco in the following. 1) Partial view of neighbor mask: Algorithm 2 shows how the neighbor mask is computed, i.e. the number of previous and subsequent steps σ and the number of upper and lower lasers λ that possibly contain points within distance  from p. As discussed in § II, ∆α and ∆θ refer to the minimum angle differences between two consecutive steps and lasers, respectively. Algorithm 2 Given point p, compute the number of previous steps σ and upper and lower lasers λ bounding at least all the points within distance  from p. 1: procedure getNeighborMask(p) 2: λ ← d| arcsin(/pd )|/∆θe 3: σ ← d| arcsin(/pd )|/∆αe 4: return λ, σ Algorithm 3 Main loop of Lisco 1: subclusters = ∅ 2: upon event reception of step s do 3: for l ∈ 1, . . . , L do 4: p = M [l, s] 5: if pd > 0 then 6: λ, σ = getNeighborMask(p) 7: cluster(p,σ,λ,subclusters) 8: upon event all steps processed do 9: for subcluster ∈ subclusters do 10: if size(subcluster) ≥ minPts then return subcluster A. Top view B. Side view C. 2D view Distance range containing points within distance ε from p ε p S steps ε p L lasers L lasers S steps p Sphere containing p’s neighbors (within distance ε) Steps that can report points within distance ε from p Lasers that can report points within distance ε from p Neighbor mask covering at least all the points within distance ε from p Figure 2: Top and side views (A and B) showing which steps and lasers, respectively, to include in the neighbor mask (C) for the latter to contain at least all the points within distance  from p. The main loop of Lisco, Algorithm 3, processes points in M , the 2D matrix of input points (described in § II), in step and laser order (Line 4). Each point is processed only if its distance is greater than 0 (that is, if the LiDAR’s pulse has been reflected for the point’s step and laser index) (Line 5). Once all the points have been processed, all the clusters containing at least minPts points are delivered. As it can be noticed, the parameter minPts does not have an effect in the complexity of the algorithm, since it is only used to filter the delivered clusters at the very end of the clustering process. We describe how clusters are discovered and managed within the function cluster in the following. Given points p1 and p2 within distance , p1 ’s neighbor mask will contain p2 and vice versa. To avoid comparing each pair of neighboring points twice, it is enough to consistently traverse half the neighbor mask. Lisco explores the half containing the λ lasers above and below p and the σ steps on p’s left side. This allows for points in p’s step to be processed as soon as they are delivered (points on p’s right side are yet to be delivered upon reception of p’s step points). Take into account that, for a minority of steps, not all the points on p’s left side lay on columns with a lower index than p’s (i.e., they are not stored on the left side of the M matrix). For instance, if a point in column 2 should be compared with 3 columns on the left (λ = 3), then it should be compared with columns S-1, S and 1. In such a case, some comparisons must be postponed until such steps are delivered. A point p0 on the left side of p and within distance  lays on a lower index than p if 0 ≤ ps − p0s ≤ σ. On the other hand, if p0s + σ > S, then p0 is on the left side and within distance  from points in steps 1, . . . , ps + σ − S. In both cases, the clustering semantics defined in § II require p and p0 to be compared, as we do in Algorithm 4. 2) Continuous cluster management: A second challenge brought by the continuous nature of Lisco is that subclusters evolve as more steps arrive. Hence, a cluster identified once all the points in a rotation are processed might be the union of several previously discovered subclusters, as seen in Figure 3. Figure 3.A shows the subclusters found when C1 C x C2 A. 4 out of 8 steps have been received, 2 clusters C1 and C2 have been identified so far B. 8 out of 8 steps have been received, 1 clusters C is eventually found Figure 3: Example showing how the points of two subclusters, that evolve concurrently in Lisco’s continuous analysis, may end up in the same cluster at a subsequent step. half of the points in the rotation have been processed. In the example, 5 points have been clustered in C1 and 4 points have been clustered in C2 . The other non-colored points have not been clustered since they had no neighbors within distance . Figure 3.B shows the clusters found when all the points in the rotation have been processed. At this stage, the points previously clustered in different subclusters are now clustered together. The point marked with x represents the point that has one neighbor in each of the two disjoint subclusters found by Lisco. Once x is processed, these two subclusters should be merged. Based on this observation, we introduce the following informal notions to facilitate the detailed description of Lisco. A subcluster is a set of points that have been clustered together during the processing of the previously received steps of the input. A cluster is a set of at least minPts points that have been clustered together once all the steps of the input have been processed, i.e. a subcluster with cardinality at least minPts is characterized as a cluster after all the steps have been processed. Finally, we consider each subcluster to have a unique identifier called head. Based on the above, one can notice that a subcluster can contain points that previously belonged to two or more subclusters. Because subclusters are found continuously while the points of a rotation are being processed, the clustering algorithm requires methods to: (1) retrieve the head of the points belonging to a subcluster and (2) merge two subclusters together in order for the final cluster to be delivered as a single item. In order to do this, we use in Algorithm 4 (overviewing the clustering process applied to each incoming point p) the functions Head H = getH(Point p) and merge(Head H1,Head H2). Once two subclusters are merged invoking function merge, we expect the function getH to return the same head for any point of the two subclusters. Without loss of generality, we assume this head to be H1 in the following. Because of this we define a third function setH(point p,Head H). Finally, we define the function createH() to allow for newly discovered subclusters to be instantiated. Algorithm 4 Given point p, cluster it together with all the points already received from the LiDAR sensor that are within distance  from it. 1: procedure cluster(p,λ,σ,subclusters) 2: for p0 |(0 ≤ ps − p0s ≤ σ ∨ 1 ≤ p0s ≤ ps + σ − S) ∧ 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: |pl − p0l | ≤ λ ∧ |pd − p0d | ≤  do H1 =getH(p) H2 =getH(p’) if H1 6= H2 ∧ ||p − p0 ||2 ≤  then if H1 = ∅ ∧ H2 = ∅ then H = createH() setH(p,H) setH(p0 ,H) subclusters.add(H) else if H1 = ∅ ∧ H2 6= ∅ then setH(p,H2 ) else if H1 6= ∅ ∧ H2 = ∅ then setH(p0 ,H1 ) else subclusters.remove(H2 ) merge(H1 ,H2 )) As shown in Algorithm 4, four different cases should be checked for two points within distance  that do not belong to the same subcluster: • Line 6: None of the two points belongs to a subcluster. In this case, a new subcluster head is created and set for both points • Line 11: Point p does not belong to a subcluster while point p0 does. In such a case, point p will refer to the same head as point p0 . 0 • Line 13: Point p belongs to a subcluster while point p 0 does not. In such a case, point p will refer to the same head as point p. 0 • Line 15: Both points p and p have been clustered but to different subclusters. In such a case these two subclusters are merged together. IV. A LGORITHMIC IMPLEMENTATION We discuss the details of the algorithmic implementation of Lisco in this section. Data points are kept in a 2D matrix, M . The number of rows and columns in the matrix is equal to the number of lasers and steps respectively. Upon reception of a column of points from LiDAR, which contains the reflected points of all lasers in one step, we store them in the corresponding column of the matrix. By using the laser and the step number, all the attributes of a point can be extracted in constant time. Each entry in M holds the attributes of the corresponding point and a pointer (initially set to NULL) to the head of its subcluster. The head of a subcluster is defined as the point with the lowest indices in lexicographical order of steps and lasers during the creation of a new subcluster. When two subclusters are merged, the head from the subcluster with the largest number of members is maintained. Subclusters is a hash map used in order to keep track of subclusters and their corresponding members. It is implemented as a linked list of arrays, where each key is the header of a subcluster and its members are stored in the array. If the size of a subcluster exceeds the size of the array, a new array is linked to the tail of the current array, so that subclusters can grow without restriction. At the end of the clustering procedure, we use subclusters to traverse through subcluster heads. Each subcluster that has more than minPts members is announced as a cluster, otherwise it is characterized as noise. To keep of Lisco’s time complexity low (discussed in § V) we aim at efficient time complexity of the main methods used in our algorithms. As shown in Algorithm 2, function getNeighborMask is executed in constant number of steps, since it boils down to a fixed number of numerical operations. Similarly, functions createH and setH can be also implemented to incur in O(1) complexity, as we discuss in the following. The algorithmic implementation of getH and merge induces the following trade-off. On the one hand, merge can be implemented to induce O(1) time-cost; this can be done by maintaining a hierarchy of subclusters being part of the same subcluster while incurring a higher cost for the getH, linear in the number of subclusters. Figure 4 shows how some of the points clustered together once all the data is processed point to head H1 via head H2 . For these points the getH method has a cost higher than that of the points directly pointing to H1 , which depends on the chains induced by the data structure to maintain the hierarchy. In the proposed implementation we opt for an O(1) cost for method getH and a higher cost for merge, as seen in Figure 5 and § V. The reason, as can be seen in Algorithm 4 and based also on our empirical evaluation in § VI, is that getH is executed twice for pairs of points being compared, while merge is executed significantly less often. In detail, the implementations of the functions used in Algorithm 4 are as follows: - createH: This function gets two points that do not belong to any subcluster, and returns the one with the lower indexpair for step and laser. Since the returned point will be head of a subcluster, a new node is created in the subclusters H1 H1 x H2 H2 A. The subclusters have different heads B. After merging, one subcluster’s head (H1) points to the other (H2) Figure 4: Possible implementation in which the merge method is made O(1) by hierarchically linking heads of subclusters belonging to the same subcluster. Notice that the complexity of getH is no longer O(1) but linear in the number of subclusters for some of the points. H1 H1 x H2 A. The subclusters have different heads B. After merging, all the points are directly linked to the head Figure 5: Possible implementation in which the getH method can run in O(1) steps by having all points directly linked with the subcluster head. Notice the head of all the points of subcluster C2 has been updated when the subcluster has been merged with subcluster C1 . and the head is mapped to it. - setH: This function sets the pointer of a point to the head point. By calling this function, we are adding a point to a subcluster with an identified head. The point also needs to be added as the last element in the array of the mapped head. - getH: This function reads the pointer to get the head point. If two points are in the same subcluster, they get the same head point as the result of this function. - merge: If two points belong to different subclusters, Lisco merges the two subclusters by calling this function. It chooses the subcluster with bigger number of members as the base subcluster, and merges the other one by changing the head of its members to the base subcluster’s head. After merging subclusters, it is also necessary to remove the merged subcluster’s head from the subclusters hash map and append its array to the base subcluster’s array. V. A NALYSIS A. Correctness Based on Lisco’s functionality and algorithmic implementation, we discuss here why Lisco’s outcome satisfies Definition 1. Claim 1. If two points p and p0 are in the -neighborhood of each other they will either be in the same cluster at the end of Lisco procedure or be characterized as noise, in the same way as given in Definition 1. Proof: (sketch) Consider that, w.l.o.g. p is processed second. As argued in paragraph getNeighborMask in the previous section, p0 will be found that it belongs in the -neighborhood of p. This implies that they will be merged/inserted in the same subcluster. Unless that subcluster in the end is found to contain fewer points than M inP ts, it will be return as a final cluster in the end of the main loop of Lisco. B. Complexity In this section, we discuss the complexity analysis of Lisco and compare it with PCL E Cluster. Regarding PCL E Cluster, the required processing work volume is similar to the DBSCAN algorithm, i.e. building a spatial index (kd-tree) and using it to execute region queries for each point, resulting in an overall expected time complexity of O(nlogn) processing steps [4], [19], [25]. Claim 2. Lisco’s time complexity is linear in the number of points, multiplied by a factor that depends on the size of the clusters in the set of data points. In the worst-case where there is a big cluster of O(n) points, it can take O(nlogn) processing steps for Lisco to complete. Proof: (sketch) Overall, the time complexity of Lisco is the number of iterations in the main loop (i.e. n, as the number of points), times the work in each iteration, i.e. for each point (i) finding its -neighborhood, and (ii) working with each point in the neighborhood. Part (i) from above, induces an asymptotically constant cost, depending on , as it is performed through the comparisons implied by the masking operation getNeighborMask. Part of the -neighborhood of a point p, i.e. the points with smaller step index, is compared with p through step 2 of Algorithm 4 on behalf of p, while for each of the remaining points p0 in p’s -neighborhood, p will be identified as part of the -neighborhood of p0 when the respective step is executed on behalf of p0 . Regarding part (ii), functions’ createH and setH algorithmic implementation incurs a constant number of processing steps each, as we explain in § IV. Moreover, as explained in the aforementioned section, getH induces O(1) timecost, as a point can identify its head in O(1) (e.g. with a direct link). This incurs a cost that is O(x) for merge, where x is the size of the smaller subcluster, since the merging itself needs to update the head for all the points of the smaller one of the subclusters being merged. Since the merge function chooses the subcluster with the bigger number of points as the base subcluster, in the worstcase the clustering has a huge subcluster of O(n) points, and an unlikely scenario for constructing it, might require Lisco to merge roughly equally-sized subclusters at each of the merge operations leading to the big subcluster – any other combination of subclusters would lead at most to an equal cost as the one described. Since halving O(n) points can be made at most O(logn) times, we can observe that the worst-case total number of merge-related processing steps will be dominated by a sequence of O(nlogn) steps, which will be the dominating cost in the worst case complexity of Lisco. VI. E VALUATION In this section, we present our experimental methodology and results of Lisco and compare them with those of PCL E Cluster algorithm. Since the clustering outcomes of PCL E Cluster and Lisco are the same, we do not need to compare the clusters and we can focus on the completion time for each approach. points (including NULL points and ground points) is the same and it is equal to 72000. After removing the ground points and eliminating NULL points, we get a number of reflected points. As shown in the table, with the same number of objects in the environment (e.g. SCEN1 and SCEN2), if we change the position of objects to near or far, we get different numbers of reflected points. Name SCEN1 SCEN2 SCEN3 SCEN4 SCEN5 # Points after removing NULL points and ground points 26891 16218 39028 18229 64518 Table I: Properties of synthetic datasets and the effect of number of objects and their distances from LiDAR on number of points after ground removal. C. Performance evaluation A. Evaluation setup we use the To run PCL E Cluster EuclideanClusterExtraction class from PCL library, which is designed to cluster 3D point clouds and is implemented in C++. We also implemented Lisco in C++11 and compiled both of algorithms with gcc-4.8.4 using the -O3 optimization flag. All the experiments have been run on the same system running Linux with 2.00GHz Intel(R) Xeon(R) E5-2650 processor and 64GB Ram. B. Data We used both synthetic and real-world datasets. The realworld dataset has been collected from the Ford Campus dataset [18] and the synthetic ones have been generated using the Webots simulator [15]. We use synthetic datasets to explore the effect of data (e.g. different total number of points that have been collected by the LiDAR, different densities and distances of objects) on the performance of algorithms. There are five scenarios for synthetic datasets. SCEN1 and SCEN2 have the same and few number of objects but we changed the position of the objects to near and far from LiDAR. Near objects reflect more points while far objects reflect fewer points with a larger gap between two nearby points. Similarly, SCEN2 and SCEN3 have the same number of objects (number of objects are more than previous scenarios) and we only changed the position of the objects. Finally, SCEN5 represents a high density environment with a lot of objects. Figure 6 shows five simulated environments for several scenarios. In all the environments, a VelodyneHDL64E is used to collect data points and generate a dataset within one physical rotation. Table I summarizes the properties of the synthetic datasets. Since we used the same specifications for the LiDAR in all scenarios, the number of steps and lasers for one physical rotation is the same, so the total number of The execution time is measured from the time-instant the first data point of the dataset is received until the time instant when the clustering algorithm has processed all data points of one full physical rotation. A higher value of  implies a larger -neighborhood of a point, hence the clustering algorithm needs more time to search in the neighborhood. 1) Synthetic datasets: Figure 7 shows the average execution time with confidence level 99% on 20 runs with different values of  and constant value of minPts = 10. Since the maximum margin of error for a confidence level 99% is small, we can not distinguish them clearly in the figure. We chose a range [0.1 − 1] meters for , so that for example if  = 0.4, all the objects that their closest points have at least 40 centimetres distance from each other, should be detected as separated objects. While clustering with smaller values of  find at least one cluster for each object, bigger values increase the probability of clustering distinct objects together. For example, with  = 1 for SCEN3, all the cars at each side of the black car, are clustered together which leads to incorrect segmentation. As expected, by increasing the value of the , the execution time increases for both algorithms. As it can be seen, when the number of points is high, regardless of the value of the , Lisco is always faster than PCL. Only for a dataset with a relatively small number of points and when  is set to a high value PCL has slightly better performance than Lisco (figure 7 SCEN2 and SCEN4). The effect of number of points is also shown in figure 8. As discussed in § V building a kd-tree and using it to find nearest neighbors in PCL becomes a bottleneck when the number of points is high. 2) Real-world dataset: This Ford Campus dataset is collected by an autonomous ground vehicle testbed, with a Velodyne HDL-64E LiDAR scanner [18]. The vehicle path trajectory in this dataset contains several large and small objects (e.g. buildings, vehicles, pedestrians, vegetation, (a) SCEN1 - The simulated sparse environment in which objects are close to the LiDAR which is located on the black car (b) SCEN2 - The simulated sparse environment in which objects are far from the LiDAR which is located on the black car (c) SCEN3 - The simulated dense environment in which objects are close to the LiDAR which is located on the black car (d) SCEN4 - The simulated dense environment in which objects are far from LiDAR which is located on the black car (e) SCEN5 - The simulated room for high density environment. The LiDAR is located on the purple column Figure 6: Different scenarios for simulated environments. SCEN1 Execution Time (sec) 0.8 SCEN2 0.2 0.6 SCEN3 4 0.15 SCEN4 0.25 0.2 3 10 0.15 0.4 0.1 2 0.1 0.2 0.05 0 1 0 0 0.5 1 SCEN5 15 0 0 0.5 (m) 1 5 0.05 0 0 (m) 0.5 1 0 0 0.5 (m) PCL 1 0 (m) 0.5 1 (m) Lisco Figure 7: The average execution time on synthetic datasets with confidence level 99% over 20 runs for minPts = 10 and different values of  1.2 PCL Lisco Execution Time (sec) Execution Time (sec) 8 6 4 2 0 2 4 #points 6 104 Figure 8: Scalability of Lisco and PCL with respect to the number of points etc.). We have tested PCL E Cluster and Lisco on 2280 rotations of this dataset and compare their execution times with confidence level 99%. Figure 9 shows the results of the comparison for  values 0.3, 0.4, and 0.7. Among all the rotations, the minimum number of reflected points after ground removal is 5000, the maximum is 75550, and the average is 50225. As shown in the figure, Lisco outperforms PCL E Cluster in real-world PCL Lisco 1 0.8 0.6 0.4 0.2 0 0.3 0.4 (m) 0.7 Figure 9: The average execution time of running PCL and Lisco over 2280 rotations of real-world dataset for minPts = 10 and different values of  datasets. In real-world data, generally there are more objects around the LiDAR and therefore there are more reflected points besides the ground. Since Lisco processes points upon receiving them from LiDAR, it can save more time and it has thus better throughput. VII. OTHER RELATED WORK Data clustering has been studied for several decades and existing algorithms have been categorized into four classes: density-based, partition-based, hierarchy-based, and grid based [9]. Due to their ability in finding arbitrarily shaped clusters without requiring to know the number of the clusters a priori, density-based methods are widely used in different applications. Well-known algorithms of this class include DBSCAN [4] and OPTICS [2]. However, to use these algorithms in big data applications and overcome their performance bottleneck in dealing with extremly large datasets, there are several attempts to parallelize DBSCAN [13], [19]. In parallel models, the clustering procedure is divided into three steps: 1) data distribution (e.g. using kdtree) 2) local clustering (which is splitted on several machines) 3) merging of local clusters. Although the efficiency is improved by splitting the clustering on several machines, pipelining of steps has not being studied yet. Rusu et al. [21] introduce an Euclidean-distance-based clustering which is a partition-based clustering method that produces arbitrarily shaped clusters. This approach is designed for unorganized data points. So, to facilitate searching for nearest neighbors, first a kd-tree is built over the dataset and then clustering is being performed. In other works [24], [27], an octree is used to identify the neighbors before starting the clustering procedure. Since LiDAR data points are implicitly ordered, organizing them (e.g. in a tree) may be avoided, similar to the spirit of this paper. Specifically, the characteristics of the sensor data can be used to establish neighborhood relations between points [12], [16]. Klasing [12] et al. proposed a clustering method for 2D laser scans that rotate with an independent motor to cover a 3D environment. While the proposed method compares points across different scans similarly to the problem studied in this work, the semantics of Definition 1 are not enforced resulting to a lower accuracy. Moosmann et al. [16] proposed an approach to turn the scan into an undirected graph to retrieve the neighborhood information of each point during clustering, but they have not studied pipelining building the graph and clustering. Zermas et al. [29] recently proposed a clustering method specific to the structure of LiDAR data points. This approach processes one scan-line (a layer (rotation) of points that are produced from the same laser) at a time and merges nearby clusters from different scan-lines. However, the entire rotation is needed as the algorithm does more than one pass over the data. Also, this approach, similarly to previous works, still relies on a kd-tree for some necessary nearest neighbor searches. Moreover, the neighborhood criterion for points clustered in the same scan-line does not take into account the distance of the point from the sensor, and thus does not guarantee the semantics of Definition 1. Clustering LiDAR data points are being used in wide range of applications [11], [14], [23]. Among all, autonomous vehicle applications are one of the most challenging since they need fast and accurate results [3], [10], [26], [29]. In [3], a set of voxelisation and meshing segmentation methods are presented. Wang et al. [26] first separates data into foreground and background. then a clustering procedure is conducted only on the foreground segments. VIII. C ONCLUSIONS AND F UTURE W ORK This work is about one of the challenges in common big data applications, namely leveraging the information carried by high-rate streams through efficient methods that can rapidly distill the valuable information from the raw measurements. A common problem in the analysis of LiDAR sensor data, that generate date at rates of megabytes per second, is clustering of the raw distance measurements, in order to facilitate detection of objects surrounding the sensor. Lisco represents a streaming approach to process the LiDAR points while the data is being collected. This characteristic helps to facilitate extraction of clusters in a continuous fashion and contribute to real-time processing. 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Hardware for Machine Learning: Challenges and Opportunities (Invited Paper) Vivienne Sze, Yu-Hsin Chen, Joel Emer, Amr Suleiman, Zhengdong Zhang arXiv:1612.07625v5 [] 17 Oct 2017 Massachusetts Institute of Technology Cambridge, MA 02139 Abstract—Machine learning plays a critical role in extracting meaningful information out of the zetabytes of sensor data collected every day. For some applications, the goal is to analyze and understand the data to identify trends (e.g., surveillance, portable/wearable electronics); in other applications, the goal is to take immediate action based the data (e.g., robotics/drones, self-driving cars, smart Internet of Things). For many of these applications, local embedded processing near the sensor is preferred over the cloud due to privacy or latency concerns, or limitations in the communication bandwidth. However, at the sensor there are often stringent constraints on energy consumption and cost in addition to throughput and accuracy requirements. Furthermore, flexibility is often required such that the processing can be adapted for different applications or environments (e.g., update the weights and model in the classifier). In many applications, machine learning often involves transforming the input data into a higher dimensional space, which, along with programmable weights, increases data movement and consequently energy consumption. In this paper, we will discuss how these challenges can be addressed at various levels of hardware design ranging from architecture, hardware-friendly algorithms, mixed-signal circuits, and advanced technologies (including memories and sensors). I. I NTRODUCTION This is the era of big data. More data has been created in the past two years than the entire history of the human race [1]. This is primarily driven by the exponential increase in the use of sensors (10 billion per year in 2013, expected to reach 1 trillion by 2020 [2]) and connected devices (6.4 billion in 2016, expected to reach 20.8 billion by 2020 [3]). These sensors and devices generate hundreds of zetabytes (1021 bytes) of data per year — petabytes (1015 bytes) per second [4]. Machine learning is needed to extract meaningful, and ideally actionable, information from this data. A significant amount of computation is required to analyze this data, which often happens in the cloud. However, given the sheer volume and rate at which data is being generated, and the high energy cost of communication and often limited bandwidth, there is an increasing need to perform the analysis locally near the sensor rather than sending the raw data to the cloud. Embedding machine learning at the edge also addresses important concerns related to privacy, latency and security. II. A PPLICATIONS Many applications can benefit from embedded machine learning ranging from multimedia to medical space. We will provide a few examples of areas that researchers have Machine Learning (Inference) Fig. 1. Dog (0.7) Cat (0.1) Bike (0.02) Car (0.02) Plane (0.02) House (0.04) Image classification. investigated; however, this paper will primarily focus on computer vision, specifically image classification, as a driving example. A. Computer Vision Video is arguably the biggest of the big data. It accounts for over 70% of today’s Internet traffic [5]. For instance, over 800 million hours of video is collected daily worldwide for video surveillance [6]. In many applications (e.g., measuring wait times in stores, traffic patterns), it would be desirable to use computer vision to extract the meaningful information from the video right at the image sensor rather than in the cloud to reduce the communication cost. For other applications such as autonomous vehicles, drone navigation and robotics, local processing is desired since the latency and security risk of relying on the cloud are too high. However, video involves a large amount of data, which is computationally complex to process; thus, low cost hardware to analyze video is challenging yet critical to enabling these applications. In computer vision, there are many different artificial intelligence (AI) tasks [7]. In this paper, we focus on image classification (Fig. 1), where the entire image is provided and the task is to determine which class of objects is in the image. B. Speech Recognition Speech recognition has significantly improved our ability to interact with electronic devices, such as smartphones. While currently most of the processing for applications such as Apple Siri and Amazon Alexa voice services is in the cloud, it is desirable to perform the recognition on the device itself to reduce latency and dependence on connectivity, and to increase privacy. Speech recognition is the first step before many other AI tasks such as machine translation, natural language processing, etc. Low power hardware for speech recognition is explored in [8, 9]. Image Trained weights (w) pixels Feature Extraction Handcrafted Features (e.g. HOG) Features (x) Classification (wTx) Scores Scores per class (select class based on max or threshold) Learned Features (e.g. DNN) extraction was designed through a hand-crafted process by experts in the field. For instance, for object recognition in computer vision, it was observed that humans are sensitive to edges (i.e., gradients) in an image. As a result, many wellknown computer vision algorithms use image gradient-based features such as Histogram of Oriented Gradients (HOG) [13] and Scale Invariant Feature Transform (SIFT) [14]. The challenge in designing these features is to make them robust to variations in illumination and noise. B. Classification Fig. 2. Inference pipeline. C. Medical There is a strong clinical need to be able to monitor patients and to collect long-term data to help either detect/diagnose various diseases or monitor treatment. For instance, constant monitoring of ECG or EEG signals would be helpful in identifying cardiovascular diseases or detecting the onset of a seizure for epilepsy patients, respectively. In many cases, the devices are either wearable or implantable, and thus the energy consumption must be kept to a minimum. Using embedded machine learning to extract meaningful physiological signal and process it locally is explored in [10–12]. III. M ACHINE L EARNING BASICS Machine learning is a form of artificial intelligence (AI) that can perform a task without being specifically programmed. Instead, it learns from previous examples of the given task during a process called training. After learning, the task is performed on new data through a process called inference. Machine learning is particularly useful for applications where the data is difficult to model analytically. Training involves learning a set of weights from a dataset. When the data is labelled, it is referred to as supervised learning, which is currently the most widely-used approach. Inference involves performing a given task using the learned weights (e.g., classify an object in an image)1 . In many cases, training is done in the cloud. Inference can also happen in the cloud; however, as previously discussed, for certain applications this is not desirable from the standpoint of communication, latency and privacy. Instead it is preferred that the inference occur locally on a device near the sensor. In these cases, the trained weights are downloaded from the cloud and stored in the device. Thus, the device needs be programmable in order to support a reasonable range of tasks. A typical machine learning pipeline for inference can be broken down into two steps as shown in Fig. 2: Feature Extraction and Classification. Approaches such as deep neural networks (DNN) blur the distinction between these steps. A. Feature Extraction Feature extraction is used to transform the raw data into meaningful inputs for the given task. Traditionally, feature 1 Machine learning can be used in a discriminative or generative manner. This paper focuses on the discriminative use. The output of feature extraction is represented by a vector, which is mapped to a score using a classifier. Depending on the application, the score is either compared to a threshold to determine if an object is present, or compared to the other scores to determine the object class. Techniques often used for classification include linear methods such as support vector machine (SVM) [15] and Softmax, and non-linear methods such as kernel-SVM [15] and Adaboost [16]. In many of these classifiers, the computation of the score is effectivelyPa dot product of the features (~x) and the weights (w) ~ (i.e., i wi xi ). As a result, much of the hardware research has been focused on reducing the cost of a multiply and accumulate (MAC). C. Deep Neural Networks (DNN) Rather than using hand-crafted features, the features can be learned directly from the data, similar to the weights in the classifier, such that the entire system is trained end-to-end. These learned features are used in a popular form of machine learning called deep neural networks (DNN), also known as deep learning [17]. DNN delivers higher accuracy than handcrafted features on a variety of tasks [18] by mapping inputs to a high-dimensional space; however, it comes at the cost of high computational complexity. There are many forms of DNN (e.g., convolutional neural networks, recurrent neural networks, etc.). For computer vision applications, DNNs are composed of multiple convolutional (CONV) layers [19] as shown in Fig. 3. With each layer, a higher-level abstraction of the input data, called a feature map, is extracted to preserve essential yet unique information. Modern DNNs are able to achieve superior performance by employing a very deep hierarchy of layers. Fig. 4 shows an example of a convolution in DNNs. The 3-D inputs to each CONV layer are 2-D feature maps (W × H) with multiple channels (C). For the first layer, the input would be the 2-D image itself with three channels, typically the red, green and blue color channels. Multiple 3-D filters (M filters with dimension R × S × C) are then convolved with the input feature maps, and each filter generates a channel in the output 3-D feature map (E × F with M channels). The same set of M filters is applied to a batch of N input feature maps. Thus there are N input feature maps and N output feature maps. In addition, a 1-D bias is added to the filtered result. The output of the final CONV layer is processed by fullyconnected (FC) layers for classification purposes. In FC layers, Modern Deep CNN: 5 – 1000 Layers CONV Layer Low-Level Features convolu'on … CONV Layer non-linearity 1 – 3 Layers High-Level FC Features Layer normaliza'on Classes pooling × Fig. 3. Deep Neural Networks are composed of several convolutional layers followed by fully connected layers. C AlexNet 16.4 5 2.3M 666M 3 58.6M 58.6M 61M 724M VGG 16 7.4 16 14.7M 15.3G 3 124M 124M 138M 15.5G GoogLeNet (v1) 6.7 21 6.0M 1.43G 1 1M 1M 7M 1.43G ResNet 50 5.3 49 23.5M 3.86G 1 2M 2M 25.5M 3.9G ImageNet M H 1 E 1 S 1 … F … … W M C C R Accuracy CONV Layers Weights MACs FC Layers Weights MACs Total Weights Total MACs LeNet 5 n/a 2 2.6k 283k 2 58k 58k 60k 341k Output fmaps C R Metrics MNIST Input fmaps Filters TABLE I S UMMARY OF POPULAR DNN S [20, 21, 24, 26, 27]. ACCURACY MEASURED BASED ON T OP -5 ERROR ON I MAGE N ET [18]. M E H S N N F W Fig. 4. Computation of a convolution in DNN. the filter and input feature map are the same size, so that there is a different weight for each input pixel. The number of FC layers has been reduced from three to one in most recent DNNs [20, 21]. In between CONV and FC layers, additional layers can be optionally added, such as the pooling and normalization layers [22]. Each of the CONV and FC layers is also immediately followed by an activation layer, such as a rectified linear unit (ReLU) [23]. Convolutions account for over 90% of the run-time and energy consumption in DNNs. Table I compares modern DNNs, with a popular neural net from the 1990s, LeNet-5 [24], in terms of number layers (depth), number of filters weights, and number of operations (i.e., MACs). Today’s DNNs are several orders of magnitude larger in terms of compute and storage. A more detailed discussion on DNNs can be found in [25]. D. Complexity versus Difficulty of Task It is important to factor in the difficulty of the task when comparing different machine learning methods. For instance, the task of classifying handwritten digits from the MNIST dataset [28] is much simpler than classifying an object into one of a 1000 classes as is required for the ImageNet dataset [18](Fig. 5). It is expected that the size of the classifier or network (i.e., number of weights) and the number of MACs will be larger for the more difficult task than the simpler task and thus require more energy. For instance, LeNet-5[24] is Fig. 5. MNIST (10 classes, 60k training, 10k testing) [28] vs. ImageNet (1000 classes, 1.3M training, 100k testing)[18] dataset. designed for digit classification, while AlexNet[26], VGG16[27], GoogLeNet[20], and ResNet[21] are designed for the 1000 class image classification. IV. C HALLENGES The key metrics for embedded machine learning are accuracy, energy consumption, throughput/latency, and cost. The accuracy of the machine learning algorithm should be measured on a sufficiently large dataset. There are many widelyused publicly-available datasets that researcher can use (e.g., ImageNet). Programmability is important since the weights need to be updated when the environment or application changes. In the case of DNNs, the processor must also be able to support different networks with varying number of layers, filters, channels and filter sizes. The high dimensionality and need for programmability both result in an increase in computation and data movement. Higher dimensionality increases the amount of data generated and programmability means that the weights also need be read and stored. This poses a challenge for energy-efficiency since data movement costs more than computation [29]. In this paper, we will discuss various methods that reduce data movement to minimize energy consumption. The throughput is dictated by the amount of computation, which also increases with the dimensionality of the data. In this paper, we will discuss various methods that the data can be transformed to reduce the number of required operations. The cost is dictated by the amount of storage required on the chip. In this paper, we will discuss various methods to reduce storage costs such that the area of the chip is reduced, while maintaining low off-chip memory bandwidth. Temporal Architecture (SIMD/SIMT) Memory Hierarchy Spatial Architecture (Dataflow Processing) Accelerator Off-Chip DRAM Memory Hierarchy Register File ALU ALU ALU ALU ALU ALU ALU ALU ALU ALU ALU ALU ALU ALU ALU ALU ALU ALU ALU ALU ALU ALU ALU ALU PE PE PE ALU PE ALU Buffer ALU ALU ALU ALU ALU RF ALU 2× 1× 1× (Reference) ALU Fig. 7. Fig. 6. 6× ALU ALU Control ALU 200× ALU PE ALU = Data Movement Energy Cost DRAM ALU Processing Engine Global Buffer Memory hierarchy and data movement energy [34]. Highly-parallel compute paradigms. Finally, training requires a significant amount of labeled data (particularly for DNNs) as well as computation for multiple iterations of back-propagation to determine the value of the weights. There is on-going research on training in the cloud using CPUs, GPUs, FPGAs and ASICs. However, this is beyond the scope of this paper. Currently, state-of-the-art DNNs consume orders of magnitude higher energy than other forms of embedded processing (e.g., video compression). We must exploit opportunities at multiple levels of hardware design to address all these challenges and close this energy gap. V. O PPORTUNITIES IN A RCHITECTURES The MAC operations in both the feature extraction (CONV layer in a DNN) and classification (for both DNN and handcrafted features) can be easily parallelized. Two common highlyparallel compute paradigms are shown in Fig. 6 with multiple arithmetic logic units (ALU). A. CPU and GPU Platforms CPUs and GPUs use temporal architectures such as SIMD or SIMT to perform the MACs in parallel. All the ALUs share the same control and memory (register file). On these platforms, all classifications are represented by a matrix multiplication. The CONV layer in a DNN can also be mapped to a matrix multiplication using the Toeplitz matrix. There are software libraries designed for CPUs (e.g., OpenBLAS, Intel MKL, etc.) and GPUs (e.g., cuBLAS, cuDNN, etc.) that optimize for matrix multiplications. The matrix multiplication is tiled to the storage hierarchy of these platforms, which are on the order of a few megabytes at the higher levels. The matrix multiplications on these platforms can be further sped up by applying transforms to the data to reduce the number of multiplications. Fast Fourier Transform (FFT) [30, 31] is a well known approach that reduces the number of multiplications from O(No2 Nf2 ) to O(No2 log2 No ), where the output size is No × No and the filter size is Nf × Nf ; however, the benefits of FFTs decrease with filter size. Other approaches include Strassen [32] and Winograd [33], which rearrange the computation such that the number of multiplications scale from O(N 3 ) to O(N 2.807 ) and 2.25× for a 3 × 3 filter, respectively, at the cost of reduced numerical stability, increased storage requirements, and specialized processing depending on the size of the filter. B. Accelerators Accelerators provide an opportunity to optimize the data movement (i.e., dataflow) in order to minimize accesses from the expensive levels of the memory hierarchy as shown in Fig. 7. In particular, for DNNs we investigate dataflows that exploit three forms of data reuse (convolutional, filter and image). We use a spatial architecture (Fig. 6) with local memory (register file) at each ALU processing element (PE) on the order of 0.5 – 1.0kB and a shared memory (global buffer) on the order of 100 – 500kB. The global buffer communicates with the off-chip memory (e.g., DRAM). Data movement is allowed between the PEs using an on-chip network (NoC) to reduce accesses to the global buffer and the off-chip memory. Three types of data movement include input pixels, filter weights and partial sums (i.e., the product of pixels and weights) that are accumulated for the output. Recent work [35–46] has proposed solutions for DNN acceleration, but it is difficult to compare their performance directly due to differences in implementation and design choices. The following taxonomy (Fig. 8) can be used to classify these existing DNN dataflows based on their data handling characteristics [34]: • Weight stationary (WS): The weights are stored in the register file at the PE and remains stationary to minimized the movement cost of the weights (Fig. 8(a)). The inputs and partial sums must move through the spatial array and global buffer. Examples are found in [35–40]. • Output stationary (OS): The outputs are stored in the register file at the PE and remains stationary to minimized the movement cost of the partial sums (Fig. 8(b)). The inputs and weights must move through the spatial array and global buffer. Examples are found in [41–43]. • No local reuse (NLR): While small register files are efficient in terms of energy (pJ/bit), they are inefficient in terms area (µm2 /bit). In order to maximize the storage capacity, and minimize the off-chip memory bandwidth, no local storage is allocated to the PE and instead all 2 1.5 psums Normalized Energy/MAC (a) Weight Stationary 1 weights pixels 0.5 0 WS OSA OSB OSC NLR RS (a) Across types of data (b) Output Stationary 2 ALU 1.5 Normalized Energy/MAC RF 1 NoC buffer 0.5 DRAM (c) No Local Reuse 0 Fig. 8. Dataflows for DNNs. Row 1 Row 2 * Row 1 Row 2 * Row 2 Row 3 * Row 3 PE 4 Row 1 * Row 2 Row 2 * Row 3 Row 3 * Row 4 PE 2 * = Fig. 9. PE 7 Row 1 * Row 3 Row 2 * Row 4 Row 3 * Row 5 PE 5 PE 3 PE 8 PE 6 * = PE 9 * Row Stationary Dataflow [34]. OSA OSB OSC NLR RS (b) Across levels of memory hierarchy Row 3 PE 1 Row 1 WS = Fig. 10. Energy breakdown of dataflows [34]. than the other dataflows for the convolutional layers. This is due to the fact that the energy of all types of data is reduced. Furthermore, both the on-chip and off-chip energy is considered. VI. O PPORTUNITIES IN J OINT A LGORITHM AND H ARDWARE D ESIGN There is on-going research on modifying the machine learning algorithms to make them more hardware-friendly while maintaining accuracy; specifically, the focus is on reducing computation, data movement and storage requirements. that area is allocated to the global buffer to increase its capacity (Fig. 8(c)). The trade-off is that there will be increased traffic on the spatial array and to the global A. Reduce Precision buffer for all data types. Examples are found in [44–46]. The default size for programmable platforms such as • Row stationary (RS): In order to increase reuse of CPUs and GPUs is often 32 or 64 bits with floating-point all types of data (weights, pixels, partial sums), a row representation. While this remains the case for training, during stationary approach is proposed in [34]. A row of the filter inference, it is possible to use a fixed-point representation and convolution remains stationary within a PE to exploit substantially reduce the bitwidth for energy and area savings, 1-D convolutional reuse within the PE. Multiple 1-D and increase in throughput. Retraining is typically required to rows are combined in the spatial array to exhaustively maintain accuracy when pushing the weights and features to exploit all convolutional reuse (Fig. 9), which reduces lower bitwidth. accesses to the global buffer. Multiple 1-D rows from In hand-crafted approaches, the bitwidth can be drastically different channels and filters are mapped to each PE to reduced to below 16-bits without impacting the accuracy. For reduce partial sum data movement and exploit filter reuse, instance, in object detection using HOG, each 36-dimension respectively. Finally, multiple passes across the spatial feature vector only requires 9-bit per dimension, and each array allow for additional image and filter reuse using the weight of the SVM uses only 4-bits [48]; for object detection global buffer. This dataflow is demonstrated in [47]. using deformable parts models (DPM) [49], only 11-bits are The dataflows are compared on a spatial array with the required per feature vector and only 5-bits are required per same number of PEs (256), area cost and DNN (AlexNet). SVM weight [50]. Fig. 10 shows the energy consumption of each approach. The Similarly for DNN inference, it is common to see accelerators row stationary approach is 1.4× to 2.5× more energy-efficient support 16-bit fixed point [45, 47]. There has been significant Fig. 11. Sparse weights after basis projection [50]. a result, compression can be applied to exploit data statistics to reduce data movement and storage cost. Various forms of lightweight compression have been explored to reduce data movement. Lossless compression can be used to reduce the transfer of data on and off chip [11, 53, 64]. Simple run-length coding of the activations in [65] provides up to 1.9× bandwidth reduction, which is within 5-10% of the theoretical entropy limit. Lossy compression such as vector quantization can also be used on feature vectors [50] and weights [8, 12, 66] such that they can be stored on-chip at low cost. Generally, the cost of the compression/decompression is on the order of a few thousand kgates with minimal energy overhead. In the lossy compression case, it is also important to evaluate the impact on performance accuracy. VII. O PPORTUNITIES IN M IXED -S IGNAL C IRCUITS research on exploring the impact of bitwidth on accuracy [51]. In fact, recently commercial hardware for DNN reportedly support 8-bit integer operations [52]. As bitwidths can vary by layer, hardware optimizations have been explored to exploit the reduced bitwidth for 2.56× energy savings [53] or 2.24× increase in throughput [54] compared to a 16-bit fixed point implementation. With more significant changes to the network, it is possible to reduce bitwidth down to 1-bit for either weights [55] or both weights and activations [56, 57] at the cost of reduced accuracy. The impact of 1-bit weights on hardware is explored in [58]. B. Sparsity For SVM classification, the weights can be projected onto a basis such that the resulting weights are sparse for a 2× reduction in number of multiplications [50] (Fig. 11). For feature extraction, the input image can be made sparse by preprocessing for a 24% reduction in power consumption [48]. For DNNs, the number of MACs and weights can be reduced by removing weights through a process called pruning. This was first explored in [59] where weights with minimal impact on the output were removed. In [60], pruning is applied to modern DNNs by removing small weights. However, removing weights does not necessarily lead to lower energy. Accordingly, in [61] weights are removed based on an energy-model to directly minimize energy consumption. The tool used for energy modeling can be found at [62]. Specialized hardware has been proposed in [47, 50, 63, 64] to exploit sparse weights for increased speed or reduced energy consumption. In Eyeriss [47], the processing elements are designed to skip reads and MACs when the inputs are zero, resulting in a 45% energy reduction. In [50], by using specialized hardware to avoid sparse weights, the energy and storage cost are reduced by 43% and 34%, respectively. C. Compression Data movement and storage are important factors in both energy and cost. Feature extraction can result in sparse data (e.g., gradient in HOG and ReLU in DNN) and the weights used in classification can also be made sparse by pruning. As Most of the data movement is in between the memory and processing element (PE), and also the sensor and PE. In this section, we discuss how this is addressed using mixedsignal circuit design. However, circuit non-idealities should also be factored into the algorithm design; these circuits can benefit from the reduced precision algorithms discussed in Section VI. In addition, since the training often occurs in the digital domain, the ADC and DAC conversion overhead should also be accounted for when evaluating the system. While spatial architectures bring the memory closer to the computation (i.e., into the PE), there have also been efforts to integrate the computation into the memory itself. For instance, in [67] the classification is embedded in the SRAM. Specifically, the word line (WL) is driven by a 5-bit feature vector using a DAC, while the bit-cells store the binary weights ±1. The bit-cell current is effectively a product of the value of the feature vector and the value of the weight stored in the bit-cell; the currents from the column are added together to discharge the bitline (BL or BLB). A comparator is then used to compare the resulting dot product to a threshold, specifically sign thresholding of the differential bitlines. Due to the variations in the bitcell, this is considered a weak classifier, and boosting is needed to combine the weak classifiers to form a strong classifier [68]. This approach gives 12× energy savings over reading the 1-bit weights from the SRAM. Recent work has also explored the use of mixed-signal circuits to reduce the computation cost of the MAC. It was shown in [69] that performing the MAC using switched capacitors can be more energy-efficient than digital circuits despite ADC and DAC conversion overhead. Accordingly, the matrix multiplication can be integrated into the ADC as demonstrated in [70], where the most significant bits of the multiplications for Adaboost classification are performed using switched capacitors in an 8-bit successive approximation format. This is extended in [71] to not only perform multiplications, but also the accumulation in the analog domain. It is assumed that 3-bits and 6-bits are sufficient to represent the weights and input vectors, respectively. This enables the computation to move closer to the sensor and reduces the number of ADC conversions by 21×. To further reduce the data movement from the sensor, [72] proposed performing the entire convolution layer (including convolution, max pooling and quantization) in the analog domain at the sensor. Similarly, in [73], the entire HOG feature is computed in the analog domain to reduce the sensor bandwidth by 96.5%. VIII. O PPORTUNITIES IN A DVANCED T ECHNOLOGIES Exponen6al 10000 VGG162 1000 Energy/ Pixel (nJ) Measured in 65nm* 1. [Suleiman, VLSI 2016] 2. [Chen, ISSCC 2016] * Only feature extrac6on. Does not include data, augmenta6on, ensemble and classifica6on energy, etc. AlexNet2 100 10 1 HOG1 0.1 Linear In the previous section, we discussed how data movement 0 20 40 60 80 can be reduced by moving the processing near the memory or Accuracy (Average Precision) the sensor using mixed-signal circuits. In this section, we will Measured in on VOC 2007 Dataset 1. DPM v5 [Girshick, 2012] discuss how this can be achieved with advanced technologies. 2. Fast R-CNN [Girshick, CVPR 2015] The use of advanced memory technologies such as embedded Fig. 12. Energy vs. accuracy comparison of hand-crafted and learned features. DRAM (eDRAM) and Hybrid Memory Cube (HMC) are explored in [46] and [74], respectively, to reduce the energy the same energy as video compression (under 1nJ/pixel [80] access cost of the weights in DNN. There has also been a lot for real-time high definition video), which servers as a good of work that investigates integrating the multiplication directly benchmark of what is acceptable for energy consumption near into advanced non-volatile memories by using them as resistive the sensor; however, DNNs currently consume several orders elements. Specifically, the multiplications are performed where of magnitude more. A more detailed comparison can be found the conductance is the weight, the voltage is the input, and in [81]. We hope that the many design opportunities that we the current is the output (note: this is the ultimate form of have highlighted in this paper will help close this gap. weight stationary, as the weights are always held in place); the addition is done by summing the current using Kirchhoff’s X. S UMMARY current law. In [75], memristors are used to compute a 16-bit Machine learning is an important area of research with dot product operation with 8 memristors each storing 2-bits; many promising applications and opportunities for innovation a 1-bit×2-bit multiplication is performed at each memristor, at various levels of hardware design. During the design process, where a 16-bit input requires 16 cycles to complete. In [76], it is important to balance the accuracy, energy, throughput and ReRAM is used to compute the product of a 3-bit input and cost requirements. 4-bit weight. Similar to the mixed-signal circuits, the precision Since data movement dominates energy consumption, the is limited, and the ADC and DAC conversion overhead must primary focus of recent research has been to reduce the data be considered in the overall cost, especially when the weights movement while maintaining accuracy, throughput and cost. are trained in the digital domain. The conversion overhead can This means selecting architectures with favorable memory be avoided by training directly in the analog domain as shown hierarchies like a spatial array, and developing dataflows that for the fabricated memristor array in [77]. increase data reuse at the low-cost levels of the memory Finally, it may be feasible to embed the computation into hierarchy. With joint design of algorithm and hardware, reduced the sensor itself. This is useful for image processing where bitwidth precision, increased sparsity and compression are used the bandwidth to read the data from the sensor accounts for to minimize the data movement requirements. With mixeda significant portion of the system energy consumption. For signal circuit design and advanced technologies, computation instance, an Angle Sensitive Pixels sensor can be used to comis moved closer to the source by embedding computation near pute the gradient of the input, which along with compression, or within the sensor and the memories. reduces the sensor bandwidth by 10× [78]. A sensor that One should also consider the interactions between these outputs gradients can also reduce the computation and energy different levels. For instance, reducing the bitwidth through consumption of subsequent processing engine [48, 79]. hardware-friendly algorithm design enables reduced precision processing with mixed-signal circuits and non-volatile memory. IX. H AND - CRAFTED VERSUS L EARNED F EATURES Hand-crafted approaches give higher energy efficiency at Reducing the cost of memory access with advanced technolothe cost of reduced accuracy as compared with learned gies could result in more energy-efficient dataflows. features such as DNNs. For hand-crafted features, the amount of computation is less, and reduced bit-width is supported. Furthermore, less data movement is required since the weights are not required for the features. The classification weights for both approaches must however remain programmable. Fig. 12 compares the energy consumption of HOG feature extraction versus the convolution layers in AlexNet and VGG-16 based measured results from fabricated 65nm chips [50] and [47], respectively. Note that HOG feature extraction consumes around ACKNOWLEDGMENT Funding provided by DARPA YFA, MIT CICS, TSMC University Shuttle, and gifts from Texas Instruments and Intel. R EFERENCES [1] B. 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M A M A D ROID: Detecting Android Malware by Building Markov Chains of Behavioral Models (Extended Version)∗ Lucky Onwuzurike1 , Enrico Mariconti1 , Panagiotis Andriotis2 , Emiliano De Cristofaro1 , Gordon Ross1 , and Gianluca Stringhini1 arXiv:1711.07477v1 [cs.CR] 20 Nov 2017 1 University College London 2 University of the West of England Abstract lenges compared to desktop/laptop computers: smartphones have limited battery life, making it impossible to use traditional approaches requiring constant scanning and complex computation [43]. Thus, Android malware detection is typically performed in a centralized fashion, i.e., by analyzing apps submitted to the Play Store using Bouncer [40]. However, many malicious apps manage to avoid detection [56, 38], and manufacturers as well as users can install apps that come from third parties, which might not perform any malware checks at all [67]. As a result, the research community has proposed a number of techniques to detect malware on Android. Previous work has often relied on the permissions requested by apps [18, 47], using models built from malware samples. This, however, is prone to false positives, since there are often legitimate reasons for benign apps to request permissions classified as dangerous [18]. Another approach, used by D ROIDAPIM INER [1], is to perform classification based on API calls frequently used by malware. However, relying on the most common calls observed during training prompts the need for constant retraining, due to the evolution of malware and the Android API alike. For instance, “old” calls are often deprecated with new API releases, so malware developers may switch to different calls to perform similar actions. In this paper, we present a novel malware detection system for Android that relies on the sequence of abstracted API calls performed by an app rather than their use or frequency, aiming to capture the behavioral model of the app. We design M A M A D ROID to abstract API calls to either the class name (e.g., java.lang.Throwable) of the call or its package name (e.g., java.lang) or its source (e.g., java, android, google), which we refer to as family. Abstraction provides resilience to API changes in the Android framework as families and packages are added and removed less frequently than single API calls. At the same time, this does not abstract away the behavior of an app: for instance, packages include classes and interfaces used to perform similar operations on similar objects, so we can model the types of operations from the package name alone. For example, the java.io package is used for system I/O and access to the file system, even though there are different classes and interfaces provided by the package for such operations. After abstracting the calls, M A M A D ROID analyzes the sequence of API calls performed by the app aiming to model As Android becomes increasingly popular, so does malware targeting it, this motivating the research community to propose many different detection techniques. However, the constant evolution of the Android ecosystem, and of malware itself, makes it hard to design robust tools that can operate for long periods of time without the need for modifications or costly re-training. Aiming to address this issue, we set to detect malware from a behavioral point of view, modeled as the sequence of abstracted API calls. We introduce M A M A D ROID, a staticanalysis based system that abstracts app’s API calls to their class, package, or family, and builds a model from their sequences obtained from the call graph of an app as Markov chains. This ensures that the model is more resilient to API changes and the features set is of manageable size. We evaluate M A M A D ROID using a dataset of 8.5K benign and 35.5K malicious apps collected over a period of six years, showing that it effectively detects malware (with up to 0.99 F-measure) and keeps its detection capabilities for long periods of time (up to 0.87 F-measure two years after training). We also show that M A M A D ROID remarkably improves over D ROIDAPIM INER, a state-of-the-art detection system that relies on the frequency of (raw) API calls. Aiming to assess whether M A M A D ROID’s effectiveness mainly stems from the API abstraction or from the sequencing modeling, we also evaluate a variant of it that uses frequency (instead of sequences), of abstracted API calls. We find that it is not as accurate, failing to capture maliciousness when trained on malware samples including API calls that are equally or more frequently used by benign apps. 1 Introduction Malware running on mobile devices can be particularly lucrative, as it may enable attackers to defeat two-factor authentication for financial and banking systems [52] and/or trigger the leakage of sensitive information [25]. As a consequence, the number of malware samples has skyrocketed in recent years, and, due to its increased popularity, cybercriminals have increasingly targeted the Android ecosystem [15]. Detecting malware on mobile devices presents additional chal∗ A preliminary version of this paper appears in the 24th Network and Distributed System Security Symposium (NDSS 2017) [36]. 1 the app’s behavior using Markov chains. Our intuition is that malware may use calls for different operations, and in a different order, than benign apps. For example, android.media.MediaRecorder can be used by any app that has permission to record audio, but the call sequence may reveal that malware only uses calls from this class after calls to getRunningTasks(), which allows recording conversations [65], as opposed to benign apps where calls from the class may appear in any order. Relying on the sequence of abstracted calls allows us to model behavior in a more complex way than previous work, which only looked at the presence or absence of certain API calls or permissions [1, 4], while still keeping the problem tractable [30]. M A M A D ROID then builds a statistical model to represent the transitions between the API calls performed by an app as Markov chains, and use them to extract features. Finally, it classifies an app as either malicious or benign using the features it extracts from the app. We present a detailed evaluation of the classification accuracy (using F-measure, precision, and recall) and runtime performance of M A M A D ROID, using a dataset of almost 44K apps (8.5K benign and 35.5K malware samples). We include a mix of older and newer apps, from October 2010 to May 2016, verifying that our model is robust to changes in Android malware samples and APIs. To the best of our knowledge, this is the largest malware dataset used to evaluate an Android malware detection system in a research paper. Our experimental analysis shows that M A M A D ROID can effectively model both benign and malicious Android apps, and efficiently classify them. Compared to other systems such as D ROIDAPIM INER [1], our approach allows us to account for changes in the Android API, without the need to frequently retrain the classifier. Moreover, to assess the impact of abstraction and Markov chain modeling on M A M A D ROID, we not only compare to D ROIDAPIM INER, but also build a variant (called FAM) that still abstracts API calls but instead of building a model from the sequence of calls it does so on the frequency, similar to D ROIDAPIM INER. Overall, we find that M A M A D ROID can effectively detect unknown malware samples not only in the “present,” (with Fmeasure up to 0.99) but also consistently over the years (i.e., when the system is trained on older samples and evaluated over newer ones), as it keeps an average detection accuracy, evaluated in terms of F-measure, of 0.87 after one year and 0.75 after two years (as opposed to 0.46 and 0.42 achieved by D ROIDAPIM INER [1] and 0.81 and 0.76 by FAM). We also highlight that when the system is not efficient anymore (when the test set is newer than the training set by more than two years), it is as a result of M A M A D ROID having low recall, but maintaining high precision. We also do the opposite, i.e., training on newer samples and verifying that the system can still detect old malware. This is particularly important as it shows that M A M A D ROID can detect newer threats, while still identifying malware samples that have been in the wild for some time. in a tool called M A M A D ROID, to detect Android malware by abstracting API calls to their class, package, and family, and model the behavior of the apps through the sequences of API calls as Markov chains. Second, we can detect unknown samples on the same year of training with an F-measure of 0.99, but also years after training the system, meaning that M A M A D ROID does not need continuous re-training. Compared to previous work [1], M A M A D ROID achieves higher accuracy with reasonably fast running times, while also being more robust to evolution in malware development and changes in the Android API. Third, by abstracting API calls and using frequency analysis we still perform better than a system that also uses frequency analysis but without abstraction (D ROIDAPIM INER). Finally, we explore the detection performance of a finer-grained abstraction and show that abstracting to classes does not perform better than abstracting to packages. Paper Organization. The rest of the paper is organized as fol- lows. The next section presents M A M A D ROID, then, Section 3 introduces the datasets used throughout the paper. In Section 4, we evaluate M A M A D ROID in family and package modes, while, in Section 5, we explore the effectiveness of finer-grained abstraction (i.e., class mode). In Section 6, we present and evaluate the variant using a frequency analysis model (FAM), while we analyze runtime performances in Section 7. Section 8 further discusses our results as well as its limitations. After reviewing related work in Section 9, the paper concludes in Section 10. 2 The MaMaDroid System In this section, we introduce M A M A D ROID, an Android malware detection system that relies on the transitions between different API calls performed by Android apps. 2.1 Overview M A M A D ROID builds a model of the sequence of API calls as Markov chains, which are in turn used to extract features for machine learning algorithms to classify apps as benign or malicious. Abstraction. M A M A D ROID does not use the raw API calls, but abstracts each call to its family, package, or class. For instance, the API call getMessage() in Figure 1 is parsed to, respectively, java, java.lang, and java.lang.Throwable. package z }| { java.lang.Throwable: String getMessage() |{z} family | {z Class } Figure 1: Example of an API call and its family, package, and class. Given the three different types of abstractions, M A M A D ROID operates in one of three modes, each using one of the types of abstraction. Naturally, we expect that the higher Summary of Contributions. This paper makes several contri- butions. First, we introduce a novel approach, implemented 2 ? Call Graph Extraction (1) Sequence Extraction (2) Markov Chain Modeling (3) Classification (4) Figure 2: Overview of M A M A D ROID operation. In (1), it extracts the call graph from an Android app, next, it builds the sequences of (abstracted) API calls from the call graph (2). In (3), the sequences of calls are used to build a Markov chain and a feature vector for that app. Finally, classification is performed in (4), labeling the app as benign or malicious. package com.fa.c; import import import import import import import android.content.Context; android.os.Environment; android.util.Log; com.stericson.RootShell.execution.Command; com.stericson.RootShell.execution.Shell; com.stericson.RootTools.RootTools; java.io.File; public class RootCommandExecutor { public static boolean Execute(Context paramContext) { paramContext = new Command(0, new String[] { "cat " + Environment.getExternalStorageDirectory().getAbsolutePath() + File.separator + Utilities.GetWatchDogName( paramContext) + " > /data/" + Utilities.GetWatchDogName(paramContext), "cat " + Environment.getExternalStorageDirectory().getAbsolutePath() + File.separator + Utilities.GetExecName(paramContext) + " > /data/" + Utilities.GetExecName(paramContext), "rm " + Environment.getExternalStorageDirectory().getAbsolutePath() + File.separator + Utilities.GetWatchDogName(paramContext), "rm " + Environment.getExternalStorageDirectory().getAbsolutePath() + File.separator + Utilities. GetExecName(paramContext), "chmod 777 /data/" + Utilities.GetWatchDogName(paramContext), "chmod 777 /data/" + Utilities.GetExecName(paramContext), "/data/" + Utilities.GetWatchDogName(paramContext) + " " + Utilities.GetDeviceInfoCommandLineArgs(paramContext) + " /data/" + Utilities.GetExecName(paramContext) + " " + Environment.getExternalStorageDirectory().getAbsolutePath() + File.separator + Utilities.GetExchangeFileName(paramContext) + " " + Environment. getExternalStorageDirectory().getAbsolutePath() + File.separator + " " + Utilities.GetPhoneNumber(paramContext) }); try { RootTools.getShell(true).add(paramContext); return true; } catch (Exception paramContext) { Log.d("CPS", paramContext.getMessage()); } return false; } } Figure 3: Code from a malicious app (com.g.o.speed.memboost) executing commands as root. the abstraction, the lighter the system is, although possibly less accurate. Building Blocks. M A M A D ROID’s operation goes through four phases, as depicted in Figure 2. First, we extract the call graph from each app by using static analysis (1), then, we obtain the sequences of API calls using all unique nodes after which we abstract each call to class, package, or family (2). Next, we model the behavior of each app by constructing Markov chains from the sequences of API calls for the app (3), with the transition probabilities used as the feature vector to classify the app as either benign or malware using a machine learning classifier (4). In the rest of this section, we discuss each of these steps in detail. lists a class extracted from the decompiled apk of malware disguised as a memory booster app (with package name com.g.o.speed.memboost), which executes commands (rm, chmod, etc.) as root.1 To ease presentation, we focus on the portion of the code executed in the try/catch block. The resulting call graph of the try/catch block is shown in Figure 4. For simplicity, we omit calls for object initialization, return types and parameters, as well as implicit calls in a method. Additional calls that are invoked when getShell(true) is called are not shown, except for the add() method that is directly called by the program code, as shown in Figure 3. 2.2 In its second phase, M A M A D ROID extracts the sequences of API calls from the call graph and abstract the calls to one of three mode. 2.3 Call Graph Extraction The first step in M A M A D ROID is to extract the app’s call graph. We do so by performing static analysis on the app’s apk, i.e., the standard Android archive file format containing all files, including the Java bytecode, making up the app. We use a Java optimization and analysis framework, Soot [51], to extract call graphs and FlowDroid [5] to ensure contexts and flows are preserved. To better clarify the different steps involved in our system, we employ, throughout this section, a “running example,” using a real-world malware sample. Figure 3 Sequence Extraction and Abstraction Sequence Extraction. Since M A M A D ROID uses static analy- sis, the graph obtained from Soot represents the sequence of functions that are potentially called by the app. However, each execution of the app could take a specific branch of the graph and only execute a subset of the calls. For instance, when running the code in Figure 3 multiple times, the Execute method could be followed by different calls, e.g., getShell() in the try 1 https://www.hackread.com/ghost-push-android-malware/ 3 com.fa.c.RootCommandExecutor: Execute() android.util.Log: d() com.stericson.RootTools.RootTools: getShell() java.lang.Throwable: getMessage() com.stericson.RootShell.execution.Shell: add() com.fa.c.RootCommandExecutor: Execute() [self-defined, self-defined, self-defined] com.stericson.RootTools.RootTools: getShell() [self-defined, self-defined, self-defined] com.fa.c.RootCommandExecutor: Execute() [self-defined, self-defined, self-defined] android.util.Log: d() [android.util.Log, android.util, android] com.fa.c.RootCommandExecutor: Execute() [self-defined, self-defined, self-defined] java.lang.Throwable: getMessage() [java.lang.Throwable, java.lang, java] com.stericson.RootShell. execution.Shell: add() [self-defined, self-defined, self-defined] Figure 5: Sequence of API calls extracted from the call graphs in Figure 4, with the corresponding class/package/family abstraction in square brackets. Figure 4: Call graph of the API calls in the try/catch block of Figure 3. (Return types and parameters omitted to ease presentation). identifier mangling. Overall, there are 11 (9+2) families, 340 (243+95+2) possible packages, and 5,973 (4,855+1,116+2) possible classes. block only or getShell() and then getMessage() in the catch block. Thus, in this phase, M A M A D ROID operates as follows. First, it identifies a set of entry nodes in the call graph, i.e., nodes with no incoming edges (for example, the Execute method in the snippet from Figure 3 is the entry node if there is no incoming edge from any other call in the app). Then, it enumerates the paths reachable from each entry node. The sets of all paths identified during this phase constitutes the sequences of API calls which will be used to build a Markov chain behavioral model and to extract features. In Figure 5, we show the sequence of API calls obtained from the call graph in Figure 4. We also report in square brackets, the family, package, and class to which the call is abstracted. 2.4 Markov-chain Based Modeling Next, M A M A D ROID builds feature vectors, used for classification, based on the Markov chains representing the sequences of abstracted API calls for an app. Before discussing this in detail, first review the basic concepts of Markov chains. Markov Chains. Markov Chains are memoryless models where the probability of transitioning from a state to another only depends on the current state [39]. They are often represented as a set of nodes, each corresponding to a different state, and a set of edges connecting one node to another labeled with the probability of that transition. The sum of all probabilities associated to all edges from any node (including, if present, an edge going back to the node itself) is exactly 1. The set of possible states of the Markov chain is denoted as S. If Sj and Sk are two connected states, Pjk denotes the probability of transition from Sj to Sk . Pjk is given by the number of occurrences (Ojk ) of state Sk after state Sj , divided by Oji O for all states i in the chain, i.e., Pjk = P jkOji . API Call Abstraction. Rather than analyzing raw API calls from the sequence of calls, we build M A M A D ROID to work at a higher level, and operate in one of three modes by abstracting each call to its family, package, or class. The intuition is to make M A M A D ROID resilient to API changes and achieve scalability. In fact, our experiments, presented in Section 3, show that, from a dataset of 44K apps, we extract more than 10 million unique API calls, which, depending on the modeling approach used to model each app, may result in the feature vectors being very sparse. When operating in family mode, we abstract an API call to one of the nine Android families, i.e., android, google, java, javax, xml, apache, junit, json, dom, which correspond to the android.*, com.google.*, java.*, javax.*, org.xml.*, org.apache.*, junit.*, org.json, and org.w3c.dom.* packages. Whereas, in package mode, we abstract the call to its package name using the list of Android packages from the documentation2 consisting of 243 packages as of API level 24 (the version as of September 2016), as well as 95 from the Google API.3 In class mode, we abstract each call to its class name using a whitelist of all class names in the Android and Google APIs, which consists respectively, 4,855 and 1116 classes.4 In all modes, we abstract developer-defined (e.g., com.stericson.roottools) and obfuscated (e.g. com.fa.a.b.d) API calls respectively, as self-defined and obfuscated. Note that we label an API call as obfuscated if we cannot tell what its class implements, extends, or inherits, due to i∈S Building the model. For each app, M A M A D ROID takes as in- put the sequence of abstracted API calls of that app (classes, packages or families, depending on the selected mode of operation), and builds a Markov chain where each class/package/family is a state and the transitions represent the probability of moving from one state to another. For each Markov chain, state S0 is the entry point from which other calls are made in a sequence. As an example, Figure 6 illustrates the Markov chains built using classes, packages, and families, respectively, from the sequences reported in Figure 5. We argue that considering single transitions is more robust against attempts to evade detection by inserting useless API calls in order to deceive signature-based systems [33]. In fact, M A M A D ROID considers all possible calls – i.e., all the branches originating from a node – in the Markov chain, so adding calls would not significantly change the probabilities of transitions between nodes (specifically, families, packages, or classes depending on the operational mode) for each app. Feature Extraction. Next, we use the probabilities of tran- 2 https://developer.android.com/reference/packages.html sitioning from one state (abstracted call) to another in the Markov chain as the feature vector of each app. States that are 3 https://developers.google.com/android/reference/packages 4 https://developer.android.com/reference/classes.html 4 self-defined 0.5 0.25 self-defined 0.25 0.25 java.lang.Throwable android.util.Log self-defined 0.5 0.5 java.lang (a) 0.25 android.util (b) 0.25 0.25 java android (c) Figure 6: Markov chains originating from the call sequence in Figure 5 when using classes (a), packages (b) or families (c). 3 not present in a chain are represented as 0 in the feature vector. The vector derived from the Markov chain depends on the operational mode of M A M A D ROID. With families, there are 11 possible states, thus 121 possible transitions in each chain, while, when abstracting to packages, there are 340 states and 115,600 possible transitions and with classes, there are 5,973 states therefore, 35,676,729 possible transitions. In this section, we introduce the datasets used in the evaluation of M A M A D ROID (presented later in Section 4), which include 43,940 apk files, specifically, 8,447 benign and 35,493 malware samples. We include a mix of older and newer apps, ranging from October 2010 to May 2016, as we aim to verify that M A M A D ROID is robust to changes in Android malware samples as well as APIs. To the best of our knowledge, we are leveraging the largest dataset of malware samples ever used in a research paper on Android malware detection. We also apply Principal Component Analysis (PCA) [29], which performs feature selection by transforming the feature space into a new space made of components that are a linear combination of the original features. The first components contain as much variance (i.e., amount of information) as possible. The variance is given as percentage of the total amount of information of the original feature space. We apply PCA to the feature set in order to select the principal components, as PCA transforms the feature space into a smaller one where the variance is represented with as few components as possible, thus considerably reducing computation/memory complexity. Also, PCA could improve the accuracy of the classification by removing, from the feature space, features that make the classifier perform worse. 2.5 Dataset Benign Samples. Our benign datasets consist of two sets of samples: (1) one, which we denote as oldbenign, includes 5,879 apps collected by PlayDrone [54] between April and November 2013, and published on the Internet Archive5 on August 7, 2014; and (2) another, newbenign, obtained by downloading the top 100 apps in each of the 29 categories on the Google Play store as of March 7, 2016, using the googleplay-api tool.6 Due to errors encountered while downloading some apps, we have actually obtained 2,843 out of 2,900 apps. Note that 275 of these belong to more than one category, therefore, the newbenign dataset ultimately includes 2,568 unique apps. Android Malware Samples. The set of malware samples in- Classification cludes apps that were used to test D REBIN [4], dating back to October 2010 – August 2012 (5,560), which we denote as drebin, as well as more recent ones that have been uploaded on the VirusShare7 site over the years. Specifically, we gather from VirusShare, respectively, 6,228, 15,417, 5,314, and 2,974 samples from 2013, 2014, 2015, and 2016. We consider each of these datasets separately for our analysis. The last step is to perform classification, i.e., labeling apps as either benign or malware. To this end, we test M A M A D ROID using different classification algorithms: Random Forests, 1Nearest Neighbor (1-NN), 3-Nearest Neighbor (3-NN), and Support Vector Machines (SVM). Note that since both accuracy and speed are worse with SVM, we omit results obtained with it. API Calls. For each app, we extract all API calls, using An- droguard8 , since, as explained in Section 4.5, these constitute the features used by D ROIDAPIM INER [1] (against which we compare our system) as well as a variant of M A M A D ROID that is based on frequency analysis (see Section 6). Due to Androguard failing to decompress some of the apks, bad CRC-32 redundancy checks, and errors during unpacking, we are not able to extract the API calls for all the samples, but only for Each model is trained using the feature vector obtained from the apps in a training sample. Results are presented and discussed in Section 4, and have been validated by using 10-fold cross validation. Also note that, due to the different number of features used in different modes, we use two distinct configurations for the Random Forests algorithm. Specifically, when abstracting to families, we use 51 trees with maximum depth 8, while, with classes and packages, we use 101 trees of maximum depth 64. To tune Random Forests we follow the methodology applied in [6]. 5 https://archive.org/details/playdrone-apk-e8 6 https://github.com/egirault/googleplay-api 7 https://virusshare.com/ 8 https://github.com/androguard/androguard 5 Category Name Date Range #Samples Benign oldbenign newbenign Apr 2013 – Nov 2013 Mar 2016 – Mar 2016 Total Benign: Malware drebin 2013 2014 2015 2016 Oct 2010 – Aug 2012 Jan 2013 – Jun 2013 Jun 2013 – Mar 2014 Jan 2015 – Jun 2015 Jan 2016 – May 2016 Total Malware: 5,879 2,568 8,447 #Samples (API Calls) 5,837 2,565 8,402 #Samples (Call Graph) 5,572 2,465 8,037 5,560 6,228 15,417 5,314 2,974 35,493 5,546 6,146 14,866 5,161 2,802 34,521 5,512 6,091 13,804 4,451 2,555 32,413 1.0 0.8 0.8 0.6 0.6 0.4 0.2 0.00 10000 20000 #API Calls (a) API calls 2015 newbenign oldbenign drebin 2013 2014 2016 30000 40000 1.0 2016 2015 2014 2013 drebin newbenign oldbenign 0.8 0.6 CDF 1.0 CDF CDF Table 1: Overview of the datasets used in our experiments. 0.4 0.4 0.2 0.2 0.0 0.0 0.2 0.4 0.6 Fraction of Calls (b) android 0.8 1.0 0.0 0.0 0.2 0.4 0.6 Fraction of Calls 2016 2015 2014 2013 drebin newbenign oldbenign 0.8 1.0 (c) google Figure 7: CDFs of the number of API calls per app in each dataset (a), and of the percentage of android (b) and google (c) family calls. ure 7(b)), both in benign and malicious datasets, while google calls become more common in newer apps (Figure 7(c)). In general, we conclude that benign and malicious apps show the same evolutionary trends over the years. Malware, however, appears to reach the same characteristics (in terms of level of complexity and fraction of API calls from certain families) as legitimate apps with a few years of delay. 40,923 (8,402 benign, 34,521 malware) out of the 43,940 apps in our datasets. Call Graphs. To extract the call graph of each apk, we use Soot. Note that for some of the larger apks, Soot requires a non-negligible amount of memory to extract the call graph, so we allocate 16GB of RAM to the Java VM heap space. We find that for 2,472 (364 benign + 2,108 malware) samples, Soot is not able to complete the extraction due to it failing to apply the jb phase as well as reporting an error in opening some zip files (i.e., the apk). The jb phase is used by Soot to transform Java bytecode into jimple intermediate representation (the primary IR of Soot) for optimization purposes. Therefore, we exclude these apps in our evaluation and discuss this limitation further in Section 8.3. In Table 1, we provide a summary of our seven datasets, reporting the total number of samples per dataset, as well as those for which we are able to extract the API calls (second-to-last column) and the call graphs (last column). Dataset Characterization. Aiming to shed light on the evolution of API calls in Android apps, we also performed some measurements over our datasets. In Figure 7(a), we plot the Cumulative Distribution Function (CDF) of the number of unique API calls in the apps in different datasets, highlighting that newer apps, both benign and malicious, use more API calls overall than older apps. This indicates that as time goes by, Android apps become more complex. When looking at the fraction of API calls belonging to specific families, we discover some interesting aspects of Android apps developed in different years. In particular, we notice that API calls to the android family become less prominent as time passes (Fig- Principal Component Analysis. Finally, we apply PCA to se- lect the two most important PCA components. We plot and compare the positions of the two components for benign (Figure 8(a)) and malicious samples (Fig. 8(b)). As PCA combines the features into components, it maximizes the variance of the distribution of samples in these components, thus, plotting the positions of the samples in the components shows that benign apps tend to be located in different areas of the components space, depending on the dataset, while malware samples occupy similar areas but with different densities. These differences highlight a different behavior between benign and malicious samples, and these differences should also be found by the machine learning algorithms used for classification. 4 M A M A D ROID Evaluation We now present an experimental evaluation of M A M A D ROID run in family and package mode. Later in Section 5, we evaluate it in class mode. We use the datasets summarized in Table 1, and evaluate M A M A D ROID, as per (1) its accuracy on benign and malicious samples developed around the same 6 1.0 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 PCA2 PCA2 1.0 0.0 0.0 0.2 0.2 0.4 0.4 0.6 0.8 1.0 0.5 0.0 PCA1 0.5 1.0 0.6 oldbenign newbenign 0.8 1.0 1.5 (a) benign 0.5 0.0 PCA1 0.5 1.0 drebin 2013 2014 2015 2016 1.5 (b) malware Figure 8: Positions of benign vs malware samples in the feature space of the first two components of the PCA (family mode). time; and (2) its robustness to the evolution of malware as well as of the Android framework by using older datasets for training and newer ones for testing and vice-versa. 4.1 smaller one. When operating in package mode, PCA could be particularly beneficial to reduce computation and memory complexity, since M A M A D ROID originally has to operate over 116,281 features. Hence, in Table 2 we report the precision, recall, and F-measure achieved by M A M A D ROID in both modes with and without the application of PCA using Random Forest classifier. We report the results for Random Forest only because it outperforms both 1-NN and 3-NN (Figure 9) while also being very fast. In package mode, we find that only 67% of the variance is taken into account by the 10 most important PCA components, and, in family mode, at least 91% of the variance is included by the 10 PCA Components. As shown in Table 2, the F-measure using PCA is only slightly lower (up to 3%) than using the full feature set. In general, M A M A D ROID performs better in package mode in all datasets with F-measure ranging from 0.92 – 0.99 compared to 0.88 – 0.98 in family mode. This is as a result of the increased granularity which enables M A M A D ROID identify more differences between benign and malicious apps. On the other hand, however, this likely reduces the efficiency of the system, as many of the states derived from the abstraction are used only a few times. The differences in time performance between the two modes are analyzed in details in Section 7. Experimental Settings To assess the accuracy of the classification, we use the standard F-measure metric, calculated as F = 2 · (precision · recall)/(precision + recall), where precision = TP/(TP+FP) and recall = TP/(TP+FN). TP denotes the number of samples correctly classified as malicious, while FP an FN indicate, respectively, the number of samples mistakenly identified as malicious and benign. Note that all our experiments perform 10-fold cross validation using at least one malicious and one benign dataset from Table 1. In other words, after merging the datasets, the resulting set is shuffled and divided into ten equal-size random subsets. Classification is then performed ten times using nine subsets for training and one for testing, and results are averaged out over the ten experiments. When implementing M A M A D ROID in family mode, we exclude json and dom families because they are almost never used across all our datasets, and junit, which is primarily used for testing. In package mode, in order to avoid mislabeling when self-defined APIs have “android” in the name, we split the android package into its two classes, i.e., android.R and android.Manifest. Therefore, in family mode, there are 8 possible states, thus 64 features, whereas, in package mode, we have 341 states and 116,281 features (cf. Section 2.4). 4.2 4.3 Detection Over Time As Android evolves over the years, so do the characteristics of both benign and malicious apps. Such evolution must be taken into account when evaluating Android malware detection systems, since their accuracy might significantly be affected as newer APIs are released and/or as malicious developers modify their strategies in order to avoid detection. Evaluating this aspect constitutes one of our research questions, and one of the reasons why our datasets span across multiple years (2010– 2016). Recall that M A M A D ROID relies on the sequence of API calls extracted from the call graphs and abstracted to either the package or the family level. Therefore, it is less susceptible to changes in the Android API than other classification systems such as D ROIDAPIM INER [1] and D REBIN [4]. Since these rely on the use, or the frequency, of certain API calls to M A M A D ROID’s Performance (Family and Package Mode) We start by evaluating the performance of M A M A D ROID when it is trained on dataset from the same year. In Figure 9, we plot the F-measure achieved by M A M A D ROID in family and package modes using datasets from the same year for training and testing and the three different classifiers. As already discussed in Section 2.4, we apply PCA as it allows us transform a large feature space into a 7 1.0 RF 1-NN 3-NN 0.8 0.8 0.6 0.6 F-measure F-measure 1.0 0.4 0.2 0.0 RF 1-NN 3-NN 0.4 0.2 Drebin & OldBenign 2013 & OldBenign 2014 & OldBenign 2014 & Newbenign 2015 & Newbenign 0.0 2016 & Newbenign Drebin & OldBenign (a) family mode 2013 & OldBenign 2014 & OldBenign 2014 & Newbenign 2015 & Newbenign 2016 & Newbenign (b) package mode Figure 9: F-measure of M A M A D ROID classification with datasets from the same year using three different classifiers. XXX X Mode Dataset XXX X drebin, oldbenign 2013, oldbenign 2014, oldbenign 2014, newbenign 2015, newbenign 2016, newbenign 0.82 0.91 0.88 0.97 0.89 0.87 Family 0.95 0.93 0.96 0.99 0.93 0.91 0.88 0.92 0.92 0.98 0.91 0.89 0.95 0.98 0.93 0.98 0.93 0.92 [Precision, Recall, F-measure] Package Family (PCA) 0.97 0.96 0.84 0.92 0.88 0.95 0.97 0.93 0.90 0.92 0.97 0.95 0.87 0.94 0.90 1.00 0.99 0.96 0.99 0.97 0.98 0.95 0.87 0.93 0.90 0.92 0.92 0.86 0.88 0.87 Package (PCA) 0.94 0.95 0.94 0.97 0.95 0.96 0.92 0.96 0.94 0.97 1.00 0.99 0.91 0.97 0.94 0.88 0.89 0.89 Table 2: Precision, Recall, and F-measure obtained by M A M A D ROID when trained and tested with dataset from the same year in package and family mode, using Random Forests, and with and without PCA. classify malware vs benign samples, they need to be retrained following new API releases. On the contrary, retraining is not needed as often with M A M A D ROID, since families and packages represent more abstract functionalities that change less over time. Consider, for instance, the android.os.health package: released with API level 24, it contains a set of classes helping developers track and monitor system resources.9 Classification systems built before this release – as in the case of D ROIDAPIM INER [1] (released in 2013, when Android API was up to level 20) – need to be retrained if this package is more frequently used by malicious apps than benign apps, while M A M A D ROID only needs to add a new state to its Markov chain when operating in package mode, while no additional state is required when operating in family mode. when classifying future samples and using drebin (with samples from 2010 to 2012) or 2013 as the malicious training set and oldbenign (late 2013/early 2014) as the benign training set. More specifically, we observe that M A M A D ROID correctly detects benign apps, while it starts missing true positives and increasing false negatives – i.e., achieving lower recall. Newer training, older testing. We also set to verify whether older malware samples can still be detected by the system— if not, this would obviously become vulnerable to older (and possibly popular) attacks. Therefore, we also perform the “opposite” experiment, i.e., training M A M A D ROID with newer benign (March 2016) and malware (early 2014 to mid 2016) datasets, and checking whether it is able to detect malware developed years before. Specifically, Figure 11(a) and 11(b) report results when training M A M A D ROID with samples from a given year, and testing it with others that are up to 4 years older: M A M A D ROID retains similar F-measure scores over the years. Specifically, in family mode, it varies from 0.93 to 0.96, whereas, in package mode, from 0.95 to 0.97 with the oldest samples. Older training, newer testing. To verify this hypothesis, we test M A M A D ROID using older samples as training sets and newer ones as test sets. Figure 10(a) reports the average F-measure of the classification in this setting, with M A M A D ROID operating in family mode. The x-axis reports the difference in years between training and test malware data. We obtain 0.86 F-measure when we classify apps one year older than the samples on which we train. Classification is still relatively accurate, at 0.75, even after two years. Then, from Figure 10(b), we observe that the average F-measure does not significantly change when operating in package mode. Both modes of operations are affected by one particular condition, already discussed in Section 3: in our models, benign datasets seem to “anticipate” malicious ones by 1–2 years in the way they use certain API calls. As a result, we notice a drop in accuracy 4.4 Case Studies of False Positives and Negatives The experiment analysis presented above show that M A M A D ROID detects Android malware with high accuracy. As in any detection system, however, the system makes a small number of incorrect classifications, incurring some false positives and false negatives. Next, we discuss a few case studies aiming to better understand these misclassifications. We focus on the experiments with newer datasets, i.e., 2016 and 9 https://developer.android.com/reference/android/os/health/ package-summary.html 8 1.0 0.8 0.6 0.4 0.2 0.0 RF 1-NN 3-NN 0.8 F-measure F-measure 1.0 RF 1-NN 3-NN 0.6 0.4 0.2 0 1 2 Years 3 0.0 4 0 1 (a) family mode 2 Years 3 4 (b) package mode 1.0 1.0 0.8 0.8 0.6 0.6 F-measure F-measure Figure 10: F-measure achieved by M A M A D ROID using older samples for training and newer samples for testing. The x-axis shows the difference in years between the training and test data. 0.4 0.2 0.0 0.2 RF 1-NN 3-NN 0 1 2 Years 3 0.4 0.0 4 RF 1-NN 3-NN 0 (a) family mode 1 2 Years 3 4 (b) package mode Figure 11: F-measure achieved by M A M A D ROID using newer samples for training and older samples for testing. The x-axis shows the difference in years between the training and test data. 45% of M A M A D ROID’s false negatives are adware, typically, repackaged apps in which the advertisement library has been substituted with a third-party one, which creates a monetary profit for the developers. Since they are not performing any clearly malicious activity, M A M A D ROID is unable to identify them as malware. Finally, we find that 16% of the false negatives reported by M A M A D ROID are samples sending text messages or starting calls to premium services. We also do a similar analysis of false negatives when abstracting to packages (74 samples), with similar results: there a few more adware samples (53%), but similar percentages for potentially benign apps (15%) and samples sending SMSs or placing calls (11%). newbenign. False Positives. We analyze the manifest of 164 apps mistak- enly detected as malware by M A M A D ROID, finding that most of them use “dangerous” permissions [3]. In particular, 67% write to external storage, 32% read the phone state, and 21% access the device’s fine location. We further analyzed apps (5%) that use the READ SMS and SEND SMS permissions, i.e., even though they are not SMS-related apps, they can read and send SMSs as part of the services they provide to users. In particular, a “in case of emergency” app is able to send messages to several contacts from its database (possibly added by the user), which is a typical behavior of Android malware in our dataset, ultimately leading M A M A D ROID to flag it as malicious. 4.5 False Negatives. We also check 114 malware samples missed M A M A D ROID vs D ROIDAPIM INER We also compare the performance of M A M A D ROID to previous work using API features for Android malware classification, specifically, to D ROIDAPIM INER [1], because: (i) it uses API calls and its parameters to perform classification; (ii) it reports high true positive rate (up to 97.8%) on almost 4K malware samples obtained from McAfee and G ENOME [66], and 16K benign samples; and (iii) its source code has been by M A M A D ROID when operating in family mode, using VirusTotal.10 We find that 18% of the false negatives are actually not classified as malware by any of the antivirus engines used by VirusTotal, suggesting that these are actually legitimate apps mistakenly included in the VirusShare dataset. 10 https://www.virustotal.com 9 made available to us by the authors. the future and past respectively. Likewise, we use the same datasets for M A M A D ROID, with the best results achieved on the same dataset as D ROIDAPIM INER. In package mode, M A M A D ROID achieves an F-measure of 0.99 which is maintained more than two years into the past, but drops to respectively, 0.85 and 0.81 one and two years into the future As summarized in Table 3, M A M A D ROID achieves significantly higher performance than D ROIDAPIM INER in all but one experiment. This case occurs when the malicious training set is much older than the malicious test set. In D ROIDAPIM INER, permissions that are requested more frequently by malware samples than by benign apps are used to perform a baseline classification. Then, the system also applies frequency analysis on the list of API calls after removing API calls from ad libraries, using the 169 most frequent API calls in the malware samples (occurring at least 6% more in malware than benign samples). Finally, data flow analysis is applied on the API calls that are frequent in both benign and malicious samples, but do not occur by at least 6% more in the malware set. Using the top 60 parameters, the 169 most frequent calls change, and authors report a precision of 97.8%. 5 After obtaining D ROIDAPIM INER’s source code, as well as a list of packages (i.e., ad libraries) used for feature refinement, we re-implement the system by modifying the code in order to reflect recent changes in Androguard (used by D ROIDAPIM INER for API call extraction), extract the API calls for all apps in the datasets listed in Table 1, and perform a frequency analysis on the calls. Recall that Androguard fails to extract calls for about 2% (1,017) of apps, thus D ROIDAPIM INER is evaluated over the samples in the second-to-last column of Table 1. We also implement classification, which is missing from the code provided by the authors, using k-NN (with k=3) since it achieves the best results according to the paper. We use 2/3 of the dataset for training and 1/3 for testing as implemented by the authors. Finer-grained Abstraction In Section 4, we have showed that building models from abstracted API calls allows M A M A D ROID to obtain high accuracy, as well as to retain it over the years, which is crucial due to the continuous evolution of the Android ecosystem. Our experiments have focused on operating M A M A D ROID in family and package mode (i.e., abstracting calls to family or package). In this section, we investigate whether a finer-grained abstraction – namely, to classes – performs better in terms of detection accuracy. Recall that our system performs better in package mode than in family mode due to the system using in the former, finer and more features to distinguish between malware and benign samples, so we set to verify whether one can trade-off higher computational and memory complexities for better accuracy. To this end, as discussed in Section 2.3, we abstract each API call to its corresponding class name using a whitelist of all classes in the Android API, which consists of 4,855 classes (as of API level 24), and in the Google API, with 1,116 classes, plus self-defined and obfuscated. In Table 3, we report the results of D ROIDAPIM INER compared to M A M A D ROID on different combination of datasets. Specifically, we report results for experiments similar to those carried out in Section 4.3 as we evaluate its performance on dataset from the same year and over time. First, we train it using older dataset composed of oldbenign combined with one of the three oldest malware datasets each (drebin, 2013, and 2014), and test on all malware datasets. Testing on all datasets ensures the model is evaluated on dataset from the same year and newer. With this configuration, the best result (with 2014 and oldbenign as training sets) is 0.62 F-measure when tested on the same dataset. The F-measure drops to 0.33 and 0.39, respectively, when tested on samples one year into the future and past. If we use the same configurations in M A M A D ROID, in package mode, we obtain up to 0.97 Fmeasure (using 2013 and oldbenign as training sets), dropping to 0.73 and 0.94 respectively, one year into the future and into the past. For the datasets where D ROIDAPIM INER achieves its best result (i.e., 2014 and oldbenign), M A M A D ROID achieves an F-measure of 0.95, which drops to respectively, 0.78 and 0.93 one year into the future and the past. The F-measure is stable even two years into the future and the past at 0.75 and 0.92 respectively. As a second set of experiments, we train D ROIDAPIM INER using a dataset composed of newbenign (March 2016) combined with one of the three most recent malware datasets each (2014, 2015, and 2016). Again, we test D ROIDAPIM INER on all malware datasets. The best result is obtained with the dataset (2014 and newbenign) used for both testing and training, yielding a F-measure of 0.92, which drops to 0.67 and 0.75 one year into 5.1 Reducing the size of the problem Since there are 5,973 classes, processing the Markov chain transitions that results in this mode increases the memory requirements. Therefore, to reduce the complexity, we cluster classes based on their similarity. To this end, we build a cooccurrence matrix that counts the number of times a class is used with other classes in the same sequence in all datasets. More specifically, we build a co-occurrence matrix C, of size (5,973·5,973)/2, where Ci,j denotes the number of times the ith and the j-th class appear in the same sequence, for all apps in all datasets. From the co-occurrence matrix, we compute the y ·y x, y ) = ||xxx||·||y cosine similarity (i.e., cos(x y || ), and use k-means to cluster the classes based on their similarity into 400 clusters and use each cluster as the label for all the classes it contains. Since we do not cluster classes abstracted to self-defined and obfuscated, we have a total of 402 labels. 5.2 Class Mode Accuracy In Table 4, we report the resulting F-measure in class mode using the above clustering approach. Once again, we also report the corresponding results from package mode for comparison 10 drebin, oldbenign Training Sets Droid MaMa drebin & oldbenign 0.32 0.96 2013 & oldbenign 0.33 0.94 2014 & oldbenign 0.36 0.92 drebin, newbenign Training Sets Droid MaMa 2014 & newbenign 0.76 0.98 2015 & newbenign 0.68 0.97 2016 & newbenign 0.33 0.96 2013, oldbenign Droid MaMa 0.35 0.95 0.36 0.97 0.39 0.93 2013, newbenign Droid MaMa 0.75 0.98 0.68 0.97 0.35 0.98 Testing Sets 2014, oldbenign Droid MaMa 0.34 0.72 0.35 0.73 0.62 0.95 2014, newbenign Droid MaMa 0.92 0.99 0.69 0.99 0.36 0.98 2015, oldbenign Droid MaMa 0.30 0.39 0.31 0.37 0.33 0.78 2015, newbenign Droid MaMa 0.67 0.85 0.77 0.95 0.34 0.92 2016, oldbenign Droid MaMa 0.33 0.42 0.33 0.28 0.37 0.75 2016, newbenign Droid MaMa 0.65 0.81 0.65 0.88 0.36 0.92 Table 3: Classification performance of D ROIDAPIM INER (Droid) [1] vs M A M A D ROID (MaMa) in package mode using Random Forest. PP Mode P Dataset PP P drebin, oldbenign 2013, oldbenign 2014, oldbenign 2014, newbenign 2015, newbenign 2016, newbenign 0.95 0.98 0.93 0.98 0.93 0.91 [Precision, Recall, F-measure] Class Package 0.97 0.96 0.95 0.97 0.95 0.97 0.98 0.95 0.97 0.95 0.93 0.97 1.00 0.99 0.98 1.00 0.98 0.95 0.93 0.98 0.92 0.92 0.92 0.92 when the training samples are two years newer, M A M A D ROID achieves an F-measure of 0.99, 0.97, and 0.95 respectively, in class, package, and family modes. Whereas, when they are three years newer, the F-measure is respectively, 0.97, 0.97, and 0.96 in class, package, and family modes. 0.96 0.97 0.95 0.99 0.95 0.92 Table 4: M A M A D ROID’s Precision, Recall, and F-measure when trained and tested on dataset from the same year using Random Forests and API calls abstracted to classes and packages. 6 M A M A D ROID mainly relies on (1) API call abstraction, and (2) behavioral modeling via sequence of calls. As shown, it outperforms state-of-the-art Android detection techniques, such as D ROIDAPIM INER [1], that are based on the frequency of non-abstracted API calls. In this section, we aim to assess whether M A M A D ROID’s effectiveness mainly stems from the API abstraction, or from the sequence modeling. To this end, we implement and evaluate a variant that uses frequency, rather than sequences, of abstracted API calls. More precisely, we perform frequency analysis on the API calls extracted using Androguard after removing ad libraries, as also done in D ROIDAPIM INER. In the rest of the section, we denote this variant as FAM (Frequency Analysis Model). We again use the datasets in Table 1 to evaluate FAM’s accuracy when training and testing on datasets from the same year and from different years. We also evaluate how it compares to standard M A M A D ROID. Although we have also implemented FAM in class mode, we do not discuss/report results here due to space limitation. (cf Section 4.2). Overall, we find that class abstraction does not provide significantly higher accuracy. In fact, compared to package mode, abstraction to classes only yields an average increase in F-measure of 0.0012. 5.3 Frequency Analysis Model (FAM) Detection over time We also compare accuracy when M A M A D ROID is trained and tested on dataset from different years (Figure 12). We find that, when M A M A D ROID operates in class mode, it achieves an F-measure of 0.95 and 0.99, respectively, when trained with datasets one and two years newer than the test sets, as reported in Figure 12(a)). Likewise, when trained on datasets one and two years older than the test set, F-measure reaches 0.84 and 0.59, respectively (see Figure 12(b)). Overall, comparing results from Figure 10 to Figure 12(b), we find that finer-grained abstraction actually performs worse with time when older samples are used for training and newer for testing. We note that this is due to a possible number of reasons: 1) newer classes or packages in recent API releases cannot be captured in the behavioral model of older tools whereas, families are; and 2) evolution of malware either as a result of changes in the API or patching of vulnerabilities or presence of newer vulnerabilities that allows for stealthier malicious activities. On the contrary, Figure 11 and 12(a) show that finer-grained abstraction performs better when the training samples are more recent than the test samples. This is because from recent samples, we are able to capture the full behavioral model of older samples. However, our results indicate there is a threshold for the level of abstraction which when exceeded, finer-grained abstraction will not yield any significant improvement in detection accuracy. This is because API calls in older releases are subsets of subsequent releases. For instance, 6.1 FAM Accuracy We start our evaluation by measuring how well FAM detects malware by training and testing using samples that are developed around the same time. Figure 13 reports the F-measure achieved in family and package modes using three different classifiers. Also, Table 5 reports precision, recall, and Fmeasure achieved by FAM on each dataset combination, when operating in family and package mode, using Random Forests. We only report the results from the Random Forest classifier because it outperforms both the 1-NN and 3-NN classifiers. Family mode. Due to the number of possible families (i.e., 11), FAM builds a model from all families that occur more in our malware dataset than the benign dataset. Note that in this modeling approach, we also remove the junit family as it is mainly used for testing. When the drebin and 2013 mal11 1.0 0.8 0.8 0.6 0.6 F-measure F-measure 1.0 0.4 0.2 0.0 0 Class Package 0.4 0.2 Class Package 1 2 3 Years 0.0 4 0 1 (a) Newer/Older 2 Years 3 4 (b) Older/Newer Figure 12: F-measure achieved by M A M A D ROID in class mode when using newer (older) samples for training and older (newer) samples for testing. The x-axis shows the difference in years between the training and test data. 0.6 0.4 0.2 0.6 0.4 0.2 Drebin & 2013 & 2014 & 2014 & 2015 & 2016 & OldBenign OldBenign OldBenign Newbenign Newbenign Newbenign (a) family mode 0.0 1.0 RF 1-NN 3-NN 0.8 F-measure F-measure 0.8 0.0 1.0 RF 1-NN 3-NN RF 1-NN 3-NN 0.8 F-measure 1.0 0.6 0.4 0.2 Drebin & 2013 & 2014 & 2014 & 2015 & 2016 & OldBenign OldBenign OldBenign Newbenign Newbenign Newbenign (b) package mode 0.0 Drebin & 2013 & 2014 & 2014 & 2015 & 2016 & OldBenign OldBenign OldBenign Newbenign Newbenign Newbenign (c) class mode Figure 13: F-measure for the FAM variant, over same-year datasets, with different classifiers. ware datasets are used in combination with the oldbenign dataset, there are no families that are more frequently used in these datasets than the benign dataset. As a result, FAM does not yield any result with these datasets as it builds a model only from API calls that are more frequently used in malware than benign samples. With the other datasets, there are two (2016), four (2014), and five families (2015) that occur more frequently in the malware dataset than the benign one. Classification performance improves in package mode, with F-measure ranging from 0.53 with 2016 and newbenign to 0.89 with 2014 and newbenign, using Random Forests. Figure 13(b) shows that Random Forests generally provides better results also in this case. Similar to family mode, the drebin and 2013 datasets respectively, have only five and two packages that occur more than in the oldbenign dataset. Hence, the results when these datasets are evaluated is poor due to the limited amount of features. From Figure 13(a), we observe that F-measure is always at least 0.5 with Random Forests, and, when tested on the 2014 (malware) dataset, it reaches 0.87. In general, lower Fmeasures are due to increased false positives. This follows a similar trend observed in Section 4.3. Class mode. When FAM operates in class mode, we build a model using the minimum of, all API calls that occur more frequently in malware than benign samples or the top 336 API calls that occur more in malware than benign apps. We use the top 336 classes so as to build a model with classes from at least two different packages as the highest number of classes in a single package (java.util) is 335. In our datasets, there is at least three classes that occur more in malware (2013) than benign samples and at most, 862 classes that occur more frequently in malware (2015) than the benign dataset. When FAM operates in package mode, it builds a model using the minimum of, all API calls that occur more frequently in malware or the top 172 API calls used more frequently in malware than benign apps. We use the top 172 API calls as we attempt to build the model where possible, with packages from at least two families (the android family has 171 packages). In our dataset, there are at least two (2013) and at most 39 (2016) packages that are used more frequently in malware than in benign samples. Hence, all packages that occur more in malware than benign apps are always used to build the model. Package mode. In Figure 13(c), we report precision, recall, and F-measure achieved by FAM. It achieves its best result (0.89 F-measure) when trained and tested with the 2014 and newbenign datatsets. Also, we show in Table 5 how the three modes compare when trained and tested on dataset from the same year. Over12 PP Mode P Dataset PP P drebin, oldbenign 2013, oldbenign 2014, oldbenign 2014, newbenign 2015, newbenign 2016, newbenign 0.71 0.85 0.64 0.51 Family 0.76 0.90 0.70 0.49 [Precision, Recall, F-measure] Package - 0.51 0.57 0.54 0.50 - 0.53 0.57 0.55 0.58 0.73 0.73 0.73 0.73 0.73 0.87 0.88 0.89 0.89 0.88 0.67 0.68 0.66 0.67 0.68 0.50 0.53 0.52 0.53 0.54 Class 0.47 0.57 0.76 0.89 0.67 0.54 0.49 0.58 0.75 0.89 0.68 0.54 Table 5: Precision, Recall, and F-measure (with Random Forests) of FAM when trained and tested on dataset from the same year in family, package, and class modes. all, we find that abstraction to classes only slightly improves over package mode. Take-Away. Although we discuss in more detail the performance of the FAM variant vs the standard M A M A D ROID in Section 6.3, we can already observe that the former does not yield as robust of a model, mostly due to the fact that in some cases no abstracted API calls occur more in malware than benign samples. 6.2 Newer training, older testing. We also evaluate the opposite setting, i.e., training FAM with newer datasets, and checking whether it is able to detect malware developed years before. Specifically, Figure 15 report results when training FAM with samples from a given year, and testing it with others that are up to 4 years older showing that F-measure range from 0.69 to 0.92 in family mode, 0.65 to 0.94 in package mode, and 0.66 to 0.97 in class mode. Recall that in family mode, FAM is unable to build a model when drebin and 2013 are used for training, thus, effecting the overall result. This effect is minimal in this setting since the training sets are newer than the test sets, thus, the drebin dataset is not used to evaluate any dataset while the 2013 dataset is used in only one setting i.e., when the training set is one year newer than the test set. Detection Over Time Once again, we evaluate the detection accuracy over time, i.e., we train FAM using older samples and test it with newer samples and vice versa. We report the F-measure as the average of the F-measure over multiple dataset combinations; e.g., when training with newer samples and testing on older samples, the F-measure after three years is the average of the Fmeasure when training with (2015, newbenign) and (2016, newbenign) respectively, and testing on drebin and 2013. Older training, newer testing. In Figure 14(a), we show the Fmeasure when FAM operates in family mode and is trained with datasets that are older than the classified datasets. The x-axis reports the difference in years between training and test data. We obtain an F-measure of 0.97 when training with samples that are one year older than the samples in the testing set. As mentioned in Section 6.1, there is no result when the drebin and 2013 datasets are used for training, hence, after 3 years the F-measure is 0. In package mode, the F-measure is 0.81 after one year, and 0.76 after two (Figure 14(b)). Whereas in class mode, the F-measure after one and two years are 0.85 and 0.70 respectively (Figure 14(c)). While FAM appears to perform better in family mode than in package and class modes, note that the detection accuracy after one and two years in family mode does not include results when the training set is (drebin, oldbenign) or (2013, oldbenign) (cf Section 6.1). We believe this is as a result of FAM performing best when trained on the 2014 dataset in all modes and performing poorly in package mode when trained with (drebin, oldbenign) and (2013, oldbenign) due to limited features. For example, accuracy after two years is the average of the F-measure when training with (2014, oldbenign/newbenign) datasets and testing on the 2016 dataset. Whereas, in package mode, accuracy is the average Fmeasure obtained from training with (drebin, oldbenign), (2013, oldbenign), and (2014, oldbenign/newbenign) datasets and testing with respectively, 2014, 2015, and 2016. 6.3 Comparing Frequency Markov Chain Model Analysis vs. We now compare the detection accuracy of FAM– a variant of M A M A D ROID that is based on a frequency analysis model – to the standard M A M A D ROID, which is based on a Markov chain model using the sequence of abstracted API calls. Detection Accuracy of malware from same year. In Table 6, we report accuracy of FAM and M A M A D ROID when they are trained and tested on samples from the same year using Random Forests in all modes. For completeness, we also report results from D ROIDAPIM INER, showing that M A M A D ROID outperforms FAM and D ROIDAPIM INER in all tests. Both FAM and D ROIDAPIM INER performs best when trained and tested with (2014 and newbenign) with F-measures of 0.89 (package mode) and 0.92, respectively. Overall, M A M A D ROID achieves higher F-measure compared to FAM and D ROIDAPIM INER due to both API call abstraction and Markov chain modeling of the sequence of calls, which successfully captures the behavior of the app. In addition, M A M A D ROID is more robust as with some datasets, frequency analysis fails to build a model with abstracted calls when the abstracted calls occur equally or more frequently in benign samples. Detection Accuracy of malware from different years. We also compare FAM with M A M A D ROID when they are trained and tested with datasets across several years. In Table 7, we report the F-measures, achieved by M A M A D ROID and FAM in package mode using Random Forests, and show how they compare with D ROIDAPIM INER using two different sets of 13 1.0 0.8 0.8 0.8 0.6 0.6 0.6 0.4 0.2 0.0 0 2 Years 3 0.4 0.2 RF 1-NN 3-NN 1 F-measure 1.0 F-measure F-measure 1.0 0.0 4 0 (a) family mode 0.2 RF 1-NN 3-NN 1 2 Years 3 0.4 0.0 4 0 RF 1-NN 3-NN 1 (b) package mode 2 Years 3 4 (c) class mode 1.0 1.0 0.8 0.8 0.8 0.6 0.6 0.6 0.4 0.2 0.0 0 2 Years (a) family mode 3 0.4 0.2 RF 1-NN 3-NN 1 F-measure 1.0 F-measure F-measure Figure 14: F-measure achieved by FAM using older samples for training and newer samples for testing. The x-axis shows the difference in years between the training and test data. 4 0.0 0 0.2 RF 1-NN 3-NN 1 2 Years (b) package mode 3 0.4 4 0.0 0 RF 1-NN 3-NN 1 2 Years 3 4 (c) class mode Figure 15: F-measure achieved by FAM using newer samples for training and older samples for testing. The x-axis shows the difference in years between the training and test data. experiments. We report the results for package mode only because with both systems, when the training sets are older than the test sets, package mode performs better than class mode (although, less than family mode, there are no results for FAM with two of our training datasets in family mode) and only slightly lower when the training sets are newer than the test sets. using samples comprising the newbenign and one of the three recent malware datasets (2014, 2015, 2016) each, and testing on all malware datasets. In this setting, M A M A D ROID outperforms both FAM and D ROIDAPIM INER in all but one experiment where FAM is only slightly better. Comparing D ROIDAPIM INER and FAM shows that D ROIDAPIM INER only performs better than FAM in two out of 15 experiments. In these two experiments, FAM was trained and tested on samples from the same year and resulted in a slightly lower precision, thus, increasing false positives. Overall, we find that the Markov chain based model achieves higher detection accuracy in both family and package modes when M A M A D ROID is trained and tested on dataset from the same year (Table 6) and across several years (Table 7). In the first set of experiments, we train M A M A D ROID, FAM, and D ROIDAPIM INER using samples comprising the oldbenign and one of the three oldest malware datasets (drebin, 2013, 2014) each, and testing on all malware datasets. M A M A D ROID and FAM both outperform D ROIDAPIM INER in all experiments in this set, showing that abstracting the API calls improves the detection accuracy of our systems. FAM outperforms M A M A D ROID in nine out of the 15 experiments, largely, when the training set comprises the drebin/2013 and oldbenign datasets. Recall that when drebin and 2013 malware datasets are used for training FAM in package mode, only five and two packages respectively, are used to build the model. It is possible that these packages are the principal components (as in PCA) that distinguishes malware from benign samples. In the second set of experiments, we train M A M A D ROID, FAM, and D ROIDAPIM INER 7 Runtime Performance We now analyze the runtime performance of M A M A D ROID and and the FAM variant, when operating in family, package, or class mode, as well as D ROIDAPIM INER. We run our experiments on a desktop with an 40-core 2.30GHz CPU and 14 F-measure XX Mode Family Package Class Dataset XXX X FAM M A M A D ROID FAM M A M A D ROID FAM M A M A D ROID D ROIDAPIM INER XX drebin, oldbenign 2013, oldbenign 2014, oldbenign 2014, newbenign 2015, newbenign 2016, newbenign 0.73 0.87 0.67 0.50 0.88 0.92 0.92 0.98 0.91 0.89 0.96 0.97 0.95 0.99 0.95 0.92 0.54 0.55 0.73 0.89 0.67 0.53 0.49 0.58 0.75 0.89 0.68 0.54 0.96 0.97 0.95 0.99 0.95 0.92 0.32 0.36 0.62 0.92 0.77 0.36 Table 6: F-measure of FAM and M A M A D ROID in all modes using Random Forests as well as, D ROIDAPIM INER [1] when trained and tested on dataset from the same year. Testing Sets drebin, oldbenign 2013, oldbenign 2014, oldbenign 2015, oldbenign 2016, oldbenign Training Sets Droid FAM MaMa Droid FAM MaMa Droid FAM MaMa Droid FAM MaMa Droid FAM MaMa drebin,oldbenign 0.32 0.54 0.96 0.35 0.50 0.96 0.34 0.50 0.79 0.30 0.50 0.42 0.33 0.51 0.43 2013,oldbenign 0.33 0.90 0.93 0.36 0.55 0.97 0.35 0.95 0.74 0.31 0.87 0.36 0.33 0.82 0.29 2014,oldbenign 0.36 0.95 0.92 0.39 0.99 0.93 0.62 0.73 0.95 0.33 0.81 0.79 0.37 0.82 0.78 drebin, newbenign 2013, newbenign 2014, newbenign 2015, newbenign 2016, newbenign Training Sets Droid FAM MaMa Droid FAM MaMa Droid FAM MaMa Droid FAM MaMa Droid FAM MaMa 2014,newbenign 0.76 0.99 0.99 0.75 0.99 0.99 0.92 0.89 0.99 0.67 0.86 0.89 0.65 0.82 0.83 2015,newbenign 0.68 0.92 0.98 0.68 0.84 0.98 0.69 0.95 0.99 0.77 0.67 0.95 0.65 0.91 0.90 2016,newbenign 0.33 0.83 0.97 0.35 0.69 0.97 0.36 0.91 0.99 0.34 0.86 0.93 0.36 0.53 0.92 Table 7: F-Measure of M A M A D ROID (MaMa) vs our variant using frequency analysis (FAM) vs D ROIDAPIM INER (Droid) [1]. M A M A D ROID’s third step includes Markov chain modeling and feature vector extraction. With malicious samples, it takes on average 0.2s±0.3, 2.5s±3.2, and 1.49s±2.39 (and at most 2.4s, 22.1s, and 46.10s), respectively, with families, packages, and classes, whereas, with benign samples, it takes 0.6s±0.3, 6.7s±3.8, and 2.23s±2.74 (at most 1.7s, 18.4s, and 43.98s). Finally, the last step is classification, and performance depends on both the machine learning algorithm employed and the mode of operation. More specifically, running times are affected by the number of features for the app to be classified, and not by the initial dimension of the call graph, or by whether the app is benign or malicious. Regardless, in family mode, Random Forests, 1-NN, and 3-NN all take less than 0.01s. With packages, it takes, respectively, 0.65s, 1.05s, and 0.007s per app with 1-NN, 3-NN, Random Forests. Whereas, it takes, respectively, 1.02s, 1.65s, and 0.05s per app with 1NN, 3-NN, and Random Forests in class mode. Overall, when operating in family mode, malware and benign samples take on average, 10.7s and 27.3s respectively to complete the entire process, from call graph extraction to classification. In package mode, the average completion times for malware and benign samples are 13.37s and 33.83s respectively. Whereas, in class mode, the average completion times are respectively, 21.7s and 41.12s for malware and benign apps. In both modes of operation, time is mostly (>80%) spent on call graph extraction. 128GB of RAM, but only use 1 core and allocate 16GB of RAM for evaluation. 7.1 M A M A D ROID We envision M A M A D ROID to be integrated in offline detection systems, e.g., run by the app store. Recall that M A M A D ROID consists of different phases, so in the following, we review the computational overhead incurred by each of them, aiming to assess the feasibility of real-world deployment. M A M A D ROID’s first step involves extracting the call graph from an apk and the complexity of this task varies significantly across apps. On average, it takes 9.2s±14 (min 0.02s, max 13m) to complete for samples in our malware sets. Benign apps usually yield larger call graphs, and the average time to extract them is 25.4s±63 (min 0.06s, max 18m) per app. Next, we measure the time needed to extract call sequences while abstracting to families, packages or classes depending on M A M A D ROID’s mode of operation. In family mode, this phase completes in about 1.3s on average (and at most 11.0s) with both benign and malicious samples. Abstracting to packages takes slightly longer, due to the use of 341 packages in M A M A D ROID. On average, this extraction takes 1.67s±3.1 for malicious apps and 1.73s±3.2 for benign samples. Recall that in class mode, after abstracting to classes, we cluster the classes to a smaller set of labels due to its size. Therefore, in this mode it takes on average, 5.84s±2.1 and 7.3s±4.2 respectively, to first abstract the calls from malware and benign apps to classes and 2.74s per app to build the co-occurrence matrix from which we compute the similarity between classes. Finally, clustering and abstracting each call to its corresponding class label takes 2.38s and 3.4s respectively, for malware and benign apps. In total, it takes 10.96s to abstract calls from malware apps to their corresponding class labels and 13.44s for benign apps. 7.2 FAM Recall that FAM is a variant of M A M A D ROID including three phases. The first one, API calls extraction, takes 0.7s±1.5 (min 0.01s, max 28.4s) per app in our malware datasets and 13.2s±22.2 (min 0.01s, max 222s) per benign app. The second phase includes API call abstraction, frequency analysis, and feature extraction. While API call abstraction is depen15 dent on the dataset and the mode of operation, frequency analysis and feature extraction are only dependent on the mode of operation and are very fast in all modes. In particular, it takes on average, 1.32s, 1.69s±3.2, and 5.86s±2.1 respectively, to complete a malware app in family, package, and class modes. Whereas, it takes on average, 1.32s±3.1, 1.75s±3.2, and 7.32s±2.1 respectively, for a benign app in family, package, and class modes. The last phase which is classification is very fast regardless of dataset, mode of operation, and classifier used. Specifically, it takes less than 0.01s to classify each app in all modes using the three different classifiers. Overall, it takes in total 2.02s, 2.39s, and 6.56s respectively, to classify a malware app in family, package, and class modes. While with benign apps, the total is 14.52s, 14.95s, and 20.52s respectively, in family, package, and class modes. 7.3 and Markov chains, discuss possible evasion techniques, and highlight some limitations of our approach. 8.1 Our work yields important insights around the use of API calls in malicious apps, showing that, by abstracting the API calls to higher levels and modeling these abstracted calls, we can obtain high detection accuracy and retain it over several years which is crucial due to the continuous evolution of the Android ecosystem. As discussed in Section 3, the use of API calls changes over time, and in different ways across malicious and benign samples. From our newer datasets, which include samples up to Spring 2016 (API level 23), we observe that newer APIs introduce more packages, classes, and methods, while also deprecating some. Figure 7(a) shows that benign apps use more calls than malicious ones developed around the same time. We also notice an interesting trend in the use of Android (Figure 7(b)) and Google (Figure 7(c)) APIs: malicious apps follow the same trend as benign apps in the way they adopt certain APIs, but with a delay of some years. This might be a side effect of Android malware authors repackaging benign apps, adding their malicious functionalities onto them. Given the frequent changes in the Android framework and the continuous evolution of malware, systems like D ROIDAPIM INER [1] – being dependent on the presence or the use of certain API calls – become increasingly less effective with time. As shown in Table 3, malware that uses API calls released after those used by samples in the training set cannot be identified by these systems. On the contrary, as shown in Figure 10, M A M A D ROID detects malware samples that are 1 year newer than the training set obtaining an Fmeasure of 0.86 (as opposed to 0.46 with D ROIDAPIM INER) when the apps are modeled as Markov chains. After 2 years, the value is still at 0.75 (0.42 with D ROIDAPIM INER), dropping to 0.51 after 4 years. We argue that the effectiveness of M A M A D ROID’s classification remains relatively high “over the years” owing to the Markov models capturing app’s behavior. These models tend to be more robust to malware evolution because abstracting to e.g., packages makes the system less susceptible to the introduction of new API calls. To verify this, we developed a variant of M A M A D ROID named FAM that abstracts API calls and is based on frequency analysis similar to D ROIDAPIM INER. Although, the addition of API call abstraction results in an improvement of the detection accuracy of the system (an Fmeasure of 0.81 and 0.76 after one and two years respectively), it also resulted in scenarios where there are no API calls that are more frequently used in malware than benign apps. In general, abstraction allows M A M A D ROID to capture newer classes/methods added to the API, since these are abstracted to already-known families or packages. In case newer packages are added to the API, and these packages start being used by malware, M A M A D ROID only requires adding a new state to the Markov chains, and probabilities of a transition from a state to this new state in old apps would be 0. In D ROIDAPIM INER Finally, we evaluate the runtime performance of D ROIDAPIM INER [1]. Its first step, i.e., extracting API calls, takes 0.7s±1.5 (min 0.01s, max 28.4s) per app in our malware datasets. Whereas, it takes on average 13.2s±22.2 (min 0.01s, max 222s) per benign app. In the second phase, i.e., frequency and data flow analysis, it takes, on average, 4.2s per app. Finally, classification using 3-NN is very fast: 0.002s on average. Therefore, in total, D ROIDAPIM INER takes respectively, 17.4s and 4.9s for a complete execution on one app from our benign and malware datasets, which while faster than M A M A D ROID, achieves significantly lower accuracy. In comparison to M A M A D ROID, D ROIDAPIM INER takes 5.8s and 9.9s less on average to analyze and classify a malicious and benign app when M A M A D ROID operates in family mode and 8.47s and 16.43s less on average in package mode. 7.4 Take Away In conclusion, our experiments show that our prototype implementation of M A M A D ROID is scalable enough to be deployed. Assuming that, everyday, a number of apps in the order of 10,000 are submitted to Google Play, and using the average execution time of benign samples in family (27.3s), package (33.83s), and class (41.12s) modes, we estimate that it would take less than an hour and a half to complete execution of all apps submitted daily in both modes, with just 64 cores. Note that we could not find accurate statistics reporting the number of apps submitted everyday, but only the total number of apps on Google Play.11 On average, this number increases of a couple of thousands per day, and although we do not know how many apps are removed, we believe 10,000 apps submitted every day is likely an upper bound. 8 Lessons Learned Discussion We now discuss the implications of our results with respect to the feasibility of modeling app behavior using static analysis 11 http://www.appbrain.com/stats/number-of-android-apps 16 reality though, methods and classes are more frequently added than packages with new API releases. Hence, we also evaluate whether M A M A D ROID still performs as well as in package mode when we abstract an API call to class and measure the overall overhead increase. Results from Figure 14, 15, and 12 indicate that finer-grained abstraction is less effective as time passes when older samples are used for training and newer samples for testing, while they are more effective when samples from the same year or newer than the test sets are used for training. However, while all three modes of abstraction performs relatively well, we believe abstraction to packages is the most effective as it generally performs better than family – though less lightweight – and as well as class but more efficient. 8.2 modeled, as it is not Java and is not processed by Soot. These limitations are not specific of M A M A D ROID, but are a problem of static analysis in general, which can be mitigated by using M A M A D ROID alongside dynamic analysis techniques. Another approach could be using dynamic dispatch so that a class X in package A is created to extend class Y in package B with static analysis reporting a call to root() defined in Y as X.root(), whereas, at runtime Y.root() is executed. This can be addressed, however, with a small increase in M A M A D ROID’s computational cost, by keeping track of self-defined classes that extend or implement classes in the recognized APIs, and abstract polymorphic functions of this self-defined class to the corresponding recognized package, while, at the same time, abstracting as self-defined overridden functions in the class. Finally, identifier mangling and other forms of obfuscation could be used aiming to obfuscate code and hide malicious actions. However, since classes in the Android framework cannot be obfuscated by obfuscation tools, malware developers can only do so for self-defined classes. M A M A D ROID labels obfuscated calls as obfuscated so, ultimately, these would be captured in the behavioral model (and the Markov chain) for the app. In our sample, we observe that benign apps use significantly less obfuscation than malicious apps, indicating that obfuscating a significant number of classes is not a good evasion strategy since this would likely make the sample more easily identifiable as malicious. Malware developers might also attempt to evade M A M A D ROID by naming their selfdefined packages in such a way that they look similar to that of the android or google APIs, e.g., java.lang.reflect.malware is easily prevented by first abstracting to classes before abstracting to any further modes as we already do in Section 5. Evasion Next, we discuss possible evasion techniques and how they can be addressed. One straightforward evasion approach could be to repackage a benign app with small snippets of malicious code added to a few classes. However, it is difficult to embed malicious code in such a way that, at the same time, the resulting Markov chain looks similar to a benign one. For instance, our running example from Section 2 (malware posing as a memory booster app and executing unwanted commands as root) is correctly classified by M A M A D ROID; although most functionalities in this malware are the same as the original app, injected API calls generate some transitions in the Markov chain that are not typical of benign samples. The opposite procedure, i.e., embedding portions of benign code into a malicious app, is also likely ineffective against M A M A D ROID, since, for each app, we derive the feature vector from the transition probability between calls over the entire app. A malware developer would have to embed benign code inside the malware in such a way that the overall sequence of calls yields similar transition probabilities as those in a benign app, but this is difficult to achieve because if the sequences of calls have to be different (otherwise there would be no attack), then the models will also be different. An attacker could also try to create an app with a similar Markov chain to that of a benign app. Because this is derived from the sequence of abstracted API calls in the app, it is actually very difficult to create sequences resulting in Markov chains similar to benign apps while, at the same time, actually engaging in malicious behavior. Moreover, attackers could try using reflection, dynamic code loading, or native code [42]. Because M A M A D ROID uses static analysis, it fails to detect malicious code when it is loaded or determined at runtime. However, M A M A D ROID can detect reflection when a method from the reflection package (java.lang.reflect) is executed. Therefore, we obtain the correct sequence of calls up to the invocation of the reflection call, which may be sufficient to distinguish between malware and benign apps. Similarly, M A M A D ROID can detect the usage of class loaders and package contexts that can be used to load arbitrary code, but it is not able to model the code loaded; likewise, native code that is part of the app cannot be 8.3 Limitations M A M A D ROID requires a sizable amount of memory in order to perform classification, when operating in package mode, working on more than 100,000 features per sample. The quantity of features, however, can be further reduced using feature selection algorithms such as PCA. As explained in Section 6 when we use 10 components from the PCA the system performs almost as well as the one using all the features; however, using PCA comes with a much lower memory complexity in order to run the machine learning algorithms, because the number of dimensions of the features space where the classifier operates is remarkably reduced. Soot [51], which we use to extract call graphs, fails to analyze some apks. In fact, we were not able to extract call graphs for a fraction (4.6%) of the apps in the original datasets due to scripts either failing to apply the jb phase, which is used to transform Java bytecode to the primary intermediate representation (i.e., jimple) of Soot or not able to open the apk. Even though this does not really affect the results of our evaluation, one could avoid it by using a different/custom intermediate representation for the analysis or use different tools to extract the call graphs. In general, static analysis methodologies for malware detection on Android could fail to capture the runtime environment 17 context, code that is executed more frequently, or other effects stemming from user input [4]. These limitations can be addressed using dynamic analysis, or by recording function calls on a device. Dynamic analysis observes the live performance of the samples, recording what activity is actually performed at runtime. Through dynamic analysis, it is also possible to provide inputs to the app and then analyze the reaction of the app to these inputs, going beyond static analysis limits. To this end, we plan to integrate dynamic analysis to build the models used by M A M A D ROID as part of future work. 9 this still takes a non-negligible toll on battery life. Recently, hybrid systems like IntelliDroid [55] have also been proposed that use input generators, producing inputs specific to dynamic analysis tools. Other work combining static and dynamic analysis include [23, 28, 58, 7]. 9.2 A number of techniques have used signatures for Android malware detection. NetworkProfiler [16] generates network profiles for Android apps and extracts fingerprints based on such traces, ASTROID [20] used common subgraphs among samples as signatures for identifying unknown malware samples, while work in [10] obtains resource-based metrics (CPU, memory, storage, network) to distinguish malware activity from benign one. [13] extract statistical features, such as permissions and API calls, and extend their vectors to add dynamic behavior-based features. While their experiments show that their solution outperforms, in terms of accuracy, other antivirus systems, [13] indicate that the quality of their detection model critically depends on the availability of representative benign and malicious apps for training. MADAM [46] also extract features at four layers which are used to build a behavioral model for apps and uses two parallel classifier to detect malware. Similarly, ScanMe Mobile [64] uses the Google Cloud Messaging Service (GCM) to perform static and dynamic analysis on apks found on the device’s SD card. While, TriFlow [37] rank apps based on potential risks by using the observed and possible information flows in the apps. The sequences of system calls have also been used to detect malware in both desktop and Android environments. Hofmeyr et al. [27] demonstrate that short sequences of system calls can be used as a signature to discriminate between normal and abnormal behavior of common UNIX programs. Signaturebased methods, however, can be evaded using polymorphism and obfuscation, as well as by call re-ordering attacks [33], even though quantitative measures, such as similarity analysis, can be used to address some of these attacks [48]. M A M A D ROID inherits the spirit of these approaches, proposing a statistical method to model app behavior that is more robust against evasion attempts. In the Android context, [9] use the sequences of three system calls (extracted from the execution traces of apps under analysis) to detect malware. This approach models specific malware families, aiming to identify additional samples belonging to such families. In contrast, M A M A D ROID’s goal is to detect previously-unseen malware, and we also show that our system can detect new malware families that even appear years after the system has been trained. In addition, using strict sequences of system or API calls can be easily evaded by malware authors who could add unnecessary calls to effectively evade detection. Conversely, M A M A D ROID builds a behavioral model of an Android app, which makes it robust to this type of evasion. Dynamic analysis has also been applied to detect Android malware by using predefined scripts of common inputs that will be performed when the device is running. However, this Related Work Over the past few years, Android security has attracted a wealth of work by the research community. In this section, we review (i) program analysis techniques focusing on general security properties of Android apps, and then (ii) systems that specifically target malware on Android. 9.1 Android Malware Detection Program Analysis Previous work on program analysis applied to Android security has used both static and dynamic analysis. With the former, the program’s code is decompiled in order to extract features without actually running the program, usually employing tools such as Dare [41] to obtain Java bytecode. The latter involves real-time execution of the program, typically in an emulated or protected environment. Static analysis techniques include work by Felt et al. [19], who analyze API calls to identify over-privileged apps, while Kirin [18] is a system that examines permissions requested by apps to perform a lightweight certification, using a set of security rules that indicate whether or not the security configuration bundled with the app is safe. RiskRanker [26] aims to identify zero-day Android malware by assessing potential security risks caused by untrusted apps. It sifts through a large number of apps from Android markets and examines them to detect certain behaviors, such as encryption and dynamic code loading, which form malicious patterns and can be used to detect stealthy malware. Other methods, such as CHEX [34], use data flow analysis to automatically vet Android apps for vulnerabilities. Static analysis has also been applied to the detection of data leaks and malicious data flows from Android apps [5, 32, 62, 31]. DroidScope [59] and TaintDroid [17] monitor run-time app behavior in a protected environment to perform dynamic taint analysis. DroidScope performs dynamic taint analysis at the machine code level, while TaintDroid monitors how thirdparty apps access or manipulate users’ personal data, aiming to detect sensitive data leaving the system. However, as it is unrealistic to deploy dynamic analysis techniques directly on users’ devices, due to the overhead they introduce, these are typically used offline [45, 67, 49]. ParanoidAndroid [44] employs a virtual clone of the smartphone, running in parallel in the cloud and replaying activities of the device – however, even if minimal execution traces are actually sent to the cloud, 18 might be inadequate due to the low probability of triggering malicious behavior, and can be side-stepped by knowledgeable adversaries, as suggested by Wong and Lie [55]. Other approaches include random fuzzing [35, 63] and concolic testing [2, 24]. Dynamic analysis can only detect malicious activities if the code exhibiting malicious behavior is actually running during the analysis. Moreover, according to [53], mobile malware authors often employ emulation or virtualization detection strategies to change malware behavior and eventually evade detection. Machine learning techniques have also been applied to assist Android malware detection. Droidmat [57] uses API call tracing and manifest files to learn features for malware detection, Teufl et al. [50] apply knowledge discovery processes and lean statistical methods on app metadata extracted from the app market, while [22] rely on embedded call graphs. DroidMiner [60] studies the program logic of sensitive Android/Java framework API functions and resources, and detects malicious behavior patterns. MAST [12] statically analyzes apps using features such as permissions, presence of native code, and intent filters and measures the correlation between multiple qualitative data. Crowdroid [8] relies on crowdsourcing to distinguish between malicious and benign apps by monitoring system calls. AppContext [61] models security-sensitive behavior, such as activation events or environmental attributes, and uses SVM to classify these behaviors, while RevealDroid [21] employs supervised learning and obfuscation-resilient methods targeting API usage and intent actions to identify their families. D REBIN [4] deduces detection patterns and identifies malicious software directly on the device, performing a broad static analysis. This is achieved by gathering numerous features from the manifest file as well as the app’s source code (API calls, network addresses, permissions). Malevolent behavior is reflected in patterns and combinations of extracted features from the static analysis: for instance, the existence of both SEND SMS permission and the android.hardware.telephony component in an app might indicate an attempt to send premium SMS messages, and this combination can eventually constitute a detection pattern. In Section 4.5, we have compared against D ROIDAPIM INER [1]. This system relies on the top169 API calls that are used more frequently in the malware than in the benign set, along with data flow analysis on calls that are frequent in both benign and malicious apps, but occur up to 6% more in the latter. As shown in our evaluation, using the most common calls observed during training requires constant retraining, due to the evolution of both malware and the Android API. On the contrary, M A M A D ROID can effectively model both benign and malicious Android apps, and perform an efficient classification on them. Compared to D ROIDAPIM INER, our approach is more resilient to changes in the Android framework than D ROIDAPIM INER, resulting in a less frequent need to re-train the classifier. Overall, compared to state-of-the-art systems like D REBIN [4] and D ROIDAPIM INER [1], M A M A D ROID is more generic and robust as its statistical modeling does not depend on specific app characteristics, but can actually be run on any app created for any Android API level. Finally, also related to M A M A D ROID are Markov-chain based models for Android malware detection. [14] dynamically analyze system- and developer-defined actions from intent messages (used by app components to communicate with each other at runtime), and probabilistically estimate whether an app is performing benign or malicious actions at run time, but obtain low accuracy overall. Canfora et al. [11] use a Hidden Markov model (HMM) to identify malware samples belonging to previously observed malware families, whereas, M A M A D ROID can detect previously unseen malware, not relying on specific malware families. 10 Conclusion This paper presented M A M A D ROID, an Android malware detection system that is based on modeling the sequences of API calls as Markov chains. Our system is designed to operate in one of three modes, with different granularities, by abstracting API calls to either families, packages or classes. We ran an extensive experimental evaluation using, to the best of our knowledge, the largest malware dataset in an Android malware detection research paper, and aiming at assessing both the accuracy of the classification (using F-measure, precision, and recall) and runtime performances. We showed that M A M A D ROID effectively detects unknown malware samples developed around the same time as the samples on which it is trained (F-measure up to 0.99). It also maintains good detection performance: one year after the model has been trained with a F-measure of 0.86, and 0.75 after two years. We compared M A M A D ROID to D ROIDAPIM INER [1], a state-of-the-art system based on API calls frequently used by malware, showing that, not only does M A M A D ROID outperforms D ROIDAPIM INER when trained and tested on datasets from the same year, but that it is also much more resilient over the years to changes in the Android API. We also developed a variant of M A M A D ROID, called FAM, that performs API call abstraction but is based on frequency analysis to evaluate whether M A M A D ROID’s high detection accuracy is based solely on the abstraction. We found that FAM improves on D ROIDAPIM INER but, while abstraction is important for high detection rate and resilience to API changes, abstraction and a modeling approach based on frequency analysis is not as robust as M A M A D ROID, especially in scenarios where API calls are not more frequent in malware than in benign apps. Overall, our results demonstrate that statistical behavioral models introduced by M A M A D ROID—in particular, abstraction and Markov chain modeling of API call sequence—are more robust than traditional techniques, highlighting how our work can form the basis of more advanced detection systems in the future. As part of future work, we plan to further investigate the resilience to possible evasion techniques, focusing on repackaged malicious apps as well as injection of API calls to maliciously alter Markov models. We also plan to explore 19 the possibility of seeding the behavioral modeling performed by M A M A D ROID with dynamic instead of static analysis. Acknowledgments. We wish to thank Yousra Aafer for sharing the D ROIDAPIM INER source code and Yanick Fratantonio for his comments on an early draft of the paper. This research was supported by the EPSRC under grant EP/N008448/1, by an EPSRC-funded “Future Leaders in Engineering and Physical Sciences” award, a Xerox University Affairs Committee grant, and by a small grant from GCHQ. Enrico Mariconti was supported by the EPSRC under grant 1490017, while Lucky Onwuzurike was funded by the Petroleum Technology Development Fund (PTDF). [12] S. Chakradeo, B. Reaves, P. Traynor, and W. Enck. 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J. Comput. Phys. Journal of Computational Physics 00 (2017) 1–33 arXiv:1708.08741v1 [] 29 Aug 2017 Coupled Multiphysics Simulations of Charged Particle Electrophoresis for Massively Parallel Supercomputers Dominik Bartuschata,∗, Ulrich Rüdea,b a Lehrstuhl für Systemsimulation, Friedrich-Alexander Universität Erlangen-Nürnberg, Cauerstrasse 11, 91058 Erlangen, Germany b Parallel Algorithms Group, CERFACS, 42 Avenue Gaspard Coriolis, 31057 Toulouse, France Abstract The article deals with the multiphysics simulation of electrokinetic flows. When charged particles are immersed in a fluid and are additionally subjected to electric fields, this results in a complex coupling of several physical phenomena. In a direct numerical simulation, the dynamics of moving and geometrically resolved particles, the hydrodynamics of the fluid, and the electric field must be suitably resolved and their coupling must be realized algorithmically. Here the two-relaxation-time variant of the lattice Boltzmann method is employed together with a momentum-exchange coupling to the particulate phase. For the electric field that varies in time according to the particle trajectories, a quasistatic continuum model and its discretization with finite volumes is chosen. This field is coupled to the particulate phase in the form of an acceleration due to electrostatic forces and conversely via the respective charges as boundary conditions for the electric potential equation. The electric field is also coupled to the fluid phase by modeling the effect of the ion transport on fluid motion. With the multiphysics algorithm presented in this article, the resulting multiply coupled, interacting system can be simulated efficiently on massively parallel supercomputers. This algorithm is implemented in the waLBerla framework, whose modular software structure naturally supports multiphysics simulations by allowing to flexibly combine different models. The largest simulation of the complete system reported here performs more than 70 000 time steps on more than five billion (5 × 109 ) mesh cells for both the hydrodynamics, as represented by a D3Q19 lattice Boltzmann automaton, and the scalar electric field. The computations are executed in a fully scalable fashion on up to 8192 processor cores of a current supercomputer. c 2017 Published by Elsevier Ltd. Keywords: Parallel simulation; Electrokinetic flow; Electrophoresis; Fluid-particle interaction; MPI. 1. Introduction 1.1. Motivation The motion of charged particles in fluids under the influence of electric fields occurs in a wide range of industrial, medical, and biological processes. When the charged particles are immersed in liquids, their migration caused by electric fields is termed electrophoresis. Due to the complex interplay of the physical effects involved in such particle-laden electrokinetic flows, numerical simulations are required to analyze, ∗ Corresponding author Email address: [email protected] (Dominik Bartuschat) 1 D. Bartuschat and U. Rüde / Journal of Computational Physics 00 (2017) 1–33 2 predict, and optimize the behavior of these processes. To this end, we present a parallel multiphysics algorithm for direct numerical simulations of electrophoretic particle motion. Industrial applications that involve electrophoretic effects are electrofiltration [1, 2, 3] and electrodewatering [4]. Moreover, electrophoresis is utilized in electrophoretic deposition techniques for fabricating advanced materials [5] and especially ceramic coatings [6, 7] in material science. Electrophoresis and electric fields are also applied in many medical and biological applications. The trend towards miniaturization of analysis processes has lead to the development of micro total analysis systems. Due to their high portability, reduced costs, fast operation, and high sensitivity [8, 9], the design of such lab-on-a-chip systems has been a highly active area of research for many years. These microfluidic systems require only small samples of liquid and particles, which are transported, manipulated, and analyzed in structures of length scales from several nm to 100 µm. Therefore, microfluidic separation and sorting of particles and cells are important steps of diagnostics in such systems [10, 11]. Many of the employed techniques utilize electric fields to manipulate, separate, and sort biological particles and macromolecules [8], such as cells [9, 10] or DNA [12]. At the small scales of microfluidic analysis systems, flow measurements are difficult or even impossible. Moreover, the complex coupling of hydrodynamic and electrostatic effects involved in electrophoretic processes make predictions of electrophoretic motion challenging, especially for large numbers of particles. Therefore, numerical simulations are essential to aid the design and optimization of electrophoretic systems. The different physical effects in electrophoretic deposition can be better understood from insight gained in simulations. By means of such simulations, electrophoretic sorting in lab-on-a-chip systems can be optimized for maximal throughput, sorting efficiency, and sorting resolution. A review of simulation methods for electrophoretic separation of macromolecules is given in [13]. Also industrial applications of electrophoretic deposition can be optimized with the help of simulations, as presented in [14] for a coating process. 1.2. Multiphysics Coupling Strategy For simulations of electrokinetic flows with electrophoretic particle motion, the coupling between three system components must be modeled: charged objects, fluid flow, and electric effects. The interacting physical effects are sketched in Fig. 1. In the simulation method introduced in this article, the motion of the ce on io ty ns io ot e si m rg n de n n a ch r fo a st e rc fo io el tro ec tic Electroquasi-statics object motion Rigid body dynamics Fluid dynamics hydrodynamic force Figure 1: Coupled physical effects of electrophoresis simulated with waLBerla and pe. rigid, charged particles is modeled with Newtonian mechanics. The motion of the surrounding fluid, which exerts hydrodynamic forces on the particles, is described by the incompressible Navier-Stokes equation. To capture fluid-particle interactions, the hydrodynamic forces and the influence of the particle motion on the fluid are modeled, based on the momentum exchange between fluid and particles. In this way, long-range hydrodynamic interactions between individual particles and between particles and walls are recovered. Moreover, electrostatic forces exerted by applied electric fields on the charged particles are modeled, which cause the electrophoretic motion. The varying positions of the charged particles in return affect the electric 2 D. Bartuschat and U. Rüde / Journal of Computational Physics 00 (2017) 1–33 3 potential distribution in the simulation domain, based on their surface charge. Such a charge is carried by most biomolecules such as cells, proteins, and DNA [15]. In fact, most substances acquire surface charges when they get into contact with an aqueous medium [16] or electrolyte solution [17]. Electrostatic forces additionally act on the ions in the fluid and affect the fluid motion via body forces present in locations of net charge. This net charge originates from the repulsion of co-ions in the fluid of the same polarity as the surface charge and from the attraction of counter-ions. The ions in the fluid are transported with the fluid flow, which in turn alters the electric potential distribution. In general, the motion of ions in the fluid due to an electric field is governed by the Nernst-Planck equation, an advection-diffusion equation that employs a continuum description and treats the ions as point charges. For flows in which diffusion strongly dominates over advection and therefore quasi-thermodynamic equilibrium can be assumed to hold, as considered in this paper, the electric potential due to the ion charge distribution is governed by the Poisson-Boltzmann equation. The region of the particle’s surface charge and of excess counter-ions in the fluid is denoted as electric double layer (EDL). According to the Stern model [18] employed in this article, this double layer comprises a region of ions attached to the surface and a diffuse part in which the ion concentration follows a Boltzmann distribution. At the surface of shear between the particle and the surrounding diffuse double layer, the characteristic ζ-potential is defined. The employed equilibrium considerations based on the PoissonBoltzmann equation capture the dominant retardation effect of electrophoretic motion. This effect describes the retardation of the charged particle motion by the action of the applied electric field on the opposite net charge in the surrounding EDL. At high ζ-potentials, additionally the weaker relaxation effect occurs [16] that is caused by a distortion of the EDL and can be captured by the Nernst-Planck equation. In this article, we present an efficient parallel multiphysics simulation algorithm for electrophoresis on a fixed Eulerian grid with a Lagrangian representation of moving particles. The particles are represented by the physics engine pe [19, 20] as geometrically fully resolved three-dimensional objects. Dependent on the electrostatic and hydrodynamic forces acting on the particles, the pe computes their trajectories by rigid body dynamics and additionally resolves particle collisions. The pe is coupled to waLBerla [21, 22, 23], a massively parallel simulation framework for fluid flow applications that employ the lattice Boltzmann method (LBM) [24, 25]. By means of a modular software structure that avoids dependencies between modules, functionality from different modules can be combined flexibly. To model fluid-particle interactions, the LBM momentum exchange method [26, 27] implemented in waLBerla [28] is applied. For the electrophoresis simulations, the LBM is performed with the two-relaxation time collision operator [29, 30] and an appropriate forcing term for the electric body force due to the ions in the fluid. The electric potential is represented by the Debye-Hückel approximation of the Poisson-Boltzmann equation that is discretized with finite volumes whose mesh structure naturally conforms to the lattice Boltzmann (LB) grid. This discretization also facilitates accommodating variable and discontinuous dielectricity values that vary in time according to the particle positions, as required for simulating dielectrophoretic effects. By means of the waLBerla solver module introduced in [31] together with the cell-centered multigrid solver implemented therein, the resulting linear system of equations is solved. Since the counter-ions lead to a quicker decay of the electric potential compared to the long-range electric potentials modeled in [31], the parallel successive over-relaxation (SOR) method implemented in this module is an adequate choice. In a previous article, we have shown that the implemented fluid-particle interaction algorithm for arbitrarily shaped objects efficiently recovers hydrodynamic interactions also for elongated particles [32]. Moreover, we presented a parallel multiphysics algorithm for charged particles in the absence of ions in the fluid in [31]. We have therein shown that several millions of charged particles with long-range hydrodynamic and electrostatic interactions can be simulated with excellent parallel performance on a supercomputer. The present paper extends these simulation algorithms by also considering ions in the EDL around the particles and their effect on the fluid motion, and presenting suitable parallel coupling techniques. Together with the full four-way coupling of the fluid-particle interaction [31], the coupling with the quasi-equilibrium representation of the electric effects results in a 7.5-way interaction. 3 D. Bartuschat and U. Rüde / Journal of Computational Physics 00 (2017) 1–33 4 1.3. Related Work In the following, we give an overview of numerical methods for simulations of electrophoretic phenomena that have been developed for different resolution levels. At the coarsest modeling scale, both fluid and solid phase are described by an Eulerian approach. These continuum models represent the charged species in terms of concentrations and are well suited for simulations with large numbers of unresolved particles. In [33] two-dimensional electrophoresis simulations of biomolecule concentrations are presented. These finite element simulations consider the effect of reactive surfaces and the electric double layer is included through the biomolecule charge. Also three-dimensional parallel simulations of electrophoretic separation with continuum approaches have been reported. The finite element simulations in [34] consider the buffer composition and the ζ-potential at channel walls. In [35, 36] mixed finite element and finite difference simulations of protein separation were performed on up to 32 processes. At the finest level of resolution, fluid, ions, and particles are simulated by Lagrangian approaches. These explicit solvent methods [37] typically apply coarse-grained molecular dynamics (MD) models to describe the motion of fluid molecules and incorporate Brownian motion [38]. The mesoscale dissipative particle dynamics method is applied in [39] to simulate electrophoresis of a polyelectrolyte in a nanochannel. Another explicit solvent method is presented in [40] for simulating DNA electrophoresis, modeling DNA as a polymer. In both methods the polymer is represented by bead-spring chains with beads represented by a truncated Lennard-Jones potential and connected by elastic spring potentials. Explicit solvent models, however, are computationally very expensive, especially for large numbers of fluid molecules due to pairwise interactions [40]. Moreover, the resolution of solvent, macromolecules, and ions on the same scale limits the maximal problem sizes that can be simulated [41]. Also the mapping of measurable properties from colloidal suspensions to these particle-based methods is problematic [37]. The high computational effort is significantly reduced in implicit particle-based methods that incorporate hydrodynamic interactions into the inter-particle forces. Such methods are applied in [37] and [42] to simulate electrophoretic deposition under consideration of Brownian motion and van der Waals forces. Nevertheless, these methods are restricted to few particle shapes and hydrodynamic interactions in Stokes flow. Euler-Lagrange methods constitute the intermediate level of resolution. These approaches employ Eulerian methods to simulate the fluid phase, whereas the motion of individual particles is described by Newtonian mechanics. For simulations of particles in steady-state motion, the resolved particles can be modeled as fixed while the moving fluid is modeled by an Eulerian approach. In [43] the finite volume method is applied to simulate electrophoresis of up to two stagnant toroids in a moving fluid, employing the Hückel approximation for a fixed ζ-potential and different electrical double layer thicknesses. The steady-state electrophoretic motion of particles with low surface potentials under a weak applied electric field in a charged cylindrical pore is simulated in [44] for a single cylinder and in [45] for two identical spheres. In both cases, a two-dimensional simulation with a finite element method for Stokes flow and Hückel approximation is performed, exploiting the axial symmetry of the problem. For electrophoresis at steady state perturbation approaches can be employed that are based on the assumption that the double layer is only slightly distorted from the equilibrium distribution for weak applied electric fields (w.r.t. the field in the EDL). In addition to the equilibrium description based on the PoissonBoltzmann equation, small perturbations in the equilibrium EDL are considered in terms of linear correction terms in the applied electric field for the ion distribution and the electric potential (see e. g. [46]). Using a perturbation approach with finite elements, the electrophoresis of two identical spheres along the symmetry axis of a cylindrical domain at pseudo steady-state were studied in [47]. Additionally to the hydrodynamic and electric interactions, these axisymmetric simulations consider van der Waals forces for particles in close proximity. In [48] a perturbation approach is applied to simulate a single colloid in a rest frame with periodic boundary conditions. The zeroth-order perturbation corresponds to the Poisson-Boltzmann equation, which is solved by a constrained variational approach suggested in [49]. For the first-order perturbation, the stationary Stokes equation is solved by a surface element method, the convection-diffusion for ionic concentrations by a finite volume solver, and the Poisson equation by a fast Fourier Transform method. More sophisticated Euler-Lagrange methods include direct numerical simulation (DNS) models that represent the moving particles as geometrically fully resolved objects. These methods for particulate flows 4 D. Bartuschat and U. Rüde / Journal of Computational Physics 00 (2017) 1–33 5 include approaches with body-fitted moving meshes and fixed meshes. Moving meshes can be represented by the Arbitrary-Lagrangian-Eulerian (ALE) formulation [50, 51] that employs moving unstructured meshes for fluid-particle interaction problems. Such an ALE method is applied in [52] to simulate electrophoresis of a single particle surrounded by a thin electrical double layer. The moving-mesh techniques require re-meshing when the distortion of the updated mesh becomes too high, and subsequent projection of the solution onto the new mesh. This overhead is circumvented in fixed-mesh techniques that allow the use of regular grids and therefore the application of efficient solvers. The fluid particle dynamics method [53] falls into the latter category, solving the Navier-Stokes and continuity equation on a fixed lattice, and representing the moving solid particles by fluid particles of very high viscosity. By means of a concentration field that represents the particle distribution, the particles affect the fluid viscosity and, together with forces acting on the particles, the body force term of the Navier-Stokes equation. The rigid particles are geometrically modeled by the Lennard-Jones potential and their motion is described by Newtonian mechanics [53]. This technique is applied to simulate electrophoretic deposition of two charged particles in [54] and electrophoretic separation in [55]. In these simulations the electrostatic interactions of the particles are modeled in terms of the body force field, together with the advection and diffusion of ions and the resulting effect of the applied electric field on the fluid motion. With this method, however, the particle rigidity is imposed by very high viscosity values that restrict the time step size [56] and the Lennard-Jones potential restricts the particles shapes to spheres. The smoothed profile method (SPM) [56] circumvents this time-step constraint by directly modeling the particles as solid objects. Inside the particles and at solid-fluid boundaries that are represented by diffuse interfaces, a body force is imposed on the fluid to model the effect of the particle motion on the fluid. The fluid is again modeled on a fixed Cartesian grid and the particle motion with Newtonian mechanics, where particle overlaps are typically prevented by a truncated Lennard-Jones potential. With this method, electrophoresis of charged spherical particles is simulated in [57] for a constant, uniform electric field and in [58] for an oscillating electric field. In both articles, the ion number concentration is modeled by an advection-diffusion equation to recover non-equilibrium double layer effects. The SPM is also applied in [59] to simulate electrophoresis of single cylinders and microtubules, employing the equilibrium representation of the EDL. A further fixed-mesh technique is the immersed boundary method [60, 61, 62], where the rigid body motion is imposed on the flow by body forces applied at the particle boundaries. This method, combined with a finite volume method for solving the steady-state Poisson-Nernst-Planck equation system, is applied in [63] to simulate the electrophoretic motion of up to three spherical particles in a two-dimensional setup. Lattice-Boltzmann based methods are very well suited for parallel direct numerical simulations of fluidparticle interactions on fixed Cartesian grids. Both the Lagrangian particles and ions are often explicitly modeled by molecular dynamics approaches that represent the rigid objects by repulsive potentials. In [64] the electrophoresis of a colloidal sphere immersed in a fluid with counter-ions is simulated, modeling the solvent by a lattice Boltzmann method. The charged sphere modeled with molecular dynamics is represented by a raspberry model that comprised several beads connected by the finitely extensible nonlinear-elastic (FENE) potential. Using a modified raspberry model with two spherical shells of beads solidly attached to a larger spherical particle, this method is extended in [65] to simulate the electrophoresis of a spherical Janus particle. The partially uncharged particle is surrounded by anions and cations represented by charged beads. Electrophoresis simulations for a single highly charged spherical macro-ion in an electrolyte solution with explicitly modeled positive and negative micro-ions are presented in [66]. Since the coupling of fluid and macro-ion is performed via several particle boundary points, a single spherical particle is sufficient to represent the macro-ion. A similar LB-MD method is applied in [67] to simulate the stretching of a charged polyelectrolyte between parallel plates. The polyelectrolyte immersed in a liquid with explicitly modeled counter- and co-ions is modeled by beads bonded together by the FENE potential. In all these LB-MD simulations with explicitly modeled ions, hydrodynamic interactions are simulated with the LBM and thermal fluctuations are added to both the fluid and the MD objects. The high computational effort for modeling each individual ion by means of molecular dynamics, however, restricts the maximum feasible problem size. With these approaches, only a limited number of ions per colloidal particle can be simulated and the colloid radius is typically restricted to one order of magnitude larger than the ion size [48]. Therefore, approaches based on continuum descriptions of the suspended ions are more practical for simulations of many or for larger charged particles. 5 D. Bartuschat and U. Rüde / Journal of Computational Physics 00 (2017) 1–33 6 Alternatively to the continuum approach based on the Poisson-Boltzmann equation employed in this article, electrophoresis can be simulated with the link-flux method [68] that models the advection and diffusion of ions in terms of Nernst-Planck equations. The link-flux method employs the LBM for fluid dynamics and models ion motion in terms of fluxes between lattice cells. In [69] this method is compared to a LB-Poisson-Boltzmann approach for a fixed spherical particle in a periodic three-dimensional domain. The aim is to examine the influence of particle motion and counter-ion concentration on the ζ-potential, leading to the conclusion that for weakly perturbing electric fields or low Péclet numbers the equilibrium and dynamic ζ-potentials are indistinguishable. The link-flux method is extended in [41] to support moving particles in combination with the LB momentum exchange method. To ensure charge conservation in electrophoresis simulations, appropriate moving boundary conditions for the solute fluxes are introduced. This method is verified in [41] by electrophoresis simulations of up to eight particles. 1.4. Objectives and Outline The primary goal of this paper is the introduction of a parallel multiphysics algorithm for electrophoresis simulations together with validations of the physical correctness of the coupled algorithm for different particle sizes. For this algorithm, waLBerla is augmented by an efficient boundary handling method that is able to treat electric potential boundary conditions on the moving particles. Moreover, a joint parameterization for the different coupled numerical methods is introduced. To achieve excellent computational performance, a matrix-free representation of the linear system based on a stencil paradigm is used in the solver module [31]. For the linear Debye-Hückel approximation it is systematically exploited that these stencils are almost uniformly identical throughout the simulation domain. The validation runs were performed on up to 8192 parallel processes of a modern supercomputer. Moreover, simulation results for the electrophoretic motion of a single particle in a microchannel are presented, including visualizations of the electric potential distribution and of the resulting flow field around the particle. The equilibrium considerations in the present paper recover the predominant retardation effect due to an opposing electrostatic force on the net opposite charge in the electrical double layer that counteracts the particle motion. For the presented method, a computationally cheap and flexible SOR method is sufficient to solve the electric potential equations. With our approach we aim for simulations of millions of charged particles as in [31]. For these large numbers of particles the dynamics of an electrical double layer as in [41] is computationally too expensive, even on modern supercomputers. This paper is structured as follows: The physical background of fluid-particle interactions and electrophoresis are described in Sec. 2 and Sec. 3, respectively. In Sec. 4, the employed LB-momentum exchange method for fluid-particle interactions is outlined, together with the finite volume discretization and the common parameterization concept for the coupled multiphysics methods. Then the extension of the waLBerla framework for the electrophoresis algorithm is described in Sec. 5. Finally validation results for the electrophoretic motion of a spherical particle and visualizations of the resulting flow field and electric potential distribution are presented in Sec. 6 before conclusions are drawn in Sec. 7. 2. Fluid-Particle Interaction The macroscopic description of fluid behavior is based on the continuum hypothesis (cf. Batchelor [70]) that allows to consider a fluid as continuum, irrespective of the underlying molecular structure. In this case, fluid properties can be represented by macroscopic quantities like density ρf , velocity ~u, and pressure p, as functions of space and time. In terms of these quantities, fluid dynamics is described by conservation laws for mass, momentum, and energy. In this article isothermal flows are considered, and therefore the energy equation does not have to be solved. Moreover, non-continuum effects that become relevant for gas flows at very small scales [71] are assumed to be negligible. Therefore, no-slip boundary conditions are assumed to hold at solid-fluid interfaces, and slip velocities due to non-continuum Knudsen layer effects are not considered. Conservation of mass is described by the continuity equation. This equation can be derived by considering a fixed control volume in the fluid relative to a stationary observer (Eulerian view). For incompressible fluids 6 D. Bartuschat and U. Rüde / Journal of Computational Physics 00 (2017) 1–33 7 the density of a fluid element is not affected by pressure changes [70, 72]. Then the density is spatially and temporally constant (i. e. ρf = const), and the continuity equation reads as ∇· ~u = 0. (1) Conservation of momentum in a viscous, compressible fluid can be described in terms of the momentum flux density tensor Π [72]. The temporal change of momentum in a control volume is balanced by the net momentum flux through the surface of this volume and by external body forces f~b acting on the volume as ∂ (ρf ~u) = −∇· Π + f~b . ∂t (2) The second-order tensor Π comprises a term for the convective transport of momentum and the total stress tensor σ for the momentum transfer due to pressure and viscosity Π = ρf ~u~u> − σ. (3) The total stress tensor σ can be decomposed into a part representing normal stresses related to the pressure and a viscous part related to shear stresses. For incompressible Newtonian fluids, the stress tensor reads as   (4) σ = −pI + µf ∇~u + (∇~u)> , where the first term with the second-rank identity tensor I contains the thermodynamic pressure p defined according to Landau & Lifshitz [72] as used in the LBM literature [73]. The second term with dynamic viscosity µf represents the shear stresses that are proportional to the rate of deformation [74] and result from molecular transport of momentum [75]. With this stress tensor the incompressible Navier-Stokes equation results from Eqn. (2), together with basic vector calculus and the continuity equation for compressible fluids [76], as   ∂~u ρf + (~u · ∇)~u = −∇p + µf ∆ ~u + f~b . (5) | {z } |{z} ∂t | {z } | {z } pressure stress viscous stress external body force inertial forces This equation describes the balance of momentum change and the net force acting on a control volume in terms of Newton’s second law. The left-hand side represents inertial forces acting on a fluid volume. It comprises a term for local change of velocity and a term for convective acceleration i. e. the change of velocity in space [77]. The right-hand side represents surface forces and body forces acting on the fluid volume. Surface forces are short-range forces acting on the surface of the fluid element and are equivalent to stress in the fluid [70]. Body forces, such as gravity or electrostatic force, act on the center of mass and are represented by the force per unit volume of fluid element (or force density) f~b . An important dimensionless quantity to characterize fluid flows is the Reynolds number Re = UνfL , with µ kinematic viscosity νf = ρff , characteristic velocity U , and characteristic length scale L. Flows in the regime of creeping motion, where Re  1 and thus inertial forces are negligible, are termed Stokes flow. The Stokes equations for incompressible Newtonian fluids resulting from Eqns. (5) and (1) read as −∇p + µf ∆ ~u + f~b = 0, ∇· ~u = 0. (6) These equations are linear in both, velocity and pressure. Due to the linearity the superposition principle holds, which is often utilized for fluid-particle interaction in Stokes flow and is employed for the validations in Sec. 6.3. Particles immersed in a fluid experience a force in case of relative fluid motion or a pressure gradient in the fluid (e. g. due to gravity). This force exerted by the surrounding fluid can be calculated from the 7 D. Bartuschat and U. Rüde / Journal of Computational Physics 00 (2017) 1–33 8 stresses in the fluid next to the particle by integrating the stress tensor σ over the particle surface Γp [74, 46] Z F~part = σ · ~n dA. (7) Γp Here, dA denotes the surface area elements and ~n the associated normal vector pointing into the fluid. For simple cases, such as spherical bodies moving in unbounded Stokes flow, or single bodies moving in a confined domain, analytical solutions for the particle motion are known as described in the following. More complex fluid-structure interaction problems must be solved numerically, e. g. by the LBM. For the computation of particle motion in incompressible fluids, hydrostatic effects that result e. g. from gravitation acting on the fluid, do not have to be considered explicitly in the momentum equation (if the buoyancy and gravitational force are directly applied to the particle). In this case, the force exerted by the fluid on the particle according to Eqns. (7) and (4) is the drag force given by Z Z ~ ~ ~ Fd = − p dA + µf ∇~u + (∇~u)> dA, Γp Γp where p is the hydrodynamic part of the total pressure. The resistance to the motion of a sphere in an unbounded fluid at very low Reynolds numbers can be calculated analytically from the above expression for the drag force and the Stokes equations (6) that govern the fluid flow w.r.t. the imposed boundary conditions (BCs). For a sphere located at ~x = ~0, the unbounded fluid is represented by the BC ~u → 0 as ~x → ∞ imposed on the fluid velocity in the Stokes equations (6). The resulting drag force acting on a rigid sphere ~ was derived by Stokes [78] as of radius R that moves at constant velocity U ~. F~d = −6πµf RU (8) This equation is commonly referred to as Stokes’ law. For a sphere moving in a fluid subject to a constant force F~ , Stokes’ law relates the terminal steady-state velocity of the sphere to the drag force exerted by the fluid. Such a constant force may e. g. be the Coulomb force that acts on a charged particle in an electric field. The terminal sphere velocity is then obtained from the balance of the external force and the drag force F~ + F~d = ~0 as ~ = U 1 F~ . 6πµf R (9) A particle moving in a confined domain experiences a retardation caused by surrounding walls. Consequently, the drag force on a sphere that moves in a viscous fluid limited by walls is higher than the force according to Stokes’ law. The effect of walls on a moving particle in Stokes regime can be determined by means of the method of reflections, as described in detail in [74]. Happel & Bart [79] employed this method to obtain a first-order correction to the drag force on a sphere settling in a long square duct with no-slip walls. Miyamura et al. [80] found polynomial expressions for the increased drag by fitting the coefficients to experimentally obtained settling velocities of spheres in different confining geometries. The correctness of the wall effect recovered in LBM simulations with the fluid-particle interaction algorithm employed in this article was verified in [76] against these expressions. 3. Electrokinetic Flows The transport of ions in fluids subject to electric fields that occurs in electrokinetic flows can be modeled by means of a continuum theory, similar to the description of fluid dynamics by the Navier-Stokes equation. 8 D. Bartuschat and U. Rüde / Journal of Computational Physics 00 (2017) 1–33 9 Instead of modeling individual ions and their interactions, local ion concentrations ni and fluxes ~ji of the different ionic species i are considered. Based on these macroscopic quantities the ion transport in dilute electrolyte can be described by the Nernst-Planck equation combined with the law for the conservation of ionic species in a solution ∂ni = −∇· ~ji (10) ∂t in the absence of chemical reactions. Here ni denotes the number density or concentration that is related to the molar concentration as ci = NnAi with Avogadro’s number NA . The total ionic flux ~ji of species i comprises an advective flux with a common mass average velocity ~u for all species and fluxes relative to the advective flux due to diffusion and electric migration [46]. This relation is expressed by the Nernst-Planck equation ~ji = ni ~u − Di ∇ni − ni µ∗i ∇Φ, (11) |{z} | {z } | {z } advective flux diffusive flux migration flux where Di and represent the spatially homogeneous diffusion coefficient and ionic mobility of species i, respectively, and ∇Φ the local electric potential gradient. The ionic mobility is defined as µ∗i =: DkiBzTi e , where e denotes the elementary charge, zi the valence of a given ionic species, kB the Boltzmann constant and T the temperature. To model the influence of the charged ions on the electric potential governed by the Poisson equation µ∗i − ∆ Φ(~x) = ρe εe (12) for spatially uniform fluid permittivity εe , the ion charge distribution is considered in terms of the local mean macroscopic charge density as X ρe = e zi ni . (13) i The Poisson-Nernst-Planck equation system Eqns. (10)–(12) is highly nonlinear, and solving the overall system is computationally very expensive, especially for electrophoresis of many particles. Therefore the problem is simplified by restriction to equilibrium considerations based on the Boltzmann distribution that capture the dominant electrophoretic effects. The resulting Poisson-Boltzmann equation holds for (quasi-)thermodynamic equilibrium when the ion distribution is not affected by fluid flow or by externally applied electric fields. Therefore the electric potential ψ resulting from the non-uniform ion distribution in the EDL is considered in the following, instead of the total electric potential Φ = ψ + ϕ that additionally comprises the potential ϕ of the externally applied electric field. The Boltzmann distribution for ions can be derived from the Nernst-Planck equation, as outlined in [46]. Considering the Nernst-Planck equation (11) in one dimension and at equilibrium, i. e., for zero macroscopic fluid velocity u = 0 and ionic flux ji = 0, results in d ni zi e d ψ = −ni dx kB T dx (14) for the above definition of µ∗i . In this case, the Péclet number P e = UDL for mass transfer relating the advection rate to diffusion rate becomes zero. Here U is the fluid speed, L a characteristic length scale, and D the diffusion coefficient. Applying the chain rule to the left-hand side of Eqn. (14) and integrating from a reference point in the bulk with potential ψ∞ and concentration ni∞ , yields zi e(ψ − ψ∞ ) kB T ni = ni∞ e . − (15) Setting the reference potential ψ∞ in the electroneutral bulk solution to zero recovers the Boltzmann distribution with the number density ni∞ at the location of the neutral state. 9 D. Bartuschat and U. Rüde / Journal of Computational Physics 00 (2017) 1–33 10 From Poisson’s equation (12) with net charge density according to Eqn. (13) and the obtained Boltzmann distribution, the Poisson-Boltzmann equation follows as zi e ψ − e X − ∆ψ = zi ni∞ e kB T , εe i (16) relating the electric potential ψ to the ion concentrations at equilibrium. For binary, symmetric electrolyte solutions comprising two species of valence z = −z− = z+ , the Poisson-Boltzmann equation takes the form   2 z e n∞ zeψ − ∆ψ = − sinh . (17) εe kB T For low ζ-potentials compared to the thermal voltage kB T /e, the term kzBe ψT in Eqn. (17) becomes smaller than unity. At room temperature this is fulfilled for ζ  25.7zmV [81]. In this case the approximation sinh(x) ≈ x is accurate, up to a small error of order O(x3 ) (cf. Taylor’s expansion). With this linearization, the symmetric Poisson-Boltzmann equation simplifies to the Debye-Hückel approximation (DHA) − ∆ψ = − 2 e2 z 2 n∞ ψ = −κ2 ψ. εr ε0 kB T (18) This equation was originally derived by Debye & Hückel [82] for strong electrolytes [81]. The parameter κ, defined by r εr ε0 kB T κ := , (19) 2 e2 z 2 n∞ is commonly referred to as Debye-Hückel parameter. Moreover, the charge density in the fluid is then given by ρe (ψ) = −κ2 εe ψ. (20) In this article, we consider spherical particles with uniform ζ-potential distribution as depicted in Fig. 2. The electric potential ψ for such a particle of radius R is represented by the Debye-Hückel equation in E∞ r ψ =ζ ~u = −U θ R x λD ~ex Figure 2: Electrophoresis setup of a stationary (negatively) charged sphere of radius R, surrounded by a double layer and subject to an applied electric field in opposing fluid flow. Similar to [16]. spherical-polar coordinates as 1 d r2 dr  r2 dψ dr 10  = κ2 ψ, (21) D. Bartuschat and U. Rüde / Journal of Computational Physics 00 (2017) 1–33 11 with radial distance r from the sphere center and subject to the Dirichlet BCs ψ=ζ ψ→0 at as r = R, r → ∞. (22) Solving this equation subject to these BCs results in the electric potential distribution in the surrounding EDL (and beyond) as [81] R ψ(r) = ζ e−κ(r−R) for r ≥ R. (23) r For the Debye-Hückel approximation, an analytical solution for the electrophoretic velocity of a single spherical particle at steady state and arbitrary EDL thickness has been derived by Henry [83]. Initially the problem was formulated to account for finite conductivities of particle and medium, and the potential in the EDL was described by the Poisson-Boltzmann equation. The final results, however, were provided for insulating spheres, and sufficiently low ζ-potentials for the Debye-Hückel approximation to hold. Therefore the Debye-Hückel approximation is employed in the following to represent the ion distribution around the particles under electrophoretic motion. Instead of modeling a sphere moving at terminal speed U , Henry [83] considered the analogous problem of a stationary sphere in a steadily moving liquid. To this end, the opposing velocity −U was imposed on the overall system by setting ~u r→∞ = −U ~ex in the liquid far from the particle. As shown in Fig. 2, a sphericalpolar coordinate system fixed at the particle center with radial distance r and polar angle θ was used. Under the assumption that the electric potential in the EDL is not distorted from its equilibrium distribution by the applied field and the fluid flow, the potentials ϕ and ψ were linearly superimposed. Therefore, the electric potential ψ in the diffuse double layer is described by the Poisson-Boltzmann equation and the applied potential ϕ by a Laplace equation. The BCs for the Laplace equation applied by Henry represent the insulating particle by homogeneous Neumann BCs at the particle surface and impose the applied field by the inhomogeneous Neumann condition ∂ϕ/∂x r→∞ = −E∞ . For the Poisson-Boltzmann equation, the ζ-potential at the hydrodynamic radius R and the decaying potential were imposed as given in Eqn. (22). Making use of the equations for the electric potential, the Stokes equations for steady-state creeping flow with body force term on the right-hand side −µf ∆ u + ∇p = −ρe ∇ (ϕ + ψ) ∇· ~u = (24) 0, (25) were solved by Henry [83]. In addition to the BC imposing the opposing velocity far from the particle to bring the whole system to rest, the no-slip condition ~u r=R = 0 was applied at the particle surface. From the flow field around the particle, the force acting on the particle was obtained by integrating the normal stresses over the sphere surface. To the resulting force that comprises Stokes drag and electric components, the electrostatic force on the particle due to its fixed surface charge was added. The total force must vanish at steady motion and was thus equated with zero, resulting in the electrophoretic velocity    ZR ZR ψ ψ ε ~ ext ~ EP = e ψR + R3 5R2 dr − 2 dr E (26) U µf r6 r4 ∞ ∞ | {z } =ζ f (κR) obtained by Henry for an insulating particle. The function f (κR) introduced in [83] is usually referred to as Henry’s function. In [84] the following expression is derived that approximates the integral equations as   f (κR) = 2  1 + 3 1  2 1+ 11 2.5 κR (1 + 2e−κR )   3  ,  (27) D. Bartuschat and U. Rüde / Journal of Computational Physics 00 (2017) 1–33 12 with a relative error below 1 % for all values of κR [84]. In terms of Henry’s function f (κR) [83], the electrophoretic mobility of a spherical, non-conducting particle reads as εe ζ f (κR) . (28) µEP = µf This solution is correct to the first order of the ζ-potential, since the relaxation effect is neglected [85]. With the definition of the electrophoretic mobility µEP := UEP /Eext as electrophoretic speed per unit applied field the electrophoretic velocity of the particle can be obtained. Henry’s analytical solution for the electrophoretic velocity of a spherical particle of radius R in an unbounded electrolyte solution of dynamic viscosity µf with Debye-Hückel parameter κ, subject to an ~ ext reads as applied field of strength E   ~ EP = 2εe ζ  U 1 + 3µf 1 h 2 1+ 2.5 κR(1+2e−κR )  ~ i3  E ext , (29) according to the expression for the electrophoretic mobility of Ohshima given in Eqn. (27). For the electrophoresis simulations in this article, the particle charge must be known to compute the electrostatic force on the particle. Analytical solutions for electrophoretic motion such as Henry’s equation, however, are typically given in terms of the ζ-potential, which is defined at the slip surface between the compact and diffuse EDL layer. Since the particle charge is acquired as a surface charge, for a given ζpotential the surface charge (density) enclosed by the slip surface is therefore needed in the simulations. The surface charge density is hereby obtained from the overall surface charge bound at the fluid-particle interface and in the Stern layer. This approach is justified by the fact that the electric potential at the Stern surface and the ζ-potential can in general be assumed to be identical [86]. The relation of the surface density σs to the ζ-potential is obtained from the Neumann BC on the surface of the insulating particle [85] dψ σs = −εe (30) d~r r=R in case these electrical properties do not vary in angular direction. This condition holds for insignificant permittivity of the insulating particle compared to the fluid permittivity εe . Alternatively, this relation can be derived from the electroneutrality condition [46]. With the spatial distribution of ψ around the spherical particle according to Eqn. (23) the ζ–σs relationship follows from Eqn. (30) as   qs 1 + κR σs = = ζ ε . (31) e 4πR2 R For a spherical particle with an EDL potential ψ described by the spherical symmetric Poisson-Boltzmann equation, the more general ζ–σs relationship v      u zeζ u 8 ln cosh   zeζ u 2 1 2εe κkB T 1 4kB T u   +    u1 + sinh  (32) σs =  2 zeζ zeζ ze κR  2kB T t  (κR) 2 2 cosh sinh 4kB T 2kB T for 1-1 and 2-1 electrolyte solutions is derived in [87]. The relative error of this approximation w.r.t. the exact numerical results computed by [88] is below 1 % for 0.5 ≤ κR < ∞ [85]. This relationship is applied in the electrophoresis simulation validation in Sec. 6.3 to compute the particle charge for a given ζ-potential. The applied ζ–σs relationship is derived for electric potentials governed by the spherical Poisson-Boltzmann equation and is thus more general than the ζ–σs relationship (31) for the Debye-Hückel approximation. For the simulation parameters used in Sec. 6.2, the deviation of σs for the general relationship w.r.t. the exact value of the Debye-Hückel approximation is about 0.2 % and hence negligible. 12 D. Bartuschat and U. Rüde / Journal of Computational Physics 00 (2017) 1–33 13 4. Numerical Modeling 4.1. Lattice Boltzmann Method with Forcing The LBM is a mesoscopic method for the numerical simulation of fluid dynamics based on kinetic theory of gases. This method statistically describes the dynamics of ensembles of fluid molecules in terms of particle distribution functions (PDFs) that represent the spatial and velocity distribution of molecules in phase space over time. The temporal and spatial variation of PDFs, balanced by molecular collisions, is described by the Boltzmann equation. The solution of this equation converges towards the Maxwell-Boltzmann velocity distribution of molecules in local thermodynamic equilibrium. For small deviations from this equilibrium, the Navier-Stokes equations can be derived from the Boltzmann equation by means of a Chapman-Enskog expansion (cf. [89, 90]). In the LBM, the phase space is discretized into a Cartesian lattice Ωdx ⊂ RD of dimension D with spacing dx, and a set of Q discrete velocities ~cq ∈ RD , q ∈ {1, . . . , Q}. Moreover, time is discretized as cq are chosen Tdt = {tn : n = 0, 1, 2, . . .} ⊂ R+ 0 , with a time increment of dt = tn+1 − tn . The velocities ~ such that within a time increment, molecules can move to adjacent lattice sites or stay at a site. Associated with each of these velocities is a PDF fq : Ωdx × Tdt 7→ R. A forward-difference discretization in time and an upwind discretization in space [91] result in the discrete lattice Boltzmann equation fq (~xi + ~cq dt, tn + dt) − fq (~xi , tn ) = dtCq + dtFq , (33) with lattice site positions ~xi , discrete collision operator dtCq , and discrete body-force term Fq . This equation describes the advection of PDFs between neighboring lattice sites and subsequent collisions. In general, the collision operator can be represented in terms of the collision matrix S as dtCq = P eq
(cf. [92]), with the vector f~ := (f0 , f1 , . . . , fQ−1 )> of the PDFs fq , and f~eq of the equij Sqj fj − fj librium distributions fqeq . The latter are obtained from a low Mach number expansion of the MaxwellBoltzmann distribution [89]. With a representation of the macroscopic fluid density as ρf = ρ0 + δρ in terms of a reference density ρ0 and a density fluctuation δρ, the equilibrium distribution function for the incompressible LBM is derived in [93] as    (~u · ~u) (~cq · ~u) (~cq · ~u)2 + − , (34) fqeq (ρ0 , ~u) = wq ρf + ρ0 c2s 2c4s 2c2s where ‘·’ denotes the standard Euclidean scalar product. This distribution function depends on ρf , the macroscopic fluid velocity ~u, and lattice-model dependent weights wq . At each instant of time, ρf and ~u are given by moments of PDFs as P ρf (~xi , t) = fq (~xi , t), qP (35) ~cq fq (~xi , t). ~u(~xi , t) = ρ10 q c2s ρf (~xi , t) according to the equation of √ state for an ideal Moreover, the pressure p is given as p(~xi , t) = gas. We employ the D3Q19 model of [94] with thermodynamic speed of sound cs = c/ 3 for the lattice velocity c = dx/dt. For this model, the weights wq are: w1 = 1/3, w2,...,7 = 1/18, and w8,...,19 = 1/36. As discussed in [93], fqeq recovers the incompressible Navier-Stokes equation with at least second-order accuracy u| 2 of the Mach number Ma := |~ cs , O(Ma ). In LBM simulations, the kinematic fluid viscosity νf is generally determined by a dimensionless relaxation time τ of the collision operator as   1 2 c dt. (36) ν= τ− 2 s As shown in [95], with this definition the LBM is second-order accurate in space and time. Among the different collision operators available for the LBM, we employ the two-relaxation-time (TRT) collision operator of [29, 30] X


Sqj fj − fjeq = λe fqe − fqeq,e + λo fqo − fqeq,o , (37) j 13 D. Bartuschat and U. Rüde / Journal of Computational Physics 00 (2017) 1–33 14 with the relaxation parameters λe and λo for even- and odd-order non-conserved moments, respectively. Alternative collision operators either have disadvantages regarding stability and accuracy [96] such as the BGK model [97] or are computationally more costly such as the MRT operator [92] or the cumulant operator [98]. To ensure stability, both relaxation parameters should be within the interval ] − 2, 0[, cf. [99, 30]. The even relaxation parameter is related to the dimensionless relaxation time by λe = − τ1 and therefore determines the fluid viscosity. The free parameter λo is set to λo = −8(2 − 1/τ )/(8 − 1/τ ) in this article, which prevents τ -dependent boundary locations for no-slip BCs as they arise for the BGK operator. Instead, walls aligned with the lattice dimensions are fixed half-way between two lattice sites, as shown in [100]. For the TRT operator, the PDFs are decomposed as fq = fqe + fqo into their even and odd components fqe = 12 (fq + fq̄ ) fqo = 12 (fq − fq̄ ) and and fqeq,e = 12 (fqeq + fq̄eq ) fqeq,o = 12 (fqeq − fq̄eq ), with ~cq = −~cq̄ . The local equilibrium distribution function in Eqn. (34) is then given by   ρ0 ρ0 u · ~u) + 2c cq · ~u)2 fqeq,e = wq ρf − 2c 2 (~ 4 (~ s s fqeq,o = wq ρc20 (~cq · ~u). (38) (39) s At each time step tn ∈ Tdt the lattice Boltzmann method performs a collide– and a stream step f˜q (~xi , tn ) = fq (~xi , tn ) +λe [fqe (~xi , tn ) − fqeq,e (~xi , tn )] +λo [fqo (~xi , tn ) − fqeq,o (~xi , tn )] (40) fq (~xi + ~eq , tn + dt) = f˜q (~xi , tn ) + dtFq , (41) where f˜q denotes the post-collision state and ~eq = ~cq dt a discrete lattice direction. In the stream step, the product of dt and the forcing term Fq is added to the post-collision PDFs. The term Fq considers the external effect of body forces acting on the fluid. In this article, the discrete forcing term according to Luo [101] is employed as   (~cq − ~u) (~cq · ~u) ~ + ~ c Fq = w q (42) q · fb , c2s c4s with f~b representing the electrical body force per unit volume. The forcing terms lead to an additional term in the continuity equation that arises for spatially varying external forces, as shown in [102, 103]. This additional term can be removed by incorporating the external force in the momentum density definition as ! 1 X dt ~ ~u = fq~cq + fb . (43) ρ0 2 q Thus, [102] suggest to use the forcing term (42) in combination with the modified momentum density definition in Eqn. (43) for the BGK. We therefore use this forcing term with second-order accuracy, together with the modified momentum density for the resulting velocity ~u. To increase the computational efficiency of the implementation, the compute-intensive collide step and the memory-intensive stream step are fused to a stream-collide step. In the simulations presented in this article, no-slip and free-slip BCs are applied as described in [32]. 4.2. Momentum Exchange Approach To model the fluid-particle interaction and the resulting hydrodynamic interactions of the particles, the momentum exchange approach suggested by [27] is employed in this article. This approach computes the momentum exchange between the fluid and the suspended rigid particles from PDFs surrounding these solid objects. The implementation of this approach in waLBerla has recently been applied to simulate fluid-particle interactions also at Reynolds numbers beyond the Stokes regime, as presented in [104, 105, 106]. 14 D. Bartuschat and U. Rüde / Journal of Computational Physics 00 (2017) 1–33 15 For the momentum exchange approach, the solid particles are mapped onto the lattice by considering each cell whose center is overlapped by a particles a moving obstacle cell. All other lattice cells are fluid cells, resulting in a staircase approximation of the particle surfaces. These surfaces are represented by surface cells indicated by subscript s in the following. On the fluid cells denoted by subscript F , the LBM is applied. To model the momentum transfer from the particles to the fluid, the velocity bounce-back BC ωq fq̄ (~xF , tn + dt) = f˜q (~xF , tn ) − 2 2 ρ0~cq · ~us cs (44) is applied at fluid cells with position ~xF = ~xs + ~eq̄ adjacent to a surface cell at ~xs . This boundary condition introduced in [26] matches the fluid velocity to the local particle surface velocity ~us . From the sum of all force contributions due to the momentum transfer from fluid cells to neighboring surface cells (cf. [32]), the overall hydrodynamic force on the particle can be obtained according to [27] as  X X  dx3 ωq F~h = 2f˜q (~xF , tn ) − 2 2 ρ0~cq · ~us ~cq . (45) cs dt s q∈Ds Here, Ds is the set of direction indices q in which a given particle surface cell s is accessed from an adjacent ~ h is given by substituting ~cq × (~xs − ~xC ) for the last ~cq term in fluid cell. Analogously, the overall torque M Eqn. (45), with ~xC representing the particle’s center of mass. The mapping of the solid particles onto the lattice results in fluid cells appearing and disappearing as the particles move. Therefore, at uncovered lattice sites the PDFs must be reconstructed. We set the PDFs at those fluid cells to the equilibrium distribution f eq (ρ0 , ~us (~xs (tn − dt)) according to Eqn. (34) dependent on the local particle surface velocity from the previous time step. 4.3. Finite Volume Discretization for Electric Potential Equations To solve the Debye-Hückel approximation a cell-centered finite volume scheme is applied on the Cartesian lattice Ωdx introduced in Sec. 4.1. Associated with this lattice of spacing dx that represents the computational domain Ω ⊂ R3 is the three-dimensional cell-centered grid Gdx defined (cf. [107]) as     j − 1/2  V V  Gdx := ~xi ∈ Ω ~xi =  k − 1/2  dx i ∈ i , (j, k, m) ∈ J , (46)   m − 1/2 with Ωdx = Ω ∩ Gdx . For V indexing of lattice cells by tuples (j, k, m) of cell indices in the three spatial dimensions, the index set J := {(j, k, m) | j = 1, . . . , lx ; k = 1, . . . , ly ; m = 1, . . . , lz } is introduced. Here V lx , ly , and lz represent the numbers of cells in x, y, and z-direction, respectively. The set is related J V to the V setVof single cell indices i := {i | i = 1, . . . , lx · ly · lz } used for the LBM by a bijective mapping g : i → J , according to Eqn. (46). The finite volume discretization of the Debye-Hückel approximation Eqn. (18) includes volume integration over each lattice cell Ωi = Ωklm := [~xj−1,k,m , ~xj,k,m ] × [~xj,k−1,m , ~xj,k,m ] × [~xj,k,m−1 , ~xj,k,m ] and applying the divergence theorem to the Laplace operator ∆ = ∇· ∇, resulting in I Z 2 ~ − ∇ψ(~x) dΓi + κ ψ(~x) d~x = 0 ∀ Ωi ∈ Ωdx . (47) ∂Ωi Ωi Here, ∂Ωi denotes the closed surface of the cell, and ~Γi is a surface directed element. The cell surface consists of six planar faces with constant outward unit normal vectors ~niq (q = 1, . . . , 6). Therefore, the surface integral can be decomposed into a sum of integrals [108] as I − ∂Ωi ∇ψ(~x) d~Γi = − Z 6 X q=1 ∂Ω iq 15 ∇ψ(~x) · ~niq dΓiq , (48) D. Bartuschat and U. Rüde / Journal of Computational Physics 00 (2017) 1–33 16 where q represents LBM direction indices introduced in Sec. 4.1, and ∂Ωiq is the common face with the neighboring cell in direction q. The gradients ∇ψ(~xi ) · ~niq in normal direction of the faces ∂Ωiq are approximated at the face centers by central differences of ψ from a neighboring and the current cell as ∇ψ(~xi ) · ~niq ~ xi + 12 ~ eq ≈ ψ(~xi + ~eq ) − ψ(~xi ) . dx (49) Here, ~eq represents the corresponding lattice direction introduced in Sec. 4.1. Substituting the approximation of the normal fluxes across the surfaces of area dΓiq = dx2 , Eqn. (49) into Eqn. (48) results in I 6 X ψ(~xi + ~eq ) − ψ(~xi ) 2 − dx . (50) ∇ψ(~x) d~Γi ≈ − dx q=1 ∂Ωi Applying the above finite volume discretization to the linear term of the Debye-Hückel approximation results in Z κ2 ψ(~x) d~x ≈ κ2 ψ(~xi ) dx3 . (51) Ωi DHA This additional term enters the central element of the resulting seven-point stencil Ξdx as     0 −1 0 0 0 0 0 0 0 1   −1 6 + κ2 dx2 −1 0 −1 0 , 0 −1 0 dx2 0 0 0 0 0 0 0 −1 0 (52) and the right-hand side for each unknown is zero. 4.4. Parametrization for Electrokinetic Flows Numerical simulations are typically performed in terms of dimensionless parameters. To ensure consistently good numerical accuracy independent of the simulated scales, the physical quantities are mapped to a computationally reasonable numerical value range. In LBM simulations usually the quantities are expressed in lattice units. Therefore, physical values must be converted to lattice values before the simulation and vice versa to obtain physical values from simulation results. In the following the lattice unit (LU) system employed in this article is presented, providing a common parameterization also for further (electrokinetic) flow scenarios [76]. Physical simulation parameters are given in terms of the international system of quantities (ISQ) associated to the international system of units (Système International d’Unités, SI [109]) [110]. The value of a quantity is defined as product of a numerical value and a unit, e. g. 10−6 m. The SI system comprises the base units displayed in Tab. 1, together with the corresponding mutually independent base quantities length, time, mass, electric current, thermodynamic temperature, amount of substance, and luminous intensity. Moreover, derived units such as N = kgs2m (Newton) are defined in the SI system as products of powers of base units [109]. For LBM simulations the base quantities are length, time, and mass density. The corresponding lattice Table 1: Physical and LBM base quantities for electrokinetic simulations. kind of quantity length time mass electr. thermodyn. chem. photometr. ISQ quantity SI unit x m t s m kg I A T K n mol Iv cd LBM quantity lattice unit x dx t dt ρ ρ0 Φ V – – – – – – 16 D. Bartuschat and U. Rüde / Journal of Computational Physics 00 (2017) 1–33 17 base units are the (physical) lattice spacing dx, time increment dt, and fluid reference density ρ0 . Therefore, the numerical values of these quantities in lattice units become unity, as shown in Tab. 2. The lattice parameters representing these dimensionless numerical values in LUs are indicated with subscript L in the following, e. g. dxL for the lattice spacing. Performing LBM computations on such normalized lattice parameters saves numerical operations: Additional scaling factors are avoided, e. g. in the LBM streamcollide step, when computing the equilibrium distribution function given in Eqn. (34) (cf. cL = dxL /dtL = 1) or the macroscopic velocity according to Eqn. (35) (cf. ρ0,L = 1). For the electrokinetic simulations, the electric potential Φ is chosen as electric base quantity. This quantity, however, is not scaled on the lattice but keeps its numerical value that typically lies in a range not too far from unity. In contrast to the SI system, the LU system for electrokinetic simulations requires no base units corresponding to the thermodynamic temperature T or the amount of substance n: In the simulations, temperature appears only in combination with the Boltzmann constant as energy, i. e., E = kB T , with the derived unit [E] = [m] [x]2 /[t]2 . With the relation of mass and mass density [m] = [ρ0 ] [x]3 , this unit can be represented in lattice base units (see Tab. 1). Moreover, by representing the molar concentration with unit [ci ] = mol l in terms of the number density as 1 ni = ci NA with Avogadro’s number NA = 6.022 14 · 1023 mol , the unit ‘mol’ cancels out, yielding [ni ] = [x]−3 . With the choice of the potential Φ as base quantity, the electric current I becomes a derived quantity in the LU system. The unit of I can be derived from the energy in electrical units [E] = [I] [Φ] [t] and the above definition of energy in terms of lattice units [E] = [ρ0 ] [x]5 /[t]2 . Equating both relations results in the derived lattice unit [ρ0 ] [x]5 . (53) [I] = [Φ] [t]3 In Tab. 2 different physical quantities and their SI units are displayed, as well as their representation by (dimensionless) lattice parameters and their lattice units. The conversion of physical quantities to lattice Table 2: Relation of physical quantities and lattice parameters for electrokinetic simulations. physical quantity SI unit dx (lattice spacing) dt (time increment) m s ρ0 (fluid density) Φ (electr. potential) kg m3 L (length) ν (kinem. viscosity) ~u (velocity) m (mass) F~ (force) I (electr. current) V m m2 s m s kg kg m s2 A lattice parameter dxL dtL ρ0,L ΦL LL νL ~uL mL F~L IL e (elem. charge) As eL ε0 (vacuum permittivity) ~ (electr. field ) E As Vm V m ε0,L ~L E J EL E (energy) numerical value 1 dx dx (= 1) 1 dt dt (= 1) 1 ρ0 ρ0 (= 1) 1 V Φ 1 dx L dt ν dx2 dt u dx ~ 1 m ρ0 dx3 dt2 F~ ρ0 dx4 3 V dt I ρ0 dx5 V dt2 e ρ0 dx5 V dt2 ε ρ0 dx5 0 dx ~ V E dt2 E ρ0 dx5 lattice unit dx dt ρ0 V dx dx2 dt dx dt ρ0 dx3 ρ0 dx4 dt2 ρ0 dx5 dt3 V ρ0 dx5 dt2 V ρ0 dx5 V dt2 V dx ρ0 dx5 dt2 units requires their division by powers of the LBM base quantities with the corresponding SI units. Since the potential Φ has the same numerical value in physical and lattice units, the LBM base unit of the potential is dx5 simply ‘1 V’. Therefore, the derived lattice unit ‘Ampere’ for the electric current is given by A = ρV0 dt 3 (see Eqn. (53)). The corresponding scale factors for converting the physical parameters (e. g. ν) to the associated 17 D. Bartuschat and U. Rüde / Journal of Computational Physics 00 (2017) 1–33 18 lattice parameters (e. g. νL ) are shown in Tab. 2. Multiplication with the inverse scale factors converts the lattice parameters back to physical quantities. 5. Extension of waLBerla for Electrophoresis Simulations Electrophoresis simulations require the mutual coupling of fluid dynamics, rigid body dynamics, and electro-statics, as shown in Fig. 1. In addition to electrostatic and hydrodynamic interactions, the applied field acts on the EDL charge and thereby affects fluid flow and particle motion. For the electrophoresis algorithm presented below, the equilibrium description of the EDL potential in terms of the linear DebyeHückel equation is employed. Therefore, the predominant retardation effect is recovered in the simulations. The applied potential ϕ and the EDL potential ψ are linearly superimposed (cf. Henry’s equation, Sec. 3), which is valid for weak applied fields when the EDL distortion by the field is negligible. Thus, the applied electric field can be imposed directly, without solving the associated Laplace equation. In the following the main concepts of the waLBerla framework are described, together with the functionality for electrophoresis simulations implemented therein. 5.1. Design Concepts of waLBerla WaLBerla is a framework for massively parallel multiphysics simulations with a MPI-based distributed memory parallelization that is specifically designed for supercomputers. The main software design goals of this framework are flexibility to combine models of different effects, extensibility to allow the incorporation of further effects and details, e. g., for electrokinetic flows, and generality to support further applications [76]. These goals are reached by integrating the coupled simulation models into waLBerla in a modular fashion that avoids unnecessary dependencies between the modules. This way, the modules can be augmented by more sophisticated models or models tailored to a certain application, and functionality from different modules can be combined flexibly. The modular code structure also provides excellent maintainability, since modifications of the code in one module do not affect other modules. Developers can therefore efficiently locate faulty modules and find bugs inside these modules, also by systematically utilizing automatic tests. In addition to modules, the waLBerla code structure comprises a core for sequence control that initializes data structures, performs the time stepping, and finalizes the simulation on each parallel process. By means of applications, multiphysics simulations can be defined by assembling the associated functionality and coupled models from the modules. The coupling strategy for multiphysics simulations is based on accessing mutually dependent data structures (see [31] for more details). These data strucutres are defined in the modules that implement models for the different physical effects. Also infrastructural and utility functionality is encapsulated in modules, e. g., for domain setup, MPI communication, BC handling, parameterization, or simulation data output. For parallel simulations the discretized simulation domain is decomposed into equally sized blocks of cells that can be assigned to different parallel processes. In this block concept, each block contains a layer of surrounding ghost cells that is needed for BC treatment and parallelization. For parallelization, cell data of neighboring processes is copied to the ghost layer, dependent on the data dependencies of the unknowns located on a given block. Moreover, metadata of a block specifies its location in the simulation domain or its rank for MPI communication. The communication concept provides a simple and flexible communication mechanism tailored to simulations on Cartesian grids and facilitates various communication patterns. Individual work steps of a simulation algorithm are specified as sweeps that are executed on a block-parallel level. The sweep concept defines a structure in which callable objects (i. e. kernels) implemented in the modules can be specified at compile time. By means of dynamic application switches, specific kernels tailored to a given computer architecture or implementing a desired model variant, can be selected at run time. For time-dependent simulations, the sweeps are organized in a timeloop that specifies the order in which the individual work steps are executed at each time step. To facilitate iterative solvers, sweeps can also be nested to repeatedly perform a grid traversal until a termination criterion is met. The boundary condition concept for handling multiple physical fields, numerical methods, and governing equations, introduced in [31], is applied in this article for moving obstacles with electric BCs. This concept relies on flags to indicate for each boundary cell the kind of boundary treatment that to is be performed, 18 D. Bartuschat and U. Rüde / Journal of Computational Physics 00 (2017) 1–33 19 with an individual BC flag for each condition. Moreover, cells adjacent to a boundary are indicated with a nearBC flag and non-boundary cells with a nonBC flag. Individual sets of these flags are defined for each governing equation. Due to specific LBM requirements, the boundary handling is performed such that the BCs are fulfilled when a boundary cell is accessed from a nearBC cell in the susequent sweep. The abstract boundary handling is implemented in the bc module and provides functionality for all BCs are handled either as direct or direction-dependent treatment. In direction-dependent treatment the BC value is set at a boundary cell dependent on the value at a neighboring cell, whereas in direct BC treatment the BC value is directly set at a boundary cell [31]. The actual boundary handling functionality is implemented in corresponding BC classes whose functions are executed when the associated BC flag is found. The parameterization concept for multiphysics simulations introduced in [76] is based on the conversion of physical parameters to lattice units before the simulation, as described in Sec. 4.4. This approach ensures consistent parameters in all waLBerla modules, independent of the underlying physics. Individual modules can therefore be developed independently w.r.t. the common lattice unit system. Simulation parameters and BCs are typically provided to waLBerla through an input file. To ensure a consistent parameter set and a correct mapping of the physical quantities to lattice units, the class PhysicalCheck has been introduced in waLBerla in [111]. This class checks the simulation parameter set specified in the input file for completeness and physical validity and converts the parameters to lattice units. Since the quantities are converted based on the SI system, the unit Ampere is re-defined for PhysicalCheck according to Eqn. (53) to support simulations including electric effects. 5.2. Overview of waLBerla Modules for Electrophoresis Simulations In the following an overview of the modules relevant for electrophoresis simulations is given. For fluid simulations with the LBM, the lbm module implements various kernels for the stream-collide step with the different collision operators and forcing terms described in Sec. 4.1. The classes for treating the corresponding BCs are provided by the associated lbm bc module. In the lbm module block-fields of cells are provided for the PDFs, the velocity, the density, and an external force. The external force field is used for coupling the LBM to other methods, e. g. via the forces exerted by electric fields on the EDL and the fluid in electrophoresis simulations. The PDF and velocity field are accessed by moving obstacle module functions for the simulation of moving particles. The moving obstacle module facilitates simulations of fluid-particle interactions with the momentum exchange method by implementing kernels for moving obstacle sweeps and providing the corresponding data structures. This module furthermore provides setup functions for initializing and connecting the pe to waLBerla. For the moving obstacle handling, an obstacle-cell relation field is provided that stores for each lattice cell overlapped by a pe object a pointer to this object. Moreover, from the lbm module the PDF source field is accessed for the hydrodynamic force computation and the reconstruction of PDFs. In the lbm velocity field, body velocities are stored and accessed in the moving boundary treatment of the LBM. For the computation of the electric potential distribution, the lse solver module described in Sec. 5.3 is employed. This module has been implemented for solving large sparse linear systems of equations as they arise from the discretization of the electric potential equations (see Sec. 4.3). The corresponding BC classes are implemented in the pot bc module described in Sec. 5.4. The data structures defined in the lse solver module, accessed by the application and other modules, include the stencil field representing the system matrix as well as the scalar fields for the solution and for the RHS. The electrokin flow module was designed for facilitating electrokinetic flow simulations. This module provides kernels for coupling the involved methods as well as the setup and parameterization of simulations such as electrophoresis. The setup includes initializing the stencils and RHS from the lse solver module according to the finite volume discretization of the Debye-Hückel equation presented in Sec. 4.3. The kernels for imposing the electric potential BCs on the moving particles and for computing the electrostatic forces on fluid and particles are described in Sec. 5.5 and Sec. 5.6, respectively. Finally, the coupled algorithm for electrophoresis provided by the electrokin flow module is presented in Sec. 5.7. 19 D. Bartuschat and U. Rüde / Journal of Computational Physics 00 (2017) 1–33 20 5.3. Solver for Linear Elliptic PDEs with Moving Boundaries The solver module lse solver in waLBerla for large linear systems of equations has been designed as an efficient and robust black-box solver on block-structured grids [112, 31, 76]. To specify the problem to be solved, the application sets up the system matrix, right-hand side, and BC handling. In the lse solver module, solver sweeps are pre-defined that perform the iterations at the position where they are added to the timeloop. For a given simulation setup, the employed solver is selected via the input file where also the solver parameters and BCs are specified. An iterative solver requires a nested sweep that is executed until a specific convergence criterion is satisfied. For all implemented solvers, these sweeps share a common structure. This structure is displayed for the SOR sweep solveTimeVaryingBCSOR for moving boundaries and linear PDEs such as the Debye-Hückel approximation in Fig. 3 as an activity diagram. For Adapt stencils and RHS to BCs Maximum iterations reached? yes no Compute residual and norm Termination criterion met? yes no Update ‘red’ unknowns Update ‘black’ unknowns Figure 3: Activity diagram for SOR sweep solveTimeVaryingBCSOR with time varying boundary conditions (BCs), e. g., due to moving particles the employed discretization of the electric potential equations, the system matrix is represented by D3Q7 stencils. These stencils comprise an entry for each, the center and the six cardinal directions. In the setup function for this SOR sweep, the communication for the ghost layer exchange of the solution field in the initialization phase is set up first. Then the SOR solver sweep is added to the timeloop, and the kernels for relaxation, communication, and BC treatment are specified as solver sub-sweeps. For parallel execution, the SOR algorithm is implemented in red-black order. The filled circle at the top of the diagram in Fig. 3 indicates the starting point of the sweep in the timeloop. At the beginning of the sweep solveTimeVaryingBCSOR for moving boundaries, the stencils are constructed to incorporate the BCs according to the present boundary locations. Furthermore, the RHS is adapted to these BCs before the solver iterations start. For this purpose, a sub-sweep with a kernel for re-setting the stencils and RHS is executed before the iteration loop, followed by the BC treatment functions adaptStencilsBC and adaptRHSBC described in Sec. 5.4. Then the standard parallel Red-Black SOR sweep is performed until the termination criterion is met. This sweep comprises a sub-sweep for computing the residual and its L2 -norm for the termination criterion, as well as two solver sub-sweeps for the SOR update of the ‘red’ and the ‘black’ unknowns, respectively. In these sub-sweeps, the quasi-constant stencil optimization technique introduced in [31] is employed. Based on the residual L2 -norm, the termination 20 D. Bartuschat and U. Rüde / Journal of Computational Physics 00 (2017) 1–33 21 criterion for the simulations performed in this article is provided as residual reduction factor RESRF w.r.t. the initial norm of the simulation. For multigrid solvers as applied in [31] to charged particle simulations in absence of ions in the fluid, the red-black update sub-sweeps in Fig. 3 are replaced by solver sub-sweeps of a V-cycle. To apply the SOR sweep to varying stencils of linearized PDEs, such as the symmetric Poisson-Boltzmann equation, only the sub-sweep for adapting the stencils and RHS to the BCs is performed at the beginning of each iteration instead of once before the iterations start (see [76]). 5.4. Boundary Condition Handling for Solver Module The implicit BC handling used and initiated by the solver module has been introduced in [31]. This boundary handling is based on incorporating the BCs into the stencils and right-hand side of the finite volume discretization. That way, at an iterative update of a near-boundary value, the method implicitly uses the new values for the BCs. For Dirichlet boundaries, the boundary values are linearly extrapolated to the boundary cell and for Neumann BCs the boundary values are approximated by central differences. For both the stencils and the right-hand side, a direction-dependent BC treatment is used. The functions for this BC treatment are implemented in the pot bc module. This module employs own nonBC and nearBC flags for the BC handling of scalar potentials. Moreover, for each BC class in this module an associated BC flag is defined. For the employed cell-centered discretization, the module contains one class for each, Neumann and Dirichlet BCs. For incorporating the BCs into the stencils the kernel adaptStencilsBC is implemented. This kernel iterates over all lattice cells to find scalar potential nearBC cells. At each cell with nearBC flag, the kernel employs the D3Q7 stencil directions to iterate over the neighboring cells. In directions of a cell with scalar potential BC flag, the stencil entry of the nearBC cell, associated with the direction of the BC flag, is adapted accordingly. The function adaptRHSBC employs the standard boundary handling of the bc module to invoke the pot bc kernels for adapting the RHS to the BCs. To this end, the direction-dependent BC treatment kernels in the corresponding BC classes implement the RHS adaption depending on the BC value. The BC value is specified in the input file for static BCs, or in a previous time step for BCs of moving particles (see Sec. 5.5). To facilitate such complex boundaries, the BC classes store the BC values and the corresponding boundary cell ranges. The latter are stored in a memory-efficient way either as cell intervals or as cell vectors. Moreover, to allow the computation of scalar potential gradients directly from the solution field, the BC values are set in this field at boundary cells when the RHS is adapted. From these BC values and from the solution at the nearBC cell, the value at the boundary cell required for the gradient can be extrapolated. 5.5. Electric Potential Boundary Condition Handling for Moving Particles Prior to the EDL potential distribution computation by the lse solver module, uniform ζ-potentials or surface charge densities are imposed at the moving particles by means of scalar potential BCs. These electrical surface properties are specified in the input file for different particle types with a common uid (unique identifier) defined in the pe. To this end, the sweep function setPotBC_ChargParticles is implemented in the electrokin flow module that maps the charged particles onto the lattice and sets the electric potential BC values and the associated BC handling flags at the corresponding cells. The function first overwrites the values of all cells in the RHS field with zero to remove the values from the previous BC treatment. Then the mapping is performed for all movable rigid bodies located in the subdomain of each process. For each particle the mapping is conducted in an extended axis-aligned bounding box that surrounds this rigid body and the associated nearBC flags. The mapping is realized in three steps: 1) First, all scalar potential nearBC and BC flags from the previous time step are removed, and the scalar potential nonBC flags are set. Moreover, the previous BC values and the associated cells in the BC class instances are removed. 2) Then, for particles with prescribed electric BCs, the BC handling for the lse solver module (see Sec. 5.4) is prepared. For each rigid body with a uid for which a surface property is specified in the input file, the associated BC is obtained. The cells overlapped by this particle are gathered and are added 21 D. Bartuschat and U. Rüde / Journal of Computational Physics 00 (2017) 1–33 22 together with the BC value to the corresponding BC class instance. Moreover, the BC flag is set at the overlapped cells. 3) Finally, the nearBC flag is set at all cells adjacent to a BC cell. Each step is performed for all bodies on a process before the next step begins, to prevent that for particles with overlapping bounding boxes the flags from a previous step are overwritten. 5.6. Computing Electrostatic Forces on Fluid and Particles The electric forces acting on the ions in the fluid are incorporated into the incompressible NavierStokes equation by the body force term f~b = −ρe (ψ) ∇ (ϕ + ψ), which coincides with the corresponding term employed in Henry’s solution (see Sec. 3). Due to the linear superposition of the electric potential ~ ext = −∇ϕ is components, the gradient is applied to both components separately. Since the applied field E given, only the EDL potential gradient must be computed. For the computation of the electric field due to the EDL, the electrokin flow module provides a kernel that performs the gradient computation at all scalar potential nonBC cells. The gradient of the electric potential is computed, as previously introduced in [31], by means of finite differences that provide O(dx2 ) accuracy. Where possible, an isotropy-preserving D3Q19 stencil is used (cf. [113]) instead of a D3Q7 stencil. With the LBM D3Q19 stencil, the gradient can be computed using wq -weighted differences of neighboring values in 18 directions ~eq as 19 ~eq 1 X wq ψ(~xb + ~eq ) · 2 . (54) ∇ψ(~xb ) ≈ w1 q=2 dx At nearBC cells the D3Q7 stencil is applied to compute the gradient of ψ from the BC values stored at particle and boundary cells in the BC treatment (see Sec. 5.4). The obtained electric field is stored in a field of cells that is accessed in the body force computation. For the computation of the electric body force and of the electrostatic force exerted on the particles, a further kernel is implemented in the electrokin flow module: The kernel first iterates on each parallel process over all lattice cells to compute the body force at scalar potential nonBC cells. This force is computed as product of charge density ρe (ψ) and total electric field ~ total = E ~ ext − ∇ψ. The relation of charge density and EDL potential follows Eqn. (20). The obtained E electric body force is written to the external force field of the lbm module that is accessed by the LBM kernels with forcing. Then the kernel iterates over all non-fixed particles residing on the current parallel process to compute the electrostatic force. For each of these particles the force is computed from the particle ~ ext and is then added to the particle. charge and the applied field as F~C = qe E 5.7. Algorithm for Electrophoresis The overall parallel algorithm for electrophoresis simulations with waLBerla is shown in Alg. 1. After the setup and initialization phase, the electric BCs for the EDL potential are set at the moving charged particles by means of setPotBC_ChargParticles at each time step. Then the Debye-Hückel approximation is solved by means of the SOR sweep solveTimeVaryingBCSOR. The iterations are performed until the specified termination criterion for the residual is met. From the obtained EDL potential distribution and the applied field the electric body force exerted on the fluid is computed, as described in Sec. 5.6. First the kernel computing the electric field caused by the EDL is applied. Then the kernel for the electric body force computation from the charge density distribution in the fluid and the total electric field is invoked. This kernel additionally applies the electrostatic force exerted by the applied field to the particles. Then the rigid body mapping sweep described in Sec. 4.2 is performed, imposing the particle velocities for the subsequent LBM sweep. In that parallel sweep, an LBM kernel with fused stream-collide step and forcing term (see Sec. 4.1) is employed to compute the fluid motion influenced by the moving particles and by the electrostatic force exerted on ions in the EDL. After the LBM sweep, the hydrodynamic forces on the particles are computed by the momentum exchange method. The obtained hydrodynamic force contributions and the electrostatic forces are then aggregated by 22 D. Bartuschat and U. Rüde / Journal of Computational Physics 00 (2017) 1–33 23 Algorithm 1: Electrophoresis Algorithm foreach time step, do // solve Debye-Hückel approximation (DHA): set electric BCs of particles while residual ≥ tol do apply SOR iteration to DHA // couple potential solver and LBM: begin compute electric field due to EDL compute charge density in fluid compute electric body force apply electrostatic force to particles // solve lattice Boltzmann equation with forcing, considering particle velocities: set velocity BCs of particles begin perform fused stream-collide step // couple potential solver and LBM to pe: begin apply hydrodynamic force to particles pe moves particles depending on forces the pe. From the resulting forces and torques, the new particle velocities and positions are computed in the subsequent pe simulation step by the PFFD algorithm [114] that additionally resolves rigid body collisions. 6. Electrophoresis Simulations In the following, the electric potential computation is validated for a sphere with uniform surface charge surrounded by an EDL. This sphere is placed in a micro-channel subject to an applied electric field. Moreover the flow field caused by the electrophoretic motion of the sphere in the micro-channel is visualized, together with the electric potential and the surrounding ions to qualitatively show the correctness of the simulations. Then the electrophoretic velocity and the retardation by the counter-ions in the EDL is validated w.r.t. Henry’s solution for different sphere sizes and values of κR. The simulation setup for the validation experiments is depicted in Fig. 4. A sphere is placed on the 2 L x/ BC Fy BC C B z x y BC BC Lz/2 Lz/2 2 L x/ C B Ly Figure 4: Setup for electrophoresis of spheres in square duct with different BCs. longitudinal axis of a cuboid domain of size Lx × Ly × Lz at an initial position of yinit , and an electrostatic force in y-direction acts on the sphere. The sphere is suspended in a symmetric aqueous electrolyte solution 23 D. Bartuschat and U. Rüde / Journal of Computational Physics 00 (2017) 1–33 24 kg of kinematic viscosity νf = 1.00 · 10−6 ms , density ρf = 1.00 · 103 m 3 , permittivity εe = 78.54 · ε0 , and ions of valence z = 1 at a temperature of T = 293 K. In all simulations, the EDL thickness λD is in the order of the particle diameter. The electric potential around the sphere due to the EDL is represented by the Debye-Hückel approximation. For this approximation, analytical solutions are known for the potential distribution and for the electrophoretic velocity. The LBM is employed with TRT operator and second-order forcing term by Luo and re-defined momentum density (see Sec. 4.1) in all simulations. The linear system of equations resulting from the finite volume discretization of the Debye-Hückel equation is solved by the SOR method that is sufficient for the quickly decaying electric potential due to the counter-ions in the EDL. For the SOR, a relaxation parameter of ωSOR = 1.7 is applied in all simulations. 2 6.1. Electric Potential in an EDL around a Sphere To validate the computation of the EDL potential ψ around a charged particle, a sphere of radius R with uniform ζ-potential is simulated in a large domain. The analytical solution of the Debye-Hückel equation (21) representing the EDL potential around the sphere is given by Eqn. (23). For the validation, a spherical particle of radius RL = 12 with initial position yinit = 64 dx is chosen and a domain size of 128 dx × 256 dx × 128 dx. The lattice spacing dx is displayed in Tab. 3 with the simulation parameters as employed related to the electric potential. The ζ-potential is chosen sufficiently low to approximate the Table 3: Parameters for sphere with uniform surface charge in micro-channel. c∞ / mol l ζ/mV −10.0 5.00 · 10 1 κ/ m −6 7.41 · 10 dx/m 6 10 · 10 λD,L −9 13.49 Poisson-Boltzmann equation by the Debye-Hückel approximation. To obtain the displayed values of κ and of the characteristic EDL thickness λD , a symmetric aqueous electrolyte solution with ions of the valence z and the bulk concentration c∞ shown in Tab. 3 is simulated. The EDL thickness is greater than the 12 lattice sites required for a sufficient resolution as observed in the electro-osmotic flow simulations in [76]. Despite its quick decay, the analytical solution of the electric potential at the domain boundaries differs from zero. Thus, the values of ψ at these boundaries are set to the analytical solution by means of Dirichlet conditions. To solve the Debye-Hückel equation subject to these BCs, a residual reduction factor of RESRF = 2 · 10−7 is employed as termination criterion for the SOR method. The analytical (ψ) and numerical (ψ ∗ ) solution at the initial particle position are depicted in Fig. 5 along ψ/V 0 ·10−3 Analytical solution −5 −10 Numerical solution 0 20 40 60 xL 80 100 120 Figure 5: Analytical and numerical solution for EDL potential of sphere with uniform surface charge. 24 D. Bartuschat and U. Rüde / Journal of Computational Physics 00 (2017) 1–33 25 a line in x-direction through the sphere center. Both graphs agree very well, showing the correctness of the finite volume discretization and the applied SOR solver, as well as the boundary handling at the particle surface. Inside the insulating particle, the electric potential is not computed explicitly, due to the uniform surface potential and the resulting symmetric distribution of ψ in the sphere. The electrostatic force needed to compute the particle motion is computed directly from the applied field and the particle charge, instead of the electric potential gradient as in [31]. 6.2. Electrophoresis of a Sphere in a Micro-Channel The application of the external electric field in the micro-channel setup described in Sec. 6.1 gives rise to an electrophoretic motion of the sphere. For the simulation of this scenario, the parameters listed in Tab. 4 are employed in addition to the parameters in Tab. 3. The particle is suspended in an electrolyte solution Table 4: Parameters for electrophoretic motion of sphere in micro-channel. V Ey / m qs /A s −18 −19.9 · 10 −4.7 · 10 dt/s 6 −12 200 · 10 UEP,L 4.49 · 10−3 with the parameters introduced at the beginning of Sec. 6. From the ζ-potential and the parameters in Tab. 3, the sphere’s charge qs in Tab. 4 is obtained from the surface charge density σs according to the ζ–σs relationship (32), multiplied by the surface area of the sphere, as qs = 4πR2 σs [81]. The high electric field strength Ey in Tab. 4 is chosen to keep the number of simulation time steps at a minimum. Due to the applied field and the resulting electrostatic force FC,y = 933 · 10−12 N the particle moves in y-direction, retarded by the channel walls and the opposing force on the EDL. For the chosen parameters, the terminal particle speed of UEP = 224.5 mm s is obtained for free space according to Henry’s solution (see Eqn. (29)), corresponding to a particle Reynolds number of Rep,d = 0.054. In the simulation, periodic BCs are applied in y-direction. At all other walls, no-slip conditions are applied for the LBM and homogeneous Neumann conditions for the electric potential. At each time step, the Debye-Hückel equation is solved by SOR with the termination criterion from Sec. 6.1. The LBM is run with τ = 6.5, resulting in the time increment dt listed in Tab. 4 for the chosen viscosity νf and dx (cf. Eqn. (36)). With dt and dx, the electrophoretic speed in lattice units UEP,L attains the values given in Tab. 4. Gravitational effects are neglected in the simulation to ensure that the particle motion is driven kg solely by electric forces. Thus, the employed density ρp = 1195 m 3 of the particle only has an impact on the time required to reach steady-state. In Fig. 6, the results of the electrophoresis simulation are visualized at different time steps. The EDL potential ψ in the y-z plane through the domain center is displayed, together with semi-transparent equipotential surfaces representing the excess counter-ions in the EDL. The flow field around the moving sphere is visualized in the x-z plane through the domain center. Arrows of uniform length indicate the flow direction, while the velocity magnitude is represented by the shown color-scale and by twelve white isosurface contour lines with logarithmic intervals in the range of 13.67 · 10−3 ms to 6.08 · 10−6 ms . The flow field around the moving particle shown in Fig. 6(a) is fully developed after 5001 time steps, and the particle has already attained its terminal velocity. Due to the periodicity of the domain, a channel flow in axial direction has emerged from the particle motion, as indicated by the contour lines. Moreover, a vortex has formed between the sphere and the surrounding no-slip walls. The flow field moves with the particle that translates along the channel centerline (see Fig. 6(b)). Because of the equilibrium representation of the EDL, the distribution of ψ is at all time steps symmetric w.r.t. the sphere center, almost up to the boundary. 6.3. Validation of Electrophoretic Motion of a Sphere To quantitatively validate the overall electrophoresis simulation algorithm, the electrophoretic velocity of spheres with uniform surface charge is compared to Henry’s solution Eqn. (29) for a spherical particle in an unbounded electrolyte solution. 25 D. Bartuschat and U. Rüde / Journal of Computational Physics 00 (2017) 1–33 26 (a) Results after 5001 time steps. (b) Results after 30 001 time steps. Figure 6: Electrophoresis of spherical particle in micro-channel with insulating no-slip walls. Visualization of flow field in x-y plane, EDL potential in y-z plane, and ion charge distribution around charged particle. In the simulation experiments, a sphere with uniform ζ-potential is moving under the influence of an ~ ext in a large domain filled with an electrolyte solution. The simulations are applied field of strength E performed for different values of κR by varying the sphere radius while keeping the EDL thickness constant. For validation, the relative deviation ∆r U of the obtained terminal sphere velocity from the theoretical velocity in Eqn. (29) is evaluated. The simulation parameters are chosen such that the electrophoretic motion is in the Stokes regime, and thus the nonlinear inertial term of Navier-Stokes equation can be neglected. Moreover, the EDL potential distribution is governed by the linear Debye-Hückel approximation. Therefore, the superposition principle is assumed to hold. Thus, from the obtained relative deviation from the analytical solution in an unbounded domain ∆r U , the relative deviation due to wall effects and volume mapping errors ∆r UStokes will be subtracted. These relative deviations were examined in [76] for several domain sizes and sphere radii. For no-slip BCs, the wall effect was shown to comply with analytical and experimental results. In the experiments, domain sizes are used for which the relative deviations ∆r UStokes for free-slip BCs are close to −3 %. For all simulations, the parameters in Tab. 5 are used. In each experiment, a sphere is suspended in a symmetric aqueous electrolyte solution with the parameters introduced at the beginning of Sec. 6 and a bulk concentration of c∞ = 1.60 · 10−5 mol l . These parameters result in the Debye-Hückel parameter κ shown in Tab. 5 and, together with the displayed lattice spacing dx, in the EDL thickness λD of approximately 15 26 D. Bartuschat and U. Rüde / Journal of Computational Physics 00 (2017) 1–33 27 Table 5: Parameters for electrophoretic velocity validation w.r.t. Henry’s solution. ζ/mV 1 κ/ m V Ey / m dx/m λD,L 10.0 13.3 · 106 99.0 · 106 5.00 · 10−9 15.08 lattice sites. The ζ-potential has the same absolute value as in the electrophoresis simulations in a microchannel from Sec. 6.2. Moreover, as in Sec. 6.2, a high electric field strength Ey is chosen to keep the number of simulation time steps low. The resulting electrostatic forces FC = qs Ey for the different sphere radii are displayed in Tab. 6, together with the electrophoretic charges qs on the spheres. These surface charges associated with the ζ-potential are obtained from the general ζ–σs relationship (32) as in Sec. 6.2. For all simulation parameters, the maximum relative deviation of σs for the general relationship from the exact value for the Debye-Hückel approximation is below 0.2 %. These charges are applied as particle charges to drive the spheres by the electrostatic force due to the electric field. For the different sphere radii and the associated values of κR, the electrophoretic velocities according to Henry’s solution are displayed in Tab. 6. These velocities correspond to particle Reynolds numbers Rep,d from 0.018 to 0.057 for the particle diameters of 40 nm to 120 nm. EM is introduced. To quantify the retardation by the opposing force on the EDL, the variable EPRet = UEPU−U EM This variable represents the relative deviation of the electrophoretic velocity of a particle with charge qs in presence of the electric double layer from the migration velocity UEM of a particle with the same charge in absence of surrounding ions. For the examined sphere radii, this retardation is in the range of 20 % to 42 %. Table 6: Electrophoresis parameters and domain sizes dependent on sphere size. Shown are the sphere radii RL in lattice units, the corresponding charges qs , and the electrostatic forces FC . For relation of sphere radius to EDL thickness κR, theoretical electrophoretic velocities UEP , Reynolds numbers Rep,d , and electrophoretic retardation parameter EPRet are given. Listed in lower part are domain sizes per dimension Lx,y,z , initial sphere positions yinit , process numbers per dimension #procx,y,z , and relative deviations of sphere velocities in free-slip domain from Stokes velocity ∆r UStokes . 4 6 8 9 12 qs /(10 A s) FC /(10−12 N) 2.21 219 3.67 363 5.36 530 6.29 622 9.43 934 κR UEP /(10−3 Rep,d EPRet /% 0.265 461 0.018 -20.6 0.398 464 0.028 -27.7 0.530 466 0.037 -33.6 0.597 468 0.042 -36.2 0.796 471 0.057 -42.8 864 1248 392 1280 1536 588 1632 2048 784 1632 2048 882 1632 2048 1176 8 × 16 × 8 8 × 16 × 16 16 × 16 × 32 16 × 16 × 32 16 × 16 × 32 -3.04 -2.65 -2.98 -2.89 -3.00 RL −18 m s ) Lx,z /dx Ly /dx yinit /dx #procx,y,z ∆r UStokes /% Then the domain sizes for which the relative deviation from Stokes velocity is about ∆r UStokes = −3 % are listed in Tab. 6, together with the initial position yinit in movement direction that correspond to 98 × RL . The LBM with TRT collision operator is employed with the relaxation time τ = 6, resulting in the time increment dt = 45.8 · 10−12 s. As in the micro-channel electrophoresis simulations, gravitational are effects kg neglected, and the insulating particles have a density of ρp = 1195 m 3. For the electric potential, homogeneous Neumann BCs are applied at the walls in x- and z-direction as 27 D. Bartuschat and U. Rüde / Journal of Computational Physics 00 (2017) 1–33 28 = 0. To improve convergence [31], homogeneous Dirichlet BCs are applied at the walls in y-direction as Φ Γ ∪Γ = 0 V. The particle is located at sufficient distance from these walls for the N S EDL potential to decay to approximately zero, and therefore these BCs do not affect the electric potential distribution. As termination criterion of the SOR solver for the Debye-Hückel equation, a residual reduction factor of RESRF = 1 · 10−6 is used. The parallel simulations are run on SuperMUC1 of the Leibniz Supercomputing Centre LRZ in Garching (Germany) on 64 to 512 nodes with the numbers of processes listed in Tab. 6. Within the execution time of 37 h to 48 h, a number of 70 296 to 74 000 time steps are performed, and the spheres cover a distance of about 300 dx. The high numbers of time steps are chosen to ensure that the spheres reach steady-state motion, and that large numbers of sampling values are available to compute the terminal velocities. The sphere velocities are sampled every 20 time steps during the simulation. In the second half of the simulation, the terminal particle velocity is reached. Thus, the mean particle velocity U ∗ and the velocity U ∗ −U ∗ fluctuations δU = maxU ∗ min due to volume mapping effects are computed from the last 50 % output values. In the considered range of time steps additionally the number of time steps between two SOR calls and the number of SOR iterations is monitored. The average number of time steps between two SOR calls decreases from ∆TSSOR = 24 to ∆TSSOR = 3 as RL increases from 4 to 12. Likewise, the average number of SOR iterations per solver call decreases from 451 iterations to 198 iterations for the respective sphere radii. The obtained terminal velocity in lattice units UL∗ and the fluctuations are displayed in Tab. 7. As expected, ∂Φ ∂~ n ΓW ∪ΓE ∪ΓB ∪ΓT Table 7: Simulation results of electrophoretic velocity validation for different sphere sizes. Shown are the theoretical velocities ∗ and fluctuations δ , the relative deviations ∆ U of UEP,L for unbounded domains in lattice units, the obtained velocities UL r U ∗ from U UL EP,L , and the relative deviations ∆r UEP corrected by hydrodynamic wall and mapping effects. 4 6 8 9 12 UEP,L /10 4.227 4.249 4.274 4.286 4.320 UL∗ /10−3 4.119 2.52 -2.56 0.5 4.144 1.52 -2.48 0.2 4.120 0.83 -3.59 -0.6 4.135 0.68 -3.51 -0.6 4.151 0.29 -3.91 -0.9 RL −3 δU /% ∆r U/% ∆r UEP /% the fluctuations decrease with increasing sphere resolution. Moreover, the relative deviation of the obtained velocity U ∗ from the theoretical electrophoretic velocity UEP given by ∆r U = (U ∗ − UEP )/UEP is listed in Tab. 7. For all examined sphere radii, the obtained velocities in the confined domain are by 2.5 % to 3.9 % lower than the theoretical values of UEP for a particle in an unbounded electrolyte solution. The effect of the confinement on the particle velocity is deducted by subtracting the relative deviation from Stokes velocity ∆r UStokes in Tab. 6 for the corresponding domain sizes from the relative electrophoretic velocity deviation ∆r U obtained in the electrophoresis simulation. From the resulting relative deviations ∆r UEP = ∆r U − ∆r UStokes , the inaccuracies due to electric effects in the electrophoresis simulation are assessed. As can be seen from the values of ∆r UEP in Tab. 7, the simulations results agree with the theoretical values with relative deviations of less than 1 %. 7. Conclusion In this article, a coupled multiphysics algorithm for parallel simulations of the electrophoretic motion of geometrically resolved particles in electrolyte solutions is presented. The physical effects are simulated by means of the lattice Boltzmann method for fluid dynamics, a physics engine for rigid body dynamics, and a scalar iterative solver for electric potentials. These components are integrated into the parallel software framework waLBerla to simulate the electrophoretic motion of fully resolved charged particles in 1 www.lrz.de/services/compute/supermuc/ 28 D. Bartuschat and U. Rüde / Journal of Computational Physics 00 (2017) 1–33 29 microfluidic flow. The simulations include fluid-particle and electrostatic particle interactions. Additionally electric effects on ions around the homogeneously charged particles are recovered. The current work is an extension of [31], where the electrical migration of charged particles without ions in the fluid was validated and excellent parallel performance was shown for more than seven millions of interacting charged particles. In the present article, the opposite net charge in the electric double layer (EDL) around the charged particles due to ions in the fluid is considered, together with its effect on the fluid motion that counteracts particle motion. To this end, the electric potential distribution in the fluid due to the EDLs is computed that causes an electric body force on the fluid. This quasi-equilibrium distribution recovers the motion of ions in the fluid along with the charged particles while neglecting EDL distortion. The overall electrophoresis algorithm is introduced and an overview of the coupled functionality implemented in the involved waLBerla modules is given. For the simulations, a solver sweep for time-varying boundary conditions has been developed that is presented here for the parallel SOR method employed to solve the EDL potential equation. Based on the multiphysics boundary handling concept [31] an efficient parallel algorithm is implemented to impose electric potential boundary conditions on the moving particles. These methods can also be employed for other governing equations with spatially varying boundary conditions that model physical effects different from electric fields. The presented parallel electrophoresis simulations are also facilitated by a joint parameterization concept for the different coupled governing equations and numerical methods implemented in waLBerla. This concept is based on lattice Boltzmann requirements and is applicable and extensible to further multiphysics simulations. For the electrophoresis simulations in this article, the electric potential in the double layer is shown to coincide with analytical solutions. The obtained terminal electrophoretic velocities comply with analytical solutions for different proportions of the particle radii to double layer thickness. These validation results verify the correctness of the implementation and the coupling of the different methods. Moreover, the observed relative errors in the modeling of electric effects are below 1 %. The retardation effect caused by the presence of the EDL is shown to be significant for the examined sphere radii, reducing the sphere velocity up to 42 %. For the electrophoretic motion in a micro-channel, the flow field and the electric potential distribution are visualized, including the ion charge distribution in the EDL surrounding the particle. The presented algorithm can be applied to find design parameters in industrial and medical applications, e. g., for optimal separation efficiency of charged biological particles in lab-on-a-chip devices, depending on fluid, particle, and electrolyte properties. Our algorithms were shown to correctly recover fluid-particle interactions for elongated particles in [32]. These future simulations may therefore include suspended, possibly charged particles of various shapes including spheres, spherocylinders, and particles of more complex shapes, e. g., to represent different biological particles. Also pairwise van-der-Waals forces can be added easily, to facilitate simulations of electrophoretic deposition in material science applications. The electrophoresis algorithm introduced here is well suited for massively parallel simulations. In the current implementation of this algorithm, the EDL thickness is restricted to values in the order of the particle radius. Therefore, adaptive lattice refinement as in [23] may be employed to allow for thinner double layers relative to the particle size. For the incorporation of transient effects in the simulations including EDLs, the link-flux method implemented into waLBerla in [115] and [41] may be employed. This method was extended in [41] to simulate electrophoresis, enabling the simulation of non-equilibrium ion distributions in the EDL. Due to the higher computational complexity of the link-flux method compared to the equilibrium approach in this article, the maximum number of particles will be lower than in our approach. Acknowledgements The authors are grateful to the LRZ for providing the computational resources on SuperMUC. References [1] K. J. Ptasinski, P. J. A. M. 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arXiv:1801.10562v1 [q-bio.QM] 31 Jan 2018 Feature Decomposition Based Saliency Detection in Electron Cryo-Tomograms ∗ Bo Zhou1 , Qiang Guo2 , Xiangrui Zeng3 , and Min Xu3 1 Robotics Institute, Carnegie Mellon University, Pittsburgh, USA 2 Max Planck Institute for Biochemistry, Martinsried, Germany 3 Computational Biology Department, Carnegie Mellon University, Pittsburgh, USA ∗ Corresponding author Abstract Electron Cryo-Tomography (ECT) allows 3D visualization of subcellular structures at the submolecular resolution in close to the native state. However, due to the high degree of structural complexity and imaging limits, the automatic segmentation of cellular components from ECT images is very difficult. To complement and speed up existing segmentation methods, it is desirable to develop a generic cell component segmentation method that is 1) not specific to particular types of cellular components, 2) able to segment unknown cellular components, 3) fully unsupervised and does not rely on the availability of training data. As an important step towards this goal, in this paper, we propose a saliency detection method that computes the likelihood that a subregion in a tomogram stands out from the background. Our method consists of four steps: supervoxel over-segmentation, feature extraction, feature matrix decomposition, and computation of saliency. The method produces a distribution map that represents the regions’ saliency in tomograms. Our experiments show that our method can successfully label most salient regions detected by a human observer, and able to filter out regions not containing cellular components. Therefore, our method can remove the majority of the background region, and significantly speed up the subsequent processing of segmentation and recognition of cellular components captured by ECT. Keywords: saliency detection, Electron Cryo-Tomography, super-voxel segmentation, 3D Gabor filter, robust PCA 1 Introduction The development of Electron Cryo-Tomography (ECT) has enabled the detailed inspection and visualization of subcellular structures at sub-molecular resolution and in near-native state[19, 24]. With this cellular imaging technique, subcellular components can be systematically analyzed at unprecedented level of detail and faithfulness. Recent studies have shown this in situ visualization enables the discovery of numerous important structural features in complex virus[10, 11], as well as in prokaryotic and eukaryotic cells[4, 9, 14]. ECT has established its position as one of the leading techniques for visualizing the subcellular and macromolecular organization of single cells[5]. In principle, an ECT tomogram captures structural information of all cellular components in the field of view. However, several factors, such as the low signal-to-noise ratio (SNR), the limited 1 tilt projection range (missing wedge effect) and the crowded nature of intracellular structures, make the systematic structural analysis of cellular components captured by ECT very difficult. In addition, tomograms are large size gray scale 3D images. A typical raw tomogram can have a size of 6000 × 6000 × 1500 voxels, which is very computationally expensive to process. In order to analyze the cellular components captured by ECT tomograms, these cellular components must first be segmented and recognized from these tomograms. Currently, most of the segmentation is performed either manually, or automatically. The manual segmentation of cellular components in such 3D images is very laborious, even with the aid of computational segmentation tools like watershed transform and thresholding [25]. On the other hand, most of the automatic segmentation is very computationally intensive and often limited to specific types of objects. Many of these automatic segmentation methods rely on matching of the geometrical model of the specific cellular structure, such as filaments, microtubules and membranes [18, 23, 16]. Recently, supervised methods have been developed that segments specific cellular structures using classification models trained on annotated training images of the specific cellular structures of interest [8, 17]. Such supervised segmentation methods are restricted to segmenting specific structures in tomograms that are already characterized by human annotator. A generic segmentation algorithm that is able to segment general cellular components including unknown cellular components, without considering the availability of training data is needed to complement the existing segmentation methods. As an important step towards this goal, we propose a saliency detection method and address the problem of automatically extracting the salient region from an unknown background in ECT. The saliency of an image subregion is the likelihood that it stands out relative to its background. Given that the perceptual, cognitive and computational resources are limited, to facilitate such segmentation, it is important to ignore the background regions which occupy the majority portion of the tomogram. To solve the saliency detection problem, we propose a method that consists of following steps: supervoxel over-segmentation, feature extraction, feature matrix decomposition, and computation of saliency. The method produces a map that represents the saliency of subregions of a tomogram. Our experiments show that 1) our method can successfully perform saliency detection compared with human annotation, and 2) the number of detected salient voxels is significantly smaller than the total number of voxels. As a result, our approach can substantially reduce the background region, and accelerate the succeeding segmentation and analysis of the cellular components in the ECT. Our main contributions are summarized as follows: • We design an easy-to-implement salient region detection method based on feature decomposition. • With the candidate regions of cellular objects in ECT, the computational cost of tasks like object detection and segmentation of specific structure is greatly reduced, facilitating the isolation of different cellular structure of interest. 2 Methods Our method comprises five major steps, including data pre-processing for de-noising, supervoxel over-segmentation, feature extraction, feature matrix construction, and computation of saliency. The flow diagram of the method is shown in Figure 1. The 1st stage is intended to enhance the Figure 1: General flow diagram of the method for salient object detection and segmentation. image contrast to improve the performance of supervoxel over-segmentation in the 2nd stage, as well as the quality of the extracted feature in the 3rd stage. The different stages are described in detail in the following sections. The procedure takes the intuitive assumption that, in terms of image features, the salient regions are the minority regions in the tomograms. The details of the de-noising stage are described in Appendix 6.1. In this paper, we assume that all the salient objects have a size bigger than the assigned kernel scale. The scale σ is chosen by visually evaluating the processed data using different scales. Given the majority of noise is smaller than the scale selected, this step can filter out the noise and preserve the useful information in the tomogram. We chose σ = 1.8 for all experiment in this paper. 2.1 Supervoxel over-segmentation The geometrical features are important information for analyzing the tomograms. Geometry is the primary feature to be considered when researchers choose the interested region in a tomogram to analyze. In this step, we use a supervoxel over-segmentation method, simple linear iterative clustering (SLIC)[3], to over-segment the tomogram into sub-volumes which generate 3D geometric boundaries and clusters that enclose small volumes with similar density within a certain distance of the neighborhood. In ECT tomogram, low gray-scale levels represent the electron-dense region. We generalize the method in R. Achanta’s paper[3] to three dimensions (x, y, z) and gray-scale (I). Each voxel corresponds to a vector in R4 . The vectors are used for clustering. One of the method’s parameter is n, the desired number of equally sized supervoxels. The clustering procedure begins by initializing n cluster centers at points spaced S voxels apart q on a regular grid, where: S = N n , and N is the number of the voxels in the tomogram. There are approximately S voxels in each supervoxel. Then each voxel is assigned to the nearest cluster center whose search region overlaps its spatial location. The raw distance measurement D0 calculates the distance between a voxel and cluster center. D0 is defined as r dI ds 0 (1) D = wI ( )2 + ws ( )2 m S , where spatial distance ds = and intensity distance p (xc − xi )2 + (yc − yi )2 + (zc − zi )2 dI = p (Ic − Ii )2 (2) (3) . D0 is further simplified as r ds 2 2 ) m (4) S , where weight coefficient w is the second parameter in the 3D SLIC clustering, the compactness of clusters. The voxel clusters for tomograms can be generated by iterating every voxel in it. In short, there are two parameters we need to specify, the number of clusters n and the compactness of cluster w. A higher value of n and lower value of w should be chosen if the potential content is small and morphological complex, which can produce higher spatial density cluster and more deformable shape to enclose the region containing potential objects, vice versa. D= 2.2 dI 2 + w( Feature extraction After the supervoxel over-segmentation, for each supervoxel, we perform feature extraction and construct a feature vector with a size of 30. Two groups of the features are calculated: 2.2.1 3D Gabor filter based features Previous studies have shown that the Gabor function provides a useful and reasonably accurate description of most spatial aspects of simple receptive fields[15]. Gabor function is a product of a Gaussian kernel and a complex sinusoid function. The derivation of extending 2D to 3D Gabor function is shown in Appendix 6.2. We assume the Gaussian envelop have the same scale in three axes; Thus the shape of this Gaussian can only be spherical but can have different scales. By modifying sinusoid frequency in three orientations and scale parameters, we generate 24 3D Gabor filters by rotating 0, 45, 90 and 135 degrees about the x, y, and z-axes, as well as two different Gaussian scales. Each Gabor feature is calculated by taking the mean of the filter response inside the supervoxels. 2.2.2 Density features A tomogram is reconstructed from multiple electron beam projections at different angles. Higher density material causes more energy attenuation of the electron beam when arriving at the energy detector. This results in a lower voxel intensity corresponding to a denser cellular structure. Cellular structure density features are inversely related to the intensity value in the tomogram. Given this information, the second group of features consists of the intensity distribution in the supervoxel. The intensity’s dynamic range could vary between ECT tomograms due to different settings of the imaging system and reconstruction parameters. With that, we first normalize the dynamic range into [0 600] and generate a six bins histogram range from the 0 to 600. The number of voxels counted in each bin is used as one density feature. There are six density features extracted from the histogram of each supervoxel. After segmenting the tomogram into candidate supervoxels, we compute the feature vector for each supervoxel, and construct a feature matrix with each row corresponding to the feature vector of a supervoxel. The feature matrix with the size of n by 30 will be used in the next step, where n is the number of supervoxels. 2.3 Feature matrix decomposition Previous studies have shown that subspace estimation by sparse representation and rank minimization is an excellent unsupervised method for separating the salient image regions from the image background [6, 22]. We decompose the feature matrix constructed from the last section into a lowrank matrix and a sparse matrix using robust principal component analysis (RPCA) via principal component pursuit (PCP)[7]. Each row of the sparse matrix represents the saliency of individual supervoxels, whereas the low-rank matrix represents the common background information. The feature matrix F is represented as F = L + S, where L is a low-rank matrix, and ||S||0 is the L0 norm of the matrix S. The minimization of ||S||0 enforces S to be a sparse matrix with a small fraction of nonzero entries. The decomposition is an optimization process in the following form: min rank(L) + β||S||0 L,S s.t. F −L−S =0 (5) , where β ≥ 0 is a hyperparameter. The minimization of Equation 5 is NP-hard. Instead of directly solving Equation 5, PCP transforms Equation 5 to an equivalent convex optimization problem[7]: min ||L||∗ + β||S||1 L,S s.t. F −L−S =0 (6) , where ||L||∗ is the nuclear norm of L, calculated from the singular values of L, ||S||1 is the L1 norm of S. The RPCA-PCP method can effectively recover the low-rank and the sparse matrix for our feature matrix by optimizing Equation 6 [7]. In an ideal decomposition of the feature space, most of the image background should lie in a low dimensional subspace so that they can be represented as a low-rank matrix L shown above. 2.4 Calculation of saliency map and salient region segmentation Given the sparse matrix S decomposed from the feature matrix F , each supervoxel’s saliency can be represented by corresponding row in the sparse matrix S. In this section, we refer it as saliency vector of supervoxel. After obtaining the saliency vector of the corresponding supervoxel, we calculate the representation of saliency by taking the mean of the saliency vector, and the saliency is assigned to the corresponding supervoxel’s spatial location which generates a saliency volume. 2.5 2.5.1 Evaluation of salient region segmentation Obtaining ground truth through manual annotation Ground truth salient region annotation is obtained by using the human annotator selected supervoxel regions where the annotator believe the regions stand out from background. The anchor points are dropped by the annotator to select regions of such supervoxels. The annotator has no previous knowledge about the salient map and any output produced by our method. 2.5.2 ROC based performance measure Binary salient region segmentation of the saliency region is generated by setting a threshold value to the saliency map volume (Figure 3). The performance of the segmentation using the saliency volume is evaluated using receptor operating curve (ROC) analysis[21]. A true positive in our ROC analysis is if the manually annotated anchor point falls within the salient region segmentation. Three tomogram slices in three different datasets are evaluated. The mean area under the curve (AUC) are also calculated for each dataset (Figure 4). 3 Results Three different experimental tomograms are included to test the method’s performance. Tomogram 1 and tomogram 3 are obtained from the EMPIAR public image archive [13]. Tomogram 1 (EMPIAR ID: 10045) captures a sub-region of the purified S. cerevisiae ribosomes[2]. Tomogram 2 contains a sub-region of a rat’s primary neuron, which was collected at Max Planck Institute of Biochemistry[12]. Tomogram 3 (EMPIAR ID: 10048) captures a sub-region of the Chlamydia trachomatis secretion system[1]. 3.1 Supervoxel over-segmentation via SLIC clustering The optimal scale space for different tomograms can vary due to the noise introduced by different imaging data acquisition parameters and reconstruction methods. Using a fixed imaging and reconstruction protocol, an optimal scale can be chosen. A de-noised ECT tomogram using different scale 3D Gaussian filters are shown in Appendix 6.1 and Figure A.1. In this tomogram, we choose scale σ = 1.8 voxels, which best preserves the cellular structural information and rule out noise. Bigger or smaller filter scale will over-blur or under-de-noise the tomogram. The tomogram de-noised with optimal Gaussian filter is used for all results shown later. Using the iterative clustering method discussed in Section 2.1, we cluster a tomogram into different number of supervoxel. The compactness of the supervoxel is set by w, which means bigger w will constrain the deformation of supervoxel. Figure 2 shows the supervoxel over-segmentation using the different combination of n and w. In principle, to avoid missing small and complex salient region in tomogram, we choose a bigger number of supervoxel n and smaller compactness coefficient w to make sure these complex regions are clustered and enclosed in supervoxels so that the saliency can be detected in later steps. We choose n = 10000 and w = 0.025 as the optimal parameters for SLIC clustering in the three test tomograms. 3.2 Extraction of feature and construction of saliency map By applying the 3D Gabor filters with different orientations to the tomogram, we extract the corresponding feature in each voxel. Given the supervoxels’ spatial information, voxel’s 3D Gabor features are assigned to the corresponding supervoxel. Example of feature extraction results with different 3D Gabor filters is shown in Appendix 6.2 and Figure B.1. The supervoxel’s density features are extracted by generating the six bins histogram for each supervoxel; each bin represents one density feature. The cellular structure, like ribosome, membrane, microtubule usually have higher density, so the first three density features of the region contains such cellular structure will have higher values. Whereas, the last three density features will be high if supervoxel contains low-density cellular structures. Given the feature vectors of every supervoxel in the tomogram, we construct the feature matrix and apply RPCA to decompose it into the low-rank matrix and the saliency matrix. The saliency Figure 2: SLIC Supervoxel over-segmentation in tomogram 2 with different number of supervoxel and compactness setting. First column: one slice from the de-noised tomogram. Second to Fourth column: visualization of supervoxels’ margins in this slice with n=1000, 5000, 10000 (columns) and w=0.025, 0.05 (rows). of supervoxel is calculated using the rows in the saliency matrix and assigned to the spatial location of the supervoxels. 3.3 Evaluation of saliency detection Our method is tested and evaluated in three different ECT tomograms consisted of simple and complex cellular contents. Figure 3 shows the saliency results generated using the tomograms from these three tomograms. The structural composition of tomogram 1 and tomogram 3 are relatively simple. As we can see, our method can efficiently detect the saliency regions. Tomogram 2 contains cellular structures with various size, density, and shape. More details of salient region visualization are shown in Appendix 6.3. Our method can effectively represent the saliency of different subregions enriched with compact cellular structures in this tomogram. The performance of salient region segmentation using saliency map is evaluated using the three tomograms mentioned above. Three slices with ground truth anchor annotation in these three tomograms are used to generate the ROC analysis. ROC curve for these tomogram slices is shown in Figure 4. The average AUC for tomogram 1 and tomogram 2 are approximately equal to 0.89 and 0.863. The average AUC for tomogram 3 is about 0.815. The performance of salient region segmentation is better in the tomogram 1, compared to segmentation in tomogram 2 and 3, due to its simple and uniform structural content. The number of detected salient supervoxels and voxels are recorded using the optimal cutoff threshold obtain from the ROC experiment shown above. The results are shown in Table 1. The number of salient supervoxel and salient voxels are significantly smaller than the total number of supervoxels and voxels. The background voxels are filtered out using this method. Therefore the automatic segmentation and consequent processing steps can be significantly sped up by only processing such a small number of salient voxels. Figure 3: Visualization of example slices of saliency map (saliency level below 21 (Smax +Smin ) set to 0) along with corresponding tomogram slices and human annotated salient region anchor in 3 different tomograms. Figure 4: Segmentation evaluation with ROC curve using saliency volume. Table 1: Salient supervoxel selected from tomogram 4 Tomogram 1 Tomogram 2 Tomogram 3 Number of Supervoxel 10000 10000 10000 Number of Salient Supervoxel 869 1324 330 Percentage of Selected Supervoxel 8.69% 13.24% 3.30% Number of voxel 321,553,500 134,522,880 37,796,240 Number of Salient voxel 25,768,154 14,756,210 1,564,276 Percentage of Selected voxel 8.01% 10.96% 4.14% Discussion Efficient and fully unsupervised automatic segmentation of all cellular components in a tomogram is extremely challenging, because of the high degree of structural complexity and the imaging limitations in the cellular electron cryo-tomograms. As an important step towards this goal, we have proposed an efficient salient region detection method in ECT that mimics the human visual system for detecting a tomogram’s salient sub-regions that contain cellular components. The quantitative results obtained suggest that our method is useful in salient region detection in studies using ECT technique. A direct application of our saliency detection is for template free particle picking in ECT tomograms. A tomogram usually contain macromolecules of diverse structures of diverse size. Currently, the main computational approach for selecting such diverse macromolecules without the use of a structural template is through Difference-of-Gaussian(DoG) filtering based template-free particle picking [26, 20]. However, such approach tend to select macromolecules with globular structure and with certain size range. By contrast, the connected salient supervoxels in the binarized saliency map generated from our approach can directly serve as candidate macromolecules or other subcellular components without structure and size constrains. Therefore our saliency detection approach can potentially be used as a more powerful and flexible template-free particle selector. Future efforts to combine our method with the automatic analysis system for systematically analyzing the object using adapted structural pattern mining methods [e.g., 29, 28, 30]. Such combination will allow us to tremendously reduce the time for laborious manual annotation of tomogram and discover the underlying correlation between structures. 5 Acknowledgements We thank Dr. Robert F. Murphy for suggestions. This work was supported in part by U.S. National Institutes of Health (NIH) grant P41 GM103712. MX acknowledge support from Samuel and Emma Winters Foundation. 6 Appendix 6.1 De-noising The noise in a tomogram can directly impact every subsequent step in our method. Thus, denoising and preserving useful information for the input tomogram is critical. According to the scale-space theory, we can isolate information according to the spatial scale. Given a scale, all the features with a size smaller than the designed scale can be filtered out, whereas the others are preserved[27]. In this step, we implement the scale-space theory with a sampled 3D Gaussian kernel, which aims for isolating useful information from noise by directly convolving the Gaussian kernel with tomogram. In three dimension, continuous 3D Gaussian kernel with scale σ is defined 2 2 2 by G[x; σ] = 3 1 3 exp(− x1 +x2σ2 2+x3 ). The tomogram T is then convolved with G, and generates σ (2π) 2 the de-noised volume V : V [x; σ] = G[x; σ] ∗ T Figure A.1: Gaussian filtered tomogram with different scale 3D Gaussian filter. Three de-noised tomograms are produced by convolving different scale 3D Gaussian filter to the original tomogram. De-noised tomogram with scale σ=1.8 better preserve detailed structural information compared to de-noised tomogram with scale σ=4 and scale σ=8. 6.2 3D Gabor filter and feature Here, we extend the 2-D Gabor function to a 3-D function. The 3D Gabor function (H) can be written as H[x, y, z] = G[x, y, z] ∗ S[x, y, z], where G is a 3D Gaussian envelope G[x, y, z; σ] = x2 +y 2 +z 2 1 ), and S is a complex sinusoid function S[x, y, z] = exp[2πi(U x + V y + W z)], 3 exp(− 2σ 2 3 σ (2π) 2 where U , V , and W are the 3D frequencies of the complex sinusoid. They determine Gabor filter’s orientation and spacing in the spatial domain. 6.3 Visualization of salient region with saliency Examples of the salient region with the corresponding saliency in different slices of the tomogram that captures a sub-region of rat’s primary neuron are shown in Figure C.1. It can be seen that the regions containing bigger cellular structure with higher material density normally present higher saliency, compared to the structure with smaller size, simpler structure, and lower density. This behavior of our method matches the human behavior when observes imaging data. Human observer normally first detect big object with higher density and complex shape, and then simpler and smaller object. Figure B.1: Visualization of extracted features from tomogram by applying different 3D Gabor filters. First row: Gabor response in x, y and z direction with 0 degree of rotation along the axes. Second row: Gabor response in x, y and z direction with 45 degree of rotation along the axes. Figure C.1: Visualization of saliency map (second row) along with the corresponding tomogram slices (first row). saliency level below the 12 (Smax + Smin )) is set to 0 in this visualization. Different regions present different level of saliency due to the various complexity of content. References [1] Cryo-electron tomogram of Chlamydia trachomatis with type III secretion system in contact with HeLa cell. https://www.ebi.ac.uk/pdbe/emdb/empiar/entry/10048/, 2015. [Online; accessed 11-06-2017]. 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arXiv:1802.01806v1 [q-bio.MN] 6 Feb 2018 Detection of sustained signals and its relation to coherent feedforward loops Chun Tung Chou School of Computer Science and Engineering, University of New South Wales, Sydney, Australia. E-mail: [email protected] February 7, 2018 Abstract Many studies have shown that cells use temporal dynamics of signaling molecules to encode information. One particular class of temporal dynamics is sustained and transient signals, i.e. signals of long and short durations respectively. This paper formulates a detection problem for distinguishing sustained signals from transient ones. The solution of the detection problem is to compute the likelihood ratio of observing a sustained signal to a transient signal. We show that, if the logarithm of this likelihood ratio is positive (i.e. when the signal is likely to be sustained), then this log-likelihood ratio can be approximately computed by a coherent feedforward loop. Although the capability of coherent feedforward loops to discriminate sustained signals is known, its statistical information processing interpretation has not been pointed out before. Living cells use many different strategies to encode information. A strategy is to encode information using the temporal dynamics of signaling molecules [9, 2]. One particular class of temporal dynamics is sustained (also known as persistent) and transient signals, i.e. signals of long and short durations respectively. The study in [8] found that PC-12 cells proliferated after a transient ERK activation but differentiated after a sustained ERK activation. This shows that the duration of ERK signaling affects the cell fate. Ref. [7] shows that coherent feedforward loops (FFL) can act as persistence detectors to differentiate sustained signals from transient ones. A particular type of FFL is the coherent type-1 feedforward loop with an AND logic at the output, or C1FFL for short. Fig. 1a depicts the structure of the C1FFL which consists of a short arm, a long arm and an AND logic combining the signals of the two arms. By modelling the reaction dynamics of C1FFL with ordinary differential equations (ODEs), Ref. [7] shows that a short and a long pulse applied to the input of the C1FFL, give, respectively a zero and a non-zero output. Instead of starting from the structure of C1FFL and show that it can detect sustained signals, we ask the 1 Input s(t) Input s(t) x*(t) X x*(t) X x*(t) Y Loglikelihood ratio computation y(t) AND Log-likelihood ratio Output z(t) (a) (b) Figure 1: (a) The coherent type-1 feedforward loop with AND logic. (b) The detection theory framework. question in the opposite direction. In this Letter, we start from the requirement of detecting a sustained signal and use detection theory (DT) [6] to determine the detector. We then show that the resulting detector can be approximately realized by a C1FFL. DT is a branch of statistical signal processing. Its aim is to use the measured data to decide whether an event of interest has occurred. For example, DT is used in radar signal processing to determine whether a target is present or not. In the context of this Letter, the events are whether the signal is transient or sustained. A detection problem is often formulated as a hypothesis testing problem, where each hypothesis corresponds to a possible event. Let us consider a detection problem with two hypotheses, denoted by H0 and H1 , which correspond to respectively, the events of transient and sustained signals. Our aim is to decide which hypothesis is more likely to hold. We define the log-likelihood ratio (LLR) R:   P[measured data|H1 ] R = log (1) P[measured data|H0 ] where P[measured data|Hi ] is the conditional probability that the measured data is generated by the signal specified in hypothesis Hi . Intuitively, if the LLR R is positive, then the measured data is more likely to have been generated by a sustained signal or hypothesis H1 , and vice versa. Therefore, the key idea of DT is to use the measured data to compute the LLR and then use it to make a decision. We will now present a big picture explanation of how we will connect DT with C1FFL. The signal x∗ (t) in Fig. 1a is the output signal of Node X in the C1FFL. We can view the C1FFL as a 2-stage signal processing engine. In the first stage, the input signal is processed by Node X to obtained x∗ (t) and this is the part within the dashed box in Fig. 1a. In the second stage, the signal 2 x∗ (t) is processed by the rest of the C1FFL to produce the output signal z(t). We will now make a connection to DT. We apply DT to the dashed box in Fig. 1a. We consider x∗ (t) as the measured data and use them to determine whether the input signal is transient or sustained. DT tells us that we should use x∗ (t) to compute the LLR. This means that we can consider the 2-stage signal processing depicted in Fig. 1b where the input signal generates x∗ (t) and the measured data x∗ (t) are used to calculate the LLR. If we can identify the LLR calculation in Fig. 1b with the processing by the part of C1FFL outside of the dashed box, then we can identify the signal z(t) with the LLR. We will now define the problem for detecting a sustained signal using DT. Our first step is to specify the signaling pathway in Node X, which consists of three chemical species: signaling molecule S, molecular type X in inactive form and its active form X* . The activation and inactivation reactions are: k+ S + X −−→ S + X∗ k− X∗ −−→ X (2a) (2b) where k+ and k− are reaction rate constants. Let x(t) and x∗ (t) denote, respectively, the number of X and X* molecules at time t. Note that both x(t) and x∗ (t) are piecewise constant because they are molecular counts. We assume that x(t) + x∗ (t) is a constant for all t and we denote this constant by M . We assume that the input signal s(t), which is the concentration of the signaling molecules S at time t, is a deterministic signal. We also assumed that the signal s(t) cannot be observed, so any characteristics of s(t) can only be inferred. We model the dynamics of the chemical reactions by using chemical master equation [4]. This means that x∗ (t) is a realisation of a continuous-time Markov chain. This also means that the same input signal s(t) can result in different x∗ (t). The measured datum at time t is x∗ (t). However, in the formulation of the detection problem, we will assume that at time t, the data available to the detection problem are x∗ (τ ) for all τ ∈ [0, t]; in other words, the data are continuous in time and are the history of the counts of X* up to time t. We will use X∗ (t) to denote the continuous-time history of x∗ (t) up to time t. Note that even though we assume that the entire history X∗ (t) is available for detection, we will show that the calculation of the LLR at time t does not require the storage of the past history. The last step in defining the detection problem is to specify the hypotheses Hi (i = 0, 1). Later on, we will identify H0 and H1 with, respectively, transient and sustained signals. However, at this stage, we want to solve the detection problem in a general way. We assume that the hypothesis H0 (resp. H1 ) is that the input signal s(t) is the signal c0 (t) (c1 (t)) where c0 (t) and c1 (t) are two different deterministic signals. Intuitively, the aim of the detection problem is to use the history X∗ (t) to decide which of the two signals c0 (t) and c1 (t) is more likely to have produced the observed history. 3 Based on the definition of the detection problem, the LLR L(t) at time t is given by:   P[X∗ (t)|H1 ] L(t) = log (3) P[X∗ (t)|H0 ] where P[X∗ (t)|Hi ] is the conditional probability of observing the history X∗ (t) given hypothesis Hi . We show in the Appendix that the L(t) obeys the following ODE:     dL(t) dx∗ (t) c1 (t) = log − dt dt c0 (t) + k+ (M − x∗ (t))(c1 (t) − c0 (t)) (4) where [a]+ = max(a, 0). We also assume that the two hypotheses are equally likely, so L(0) = 0. Since x∗ (t) is a piecewise constant function counting the number of X* molecules, its derivative is a sequence of Dirac deltas at the time instants that X is activated or X* is deactivated. Note that the Dirac deltas corresponding to the activation of X carries a positive sign and the [ ]+ operator keeps only these. Note that a special case of Eq. (4) with constant ci (t) and M = 1 appeared in [10]. A more general form of Eq. (4) which includes the diffusion of signaling molecules can be found in [3]. The importance of Eq. (4) is that, given the measured data x∗ (t), we can use Eq. (4) together with ci (t) to compute the LLR L(t). If s(t) is similar to c1 (t) (resp. c0 (t)), then the LLR L(t) generally increases (decreases) over time and becomes more positive (negative). If our aim is to distinguish a sustained signal from a transient one accurately, then we want the sustained signal to produce a large positive L(t). Since the positive contribution of L(t) comes from the first term in Eq. (4), we can get a large positive L(t) by making sure that a sustained signal will produce many activations. This occurs when a sustained signal has a duration which is long compared to time scale of the activation and deactivation reactions (2) and we will make use of this condition later. The LLR L(t) can certainly be computed numerically by a computer but a more interesting question is whether it can be computed by using a set of chemical reactions. We will address this next. In order to build a connection with C1FFL, we need to specify the form of ci (t). We now assume that the transient signal c0 (t) and sustained signal c1 (t) are of the form of a rectangular pulse. The durations for c0 (t) and c1 (t) are, respectively, d0 and d1 with d1 > d0 . We define two concentration levels for the signaling molecules: basal concentration a0 and reference concentration a1 , with a1 > a0 . The temporal profile of ci (t) is: for 0 ≤ t < di , ci (t) = a1 , otherwise ci (t) = a0 . Furthermore, we assume the input s(t) is a pulse of duration d. The temporal profile of s(t) is: for 0 ≤ t < d, s(t) = a, otherwise s(t) = a0 . There are two difficulties why the computation in Eq. (4) cannot be carried out by chemical reactions. These are: (1) The LLR can take any real value but chemical concentration can only be non-negative; (2) It is difficult to calculate derivatives and M − x∗ (t) using chemical reactions. Instead of using L(t), we 4 will derive an approximation L̂(t) which has the properties: L̂(t) ≈ L(t) for sustained signals and L̂(t) = 0 for transient signals. We first consider the case when the input s(t) is a sustained signal which results in a positive L(t). We assume that d is long compared with the time scale of the activation and deactivation reactions. Since these reactions are fast, we can assume quasi-steady state: x∗ (t) ≈ M k+ s(t) k+ s(t) + k− (5) By using the form of ci (t) and s(t) in Eq. (4), we see that, for t ∈ [d0 , min{d, d1 }], the contribution of the first term on the right-hand side (RHS) of (4) to L(t) equals to the number of times that X is activated in the time interval t ∈ [d0 , t]. If t is sufficiently large and if the activation and deactivation reactions are fast, then a large number of activations take place. In this case, we can get approximately the same L(t) by approximating the positive derivative of x∗ (t) by its mean value:   dx∗ (t) ≈ k+ (M − x∗ (t))s(t) (6) dt + After substituting Eqs. (5) and (6) into Eq. (4), and after some manipulation, we arrive at: dL̂(t) = x∗ (t) × {k− π(t) [φ(s(t))]+ } where dt   a1 a1 − a0 φ(u) = log , − a0 u (7) (8) and π(t) = 1 for d0 ≤ t < d1 and zero otherwise. We first verify that L̂(t) ≈ L(t) for t ∈ [0, min{d, d1 }) for long d and d1 . We use the Stochastic Simulation Algorithm (SSA) [5] to obtain 100 realisations of x∗ (t). We then use these time series and Eq. (4) to compute the mean and standard deviation of L(t). We numerically integrate Eq. (7) to obtain L̂(t) for comparison. We use k+ = 0.02, k− = 0.5, d0 = 5, d1 = 60, a0 = 0.25, a1 = a = 10.7. The top and bottom plots in Fig. 2a show the results for d = 70 and d = 40. We can see that the approximation is good. In general, the approximation is good if d and d1 are long compared to the time scale of the activation and deactivation reactions. We note from Fig. 2a that when the approximation holds, the maximum value of L̂(t) and L(t) match very well. There are a few noteworthy features on the RHS of Eq. (7). The RHS consists of the multiplication of x∗ (t) with some other functions. We will map x∗ (t) to the short arm of the C1FFL and the other functions to the long arm. The multiplication is to implement the AND logic. Next, we note that the function π(t) is zero in the time interval [0, d0 ) which means the RHS of Eq. (7) is zero t < d0 independent of the input. This shows how the detector is making use of the prior knowledge of c0 (t) by not making use of any observations before time d0 . 5 The form of Eq. (7) appears to be robust to how we define Hi . Instead of assuming that Hi is a pulse of a specific duration di , we consider an alternative where we define hypothesis H0 (resp. H1 ) to be pulses whose duration is uniformly distributed in [0, d0 ] ([d0 , d1 ]). These are composite hypotheses and can be handled by using the Bayesian approach in [6, p.198]. We can show that these hypotheses give an ODE in the same form as Eq. (7) except that π(t) is slightly different. Thus the mapping to C1FFL also holds for these alternative hypotheses. Having shown that L̂(t) ≈ L(t) for long pulses, we now show that L̂(t) = 0 for short pulses. For a small d, the LLR L(t) becomes negative. In Eq. (7), the [ ]+ operator is included so that L̂(t) ≥ 0 for all t. In particular, for transient inputs with duration d < d0 , L̂(t) = 0 ∀t ≥ 0. This is consistent with the behaviour of an ideal C1FFL which gives a zero output for short pulses [1]. Our next step is to show that Eq. (7) can be approximately realized by the following reaction system, which models Node Y and the AND logic in the C1FFL in Fig. 1a: dy(t) dt = hy x∗ (t)ny ny Ky + x∗ (t)ny | dz(t) dt = {z Hy (x∗(t)) x∗ (t) × −dy y(t) (9a) } hz y(t)nz Kznz + y(t)nz {z } | (9b) Hz (y(t)) where hy , ny etc. are coefficients of the Hill functions. Our aim is to make z(t) in Eq. (9b) to be approximately equal to L̂(t), and this can be achieved by approximating k− π(t)[φ(s(t))]+ (= η(t)) in Eq. (7) by Hz (y(t)) in Eq. (9b). If the input s(t) is a long pulse, then the time profiles of both η(t) and y(t) contain a period of time that they plateau. The plateau in η(t) contributes to the ramplike increase in L̂(t) in Fig. 2a. This means we need to match the values of η(t) and Hz (y(t)) at their plateau. For a pulse s(t) with an amplitude a, the heights of the plateau of η(t) and Hz (y(t)) are, respectively, k− [φ(a)]+ (= f1 (a)) and k+ a Hz ( d1y Hy ( kM ))(= f2 (a)), and we want f1 (a) ≈ f2 (a) for as large a range of + a+k− a as possible. Note that for all a such that f1 (a) > 0, both functions f1 (a) and f2 (a) are strictly increasing, strictly concave and both f1 (∞) and f2 (∞) are constants. Therefore, we can choose the Hill function coefficients to fit f2 (a) to f1 (a). This argument takes care of the case when s(t) is a long pulse. For short pulses, we need to realise π(t) whose purpose is to make the initial part of L̂(t) zero. This is a feature shared by the ideal C1FFL model in [1]. Ref. [1] shows that this can be realized by choosing a big enough Kz in Eq. (9b) so that the production rate of z(t) is small initially. We now present two numerical examples. We use the same k+ , k− , M , a0 and a1 values as before. We choose d0 = 10, d1 = 80. We use parameter estimation to determine the reaction constants in Eq. (9) so that the C1FFL output z(t) matches L̂(t) for a range of a. Fig. 2b compares L̂(t) and z(t) for 6 input s(t) with d = 10, 30 and 70 for a = 10.7. When d = 10, the output of the C1FFL is small. For d = 30 and 70, the C1FFL output matches well with the approximate LLR L̂(t). In the second example, we use inputs of a fixed duration of 70 but vary the amplitude a from 2.7 to 85.7. Fig. 2b compares L̂(t) and z(t) at t = 70. It can be seen that the C1FFL approximation works for a wide range of a. These examples show that we can match the C1FFL output z(t) to approximate LLR L̂(t) for input s(t) of different durations and amplitudes. Although we have only shown that z(t) matches L̂(t) for pulse inputs, the match also extends to other slowly-varying inputs. Conclusions: This Letter considers the problem of detecting sustained signals from transient ones. It makes use of detection theory and turns the problem into one of computing the log-likelihood ratio. It further shows that the log-likelihood ratio can be approximately computed by a coherent type-1 feedforward loop. Although the capability of coherent feedforward loops to discriminate sustained signals is known, its statistical information processing interpretation has not been pointed out before. A Proof of (4) In order to derive (4), we consider the history X∗ (t + ∆t) as a concatenation of X∗ (t) and x∗ (t) in the time interval (t, t+∆t]. We assume that ∆t is chosen small enough so that no more than one activation or deactivation reaction can take place in (t, t + ∆t]. Given this assumption and right continuity of continuoustime Markov Chain, we can use x∗ (t + ∆t) to denote the history in (t, t + ∆t]. By using the Markov property, we can show that:   P[x∗ (t + ∆t)|H1 , x∗ (t)] (10) L(t + ∆t) = L(t) + log P[x∗ (t + ∆t)|H0 , x∗ (t)] By using the propensity functions of the reactions, we have: P[x∗ (t + ∆t)|Hi , x∗ (t)] = δx∗ (t+∆t),x∗ (t)+1 k+ (M − x∗ (t)) ∆t ci (t)(t)+ δx∗ (t+∆t),x∗ (t)−1 k− b(t) ∆t + δx∗ (t+∆t),x∗ (t) (1 − k+ (M − x∗ (t))ci (t) ∆t − k− b(t) ∆t) (11) where δa,b is the Kronecker delta which is 1 when a = b. By substituting Eq. (11) into Eq. (10) and taking the limit ∆t → 0, we have after some manipulations:   δx∗ (t+∆t),x∗ (t)+1 dL(t) c1 (t) = lim log − ∆t→0 dt ∆t c0 (t) δx∗ (t+∆t),x∗ (t) k+ (M − x∗ (t)) (c1 (t) − c0 (t)) (12) In order to obtain Eq. (4), we use the following reasonings. First, the term δ ∗ (t)+1 lim∆t→0 x∗ (t+∆t),x is a Dirac delta at the time instant that an X molecule ∆t 7 is activated. Second, the term δx∗ (t+∆t),x∗ (t) is only zero when the number of X∗ molecule changes but the number of such changes is countable. In other words, δx∗ (t+∆t),x∗ (t) = 1 with probability one. This allows us to drop δx∗ (t+∆t),x∗ (t) . References [1] U. Alon. An Introduction to Systems Biology: Design Principles of Biological Circuits. Chapman & Hall, 2006. [2] Marcelo Behar and Alexander Hoffmann. Understanding the temporal codes of intra-cellular signals. Current opinion in genetics & development, 20(6):684–693, December 2010. [3] Chun Tung Chou. Maximum a-posteriori decoding for diffusion-based molecular communication using analog filters. Nanotechnology, IEEE Transactions on, 14(6):1054–1067, 2015. [4] C. Gardiner. Stochastic methods. Springer, 2010. [5] D Gillespie. Exact stochastic simulation of coupled chemical reactions. The journal of physical chemistry, 1977. [6] Steve M. Kay. Fundamentals of Statistical Signal Processing, Volume II: Detection Theory. Prentice Hall, 1998. [7] S Mangan and U Alon. Structure and function of the feed-forward loop network motif. Proceedings of the National Academy of Sciences of the United States of America, 100(21):11980–11985, October 2003. [8] C J Marshall. Specificity of receptor tyrosine kinase signaling: transient versus sustained extracellular signal-regulated kinase activation. Cell, 80:179– 185, January 1995. [9] Jeremy E Purvis and Galit Lahav. Encoding and Decoding Cellular Information through Signaling Dynamics. Cell, 152(5):945–956, February 2013. [10] Eric D Siggia and Massimo Vergassola. Decisions on the fly in cellular sensory systems. Proceedings of the National Academy of Sciences, 110(39):E3704–12, September 2013. 8 LLR 2500 1250 0 0 20 40 60 80 60 80 time LLR 1600 800 0 0 20 40 time (a) Comparing L(t) to L̂(t). Approx LLR / C1FFL output 1000 500 0 0 40 80 time Approx LLR / C1FFL output (b) Comparing L̂(t) to C1FFL output. 8000 6000 4000 2000 0 0 20 40 60 80 (c) Comparing L̂(t) and C1FFL output for different pulse input amplitude a. Figure 2: Numerical results. (Best view in color.) 9 

FINITE ELEMENT MODEL UPDATING USING RESPONSE SURFACE METHOD Tshilidzi Marwala School of Electrical and Information Engineering University of the Witwatersrand P/Bag 3, Wits, 2050, South Africa [email protected] This paper proposes the response surface method for finite element model updating. The response surface method is implemented by approximating the finite element model surface response equation by a multi-layer perceptron. The updated parameters of the finite element model were calculated using genetic algorithm by optimizing the surface response equation. The proposed method was compared to the existing methods that use simulated annealing or genetic algorithm together with a full finite element model for finite element model updating. The proposed method was tested on an unsymmetrical H-shaped structure. It was observed that the proposed method gave the updated natural frequencies and mode shapes that were of the same order of accuracy as those given by simulated annealing and genetic algorithm. Furthermore, it was observed that the response surface method achieved these results at a computational speed that was more than 2.5 times as fast as the genetic algorithm and a full finite element model and 24 times faster than the simulated annealing. proximation model, which traditionally is a polynomial and therefore is less expensive to evaluate. This makes RSM very useful to FE model updating because optimizing the FE model to match measured data to FE model generated data is a computationally expensive exercise. Furthermore, the calculation of the gradients that are essential when traditional optimization methods, such as conjugate gradient methods, are used is computationally expensive and often encounters numerical problems such as ill-conditioning. RSM tends to be immune to such problems when used for FE model updating. This is largely because RSM solves a crude approximation of the FE model rather than the full FE model which is of high dimensional order. The multi-layer perceptron (MLP)6 is used to approximate the response equation. The RSM is particularly useful for optimizing systems that are evolving as a function of time, a situation that is prevalent in model-based fault diagnostics found in the manufacturing sector. To date, RSM has been used extensively to optimize complex models and processes7,8. In summary, the RSM is used because of the following reasons: (1) the ease of implementation that includes low computational time; (2) the suitability of the approach to the manufacturing sector where model-based methods are often used to monitor structures that evolve as a function of time. FE model updating has been used widely to detect damage in structures9. When implementing FE updating methods for damage identification, it is assumed that the FE model is a true dynamic representation of the structure and this is achieved through FE model updating. This means that changing any physical parameter of an element in the FE model is equivalent to introducing damage in that region. There are two approaches that are used in FE updating: direct methods and iterative Introduction Finite element (FE) models are widely used to predict the dynamic characteristics of aerospace structures. These models often give results that differ from the measured results and therefore need to be updated to match the measured data. FE model updating entails tuning the model so that it can better reflect the measured data from the physical structure being modeled1. One fundamental characteristic of an FE model is that it can never be a true reflection of the physical structure but it will forever be an approximation. FE model updating fundamentally implies that we are identifying a better approximation model of the physical structure than the original model. The aim of this paper is to introduce updating of finite element models using Response Surface Method (RSM)2. Thus far, the RSM method has not been used to solve the FE updating problem1. This new approach to FE model updating is compared to methods that use simulated annealing (SA) or genetic algorithm (GA) together with full FE models for FE model updating. FE model updating methods have been implemented using different types of optimization methods such as genetic algorithm and conjugate gradient methods3-5. Levin and Lieven5 proposed the use of SA and GA for FE updating. RSM is an approximate optimization method that looks at various design variables and their responses and identify the combination of design variables that give the best response. The best response, in this paper, is defined as the one that gives the minimum distance between the measured data and the data predicted by the FE model. RSM attempts to replace implicit functions of the original design optimization problem with an ap* Associate Professor 1 methods1. Direct methods, which use the modal properties, are computationally efficient to implement and reproduce the measured modal data exactly. Furthermore, they do not take into account the physical parameters that are updated. Consequently, even though the FE model is able to predict measured quantities, the updated model is limited in the following ways: it may lack the connectivity of nodes - connectivity of nodes is a phenomenon that occurs naturally in finite element modeling because of the physical reality that the structure is connected; the updated matrices are populated instead of banded - the fact that structural elements are only connected to their neighbors ensures that the mass and stiffness matrices are diagonally dominated with few couplings between elements that are far apart; and there is a possible loss of symmetry of the systems matrices. Iterative procedures use changes in physical parameters to update FE models and produce models that are physically realistic. Iterative methods that use modal properties and the RSM for FE model updating are implemented in this paper. The FE models are updated so that the measured modal properties match the FE model predicted modal properties. The proposed RSM updating method is tested on an unsymmetrical H-shaped structure. and stiffness matrices. The frequency response functions (FRFs) are defined as the ratio of the Fourier transformed response to the Fourier transformed force. The FRFs may be expressed in receptance and inertance form. On the one hand, receptance expression of the FRF is defined as the ratio of the Fourier transformed displacement to the Fourier transformed force. On the other hand, inertance expression of the FRF is defined as the ratio of the Fourier transformed acceleration to the Fourier transformed force. The inertance FRF (H) may be written in terms of the modal properties by using the modal summation equation as follows10: N − ω 2φkiφli H kl ( ω ) = ∑ (3) 2 2 i =1 − ω + 2ζ i ω i ωj + ω i Equation 3 is an FRF due to excitation at position k and response measurement at position l, ω is the frequency point, ω i is the ith natural frequency, N is the number of modes and ζi is the damping ratio of mode i. The excitation and response of the structure and Fourier transform method10 can be used to calculate the FRFs. Through equation 3 and a technique called modal analysis10, the natural frequencies and mode shapes can be indirectly calculated from the measured FRFs. The modal properties of a dynamic system depend on the mass and stiffness matrices of the system as indicated by equation 2. Therefore, the measured modal properties can be reproduced by the FE model if the correct mass and stiffness matrices are identified. FE model updating is achieved by identifying the correct mass and stiffness matrices. The correct mass and stiffness matrices, in the light of the measured data, can be obtained by identifying the correct moduli of elasticity for various sections of the structure under consideration1. In this paper, to correctly identify the moduli of elasticity of the structure, the following cost function that measures the distance between measured data and FE model calculated data, is minimized: Mathematical Background In this study, modal properties, i.e. natural frequencies and mode shapes, are used as a basis for FE model updating. For this reason these parameters are described in this section. Modal properties are related to the physical properties of the structure. All elastic structures may be described in terms of their distributed mass, damping and stiffness matrices in the time domain through the following expression10: [ M ]{ X ' ' } + [ C ]{ X ' } + [ K ]}{ X } = { F } (1) where [M], [C] and [K] are the mass, damping and stiffness matrices respectively, and {X}, {X′} and {X′′} are the displacement, velocity and acceleration vectors respectively while {F} is the applied force vector. If equation 1 is transformed into the modal domain to form an eigenvalue equation for the ith mode, then10:  ω m − ω calc E = ∑ γ i  i m i ωi i =i  N N ( 2   ...   + β ∑ 1 − diag( MAC({ φ }icalc ,{ φ }im )) (4) ) i 2 ( −ω i [ M ] + jω i [ C ] + [ K ]){ φ }i = { 0 } (2) th where j = − 1 , ω i is the i complex eigenvalue, with its imaginary part corresponding to the natural frequency ωi, { 0 } is the null vector and { φ }i is the ith complex mode shape vector with the real part corresponding to the normalized mode shape {φ}i. From equation 2, it may be deduced that the changes in the mass and stiffness matrices cause changes in the modal properties of the structure. Therefore, the modal properties can be identified through the identification of the correct mass Here m is for measured, calc is for calculated, N is the number of modes; γ i is the weighting factor that measures the relative distance between the initial estimated natural frequencies for mode i and the target frequency of the same mode; the parameter β is the weighting function on the mode shapes; the MAC is the modal assurance criterion11; and the diag(MAC)i stands for the ith diagonal element of the MAC matrix. The MAC is a measure of the correlation between two sets of mode shapes of the same dimension. In equation 4 the first 2 part has a function of ensuring that the natural frequencies predicted by the FE model are as close to the measured ones as possible while the second term ensures that the mode shapes between measurements and those predicted by the FE model are correlated. When two sets of mode shapes are perfectly correlated then the MAC matrix is an identity matrix. The updated model is evaluated by comparing the natural frequencies and mode shapes from the FE models before and after updating to the measured ones. Initial Conditions Updating Parameters Updating Objective Updating Space Generation of surface response data using the FE model Response Surface Method Functional approximation RSM method is a procedure that operates by generating a response for a given input. The inputs are the parameters to be updated and the response is the error between the measured data and the FE model generated data. Then an approximation model of the input parameters and the response, called a response surface equation, is constructed. As a consequence of this, the optimization method operates on the surface response. This equation is usually simple and not computationally intensive as opposed to a full FE model. RSM has other advantages such as the ease of implementation through parallel computation and the ease at which parameter sensitivity can be calculated. The proposed RSM consists of these essential components: (1) the response surface approximation equation; and (2) the optimization procedure. There are many techniques that have been used for response surface approximation such as polynomial approximation12 and neural networks13. A multi-layer perceptron is used as a response surface approximation equation6. Further understanding of different approaches to response surface approximation may be found in the literature14-19. In this paper, MLP is used because it has been successfully used to solve complicated regression problems. The details of the MLP are described in the next section. The second component of the RSM is the optimization of the response surface. There are many types of optimization methods that can be used to optimize the response surface equation and these include the gradient based methods20 and evolutionary computation methods21. The gradient based methods have a shortcoming of identifying local optimum solutions while evolutionary computing methods are better able to identify global optimum solution. As a result of the global optimum advantage of evolutionary methods, in this study the GA is used to optimize the response surface equation. The manner in which the RSM is implemented is shown in Figure 1. In this figure it shown that the RSM is implemented by following these steps: 1) Setting initial conditions which are: updating parameters, updating objective, which is in equation 4, Global optimization Functional evaluation at the optimum solution on the FE model Updating criteria satisfied? N Replace the worst surface response data with the optimum point and its FE response Y Stop Figure 1. The flowchart of the RSM. Here N stands for no and Y stands for yes. and updating space. 2) The FE model is then used to generate sample response surface data 3) MLP is used to approximate the response surface approximation equation from the data generated in Step 2. 4) GA is used to find a global optimum solution. 5) The new optimum solution is used to evaluate the response from the full FE model. 6) If the optimum solution does not satisfy the objective, then the new optimum and the corresponding FE model calculated response replaces the candidate with the worst response in data set generated in Step 2 and then steps 3 to 5 are repeated. If the objective is satisfied then stop and the optimum solution becomes the ultimate solution. Step 6 ensures that the simulation always operates in the region of the optimum solution. The next section 3 sic dimensionality of the data. Models of this form can approximate any continuous function to arbitrary accuracy if the number of hidden units M is sufficiently large. The relationship between the output y, representing error between the model and measured data, and input, x, representing updating parameters may be written as follows6: describes an MLP, which is used for functional approximation. Multi-layer Perceptron Multi-layer perceptron is a type of neural networks which used in the present study. This section gives the over-view of the MLP in the context of functional approximation. The MLP is viewed in this paper as parameterized graphs that make probabilistic assumptions about data. Learning algorithms are viewed as methods for finding parameter values that look probable in the light of the data. Supervised learning is used to identify the mapping function between the updating parameters (x) and the response (y). The response is calculated using equation 4. The reason why the MLP is used is because it provides a distributed representation with respect to the input space due to cross-coupling between input, hidden and output layers. The MLP architecture contains a hyperbolic tangent basis function in the hidden units and linear basis functions in the output units6. A schematic illustration of the MLP is shown in Figure 2. This network architecture contains hidden units and output units and has one hidden layer. The bias parameters in the first layer are shown as weights from an extra M   d  y k =  ∑ wkj( 2 ) tanh ∑ w (ji1 ) x i + w (j 01 )  + wk( 02 )  (5)  i =1   j =1  Here, w (ji1 ) and w(ji2 ) indicate weights in the first and second layers, respectively, going from input i to hidden unit j, M is the number of input units, d is the number of output units while w (j 01 ) indicates the bias for the hidden unit j. Training the neural network identifies the weights in equations 5 and a cost function must be chosen to identify these weights. A cost function is a mathematical representation of the overall objective of the problem. The main objective, this is used to construct a cost function, is to identify a set of neural network weights given updating parameters and the error between the FE model and the measured data. If the training set D = { x k , t k }kN=1 is used and assuming that the targets t are sampled independently given the inputs xk and the weight parameters, wkj, the cost function, E, may be written as follows using the sum-of-square error func6 tion : Output Units N zM The sum-of-square error function is chosen because it 6 has been found to be suited to regression problems . In equation 6, N is the number of training examples and K is the number of output units. In this paper, N is equal to 150, while K is equal to 1. Before the MLP is trained, the network architecture needs to be constructed by choosing the number of hidden units, M. If M is too small, the MLP will be insufficiently flexible and will give poor generalization of the data because of high bias. However, if M is too large, the neural network will be unnecessarily flexible and will give poor generalization due to a phenomenon known as over-fitting caused by high variance. In this study, we choose M such that the number of weights is at most fewer than the number of response data. This is in line with the basic mathematical principle which states that in order to solve a set of equations with n variables you need at least n independent data points. The next section describes the GA, which is a method that is used to solve for the optimum solution of the response surface approximation equation. Hidden Units bias xd (6) n =1 k =1 z0 z1 K E = ∑ ∑ { t nk − y nk } 2 yc y1 x1 x0 Input Units Figure 2. Feed-forward network having two layers of adaptive weights input having a fixed value of x0=1. The bias parameters in the second layer are shown as weights from an extra hidden unit, with the activation fixed at z0=1. The model in Figure 2 is able to take into account the intrinsic di- Genetic Algorithms GA was inspired by Darwin’s theory of natural evolution. Genetic algorithm is a simulation of natural evolu- 4 tion where the law of the survival of the fittest is applied to a population of individuals. This natural optimization method is used to optimize either the response surface approximation equation or the error between the FE model and the measured data. GA is implemented by generating a population and creating a new population by performing the following procedures: (1) crossover; (2) mutation; (3) and reproduction. The details of these procedures can be found in Holland21 and Goldberg22. The crossover operator mixes genetic information in the population by cutting pairs of chromosomes at random points along their length and exchanging over the cut sections. This has a potential of joining successful operators together. Arithmetic crossover technique22 is used in this paper. Arithmetic crossover takes two parents and performs an interpolation along the line formed by the two parents. For example if two parents p1 and p2 undergo crossover, then a random number a which lies in the interval [0,1] is generated and the new offsprings formed are p1(a-1) and pa. The next section describes simulated annealing which is used to update an FE model using a FE model. Simulated Annealing Simulated Annealing is a Monte Carlo method that is used to investigate the equations of state and frozen states of n degrees of freedom system23. SA was inspired by the process of annealing where objects, such as metals, re-crystallize or liquids freeze. In the annealing process the object is heated until it is molten, then it is slowly cooled down such that the metal at any given time is approximately in thermodynamic equilibrium. As the temperature of the object is lowered, the system becomes more ordered and approaches a frozen state at T=0. If the cooling process is conducted insufficiently or the initial temperature of the object is not sufficiently high, the system may become quenched forming defects or freezing out in metastable states. This indicates that the system is trapped in a local minimum energy state. The process that is followed to simulate the annealing process was proposed by Metropolis et al.24 and it involves choosing the initial state with energy Eold (see equation 4) and temperature T and holding T constant and perturbing the initial configuration and computing Enew at the new state. If Enew is lower than Eold, then accept the new state, otherwise if the opposite is the case then accept this state with a probability of exp -(dE/T) where dE is the change in energy. This process can be mathematically represented as follows: Mutation is a process that introduces to a population, new information. Non-uniform mutation22 was used and it changes one of the parameters of the parent based on a non-uniform probability distribution. The Gaussian distribution starts with a high variance and narrows to a point distribution as the current generation approaches the maximum generation. Reproduction takes successful chromosomes and reproduces them in accordance to their fitness functions. In this study normalized geometric selection method was used22. This method is a ranking selection function which is based on the normalized geometric distribution. Using this method the least fit members of the population are gradually driven out of the population. The basic genetic algorithm was implemented in this paper as follows: if E new < E old accept state E new  E − E old  (7)  else accept E new with probability exp new T   This processes is repeated such that the sampling statistics for the current temperature is adequate, and then the temperature is decreased and the process is repeated until a frozen state where T=0 is achieved. SA was first applied to optimization problems by Kirkpatrick, et al. 23. The current state is the current updating solution, the energy equation is the objective function in equation 4, and the ground state is the global optimum solution. 1) Randomly create an initial population of a certain size. 2) Evaluate all of the individuals in the population using the objective function in equation 4. 3) Use the normalized geometric selection method to select a new population from the old population based on the fitness of the individuals as given by the objective function. Example: Asymmetrical H-structure An unsymmetrical H-shaped aluminum structure shown in Figure 3 was used to validate the proposed method. This structure was also used by Marwala and Heyns4 as well as Marwala25. This structure had three thin cuts of 1mm that went half-way through the crosssection of the beam. These cuts were introduced to elements 3, 4 and 5. The structure with these cuts was used so that the initial FE model gives data that are far from the measured data and, thereby test the proposed proce- 4) Apply some genetic operators, non-uniform mutation and arithmetic crossover, to members of the population to create new solutions. 5) Repeat steps 2-6, which is termed one generation, until a certain fixed number of generations has been achieved 5 dure on a difficult FE model updating problem. The structure was suspended using elastic rubber bands. The structure was excited using an electromagnetic shaker and the response was measured using an accelerometer. The structure was divided into 12 elements. It was excited at a position indicated by double-arrows, in Figure 3, and acceleration was measured at 15 positions indicated by single-arrows in Figure 3. The structure was tested freely-suspended, and a set of 15 frequency response functions were calculated. A roving accelerometer was used for the testing. The mass of the accelerometer was found to be negligible compared to the mass was run 150 times to generate the data for functional approximation. The MLP implemented had 12 input variables corresponding to the 12 elements in the FE model, 8 hidden units and one output unit corresponding to the error in equation 4. As described before, the MLP had a hyperbolic tangent activation function in the hidden layer and linear activation function in the output layer. The RSM functional approximation via the MLP was evaluated 10 times (iterations) each time using the GA to calculate the optimum point and evaluating this optimum point on the FE model and then storing the previous optimum point in the data set for the current A 200mm 400mm B 600mm y Cross-section indicated by line AB 9.8mm x 32.2mm Figure 3. Irregular H-shaped structure of the structure. The number of measured coordinates functional approximation. The scaled conjugate gradiwas 15. ent method was used to train the MLP, primarily beThereafter, the finite element model was constructed cause of its computational efficiency27. The initial funcusing the Structural Dynamics Toolbox26. The FE tional approximation was obtained by training the MLP model used Euler-Bernoulli beam elements. The FE for 150 training cycles and on a subsequent functional model contained 12 elements. The moduli of elasticity of these elements were used as updating parameTable 1. Results showing measured frequencies, the initial freters. When the FE updating was implemented the quencies and the frequencies obtained when the FE model is moduli of elasticity was restricted to vary from updated using the RSM, SA and GA 10 10 -2 6.00x10 to 8.00x10 N.m . The weighting factors, Measured Initial Frequencies Frequencies Frequencies in the first term in equation 4, were calculated for Freq Freq from RSM from SA from GA each mode as the square of the error between the (Hz) (Hz) Updated Updated Updated measured natural frequency and the natural frequency Model (Hz) Model (Hz) Model (Hz) calculated from the initial model and the weighting 53.9 56.2 52.2 54.0 53.9 function for the second term in equation 4 was set to 117.3 127.1 118.4 118.8 120.1 0.75. When the RSM, SA and GA were implemented 228.4 209.4 209.7 211.3 for model updating the results shown in Table 1 were 208.4 254.0 263.4 251.1 253.8 253.4 obtained. 445.1 452.4 432.7 435.8 438.6 On implementing the proposed RSM, the FE model 6 and that from the initial model was 8.4%. When the RSM was used, this error was reduced to 0.9% while using SA it was reduced to 1.3% and using the GA it was reduced to 2.4%. The error of the third natural frequencies between the measured data and the initial FE model was 9.6%. When the RSM was used, this error was reduced to 0.5% while using SA reduced it to 0.6% and using the GA and a full FE model reduced it to 1.4%. The error between the fourth measured natural frequency and that from the initial model was 3.7%. When the RSM was used for FE updating, this error was reduced to 1.1% while using the SA reduced it to 0.1% and using the GA and a full FE model reduced it to 0.2%. The error between the fifth measured natural frequency and that from the initial model was 1.6%. When the RSM was used, this error was increased to 2.8% while using SA increased it to 2.1% and using the GA and a full FE model the error was reduced to 1.5%. Overall, the SA gave the best results with an average error, calculated over all the five natural frequencies, of 0.9% followed by the GA with an average error of 1.1% and then RSM with an average error of 1.7%. All the three methods on average improved when compared to the average error between the initial FE model and the measured data, which was 5.5%. The updated FE models implemented were 9 also validated on the mode shapes they pre8 dicted. To make this assessment possible the 7 MAC11 was used. The mean of the diagonal of 6 the MAC vector was used to compare the mode 5 shapes predicted by the updated and initial FE models to the measured mode shapes. The av4 erage MAC calculated between the mode shapes 3 from an initial FE model and the measured 2 mode shapes was 0.8394. When the average 1 MAC was calculated between the measured data 0 and data obtained from the updated FE models, it was observed that the RSM, SA and GA up1 2 3 4 5 6 7 8 9 10 11 12 dated FE model gave the improved average of Element the diagonal of the MAC matrix of 0.8413, Initial RSM GA SA 0.8430 and 0.8419, respectively. Therefore, the Figure 4. Graph showing the initial moduli of elasticity and the SA gave the best MAC followed by the GA moduli of elasticity obtained when the FE model is updated using which was followed by the RSM. However, the RSM, GA and SA. Here e10 indicates 10 to the power 10 and these differences in accuracies of the MAC and the units are Nm-2 natural frequencies were not significant. The computational time taken to run the complete RSM method was 46 CPU seconds, while measured natural frequency and that from the initial FE the SA and a full FE model took 19 CPU minutes to run model, which was obtained when the modulus of elasticand the GA and a full FE model took 117 CPU seconds. 10 -2 ity of 7.00x10 N.m was assumed, was 4.3%. When The RSM was found to be faster than the GA which was the RSM was used for FE updating, this error was rein turn much faster than the SA which was faster that the duced to 3.1% while using SA it was reduced to 0.2% GA. On implementing the RSM, 160 FE model evaluaand using the GA approach it was reduced to 0%. The tions were made, while on implementing the SA 19485 error between the second measured natural frequency FE model evaluations were made and on implementing Modulus of Elasticity (e10) approximation, where the data set had the previous optimum solution added to it, used 5 training cycles. On using the RSM, the MLP was only initialized once. The GA was implemented on a population size of 50 and 200 generations. The normalized geometric distribution was implemented with a probability of selecting the best candidate set to 8%, mutation rate of 0.3% and crossover rate of 60%. When SA and a full FE model was implemented for FE updating, the scale of the cooling schedule was set to 4 and the number of individual annealing runs was set to 3. When the simulation was run, the first run involved 7008 FE model calculations, in the second run 6546 FE model calculations and in the third run 5931 FE model calculations were made. On implementing the GA and a full FE model, the same options as those that were used in the implementation of the RSM were used. The results showing the moduli of elasticity of the initial FE model, RSM updated FE model, SA updated FE model and GA updated FE model are shown in Figure 4. Table 1 shows the measured natural frequencies, initial natural frequencies and natural frequencies obtained by the RSM, SA and GA updated FE models. The error between the first 7 the GA 10000 FE model calculations were made. In this paper, a simple FE model with 39 degrees of freedom is updated. It can, therefore, be concluded that if the FE model had several thousand degrees of freedom, the RSM will be substantially faster than the other methods. This conclusion should be understood in the light of the fact that FE models usually have many degrees of freedom. changes in their vibration characteristics: a literature review” Los Alamos National Laboratory Report LA13070-MS, 1996. 10 Ewins, D.J., Modal testing: theory and practice, Research Studies Press, Letchworth, U.K, 1995. 11 Allemang, R.J. and Brown, D.L., “A correlation coefficient for modal vector analysis” Proceedings of the 1st International Modal Analysis Conference, 1-18, 1982. 12 Sacks, J., Welch, W.J., Mitchell, T.J., and Wynn, H.P., “Design and analysis of computer experiments” Statistical Science, 4(4), 1989, pp. 409-435. 13 Varaajan, S., Chen, W., and Pelka, C.J., “Robust concept exploration of propulsion systems with enhanced model approximation” Engineering Optimization, 32(3), 2000, pp. 309-334. 14 Giunta, A. A., and Watson, L. T., “A Comparison of Approximation Modeling Techniques: Polynomial Versus Interpolating Models,” AIAA-98-4758, American Institute of Aeronautics and Astronautics, Inc., 1998, pp.392-401. 15 Koch, P. N., Simpson, T. W., Allen, J. K., and Mistree, F., 1999, “Statistical Approximations for Multidisciplinary Design Optimization: The Problem of Size,” Journal of Aircraft, 36(1), 1999, pp. 275-286. 16 Jin, R., Chen, W., and Simpson, T., 2000, “Comparative Studies of Metamodeling Techniques under Multiple Modeling Criteria,” 8th AIAA/NASA/USAF/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Long Beach, CA, September 6-8, 2000. 17 Lin, Y., Krishnapur, K., Allen, J. K., and Mistree, F., 2000, “Robust Concept Exploration in Engineering Design: Metamodeling Techniques and Goal Formulations,” Proceedings of the 2000 ASME Design Engineering Technical Conferences, DETC2000/DAC14283, September 10-14, 2000, Baltimore, Maryland. 18 Wang, G.G., 2003, “Adaptive response surface method using inherited Latin hypercube design points,” Transactions of the ASME, Journal of Mechanical Design,125, pp. 210-220. 19 Simpson, T. W., Peplinski, J. D., Koch, P. N., and Allen, J. K., “Metamodels for Computer-based Engineering Design: Survey and Recommendations,” Engineering with Computers, 17, 2001, pp. 129-150. 20 Fletcher, R., Practical Methods of Optimization. 2nd edition, New York: Wiley, 1987. 21 Holland, J, Adaptation in Natural and Artificial Systems, Ann Arbor: University of Michigan Press, 1975. 22 Goldberg, D.E., Genetic algorithms in search, optimization and machine learning, Addison-Wesley, Reading, MA, 1989. 23 Metropolis, N, Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., and Teller, E., “Equations of state Conclusion In this study, RSM is proposed for FE model updating. The proposed RSM was implemented within the framework of the MLP for functional approximation and GA for optimization of the MLP response surface function. This procedure was compared to the GA and SA. When these techniques were tested on the unsymmetrical H-shaped structure, it was observed that the RSM was faster than the SA and GA without much compromise on the accuracy of the predicted modal properties. Acknowledgment The author would like to thank Stefan Heyns, the now National Research Foundation as well as the AECI, Ltd for their assistance in this work. 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Non-Asymptotic Bounds and a General Formula for the Rate-Distortion Region of the Successive Refinement Problem ∗ Tetsunao Matsuta† arXiv:1802.07458v1 [] 21 Feb 2018 source symbol X encoder 1 ✲ f1 encoder 2 ✲ f2 decoder 1 ✲ ϕ1 reprod. symbol 1 decoder 2 reprod. symbol 2 ✲ϕ ✲ 2 Tomohiko Uyematsu‡ rate-distortion region. Since a codeword is used in both decoders, we cannot always optimize rates like the case where each codeword is used for each reconstruction separately. However, there are some cases where we can achieve the optimum rates. Necessary and sufficient conditions for such cases were independently given by Koshelev [3], [4] and Equitz and Cover [5]. The complete characterization of the rate-distortion region for discrete stationary memoryless sources was given by Rimoldi [6]. Yamamoto [7] also gave the rate-distortion region as a special case of a more general coding problem. Later, Effros [8] characterized the rate-distortion region for discrete stationary ergodic and non-ergodic sources. Recently, the asymptotic analysis of the second-order rates to the blocklength becomes an active target of the study. Especially, for the successive refinement problem, No et al. [9] and Zhou et al. [10] gave a lot of results to the set of second-order rates for discrete and Gaussian stationary memoryless sources. No et al. [9] considered separate excess-distortion criteria such that a probability that a distortion exceeds a given distortion level is less than a given probability level separately for each reconstruction. On the other hand, Zhou et al. [10] considered the joint excess-distortion criterion such that a probability that either of distortions exceeds a given distortion level is less than a given probability level. Although they also gave several nonasymptotic bounds on the set of pairs of rates, they mainly focus on the asymptotic behavior of the set. On the other hand, in this paper, we consider non-asymptotic bounds on pairs of rates in finite blocklengths. Especially, since a rate is easily calculated by a number of codewords, we focus on pairs of two numbers of codewords. Although we adopt separate excess-distortion criteria, our result can be easily applied to the joint excess-distortion criterion. We give inner and outer bounds on pairs of numbers of codewords. These bounds are characterized by using the smooth max Rényi divergence introduced by Warsi [11]. For the point-to-point lossy source coding problem, we also used the smooth max Rényi divergence to characterize the rate-distortion function which is the minimum rate when the blocklength is unlimited [12]. Proof techniques are similar to that of [12], but we employ several extended results for the successive refinement problem. The inner bound is derived by using an extended version of the previous lemma [12, Lemma 2]. We give this lemma as a special case of an extended version of the previous generalized covering lemma [13, Lemma 1]. The outer bound is derived by using an extended version of the previous converse bound [12, Lemma 4]. In this paper, we also consider the rate-distortion region for ✲ Y ✲ Z Figure 1: Successive refinement problem SUMMARY In the successive refinement problem, a fixed-length sequence emitted from an information source is encoded into two codewords by two encoders in order to give two reconstructions of the sequence. One of two reconstructions is obtained by one of two codewords, and the other reconstruction is obtained by all two codewords. For this coding problem, we give non-asymptotic inner and outer bounds on pairs of numbers of codewords of two encoders such that each probability that a distortion exceeds a given distortion level is less than a given probability level. We also give a general formula for the rate-distortion region for general sources, where the rate-distortion region is the set of rate pairs of two encoders such that each maximum value of possible distortions is less than a given distortion level. Key words: general source, information spectrum, non-asymptotic bound, rate-distortion region, successive refinement 1 Introduction The successive refinement problem is a fixed-length lossy source coding problem with many terminals (see Fig. 1). In this coding problem, a fixed-length sequence emitted from an information source is encoded into two codewords by two encoders in order to give two reconstructions of the sequence. One of two reconstructions is obtained by one of two codewords by using a decoder, and the other reconstruction is obtained by all two codewords by using the other decoder. An important parameter of the successive refinement problem is a pair of rates of two encoders such that each distortion between the source sequence and a reconstruction is less than a given distortion level. The set of these pairs when the length (blocklength) of the source sequence is unlimited is called the ∗ Portions of this paper were presented at the 38th Symposium on Information Theory and Its Applications [1], and at the 2016 IEICE Society Conference [2]. † [email protected] ‡ [email protected] The authors are with Dept. of Information and Communications Engineering, Tokyo Institute of Technology, Tokyo, 152-8552 Japan. 1 general sources. In this case, we adopt the maximum-distortion criterion such that the maximum value of possible distortion is less than a given distortion level for each reconstruction. By using the spectral sup-mutual information rate (cf. [14]) and the non-asymptotic inner and outer bounds, we give a general formula for the rate-distortion region. We show that our ratedistortion region coincides with the region obtained by Rimoldi [6] when a source is discrete stationary memoryless. Furthermore, we consider a mixed source which is a mixture of two sources and show that the rate-distortion region is the intersection of those of two sources. The rest of this paper is organized as follows. In Section 2, we provide some notations and the formal definition of the successive refinement problem. In Section 3, we give several lemmas for an inner bound on pairs of numbers of codewords and the rate-distortion region. These lemmas are extended versions of our previous results [12, Lemma 2] and [13, Lemma 1]. In Section 4, we give outer and inner bounds using the smooth max Rényi divergence on pairs of numbers of codewords. In Section 5, we give a general formula for the rate-distortion region. In this section, we consider the rate-distortion region for discrete stationary memoryless sources and mixed sources. In Section 6, we conclude the paper. note numbers of codewords. Two decoders decoder 1 and decoder 2 are represented as functions ϕ1 : [1 : M1 ] → Y and ϕ2 : [1 : M1 ] × [1 : M2 ] → Z, respectively. We refer to a tuple of encoders and decoders ( f1, f2, ϕ1, ϕ2 ) as a code. In order to measure distortions between the source symbol and reconstruction symbols, we introduce distortion measures defined by functions d1 : X × Y → [0, +∞) and d2 : X × Z → [0, +∞). We define two events of exceeding given distortion levels D1 ≥ 0 and D2 ≥ 0 as follows: E1 (D1 ) , {d1 (X, ϕ1 ( f1 (X))) > D1 }, E2 (D2 ) , {d2 (X, ϕ2 ( f1 (X), f2 (X))) > D2 }. Then, we define the achievability under the excess-distortion criterion. Definition 1. For positive integers M1, M2 , real numbers D1, D2 ≥ 0, and ǫ1, ǫ2 ∈ [0, 1], let M = (M1, M2 ), D = (D1, D2 ), and ǫ = (ǫ1, ǫ2 ). Then, for a source X, we say (M, D) is ǫachievable if and only if there exists a code ( f1, f2, ϕ1, ϕ2 ) such that numbers of codewords of encoder 1 and encoder 2 are M1 and M2 , respectively, and Pr{Ei (Di )} ≤ ǫi, ∀i ∈ {1, 2}. In what follows, for constants M1, M2, D1, D2, ǫ1, and ǫ2 , we often use the above simple notations: M = (M1, M2 ), D = (D1, D2 ), and ǫ = (ǫ1, ǫ2 ). In this setting, we consider the set of all pairs (M1, M2 ) of numbers of codewords under the excessdistortion criterion. According to the ǫ-achievability, this set is defined as follows: 2 Preliminaries Let N, R, and R ≥0 be sets of positive integers, real numbers, and non-negative real numbers, respectively. Unless otherwise stated, we use the following notations. For a pair of integers i ≤ j, the set of integers {i, i + 1, · · · , j} is denoted by [i : j]. For finite or countably infinite sets X and Y, the set of all probability distributions over X and X × Y are denoted by PX and PX Y , respectively. The set of all conditional probability distributions over X given Y is denoted by PX |Y . The probability distribution of a random variable (RV) X is denoted by the subscript notation PX , and the conditional probability distribution for X given an RV Y is denoted by PX |Y . The n-fold Cartesian product of a set X is denoted by X n while an n-length sequence of symbols (a1, a2, · · · , an ) is denoted by ∞ is denoted by the bold-face an . The sequence of RVs {X n }n=1 ∞ and letter X. Sequences of probability distributions {PX n }n=1 ∞ conditional probability distributions {PX n |Y n }n=1 are denoted by bold-face letters PX and PX |Y , respectively. For the successive refinement problem (Fig. 1), let X, Y, and Z be finite or countably infinite sets, where X represents the source alphabet, and Y and Z represent two reconstruction alphabets. Let X over X be an RV which represents a single source symbol. Since the source can be characterized by X, we also refer to it as the source. When we consider X as an n-fold Cartesian product of a certain finite or countably infinite set, we can regard the source symbol X as an n-length source sequence. Thus, for the sake of brevity, we deal with the single source symbol unless otherwise stated. Two encoders encoder 1 and encoder 2 are represented as functions f1 : X → [1 : M1 ] and f2 : X → [1 : M2 ], respectively, where M1 and M2 are positive integers which de- Definition 2. For a source X, real numbers D1, D2 ≥ 0, and ǫ1, ǫ2 ∈ [0, 1], we define M(D, ǫ |X) , {(M1, M2 ) ∈ N2 : (M, D) is ǫ-achievable}. Basically, this paper deals with a coding for a single source symbol. However, in Section 5, we deal with the coding for an n-length source sequence. Hence in that section, by abuse of notation, we regard the above sets X, Y, and Z as n-fold Cartesian products X n , Y n , and Z n , respectively. We also regard source symbol X on X as an n-length source sequence ∞ X n on X n . Then we call the sequence X = {X n }n=1 of source sequences the general source that is not required to satisfy the consistency condition. We use the superscript (n) for a code, distortion measures, (n) (n) (n) (n) and numbers of codewords (e.g., ( f1 , f2 , ϕ1 , ϕ2 )) to make clear that we are dealing with source sequences of length n. For a code, we define rates R1(n) and R2(n) as 1 log Mi(n), ∀i ∈ {1, 2}. n Hereafter, log means the natural logarithm. We introduce maximum distortions for a sequence of codes. To this end, we define the limit superior in probability [14]. Ri(n) , Definition 3 (Limit superior in probability). For an arbitrary ∞ of real-valued RVs, we define the limit sequence S = {S n }n=1 superior in probability by n o p-lim sup Sn , inf α : lim Pr {Sn > α} = 0 . n→∞ 2 n→∞  h    M2 i M1 = E E E 1{(A, B̃, C̃) < F } A, B̃ A , Now we introduce the maximum distortions: p-lim sup d1(n) (X n, ϕ1(n) ( f1(n) (X n ))), n→∞ (n) p-lim sup d2 (X n, ϕ2(n) ( f1(n) (X n ), f2(n) (X n ))). n→∞ where 1{·} denotes the indicator function. Proof. We have Then, we define the achievability under the maximum distortion criterion. Pr Definition 4. For real numbers R1, R2 ≥ 0, let R = (R1, R2 ). Then, for a general source X, and real numbers D1, D2 ≥ 0, we say a pair (R, D) is fm-achievable if and only if there exists a sequence {( f1(n), f2(n), ϕ1(n), ϕ2(n) )} of codes satisfying p-lim sup d1(n) (X n, ϕ1(n) ( f1(n) (X n ))) n→∞ (n) p-lim sup d2 (X n, ϕ2(n) ( f1(n) (X n ), f2(n) (X n ))) n→∞ = ≤ D1, ≤ D2, = ∀i ∈ {1, 2}. M1 Ö PB̃ (b̃i ) M2 Ö Õ PC̃ | B̃ (c̃i, j | b̃i ) j=1 × 1{(a, b̃i, c̃i, j ) < F } = Õ P A (a) a ∈A M1 Õ Ö PB̃ (b̃i ) i=1 b̃i ∈B × 1{(a, b̃i, c̃i, j ) < F } Remark 1. We can show that the rate-distortion region R(D|X) is a closed set by the definition and using the diagonal line argument (cf. [14]). = Õ a ∈A P A (a) M1 Õ Ö i=1 b̃ ∈B PB̃ (b̃) M2 Õ Ö PC̃ | B̃ (c̃i, j | b̃i ) j=1 c̃i, j ∈C M2 Ö j=1 Õ PC̃ | B̃ (c̃| b̃) c̃ ∈C: (a, b̃, c̃)<F ª © Õ ©Õ ® ­ ­ PC̃ | B̃ (c̃| b̃)® PB̃ (b̃) ­ = P A (a) ­ ­ ­ ® c̃ ∈C: a ∈A b̃ ∈B ¬ «(a, b̃, c̃)<F « Õ We note that when we regard X as n-length sequence in the definition of M(D, ǫ |X), it gives a non-asymptotic region of pairs of rates for a given finite blocklength. 3 Covering Lemma M2 M1 ª ® ® ® ¬ . By recalling that ( B̃, C̃) is independent of A, this coincides with the right side of (1).  In this section, we introduce some useful lemmas and corollaries for an inner bound on the set M(D, ǫ |X) and R(D|X). The next lemma is the most basic and important result in the sense that all subsequent results in this section are given by this lemma. This lemma implies an exact analysis of the error probability of covering a set A in terms of a given condition F by codewords { B̃i } and {C̃i, j } of random coding. Hence, this lemma can be regarded as an extended version of [15, Theorem 9]. Although the above lemma gives an exact analysis, it is difficult to use it for characterizing an inner bound on pairs of numbers of codewords and the rate-distortion region. Instead of it, we will use the next convenient lemma. Lemma 1. Let A ∈ A be an arbitrary RV, and B̃ ∈ B and C̃ ∈ C be RVs such that the pair ( B̃, C̃) is independent of A. For an integer M1 ≥ 1, let B̃1, B̃2, · · · , B̃ M1 be RVs which are independent of each other and of A, and each distributed according to PB̃ . For an integer i ∈ [1 : M1 ] and M2 ≥ 1, let C̃i,1, C̃i,2, · · · , C̃i, M2 be RVs which are independent of each other and of A, and each distributed according to PC̃ | B̃ (·| B̃i ). Then, for any set F ⊆ A × B × C, we have j=1   M2 i=1 R(D|X) , {(R1, R2 ) ∈ R2≥0 : (R, D) is fm-achievable}.   i=1 Õ Ö Ö © ª × ­P A(a) PC̃ | B̃ (c̃i, j | b̃i )® PB̃ (b̃i ) j=1 i=1 « ¬ M M 2 1 ©Ö Ö ª ×­ 1{(a, b̃i, c̃i, j ) < F }® « i=1 j=1Õ ¬ Õ Õ × P A(a) Definition 5 (Rate-distortion region). For a general source X and real numbers D1, D2 ≥ 0, we define {(A, B̃i, C̃i, j ) < F } j=1    a ∈A (b̃1 ··· , b̃ M )∈B M1 (c̃1,1, ··· , c̃1, M ,c2,1 ··· , c̃ M , M )∈C M1 M2 2 1 2 1 In what follows, for constants R1 and R2 , we often use the above simple notation: R = (R1, R2 ). In this setting, we consider the set of all rate pairs under the maximum distortion criterion. According to the fm-achievability, this set, usually called the rate-distortion region, is defined as follows: M2 M1 Ù  Ù    i=1 Õ {(A, B̃i, C̃i, j ) < F } M1 n→∞ Pr M2 M1 Ù   Ù a ∈A (b̃1 ··· , b̃ M )∈B M1 (c̃1,1, ··· , c̃1, M ,c2,1 ··· , c̃ M , M )∈C M1 M2 2 1 2 1 and lim sup Ri(n) ≤ Ri, (1) Lemma 2. Let A ∈ A, B ∈ B, and C ∈ C be arbitrary RVs, and B̃ ∈ B and C̃ ∈ C be RVs such that the pair ( B̃, C̃) is independent of A. Let ψ1 : A × B → [0, 1] be a function and α1 ∈ [0, 1] be a constant such that      P A(a)PB̃ (b) ≥ α1 ψ1 (a, b)P AB (a, b), ∀(a, b) ∈ A × B. 3 (2) Furthermore, let ψ2 : A × B × C → [0, 1] be a function and α2 ∈ [0, 1] be a constant such that × PBC | A (b, c|a) − e−α2 M2 −log α1 (3) Then, we have  h    M2 i M1 E E E 1{(A, B̃, C̃) < F } A, B̃ A © ­ ­ P A(a) ­­1 − α1 ≤ ­ a ∈A: ­ P A (a)>0 « Õ ≤ 1 − E[ψ1 (A, B)ψ2 (A, B, C)] + Pr {(A, B, C) < F } + e−α2 M2 −log α1 + e−α1 M1 . (4) Proof. We have   h  M i M1  E E E 1{(A, B̃, C̃) < F } A, B̃ 2 A = Õ P A(a) a ∈A: P A (a)>0 Õ a ∈A: P A (a)>0 © ­ ­ P A(a) ­­1 − ≤ ­ a ∈A: ­ P A (a)>0 « (a) b ∈B b ∈B M2 M1 ª ® ® ® ¬ (c) ≤ Õ ψ1 (a, b)ψ2 (a, b, c) (b,c)∈B×C: (a,b,c)∈F, PB| A (b |a)>0 + e−α2 M2 −log α1 + e−α1 M1 M2 M1 Õ (b) ª ® ® ® ® ® ® ¬ ≤ 1 − E[ψ1 (A, B)ψ2 (A, B, C)] + Pr {(A, B, C) < F } + e−α2 M2 −log α1 + e−α1 M1 , where |x| + = max{0, x}, (a) follows since (1 − x y) M ≤ 1 − y + e−x M for 0 ≤ x, y ≤ 1 and M > 0, and (b) comes from the fact that ψ1 (a, b)ψ2 (a, b, c) ≤ 1.  The importance of this lemma is to be able to change RVs from (A, B̃, C̃) to arbitrary correlated RVs (A, B, C). This makes it possible to characterize an inner bound on pairs of numbers of codewords and the rate-distortion region. Lemma 2 can be regarded as an extended version of our previous lemma [13, Lemma 1] to multiple correlated RVs. Hence, like the previous lemma, by changing functions and constants, it gives many types of bounds such as the following two corollaries. © Õ ­ α1 ψ1 (a, b)PB | A(b|a) P A(a) ­1 − ­ a ∈A: b ∈B: PB| A (b |a)>0 P A (a)>0 « Õ ª ® ψ2 (a, b, c)PC | AB (c|a, b) + e−α2 M2 ® × ® c ∈C: (a,b,c)∈F ¬ Õ  + M1 (a,b,c)∈A×B×C: (a,b,c)∈F, P AB (a,b)>0 © ª ­ ® Õ ­ ® ­ ψ2 (a, b, c)PC | AB (c|a, b)®® × ­1 − α2 ­ ® c ∈C: ­ ® (a,b,c)∈F, PB| A (b |a)>0 « ¬ Õ ψ1 (a, b)ψ2 (a, b, c) (b,c)∈B×C: (a,b,c)∈F, PB| A (b |a)>0  + ×PBC | A (b, c|a) − e−α2 M2 −log α1 + e−α1 M1 Õ ψ1 (a, b)ψ2 (a, b, c)P ABC (a, b, c) ≤ 1− © © ­Õ ­ Õ ­ ­ P A(a) ­­ ≤ PB̃ (b) ­­1 − ψ2 (a, b, c) ­b ∈B ­ a ∈A: c ∈C: ­ ­ P A (a)>0 (a,b,c)∈F, PB| A (b |a)>0 « «  M1 ×PC | AB (c|a, b) + e−α2 M2 (b) Õ × PBC | A (b, c|a) − e−α2 M2 −log α1 PB̃ (b) ª © Õ ® ­ PC̃ | B̃ (c|b)® × ­1 − ­ ® c ∈C: (a,b,c)∈F « ¬ Õ (a) Õ ≤ P A(a) PB̃ (b) , where (a) comes from (3), (b) follows since (1 − x y) M ≤ 1 − y + e−x M for 0 ≤ x, y ≤ 1 and M > 0 (cf. [16, Lemma 10.5.3]), and (c) comes from (2). Since the probability is not grater than 1, we have   h   M2 i M1 E E E 1{(A, B̃, C̃) < F } A, B̃ A P AB (a, b)PC̃ | B̃ (c|b) ≥ α2 ψ2 (a, b, c)P ABC (a, b, c), ∀(a, b, c) ∈ A × B × C.   M1 Corollary 1. For any real numbers γ1, γ2 ∈ R, and any integers M1, M2 ≥ 1 such that M1 ≥ exp(γ1 ) and M2 ≥ exp(γ2 ), we have  h    M2 i M1 E E E 1{(A, B̃, C̃) < F } A, B̃ A  PB | A (B| A) ≤ Pr log > log M1 − γ1 PB̃ (B) ) PC | AB (C| A, B) or log > log M2 − γ2 PC̃ | B̃ (C|B) M1 © © ­ ­ Õ ­ ­ P A(a) ­­1 − α1 ­­ = ψ1 (a, b)ψ2 (a, b, c) ­ ­ (b,c)∈B×C: a ∈A: ­ ­ (a,b,c)∈F, P A (a)>0 « « PB| A (b |a)>0 Õ + Pr {(A, B, C) < F } + e− exp(γ2 )−γ1 +log M1 + e− exp(γ1 ) . 4 Proof. Let α1 = exp(γ1 ) M1 , α2 = exp(γ2 ) M2 , On the other hand, let α1 and α2 be constants such that   δ1 α1 = exp − D̄∞ (P AB kP A × PB̃ |ψ1 ) ,   δ2 α2 = exp − D̄∞ (P ABC kP AB × PC̃ | B̃ |ψ2 ) .   PB | A (b|a) ≤ log M1 − γ1 , ψ1 (a, b) = 1 log PB̃ (b) ( ) PC | AB (c|a, b) ψ2 (a, b, c) = 1 log ≤ log M2 − γ2 . PC̃ | B̃ (c|b) Then, for any (a, b, c) ∈ A × B × C, we have α1 ψ1 (a, b)P AB (a, b)   P A(a)PB̃ (b) ≤ inf ψ1 (a, b)P AB (a, b) (a,b)∈A×B ψ1 (a, b)P AB (a, b) Then, we can easily check that these constants and functions satisfy (2) and (3). Plugging these functions and constants into (4), we have the desired bound.  ≤ P A(a)PB̃ (b), This corollary can be regarded as a bound in terms of the information spectrum (cf. [14]). To the best of our knowledge, this type of bound has not been reported so far (although, there are some converse bounds [10, Lemma 15] and [17, Theorem 3]). On the other hand, the next corollary gives a bound in terms δ of the smooth max Rényi divergence D∞ (PkQ) defined as δ D∞ (PkQ) where |x| + , Í inf ψ:A→[0,1]: a∈A ψ(a)P(a)≥1−δ and α2 ψ2 (a, b, c)P ABC (a, b, c)   P AB (a, b)PC̃ | B̃ (c|b) ≤ inf (a,b,c)∈A×B×C ψ2 (a, b, c)P ABC (a, b, c) × ψ2 (a, b, c)P ABC (a, b, c) ≤ P AB (a, b)PC̃ | B̃ (c|b). ψ(a)P(a) + , log sup Q(a) a ∈A Thus, ψ1 , ψ2 , α1 , and α2 satisfy (2) and (3). Plugging these functions and constants into (4), we have  h    M i M1 E E E 1{(A, B̃, C̃) < F } A, B̃ 2 A = max{0, x}. Corollary 2. For any real numbers δ1, δ2 ≥ 0, and any integers M1, M2 ≥ 1, we have  h    M i M1 E E E 1{(A, B̃, C̃) < F } A, B̃ 2 A ≤ δ1 + δ2 + Pr {(A, B, C) < F } δ2 + e− exp(−D∞ δ1 × e D∞ ≤ δ1 + δ2 + Pr {(A, B, C) < F } δ2 + e− exp(−D∞ +e δ − exp(−D∞1 (P AB k P A ×PB̃ ))M1 + e− exp(−D∞ . (5) a∈A where δ D̄∞ (PkQ|ψ) = log sup a ∈A ψ(a)P(a) Q(a) + . δ D∞ (PkQ)− can be simply defined as δ D∞ (PkQ)− , Then, we have E[ψ1 (A, B)ψ2 (A, B, C)] Í log sup inf ψ:A→[0,1]: ψ(a)P(a)≥1−δ a ∈A ψ(a)P(a) . Q(a) a∈A In this definition, it may be a negative value depending on δ. δ (PkQ). Since this case is meaningless in this study, we adopt D∞ Here we also note that (a) ≥ E[ψ1 (A, B)] + E[ψ2 (A, B, C)] − 1 ≥ 1 − δ1 − δ2, ψ(a)P(a)≥1−δ Since for non-negative real valued functions f (b) and g(b), it holds that (cf. e.g. [16, Lemma 16.7.1]) Í f (b) f (b) Íb ∈B , ≤ sup g(b) g(b) b ∈B b ∈B (6) ǫ, , Remark 2. The original definition of the smooth max Rényi divergence (cf. [12]) is as follows: Í ψ(b)P(b) δ . log sup bÍ∈B D∞ (PkQ)− , inf B⊆A b ∈B Q(b) Í ψ:A→[0,1]: and δ2 D̄∞ (P ABC kP AB × PC̃ | B̃ |ψ2 ) δ2 ≤ D∞ (P ABC kP AB × PC̃ | B̃ ) + (P AB k P A ×PB̃ )−ǫ )M1 where we use inequalities (5), (6), and (7). Since ǫ > 0 is arbitrary, this completes the proof.  Proof. For an arbitrarily fixed ǫ > 0, let ψ1 and ψ2 be functions such that E[ψ1 (A, B)] ≥ 1 − δ1 , E[ψ2 (A, B, C)] ≥ 1 − δ2 , δ1 δ1 (P AB kP A × PB̃ ) + ǫ, (P AB kP A × PB̃ |ψ1 ) ≤ D∞ D̄∞ (P AB k P A ×PB̃ )+ǫ δ1 δ (P ABC k P AB ×PC̃ | B̃ ))M2 +D∞1 (P AB k P A ×PB̃ ) (P ABC k P AB ×PC̃ | B̃ )−ǫ )M2 (7) + δ δ D∞ (PkQ) = D∞ (PkQ)− . where (a) follows since x y ≥ x + y − 1 for x, y ∈ [0, 1]. 5 (8) 4 Inner and Outer Bounds on the Set of Pairs of Numbers of Codewords Remark 3. This proof is valid even without the RV Ũ. This auxiliary RV is introduced merely for consistency with the outer bound. In this section, we give outer and inner bounds on M(D, ǫ |X) by using the smooth max Rényi divergence. First of all, we show a bound on the probability of the two events E1 (D1 ) and E2 (D2 ) for the successive refinement problem. In what follows, let U be an arbitrary set. We use the next notation for the sake of simplicity. Definition 6. For RVs (A, B, C), we define δ δ I∞ (A; B) , D∞ (P AB kP A × PB ), δ δ (A; B|C) , D∞ (P ABC kP AC × PB |C ). I∞ Theorem 1. For a source X, let (Ũ, Ỹ, Z̃) ∈ U × Y × Z be RVs such that (Ũ, Ỹ, Z̃) is independent of X. Then, for any real numbers D1 , D2 ≥ 0, there exists a code ( f1, f2, ϕ1, ϕ2 ) such that numbers of codewords of encoder 1 and encoder 2 are M1 and M2 , respectively, and We also define the following set of probability distributions for a given source X and constants D and ǫ. P(D, ǫ |X) , {PUY Z |X ∈ PU Y Z |X : Pr{d1 (X, Y ) > D1 } ≤ ǫ1, Pr{d2 (X, Z) > D2 } ≤ ǫ2 } Pr{E1 (D1 ) ∪ E2 (D2 )}  h    M2 i M1 ≤ E E E 1{(X, Ũ, Ỹ, Z̃) < D} X, Ũ, Ỹ X , Now, by using the above theorem, we give an inner bound on M(D, ǫ |X). where Theorem 2 (Inner bound). For a source X, real numbers D1 , D2 ≥ 0, and ǫ1, ǫ2 > 0, we have Ø Ø M(D, ǫ |X) ⊇ MI (δ, β, PUY Z |X ), D = {(x, u, y, z) ∈ X × U × Y × Z : d1 (x, y) ≤ D1, d2 (x, z) ≤ D2 } . Proof. We generate (ũ1, ỹ1 ), (ũ2, ỹ2 ), · · · , (ũ M1, ỹ M1 ) ∈ U × Y independently subject to the probability distribution PŨỸ , and define the set C1 , {(ũ1, ỹ1 ), (ũ2, ỹ2 ), · · · , (ũ M1, ỹ M1 )}. For any i ∈ [1 : M1 ], we generate z̃i,1, z̃i,2, · · · , z̃i, M2 ∈ Z independently subject to the probability distribution PZ̃ |Ũ Ỹ (·|ui, yi ), and define the set C2,i , { z̃i,1, z̃i,2, · · · , z̃i, M2 }. We denote {C2,1, C2,2, · · · , C2, M1 } as C2 . For a given set C1 , C2 and a given symbol x ∈ X, we choose i ∈ [1 : M1 ] and j ∈ [1 : M2 ] such that (δ,β,γ)∈S(ǫ ) PUY Z | X ∈P(D,γ |X) where δ = (δ1, δ2 ), β = (β1, β2 ), γ = (γ1, γ2 ),  S(ǫ) , (δ, β, γ) ∈ (0, 1]6 : δ1 + δ2 + γ1 + γ2 +β1 + β2 ≤ min{ǫ1, ǫ2 }} ,  MI (δ, β, PUY Z |X ) , (M1, M2 ) ∈ N2 : 1 δ1 log M1 ≥ I∞ (X; U, Y ) + log log , β1 d1 (x, ỹi ) ≤ D1 and d2 (x, z̃i, j ) ≤ D2 . δ2 (X; Z |U, Y ) log M2 ≥ I∞   1 δ1 (X; U, Y ) + log + log I∞ , β2 If there not exists such pair, we set (i, j) = (1, 1). For this pair, we define encoders f1 and f2 as f1 (x) = i and f2 (x) = j. (9) (10) and (X, U, Y, Z) is a tuple of RVs with the probability distribution PX × PUY Z |X . On the other hand, we define decoders ϕ1 and ϕ2 as Proof. We only have to show that (M, D) is ǫ-achievable for (δ, β, γ) ∈ S(ǫ), PUY Z |X ∈ P(D, γ|X), and M1, M2 ≥ 1 such that     1 δ1 M1 = exp I∞ (X; U, Y ) log , (11) β1     1 δ2 δ1 . M2 = exp I∞ (X; Z |U, Y) I∞ (X; U, Y) + log β2 (12) ϕ1 (i) = ỹi and ϕ2 (i, j) = z̃i, j . By taking the average over the random selection of C1 and C2 , the average probability of Pr{E1 (D1 ) ∪ E2 (D2 )} is as follows: E[Pr{E1 (D1 ) ∪ E2 (D2 )}] M2 M1 Ù     Ù  {d1 (X, Ỹi ) > D1 or d2 (X, Z̃i, j ) > D2 } = Pr     i=1 j=1 M M   2 1 Ù Ù     {(X, Ũi, Ỹi, Z̃i, j ) < D} , = Pr     i=1 j=1 To this end, let (Ũ, Ỹ, Z̃) ∈ U × Y × Z be RVs that is independent of X and has the same marginal distribution as (U, Y, Z), i.e., PŨỸ Z̃ = PUY Z . Then, according to Theorem 1, there exists a code ( f1, f2, ϕ1, ϕ2 ) such that numbers of codewords of encoder 1 and encoder 2 are M1 and M2 , respectively, and where {(Ũi, Ỹi, Z̃i, j )} denote randomly selected sequences in C1 and C2 . Now, by noting that Z̃i, j is generated for a given (Ũi, Ỹi ), the theorem follows from Lemma 1 by setting that A = X, B̃ = (Ũ, Ỹ ), and C̃ = Z̃.  max{Pr{E1 (D1 )}, Pr{E2 (D2 )}} 6  h    M2 i M1 ≤ E E E 1{(X, Ũ, Ỹ, Z̃) < D} X, Ũ, Ỹ X . (13) Lemma 4. For a function g : A → B × C and c ∈ C, let kgkc denote the size of the image of g when one output is fixed to c, i.e., On the other hand, according to Corollary 2, we have  h i M1   M E E E 1{(X, Ũ, Ỹ, Z̃) < D} X, Ũ, Ỹ 2 X kgkc = |{b ∈ B : g(a) = (b, c), ∃a ∈ A}|. Then, for any δ ∈ (0, 1], any RV A ∈ A, and (B, C) = g(A), we have ≤ δ1 + δ2 + Pr {(X, U, Y, Z) < D} δ2 + e− exp(−I∞ δ log sup kgkc ≥ I∞ (A; B|C) + log δ. δ (X;Z |U,Y))M2 +I∞1 (X;U,Y) c ∈C δ +e − exp(−I∞1 (X;U,Y))M1 Proof. Let M = supc ∈C kgkc . Define a subset Dδ ⊆ B × C and the function ψo : A × B × C → [0, 1] as   δ , (16) Dδ , (b, c) ∈ B × C : PB |C (b|c) ≥ M (a) ≤ δ1 + δ2 + γ1 + γ2 + β1 + β2 (b) ≤ min{ǫ1, ǫ2 }, (14) ψo (a, b, c) , 1{(a, b, c) ∈ A × Dδ }. (17) Í Since PBC (b, c) = a ∈A P A(a)1{(b, c) = g(a)}, PB |C (b|c) > 0 for c ∈ C such that PC (c) > 0 if and only if there exists a ∈ A such that (b, c) = g(a) and P A (a) > 0. Thus, for c ∈ C such that PC (c) > 0, we have where (a) follows from (11), (12) and the fact that PUY Z |X ∈ P(D, γ|X), and (b) comes from (9). This implies that (M, D) is ǫ-achievable.  Remark 4. The proof is also valid if we do not restrict (Ũ, Ỹ, Z̃) to be the same distribution as (U, Y, Z). However, for the sake of simplicity, we consider the restricted case. |{b ∈ B : PB |C (b|c) > 0} = |{b ∈ B : (b, c) = g(a), ∃a ∈ A s.t. P A (a) > 0} An outer bound on M(D, ǫ |X) is given in the next theorem. ≤ kgkc ≤ M. Theorem 3 (Outer bound). For a source X, real numbers D1 , D2 ≥ 0 and ǫ1, ǫ2 > 0, and any set U such that |U| ≥ |X|, we have Ø Ù M(D, ǫ |X) ⊆ MO (δ, PUY Z |X ), Then, by using Lemma 3, it is easy to see that Õ ψo (a, b, c)P ABC (a, b, c) PUY Z | X ∈P(D,ǫ |X) δ ∈(0,1]2   δ = Pr PB |C (B|C) ≥ M (a,b,c)∈A×B×C where  MO (δ, PUY Z |X ) = (M1, M2 ) ∈ N2 : log M1 ≥ δ1 I∞ (X; U, Y) log M2 ≥ δ2 I∞ (X; Z |U, Y ) + log δ1, > 1 − δ. o Thus, we have + log δ2 . δ I∞ (A; B|C) Before proving the theorem, we show some necessary lemmas. ≤ log Lemma 3. Suppose that a pair of RVs (A, B) on A × B satisfies = log sup (a,b,c)∈A×D δ |{a ∈ A : P A|B (a|b) > 0}| ≤ M, ∀b ∈ B s.t. PB (b) > 0, sup (a,b,c)∈A×B×C (a) (15) ≤ log sup (a,b,c)∈A×D δ for some M > 0. Then, for any ǫ ∈ (0, 1], we have ψo (a, b, c)P ABC (a, b, c) P AC (a, c)PB |C (b|c) PB | AC (b|a, c) PB |C (b|c) + PB | AC (b|a, c) δ/M + + ≤ log M − log δ, ǫ o > 1 − ǫ. Pr P A|B (A|B) ≥ M n where (a) comes from the definition (16). This completes the proof.  Proof. Since the lemma can be proved in a similar manner as [14, Lemma 2.6.2], we omit the proof.  Now, we give the proof of Theorem 3. Proof of Theorem 3. Let k f1 k be the size of the image of an encoder f1 . Since k f1 k ≤ |X| and |X| ≤ |U| by the assumption, there exists an injective function id : [1 : k f1 k] → U. For this The next lemma is an extended version of [12, Lemma 4], which gives a bound on the size of the image of a function. 7 function, let Uid ⊆ U be the image of id and id−1 : Uid → [1 : k f1 k] be the inverse function of id on Uid . Suppose that (M, D) is ǫ-achievable. Then, there exists a code ( f1, f2, ϕ1, ϕ2 ) such that Since δ1 ∈ (0, 1] and δ2 ∈ (0, 1] are arbitrary, (22) and (25) imply that Ù (M1, M2 ) ∈ MO (δ, PUY Z |X ). δ ∈(0,1]2 Pr {d1 (X, ϕ1 ( f1 (X))) > D1 } ≤ ǫ1, Pr {d2 (X, ϕ2 ( f1 (X), f2 (X))) > D2 } ≤ ǫ2 . Now, by recalling that (X, U, Y, Z) satisfy (18) and (19), for any ǫ-achievable pair (M, D), we have Ø Ù (M1, M2 ) ∈ MO (δ, PUY Z |X ). Thus, by setting U = id( f1 (X)), Y = ϕ1 ( f1 (X)), and Z = ϕ2 ( f1 (X), f2 (X)), we have Pr {d1 (X, Y) > D1 } ≤ ǫ1, Pr {d2 (X, Z) > D2 } ≤ ǫ2 . PUY Z | X ∈P(D,ǫ |X) δ ∈(0,1]2 (18) (19) Remark 5. If we do not employ the RV U which has a role in fixing the RV f1 (X) to a certain codeword, we cannot bound kg2 k(u, y) by M2 in (24). Thus in this proof, introducing U is quite important. For a constant value c, let g1 (x) = (id( f1 (x)), ϕ1 ( f1 (x)), c), A = X, B = (U, Y), and C = c. According to Lemma 4, for any δ1 ∈ (0, 1], we have δ1 log kg1 kc ≥ I∞ (X; U, Y |C) + log δ1 δ1 = I∞ (X; U, Y) + log δ1 . Remark 6. In [1], we gave inner and outer bounds on M(D,ǫ |X) by using the α-mutual information of order infinity [18], where the α-mutual information is a generalized version of the mutual information. In this paper, however, we use the smooth max Rényi divergence. This is because it is compatible with the information spectrum quantity which is well studied and useful to analyze rates of a code. (20) On the other hand, we have kg1 kc = |{(u, y) ∈ Uid × Y : g1 (x) = (u, y, c), ∃x ∈ X}| Õ Õ = 1{g1 (x) = (u, y, c), ∃x ∈ X} Õ u ∈Uid y ∈Y = 1{g1 (x) = (u, ϕ1 (id−1 (u)), c), ∃x ∈ X} 5 General Formula Distortion Region u ∈Uid ≤ M1, (21) where the last inequality follows since the size of Uid is at most k f1 k and k f1 k ≤ M1 . Combining (20) and (21), we have δ1 log M1 ≥ I∞ (X; U, Y) + log δ1 . δ2 kg2 k(u, y) ≥ I∞ (X; Z |U, Y) + log δ2 . sup (22) (23) Rate- PY n |X n (Y n |X n ) 1 log , PY n (Y n ) n→∞ n PY n |X n Z n (Y n |X n, Z n ) 1 . I(X; Y|Z) , p-lim sup log PY n |Z n (Y n |Z n ) n→∞ n I(X; Y) , p-lim sup On the other hand, for any (u, y) ∈ Uid × Y, we have Õ kg2 k(u, y) = 1{z = ϕ2 (id−1 (u), f2 (x)), z ∈Z −1 The smooth max Rényi divergence is related to the spectral sup-mutual information rate as shown in the corollary of the next lemma. id (u) = f1 (x), y = ϕ1 (id−1 (u)), ∃x ∈ X} Õ ≤ 1{∃x ∈ X, z = ϕ2 (id−1 (u), f2 (x))} Õ z ∈Z Lemma 5. Consider two sequences X and Y of RVs over the same set. Then, we have 1{∃ j ∈ [1 : M2 ], z = ϕ2 (id−1 (u), j)} Õ z ∈Z ≤ the ∞ of Definition 7. For a sequence (X, Y, Z) = {(X n, Y n, Z n )}n=1 RVs, we define (u, y)∈U×Y ≤ for In this section, we deal with the coding for an n-length source sequence and give a general formula for the rate-distortion region. First of all, we introduce the spectral (conditional) sup-mutual information rate [14]. Let g2 (x) = (ϕ2 ( f1 (x), f2 (x)), id( f1 (x)), ϕ1 ( f1 (x))), A = X, B = Z, and C = (U, Y). Then, according to Lemma 4, for any δ2 ∈ (0, 1], we have log  This completes the proof. Õ 1 δ 1 PX n (X n ) lim lim sup D∞ (PX n kPY n ) = p-lim sup log . δ↓0 n→∞ n PY n (X n ) n→∞ n 1{z = ϕ2 (id−1 (u), j)} j ∈[1:M2 ] z ∈Z = M2 . ∞ of real numbers, it holds that Proof. For a sequence {an }n=1 + lim supn→∞ |an | = | lim supn→∞ an | + . Thus, according to (8), we have (24) We note that for any (u, y) ∈ {U \ Uid } × Y, it holds that kg2 k(u, y) = 0. Combining (23) and (24), we have δ2 log M2 ≥ I∞ (X; Z |U, Y ) + log δ2 . 1 δ (PX n kPY n ) lim lim sup D∞ δ↓0 n→∞ n (25) 8 + 1 δ = lim lim sup D∞ (PX n kPY n )− . δ↓0 n→∞ n 5.1 Direct Part (26) In this section, we show Ø R(D|X) ⊇ According to [11, Lemma 3], it holds that 1 δ (PX n kPY n )− lim lim sup D∞ δ↓0 n→∞ n PX n (X n ) 1 . = p-lim sup log PY n (X n ) n→∞ n R G (PUYZ|X |X). Let PUYZ|X ∈ PG (D|X) and suppose that I(X; U, Y) < ∞ and I(X; Z|U, Y) < ∞. Then, for any ǫ, ǫ1, ǫ2 > 0 such that ǫ1 = ǫ2 = ǫ, any (δ, β, γ) ∈ S(ǫ) in (9), and every sufficiently large n, we have (27) Furthermore, according to [14, Lemma 3.2.1], the right side of (27) is non-negative. Thus, by combining (26) and (27), we have the lemma.  Pr{d1(n) (X n, Y n ) > D1 + ǫ } ≤ γ1, (n) Pr{d2 (X n, Z n ) > D2 + ǫ } ≤ γ2 . Corollary 3. For a sequence (X, Y, Z) of RVs, we have 1 δ n n lim lim sup I∞ (X ; Y ) = I(X; Y), δ↓0 n→∞ n 1 δ n n n (X ; Y |Z ) = I(X; Y|Z). lim lim sup I∞ δ↓0 n→∞ n Hence, according to Theorem 2, there exists a sequence of codes {( f1(n), f2(n), ϕ1(n), ϕ2(n) )} such that for sufficiently large n, Pr{d1(n) (X n, Ŷ n ) > D1 + ǫ } ≤ ǫ, Pr{d2(n) (X n, Ẑ n ) > D2 + ǫ } ≤ ǫ, Proof. Since this corollary immediately follows from Lemma 5 and Definition 7, we omit the proof.  and Let PUYZ|X be a sequence of conditional probability distributions PU n Y n Z n |X n ∈ PU n Y n Z n |X n . We define PG (D|X) , {PUYZ|X : D1 (X, Y) ≤ D1, D2 (X, Z) ≤ D2 }, (n) D1 (X, Y) , p-lim sup d1 (X n, Y n ), n→∞ D2 (X, Z) , p-lim sup d2(n) (X n, Z n ),     1 1 1 (n) δ1 n n n log M1 = log exp I∞ (X ; U , Y ) log , n n β1 l   1 1 δ2 log M2(n) = log exp I∞ (X n ; Z n |U n, Y n ) n n   1 δ1 n n n . × I∞ (X ; U , Y ) + log β2 Thus, we have n→∞ 1 δ1 n n n lim sup R1(n) ≤ lim sup I∞ (X ; U , Y ) n→∞ n→∞ n R G (PUYZ|X |X) , {(R1, R2 ) ∈ R2 : R1 ≥ I(X; U, Y), R2 ≥ I(X; Z|U, Y)}, where (X, U, Y, Z) is a sequence of RVs (X n, U n, Y n, (a) ≤ I(X; U, Y), Z n ) induced by PUYZ|X and a general source X. The main result of this section is the next theorem which gives a general formula for the rate-distortion region. and lim sup R2(n)    1 δ2 (X n ; Z n |U n, Y n ) ≤ lim sup log exp I∞ n→∞ n   1 δ1 +1 × I∞ (X n ; U n, Y n ) + log β2    1 δ2 ≤ lim sup log exp I∞ (X n ; Z n |U n, Y n ) n→∞ n   1 δ1 n n n (X ; U , Y ) + δ2 × n lim sup I∞ n→∞ n    (a) 1 δ2 ≤ lim sup log exp I∞ (X n ; Z n |U n, Y n ) n→∞ n   × n I(X; U, Y) + δ2 n→∞ Theorem 4. For a general source X, real numbers D1, D2 ≥ 0, and any set U such that |U| ≥ |X|, we have Ø R(D|X) = R G (PUYZ|X |X). (28) PUYZ|X ∈PG (D |X) Remark 7. We can show that the right side of (28) is a closed set by using the diagonal line argument (cf. [14, Remark 5.7.5]). Remark 8. We are not sure whether a sequence U of auxiliary RVs is really necessary or not. It may be possible to characterize the region without it. The proof of this theorem is presented in the subsequent two sections. In these sections, for a code, we denote Ŷ n = Ẑ n = (29) PUYZ|X ∈PG (D |X) 1 δ2 n n n n (X ; Z |U , Y ) = lim sup I∞ n→∞ n (b) (n) (n) ϕ1 ( f1 (X n )), ϕ2(n) ( f1(n) (X n ), f2(n) (X n )). (a) ≤ I(X; Z|U, Y), 9 This means that (M (n), D + γn ) is γn -achievable. According to ∞ Theorem 3, there exists a sequence PUYZ|X = {PU nY n Z n |X n }n=1 of conditional probability distributions such that PU n Y n Z n |X n ∈ P(D + γn, γn |X n ) and for any δ ∈ (0, 1]2 , δ is a nonwhere (a) comes from Corollary 3 and the fact that I∞ increasing function of δ, and (b) follows since I(X; U, Y) < ∞. Now, by using the usual diagonal line argument, we can construct a sequence of codes {( f1(n), f2(n), ϕ1(n), ϕ2(n) )} such that 1 1 δ n n n log M1(n) ≥ lim sup I∞ (X ; U , Y ), n→∞ n n→∞ n 1 1 δ n n n n lim sup log M2(n) ≥ lim sup I∞ (X ; Z |U , Y ). n n n→∞ n→∞ (n) lim sup R1 ≤ I(X; U, Y), lim sup n→∞ lim sup R2(n) ≤ I(X; Z|U, Y). n→∞ and for any ǫ > 0, Since this holds for any δ ∈ (0, 1]2 , we have lim Pr{d1(n) (X n, Ŷ n ) > D1 + ǫ } = 0, lim sup n→∞ n→∞ lim Pr{d2(n) (X n, Ẑ n ) > D2 + ǫ } = 0. 1 δ n n n 1 log M1(n) ≥ lim lim sup I∞ (X ; U , Y ) n δ↓0 n→∞ n (a) n→∞ = I(X; U, Y), This implies that lim sup n→∞ p-lim sup d1(n) (X n, Ŷ n ) ≤ D1, 1 δ n n n n 1 (n) log M2 ≥ lim lim sup I∞ (X ; Z |U , Y ) n δ↓0 n→∞ n (a) n→∞ p-lim sup d2(n) (X n, n→∞ (32) = I(X; Z|U, Y). n (33) Ẑ ) ≤ D2 . where (a) are comes from Corollary 3. On the other hand, since PU nY n Z n |X n ∈ P(D + γn, γn |X n ), PUYZ|X must satisfy D1 (X; Y) ≤ D1 and D2 (X; Z) ≤ D1 , i.e., PUYZ|X ∈ PG (D|X). By combining (31), (32), and (33), we can conclude that for any fm-achievable pair (R, D), Ø (R1, R2 ) ∈ R G (PUYZ|X |X). Thus, for any PUYZ|X ∈ PG (D|X) such that I(X; U, Y) ∈ R and I(X; Z|U, Y) ∈ R, we have (I(X; U, Y), I(X; Z|U, Y)) ∈ R(D|X). This implies (29). PUYZ|X ∈PG (D |X) 5.2 Converse Part This implies (30). In this section, we show that Ø R(D|X) ⊆ R G (PUYZ|X |X). 5.3 Discrete Stationary Memoryless Sources (30) In this section, we show that the rate-distortion region given in Theorem 4 coincides with the region by Rimoldi [6] when a source X is a discrete stationary memoryless source. ∞ is a Let X, Y, and Z be finite sets. Since X = {X n }n=1 discrete stationary memoryless source, we assume that X n = (X1, X2, · · · , Xn ) is a sequence of independent copies of an RV X on X. We also assume that distortion measures d1(n) and d2(n) are additive, i.e., for two functions d1 : X × Y → [0, +∞) and d2 : X × Z → [0, +∞), distortion measures are represented as PUYZ|X ∈PG (D |X) Suppose that (R, D) is fm-achievable. Then, there exists a sequence of codes satisfying p-lim sup d1(n) (X n, Ŷ n ) ≤ D1, n→∞ p-lim sup d2(n) (X n, Ẑ n ) ≤ D2, n→∞ and 1Õ d1 (xi, yi ), n i=1 n 1 lim sup log Mi(n) ≤ Ri, n→∞ n ∀i ∈ {1, 2}. d1(n) (x n, y n ) = (31) Thus, we have for any γ > 0 (n) lim Pr{d1 (X n, Ŷ n ) > D1 + γ} = 0, We define n→∞ lim Pr{d2(n) (X n, Ẑ n ) > D2 + γ} = 0. PM (D|X) , {PY Z |X ∈ PY Z |X : E[d1 (X, Y)] ≤ D1, n→∞ E[d2 (X, Z)] ≤ D2 }, ∞ This implies that there exists a sequence {γn }n=1 such that limn→∞ γn = 0 and Pr{d1(n) (X n, Ŷ n ) Pr{d2(n) (X n, Ẑ n ) 1Õ d2 (xi, zi ). n i=1 n d2(n) (x n, z n ) = and for PY Z |X ∈ PY Z |X , > D1 + γ n } ≤ γ n , R M (PY Z |X |X) , {(R1, R2 ) ∈ R2≥0 : R1 ≥ I(X; Y ), R1 + R2 ≥ I(X; Y, Z)}, > D2 + γ n } ≤ γ n . 10 (X, Ȳ, Z̄) = {(X n, Ȳ n, Z̄ n )} be a sequence of RVs such that (X1, Ȳ1, Z̄1 ), (X2, Ȳ2, Z̄2 ), · · · , (Xn, Ȳn, Z̄n ) are independent and where (X, Y, Z) is the tuple of RVs induced by a conditional probability distribution PY Z |X ∈ PY Z |X and a given RV X. Then, we have the next theorem. Theorem 5. For a discrete stationary memoryless source X, additive distortion measures, and any set U such that |U| ≥ |Z| + 1, we have Ø R G (PUYZ|X |X) Ø PXi Ȳi Z̄i (x, y, z) = PXi Yi Zi (x, y, z), where PXi Yi Zi (x, y, z) is the i-th marginal distribution of (X n, Y n, Z n ). Then, according to [14, Lemma 5.8.1 and 5.8.2], we have D1 (X, Y) ≥ D1 (X, Ȳ), D2 (X, Z) ≥ D2 (X, Z̄), I(X; Y) ≥ I(X; Ȳ), and I(X; Y, Z) ≥ I(X; Ȳ, Z̄). Thus, by introducing the set I of probability distributions for independent RVs as ( n Ö PYi Zi |Xi , I , PYZ|X = {PY n Z n |X n } : PY n Z n |X n = PUYZ|X ∈PG (D |X) = R M (PY Z |X |X). (34) PY Z | X ∈PM (D |X) Remark 9. The right side of (34) can be written as i=1 {(R1, R2 ) ∈ R2≥0 : R1 ≥ R1 (D1 ), R1 + R2 ≥ Rb (R1, D1, D2 )}, ∃PYi Zi |Xi ∈ PYZ |X, i ∈ [1 : n] , (35) we have where R1 (D1 ) is the rate-distortion function and Rb (R1, D1, D2 ) gives the boundary for a given R1 , which are defined as (see, e.g., [19, Corollary 1], [10, (22)]) R1 (D1 ) , min PY | X ∈PY |X :E[d1 (X,Y )]≤D1 Rb (R1, D1, D2 ) , min R1 + R2 ≥ I(X; Y, Z)} Ø ⊆ {(R1, R2 ) : R1 ≥ I(X; Y), I(X; Y, Z). PYZ|X ∈I: D 1 (X,Y)≤D1,D 2 (X,Z)≤D2 R1 + R2 ≥ I(X; Y, Z)}. We note that R1 (D1 ) and Rb (R1, D1, D2 ) are convex and continuous functions of a triple (R1, D1, D2 ) (see, e.g., [14, Remark 5.2.1] and [20, Lemma 4]). I(X; Y) ≥ I(X; Y ) − δ, Proof: The left side of (34) ⊆ The right side of (34). We have Ø R G (PUYZ|X |X) I(X; Y, Z) ≥ I(X; Y, Z) − δ, and PUYZ|X ∈PG (D |X) ⊆ {(R1, R2 ) : R1 ≥ I(X; U, Y), D1 ≥ E[d1 (X, Y )] − γ, PUYZ|X ∈PG (D |X) D2 ≥ E[d2 (X, Z)] − γ. R1 + R2 ≥ I(X; U, Y) + I(X; Z|U, Y)} Ø (a) ⊆ {(R1, R2 ) : R1 ≥ I(X; U, Y), Thus, we have PUYZ|X ∈PG (D |X) Ø {(R1, R2 ) : R1 ≥ I(X; Y), PYZ|X ∈I: D 1 (X,Y)≤D1,D 2 (X,Z)≤D2 R1 + R2 ≥ I(X; U, Y, Z)} Ø (b) ⊆ {(R1, R2 ) : R1 ≥ I(X; Y), PYZ|X : D 1 (X,Y)≤D1,D 2 (X,Z)≤D2 R1 + R2 ≥ I(X; Y, Z)}, (37) On the other hand, in the same way as [14, p.372], for any δ > 0, γ > 0, and any PYZ|X ∈ I such that D1 (X, Y) ≤ D1 and D2 (X, Z) ≤ D2 , there exists PY Z |X ∈ PY Z |X such that We will prove the theorem by two parts separately. Ø {(R1, R2 ) : R1 ≥ I(X; Y), PYZ|X : D 1 (X,Y)≤D1,D 2 (X,Z)≤D2 I(X; Y ), PY Z | X ∈PYZ|X : E[d1 (X,Y)]≤D1,E[d2 (X, Z)]≤D2, I (X;Y )≤R1 Ø ⊆ R1 + R2 ≥ I(X; Y, Z)} ÙÙ Ø {(R1, R2 ) : γ>0 δ>0 PY Z | X ∈PM (D+γ |X) (36) R1 ≥ I(X; Y ) − δ, R1 + R2 ≥ I(X; Y, Z) − δ} ÙÙ = {(R1, R2 ) : R1 + δ ≥ R1 (D1 + γ), where (a) and (b) respectively come from the fact that γ>0 δ>0 I(X; U, Y) + I(X; Z|U, Y) ≥ I(X; U, Y, Z), R1 + R2 + δ ≥ Rb (R1 + δ, D1 + γ, D2 + γ)}, (38) I(X; U, Y) ≥ I(X; Y). For a sequence (X, Y, Z) = {(X n, Y n, Z n )} RVs induced by PYZ|X and a given source X, where the last equality comes from (35). Since R1 (D1 ) and Rb (R1, D1, D2 ) are convex and continuous functions of a triple (R1, D1, D2 ) (see Remark 9), it holds that of let 11 we have for any ǫ > 0, there exist sufficiently small γǫ > 0 and δǫ > 0 such that ÙÙ {(R1, R2 ) : R1 + δ ≥ R1 (D1 + γ), I(X; Z |U, Y ) = H(X |U, Y) − H(X |U, Y, Z) = αH(X |Y, Z) + (1 − α)H(X |Y) − H(X |Y, Z) δ>0 γ>0 = (1 − α)I(X; Z |Y ). R1 + R2 + δ ≥ Rb (R1 + δ, D1 + γ, D2 + γ)} Ù Ù ⊆ {(R1, R2 ) : R1 ≥ R1 (D1 ) − δ − ǫ, Now by defining PUYZ|X as PU n Y n Z n |X n (un, y n, z n |x n ) = δǫ >δ>0 γǫ >γ>0 R1 + R2 ≥ Rb (R1, D1, D2 ) − δ − ǫ } Ù = {(R1, R2 ) : R1 ≥ R1 (D1 ) − δ − ǫ, (X, U, Y, Z) becomes a sequence of independent copies of RVs (X, U, Y, Z). Thus, we have (39) I(X; U, Y) = I(X; U, Y) = αI(X; Z, Y) + (1 − α)I(X; Y), I(X; Z|U, Y) = I(X; Z |U, Y ) = (1 − α)I(X; Z |Y ), By combining (36), (37), (38), and (39), we have Ø R G (PUYZ|X |X) Ù Ù D1 (X, Y) = E[d1 (X, Y)] ≤ D1, D2 (X, Z) = E[d2 (X, Z)] ≤ D2, PUYZ|X ∈PG (D |X) ⊆ ∞ , where we use the fact that for i.i.d. RVs {Ai }i=1 Í n Ai = E[A1 ]. Hence, by noting that PUYZ|X ∈ p-lim sup n1 i=1 {(R1, R2 ) : R1 ≥ R1 (D1 ) − δ − ǫ, ǫ >0 δǫ >δ>0 n→∞ R1 + R2 ≥ Rb (R1, D1, D2 ) − δ − ǫ } PG (D|X), we have for any α ∈ [0, 1], = {(R1, R2 ) : R1 ≥ R1 (D1 ), R1 + R2 ≥ Rb (R1, D1, D2 )}. According to Remark 9, this completes the proof. (αI(X; Z, Y) + (1 − α)I(X; Y), (1 − α)I(X; Z |Y )) Ø ∈ R G (PUYZ|X |X).  (40) PUYZ|X ∈PG (D |X) Proof: The left side of (34) ⊇ The right side of (34). Since |U| ≥ |Z| + 1, there exists an injective function id : Z → U. For this function, let Uid ⊆ U be the image of id and id−1 : Uid → Z be the inverse function of id on Uid . Let u∗ ∈ U be a symbol such that u∗ < Uid . For any PY Z |X ∈ PM (D|X) and any α ∈ [0, 1], we define PUY Z |X ∈ PU Y Z |X as   αP (y, z|x)   Y Z |X  PUY Z |X (u, y, z|x) , (1 − α)PY Z |X (y, z|x)   0  PUY Z |X (ui, yi, zi |xi ), i=1 δǫ >δ>0 R1 + R2 ≥ Rb (R1, D1, D2 ) − δ − ǫ }. n Ö This implies that for any (R1, R2 ) ∈ R M (PY Z |X |X), it holds that Ð (R1, R2 ) ∈ PUYZ|X ∈PG (D |X) R G (PUYZ|X |X). This is because for any (R1, R2 ) ∈ R M (PY Z |X |X) such that I(X; Y, Z) ≥ R1 , there exists α ∈ [0, 1] such that R1 = αI(X; Z |Y ) + I(X; Y ) = αI(X; Y, Z) + (1 − α)I(X; Y ). if u = id(z), if u = u∗, otherwise. Then, we have R2 ≥ I(X; Y, Z) − R1 = I(X; Y, Z) − αI(X; Y, Z) − (1 − α)I(X; Y ) Since = (1 − α)I(X; Z |Y ). (41)   αP (x, y, id−1 (u)) if u ∈ Uid,   XY Z  PXUY (x, u, y) = (1 − α)PXY (x, y) if u = u∗,   0 otherwise,    P (x| y, id−1 (u)) if u ∈ Uid,   X |Y Z  PX |UY (x|u, y) = PX |Y (x| y) if u = u∗,   0 otherwise,  According to (40), such pair (R1, R2 ) is included in the reÐ gion PUYZ|X ∈PG (D |X) R G (PUYZ|X |X). On the other hand, for any (R1, R2 ) ∈ R M (PY Z |X |X) such that R1 > I(X; Y, Z), we have R2 ≥ 0. Since it holds that (I(X; Y, Z), 0) ∈ Ð PUYZ|X ∈PG (D |X) R G (PUYZ|X |X) Ð due to (40), such pair (R1, R2 ) is also included in the region PUYZ|X ∈PG (D |X) R G (PUYZ|X |X). Therefore, we have Ø R G (PUYZ|X |X) ⊇ R M (PY Z |X |X). I(X; U, Y ) = H(X) − αH(X |Y, Z) − (1 − α)H(X |Y ) = αI(X; Y, Z) + (1 − α)I(X; Y). Since this holds for any PY Z |X ∈ PM (D|X), this completes the proof.  we have PUYZ|X ∈PG (D |X) Remark 10. Unlike the rate-distortion region by Rimoldi, our region includes a sequence U of auxiliary RVs. This comes from the fact that the time-sharing argument as in (41) cannot be employed because it holds that in general, Furthermore, since   P (x| y, z)   X |Y Z  PX |UY Z (x|u, y, z) = PX |Y Z (x| y, z)   0  if u = id(z), if u = u∗, otherwise, I(X; Y, Z) , I(X; Y) + I(X; Z|Y). 12 5.4 Mixed Sources R G (PŨỸZ̃|X |X) PU1 Y1 Z1 |X1 ,PU2 Y2 Z2 |X2 : In this section, we give the rate-distortion region for mixed sources. ∞ and X = The mixed source X is defined by X1 = {X1n }n=1 2 n ∞ {X2 }n=1 as Ø D 1 (X, Ỹ)≤D1,D 2 (X, Z̃)≤D2 (a) = {(R1, R2 ) : PU1 Y1 Z1 |X1 ,PU2 Y2 Z2 |X2 : max{D 1 (X1,Y1 ),D 1 (X2,Y2 )} ≤D1, max{D 2 (X1,Z1 ),D 2 (X2,Z2 )} ≤D2 PX n (x n ) = α1 PX1n (x n ) + α2 PX2n (x n ), R1 ≥ max{I(X1 ; U1, Y1 ), I(X; U2, Y2 )} where α1, α2 ∈ [0, 1] and α1 + α2 = 1. The next lemma shows a fundamental property of the information spectrum of mixed sources. = R2 ≥ max{I(X; Z1 |U1, Y1 ), I(X; Z2 |U2, Y2 )}} Ø Ø PU1 Y1 Z1 |X1 ∈PG (D |X1 ) PU2 Y2 Z2 |X2 ∈PG (D |X2 ) Lemma 6. For sequences of RVs (X1, Y1, Z1 ) and (X2, Y2, Z2 ), let (X, Y, Z) be defined by R G (PU1 Y1 Z1 |X1 |X1 ) ∩ R G (PU2 Y2 Z2 |X2 |X2 ) = R(D|X1 ) ∩ R(D|X2 ). PX n Y n Z n (x n, y n, z n ) = α1 PX1n Y1n Z1n (x n, y n, z n ) where (a) comes from [14, Lemma 1.4.2], Lemma 6 and the fact that + α2 PX2n Y2n Z2n (x n, y n, z n ). Then, we have PX n Ũ n Ỹ n Z̃ n (x n, un, y n, z n ) I(X; Y) = max{I(X1 ; Y1 ), I(X2 ; Y2 )}, = α1 PX1n U1n Y1n Z1n (x n, un, y n, z n ) + α2 PX2n U2n Y2n Z2n (x n, un, y n, z n ). I(X; Y|Z) = max{I(X1 ; Y1 |Z1 ), I(X2 ; Y2 |Z2 )}. Proof. Since this lemma can be proved in the same way as [14, Lemma 7.9.1] by using [14, Lemma 1.4.2], we omit the details.  This completes the proof.  6 Conclusion The next theorem shows that the rate-distortion region for a mixed source is the intersection of those of two sources. In this paper, we have dealt with the successive refinement problem. We gave inner and outer bounds using the smooth max Rényi divergence on the set of pairs of numbers of codewords. These bounds are obtained by extended versions of our previous covering lemma and converse bound. By using these bounds, we also gave a general formula using the spectral sup-mutual information rate for the rate-distortion region. Further, we showed some special cases of our rate-distortion region for discrete stationary memoryless sources and mixed sources. Theorem 6. For a mixed source X defined by X1 and X2 , and any real numbers D1, D2 ≥ 0, we have R(D|X) = R(D|X1 ) ∩ R(D|X2 ). Proof. For PU1 Y1 Z1 |X1 and PU2 Y2 Z2 |X2 , we define PŨỸZ̃|X by PŨ n Ỹ n Z̃ n |X n (un, y n, z n |x n ) = Ø = α1 PX1n (x n )PU1nY1n Z1n |X1n (un, y n, z n |x n ) α2 P + X2n PX n (x n ) n (x )PU2n Y2n Z2n |X2n (un, Acknowledgment y n, z n |x n ) PX n (x n ) . This work was supported in part by JSPS KAKENHI Grant Number 15K15935. When PU1 Y1 Z1 |X1 = PU2 Y2 Z2 |X2 = PUYZ|X , we have PŨỸZ̃ |X = PUYZ|X by the definition. This implies that for any PUYZ|X , there exist PU1 Y1 Z1 |X1 and PU2 Y2 Z2 |X2 such that PUYZ|X = PŨỸZ̃ |X . On the other hand, for any PU1 Y1 Z1 |X1 and PU2 Y2 Z2 |X2 , there trivially exists PUYZ|X such that PUYZ|X = PŨỸZ̃|X . Thus, we have Ø R G (PUYZ|X |X) Ø References [1] T. Matsuta and T. Uyematsu, “Non-asymptotic bounds on numbers of codewords for the successive refinement problem,” Proc. 38th Symp. on Inf. Theory and its Apps. (SITA2015), pp.43–48, Nov. 2015. PUYZ|X ∈PG (D |X) = [2] T. Matsuta and T. Uyematsu, “A general formula of the achievable rate region for the successive refinement problem,” Proc. IEICE Society Conference, p.37, Sep. 2016. R G (PŨỸZ̃|X |X). PU1 Y1 Z1 |X1 ,PU2 Y2 Z2 |X2 : D 1 (X, Ỹ)≤D1,D 2 (X, Z̃)≤D2 [3] V. N. Koshelev, “Estimation of mean error for a discrete successive-approximation scheme,” Problemy Peredachi Informatsii, vol.17, no.3, pp.20–33, 1981. Thus, according to Theorem 4, we have R(D|X) 13 [20] V. Kostina and E. Tuncel, “Successive refinement of abstract sources,” arXiv preprint arXiv:1707.09567, July 2017. [4] V. N. Koshelev, “Hierarchical coding of discrete sources,” Problemy Peredachi Informatsii, vol.16, no.3, pp.31–49, 1980. [5] W. Equitz and T. Cover, “Successive refinement of information,” IEEE Trans. Inf. Theory, vol.37, no.2, pp.269–275, Mar. 1991. [6] B. Rimoldi, “Successive refinement of information: characterization of the achievable rates,” IEEE Trans. Inf. Theory, vol.40, no.1, pp.253–259, Jan. 1994. [7] H. Yamamoto, “Source coding theory for a triangular communication system,” IEEE Trans. Inf. Theory, vol.42, no.3, pp.848–853, May 1996. [8] M. Effros, “Distortion-rate bounds for fixed- and variablerate multiresolution source codes,” IEEE Trans. Inf. Theory, vol.45, no.6, pp.1887–1910, Sep. 1999. [9] A. No, A. Ingber, and T. Weissman, “Strong successive refinability and rate-distortion-complexity tradeoff,” IEEE Trans. Inf. Theory, vol.62, no.6, pp.3618–3635, June 2016. [10] L. Zhou, V.Y.F. Tan, and M. Motani, “Second-order and moderate deviations asymptotics for successive refinement,” IEEE Trans. Inf. Theory, vol.63, no.5, pp.2896– 2921, May 2017. [11] N.A. Warsi, “One-shot bounds for various information theoretic problems using smooth min and max Rényi divergences,” Proc. 2013 IEEE Inf. Theory Workshop, pp.1–5, Sep. 2013. [12] T. Uyematsu and T. Matsuta, “Revisiting the rate-distortion theory using smooth max Rényi divergence,” Proc. 2014 IEEE Inf. Theory Workshop, pp.202–206, Nov. 2014. [13] T. Matsuta and T. Uyematsu, “New non-asymptotic bounds on numbers of codewords for the fixed-length lossy compression,” IEICE Trans. Fundamentals, vol.E99-A, no.12, pp.2116–2129, Dec. 2016. [14] T. S. Han, Information-Spectrum Methods in Information Theory, Springer, 2003. [15] V. Kostina and S. Verdú, “Fixed-length lossy compression in the finite blocklength regime,” IEEE Trans. Inf. Theory, vol.58, no.6, pp.3309–3338, June 2012. [16] T. M. Cover and J. A. Thomas, Elements of Information Theory, 2nd ed., Wiley, New York, 2006. [17] V. Kostina and E. Tuncel, “The rate-distortion function for successive refinement of abstract sources,” Proc. IEEE Int. Symp. on Inf. Theory, pp.1923–1927, June 2017. [18] S. Verdú, “α-mutual information,” 2015 Inf. Theory and Apps. Workshop, pp.1–6, Feb. 2015. [19] A. Kanlis and P. Narayan, “Error exponents for successive refinement by partitioning,” IEEE Trans. Inf. Theory, vol.42, no.1, pp.275–282, Jan. 1996. 14 

Shannon information storage in noisy phasemodulated fringes and fringe-data compression by phase-shifting algorithms MANUEL SERVIN,* AND MOISES PADILLA Centro de Investigaciones en Optica A. C., Loma del Bosque 115, 37150 Leon Guanajuato, Mexico. *[email protected] www.cio.mx Abstract: Optical phase-modulated fringe-patterns are usually digitized with XxY pixels and 8 bits/pixel (or higher) gray-levels. The digitized 8 bits/pixel are raw-data bits, not Shannon information bits. Here we show that noisy fringe-patterns store much less Shannon information than the capacity of the digitizing camera. This means that high signal-to-noise ratio (S/N) cameras may waste to noise most bits/pixel. For example one would not use smartphone cameras for high quality phase-metrology, because of their lower (S/N) images. However smartphones digitize high-resolution (12 megapixel) images, and as we show here, the information storage of an image depends on its bandwidth and its (S/N). The standard formalism for measuring information are the Shannon-entropy H, and the Shannon capacity theorem (SCT). According to SCT, low (S/N) images may be compensated with a larger fringe-bandwidth to obtain high-information phase measurements. So broad bandwidth fringes may give high quality phase, in spite of digitizing low (S/N) fringe images. Most reallife images are redundant, they have smooth zones where the pixel-value do not change much, and data compression algorithms are paramount for image transmission/storage. Shannon's capacity theorem is used to gauge competing image compression algorithms. Here we show that phase-modulated phase-shifted fringes are highly correlated, and as a consequence, phase-shifting algorithms (PSAs) may be used as fringe-data compressors. Therefore a PSA may compress a large number of phase-shifted fringes into a single complex-valued image. This is important in spaceborne optical/RADAR phase-telemetry where downlink is severely limited by huge distance and low-power downlink. That is, instead of transmitting M phaseshifted fringes, one only transmit the phase-demodulated signal as compressed sensing data. 2017-11 Centro de Investgaciones en Optica A. C. OCIS Codes: (120.0120) Instrumentation, measurement, and metrology; (100.2650) Fringe analysis; (120.3180) Interferometry; (120.5050) Phase measurement. References and Links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. K. J. Gasvik, Optical Metrology, 3rd ed. (John Wiley & Sons, 2003). D. Malacara, Optical Shop Testing, 3rd ed. (Wiley-Interscience, 2007). M. Servin, J. A. Quiroga, and M. Padilla, Fringe Pattern Analysis for Optical Metrology, Theory, Algorithms, and applications (Wiley-VCH, 2014). M. Schwartz, Information Transmission Modulation and Noise, 4th ed. (McGraw-Hill, 1990). A. Papoulis, Probability, Random Variables and Stochastic Processes, 4th ed. (McGraw Hill, 2000). C. E. Shannon and W. Weaver, The Mathematical Theory of Communication. (University of Illinois Press. Urbana. IL, 1949). H. Nyquist, “Certain factors affecting telegraph speed,” Bell Syst. Tech. J., 3, 412-422, (1924) R. V. L. Hartley, “Transmission of information,” Bell Syst. Tech. J., 3, 535–564 (1928). F. T. S. Yu, Optics and Information Theory, (John Wiley & Sons, 1976). T. M. Cover and J. A. Thomas, Elements of Information Theory, 2th ed. (Wiley-Interscience, 2006). A. N. Kolmogorov, " On the Shannon Theory of Information Transmission in the Case of Continuous Signals," IRE Trans. Inf. Theory, 102-108 (1956). T. J. Lim and M. Franceschetti, "Information without rolling dice," IEEE Trans. Inf. Theory, 63(3), 1349-1363 (2017). Y. C. Eldar, Compressed Sensing: Theory and Applications, (Cambridge Univ. Press, 2012). 1 1. Introduction Optical phase-metrology has productively incorporated well known results from digital signal processing, stochastic processes and telecommunications [1-5]. In particular, we have applied the frequency transfer function (FTF) paradigm to the theory of phase-shifting algorithms (PSAs) [3]. The FTF is now widely used in phase-shifting algorithm (PSA) theory to investigate their signal-to-noise-ratio (S/N), their harmonic and detuning robustness [3]. Also stochastic processes theory applied to linear system has been used in PSAs to study the signal-to-noise-ratio (S/N) of the phase demodulation processes [1,3,4,5]. By plotting the absolute value of the FTF associated to a PSA one may gauge graphically its measuring noise, detuning and harmonic rejection [3]. Shannon information theory has not been applied to wavefront phase-metrology [4-11]. Many scientific and engineering disciplines, including optics, use Shannon theory to study the information processing behavior of their systems [9,10]. Shannon built on Nyquist [7] and Hartley [8] to develop a general theory for reliable information transmission/storage over noisy channels [6]. Nowadays this is simply known as Shannon information theory [4,10]. In this work we use Shannon information theory [6] to quantify the information storage of noisy fringe-patterns and fringe-data compression by phase-shifting algorithms (PSAs) [1-3]. In digital phase-metrology one wrongly equals the amount of raw fringe data-bits with information. One normally think that a fringe-image with XxY pixels, and 8 bit/pixel, has 8(XxY) bits of information; however these are raw-data bits, not Shannon information bits. We show that noisy fringe patterns have much lower Shannon information than the capacity of their digitizing cameras. Quantifying the Shannon information storage of noisy fringeimages is important for fringe data compression for phase-measuring telemetry. For example, a spaceborne interferometer may take many phase-shifted fringe-images and send them back to earth for phase-demodulation. Or phase demodulate them within the spacecraft, and send just a single complex-valued demodulated signal as compressive sensing wave-data [13]. Knowing the Shannon information storage of noisy-fringes is also important because costly, high (S/N) CCD-cameras may not be necessary to digitize phase-modulated fringes for precision phase-metrology. That is because noisy fringe-images contains only a fraction of the information capacity of the CCD cameras which digitize them. Most bits/pixel information are wasted to speckle/electronic fringe-noise; for example an electronic speckle interferometry (ESPI) fringe-image, probably waste 7 out of 12 bits/pixel to noise. So a much cheaper, high resolution, low (S/N) camera, may be enough for ESPI metrology. In this work we show that poor (S/N) fringe-images may be compensated by large bandwidth B fringes. For example, low (S/N) smartphone cameras, compensate this with high resolution images. The SCT allows us to think seriously about low (S/N), high resolution, mass-produced smartphone cameras for high-quality wave phase-metrology. 2. Mathematical theory of communication Here we outline two important contributions on the mathematical theory of information by Claude E. Shannon: information-source entropy and channel-capacity [6]. Figure 1 shows the Shannon's abstraction of a telecommunication process [6]. The information source (the message) may be continuous or discrete data [4]. The message may then be amplitude or phase-modulated, digitally encoded/compressed, and finally transmitted. The radio signal then propagate towards the receiver. The communication channel is assumed bandlimited and noisy. The received signal is then decoded to obtain a reliable estimation of the original message. Signal decoding may consist on phase-demodulation, decompressing and error correction-algorithms [4]. We finally produce a recovered information which is a reliable copy the original message. 2 Fig. 1. Shannon abstraction of a memoryless noisy digital communication process in electrical engineering. The information source generate r symbols/second [6]. Claude Shannon proved a formula for the channel capacity (the SCT) needed to transmit a message over a memoryless, noisy-channel with negligible information degradation [6]. In this paper [6], Shannon introduced the information-entropy and channel-capacity formulas for reliable transmission over a noisy additive white Gaussian noise (AWGN) channel [6]. 2.1 Shannon information entropy of a discrete information source Shannon information entropy H of a discrete message consisting of a large sequence  xm (t  m r ) of symbols, drawn from {x0 , x1,..., xK 1} , each sampled with probability p( xk )  0 is [6], K 1 H    p ( xk ) log 2  p ( xk )  k 0 In the average, a large data-sequence bits ; symbol  x  (t  m r ) m K 1  p( x )  1.0 . . k 0 k (1) have an information entropy of H bits/symbol. If these symbols {xk } are transmitted at a rate of r symbols/second then, the information-rate is,  K 1  bits . R  rH   r   p ( xk ) log 2  p ( xk )  k 0  second (2) The prototypical example of a discrete source is the English language where the data are { A, B, C ,..., Z } , with p ( A)  ...  p ( Z )  1.0 , being { p ( A),..., p ( Z )} the probability of occurrence of a single letter in a text, then H= [ p ( A) log 2 p( A)  ...  p ( Z ) log 2 p ( Z )] . 2.2 Shannon capacity theorem (SCT) for a bandlimited AWGN-channel The message  x  (t  m r ), m is then transmitted over a AGWN-channel (see Fig. 1). The received signal is, X (t )   ( axm  nm ) (t  m r ); t  [0, T ]; T  1. (3) m Where a<1.0 is the channel attenuation. Shannon proved that the rate of information that a bandlimited AWGN-channel can transmit with negligible errors is [4-6,10], S  bits  C  B log 2  1   ; S  a 2 E xm2 ; N  E nm2  . . N second   (4) Where B is the channel's bandwidth in cycles/second, S is the received signal-power, N is the channel noise-power, and E   is the ensemble average. This is the famous Shannon capacity theorem (SCT) for a bandlimited AWGN-channel [4-6,10-11]. The relation between the source information-rate R and channel capacity C for reliable communication is, 3 R  C ; Unreliable Communication ; . (5) R  C ; Negligible Error Communication . Therefore in order to transmit reliably through an AWGN-channel the information-rate R must be below C. For example, a typical AWGN telephone line with B  3kHz and (S/N)=103 has a capacity of C  3000 log 2 (1  103 )  30,000 bits/second. The SCT also tell us that we can interchange (S/N) with bandwidth B while keeping the channel capacity C constant, as Table 1 shows, Table 1. Channel capacity C=log2[1+(S/N)] trade between (S/N) and bandwidth B Channel capacity C (bits/sec) Channel Bandwidth B (Hz). Signal-to-noise ratio (S/N) 30,000 1,500 1,100,000 30,000 3,000 1,000 30,000 6,000 31 30,000 9,000 9 30,000 12,000 5 To keep C constant, B increases linearly while (S/N) decreases exponentially. That is, for low (S/N), one may keep C constant by increasing B, and still have reliable communication; this is why spread-spectrum communication is so robust against thermal-noise, radio-jamming and low-power spacecraft communications [4]. A daily example of the use of the SCT is when one walks away from a Wi-Fi transmitter. Given that the hardware have not changed, the channel bandwidth B remains the same. However at a large distance from the Wi-Fi transmitter the signal-power decreases. This reduces the data transfer rate R towards our laptop because the channel encoding must be more redundant, less information-rate efficient. 2.3 Shannon's capacity is a mathematical existence theorem Finally keep in mind that Shannon capacity theorem (SCT) is a theoretical upper-bound limit for reliable classical communication/storage of information independently of any engineering apparatus. The SCT is an existence theorem, it say nothing on how to implement it physically or algorithmically. That is, the SCT is a mathematical abstraction disembodied of any implementation, such as Einstein's E=mc2 is not a receipt for building a nuclear reactor. In fact the SCT may be seen as a sphere-packing theorem in a K-dimensional Euclidian space [6,10,12]. In other words, no matter how advanced (non-quantum) digital-modem one may built, it cannot reliably transmit more classical information than C. This is like trying to obtain more energy than E=mc2 from a rest mass, or trying to build a perpetual moving machine bypassing the second law of thermodynamics. Modern digital communication modems using efficient block error-correcting codes approaches the SCT limit, i.e. R  C . 3. Shannon information storage of noisy fringe-patterns Many signals that we communicate/store are continuous such as voice, video, or telemetry, where the discrete-data entropy H cannot be used. The source information-rate R=R(H,r) and channel-capacity C=C(B,S,N) are both given in bits/second, however they use different parameters [6]. The entropy H, the rate r, and signal-power S are information-source variables [6]. On the other hand, the bandwidth B, and AWGN-power N are communication channel variables [6,4,10]. To estimate the source information-rate of a continuous-signal, one would need a continuous-density entropy. However a continuous-density entropy is not always welldefined [10]. In contrast, the bandwidth, the noise, and the signal-power of a noisy continuous-source are always well-defined. So it is better to have an information-rate formula for continuous-signals based on its bandwidth and signal-to-noise ratio, rather than on its 4 continuous-density entropy [10,11]. Fortunately mathematician Andrei Kolmogorov proposed an information-rate formula for continuous, noisy information-sources, the   entropy [11,12]. That is, Kolmogorov   entropy does not depend on a continuous-density entropy which is good, because it avoids some theoretical problems [10,11,12]. In wave phase-metrology the message is the continuous phase-field which modulate the fringes; it could be an optical wavefront W ( x, y ) , or a solid z  h ( x , y ) in fringe-projection profilometry [1,2]. The model for a continuous fringe-pattern is,  2  I ( x, y )  a ( x, y )  b( x, y ) cos  W ( x, y )   n ( x, y ).   (6) Here a ( x, y ) , b( x, y ) are smooth functions, and  the illumination wavelength. The probability density p ( I ) of these fringes is continuous. Shannon defined the differential entropy of a continuous random variable X with density p(x) as [6,10], H    p( x ) log 2  p( x ) dx;  p( x) dx  1.0. (7) As in every example involving an integral, or even a density, we should include the statement if it exist [10]. It is easy to construct examples of random variables for which a density function does not exist or for which the above integral does not exist [10]. Instead of trying to estimate a continuous-density entropy H of a fringe-image, we use the Kolmogorov's   entropy H  ( ) . In 1956 Kolmogorov wrote [11]: According to the proper meaning of the word, the entropy of a signal with a continuous distribution is always infinite. If the continuous signals can, nevertheless, serve to transmit finite information, then it is only because they are always observed with bounded accuracy. Consequently, it is natural to define the appropriate "   entropy " H  ( ) of the object  by giving the accuracy of observation  . Shannon did this under the designation "rate of creating information with respect to a fidelity criterion" [11]. In our notation, the noiseless object is   a  b cos[ ] , which is observed with bounded-accuracy   n . Kolmogorov proved that H  ( I ) approximates Shannon's capacity C [11, 12]. The information-rate H  ( I ) for I ( x , y ) is [11], H  ( I )  BI log 2 1  SNR K  bits ; pixel BI in fringes . pixel (8) ; s( x, y )  b cos   . (9) Being SNRK the signal-to-noise ratio of I ( x , y ) , given by, SNR K   s 2 ( x , y ) dxdy  n 2 ( x, y ) dxdy ( x , y ) I ( x, y )   2 s (u, v ) dudv ( u , v ) B      2 n (u, v ) dudv [  , ]x[  , ] Where s(u, v ) , n(u, v ) are the spectra of s( x, y ) , n( x, y ) respectively; ( x, y )   is the region with well defined fringes; (u, v)  B the fringes' spectral lobe; BI is the average bandwidth (see Fig. 2). Note that formally H  ( I ) equals C; but the difference is that now B and SNRK refer to the noisy signal, not the noisy channel. Kolmogorov H  ( I ) cannot exceed the storage-capacity of the digitizing CCD-camera [11]. For example, for a 256 gray-levels per pixel camera, then H  ( I ) <8 bits/pixel. 5 Fig. 2. A 8 bits/pixel quantized, noiseless simulated fringes I(x,y). The spectrum |I(u,v)| has a bandwidth . Let us consider the limits for H  ( I ) when the fringe's noise n( x, y ) tends to zero or infinite, lim  BI log 2 1  SNR K     ; lim  BI log 2 1  SNR K    0. n  n 0 (10) The information-rate is infinite H  ( I )   for noiseless fringes, and zero H  ( I )  0 for infinite noisy fringes; regardless of BI. One might think that a constant-signal would hold little information; however if that signal is coded into a large number with say, 1x108 significant digits (above the noise n ), then one may encode a large amount of information within that single number. As mentioned, Shannon capacity C (Eq. (3)) and Kolmogorov entropy-rate H  ( I ) (Eq. (7)) look identical, just substituting (S/N) for SNRK. However in Shannon's capacity N is a Gaussian stochastic processes [6], while the observation accuracy  , in H  ( I ) , is a deterministic error-signal [11,12]. This seems as an irrelevant theoretical nuance, but it has however practical consequences [6,10,11,12]. Lim and Franceschetti proved that C and H  ( I ) , are indeed "identical" under appropriate mathematical conditions [12]. 4. Simulated Shannon information storage of fringe-images Here we give two simple but illustrative examples of noisy fringe information storage. We use low resolution fringes to show the fringes at their Nyquist sampling rate of BI=0.5 fringes/pixel. Figure 3 shows three 200x200-pixel, computer-generated noisy fringes phasemodulated by defocus and increasing n(x,y). Fig. 3. Information storage-rate for fringes with BI= fringes/pixel. The 8 bit/pixel fringes have SNRK=70,000 (about 8-bit quantizing noise). The noise is obvious by 1.0 bit/pixel (SNRK=3.0). The noisier fringes have 0.5 bits/pixel (SNRK=1.0). 6 We then decrease the SNRK, until reaching H  ( I ) =0.5-bits/pixel. Figure 3 and Fig. 4 show the interplay between BI and SNRK stated by H  ( I ) . Figure 4 shows a similar sequence of fringes but with reduced bandwidth BI=0.125 fringes/pixel. In Fig. 4 the maximum fringefrequency is one fourth of the Nyquist rate: BI=0.25(0.5)=0.125 fringes/pixel, therefore the average information-rate H  ( I ) is reduced one fourth with respect to Fig. 3 for the same SNRk. Fig. 4. Average information-rate of fringe-images with BI=0.125 fringes/pixel. The 2-bit/pixel fringes have SNRK=70,000 (about 8-bits quantizing noise). The noisier fringes have 0.2 bits/pixel (SNRK=2.03). The reduced spectrum of the noisier fringes is at far right. The information-rate H  ( I ) is reduced by decreasing the SNRk until the noisier fringes at the far-right with 0.2-bit/pixel is obtained. Finally the spectrum of the noisiest-fringes with 0.2 bits/pixel is shown at the far right. Remember that we are finding the low information-rate H  ( I ) of noisy fringes with respect to the CCD camera used to digitize them. Kolmogorov's H  ( I ) tell us nothing about how to program a compressing algorithm to store/transmit efficiently a large number of noisy fringe images. 5. Shannon information storage of experimental fringes Here we are estimating the Shannon information content of experimental fringe-projection profilometry fringes. To estimate H  ( I ) for profilometry fringes, one possibility is to take two images: the noisy fringes, and the noisy background. The formalism for both images are [1],  2  I ( x , y )  a ( x, y )  b( x, y ) cos  tan( )h ( x , y )   n ( x, y ); p   I n ( x, y )  a ( x, y )  n ( x, y ). (11) Where h( x, y ) is the object being digitized; p the period of the projected fringes and  is the sensitivity angle between the projector and the (8 bits/pixel) CCD-camera [1]. 7 Fig. 5. High spatial frequency fringe-images (512x640, 8 bits/pixel) for obtaining an estimation of the information storage-rate for fringe-projection profilometry carrier-fringes. White-light fringe-projection patterns have relatively high SNRK. We then find the spectrum of the fringes I (u, v )  F {I ( x, y )} , and define the indicator function for the signal-plus-noise region as (u, v )  SN , and for the noise-only region as (u, v )  NR . Fig. 6. Fourier spectrum corresponding to the fringes and noise (512x640 pixels). Figure 7 shows the Fourier spectrum of the profilometry fringes and the Fourier spectrum of the background-image; both images defined in Eq. (11). Figure 7 shows the spatial filters to keep just the signal-lobes spectra within the region (u, v )  SN . Fig. 7. Fourier filtered fringe-lobes (BI=0.11), and high-pass filtered noise. Next we show the demodulated wrapped-phase obtained by taken the right lobe of the signalplus-noise region (u, v )  SN in Fig. 7. This was made to be sure that we have included all the phase signal bandwidth. This is shown in Fig. 8, 8 Fig. 8. The fringe-pattern and its wrapped demodulated phase (512x640 pixels). At the fringe self-occluding shadow the fringe-amplitude drops and the wrapped phase becomes noisier. We then find the noise-power density within the noise-only region (u, v )  NR as,  1 ANR  a ( u, v )  n (u , v ) . 2 (12) ( u , v )NR Where a (u, v )  n(u, v )  F{a ( x, y )  n( x, y )} and ANR the area of (u, v )  NR . We then assume that the noise-density within (u, v )  SN is also  (Eq. (15)). Finding (S/N) as, SNR K   ( u , v )SN  I ( u, v ) 2 512(640)    11.2 . (13) We finally estimate the information-rate (with BI=0.11, Fig. 7) of this fringe-pattern as, H  ( I )  BI log 2 1  SNR K   0.41 bits . pixel (14) The average bandwidth BI (Fig. 7) of the fringes may be estimated by looking at the spectrum the fringes I (u, v )  F {I ( x, y )} . This profilometry fringe-image has in the average 0.41 bits/pixel, much less information than the 8 bits/pixel capacity of the CCD-camera. The information storage of most fringe patterns is usually well below the capacity of the digitizing CCD-camera. This means that if we want to transmit/store fringe-images one may effectively use compression image algorithms. 6. Real-life versus fringe-pattern images data compression Real life images are highly redundant; have low information entropy. A real life photograph usually have smooth image regions, along with sparse noisy-like regions with detailed image features. Data compression algorithms may use local space-frequency wavelets expansion to reduce redundancy [13]. Image compression algorithms are broadly classified as lossy and lossless. In lossy compression the encoding/decoding algorithm decompress the original image not 100% accurately. However, subjective quality perception may tolerate large compression ratios without subjective degradation [10]. In lossless algorithms, decompressed images remains 100% accurate to the original. Therefore the most compressible images are usually smooth piece-wise continuous. On the other hand, one cannot compress a purelyrandom image, if we want to preserve exactly every pixel value; a white noise image is not compressible. Common image storage formats are: 8-bits/pixel Bit Maps (BMP); Joint Photographic Experts Group (JPEG); 8-bits/pixel Graphic Interchange Format (GIF); Tagged Image File Format (TIFF), and lossless Portable Network Graphics (PNG). These file formats have been designed to store real-life color images. These formats store noisy fringepatterns as images containing more "information" than noiseless fringes. The noise is 9 interpreted as fine image-details that the user wants to "preserve". On the contrary, noiseless fringes occupy less storage bytes; they are interpreted as smoother images (see Fig. 9). Fig. 9. Noiseless and noisy fringes with 200x200 pixel resolution and 8 bits/pixel. At right, the PNG file size is 40-Kbytes, while the noisy fringe image has a PNG file size of 180-Kbytes. We have simulated two fringe patterns shown in Fig. 9. We have store them using lossless PNG format, and Fig. 9 shows the resulting PNG file sizes in kilobytes. The noisy image file is about four times bigger because the noise is interpreted as high frequency image details. The PNG format is designed to compress color real-life images for good visual perception, not phase-modulated noisy fringes. Fringe patterns have more mathematical structure than real-life images. So noisy fringe images are far more redundant and as a consequence, more compressible. The mathematical model for fringe patterns (Eq. (6)) allow us to distinguish between phase information and noise. Due to these considerations, real-life image compressors are not suitable for efficient transmission/storage of noisy fringe patterns; they were simply not designed for that purpose. 7. Fringe-data compression by phase-shifting algorithms (PSAs) We have seen that noisy fringe patterns store much less Shannon information than the capacity of the CCD used to digitize them. In this section we show that phase-shifted fringes are even more information redundant. In remote optical/RADAR phase-metrology one may digitize M phase-shifted fringes at a spacecraft (see Fig. 10). Then, one may downlink these M fringes sequentially, and phase-demodulate them at the receiver. Or we may use a PSA to phase-demodulate them at the spacecraft, and downlink just the demodulated analytic-signal. A PSA may then be regarded as compressive-sensing, where we are "sensing" not real-valued fringes, but less noisy complex-valued wave-data [13]. So PSA compression of phase-shifted fringes is paramount in remote phase-shifting metrology where communications are limited due to large distances and few watts for radio transmission. A PSA then packs the information contained in M phase-shifted images into a single phase-demodulated complex-valued image. Fig. 10. Schematic of a spacecraft transmitting high-information, complex-valued wave-data from a set of M phase-shifted fringe-patterns. To quantify the increase of information-rate by phase-shifting algorithms (PSAs), consider a sequence of M phase-shifted noisy interferograms, 10 I ( x, y, t )  M 1   2  a  b cos   m 0 2   W  m 0   nm   (t  m ); 0  . M   (15) Here some (x,y) coordinates were omitted for clarity. We may phase-demodulate these fringes by a PSA whose impulse response is [3], h (t )  M 1 c m 0 m  (t  m ). (16) Being cm complex-valued. The Fourier transform of h(t ) ( H ( )  F [h(t )] ) must have at least the following zeroes [3], H ( 0 )  H (0)  0; and H (0 )  0. (17) Obtaining the estimated quadrature analytic signal as [3], Aei  ( x , y )  I ( x, y , t )  h (t ) t  M 1  M 1 c m 1 m I ( x, y , m ). (18) A PSA increases the (S/N) of the analytic-signal with respect to the fringes (Eq. (15)) by [3], GS / N  H (0 ) 1 2   2 (19) . H ( ) d  2  The highest GS / N is obtained by the least-squares PSA, where GS / N  M [3]. Then the SNRK of the noisy fringes increases to M(SNRK) for the complex-valued phase-demodulated signal Aei  ( x , y ) . Then the information-rate contained in the analytic signal Aei  ( x , y ) is, H  ( A; M )  BI log 2 1  M SNR K  bits . pixel (20) The information of Aei  ( x , y ) increases logarithmically with M. An interesting question is: how many phase-shifted fringes double the information content of H  ( A; M ) , or, BI log 2 1  M SNR K   2  BI log 2 1  SNR K  . (21) Solving for M one obtains, M  2  SNR K . This is shown in Fig. 11, Fig. 11. Information storage as a function of SNRK, with BI=0.5 fringes/pixel. At the green-zone, the spatial information-rate increases almost linearly from 1 to 2 bits/pixel. In contrast in the blue-zone, this rate increases logarithmically from 2 to 4 bits/pixel. 11 (22) For example with SNR K  1 one would need M=3 phase-shifted fringes to double the information-rate of Aei  ( x , y ) . In contrast for SNR K  20 , one would need M=22 phaseshifted fringes to double the information-rate of Aei  ( x , y ) from 2 to 4 bits/pixel. An example of this is shown in Fig. 12 and Fig. 13 (for M=12). Fig. 12. Four out-of 12 phase-shifted fringes degraded with AWGN. These fringes have an average information-rate of 0.4bits/pixel (BI=0.125fringes/pixel, SNRK=8.1). The M=12 phase-shifted fringes in Fig. 12 were phase-demodulated using a least-squares PSA and the resulting real and imaginary images of the analytic signal are shown in Fig. 13, Fig. 13. Real and imaginary parts of the phase-demodulated signal from the 12 noisy phaseshifted fringes in Fig. 12. The information-rate has increased about twice, from 0.4 bits/pixel to 0.82 bits/pixel (BI=0.12fringes/pixel, SNRK=97.5, M=12). As conclusion, a PSA regarded as fringe-image compressor packs into a single analyticsignal Aei ( x , y ) the phase information stored in M phase-shifted fringes. In other words, instead of transmitting/storing M phase-shifted fringes, one may send/store a single, higherinformation signal Aei ( x , y ) , or wrapped-phase arg[ Aei  ( x , y ) ] . Therefore a PSA may be regarded as an efficient compression algorithm for transmission or storage of large blocks of phase-shifted fringe-data. This is paramount for efficient transmission of optical/RADAR interference fringes from remote phase-metrology sites where the channel is severely limited by large distances (high attenuation) and low power radio link. Also a PSA may be regarded as a compressive-sensing algorithm where the collected data is complex-valued with lowernoise and as consequence higher information rate [13]. 8. Summary We have shown that noisy fringe-images in wave phase-metrology contain much less Shannon information than the capacity of the CCD camera used to digitize them. For example, it does not make sense to store electronic speckle-pattern interferometry (ESPI) fringes using a costly 14 bits/pixel cooled CCD-camera. That is because most data bits (not information bits) are wasted to fringe-noise. ESPI noisy fringes have very-low spatial information-rate. Shannon information may be used as cost-efficiency criteria for choosing a CCD camera according to the class of fringes that we are dealing with. For example, mass- 12 produced smartphone cameras may do just fine in many optical-metrology cases. Smartphone cameras have low signal-to-noise ratio, but this is compensated with high image-pixel resolution; 12 megapixels or more is nowadays available. So smartphone cameras with moderate (S/N) and very high resolution images, may be efficiently used for remote phasewave telemetry applications. We showed that low-noise fringe-images may be obtained from white-light, fringeprojection profilometry. Here we have estimated that an experimental carrier-fringe profilometry image may contain less than 0.5 bits/pixel of Shannon information (Figs. 5-6). With these small information-rates one realizes that there is plenty of room for noisy fringedata compression for efficient transmission or storage. We have seen that PSAs may be regarded as fringe-data compression algorithms. Using PSAs, all Shannon information contained in M phase-shifted fringes may be compressed into a single analytic signal. In spaceborne phase-metrology for example, one may use PSAs to compress large blocks of M phase-shifted fringes for efficient downlink. Finally we saw that PSAs compress noisy fringedata information by increasing the SNR K of the phase-demodulated complex-signal. 13 

arXiv:1708.02546v1 [] 8 Aug 2017 A PRINCIPAL IDEAL THEOREM FOR COMPACT SETS OF RANK ONE VALUATION RINGS BRUCE OLBERDING Abstract. Let F be a field, and let Zar(F ) be the space of valuation rings of F with respect to the Zariski topology. We prove that if X is a quasicompact set of rank one valuation rings in Zar(F ) whose maximal ideals do not intersect to 0, then the intersection of the rings in X is an integral domain with quotient field F such that every finitely generated ideal is a principal ideal. To prove this result, we develop a duality between (a) quasicompact sets of rank one valuation rings whose maximal ideals do not intersect to 0, and (b) one-dimensional Prüfer domains with nonzero Jacobson radical and quotient field F . The necessary restriction in all these cases to collections of valuation rings whose maximal ideals do not intersect to 0 is motivated by settings in which the valuation rings considered all dominate a given local ring. 1. Introduction Throughout this article, F denotes a field and Zar(F ) denotes the set of valuation rings of F ; i.e., the subrings V of F such that for all 0 ≠ t ∈ F , t ∈ V or t−1 ∈ V . In this article we are interested in subrings A of F which are an intersection of rank one valuation rings in a quasicompact subset of Zar(F ). The rank of a valuation ring, which coincides with its Krull dimension, is the real rank of its value group. Thus the rank one valuation rings have valuations that take values in R ∪ {∞}. The Zariski topology on Zar(F ) is the topology having as a basis of open sets the subsets of Zar(F ) of the form {V ∈ Zar(F ) ∶ t1 , . . . , tn ∈ F } for t1 , . . . , tn ∈ F . With this topology, Zar(F ) is the Zariski-Riemann space of F . The main purpose of this article is to prove the following instance of what Roquette [38] calls a Principal Ideal Theorem, that is, a theorem which guarantees a given class of integral domains has the property that every finitely generated ideal is principal. Main Theorem. If X is a quasicompact set of rank one valuation rings in Zar(F ) whose maximal ideals do not intersect to 0, then the intersection of the valuation rings in X is an integral domain with quotient field F and Krull dimension one such that every finitely generated ideal is a principal ideal. The theorem, which most of the paper is devoted to proving, asserts that the intersection of such valuation rings is a Bézout domain, a domain for which every MSC: 13A18; 13F05; 14A15. 1 2 BRUCE OLBERDING finitely generated ideal is principal. Such rings belong to the extensively studied class of Prüfer domains, those domains A for which AM is a valuation domain for each maximal ideal M of A. The problem of when an intersection of valuation rings is a Prüfer domain is a difficult but well-studied problem that has applications to real algebraic geometry (e.g., [1, 2, 3, 42]), non-Noetherian commutative ring theory (e.g., [15, 27, 39]), formally p-adic holomorphy rings [38] and the study of integer-valued polynomials [4, 24]. Using the equivalence of this problem to that of determining when a subspace of Zar(F ) yields an affine scheme, a geometric criterion involving morphisms of the Zariski-Riemann space into the projective line was given in [33]. In that approach, treating the Zariski-Riemann space as a locally ringed space, not simply a topological space, is crucial. By contrast, the main theorem shows that unlike in the general case, the geometry of the locally ringed space structure is not needed to distinguish a Bézout intersection when the valuation rings satisfy the hypotheses of the theorem. Instead, the question of whether the intersection of such a collection of rank one valuation rings is a Prüfer domain is purely topological. Moving beyond rank one valuation rings, quasicompactness of a subset X of Zar(F ) is far from sufficient to guarantee that the intersection of valuation rings in X is a Bézout domain. For example, if D is a Noetherian local domain with quotient field F , then for any t1 , . . . , tn ∈ F , U = {V ∈ Zar(F /D) ∶ x1 , . . . , xn ∈ V } is quasicompact, but the intersection of the valuation rings in U is a Bézout domain if and only if the integral closure of D[x1 , . . . , xn ] is a principal ideal domain, something that occurs only for very special choices of D and x1 , . . . , xn . In fact, the main theorem is optimal in the sense that if any one of the hypotheses “quasicompact”, “rank one”, or “the maximal ideals do not intersect to 0” is omitted, the conclusion is false; see Example 5.7. Note that the condition that the maximal ideals of the rank one valuation rings in X do not intersect to 0 occurs naturally in settings where the valuation rings in X are assumed to dominate a local ring of Krull dimension > 0. Examples of such settings of recent interest include Berkovich spaces and tropical geometry (see for example [13]) and valuative trees of regular local rings [9, 17]. Interest in compactness in the Zariski-Riemann space dates back to Zariski’s introduction of the topology on Zar(F ) in [44]. If D is a subring of F , then Zar(F /D), the subspace of Zar(F ) consisting of the valuation rings in Zar(F ) that contain D as a subring, is the Zariski-Riemann space of F /D. That Zar(F /D) is quasicompact was proved by Zariski [44] in 1944 as a step in his program for resolution of singularities of surfaces and three-folds. In more recent treatments of the topology of Zar(F /D) such as [10, 36], quasicompactness is viewed as part of a more refined topological picture that treats Zar(F /D) as a spectral space or as a locally ringed COMPACT SETS AND HOLOMORPHY RINGS 3 space that is a projective limit of projective schemes. The latter point of view also has its origins in Zariski’s work [44]. However, in restricting to the subspace of rank one valuation rings of F , key topological features of Zar(F ) are lost. For example, this subspace need not be spectral, nor even quasicompact. Yet, in passing to the space of rank one valuation rings, the main theorem shows that the topology becomes much more consequential for the ring-theoretic structure of an intersection of valuation rings. One of the key steps in proving this is first showing that in the setting of the main theorem, the intersection of valuation rings in X is a Prüfer domain. We prove this in Theorem 5.6 by establishing the following lemma. For a subset X of Zar(F ), we let A(X) = ⋂V ∈X V be the holomorphy ring1 of X and J(X) = ⋂V ∈X MV be the ideal of A(X) determined by the intersection of the maximal ideals MV of the valuation rings V . For a ring A, Max(A) denotes its set of maximal ideals. Main Lemma. The mappings X ↦ A(X) and A ↦ {AM ∶ M ∈ Max(A)} define a bijection between the quasicompact sets X of rank one valuation rings in Zar(F ) with J(X) ≠ 0 and the one-dimensional Prüfer domains A with nonzero Jacobson radical and quotient field F . Corollary 5.9 gives another version of this result in which the spaces X need not be assumed a priori to satisfy J(X) ≠ 0. In this case, “quasicompact” is replaced with “compact” (= quasicompact and Hausdorff), and the holomorphy rings are all assumed to have quotient field F . A consequence of the main lemma is that if X is quasicompact, J(X) ≠ 0 and X consists of rank one valuation rings, then X = {A(X)M ∶ M ∈ Max(A(X))}. As this suggests, the main lemma can be recast in the language of schemes, and we do this in Corollary 5.14. The applicability of the main theorem depends on whether a space of rank one valuation rings can be determined to be quasicompact. The key technical observation behind our approach is that if a subset X of Zar(F ) consists of rank one valuation rings and J(X) ≠ 0, then X is quasicompact if and only if X is closed in the patch topology of Zar(F ). (See Section 2.) On the level of proofs and examples, this reduces the issue of quasicompactness to calculation of patch limits point of X in Zar(F ), and specifically whether such limit points have rank one. In a future paper [37] we use the results of the present article along with additional methods to show how to apply these ideas to divisorial valuation overrings of a Noetherian local domain of Krull dimension two. This is discussed in Remark 6.4. The present paper is motivated by recent work such as in [10, 11, 31, 32, 33, 34, 35, 36] on understanding how topological or geometric properties of a space of 1See Roquette [38] for an explanation of this terminology. 4 BRUCE OLBERDING valuation rings are reflected in the algebraic structure of the intersection of these valuation rings. 2. Topological preliminaries In this section we outline the topological point of view needed for the later sections. Recall that throughout the paper, F denotes an arbitrary field. Notation 2.1. For each subset S of the field F , let U (S) = {V ∈ Zar(F ) ∶ S ⊆ V } and V (S) = {V ∈ Zar(F ) ∶ S ⊆/ V }. The Zariski topology on Zar(F ) has as a basis of nonempty open sets the sets of the form U (x1 , . . . , xn ), where x1 , . . . , xn ∈ F . The set Zar(F ) with the Zariski topology is the Zariski-Riemann space of F . Several authors have established independently that the Zariski-Riemann space Zar(F ) is a spectral space2, meaning that (a) Zar(F ) is quasicompact and T0 , (b) Zar(F ) has a basis of quasicompact open sets, (c) the intersection of finitely many quasicompact open sets in Zar(F ) is quasicompact, and (d) every nonempty irreducible closed set in Zar(F ) has a unique generic point. See [10] and [36] for more on the history of this central result. A spectral space admits the specialization order ≤ given by x ≤ y if and only if y is in the closure of {x}. In the Zariski topology on Zar(F ), we have for V, W ∈ Zar(F ) that V ≤ W if and only if W ⊆ V . We use the specialization order in the results in this section but not elsewhere in the paper. As a spectral space, Zar(F ) admits two other useful topologies, the inverse and patch topologies. The inverse topology on Zar(F ) is the topology that has, as a basis of closed sets, the subsets of Zar(F ) that are quasicompact and open in the Zariski topology; i.e., the nonempty closed sets are intersections of finite unions of sets of the form U (x1 , . . . , xn ), x1 , . . . , xn ∈ F . The inverse topology is useful for dealing with issues of irredundance and uniqueness of representations of integrally closed subrings of F ; for example, see [35]. In the present article, we use the inverse topology in a limited way. The most important topology on Zar(F ) for the purposes of this article is the patch topology on Zar(F ), which is given by the topology that has as a basis of open sets the subsets of Zar(F ) of the form U (x1 , . . . , xn ), V (y1 ) or U (x1 , . . . , xn ) ∩ V (y1 ) ∩ ⋯ ∩ V (ym ), where x1 , . . . , xn , y1 , . . . , ym ∈ F . The complement in Zar(F ) of any set in this basis is again open in the patch topology. Thus the patch topology has as a basis sets that are both closed and open (i.e., the patch topology is zero-dimensional). The patch topology is also spectral and hence quasicompact [23, p. 45]. Unlike the 2The terminology is motivated by a theorem of Hochster [23] that shows a space is spectral if and only if it is homeomorphic to the prime spectrum of a ring COMPACT SETS AND HOLOMORPHY RINGS 5 Zariski and inverse topologies on Zar(F ), the patch topology is always Hausdorff [23, Theorem 1]. Convention. In the article we work with all three topologies, inverse, patch and Zariski, sometimes even in the same proof. To avoid confusion, we insert the adjective “patch” before a topological property when working with it in the patch topology. For example, a “patch open set” is a set that is open in the patch topology. Similarly, we insert “inverse” as an adjective when working with the inverse topology. If no adjective is present (e.g., “the set Z is quasicompact”), this is always to be understood as indicating we are working in the Zariski topology. Thus the Zariski topology is the default topology if no other topology is specified. Recall also from the Introduction that by compact we mean both quasicompact and Hausdorff. One of our main technical devices in the paper is that of a patch limit point. Let X ⊆ Zar(F ). Then V ∈ Zar(F ) is a patch limit point of X if each patch open neighborhood U of V in X contains a point in X distinct from V ; equivalently (since the patch topology is Hausdorff), every patch open neighborhood U of V contains infinitely many valuation rings in X. Applying the relevant definitions, it follows that V is a patch limit point of X if and only if for all finite (possibly empty) subsets S of V and T of MV there is a valuation ring U in X ∖ {V } such that S ⊆ U and T ⊆ MU .3 Notation 2.2. Let ∅ ≠ X ⊆ Zar(F ). We use the following notation. ● ● ● ● lim(X) = the set of patch limit points of X in Zar(F ). patch(X) = X ∪ lim(X) = closure of X in the patch topology of Zar(F ). A(X) = ⋂V ∈X V = holomorphy ring of X. J(X) = ⋂V ∈X MV . In the next lemma we collect some properties of patch closure that are needed in later sections. More systematic treatments of Zariski, patch and inverse closure in Zar(F ) can be found in [10] and [36] and their references. Lemma 2.3. Let X be a nonempty subset of Zar(F ). Then (1) A(X) = ⋂V ∈patch(X) V and J(X) = ⋂V ∈patch(X) MV . (2) The set patch(X) is spectral in the subspace Zariski topology. Suppose in addition that X is quasicompact and consists of rank one valuation rings. (3) The set patch(X) is contained in X ∪ {F }. (4) If S is a multiplicatively closed subset of A = A(X), then AS = ⋂AS ⊆V ∈X∪{F } V . Proof. (1) The first assertion can be found in [10, Proposition 4.1] or [36, Proposition 5.6]. Since patch(X) = X ∪ lim(X), to see that the second assertion holds, it suffices to show that J(X) ⊆ MU for each U ∈ lim(X). Let 0 ≠ a ∈ J(X), and let U ∈ lim(X). 3Throughout the paper, we denote the maximal ideal of a valuation ring V by M . V 6 BRUCE OLBERDING If a ∈/ MU , then U ∈ U (a−1 ). Since U ∈ lim(X), there exists V ∈ X ∩ U (a−1 ), so that a−1 ∈ V . However, a ∈ MV by the choice of a, a contradiction. Thus J(X) ⊆ MU , which verifies (1). (2) A patch closed subspace of a spectral space is spectral in the subspace topology [18, Proposition 9.5.29, p. 433]. (3) Suppose X is quasicompact. This implies that if U ∈ patch(X), then there exists V ∈ X such that V ⊆ U [36, Proposition 2.2]. Since X consists of rank one valuation rings, we have V = U or U = F . Thus patch(X) ⊆ X ∪ {F }. (4) By [36, Corollary 5.7], AS = ⋂ V, AS ⊆V ∈Y where Y is the set of all V ∈ Zar(F ) such that V ⊇ U for some U ∈ patch(X). Thus (4) follows from (3) and the fact that every valuation ring in X has rank one.  The following proposition reinterprets for rank one valuation rings the property of compactness in the Zariski topology of Zar(F ) in terms of the patch topology. This enables us to work with the patch topology– and specifically, patch limit points– in the algebraic arguments of the next sections. The proposition also shows that the Hausdorff condition for a set X of rank one valuation rings is closely connected with the algebraic property that J(X) ≠ 0. Proposition 2.4. The following statements hold for every nonempty set X of rank one valuation rings in Zar(F ). (1) X is compact if and only if X is patch closed in Zar(F ). (2) If J(X) ≠ 0, then the Zariski and patch topologies agree on X, and hence X is Hausdorff and zero-dimensional. (3) Suppose A(X) has quotient field F . Then J(X) ≠ 0 if and only if X is Hausdorff. Proof. (1) Suppose that X is compact. By Lemma 2.3(3), patch(X) ⊆ X ∪{F }, so to show that X is patch closed it suffices to show that F /∈ patch(X). If X consists of a single valuation ring, then patch(X) = X and the claim is clear since this valuation ring must have rank one. Suppose X contains at least two distinct valuation rings V and W . Since X is Hausdorff, there exist x1 , . . . , xn , y1 , . . . , ym ∈ F such that V ∈ U (x1 , . . . , xn ), W ∈ U (y1 , . . . , ym ) and U (x1 , . . . , xn ) ∩ U (y1 , . . . , ym ) ∩ X = ∅. Let Y = V (x1 , . . . , xn ) ∪ V (y1 , . . . , ym ). Then X ⊆ Y and Y is patch closed, so patch(X) ⊆ Y . Since F ∈/ Y , we have F /∈ patch(X). Therefore, X is patch closed. COMPACT SETS AND HOLOMORPHY RINGS 7 Conversely, suppose that X is patch closed in Zar(F ). By Lemma 2.3(2), X is a spectral space with respect to the Zariski topology and hence is quasicompact. Since the valuation rings in X have rank one and F ∈/ X, the elements of X are minimal in X with respect to the specialization order. This implies the patch and Zariski topologies agree on X [41, Corollary 2.6]. Thus X is Hausdorff since the patch topology is Hausdorff. (2) Let 0 ≠ x ∈ J(X). Then X ⊆ V (x−1 ). Since the valuation rings in X have rank one and F ∈/ V (x−1 ), the elements of X are minimal in V (x−1 ) with respect to the specialization order induced by the Zariski topology. As a patch closed subset of Zar(F ), the set V (x−1 ) is, with respect to the Zariski topology, a spectral space (Lemma 2.3(2)). Thus the Zariski and patch topologies agree on the set of elements of V (x−1 ) that are minimal with respect to the specialization order [41, Corollary 2.6]. Since the patch topology is Hausdorff and zero-dimensional, the last statement of (2) now follows. (3) Suppose A = A(X) has quotient field F and X is Hausdorff. If X consists of a single valuation ring, then it is clear that J(X) ≠ 0. Assume that X has more than one valuation ring. Since X is Hausdorff and A has quotient field F , there exist nonzero a1 , . . . , an , b1 , . . . , bm , c ∈ A such that U (a1 /c, . . . , an /c) ∩ U (b1 /c, . . . , bm /c) ∩ X = ∅. Therefore, U (1/c) ∩ X = ∅. Since each V ∈ X is a valuation ring, this implies c ∈ MV , so that 0 ≠ c ∈ J(X). The converse follows from (2).  3. Residually transcendental limit points The main result of this section, Theorem 3.2, is of a technical nature and involves the existence of patch limit points that are residually transcendental over a local subring of F . For the purpose of proving the results in this section, we recall the notion of a projective model of a field; see [45, Chapter 6, §17] for more background on this topic. Let D be a subring of the field F , and let t0 , t1 , . . . , tn be nonzero elements of F . For each i, let Di = D[t0 /ti , . . . , tn /ti ], and let M = ⋃ {(Di )P ∶ P ∈ Spec(Di )}. n i=0 Then M is the projective model of F /D defined by t0 , t1 , . . . , tn . Alternatively, a projective model of F /D can be viewed as a projective integral scheme over Spec(D) whose function field is a subfield of F . Motivated by this interpretation, it is often convenient to view M as a locally ringed space; see for example [36, Section 3]. We do not need to do this explicitly in the present paper, but the notation remains helpful here when we wish to view the local rings in M as points. In particular, 8 BRUCE OLBERDING viewing the set M as a topological space with respect to the Zariski topology4, the local rings in M are points in M. For this reason, and in keeping with the locally ringed space point of view, we denote a local ring x ∈ M by OM,x , despite the redundancy in doing so. A subset Y of M is an affine submodel of M if there exists a D-subalgebra R of F such that Y = {RP ∶ P ∈ Spec(R)}. (We differ here from [45] in that we do not require R to be a finitely generated D-algebra.) For each x ∈ M, there exists a valuation ring in Zar(F ) that dominates OM,x , and for each V ∈ Zar(F /D), there exists a unique x ∈ M such that V dominates OM,x ; see [45, pp. 119–120] or apply the valuative criterion for properness [20, Theorem 4.7, p. 101]. For a nonempty subset X of M, we denote by M(X) the set of all x ∈ M such that OM,x is dominated by some V ∈ X. More formally, M(X) is the image of X under the continuous closed surjection Zar(F ) → M given by the domination mapping [45, Lemma 5, p. 119]. Lemma 3.1. Let X be a nonempty subset of Zar(F ) such that A = A(X) is a local ring. Let D be a subring of A, let t0 , . . . , tn be nonzero elements of F , and let M be the projective model of F /D defined by t0 , . . . , tn . If M(X) is finite, then t0 /ti , . . . , tn /ti ∈ A for some i = 0, 1, . . . , n. Proof. Suppose M(X) is finite. Each finite subset of the projective model M is contained in an affine open submodel of M. (This is a consequence of homogeneous prime avoidance; see for example [43, Lemma 01ZY] or the proof of [33, Corollary 3.2]). Thus M(X) is contained in an affine submodel of M, and hence there is a D-subalgebra R of F such that for each x ∈ M(X), OM,x is a localization of R. For each V ∈ X, there is x ∈ M(X) such that V dominates OM,x . Since A is a local ring, so is the subring B = ⋂ OM,x x∈M(X) of A. (Indeed, if b ∈ B is a unit in A, then since, for each x ∈ M(X), OM,x is dominated by a valuation ring in X, b is a unit in B. Thus if b, c ∈ B are nonunits in B, then b + c is a nonunit in the local ring A, so that b + c is a nonunit in B.) For each x ∈ M(X), since OM,x is a localization of R and R ⊆ B ⊆ OM,x with B a local ring, we have that OM,x is a localization of B at a prime ideal of B. Since M(X) is finite, B is a local ring that is a finite intersection of the local rings OM,x , x ∈ M(X), each of which is a localization of B at a prime ideal. It follows that B is equal to one of these localizations; i.e., B = OM,x for some x ∈ M(X). Since B ∈ M, there is i such that t0 /ti , . . . , tn /ti ∈ B ⊆ A.  The proof of the next lemma can be streamlined by using the language of schemes and morphisms into the projective line (see [33] for more on this point of view in 4The basis for this topology is given by sets of the form {R ∈ M ∶ x , . . . , x ∈ R}, where 1 n x1 , . . . , xn ∈ F . COMPACT SETS AND HOLOMORPHY RINGS 9 our context), but in order to make the proof self-contained, we develop the needed ideas in the course of the proof. Theorem 3.2. Let X be a nonempty subset of Zar(F ) such that A = A(X) is a local ring, and let 0 ≠ t ∈ F . Suppose D is a local subring of A, t ∈/ A, 1/t ∈/ A, and all but at most finitely many V ∈ X dominate D. Then there exists U ∈ lim(X) such that t, 1/t ∈ U , U dominates D and the image of t in U /MU is transcendental over the residue field of D. Proof. We first show that we can replace D with a local subring D ′ of A such that D ′ is integrally closed in D ′ [t] and all but finitely many valuation rings in X dominate D ′ . Let D denote the integral closure of D in D[t]. Since A in integrally closed in F , D ⊆ A. Let m = M ∩ D, where M is the maximal ideal of A, and let D ′ = D m . Then A dominates D ′ . Since D ⊆ D[t], we have D[t] = D[t], and so D is integrally closed in D[t]. Thus D ′ = D m is integrally closed in D ′ [t] = D m [t] since D ′ and D ′ [t] are localizations of D and D[t], respectively, at the same multiplicatively closed set. Therefore, D ′ is a local ring that is integrally closed in D ′ [t] and is dominated by A. To see next that all but finitely many valuation rings in X dominate D ′ , suppose that V ∈ X and V dominates D. We claim that V dominates D ′ . Let n denote the maximal ideal of D, and let p = MV ∩ D. Since V dominates D, p lies over n. Thus, since D is integral over D, p is a maximal ideal of D. If p ≠ m, then there exists d ∈ p ∖ m so that 1/d ∈ D m = D ′ ⊆ A ⊆ V . Since d ∈ MV , this is a contradiction that implies p = m. Thus V dominates D ′ . This shows that every valuation ring in X that dominates D also dominates D ′ . Thus, if V ∈ X does not dominate D ′ , then V does not dominate D. By assumption, there are at most finitely many valuation rings in X that do not dominate D, so there are at most finitely many valuation rings in X that do not dominate D ′ . With this in mind, we work for the rest of the proof with D ′ instead of D and draw the desired conclusion for D in the last step of the proof. The advantage of working with D ′ is that D ′ is a local ring that is integrally closed in D ′ [t]. Let M be the projective model of F /D ′ defined by 1, t. Let f ∶ M → Spec(D ′ ) be the canonical mapping that sends a local ring R in M to D ′ ∩ N , where N is the maximal ideal of R. Let C = f −1 (m′ ), where m′ is the unique maximal ideal of D ′ . Since f is continuous in the Zariski topology [45, Lemma 5, p. 119], C is the closed subset of M consisting of all the local rings in M that dominate D ′ . Since D ′ is integrally closed in D ′ [t] and t, t−1 ∈/ D ′ (indeed, by assumption, t, t−1 ∈/ A), Seidenberg’s Lemma [29, Exercise 31, pp. 43–44] implies that the rings D ′ [t]/mD ′ [t] and D ′ [t−1 ]/m′ D ′ [t−1 ] are each isomorphic to the polynomial ring (D ′ /m′ )[T ], where T is an indeterminate. These isomorphisms are induced by the mappings t ↦ T and t−1 ↦ T , respectively. Thus (t, m′ )D ′ [t] is a maximal ideal of D ′ [t], while m′ D ′ [t] is a prime ideal in D ′ [t] and m′ D ′ [t−1 ] is a prime ideal in D ′ [t−1 ]. 10 BRUCE OLBERDING Now, since m′ extends to a prime ideal in both D ′ [t] and D ′ [t−1 ], C is irreducible in the Zariski topology with generic point the local ring D ′ [t]m′ D′ [t] = D ′ [t−1 ]m′ D′ [t−1 ] . Using the fact that the rings D ′ [t]/m′ D ′ [t] and D ′ [t−1 ]/m′ D ′ [t−1 ] are each PIDs isomorphic to the polynomial ring (D ′ /m′ )[T ], it follows that the closed points in C are the local rings in the set {D ′ [t]P ∶ P is a maximal ideal in D ′ [t], m ⊆ P } ∪ {D ′ [t−1 ](m,t−1 ) }. The only point in C that is not closed is D ′ [t]m′ D′ [t] , the unique generic point of C. This accounts for all the local rings in C. The rest of the proof consists in showing that there is a valuation ring U in lim(X) that dominates D ′ [t]m′ D′ [t] . To this end, we next describe the local rings in M(X) ∩ C; i.e., we describe the local rings in C that are dominated by valuation rings in X. In particular, we show that there are infinitely many such local rings in C and that the Zariski closure of M(X) ∩ C in M is C. Let X ∗ be the set of valuation rings in X that dominate D ′ . We have established that all but finitely many valuation rings in X dominate D ′ ; that is, X ∖X ∗ is finite. The image M(X ∗ ) of X ∗ in M under the domination mapping is contained in C, so that C(X ∗ ) = M(X ∗ ). By Lemma 3.1 the fact that A is a local ring and t and 1/t are not elements of D ′ implies M(X) is infinite. Since also X ∖ X ∗ is finite, it follows that M(X ∗ ) = C(X ∗ ) is infinite. Thus, since C consists only of closed points and a unique generic point for C, there are infinitely many closed points of C in C(X ∗ ), which means there is a subset X ′ of X ∗ such that the image C(X ′ ) of X ′ in C is infinite and consists of rings of the form D ′ [t]P , where P is a maximal ideal of D ′ [t] that contains m′ . Therefore, the valuation rings in X ′ contain D ′ [t], and the image of X ′ in Spec(D ′ [t]) under the map that sends a valuation ring to its center in D ′ [t] consists of infinitely many maximal ideals of D ′ [t], all of which contain the dimension one prime ideal m′ D ′ [t]. Since D ′ [t]/m′ D ′ [t] is a PID, the intersection of these infinitely many maximal ideal is m′ D ′ [t]. Next, to see how this fact is reflected in C, let M1 be the affine submodel of M given by M1 = {D′ [t]P ∶ P ∈ Spec(D′ [t])}. Let g ∶ M1 → Spec(D ′ [t]) denote the homeomorphism that sends a local ring D ′ [t]P in M1 to its center in D ′ [t]. We have shown that the Zariski closure of C(X ′ ) in Spec(D ′ [t]) is the set of all prime ideals of D ′ [t] containing the prime ideal m′ D ′ [t]. Since g is a homeomorphism, it follows that D ′ [t]m′ D′ [t] is in the Zariski closure of C(X ′ ). Hence the Zariski closure of C(X ′ ) in M is C. Now, since the generic point of the closed set C is the local ring D ′ [t]m′ D′ [t] and C(X ′ ) is infinite and Zariski dense in C, we apply [36, Lemma 2.7(4)] to obtain that there is a valuation ring U in the patch closure of X ′ that dominates D ′ [t]m′ D′ [t] = D ′ [1/t]m′ D′ [1/t] . COMPACT SETS AND HOLOMORPHY RINGS Since 11 D ′ [t]/m′ D ′ [t] ≅ (D ′ /m′ )[T ], the image of t in the residue field of U is transcendental over the residue field of D ′ . Since all the valuation rings in X ′ are centered on maximal ideals of D ′ [t], U is not a member of X ′ . Therefore, since U is in the patch closure of X ′ but not in X ′ , it must be that U ∈ lim(X ′ ) ⊆ lim(X). Finally, since D ′ dominates D, we conclude that the image of t in U /MU is transcendental over the residue field of D, which completes the proof of the theorem.  4. Limit points of rank greater than one We show in Theorem 4.3 that if X ⊆ Zar(F ) and A(X) is local but not a valuation domain, then there is a patch limit valuation ring of X of rank > 1. This is the key result needed in the next section to prove the main results of the paper. The proof of Theorem 4.3 relies on the following lemma, which gives a criterion for the existence of a valuation ring of rank > 1 to lie in the patch closure of X. Lemma 4.1. Let A be a local integrally closed subring of F with maximal ideal M , and let X be a patch closed subset of Zar(F ) such that A = A(X). Suppose that there exist 0 ≠ t ∈ F and m ∈ M such that mt /∈ A and t−1 /∈ A. If for each i > 0 there exists Vi ∈ X such that m ∈ MVi and mti is a unit in Vi , then X contains a valuation ring of rank > 1. Proof. Let i > 0. If mti ∈ A, then (mt)i ∈ A, so that since A is integrally closed we have mt ∈ A, contrary to assumption. Thus mti ∈/ A for all i > 0. Moreover, (mti )−1 ∈/ A, since otherwise t−i = m(m−1 t−i ) ∈ A, which since A is integrally closed forces t−1 ∈ A, contrary to the choice of t. By assumption, there exists Vi ∈ X such that mti is a unit in Vi and m ∈ MVi . If t ∈ Vi , then since mti is a unit in Vi it is the case that m is a unit in Vi and an element of MVi , a contradiction. Thus t ∈/ Vi . Using Notation 2.2, let Ci = U (mti ) ∩ V (t) ∩ X. Then Vi ∈ Ci , so that Ci is a nonempty patch closed subset of X. We use compactness to show that ⋂i>0 Ci is nonempty. To this end, we claim that the collection {Ci ∶ i > 0} has the finite intersection property. Let i > 0, and let 0 < j < i. Then Vi ∈ Cj = U (mtj ) ∩ V (t) ∩ X since mti ∈ Vi and t−1 ∈ Vi implies mtj = mti (t−1 )i−j ∈ Vi . For each i > 0, it follows that C1 ∩ C2 ∩ ⋯ ∩ Ci contains the valuation ring Vi . Therefore, the collection {Ci ∶ i > 0} of patch closed subsets of X has the finite intersection property. Since X is a patch closed subset of the patch quasicompact space Zar(F ), X is patch 12 BRUCE OLBERDING quasicompact. Thus the set i ⋂ Ci = X ∩ V (t) ∩ ( ⋂ U (mt )) i>0 i>0 is nonempty. Let U be a valuation ring in this intersection. Then U ∈ X, and, for each i > 0, we have 0 ≠ m ∈ (t−1 )i U . Also, since t /∈ U , we have t−1 ∈ MU . Thus m ∈ (t−1 )i U ⊆ MU . If U has rank 1, then the radical of mU in U is MU . In this case there exists n > 0 such that (t−1 )n ∈ mU ⊆ (t−1 )n+1 U, a contradiction to the fact that t−1 is in U but is not a unit in U . We conclude that U has rank > 1.  In the proof of Theorem 4.3 we pass to a subfield K of F . In doing so, we need that features of the topology of X are preserved in the image of X in the Zariski-Riemann space of K. This is given by the next lemma. Lemma 4.2. Let K be a subfield of F . Then the mapping f ∶ Zar(F ) → Zar(K) ∶ V ↦ V ∩ K is a surjective map that is closed and continuous in the patch topology. Proof. That f is surjective follows from the Chevalley Extension Theorem [8, Theorem 3.1.1, p. 57]. To see that f is continuous in the patch topology observe that for x1 , . . . , xn , y ∈ K, the preimages under f of the subbasic patch open sets {V ∈ Zar(K) ∶ x1 , . . . , xn ∈ V } and {V ∈ Zar(K) ∶ y ∈/ V } are U (x1 , . . . , xn ) and V (y), respectively, and hence are patch open in Zar(F ). Finally, with the patch topologies on Zar(F ) and Zar(K), f is a continuous map between compact Hausdorff spaces, and so f is closed [7, Theorem 3.1.12, p. 125].  Theorem 4.3. Let X be a nonempty subset of Zar(F ) such that J(X) ≠ 0. If A(X) is a local ring that is not a valuation domain, then patch(X) contains a valuation ring of rank > 1. Proof. Let A = A(X), J = J(X), and let M denote the unique maximal ideal of A. We prove the lemma by establishing a series of claims. In the proof, for an ideal I of A, we denote by End(I) the subring of F given by {t ∈ F ∶ tI ⊆ I}. Thus End(I) is the largest ring in F in which I is an ideal. Claim 1. If A is not completely integrally closed5, then patch(X) contains a valuation ring of rank > 1. 5A domain R is completely integrally closed if for each nonzero ideal I of R and each element t of the quotient field of R, tI ⊆ I if and only if t ∈ R; equivalently, R = End(I) for each nonzero ideal I of R. COMPACT SETS AND HOLOMORPHY RINGS 13 Proof of Claim 1. If every valuation ring in patch(X) has rank ≤ 1, then A, as an intersection of completely integrally closed domains, is completely integrally closed.  Claim 2. If M is a principal ideal of A, then patch(X) contains a valuation ring of rank > 1. Proof of Claim 2. Suppose M = mA for some m ∈ M . Since A is not a valuation domain, A is not field, and hence M ≠ 0. Since M is principal, the ideal P = ⋂i>0 M i is the unique largest nonmaximal prime ideal of A and P AP = P [29, Exercise 1.5, p. 7]. If P = 0, then A is valuation ring (in fact, a DVR), contrary to assumption. Thus P ≠ 0, and, since P = P AP , it follows that AP ⊆ End(P ). Since P is a nonmaximal prime ideal, this implies A ⊊ End(P ), so that A is not completely integrally closed. By Claim 1, patch(X) contains a valuation ring of rank > 1, which proves Claim 2.  Claim 3. If all the valuation rings in X dominate A, then patch(X) contains a valuation ring of rank > 1. Proof of Claim 3. If M is a principal ideal of A, Claim 2 implies that patch(X) contains a valuation ring of rank > 1, and the proof of the claim is complete. Thus we assume M is not a principal ideal of A. Also, by Claim 1, we may assume that A is completely integrally closed and hence End(M ) = A. It remains to prove Claim 3 in the case in which End(M ) = A, M is not a principal ideal of A and all the valuation rings in X dominate A. Since A is not a valuation ring, there exists t ∈ F such that t /∈ A and t−1 ∈/ A. Since M is not an invertible ideal of A, we have M ⊆ M (A ∶F M ) ⊊ A, which forces M = M (A ∶F M ). Thus (A ∶F M ) = End(M ) = A, so t ∈/ A = (A ∶F M ). Let m ∈ M such that mt ∈/ A. Since t−1 , mt ∈/ A, we have mti , (mti )−1 ∈/ A for each i > 0, as noted at the beginning of the proof of Lemma 4.1. By Theorem 3.2 (applied in the case where “D” is A), there exists, for each i > 0, a valuation ring Vi ∈ patch(X) that dominates A and for which mti is a unit in Vi . By assumption, every valuation ring in X dominates A, so Lemma 2.3(1) implies that every valuation ring in patch(X) dominates A. Thus m ∈ MVi for all i. By Lemma 4.1, patch(X) contains a valuation ring of rank > 1, which completes the proof of Claim 3.  Claim 4. If there are valuation rings in X that do not dominate A, then there exists a two-dimensional Noetherian local subring D of A such that D is dominated by A and D contains nonzero elements a, b with a ∈ M ∖ J and b ∈ J. 14 BRUCE OLBERDING Proof of Claim 4. Since there are valuation rings in X that do not dominate A, we have 0 ≠ J ⊊ M . To prove the existence of the ring D, suppose first A contains a field k. In this case, choose a ∈ M ∖ J and 0 ≠ b ∈ J. Let C = k[a, b] and let D = CM ∩C . Then D is a Noetherian local subring of A that is dominated by A. Since C is generated by two elements over a field, D has Krull dimension at most two. To see that D has exactly Krull dimension two, observe that since a ∈ M ∖ J there is V ∈ X such that a ∈/ MV . Thus 0 ≠ b ∈ MV ∩ D ⊊ M ∩ D, so that D has Krull dimension two. This verifies Claim 4 if A contains a field. Suppose next that A does not contain a field. Then the contraction of M to the prime subring of A is a nonzero principal ideal, and hence there is a DVR W that is dominated by A. Let p be the generator of the maximal ideal of W . If p ∈ J, then let b = p and choose a ∈ M ∖ J. If p /∈ J, then let a = p and choose 0 ≠ b ∈ J. In either case, we have a ∈ M ∖ J and 0 ≠ b ∈ J. Let C = W [a, b], and let D = CM ∩C . Since W is a DVR and either a ∈ W or b ∈ W , the ring D has Krull dimension at most two. As in the case in which A contains a field, since a ∈ M ∖ J and J ≠ 0, it follows that D has Krull dimension two. This verifies Claim 4.  Claim 5. The set patch(X) contains a valuation ring of rank > 1. Proof of Claim 5. If every valuation ring in X dominates A, then, by Claim 3, patch(X) contains a valuation ring of rank > 1. It remains to consider the case where there are valuation rings in X that do not dominate A. By Claim 4, there is a two-dimensional local Noetherian subring D of A such that D is dominated by A and D contains nonzero elements a, b such that a ∈ M ∖ J and b ∈ J. The next step of the proof involves a reduction that allows us to work in the quotient field of D. Let K denote the quotient field of D. Let A′ = A ∩ K, M ′ = M ∩ K, and X ′ = {V ∩ K ∶ V ∈ patch(X)}. Then A′ is a local integrally closed overring6 of D with maximal ideal M ′ and A′ = ⋂V ∈X ′ V . Also, by Lemma 4.2, X ′ is patch closed in Zar(K). Claim 5(a). All but finitely many valuation rings in X ′ dominate D. Proof of Claim 5(a). Suppose that V ∈ X ′ and V does not dominate D. Then MV ∩ D is a nonmaximal prime ideal of D that by Lemma 2.3(1) contains b. Since D is a Noetherian ring of Krull dimension 2, there are only finitely many height one prime ideals P1 , . . . , Pn of D containing b. The valuation ring V contains the integral closure D ′ of the semilocal one-dimensional Noetherian ring DP1 ∩ ⋯ ∩ DPn . Since the ring D ′ is a semilocal PID, there are only finitely many valuation rings between D ′ and its quotient field K. Therefore, the set of valuation rings in X ′ that do not dominate D is finite.  6By an overring of a domain, we mean a ring between the domain and its quotient field. COMPACT SETS AND HOLOMORPHY RINGS 15 Returning to the proof of Claim 5, to show that patch(X) contains a valuation ring of rank > 1, it is enough to prove that X ′ contains a valuation ring of rank > 1. This is because if U ∈ X ′ has rank > 1, there is a valuation ring V ∈ patch(X) with V ∩ K = U , and V is necessarily of rank > 1 since V extends U . Thus we need only show that X ′ contains a valuation ring of rank > 1. We prove this by verifying two claims. Claim 5(b). If A′ [1/a] is a valuation ring, then X ′ contains a valuation ring of rank > 1. Proof of Claim 5(b). Since a ∈/ J, there exists V ∈ X ′ such that a ∈/ MV , and hence 1/a ∈ V . Since 0 ≠ b ∈ J ∩ A′ ⊆ MV , the valuation ring V does not have rank 0. Therefore, A′ [1/a] ⊆ V ⊊ K. As an overring of the two-dimensional Noetherian ring D, A′ has Krull dimension at most two. (This follows from the Dimension Inequality [28, Theorem 15.5, p. 118].) Since a ∈ M ′ , the Krull dimension of A′ [1/a] is less than that of A′ , and hence, since A′ [1/a] ⊊ K, it follows that A′ [1/a] has Krull dimension one (and A′ has Krull dimension two). Thus the valuation ring A′ [1/a] has rank one. Now suppose by way of contradiction that every valuation ring in X ′ has rank ≤ 1. Then A′ , as an intersection of valuation rings in X ′ , is completely integrally closed. Thus, since A′ has Krull dimension 2, A′ is not a valuation domain. By [35, (4.2)] every patch closed representation of a domain contains a minimal patch closed representation. Thus there is a patch closed subset Y of X ′ (recall that X ′ is patch closed) such that A′ = ⋂V ∈Y V and no proper patch closed subset of Y is a representation of A′ . By Lemma 2.3(4), the rank one valuation ring A′ [1/a], as a proper subring of K, is an intersection of valuation rings in Y . Since A′ [1/a] has the same quotient field as the valuation rings in X ′ , we conclude that A′ [1/a] is in Y. Since Y ⊆ X ′ and J ∩ A′ = ⋂V ∈X ′ MV , we have by Lemma 2.3(1) that 0 ≠ b ∈ J ∩ A′ ⊆ MV for all V ∈ Y . In particular, K ∈/ Y . Let W = A′ [1/a]. Since W has rank one, we conclude that {W } = U (1/a) ∩ Y , so that W is a patch isolated point in Y . We show this leads to a contradiction to the fact that Y is a minimal patch closed representation of A′ . Since W is a patch isolated point in Y , Y ∖ {W } is a patch closed set, and hence by the minimality of Y we have have A′ ⊊ ⋂ U. U ∈Y ∖{W } Let I = ⋂ U ∈Y ∖{W } MU . 16 Then BRUCE OLBERDING ⋂ U ⊆ End(I). U ∈Y ∖{W } Since A′ is completely integrally closed, it follows that I = 0 or I ⊆/ A′ . The former case is impossible, since 0 ≠ b ∈ J ∩ A′ ⊆ I. Thus we conclude that I ⊆/ A′ . Let t ∈ I ∖ A′ . Then t−1 ∈/ A′ since for any U ∈ Y ∖ {W }, we have t ∈ MU and A′ ⊆ U . Thus t, t−1 /∈ A′ . By Claim 5(a), all but finitely many valuation rings in X ′ , hence also in Y , dominate D. Applying Theorem 3.2 to D, A′ , Y and t, there exists a valuation ring U ∈ lim(Y ) such that t, t−1 ∈ U . Since Y is patch closed, U ∈ Y , and, since t is a unit in U , t /∈ MU . By the choice of t, t is in the maximal ideal of every valuation ring in Y except W . This forces W = U ∈ lim(Y ), contradicting the fact that W is a patch isolated point in Y . This contradiction shows that X ′ contains a valuation ring of rank > 1. This completes the proof of Claim 5(b).  Claim 5(c). If A′ [1/a] is not a valuation ring, then X ′ contains a valuation ring of rank > 1. Proof of Claim 5(c). Since A′ [1/a] is not a valuation ring and A′ has quotient field K, there exists 0 ≠ t ∈ K such that t ∈/ A′ [1/a] and t−1 ∈/ A′ [1/a]. Thus t, t−1 ∈/ A′ and at ∈/ A′ . With the aim of applying Lemma 4.1 to A′ and X ′ , we fix i > 0 and we show that there is a valuation ring V ∈ X ′ such that ati is a unit in V and a ∈ MV . Once this is proved, Lemma 4.1 implies that X ′ contains a valuation ring of rank > 1, and the proof of Claim 5(c) is complete. Let s = ati . Since t ∈/ A′ [1/a] and A′ [1/a] is integrally closed, it follows that i t ∈/ A′ [1/a], and hence s ∈/ A′ . If s−1 ∈ A′ , then t−i = a(a−1 t−i ) = as−1 ∈ A′ , so that, since A′ is integrally closed in K, t−1 ∈ A′ , contrary to the choice of t. Therefore, s, s−1 ∈/ A′ . By Claim 5(a), all but finitely many valuation rings in X ′ dominate D. By Theorem 3.2, with D, A′ , X ′ and s playing the roles of “D”, “A”, “X” and “t” in the theorem, we obtain U ∈ X ′ such that s, 1/s ∈ U and U dominates D. Therefore, s = ati is a unit in U and a ∈ MU since U dominates D. By Lemma 4.1, X ′ contains a valuation of rank > 1, which proves Claim 5(c).  Finally, to complete the proof of Claim 5, we note that Claim 5(b) and 5(c) show that X ′ contains a valuation ring of rank > 1. As discussed after the proof of Claim 5(a), this implies that patch(X) contains a valuation of rank > 1. Therefore, with the proof of Claim 5 complete, the proof of the theorem is complete also.  5. Compact sets and holomorphy rings The main results of the paper involve one-dimensional Prüfer domains. We collect in the next lemma some basic properties of such rings that are needed for the COMPACT SETS AND HOLOMORPHY RINGS 17 theorems in this section. We denote by J(A) the Jacobson radical of a ring A, and by Max(A) the space of maximal ideals of A endowed with the Zariski topology. Lemma 5.1. Let A be a one-dimensional Prüfer domain with quotient field F , and let X ⊆ Zar(F ) such that F ∈/ X and A = A(X). Then J(A) = J(X). If J(A) ≠ 0, then the Zariski, patch and inverse topologies all coincide on X and patch(X) = {AM ∶ M ∈ Max(A)}. Proof. It is straightforward to check that J(X) ⊆ J(A); for example, see [19, Remark 1.3]. To see that the reverse inclusion holds, let V ∈ X. Since F /∈ X and A is a one-dimensional Prüfer domain, V = AM , where M = MV ∩ A is a maximal ideal of A. Since J(A) ⊆ M ⊆ MV , we have J(A) ⊆ J(X). This proves the first assertion of the lemma. Suppose now that J(A) ≠ 0. Since A has Krull dimension one, Max(A) is homeomorphic to the spectral space Spec(A/J(A)). In a spectral space for which every point is both minimal and maximal with respect to the specialization order, the Zariski, patch and inverse topologies all agree; cf. [41, Corollary 2.6] or use the fact that A/J(A) is a von Neumann regular ring. Thus these three topologies all agree on X since X is homeomorphic to a subspace of Max(A). Finally, since A is a one-dimensional Prüfer domain represented by X, the set of all valuation overrings of A (each of which must have rank ≤ 1) is patch(X)∪{F } [10, Corollary 4.10]. Since J(A) ≠ 0, Lemma 2.3(1) implies F /∈ patch(X). Therefore, patch(X) = {AM ∶ M ∈ Max(A)}.  Remark 5.2. Topological aspects and factorization theory of one-dimensional Prüfer domains with nonzero Jacobson radical are studied in [22]. The first application of the results of the previous section is the following characterization of subsets of Zar(F ) whose holomorphy ring is a one-dimensional Prüfer domain with nonzero Jacobson radical and quotient field F . Theorem 5.3. The following are equivalent for a nonempty subset X of Zar(F ) with J(X) ≠ 0. (1) A(X) is a one-dimensional Prüfer domain with quotient field F . (2) X is contained in a quasicompact set of rank one valuation rings in Zar(F ). (3) Every valuation ring in patch(X) has rank one. Proof. Let A = A(X) and J = J(X). (1) ⇒ (2) Since A is a one-dimensional Prüfer domain with quotient field F , the subset of Zar(F ) given by Y = {AM ∶ M ∈ Max(A)} consists of rank one valuation rings in Zar(F ). The only other valuation overring of A is F . Since J ≠ 0, we have F ∈/ X, which forces X ⊆ Y . Moreover, Y is homeomorphic to Max(A) and the maximal spectrum of a ring is quasicompact, so statement (2) follows. 18 BRUCE OLBERDING (2) ⇒ (3) Let Y be a quasicompact set of rank one valuation rings in Zar(F ) such that X ⊆ Y . Since Y is quasicompact, Lemma 2.3(3) implies patch(X) ⊆ Y ∪ {F }. If F ∈ patch(X), then from Lemma 2.3(1) it follows that J = 0, contrary to assumption. Therefore, patch(X) ⊆ Y , so that patch(X) consists of rank one valuation rings. (3) ⇒ (1) By Lemma 2.3(1), A = ⋂V ∈patch(X) V and 0 ≠ J = ⋂V ∈patch(X) MV . Thus we can assume without loss of generality that X = patch(X). We claim first that A has quotient field F . Let 0 ≠ a ∈ J. By Lemma 2.3(2), X is quasicompact, so by Lemma 2.3(4) we have A[1/a] = ⋂ V = F, 1/a∈V ∈X∪{F } where the last equality follows from the fact that every valuation ring V in X has rank one and satisfies a ∈ MV . Since A[1/a] = F , we conclude that A has quotient field F . To prove that A is a one-dimensional Prüfer domain, it suffices to show that AM is a rank one valuation domain for each maximal ideal M of A. Let M be a maximal ideal of A. Let Y = {V ∈ X ∶ AM ⊆ V }. Since Y = X ∩ ( ⋂ U (t)), t∈AM Y is patch closed in X. By Lemma 2.3(2) and the fact that X is patch closed in Zar(F ), Y is a quasicompact subset of Zar(F ). Since Y is quasicompact and consists of rank one valuation rings, Lemma 2.3(4) implies that AM = ⋂V ∈Y V . Thus Y is a patch closed representation of AM consisting of rank one valuation rings. Since J ≠ 0, Theorem 4.3 implies that AM is a valuation domain. Since AM has quotient field F and Y is a representation of AM consisting of rank one valuation domains, it follows that AM ∈ Y . Hence AM is a rank one valuation domain, which proves that A is a Prüfer domain with Krull dimension one.  A domain A is an almost Dedekind domain if for each maximal ideal M of A, AM is a DVR. There exist many interesting examples of almost Dedekind domains; see for example [22, 25, 30] and their references. For the factorization theory of such rings, see [12, 22, 26]. Corollary 5.4. Let X be a nonempty set of Zar(F ) such that J(X) ≠ 0. Then X is contained in a quasicompact set of DVRs in Zar(F ) if and only if A(X) is an almost Dedekind domain with quotient field F . Proof. Let A = A(X). Suppose X is contained in a quasicompact set Y of DVRs in Zar(F ). By Theorem 5.3, A is a one-dimensional Prüfer domain with quotient field F and nonzero Jacobson radical. By Lemma 2.3(3), patch(X) ⊆ patch(Y ) = Y ∪ {F }. COMPACT SETS AND HOLOMORPHY RINGS 19 Since J(X) ≠ 0, Lemma 2.3(1) implies F ∈/ patch(X), so patch(X) ⊆ Y . Thus patch(X) consists of DVRs. By Lemma 5.1, we have that for each maximal ideal M of A, AM is in patch(X) and hence AM is a DVR. Thus A is an almost Dedekind domain. The converse follows from Lemma 5.1 and Theorem 5.3.  Remark 5.5. Let X be a nonempty quasicompact set of DVRs in Zar(F ) such that J(X) ≠ 0. By Corollary 5.4, A = A(X) is an almost Dedekind domain, and J(A) = J(X) by Lemma 5.1. If there is t ∈ J(A) such that J(A) = tA (equivalently, MV = tV for each V ∈ X), then A has the property that every proper ideal is a product of radical ideals. For this and related results on such rings, which are known in the literature as SP-domains or domains with the radical factorization property, see [12, 22, 30]. We prove next our main theorem of this section (the “main lemma” of the introduction) regarding the correspondence between quasicompact sets and holomorphy rings in the space of rank one valuation rings. Theorem 5.6. The mappings X ↦ A(X) and A ↦ {AM ∶ M ∈ Max(A)} define a bijection between the quasicompact sets X of rank one valuation rings in Zar(F ) with J(X) ≠ 0 and the one-dimensional Prüfer domains A with quotient field F and nonzero Jacobson radical. Proof. Let X be a quasicompact set of rank one valuation rings in Zar(F ) with J(X) ≠ 0. By Lemma 5.1 and Theorem 5.3, A = A(X) is a one-dimensional Prüfer domain such that J(A) ≠ 0 and A has quotient field F . By Lemma 2.3(3), patch(X) ⊆ X ∪ {F }. Since J(A) ≠ 0, Lemma 2.3(1) implies F /∈ patch(X). Thus X = patch(X), and, by Lemma 5.1, X = {AM ∶ M ∈ Max(A)}. Conversely, suppose A is one-dimensional Prüfer overring with quotient field F and 0 ≠ t ∈ J(A). Since Max(A) is quasicompact, X = {AM ∶ M ∈ Max(A)} is a quasicompact set of rank one valuation rings such that A = A(X) and 0 ≠ t ∈ J(X).  The next example shows the necessity of the hypotheses in Theorem 5.6. Example 5.7. Let X be a nonempty subset of Zar(F ). Theorem 5.6 shows that if (a) X consists of rank one valuation rings, (b) X is quasicompact and (c) J(X) ≠ 0, then A(X) is a Prüfer domain. These hypotheses are necessary in the sense that A(X) need not be a Prüfer domain if any one of (a), (b) or (c) is omitted. The following classes of examples illustrate this. (1) An example in which A(X) is not a Prüfer domain but in which (a) and (b) hold. Let D be an integrally closed Noetherian domain of Krull dimension > 1. Then the set X of localizations of D at height one prime ideals is a quasicompact set of rank one valuation rings for which A(X) = D. 20 BRUCE OLBERDING (2) An example in which A(X) is not a Prüfer domain but in which (b) and (c) hold. Let k be a field, and let V be the DVR k(S)[T ](T ) , where S and T are indeterminates for k. Let R = k + MV . Then R is a one-dimensional integrally closed local domain. With X the set of valuation overrings of R of rank > 0, we have that R = A(X) and J(X) ≠ 0. Moreover, X is quasicompact since X is the closed subset of the (quasicompact) space of valuation overrings. Indeed, X is the set of valuation overrings of R contained in V . Thus (b) and (c) hold, but A(X) is not a Prüfer domain since A(X) is a local domain contained in more than one valuation ring of Krull dimension 2. (3) An example in which A(X) is not a Prüfer domain but in which (a) and (c) hold. Let D be an integrally closed Noetherian local domain of Krull dimension > 1 with quotient field F . To exhibit the desired example, it suffices to show that D is the intersection of the DVRs in Zar(F /D) that dominate D. Let m denote the maximal ideal of D, and let x ∈ F ∖ D. If x−1 ∈ D, then x−1 ∈ m. Since every Noetherian local domain is birationally dominated by a DVR [6, p. 26], there exists a DVR V in Zar(F /D) such that x−1 ∈ MV . Thus x ∈/ V . On the other hand, if x−1 /∈ D, then the ring D[x−1 ] has a maximal ideal generated by m and x−1 [29, Exercise 31, pp. 43–44], so, again by [6, p. 26], there is a DVR V in Zar(F /D) that dominates D and for which x−1 ∈ MV and hence x ∈/ V . It follows that D is the intersection of all the DVRs in Zar(F /D) that dominate it. Restricting to DVRs in Theorem 5.6 yields a correspondence with almost Dedekind domains. Corollary 5.8. The mappings X ↦ A(X) and A ↦ {AM ∶ M ∈ Max(A)} define a bijection between the quasicompact sets X of DVRs in Zar(F ) with J(X) ≠ 0 and the almost Dedekind domains A with quotient field F and nonzero Jacobson radical. Proof. If X is a quasicompact set of DVRs in Zar(F ) with J(X) ≠ 0, then, by Corollary 5.4, A is an almost Dedekind domain. Conversely, if A is almost Dedekind domain, then clearly X = {AM ∶ M ∈ Max(A)} is a quasicompact set of DVRs. Thus the corollary follows from Theorem 5.6.  By Proposition 2.4(2) the quasicompact sets in Theorem 5.6 are compact, so the theorem alternatively can be stated for compact sets instead. Along these lines, in the case in which all the valuation rings under consideration occur as overrings of a domain with quotient field F , the restriction that J(X) ≠ 0 in Theorem 5.6 can be omitted (or, more correctly, hidden) if the quasicompact hypothesis is strengthened to that of being compact. COMPACT SETS AND HOLOMORPHY RINGS 21 Corollary 5.9. Let R be a proper subring of F with quotient field F . There is a bijection given by X ↦ A(X) and A ↦ {AM ∶ M ∈ Max(A)} between the compact sets X of rank one valuation overrings of R and the onedimensional Prüfer overrings A of R with nonzero Jacobson radical. Proof. If X is a compact set of rank one valuation overrings, then J(X) ≠ 0 by Proposition 2.4(3). By Theorem 5.6, A = A(X) is a one-dimensional Prüfer domain with J(A) ≠ 0 and X = {AM ∶ M ∈ Max(A)}. Conversely, if A is one-dimensional Prüfer overring of R with J(A) ≠ 0, then X = {AM ∶ M ∈ Max(A)} is quasicompact by Theorem 5.6, and X is Hausdorff by Proposition 2.4(2). Clearly, A = A(X).  Remark 5.10. As in Corollary 5.8, the bijection in Corollary 5.9 restricts to a bijection between compact sets of DVRs and almost Dedekind overrings A of R with nonzero Jacobson radical. In general it is difficult to determine when an intersection of two one-dimensional Prüfer domains with quotient field F is a Prüfer domain. For example, it is easy to see any Noetherian local UFD A of Krull dimension 2 can be written as an intersection A = A1 ∩ A2 where A1 is a DVR overring and A2 is a PID overring. In this example, A is not a Prüfer domain despite being an intersection of two PIDs. Significantly, the ring A2 here has J(A2 ) = 0. In our context, the topological characterization in Corollary 5.9 shows that the difficulty here is removed if J(Ai ) ≠ 0, a fact we prove in the next corollary. Corollary 5.11. Let A1 , . . . , An be one-dimensional Prüfer domains, each with quotient field F . Let J = J(A1 ) ∩ ⋯ ∩ J(An ), and let A = A1 ∩ ⋯ ∩ An . If J(Ai ) ∩ A ≠ 0 for all i, then A is a one-dimensional Prüfer domain with J(A) = J and quotient field F . If also each Ai is an almost Dedekind domain, then so is A. Proof. For each i = 1, 2, . . . , n, let Xi = {(Ai )M ∶ M ∈ Max(Ai )}. By Corollary 5.9, each Xi is compact. Thus X = X1 ∪ ⋯ ∪ Xn is quasicompact. Since J(Ai ) ∩ A ≠ 0 for all i, we have J(X) = J(A1 ) ∩ ⋯ ∩ J(An ) ≠ 0. Thus Theorem 5.6 implies A = A(X) is a one-dimensional Prüfer domain, and, by Lemma 5.1, J(A) = J(A1 ) ∩ ⋯ ∩ J(An ). If also Ai is an almost Dedekind domain, then each Xi is a quasicompact set of DVRs, so that X is also a quasicompact set of DVRs. In this case, A is an almost Dedekind domain by Corollary 5.8.  Remark 5.12. It is an open question as to whether the intersection A = A1 ∩ A2 of one-dimensional Prüfer domains A1 and A2 with quotient field F and nonzero 22 BRUCE OLBERDING Jacobson radical has quotient field F . If the answer is affirmative, then A is a one-dimensional Prüfer domain by Corollary 5.11. In any case, let D be the prime subring of A, let Si = {1 + d ∶ d ∈ J(Ai )} and let Bi = DSi + J(Ai ). Then B1 and B2 are local domains of Krull dimension one with quotient field F . A necessary condition for A to have quotient field F is that B1 ∩ B2 has quotient field F . In [16, Question 2.1] Gilmer and Heinzer ask whether the intersection of two onedimensional local domains, each with the same quotient field F , has quotient field F? Corollary 5.13. Let X be a nonempty quasicompact set of rank one valuation rings in Zar(F ). If J(X) ≠ 0, then the patch, inverse and Zariski topologies agree on X. Proof. By Theorem 5.6, A = A(X) is a one-dimensional Prüfer domain with J(A) ≠ 0 and quotient field F . By Lemma 5.1, the inverse, patch and Zariski topologies agree on X.  As the last application of this section, we describe schemes in Zar(F ) consisting of rank ≤ 1 valuation rings. Let X be a subspace of Zar(F ), and let X ∗ = X ∪ {F }. Let OX ∗ be the sheaf on X ∗ defined for each nonempty open set U of X ∗ by OX ∗ (U ) = ⋂V ∈U V . (The reason for appending F to X is to guarantee that OX ∗ is a sheaf.) We say that X ∗ is a scheme in Zar(F ) if the locally ringed space (X ∗ , OX ∗ ) is a scheme, and that X ∗ is an affine scheme in Zar(F ) if (X ∗ , OX ∗ ) is an affine scheme. Thus X ∗ is an affine scheme in Zar(F ) if and only if the set of all localizations A(X)P , P a nonzero prime ideal of A(X), is X. The question of whether a subset of Zar(F ) is an affine scheme is closely connected to the question of whether the intersection of valuation rings in the set is a Prüfer domain with quotient field F . For more on this, see [36]. A necessarily condition for X to be an affine scheme in Zar(F ) is that X is quasicompact; similarly, for X to be a scheme, X must be locally quasicompact (i.e., every point has a quasicompact neighborhood). The corollary shows that these conditions are also sufficient for sets X of valuation rings of rank ≤ 1 with J(X) ≠ 0. Corollary 5.14. Let X be a nonempty set of rank one valuation rings in Zar(R) with J(X) ≠ 0. Then X ∗ is an affine scheme in Zar(F ) if and only if X is quasicompact; X ∗ is a scheme in Zar(F ) if and only X is locally quasicompact. Proof. An affine scheme in Zar(F ) is quasicompact since the prime spectrum of a ring is quasicompact. Conversely, if X is quasicompact, then, with A = A(X), Theorem 5.6 implies that X ∗ = {AP ∶ P ∈ Spec(A)}, so that X ∗ is an affine scheme in Zar(F ). Now suppose X is locally quasicompact. Let V ∈ X, and let Z be a quasicompact neighborhood of V in X. Then there is an open subset Y of X of the form Y = COMPACT SETS AND HOLOMORPHY RINGS 23 U (x1 , . . . , xn ) ∩ X, where x1 , . . . , xn ∈ V and Y ⊆ Z. Since Z is quasicompact with J(Z) ≠ 0, Corollary 5.13 implies that the Zariski and inverse topologies agree on Z. Thus Y is a Zariski closed subset of Z. Since Z is quasicompact, Y is also quasicompact, and hence Y is an affine scheme that is open in X. This shows that X is a union of affine open schemes, so that X is a scheme in Zar(F ). Conversely, if X is a scheme in Zar(F ), then X is a union of open sets that are affine schemes in Zar(F ). Thus X is a union of quasicompact open subsets, proving that X is locally quasicompact.  6. Proof of Main Theorem In this section we prove the main theorem of the introduction. In light of Theorem 5.6, what remains to be shown is that the one-dimensional Prüfer domains with nonzero Jacobson radical are Bézout domains. In fact, we prove more generally that any one-dimensional domain with nonzero Jacobson radical has trivial Picard group. Theorem 6.1. If A is a one-dimensional domain with J(A) ≠ 0, then every invertible ideal of A is a principal ideal. Proof. Let I be an invertible ideal of A. We show that I is a principal ideal of A. After multiplying I by a nonzero element of J(A), we can assume I ⊆ J(A). Since A has Krull dimension one and I is invertible, there exist 0 ≠ a, b ∈ I such that I = (a, b)A; see [40, Corollary 4.3], or [21, Theorem 3.1] for a more general result. Let J = J(A), and let X = {M ∈ Max(A) ∶ (aA ∶A b) ⊆/ M } and Y = {M ∈ Max(A) ∶ (aA ∶A b) ⊆ M }. First we show that there is e ∈ A such that e2 − e ∈ J and X = {M ∈ Max(A) ∶ e ∈ M } and Y = {M ∈ Max(A) ∶ 1 − e ∈ M }. Since A/J is a reduced ring of Krull dimension 0, A/J is a von Neumann regular ring, and hence every finitely generated ideal of A/J is generated by an idempotent. In order to apply this observation to the image of (aA ∶A b) in A/J, we claim that (aA ∶A b) is a finitely generated ideal of A. Since I = (a, b)A, we have (aA ∶A b) = (aA ∶F I). Since I is invertible, it follows that (aA ∶A b)I = (aA ∶F I)I = aA. With I −1 = (A ∶F I), we have (aA ∶A b) = aI −1 . Since I −1 is invertible, I −1 is a finitely generated A-submodule of F , and it follows that (aA ∶A b) is a finitely generated ideal of A. Therefore, since A/J is a von Neumann regular ring, there is f ∈ (aA ∶A b) such that f A + J = (aA ∶A b) + J and f 2 − f ∈ J. Set e = 1 − f . Then e2 − e ∈ J, and, since J is contained in every maximal ideal of A, it follows that 24 BRUCE OLBERDING X = {M ∈ Max(A) ∶ e ∈ M } and Y = {M ∈ Max(A) ∶ f ∈ M }. √ √ Next, since ab ∈ J and A/ abA has Krull dimension 0, it follows that J = abA. (Recall we have assumed that I ⊆ J.) Since √ √ ef = e − e2 ∈ J = abA = abJ, there is k > 0 such that ek f k ∈ abJ. Let c = (a − ek )(b − f k ). We claim that I = cA. It suffices to check that this equality holds locally. Let M be a maximal ideal of A. Suppose first that M ∈ X. Then (aA ∶A b) ⊆/ M , so there exists d ∈ A ∖ M such that db ∈ aA. It follows that bAM ⊆ aAM , and hence IAM = aAM . Thus to show that IAM = cAM , it suffices to show that aAM = cAM . If b ∈/ M , then since d ∈/ M we have db ∈/ M . However, with b ∈/ M , the fact that ab ∈ M implies db ∈ aA ⊆ M , a contradiction. Thus b ∈ M . Now, since e ∈ M , we have f = 1 − e ∈/ M and hence (because b ∈ M ) we conclude that b − f k ∈/ M . This implies that cAM = (a − ek )AM . Since ek AM = ek f k AM ⊆ aJAM , there is j ∈ J and h ∈ A ∖ M such that hek = aj. Thus h(a − ek ) = ha − hek = ha − aj = (h − j)a. Since j ∈ M and h ∈/ M , we have h − j /∈ M . From the fact that h(a − ek ) = (h − j)a we conclude that cAM = (a − ek )AM = aAM , which proves the claim that for each M ∈ X, IAM = cAM . Now suppose that M ∈ Y , so that f ∈ M . We show that IAM = cAM in this case also. Since (aA ∶A b) ⊆ M , we have bAM ⊆/ aAM . Since I is invertible, IAM is a principal ideal of AM . The following standard argument shows that this implies that IAM = bAM . Let z ∈ I such that IAM = zAM . Then there exist x, y, s, t ∈ AM such that a = zx, b = zy and z = as + bt. If x is a unit in AM , then bAM ⊆ zAM = aAM , a contradiction. Thus x is not a unit in AM . Since a = zx = (as + bt)x, we have a(1 − sx) = btx, with 1 − sx a unit in AM since x ∈ M AM . Therefore, aAM ⊆ bAM , which proves that IAM = bAM . We have shown that for M ∈ Y , we have IAM = bAM . To complete the proof of the lemma, it suffices to show that bAM = cAM . The proof proceeds as in the case where M ∈ X. Since bAM ⊆/ aAM , it follows that a ∈ M , and hence a − ek ∈/ M . Thus cAM = (b − f k )AM . Also, f k AM = ek f k AM ⊆ bJAM , so that there is h ∈ A ∖ M such that hf k = bj for some j ∈ J. Thus h(b − f k ) = hb − bj = b(h − j), COMPACT SETS AND HOLOMORPHY RINGS 25 with h, h − j units in AM . Therefore, (b − f k )AM = bAM , which shows that cAM = (b − f k )AM = bAM = IAM . This proves that IAM = cAM for all maximal ideals M of Y . Since Max(A) = X ∪Y , we conclude that I = cA.  Corollary 6.2. If A is a one-dimensional Prüfer domain with J(A) ≠ 0, then A is a Bézout domain. Proof. This follows from Theorem 6.1 and the fact that every finitely generated ideal of a Prüfer domain is invertible [14, Theorem 22.1].  A special case of the lemma in which it is assumed in addition that A is an almost Dedekind domain for which every maximal ideal of A has finite sharp degree was proved by Loper and Lucas [26, Theorem 2.9] using different methods. With Corollary 6.2 and the results of the preceding sections, we can now prove the main theorem from the introduction. Theorem 6.3. If X is a quasicompact set of rank one valuation rings in Zar(F ) such that J(X) ≠ 0, then A(X) is a Bézout domain of Krull dimension one with quotient field F . Proof. By Theorem 5.6, A(X) is a one-dimensional Prüfer domain with quotient field F and nonzero Jacobson radical. By Corollary 6.2, A(X) is a Bézout domain, which proves the theorem.  Remark 6.4. In [37] we apply the results of this article to rank one valuation overrings of a two-dimensional Noetherian local domain D with quotient field F . We focus on the divisorial valuation overrings of D, i.e., the DVRs that birationally dominate D and are residually transcendental over D. It is shown, for example, that if n is a positive integer, then the subspace X of Zar(F /D) consisting of all divisorial valuation rings that can be reached through an iterated sequence of at most n normalized quadratic transforms of D is quasicompact. By Corollary 5.8, A(X) is an almost Dedekind domain with nonzero Jacobson radical. Acknowledgment. I thank the referee for helpful comments that improved the presentation of the article and for suggesting the version of Example 5.7(2) that is included here. References [1] E. Becker, The real holomorphy ring and sums of 2nth powers. In Real algebraic geometry and quadratic forms (Rennes, 1981), Lecture Notes in Math. 959, pages 139–181. 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Optimum Synthesis of Mechanism for single- and hybrid-tasks using Differential Evolution F. Peñuñuri,∗ R. Peón-Escalante,† C. Villanueva,‡ and D. Pech-Oy§ arXiv:1102.2017v2 [] 18 Jun 2011 Facultad de Ingenierı́a, Universidad Autónoma de Yucatán, A.P. 150, Cordemex, Mérida, Yucatán, México. The optimal dimensional synthesis for planar mechanisms using differential evolution (DE) is demonstrated. Four examples are included: in the first case, the synthesis of a mechanism for hybrid-tasks, considering path generation, function generation, and motion generation, is carried out. The second and third cases pertain to path generation, with and without prescribed timing. Finally, the synthesis of an Ackerman mechanism is reported. Order defect problem is solved by manipulating individuals instead of penalizing or discretizing the search space for the parameters. A technique that consists in applying a transformation in order to satisfy the Grashof and crank conditions to generate an initial elitist population is introduced. As a result, the evolutionary algorithm increases its efficiency. I. INTRODUCTION Dimensional synthesis of mechanisms comprises the problems of path, function and motion generation. There are three types of methods for this purpose: graphical, analytical, and those involving optimization [1]. Graphical methods offer a quick solution by sacrificing accuracy, and are rarely used since computers can do the same work faster and better. Analytical methods are based on algebraic expressions [1, 2], displacement matrix [3], complex numbers [4], or continuation methods [5] resulting in mechanisms whose error will be zero at the precision points. The problem of motion generation, in the case of a planar four-bar mechanism, can be designed based on the Burmester curve. This is one of the first proposed analytical methods for the dimensional synthesis of mechanisms. In [6] an algorithm for the robust computation of the solution of the fiveposed Burmester problem is introduced. In [7] a Matlab-based graphical user interface to the algorithm of [6] is done. Also, the general equation of the coupler curve of a four-bar linkage has attracted the attention of researchers. For a given set of points on the coupler curve, Blechschmidt and Uicker [8] have used the equation of the coupler curve to synthesize a four-bar linkage by determining the coefficients of the curve. The main disadvantage of the analytical methods lies in the maximum number of points of accuracy that can be set. The mechanisms are restricted to move exactly in a number of points equal to the number of independent parameters that define them [9, 10]. Even though the mechanisms obtained can reach the precision points, they may have other problems, known as design defects, that are not taken into account during the synthesis process, thereby preventing the mechanisms from fulfilling the task for which they were designed [9]. Optimization methods are based on numerical methods and allow a large number of design points tolerating a loss of accuracy. These are formulated in terms of nonlinear programming problems. The optimal solution is found by optimizing an objective function within an iterative procedure. The objective function can be defined as a difference between the generated and the specified movement, known as the structural error [3]. In general, it can be defined as the design error, i.e., the error ∗ † ‡ § [email protected] [email protected] [email protected] [email protected] 2 that arises when we are trying to satisfy a design equation [11] (which could be the Freudenstein equation). An interesting definition for the objective function is presented in [12] where it is defined as a kind of entropy that is maximized. The use of optimization methods is inevitable when the number of positions to be covered during the duty cycle exceeds a certain number (in the case of motion generation synthesis the classical analytical approach is limited to five specified points for a four-bar mechanism). The interest in optimum synthesis of mechanisms is not new. There have been a large number of studies on this topic using a variety of methods. For example, some local search methods have been described in references [13–20]. The main disadvantages of these methods are that the objective function must be differentiable. Also, they are very sensitive to the initial search point. Within the global search methods some of the techniques that have been used are Simulated Annealing (SA) [21], Neural Network [23, 24], Genetic Algorithm (GA) [25–30], Particle Swarm Optimization Technique (PSO) [30], and Differential Evolution (DE) [30–34]. There are works that use a combination of two optimization methods such as SA-Powell’s Method [35], GA-FL [36], Tabu-Gradient [37], Ant Colony Optimization-Gradient (AG) [38], and GA-DE [39]. The use of evolutionary algorithms has been of significant interest in recent years. For instance, Ullah and Kota solved the path generation problem by presenting an objective function based on Fourier descriptors that evaluates only the shape differences between two curves [35]. This function is first minimized using a simulated annealing followed by Powell’s method. The size, orientation and position of the desired curve are addressed at a later stage by determining analogous points on the desired and candidate curves. Similarly, Vasiliu and Yannou [23] synthesized the dimensions of a planar mechanism whose purpose is to generate a trajectory shape by using a neural network. Laribi et al. presented the combined GA-FL method to solve the problem of path generation in mechanism synthesis [36]. The FL-controller monitors the variation of the design variables during the first run of the GA and modifies the initial bounding intervals to restart a second run of the GA. Smaili and Diab (2007) apply AG to the mechanism synthesis problem for single- and hybrid-tasks [38]. Shiakolas introduced a technique called the Geometric Centroid of Precision Points for defining initial bounds for the design variables combining with DE [31]. Acharyya and Mandal carry out the path synthesis of a four-bar linkage using three different methods [30]. They found that the DE with /rand/1/exp method performs better than the two others; one being a binary-coded genetic algorithm (BGA) with multipoint crossover, and the other a PSO with the constriction factor approach. In [39] they used a GA-DE hybrid algorithm to make a path synthesis of a four-bar linkage. A real-valued genetic algorithm, where the crossover operation of GA is replaced by differential vector perturbation, is employed. The DE method is a simple yet powerful algorithm for global optimization [40]. It is not difficult to modify the main operators and try for improvements of the method. In the present work, we use DE to find optimum solutions for the dimensional synthesis problem of four mechanisms. The first three correspond to planar four-bar and the last one to a six-bar mechanism. The paper is organized as follows: In section II we present the classical DE method, which is used throughout this work; section III presents notation and conventions. In Section IV we employ the idea of hybrid-task for the synthesis of mechanisms as was introduced in [38]. The problem of this section presents us with the difficulty of mixing angles with lengths. This difficulty is addressed by introducing a factor that, on the one hand, defines consistently the objective function and on the other hand, allows for proper weighing of the involved errors. This is important in order to fulfill the task of function and motion generation. Moreover, this problem is used to show an easy and effective way to handle the order defect problem. The proposed method avoids entirely both individual penalization and space search discretization. Section V deals with the prescribed timing path generation for 18 points and 10 design variables. We introduce a transformation which constructs an elitist population, in the sense of satisfying the Grashof and crank conditions, avoiding a probabilistic or penalization approach. This problem has been presented by other authors [26, 27, 39]. To avoid some controversies related to the values of the objective function that each of them report, we have written a Fortran 90 program that evaluates the objective function. 3 Start Specify the DE parameters Initialize the population Generate a trial population: Mutation, Crossover, Selection No Termination criteria satisfied? Yes Stop FIG. 1. Flowchart for the DE algorithm. The ideas introduced in Sections IV and V allow us to solve in a single manner the path generation problem without prescribed timing for 18 target points and 27 design variables, which is the problem described in Section VI. In Section VII an Ackerman mechanism is optimized. Finally, we present our conclusions in Section VIII. II. CLASSICAL DE Below, the original version of the method is outlined [41]. 1. The population: Px,g = (xi,g ), i = 1, ...m; g = 0, ...gmax xi;g = (xji;g ), j = 1, ...D; (1) where D, m and gmax represent the dimensionality of x, the number of individuals and the number of generations respectively. In [42] it is mentioned that a good choice for m is 10D. However, to balance the speed and reliability in [43] values from 2D to 40D are suggested. 2. Initialization of population: xji;0 = randj (0, 1) · (bjU − bjL ) + bjL . Vectors bU and bL are the parameter limits and randj (0, 1) is a random number in [0, 1) generated for each parameter. 4 3. Mutation: vi;g = xr0 ;g + F · (xr1 ;g − xr2 ;g ). (2) The main difference between DE and other evolutionary algorithms like GA comes from this mutation operator. xr0 ;g is called the base vector which is perturbed by the difference of other two vectors. r0 , r1 , r2 ∈ {1, 2, ...m}, r1 6= r2 6= r3 6= i . F is a scale factor greater than zero. Even though upper limits for F do not exist, values greater than 1 are rarely chosen in the literature [40–42, 44]. 4. Crossover: A dual recombination of vectors is used to generate the trial vector: ( j if(randj (0, 1) 6 Cr or j = jrand ) vi;g j ui;g = ui;g = j xi;g otherwise. (3) The crossover probability, Cr ∈ [0, 1], is a user-defined value. 5. Selection: The selection is made according to  xi;g+1 = ui;g if f (ui;g ) 6 f (xi;g ) xi;g otherwise (4) The method just described is known as DE/rand/1/bin. There are variants of it. For example, when F is chosen to be a random number, the variant is called dither. In this work we will use the exposed method with the dither variant where F ∈ [0; 1). Fig. 1 shows the flowchart for the DE algorithm. III. MECHANISM SYNTHESIS PROBLEM: NOTATION AND CONVENTIONS The simplicity of a 4-bar mechanism, (easy to manufacture and highly reliable) makes it a very important mechanism with a large number of industrial applications. Its use ranges from simple devices such as windshield-wiping mechanisms and door-closing mechanisms to more complicated ones such as rock crushers, sewing machines, round balers, and suspension systems of automobiles [30]. In this section the notation and conventions used throughout this work are established. The only exception is in Section VII where we will deal with a 6-bar mechanism. A four-bar linkage shown in Fig. 2 consists of four rigid links and four revolute joints. The set of variables that describes the mechanism (the design variable vector) will be put into the vector X whose components will be enclosed within braces. Usually, in the synthesis of a mechanism there are two sets of points (or coordinates), desired and generated points, allocated in the vectors rd and rgen , respectively. A vector error E = rd − rgen is proposed and the objective function is defined as the square of its Euclidean norm. f ob = |E|2 . (5) If quantities are not dimensionally homogeneous, constants with appropriate units must be introduced so that equations have compatible units. In this work, there are quantities with different units, and some constants are chosen so that f ob is dimensionless. For example, in the problem of motion generation, we have to fit angles and coordinates, so the quadratic error will be X  E2 = c(xi;d − xi;gen )2 + c(yi;d − yi;gen )2 + (θi;d − θi;gen )2 , (6) i where xi;d[gen] , yi;d[gen] and θi;d[gen] are the coordinates and angles of the desired [generated] point i. 5 FIG. 2. Four-bar linkage notation. The constant c has a numerical value equal to 1 and is introduced for consistency with units. The objective function is given by X  f ob = fc2 (xi;d − xi;gen )2 + fc2 (yi;d − yi;gen )2 + (θi;d − θi;gen )2 . (7) i The fc constant is introduced for consistency with units but is not necessarily 1. Such a constant can be defined by the user as a weight factor. In this work it is chosen as the inverse of the longest distance between the coordinates. As a matter of illustration, for the points P = {(1, 1), (2, 3), (−5, −1)} we construct the set Uxy = {1, 2, 3, −5, −1} (i.e., the union of the coordinates) and take fc = 1/dr with dr = max(Uxy ) − min(Uxy ). In this case min(Uxy ) = −5, max(Uxy ) = 3 thus fc = 1/8. Notice that this definition is motivated by the curvature concept. For example, in the case of a circle with radius r, we have s/r = θ or ks = θ where k is the curvature, s the arc length subtended by the angle θ. In general, f ob 6= 0 and its minimization process is what generates values for the parameters of a possible mechanism. In the analysis of mechanisms, two conditions are important. They are known as the crank and Grashof conditions (CG): min(r1 , r2 , r3 , r4 ) = crank, 2 min(r1 , r2 , r3 , r4 ) + 2 max(r1 , r2 , r3 , r4 ) < r1 + r2 + r3 + r4 . (8) (9) In our case r2 is the crank, see Fig 2. Whenever we refer to a transformation acting on a vector, |xi is used instead of x. Usually such transformations are carried out by subroutines or functions in Fortran 90 and by functions in C++. For linear transformations, the matrix representation can be used. In this work, all the algorithms for the synthesis of mechanisms were implemented in Fortran 90. The compiler used was ifort and the calculations were made in an intel Core 2 Duo processor with velocity of 2.53 GHz, 4 GB of memory and a bus velocity of 1.07 GHz. IV. HYBRID TASK SYNTHESIS In this section we analyze the problem presented by McGarva [45]. We address the problem from the viewpoint of hybrid tasks as proposed by Smaili and Diab in [38]. The problem has three tasks: function generation, motion generation and path generation. Table I (as presented in [38]) summarizes the variables used in this study. The design variable vector is defined as X = {x0 , y0 , r1 , r2 , r3 , r4 , rcx , rcy , ψ1 , ψ2 , ψ3 , ψ4 , ψ5 , ψ6 , ψ7 , γ}. (10) 6 Desired point, i 1 2 3 4 5 6 7 8 9 10 4.67 2.6 122 * 138 4.38 2.2 137 * 143 4.04 1.67 151 * 147 Function points xid yid ψi + γ θi + γ φi + γ 7.03 5.99 21 * 108 11 6.95 5.45 36 * 110 12 6.77 5.03 50 * 113 13 6.4 4.6 65 * 117 14 5.91 4.03 79 * 121 15 5.43 3.56 93 * 126 16 4.93 2.94 108 * 132 17 Motion function xid yid ψi + γ θi + γ φi + γ 3.76 1.22 N −13 * 18 3.76 1.97 N −7 * 19 3.76 2.78 N −2 * 20 3.76 3.56 N 2 * 21 3.76 4.34 N 7 * 22 3.76 4.91 N 11 * 23 3.76 5.47 N 14 * 24 25 Path point xid yid ψi + γ θi + γ φi + γ 3.8 5.98 266 * * 4.07 6.4 281 * * 4.53 6.75 295 * * 5.07 6.85 309 * * 5.05 6.84 324 * * 5.89 6.83 338 * * 6.41 6.8 353 * * 6.92 6.58 367 * * TABLE I. Hybrid-tasks problem; N: Generated crank angle values, *: Non-prescribed values. In addition to the CG restrictions, we have the following constraints for motion generation: ψmin < ψ j < ψmax ; j = {1, 2, ..., 7} ψ k < ψ k+1 ; k = {1, 2, .., 6}. (11) In this case the objective function consists of three parts: f ob = f obf unc + fg obmot + fg obpath (12) where, in the usual approach of the least square method, the partial objective functions are defined as: X f obf unc = (θi;d − θi;gen )2f unc (13) i fg obmot X  = fc2 (xi;d − xi;gen )2mot + fc2 (yi;d − yi;gen )2mot + (θi;d − θi;gen )2mot (14) i fg obpath = X  fc2 (xi;d − xi;gen )2path + fc2 (yi;d − yi;gen )2path (15) i The evaluation of the weight factor fc is explained in Sec. III. In this case we have max(Uxy ) = 7.03, min(Uxy ) = 1.22 thus fc2 = 0.02 with units of length−2 . The following values were tested for Cr: 0.05, 0.1, ..., 0.9. It turns out that 0.3 gives the best results. The number of individuals and generations were m = 250, gmax = 15 000, respectively. The evaluation of f ob resulted in a value of 6.99 × 10−3 with the design variables shown in Table II. x0 y0 r1 r2 r3 r4 rcx rcy -8.0339 1.07673 13.2425 1.96639 7.71759 7.57298 13.4593 3.13037 ψ1 ψ2 ψ3 ψ4 ψ5 ψ6 ψ7 γ 3.5639 3.83348 4.05641 4.22857 4.48498 4.71726 4.92507 5.83047 TABLE II. Parameter values of an optimal mechanism. Hybrid-tasks synthesis. For the searching space we have used the limits: xmin = {−15, −15, 0, 0, 0, 0, 0, 0, 3, 3, 3, 3, 3, 3, 3, 0} xmax = {15, 15, 15, 15, 15, 15, 15, 15, 5.03, 5.03, 5.03, 5.03, 5.03, 5.03, 5.03, 2π}. 7 A. Constraints Management In this case the CG conditions do not play any active role. We can just verify that they are met after the minimum of f ob is obtained. Concerning the requirement of the constraints of Eq. (11), previous methods are based on the discretization of the search space for ψ angles [38], h i j j ψ j ∈ ψmin , ψmax . (16) The best fitting angle is selected from this range. In our case this discretization is not applied, and individuals xi,g are chosen so as to comply with the restriction of Eq. (11). To this end, a random vector of angles within desired limits is generated, and its coordinates written in ascending order. This idea has been used in [30, 39]. Thus the method here is not exactly a classic DE because the evolution of individuals is manipulated. However, it is clear that the results will be the same. The only thing it does is to accelerate the evolutionary process. Symbolically, if |ψi represents a vector of random d represents a transformation that puts them in ascending order, then numbers and sort h ij d ψ j = sort|ψi . (17) There are several ways to implement Eq. (17). In particular, it can be done in the crossover part. ( j ṽi;g if(randj (0, 1) 6 Cr or j = jrand ) j ui;g = ui;g = (18) j x̃i;g otherwise. where h ij d r̃j = sort|ri . (19) d will act only on those components that we choose to order. The transformation sort FIG. 3. Optimal mechanism and the corresponding coupler curve. Hybrid-tasks synthesis. For the ordering of the ψ angles, we have used the heap sort method [46–48] as it is efficient enough and easy to implement. Penalizing angles ψ j is not very efficient because the probability of having a set of size n randomly ordered is low if n is large. For example, the probability to throw in seven random numbers between 0 and 1 (or any other continuum interval) in an ordered way is 1/7!, which is about 2 × 10−4 . We thus end up with a method without individuals to evolve unless the number of individuals in the initial 8 population were extremely large, which would lead to a grossly inefficient method. Proceeding as [38] is a brilliant possibility and the results so obtained are very good. However, discretizing the searching space could prevent us from locating the minimum. Fig. 3 shows the mechanism obtained. In B a program in M athematica R that makes an animation of the mechanism is shown step by step. V. A CLASSICAL COMPARISON: PATH GENERATION FOR 18 TARGET POINTS AND 10 DESIGN VARIABLES Recently, in [39] a hybrid method (GA-DE) was proposed that can synthesize a four-bar mechanism and the problem of prescribed timing path generation for 18 points, (previously introduced by [26] and [27]) is addressed. We will optimize this problem by using a DE algorithm. The objective function value is lower than the reported values of previous references. It is worth mentioning that the values for the links of the mechanism generated are of the same order of magnitude of the generation path dimensions. FIG. 4. Optimal mechanism and the corresponding coupler curve. Prescribed timing path generation. A. The problem The target points are: xd = {0.5, 0.4, 0.3, 0.2, 0.1, 0.05, 0.02, 0, 0, 0.03, 0.1, 0.15, 0.2, 0.3, 0.4, 0.5, 0.6, 0.6} yd = {1.1, 1.1, 1.1, 1, 0.9, 0.75, 0.6, 0.5, 0.4, 0.3, 0.25, 0.2, 0.3, 0.4, 0.5, 0.7, 0.9, 1} (20) (21) The design variable vector is X = {x0 , y0 , r1 , r2 , r3 , r4 , rcx , rcy , γ, ψ 0 } (22) and the precribed timing is defined by ψk = ψ0 + π (k − 1); k = {1, 2, ..., 18}. 9 (23) 9 Figure 2 shows the design variables. This problem has been discussed in Refs. [26, 27, 39]. They do not show the explicit form of f ob, and there is a controversy concerning the numerical values of the objective function. We show in Table III the values that according to [39] the other two references should have obtained. Wen-Yi Lin [39] −2 f ob = 1.08613 × 10 Kunjur and Krishnamurty [26] Cabrera etal [27] f ob = 1.09034 × 10−2 f ob = 3.48391 × 10−2 TABLE III. Values for the objective function reported by [39]. Here we get the following values of f ob for the design variable vectors that they report: [39], f ob = 1.0306 × 10−2 ; [26], f ob = 1.0214 × 10−2 ; [27], f ob = 3.3748 × 10−2 . They are slightly different from the values of [39], perhaps because of rounding errors. With the purpose of avoiding any misunderstanding, in A we show a Fortran 90 program that evaluates f ob. Table IV shows the values for the design variable vector for which the objective function is 9.088 × 10−3 . The values 0.1, 0.2, . . . , 0.9 were tested for Cr. It turns out that 0.3 gives the best results. Figure 4 shows the optimum mechanism and its path. x0 y0 r1 r2 r3 r4 rcx rcy γ ψ0 0.27892 0.11673 1.08913 0.42259 0.96444 0.58781 0.39137 0.42950 0.32195 0.86323 TABLE IV. Parameter values of an optimal mechanism with f ob = 9.088 × 10−3 . In order to obtain the last result for f ob, we proceed in two steps. First, we choose parameter values inside the interval [−1.5, 1.5] for x0 and y0 . For the remaining parameters we choose values in [0, 1.5], and we evaluate f ob over and over until we find a design variable vector for which f ob 6 5 × 10−2 . Second, from the obtained parameters, the searching space is reduced to vxmin = {0.2, 0.1, 0.8, 0.3, 0.7, 0.4, 0.2, 0.3, 0.1, 0.7} vxmax = {0.3, 0.3, 1.1, 1.1, 1.1, 1.1, 1.1, 1.1, 1.1, 1.1}. (24) (25) The value of f ob = 9.088 × 10−3 was obtained for 200 individuals and 11 817 generations. We could reach smaller values of f ob if the generation number and/or individual number were increased, but improvements are not considerable. For example, for 200 individuals and 30 000 generations we obtain f ob = 9.06 × 10−3 . Moreover, by making a third refinement of the searching space, we obtain f ob = 9.03 × 10−3 for the design variables shown in Table V. x0 y0 r1 r2 r3 r4 rcx rcy γ ψ0 0.26439 0.16956 1.04028 0.42446 0.89397 0.60308 0.36129 0.38864 0.26873 0.90493 TABLE V. Parameter values of an optimal mechanism with f ob = 9.03 × 10−3 . B. On steps 1 and 2 In this work we subdivide the optimization task in two steps. In the first step we use an elitist population in the sense of choosing only those individuals that satisfy the CG condition. To this end we construct a transformation that takes an individual that does not satisfy the CG condition and turns it into one that does. Then, in the second stage (with the result for the possible mechanism obtained in this first stage) we refine the searching space, remove the CG condition and re-run the optimization program. The process terminates when some criteria have been met and the individual satisfies the CG condition. 10 C. On the construction of the elitist population In order to construct individuals satisfying the CG conditions we could proceed in a random way, but this would be inefficient. In this work we proceed as follows: Assuming the links of the four-bar mechanism belong to the interval (0,1), four random numbers are generated (the links) and they are sorted in ascending order. At this point, proceeding randomly would not be a bad choice since the probability of satisfying the CG condition for the sorted list is 0.5. However a better choice will be to construct a transformation T̂ that makes the CG condition fulfill the ascending order list, |xi. There are many possible forms for the transformation T̂ . We define T̂ = F̂ R̂, where R̂ is defined as the transformation that inverts the components of a vector and F̂ a reflection plus a translation. Symbolically, R̂|x1 , x2 , x3 , x4 i = |x4 , x3 , x2 , x1 i, F̂ |xi = −|xi + |1i, |1i = |1, 1, 1, 1i . (26) (27) (28) If the upper limit for the links is |Li, we replace |1i by |Li. Notice that R̂ is a linear transformation, whereas F̂ is not. Once the vector that satisfies the Grashof condition has been constructed, the crank is taken as the lesser of the elements; thus the conditions CG will be satisfied. For example, suppose that we have the four numbers xr = {0.38, 0.98, 0.25, 0.19} which do not satisfy the CG condition. After sorting them we have |xi = |0.19, 0.25, 0.38, 0.98i and R̂|xi = |0.98, 0.38, 0.25, 0.19i, F̂ R̂|xi = | − 0.98, −0.38, −0.25, −0.19i + |1, 1, 1, 1i, F̂ R̂|xi = |0.02, 0.62, 0.75, 0.81i. By choosing the crank as 0.02, we can see that xg = {0.02, 0.62, 0.75, 0.81} satisfies the CG conditions since min(xg) = 0.02, max(xg) = 0.81 and 0.02 + 0.81 < 0.62 + 0.75. In general, suppose we have four positive numbers less or equal than 1 that are sorted in ascending order, but that do not satisfy the CG conditions. Let |xi = |x1 , x2 , x3 , x4 i be the vector containing such numbers. We have x1 + x4 > x2 + x3 since the numbers are sorted and do not comply Eq. (9). Clearly −x1 − x4 < −x2 − x3 and (1 − x1 ) + (1 − x4 ) < (1 − x2 ) + (1 − x3 ). For these four constructed numbers, the minimum is (1 − x4 ) and the maximum is (1 − x1 ) so the CG conditions are satisfied if we chose the crank as (1 − x4 ). VI. PATH GENERATION WITHOUT PRESCRIBED TIMING FOR 18 TARGET POINTS AND 27 DESIGN VARIABLES It is interesting to synthesize the above mechanism without the prescribed timing Eq. (23). Finding a minimum for the objective function is now more difficult. We have 27 design variables and the order defect problem appears hard to solve. Let X ={x0 , y0 , r1 , r2 , r3 , r4 , rcx , rcy , γ, ψ 0 , ψ 1 , ψ 2 , ψ 3 , ψ 4 , ψ 5 , ψ 6 , ψ 7 , ψ 8 , ψ 9 , ψ 10 , ψ 11 , ψ 12 , ψ 13 , ψ 14 , ψ 15 , ψ 16 , ψ 17 } (29) be the design variable vector. Besides the CG conditions we also have the requirement ψ k < ψ k+1 ; k = {0, 1, .., 16} which prevents the order defect. (30) 11 The use of penalization for the restriction Eq. (30) is not effective. Practically all the individuals would be penalized as the probability of finding one that would not is 1/18! – a very small probability. If we discretize the searching space for angles then there is no guarantee that the minimum will lie in the generated intervals. However, if we adopt the approach stated in subsection IV A, the problem is easily solved and in a consistent manner. The time used for the algorithm was 110 seconds and this was the longest time for all the programs run in this study. The running times for the other cases, were between 30 and 80 seconds. The value of the objective function was f ob = 3.69 × 10−3 for the design variable vector whose components are shown in Table VI. x0 y0 r1 r2 r3 r4 rcx rcy γ 0.22922 -0.63525 2.27468 0.44667 2.18422 0.72409 1.02937 0.82440 0.58183 ψ0 ψ1 ψ2 ψ3 ψ4 ψ5 ψ6 ψ7 ψ8 0.78140 1.09985 1.34998 1.68045 2.00009 2.35036 2.70304 2.95102 3.22683 ψ9 ψ 10 ψ 11 ψ 12 ψ 13 ψ 14 ψ 15 ψ 16 ψ 17 3.58801 4.11376 4.35829 4.70801 5.07939 5.35914 5.76271 6.21586 6.49216 TABLE VI. Parameter values of an optimal mechanism. Path generation without prescribed timing. The searching interval for the angles ψ was 0 < ψ j < 2π; j = {0, 1, ..., 17}. (31) It is well known that DE can yield individuals that do not belong to the searching interval. This is the case for the last angle. However, since there is no order defect the values of table VI are an acceptable solution for the problem. Once again the result is obtained in two steps. First, we choose an elitist initial population that satisfies the CG conditions. Then, the CG condition is removed and the searching space is restricted according to the solution obtained in the first step. It is worthwhile to mention that we tried to optimize the f ob function using the DE method without the transformations of sections IV A and V C but the method was not capable of finding the minimum. VII. ACKERMAN STEERING LINKAGE SYNTHESIS In this section DE is used for the synthesis of an Ackerman steering. For the deduction of the equations used and a detailed treatment of the problem see [49]. It is known that when a vehicle is moving very slowly there is a kinematic condition between the inner and outer wheel that allows it to turn slip-free. The condition is called the Ackerman condition and is written as follows: w cot δo − cot δi = , (32) l where w and l represent the width and length of the vehicle, δo and δi are the rotation angles of the wheels (Figure 5). In general it is desirable for a mechanism to satisfy the Ackerman condition. Unfortunately, there is no four-bar mechanism that can fulfill the Ackerman condition perfectly. However, it is possible to synthesize a six-bar mechanism to work closely to the Ackerman condition and be exact at a few points. A six-bar Watt’s mechanism can be used to design the vehicle steering. The sizes In this case are w = 1 m, l = 1.8 m and the minimum radius R = 2.5 m. The position of the center of mas with respect to the rear axle is a = 0.45 m. We have p R = a2 + l2 cot2 δM , (33) 12 Outer whell Inner whell Center of rotation FIG. 5. Vehicle diagram. FIG. 6. Six-bar Watt’s mechanism. with δM = cot δo + cot δi 2 therefore δM = 37.2731◦ , R1 = l cos δM and consequently R1 = 2.36514 m. From trigonometry we have     l l δi = arctan ; δ = arctan o R1 − w2 R1 + w2 (34) (35) so we obtain that δi and δ0 must lie in the ranges −32.1387◦ 6 δi 6 43.9818◦ and −43.9818◦ 6 δo 6 32.1387◦ in order to achieve the desired turning radius. Unlike previous examples where the number of points is finite, in this case it is possible to use an arbitrary number of desired angles. Therefore, it is convenient to define the objective function as f ob = |E|2 , n (36) where E is the vector containing the n differences between δ2 and δack . The angle δack is the steering angle δo from Eq. (32). 13 h x ξ 0.298192 -0.472091 0.219837 TABLE VII. Parameter values for the multi-link Ackerman steering. 0.5 0.4 0.3 0.2 0.1 0.5 -0.5 -0.1 FIG. 7. Optimal mechanism. Ackerman steering linkage synthesis. The design variable vector is X = {h, x, ξ} . (37) Fig. 6 shows the mechanism. With a search space of 0.1 6 h 6 0.45, −0.5 6 x 6 0.2, 13◦ 6 ξ 6 30◦ , and a working range (−35◦ , 45◦ ) with steps of 0.1◦ for δ1 we obtain the values shown in Table VII for the design variables. The objective function is found to be f ob = 7.6 × 10−5 . The obtained mechanism is illustrated in Fig. 7. Table VIII shows the values of the desired angles and generated angles. VIII. CONCLUSIONS Dimensional synthesis of mechanisms is a subject of great relevance in the field of mechanical design. Among the great variety of optimization methods available, those that employ evolutionary algorithms have seen an increase in use due to the excellent results that they allow. In this work we have presented a methodology that uses differential evolution to solve the dimensional synthesis problem of four mechanisms. With the use of a heuristic deduction, we have determined a weight factor that allows us to solve the hybrid-tasks problem in an efficient manner. Two transformations were implemented in the differential evolution algorithm. The first one deals with the order defect problem and was coded in the crossover part of the differential evolution algorithm. With this transformation, the penalization approach and the use of big populations are avoided. In addition, the chance of not finding the minimum of the objective function has disappeared as the need of discretization of the search space is also avoided. The second transformation constructs elitist populations in the sense that their individuals satisfy the Grashof and crank conditions. Therefore, a random generation and/or a penalization procedure are avoided, which makes this method more efficient. Something that deserves mention is the amazing speed of convergence of the differential evolution method which for generations as large as 80 000, the total CPU time was less than two minutes in a single processor. 14 δ1 δack ◦ −30 −20◦ −10◦ 0◦ 10◦ 20◦ 30◦ 40◦ δ2 ◦ −40.471 −24.567◦ −11.069◦ 0◦ 9.117◦ 16.822◦ 23.571◦ 29.720◦ −40.254◦ −23.700◦ −10.810◦ 0◦ 9.303◦ 17.332◦ 24.174◦ 29.869◦ TABLE VIII. Desired (δack ) and generated (δ2 ) angles. ACKNOWLEDGMENTS We want to thank F. Larios for proofreading the manuscript. We also thank PROMEP and Conacyt for support. Appendix A: Fortran 90 objective function (f ob). Path generation for 18 target points and 10 design variables What follows is the code for the f ob function in Fortran 90. Double precision function fob(x0,y0,r1,r2,r3,r4,rcx,rcy,gamma,psi0) Implicit None Integer, Parameter:: Np=18 Double precision, Parameter :: Pi=3.14159265358979d0 Double precision :: x0,y0,r1,r2,r3,r4,rcx,rcy,gamma,psi0,L1,L2,L3,xd(Np), & yd(Np),psi(Np),KA(Np),KB(Np),KC(Np),theta(Np),px(Np),py(Np),Ex(Np),Ey(Np),& Ex2,Ey2 Integer :: k xd=(/0.5d0, 0.4d0, 0.3d0, 0.2d0, 0.1d0, 0.05d0, 0.02d0, 0d0, 0d0,0.03d0,& 0.1d0, 0.15d0, 0.2d0, 0.3d0, 0.4d0, 0.5d0, 0.6d0, 0.6d0/) yd = (/1.1d0, 1.1d0, 1.1d0, 1d0, 0.9d0, 0.75d0, 0.6d0,0.5d0,0.4d0,0.3d0,& 0.25d0, 0.2d0, 0.3d0, 0.4d0, 0.5d0, 0.7d0, 0.9d0, 1d0/) L3=(r4**2 - r1**2 - r2**2 - r3**2)/(2d0*r2*r3) L2=r1/r3 L1=r1/r2 Do, k=1,Np psi(k) = psi0 + (k-1)*Pi/9d0 Enddo KA = Dcos(psi) - L1 + L2*Dcos(psi) + L3 KB = -2d0*Dsin(psi) KC = L1 + (L2 - 1)*Dcos(psi) + L3 theta = 2d0*Datan2(-KB - Dsqrt(KB**2-4d0*KA*KC),2d0*KA) px = x0 + Dcos(gamma)*(r2*Dcos(psi) + rcx*Dcos(theta) - rcy*Dsin(theta)) - & 15 Dsin(gamma)*(r2*Dsin(psi) + rcx*Dsin(theta) + rcy*Dcos(theta)) py = y0 + Dsin(gamma)*(r2*Dcos(psi) + rcx*Dcos(theta) - rcy*Dsin(theta)) + & Dcos(gamma)*(r2*Dsin(psi) + rcx*Dsin(theta) + rcy*Dcos(theta)) Ex = xd-px Ey = yd-py Ex2 = Dot_Product(Ex,Ex) Ey2 = Dot_Product(Ey,Ey) fob = Ex2 + Ey2 Return End Appendix B: Mathematica R hybrid task animation nparam = {-8.0339,1.07673,13.2425,1.96639,7.71759,7.57298,13.4593,3.13037, 3.5639,3.83348,4.05641,4.22857,4.48498,4.71726,4.92507,5.83047}; vparam = {x0,y0,r1,r2,r3,r4,rcx,rcy,psi1,psi2,psi3,psi4,psi5,psi6,psi7,gamma}; supersolanima = Thread[Rule[vparam, nparam]] r0 = {x0, y0} /. supersolanima; xs = {7.03,6.95,6.77,6.4,5.91,5.43,4.93,4.67,4.38,4.04,3.76,3.76,3.76,3.76, 3.76,3.76,3.76,3.8,4.07,4.53,5.07,5.05,5.89,6.41,6.92}; ys = {5.99,5.45,5.03,4.6,4.03,3.56,2.94,2.6,2.2,1.657,1.22,1.97,2.78,3.56, 4.34,4.91,5.47,5.98,6.4,6.75,6.85,6.84,6.83,6.8,6.58}; DatT = Thread[{xs, ys}]; Dat = Take[DatT, {11, 25}]; L3 = (r4^2 - r1^2 - r2^2 - r3^2)/(2 r2 r3); L2 = r1/r3; L1 = r1/r2; KA = Cos[psi] - L1 + L2 Cos[psi] + L3; KB = -2 Sin[psi]; KC = L1 + (L2 - 1) Cos[psi] + L3; theta[psi_] = 2 ArcTan[(-KB - Sqrt[KB^2 - 4 KA KC])/ (2 KA)] /.supersolanima; Px[psi_] = (x0 + Cos[gamma] (r2 Cos[psi] + rcx Cos[theta[psi]] rcy Sin[theta[psi]]) - Sin[gamma] (r2 Sin[psi] + rcx Sin[theta[psi]] + rcy Cos[theta[psi]])) /.supersolanima; Py[psi_] = (y0 + Sin[gamma] (r2 Cos[psi] + rcx Cos[theta[psi]] rcy Sin[theta[psi]]) + Cos[gamma] (r2 Sin[psi] + rcx Sin[theta[psi]] + rcy Cos[theta[psi]])) /.supersolanima; 16 PrPl[psi_] = Thread[{Px[psi], Py[psi]}]; B[psi_] = r0 + {r2 Cos[psi + gamma], r2 Sin[psi + gamma]} /.supersolanima; Cc[psi_] = B[psi] + {r3 Cos[theta[psi] + gamma], r3 Sin[theta[psi] + gamma]} /.supersolanima; CoorD = (r0 + r1 {Cos[gamma], Sin[gamma]}) /. supersolanima; gDat = ListPlot[Dat]; linkb[psi_] linkc[psi_] linkd[psi_] linke[psi_] linkf[psi_] := := := := := Graphics[{Thick, Graphics[{Thick, Graphics[{Thick, Graphics[{Thick, Graphics[{Thick, Line[{B[psi], r0}]}]; Line[{Cc[psi], B[psi]}]}]; Line[{Cc[psi], CoorD}]}]; Line[{PrPl[psi], B[psi]}]}]; Line[{PrPl[psi], Cc[psi]}]}]; linka = Graphics[{Thickness[.01], EdgeForm[Thick],RGBColor[0.75, 0.75, 0.75], Polygon[{r0, {r0[[1]], CoorD[[2]]}, CoorD}]},PlotRange -> {{-10, 10}, {-4.8, 8}}]; gr = ListPlot[{Table[PrPl[psi], {psi, 0, 2 Pi, Pi/50}], Dat},Joined -> {True, False}, PlotStyle ->{{PointSize[Medium],AbsoluteThickness[1.1]}}, PlotRange ->{{-10, 10}, {-5.2, 7.5}}, Frame -> True, Axes -> False, FrameLabel -> {"X", "Y"}, AspectRatio -> Automatic]; Animate[Show[{linka, gr, linkb[psi], linkc[psi], linkd[psi], linke[psi], linkf[psi], gDat}, Axes -> True, AspectRatio -> Automatic], {psi, 0, 2 Pi}] [1] R. 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Graphical-model Based Multiple Testing under Dependence, with Applications to Genome-wide Association Studies Jie Liu Computer Sciences, UW-Madison Chunming Zhang Statistics, UW-Madison Catherine McCarty Essentia Institute of Rural Health Peggy Peissig Marshfield Clinic Research Foundation Elizabeth Burnside Radiology, UW-Madison David Page BMI & CS, UW-Madison Abstract Large-scale multiple testing tasks often exhibit dependence, and leveraging the dependence between individual tests is still one challenging and important problem in statistics. With recent advances in graphical models, it is feasible to use them to perform multiple testing under dependence. We propose a multiple testing procedure which is based on a Markov-random-field-coupled mixture model. The ground truth of hypotheses is represented by a latent binary Markov random field, and the observed test statistics appear as the coupled mixture variables. The parameters in our model can be automatically learned by a novel EM algorithm. We use an MCMC algorithm to infer the posterior probability that each hypothesis is null (termed local index of significance), and the false discovery rate can be controlled accordingly. Simulations show that the numerical performance of multiple testing can be improved substantially by using our procedure. We apply the procedure to a real-world genome-wide association study on breast cancer, and we identify several SNPs with strong association evidence. 1 Introduction Observations from large-scale multiple testing problems often exhibit dependence. For instance, in genome-wide association studies, researchers collect hundreds of thousands of highly correlated genetic markers (single-nucleotide polymorphisms, or SNPs) with the purpose of identifying the subset of markers associated with a heritable disease or trait. In functional magnetic resonance imaging studies of the brain, thousands of spatially correlated voxels are col- lected while subjects are performing certain tasks, with the purpose of detecting the relevant voxels. The most popular family of large-scale multiple testing procedures is the false discovery rate analysis, such as the p-value thresholding procedures (Benjamini & Hochberg, 1995, 2000; Genovese & Wasserman, 2004), the local false discovery rate procedure (Efron et al., 2001), and the positive false discovery rate procedure (Storey, 2002, 2003). However, all these classical multiple testing procedures ignore the correlation structure among the individual factors, and the question is whether we can reduce the false non-discovery rate by leveraging the dependence, while still controlling the false discovery rate in multiple testing. Graphical models provide an elegant way of representing dependence. With recent advances in graphical models, especially more efficient algorithms for inference and parameter learning, it is feasible to use these models to leverage the dependence between individual tests in multiple testing problems. One influential paper (Sun & Cai, 2009) in the statistics community uses a hidden Markov model to represent the dependence structure, and has shown its optimality under certain conditions and its strong empirical performance. It is the first graphical model (and the only one so far) used in multiple testing problems. However, their procedure can only deal with a sequential dependence structure, and the dependence parameters are homogenous. In this paper, we propose a multiple testing procedure based on a Markov-random-fieldcoupled mixture model which allows arbitrary dependence structures and heterogeneous dependence parameters. This extension requires more sophisticated algorithms for parameter learning and inference. For parameter learning, we design an EM algorithm with MCMC in the E-step and persistent contrastive divergence algorithm (Tieleman, 2008) in the M-step. We use the MCMC algorithm to infer the posterior probability that each hypothesis is null (termed local index of significance or LIS). Finally, the false discovery rate can be controlled by thresholding the LIS. Section 2 introduces related work and our procedure. Sections 3 and 4 evaluate our procedure on a variety of simulations, and the empirical results show that the numerical performance can be improved substantially by using our procedure. In Section 5, we apply the procedure to a real-world genome-wide association study (GWAS) on breast cancer, and we identify several SNPs with strong association evidence. We finally conclude in Section 6. 0 ߰ 1 0  ij 1  ij … 1 1  ik  ik 2.1 Method … 1 0 0  ik 1  ik ߰ 2 ߰ 1 1  ij  ij 1 0 0  jk 1   jk 1 1   jk  jk … Terminology and Previous Work Table 1: Classification of tested hypotheses Null Non-null Total Not rejected N00 N01 S Rejected N10 N11 R Total m0 m1 m Suppose that we carry out m tests whose results can be categorized as in Table 1. False discovery rate (FDR), defined as E(N10 /R|R > 0)P (R > 0), depicts the expected proportion of incorrectly rejected null hypotheses (Benjamini & Hochberg, 1995). False non-discovery rate (FNR), defined as E(N01 /S|S > 0)P (S > 0), depicts the expected proportion of false non-rejections in those tests whose null hypotheses are not rejected (Genovese & Wasserman, 2002). An FDR procedure is valid if it controls FDR at a nominal level, and optimal if it has the smallest FNR among all the valid FDR procedures (Sun & Cai, 2009). The effects of correlation on multiple testing have been discussed, under different assumptions, with a focus on the validity issue (Benjamini & Yekutieli, 2001; Finner & Roters, 2002; Owen, 2005; Sarkar, 2006; Efron, 2007; Farcomeni, 2007; Romano et al., 2008; Wu, 2008; Blanchard & Roquain, 2009). The efficiency issue has also been investigated (Yekutieli & Benjamini, 1999; Genovese et al., 2006; Benjamini & Heller, 2007; Zhang et al., 2011), indicating FNR could be decreased by considering dependence in multiple testing. Several approaches have been proposed, such as dependence kernels (Leek & Storey, 2008), factor models (Friguet et al., 2009) and principal factor approximation (Fan et al., 2012). Sun & Cai (2009) explicitly use a hidden Markov model (HMM) to represent the dependence structure and analyze the optimality under the compound decision framework (Sun & Cai, 2007). However, their procedure can only deal with sequential dependence, and it uses only a single dependence parameter throughout. In this paper, we replace HMM with a Markov-random-field-coupled mixture model, which allows richer and more flexible dependence structures. The Markov-random-field-coupled mixture models are Figure 1: The MRF-coupled mixture model for three dependent hypotheses Hi , Hj and Hk with observed test statistics (xi , xj and xk ) and latent ground truth (θi ,θj and θk ). The dependence is captured by potential functions parameterized by φij ,φjk and φik , and coupled mixtures are parameterized by ψ. related to the hidden Markov random field models used in many image segmentation problems (Zhang et al., 2001; Celeux et al., 2003; Chatzis & Varvarigou, 2008). 2.2 Our Multiple Testing Procedure Let x = (x1 , ..., xm ) be a vector of test statistics from a set of hypotheses (H1 , ..., Hm ). The ground truth of these hypotheses is denoted by a latent Bernoulli vector θ = (θ1 , ..., θm ) ∈ {0, 1}m , with θi = 0 denoting that the hypothesis Hi is null and θi = 1 denoting that the hypothesis Hi is non-null. The dependence among these hypotheses is represented as a binary Markov random field (MRF) on θ. The structure of the MRF can be described by an undirected graph G(V, E) with the node set V and the edge set E. The dependence between Hi and Hj is denoted by an edge connecting nodei and nodej in E, and the strength of dependence is parameterized by the potential function on the edge. Suppose that the probability density function of the test statistic xi given θi = 0 is f0 , and the density of xi given θi = 1 is f1 . Then, x is an MRFcoupled mixture. The mixture model is parameterized by a parameter set ϑ = (φ, ψ), where φ parameterizes the binary MRF and ψ parameterizes f0 and f1 . For example, if f0 is standard normal N (0, 1) and f1 is noncentered normal N (µ, 1), then ψ only contains parameter µ. Figure 1 shows the MRF-coupled mixture model for three dependent hypotheses Hi , Hj and Hk . In our MRF-coupled mixture model, x is observable, and θ is hidden. With the parameter set ϑ = (φ, ψ), the joint probability density over x and θ is P (x, θ|φ, ψ) = P (θ; φ) Ym i=1 P (xi |θi ; ψ). (1) Define the marginal probability that Hi is null given all the observed statistics x under the parameters in ϑ, Pϑ (θi = 0|x), to be the local index of significance (LIS) for Hi (Sun & Cai, 2009). If we can accurately calculate the posterior marginal probabilities of θ (or LIS), then we can use a step-up procedure to control FDR at the nominal level α as follows (Sun & Cai, 2009). We first sort LIS from the smallest value to the largest value. Suppose LIS(1) , LIS(2) , ..., and LIS(m) are the ordered LIS, and the corresponding hypotheses are H(1) , H(2) ,..., and H(m) . Let   1 Xi LIS(j) ≤ α . k = max i : j=1 i (2) Then we reject H(i) for i = 1, ..., k. Therefore, the key inferential problem that we need to solve is that of computing the posterior marginal distribution of the hidden variables θi given the test statistics x, namely Pϑ (θi = 0|x), for i = 1, ..., m. It is a typical inference problem if the parameters in ϑ are known. Section 2.3 provides possible inference algorithms for calculating Pϑ (θi = 0|x) for given ϑ. However, ϑ is usually unknown in real-world applications, and we need to estimate it. Section 2.4 provides a novel EM algorithm for parameter learning in our MRF-coupled mixture model. 2.3 Posterior Inference Now we are interested in calculating Pϑ (θi = 0|x) for a given parameter set ϑ. One popular family of inference algorithms is the sum-product family (Kschischang et al., 2001), also known as belief propagation (Yedidia et al., 2000). For loop-free graphs, belief propagation algorithms provide exact inference results with a computational cost linear in the number of variables. In our MRF-coupled mixture model, the structure of the latent MRF is described by a graph G(V, E). When G is chain structured, the instantiation of belief propagation is the forward-backward algorithm (Baum et al., 1970). When G is tree structured, the instantiation of belief propagation is the upward-downward algorithm (Crouse et al., 1998). For graphical models with cycles, loopy belief propagation (Murphy et al., 1999; Weiss, 2000) and the tree-reweighted algorithm (Wainwright et al., 2003a) can be used for approximate inference. Other inference algorithms for graphical models include junction trees (Lauritzen & Spiegelhalter, 1988), sampling methods (Gelfand & Smith, 1990), and variational methods (Jordan et al., 1999). Recent papers (Schraudolph & Kamenetsky, 2009; Schraudolph, 2010) discuss exact inference algorithms on binary Markov random fields which allow loops. In our simulations, we use belief propaga- tion when the graph G has no loops. When G has loops (e.g. in the simulations on genetic data and the real-world application), we use a Markov chain Monte Carlo (MCMC) algorithm to perform inference for Pϑ (θi = 0|x). 2.4 Parameters and Parameter Learning In our procedure, the dependence among these hypotheses is represented by a graphical model on the latent vector θ parameterized by φ, and observed test statistics x are represented by the coupled mixture parameterized by ψ. In Sun and Cai’s work on HMMs, φ is the transition parameter and ψ is the emission parameter. One implicit assumption in their work is that the transition parameter and the emission parameter stay the same for i(i = 1, ..., m). Our extension to MRFs also allows us to untie these parameters. In the second set of basic simulations in Section 3, we make φ and ψ heterogeneous and investigate how this affects the numerical performance. In the simulations on genetic data in Section 4 and the real-world GWAS application in Section 5, we have different parameters for SNP pairs with different levels of correlation. In our model, learning (φ, ψ) is difficult for two reasons. First, learning parameters is difficult by nature in undirected graphical models due to the global normalization constant (Wainwright et al., 2003b; Welling & Sutton, 2005). State-of-the-art MRF parameter learning methods include MCMC-MLE (Geyer, 1991), contrastive divergence (Hinton, 2002) and variational methods (Ganapathi et al., 2008). Several new sampling methods with higher efficiency have been recently proposed, such as persistent contrastive divergence (Tieleman, 2008), fast-weight contrastive divergence (Tieleman & Hinton, 2009), tempered transitions (Salakhutdinov, 2009), and particle-filtered MCMC-MLE (Asuncion et al., 2010). In our procedure, we use the persistent contrastive divergence algorithm to estimate parameters φ. Another difficulty is that θ is latent and we only have one observed training sample x. We use an EM algorithm to solve this problem. In the E-step, we run our MCMC algorithm in Section 2.3 to infer the latent θ based on the currently estimated parameters ϑ = (φ, ψ). In the M-step, we run the persistent contrastive divergence (PCD) algorithm (Tieleman, 2008) to estimate φ from the currently inferred θ. Note that PCD is also an iterative algorithm, and we run it until it converges in each M-step. In the M-step, we also do a maximum likelihood estimation of ψ from the currently inferred θ and observed x. We run the EM algorithm until both φ and ψ converge. Although this EM algorithm involves intensive computation in both E-step and M-step, it converges very quickly in our experiments. 3 Basic Simulations In the basic simulations, we investigate the numerical performance of our multiple testing approach on different fabricated dependence structures where we can control the ground truth parameters. We first simulate θ from P (θ; φ) and then simulate x from P (x|θ; ψ) under a variety of settings of ϑ = (φ, ψ). Because we have the ground truth parameters, we have two versions of our multiple testing approach, namely the oracle procedure (OR) and the data-driven procedure (LIS). The oracle procedure knows the true parameters ϑ in the graphical models, whereas the data-driven procedure does not and has to estimate ϑ. The baseline procedures include the BH procedure (Benjamini & Hochberg, 1995) and the adaptive p-value procedure (AP) (Benjamini & Hochberg, 2000; Genovese & Wasserman, 2004) which are compared by Sun & Cai (2009). We include another baseline procedure, the local false discovery rate procedure (localFDR) (Efron et al., 2001). The adaptive p-value procedure requires a consistent estimate of the proportion of the true null hypotheses. The localFDR procedure requires a consistent estimate of the proportion of the true null hypotheses and the knowledge of the distribution of the test statistics under the null and under the alternative. In our simulations, we endow AP and localFDR with the ground truth values of these in order to let these baseline procedures achieve their best performance. In the simulations, we assume that the observed xi under the null hypothesis (namely θi = 0) is standardnormally distributed and that xi under the alternative hypothesis (namely θi = 1) is normally distributed with mean µ and standard deviation 1.0. We choose the setup and parameters to be consistent with the work of Sun & Cai (2009) when possible. In total, we consider three MRF models, namely a chain-structured MRF, tree-structured MRF and gridstructured MRF. For chain-MRF, we choose the number of hypotheses m = 3, 000. For tree-MRF, we choose perfect binary trees of height 12 which yields a total number of 8, 191 hypotheses. For grid-MRF, we choose the number of rows and the number of columns to be 100 which yields a total number of 10, 000 hypotheses. In all the experiments, we choose the number of replications N = 500 which is also the same as the work of Sun & Cai (2009). In total, we have three sets of simulations with different goals as follows. Basic simulation 1: We stay consistent with Sun & Cai (2009) in the simulations except that we use the three MRF models. In all three structures, (θi )m 1 is generated whose potentials on the  from the MRFs  φ 1−φ edges are . Therefore, φ only contains 1−φ φ parameter φ, and ψ only includes parameter µ. Basic simulation 2: One assumption in basic simulation 1 is that the parameters φ and µ are homogeneous in the sense that they stay the same for i(i = 1, ..., m). This assumption is carried down from the work of Sun & Cai (2009). However in many real-world applications, the transition parameters can be different across the multiple hypotheses. Similarly, the test statistics for the non-null hypotheses, although normally distributed and standardized, could have different µ values. Therefore, we investigate the situation where the parameters can vary in different hypotheses. The simulations are carried out for all three different dependence structures aforementioned. In the first set of simulations, instead of fixing φ, we choose φ’s uniformly distributed on the interval (0.8 − ∆(φ)/2, 0.8 + ∆(φ)/2). In the second set of simulations, instead of fixing µ, we choose µ’s uniformly distributed on the interval (2.0−∆(µ)/2, 2.0+∆(µ)/2). The oracle procedure knows the true parameters. The data-driven procedure does not know the parameters, and assumes the parameters are homogeneous. Basic simulation 3: Another implicit assumption in basic simulation 1 is that each individual test in the multiple testing problem is exact. Many widely used hypothesis tests, such as Pearson’s χ2 test and the likelihood ratio test, are asymptotic in the sense that we only know the limiting distribution of the test statistics for large samples. As an example, we simulate the two-proportion z-test in this section and show how the sample size affects the performance of the procedures when the individual test is asymptotic. Suppose that we have n samples (half of them are positive samples and half of them are negative samples). For each sample, we have m Bernoulli distributed attributes. A fraction of the attributes are relevant. If the attribute A is relevant, then the probability of “heads” in the positive samples (p+ A ) is different from that in the neg− ative samples (p− ). p+ A A and pA are the same if A is nonrelevant. For each individual test, the null hypothesis is that the attribute is not relevant, and the alternative hypothesis is otherwise. The two-proportion z-test can − be used to test whether p+ A −pA is zero, which yields an asymptotic N (0, 1) under the null and N (µ, 1) under the alternative (µ is nonzero). In the simulations, we fix µ, but vary the sample size n, and apply the aforementioned tree-MRF structure (m = 8, 191). The oracle procedure and localFDR only know the limiting distribution of the test statistics and assume the test statistics exactly follow the limiting distributions even when the sample size is small. Figure 2 shows the numerical results in basic simulation 1. Figures (1a)-(1f) are for the chain structure. Figures (2a)-(2f) are for tree structure. Figures (3a)(3f) are for the grid structure. In Figures (1a)-(1c), 1500 0 ATP 500 1000 FNR 0.0 0.1 0.2 0.3 0.4 0.5 0.10 0.04 ATP 600 800 1000 FDR 0.06 0.08 0.35 0.15 FNR 0.25 0.09 FDR 0.07 0.05 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1.0 1.5 2.0 2.5 3.0 3.5 4.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 (1a) (1b) (1c) (1d) (1e) (1f)  1000 ATP 3000  0 FDR 0.06 0.08 FNR 0.0 0.1 0.2 0.3 0.4 0.5 0.10  0.04 1500 0.15 ATP 2500 FNR 0.25 0.09 FDR 0.07 0.05  3500  0.35  0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1.0 1.5 2.0 2.5 3.0 3.5 4.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 (2a) (2b) (2c) (2d) (2e) (2f)    ATP 2000 4000 0.4 FNR 0.2 FDR 0.10 0.14 0 0.0 0.06 0.0 2000 0.1 ATP 3000 4000 FNR 0.2 0.3 0.09 FDR 0.07 0.05  5000  0.4  0.2 0.3 0.4 0.5 0.6 0.7 0.8  0.2 0.3 0.4 0.5 0.6 0.7 0.8  0.2 0.3 0.4 0.5 0.6 0.7 0.8  1.0 1.5 2.0 2.5 3.0 3.5 4.0  1.0 1.5 2.0 2.5 3.0 3.5 4.0  1.0 1.5 2.0 2.5 3.0 3.5 4.0 (3a) (3b) (3c) (3d) (3e) (3f)  Figure 2: Comparison of BH( ), AP(4), localFDR(×), OR (+), and LIS () in basic simulation 1: (1) chainMRF, (2) tree-MRF, (3) grid-MRF; (a) FDR vs φ, (b) FNR vs φ, (c) ATP vs φ, (d) FDR vs µ, (e) FNR vs µ, (f) ATP vs µ. (2a)-(2c) and (3a)-(3c), we set µ = 2 and plot FDR, FNR and the average number of true positives (ATP) when we vary φ between 0.2 and 0.8. In Figures (1d)(1f), (2d)-(2f) and (3d)-(3f), we set φ = 0.8 and plot FDR, FNR and ATP when we vary µ between 1.0 and 4.0. The nominal FDR level is set to be 0.10. From Figure 2, we can observe comparable numerical results between the chain structure and tree structure. The FDR levels of all five procedures are controlled at 0.10 and BH is conservative. From the plots for FNR and ATP, we can observe that the data-driven procedure performs almost the same as the oracle procedure, and they dominate the p-value thresholding procedures BH and AP. The oracle procedure and the data-driven procedure also dominate localFDR except when φ = 0.5, when they perform comparably. This is to be expected because the dependence structure is no longer informative when φ is 0.5. In this situation when the hypotheses are independent, our procedure reduces to the localFDR procedure. As φ departs from 0.5 and approaches either 0 or 1.0, the difference between OR/LIS and the baselines gets larger. When the individual hypotheses are easy to test (large µ values), the differences between them are not substantial. When we turn to the grid structure, the numerical performance is similar to that in the chain structure and the tree structure except for two observations. First, the data-driven procedure does not appear to control the FDR at 0.1 when µ is small (e.g. µ = 1.0), although the oracle procedure does, which indicates the parameter estimation in the EM algorithm is difficult when µ is small. In other words, with a limited number of hypotheses, it is difficult to estimate the pairwise potential parameters if the test statistics of the non-nulls do not look much different from the test statistics of the nulls. The second observation is that the slopes of the FNR curve and ATP curve for the grid structure are different from those in the chain and tree structures. The reason is that the connectivity in the grid structure is higher than that in the chain and tree. Therefore we can observe that even when the individual hypotheses are difficult to test (small µ values), the FNR is still low because each individual hypothesis has more neighbors in the grid than in the chain or tree, and the neighbors are informative. Figure 3 shows the numerical performance in basic simulation 2. Figures (1a)-(1f), (2a)-(2f), and (3a)(3f) correspond to the chain structure, the tree structure and the grid structure respectively. In Figures (1a)-(1c), (2a)-(2c), and (3a-3c), we set µ = 2 and vary ∆(φ) between 0 and 0.4. In Figures (1d)-(1f), (2d)-(2f), and (3d)-(3f), we set φ = 0.8 and vary ∆(µ) between 0 and 4.0. Again, the nominal FDR level is set to be 0.10. From Figure 3, we observe that all five procedures control FDR at the nominal level and BH is conservative when the transition parameter φ is heterogeneous. However, the data-driven procedure becomes more and more conservative as we increase the variance of φ in the grid-structure. Nevertheless, the data-driven procedure does not lose much efficiency compared with the oracle procedure based on FNR and ATP. Both the data-driven procedure and the oracle procedure dominate the three baselines. When the µ parameter is heterogeneous, all five procedures 0.4 0.4 0.1 0.2  0.3 0.4 ATP 600 800 1000 0.35 FNR 0.25 0.05 0.15 1 3 4 1 2  3 4 0.2  0.3 0.4 1 (3c) 2  (3d) 4 4 3 4 3 4 ATP 2500 2  3 4 (2e) 3 3 1500 1 (2d) 0.0 0.1 (3b) 2  3500 0.35 0.15 0.4 1 (1f) FNR 0.25 0.09 0.3 2  (1e) FDR 0.07 0.2  (2c) 2000 0.1 0.0 0.4 4 0.05 0.1 ATP 3000 4000 FNR 0.2 0.3 0.09 0.3 (3a) 3 1 2  (2f) 5000 0.3 2  0.4 0.2  1 (1d) ATP 2500 0.1 (2b) FDR 0.07 0.2  0.4 1500 0.4 0.05 0.1 0.3 3500 0.40 0.3 (2a) 5000 0.2  0.2  (1c) 0.10 0.05 0.1 0.1 ATP 3000 4000 0.4 2000 0.3 FNR 0.20 0.30 FDR 0.07 0.09 0.2  (1b) FNR 0.2 0.3 0.1 0.1 0.4 FDR 0.04 0.06 0.08 0.10 0.3 FDR 0.07 ATP 600 800 0.10 0.2  (1a) 0.09 1200 0.40 FNR 0.20 0.30 FDR 0.07 0.09 0.05 0.1 1 2  (3e) 3 4 1 2  (3f) Figure 3: Comparison of BH( ), AP(4), localFDR(×), OR (+), and LIS () in basic simulation 2: (1) chainMRF, (2) tree-MRF, (3) grid-MRF; (a) FDR vs ∆(φ), (b) FNR vs ∆(φ), (c) ATP vs ∆(φ), (d) FDR vs ∆(µ), (e) FNR vs ∆(µ), (f) ATP vs ∆(µ). 100 200 300 Sample size (a) 400 500 3500 ATP 2500 1500 0.05 0.10 FDR 0.07 0.09 FNR 0.20 0.30 0.11 0.40 are still valid, but the data-driven procedure becomes more and more conservative as we increase the variance of µ. The data-driven procedure can be more conservative than the BH procedure when ∆(µ) is large enough. The conservativeness appears most severe in the grid-structure. However when we look at the FNR and ATP, the data-driven procedure still dominates BH, AP and localFDR substantially in all the situations, although the data-driven procedure loses a certain amount of efficiency compared with the oracle procedure when the variance of µ gets large. 100 200 300 Sample size (b) 400 500 100 200 300 400 500 Sample size (c) Figure 4: Comparing BH( ), AP(4), localFDR(×), OR(+), and LIS() in basic simulation 3: (a)FDR vs n, (b)FNR vs n, (c)ATP vs n. Figure 4 shows the results from basic simulation 3. The oracle procedure and localFDR are liberal when the sample size is small. This is because when the sample size is small, there exists a discrepancy between the true distribution of the test statistic and the limiting distribution. Quite surprisingly, the data-driven procedure stays valid. The reason is that the data-driven procedure can estimate the parameters from data. The data-driven procedure and the oracle procedure still have comparable performance and enjoy a much lower level of FNR compared with the baselines. For all the basic simulations, we set the nominal FDR level to be 0.10. We have also replicated the basic simulations by setting the nominal level to be 0.05, and similar conclusions can be made. 4 Simulations on Genetic Data Unlike the fabricated dependence structures in the basic simulations in Section 3, the dependence structure in the simulations on genetic data in this section is real. We simulate the linkage disequilibrium structure of a segment on human chromosome 22, and treat a test of whether a SNP is associated as one individual test. We follow the simulation settings in the work of Wu et al. (2010). We use HAPGEN2 (Su et al., 2011) and the CEU sample of HapMap (The International HapMap Consortium, 2003) (Release 22) to generate SNP genotype data at each of the 2, 420 loci between bp 14431347 and bp 17999745 on Chromosome 22. A total of 685 out of 2, 420 SNPs can be genotyped with the Affymetrix 6.0 array. These are the typed SNPs that we use for our simulations. Within the overall 2, 420 SNPs, we randomly select 10 SNPs to be the causal SNPs. All the SNPs on the Affymetrix 6.0 array whose r2 values, according to HapMap, with any of the causal SNPs are above t are set to be the associated SNPs. In the simulations, we report results for three different t values, namely 0.8, 0.5 and 0.25. We also simulate three different genetic models (additive model, dominant model, and recessive model) with different levels of relative risk (1.2 and 1.3). In total, we simulate 250 cases and 250 controls. The experiment is replicated for 100 times and the average result is provided. With the simulated data, we apply our multiple testing procedure (LIS) and three baseline procedures: the BH procedure, the adaptive p-value procedure (AP), and the local false discovery rate procedure (localFDR). Because the dependence structure is real and the ground truth parameters are unknown to us, we do not have the oracle procedure in the simulations on genetic data. With the simulated genetic data, we use two commonly used tests in genetic association studies, namely two-proportion z-test and Cochran-Armitage’s trend test (CATT) (Cochran, 1954; Armitage, 1955; Slager & Schaid, 2001; Freidlin et al., 2002) as the individual tests for the association of each SNP. CATT also yields an asymptotic N (0, 1) under the null and N (µ, 1) under the alternative (µ is nonzero). Therefore, we parameterize ψ = (µ1 , σ12 ) where µ1 and σ12 are the mean and variance of the test statistics under alternative. The graph structure is built as follows. Each SNP becomes a node in the graph. For each SNP, we connect it with the SNP with the highest r2 value with it. There are in total 490 edges in the graph. We further categorize the edges into a high correlation edge set Eh (r2 above 0.8), medium correlation edge set Em (r2 between 0.5 and 0.8) and low correlation edge set El (r2 between 0.25 and 0.5). We have three different parameters (φh , φm , and φl ) for the three sets of edges. Then the density of θ in formula (1) takes the form P (θ; φ) ∝ exp{ X φh I(θi = θj )+ (i,j)∈Eh X (i,j)∈Em φm I(θi = θj ) + X (3) φl I(θi = θj )}, (i,j)∈El where I(θi = θj ) is an indicator variable that indicates whether θi and θj take the same value. In the MCMC algorithm, we run the Markov chain for 20, 000 iterations with a burn-in of 100 iterations. In the PCD algorithm, we generate 100 particles. In each iteration of PCD learning, the particles move forward for 5 iterations (the n parameter in PCD-n). The learning rate in PCD gradually decreases as suggested by Tieleman (2008). The EM algorithm converges after about 10 to 20 iterations, which usually take less than 10 minutes on a 3.00GHz CPU. Figure 5 shows the performance of the procedures in the additive models with the homozygous relative risk set to 1.2 and 1.3. The test statistics are from a twoproportion z-test. We have also replicated the simulations on Cochran-Armitage’s trend test, and the results are almost the same. In Figure 5, table (1) summarizes the empirical FDR and the total number of true positives (#TP) of our LIS procedure, BH, AP and localFDR (lfdr), in the additive models with different (homozygous) relative risk levels, when we vary t and when we vary the nominal FDR level α. We regard a SNP having r2 above t with any causal SNP as an associated SNP, and we regard a rejection of the null hypothesis for an associated SNP as a true positive. Our LIS procedure and localFDR are valid while being conservative. BH and AP appear liberal in some of the configurations. In any of the circumstances, our LIS procedure can identify more associated SNPs than the baselines. We can find a clue to why our procedure LIS is being conservative from the results in Figure 3. In basic simulation 2, we observe that when the parameters µ and φ are heterogeneous and we carry out the data-driven procedure under the homogeneous parameter assumption, the data-driven procedure is conservative. The discrepancy between the nominal FDR level and the empirical FDR level increases as the parameters move further away from homogeneity. Although we assign three different parameters φh , φm , and φl to Eh , Em and El respectively, the edges within the same set (e.g. El ) may still be heterogeneous. The fact that the LIS procedure recaptures more true positives than the baselines while remaining more conservative in many configurations indicates that the local indices of significance provide a ranking more efficient than the ranking provided by the p-values from the individual tests. Therefore, we further plot the ROC curves and precision-recall (PR) curves when we rank SNPs by LIS and by the p-values from the two-proportion z-test. The ROC curve and PR curve are vertically averaged from 100 replications. Subfigures (2a)-(2f) are for the additive model with homozygous relative risk level set to be 1.2. Subfigures (3a)-(3f) are for the additive model with homozygous relative risk level set to be 1.3. It is observed that the curves from LIS dominate those from the p-values from individual tests in most places, which further suggests that LIS provides a more efficient ranking of the SNPs than the individual tests. Figure 6 shows the performance of the procedures in the dominant model and the recessive model with the homozygous relative risk set to be 1.2. The test statistics are from a two-proportion z-test. In Figure 6, table (1) summarizes the empirical FDR and the total number of true positives (#TP) of our LIS procedure, BH, AP and localFDR (lfdr) in the dominant model and the recessive model when we vary t and when we vary the nominal FDR level α. Our LIS procedure and localFDR are valid while being conservative in all configurations, and they appear more conservative in the recessive model than in the dominant model. On the other hand, BH and AP appear liberal in the recessive model. Our LIS procedure still confers an advantage t = 0.8 LIS α = 0.05 rr = 1.2 α = 0.10 α = 0.05 rr = 1.3 α = 0.10 FDR: #TP: FDR: #TP: FDR: #TP: FDR: #TP: BH t = 0.5 AP lfdr LIS BH t = 0.25 AP lfdr LIS BH AP lfdr 0.018 12 0.077 13 0.059 11 0.089 11 0.059 11 0.089 11 0.010 1 0.010 1 0.018 12 0.077 13 0.059 11 0.089 11 0.059 11 0.089 11 0.010 1 0.010 1 0.018 20 0.076 21 0.058 18 0.079 20 0.058 19 0.079 20 0.009 7 0.009 8 0.047 16 0.067 18 0.044 4 0.104 15 0.054 4 0.104 15 0.015 1 0.015 1 0.047 16 0.067 18 0.044 4 0.104 15 0.064 4 0.104 15 0.005 1 0.005 1 0.046 22 0.066 27 0.044 10 0.103 21 0.064 10 0.103 21 0.014 6 0.014 6 0.8 1.0 0.0 0.4 0.6 Recall 0.8 1.0 0.4 0.6 FPR 0.8 (3a) ROC t=0.8 1.0 0.8 0.2 0.0 0.2 0.4 0.6 Recall 0.8 1.0 (3b) PR t=0.8 Precision 0.2 0.3 0.1 0.2 0.4 0.6 FPR 0.8 1.0 0.0 (2e) ROC t=0.25 0.0 0.2 0.0 (2d) PR t=0.5 TPR 0.4 0.6 Precision 0.2 0.3 0.1 0.2 0.2 0.4 0.6 Recall 0.8 1.0 (2f) PR t=0.25 0.4 0.4 0.6 FPR 1.0 0.4 1.0 0.8 TPR 0.4 0.6 0.2 0.2 (2c) ROC t=0.5 0.0 0.0 0.4 1.0 0.0 Precision 0.2 0.3 1.0 0.1 0.8 1.0 0.4 0.6 Recall (2b) PR t=0.8 0.8 0.2 TPR 0.4 0.6 0.0 0.2 1.0 0.0 0.8 0.4 0.4 0.6 FPR (2a) ROC t=0.8 Precision 0.2 0.3 0.2 0.1 0.0 0.0 0.1 0.2 TPR 0.4 0.6 Precision 0.2 0.3 0.8 0.4 1.0 0.8 TPR 0.4 0.6 0.2 0.0 0.0 0.1 0.2 TPR 0.4 0.6 Precision 0.2 0.3 0.8 0.4 1.0 (1) 0.0 0.2 0.4 0.6 FPR 0.8 1.0 (3c) ROC t=0.5 0.0 0.2 0.4 0.6 Recall 0.8 (3d) PR t=0.5 1.0 0.0 0.2 0.4 0.6 FPR 0.8 (3e) ROC t=0.25 1.0 0.0 0.2 0.4 0.6 Recall 0.8 1.0 (3f) PR t=0.25 Figure 5: Comparison of BH, AP, localFDR and LIS in the additive models when we vary relative risk rr, t and the nominal FDR level α. Table (1) summarizes results. Subfigures (2a)-(2f) shows ROC and PR curves of LIS (solid red lines) and individual p-values (dashed green lines) with rr = 1.2. Subfigures (3a)-(3f) shows ROC and PR curves of LIS (solid red lines) and individual p-values (dashed green lines) with rr = 1.3. over the baselines in the dominant model. The LIS procedure also recaptures almost the same number of true positives as BH and AP while maintaining a much lower FDR in the recessive model. Again, we further plot the ROC curves and precision-recall curves when we rank SNPs by LIS and by the p-values from individual tests. Subfigures (2a)-(2f) are for the dominant model. Subfigures (3a)-(3f) are for the recessive model. It is also observed that the curves from LIS dominate those from the p-values from individual tests in most places, which also suggests that LIS provides a more efficient ranking. 5 Real-world Application Our primary GWAS dataset on breast cancer is from NCI’s Cancer Genetics Markers of Susceptibility (CGEMS) (Hunter et al., 2007). 528, 173 SNPs for 1, 145 cases and 1, 142 controls are genotyped on the Illumina HumanHap500 array. Our secondary GWAS dataset comes from Marshfield Clinic. The Personalized Medicine Research Project (McCarty et al., 2005), sponsored by Marshfield Clinic, was used as the sampling frame to identify 162 breast cancer cases and 162 controls. The project was reviewed and approved by the Marshfield Clinic IRB. Subjects were selected using clinical data from the Marshfield Clinic Cancer Registry and Data Warehouse. Cases were defined as women having a confirmed diagnosis of breast cancer. Both the cases and controls had to have at least one mammogram within 12 months prior to having a biopsy. The subjects also had DNA samples that were genotyped using the Illumina HumanHap660 array, as part of the eMERGE (electronic MEdical Records and Genomics) network by McCarty et al. (2011). We apply our multiple testing procedure on the CGEMS data. The settings of the procedure are the same as in the simulations on genetic data in Section 4. The individual test is two-proportion z-test. Our procedure reports 32 SNPs with LIS value of 0.0 (an estimated probability 1.0 of being associated). We further calculate the per-allele odds-ratio of these SNPs on the Marshfield data, and 14 of them have an oddsratio around 1.2 or above. The details about the 14 SNPs are given in supplementary material. There are two clusters among them. First, rs3870371, rs7830137 and rs920455 (on chromosome 8) locate near each other and near the gene hyaluronan synthase 2 (HAS2) which has been shown to be associated with invasive breast cancer by many studies (Udabage et al., 2005; t = 0.8 LIS α = 0.05 Dominant α = 0.10 α = 0.05 Recessive α = 0.10 FDR: #TP: FDR: #TP: FDR: #TP: FDR: #TP: BH t = 0.5 AP lfdr LIS BH t = 0.25 AP lfdr LIS BH AP lfdr 0.026 14 0.051 20 0.040 4 0.079 12 0.040 4 0.089 12 0.010 2 0.010 3 0.026 14 0.048 22 0.040 4 0.079 12 0.040 4 0.109 12 0.010 2 0.010 3 0.025 21 0.044 33 0.039 10 0.079 19 0.039 10 0.109 29 0.009 7 0.009 18 0.009 11 0.018 11 0.079 11 0.104 12 0.079 11 0.104 12 0.009 11 0.009 11 0.009 11 0.018 11 0.079 11 0.104 12 0.079 11 0.114 12 0.009 11 0.009 11 0.009 18 0.017 22 0.079 17 0.104 21 0.079 18 0.114 21 0.009 17 0.009 17 0.4 0.6 FPR 0.8 1.0 0.2 0.4 0.6 Recall 0.8 1.0 0.4 0.6 FPR 0.8 1.0 (3a) ROC t=0.8 1.0 0.8 0.2 0.0 0.2 0.4 0.6 Recall 0.8 (3b) PR t=0.8 1.0 Precision 0.2 0.3 0.1 0.2 0.4 0.6 FPR 0.8 1.0 0.0 (2e) ROC t=0.25 0.0 0.2 0.0 (2d) PR t=0.5 TPR 0.4 0.6 Precision 0.2 0.3 0.1 0.0 0.0 0.4 1.0 0.0 (2c) ROC t=0.5 0.4 1.0 0.8 TPR 0.4 0.6 0.2 0.2 0.2 0.4 0.6 Recall 0.8 1.0 (2f) PR t=0.25 0.4 0.0 Precision 0.2 0.3 1.0 0.1 0.8 1.0 0.4 0.6 Recall (2b) PR t=0.8 0.8 0.2 TPR 0.4 0.6 0.0 0.2 1.0 0.0 0.8 0.4 0.4 0.6 FPR (2a) ROC t=0.8 Precision 0.2 0.3 0.2 0.1 0.0 0.0 0.1 0.2 TPR 0.4 0.6 Precision 0.2 0.3 0.8 0.4 1.0 0.8 TPR 0.4 0.6 0.2 0.0 0.0 0.1 0.2 TPR 0.4 0.6 Precision 0.2 0.3 0.8 0.4 1.0 (1) 0.0 0.2 0.4 0.6 FPR 0.8 1.0 (3c) ROC t=0.5 0.0 0.2 0.4 0.6 Recall 0.8 (3d) PR t=0.5 1.0 0.0 0.2 0.4 0.6 FPR 0.8 (3e) ROC t=0.25 1.0 0.0 0.2 0.4 0.6 Recall 0.8 1.0 (3f) PR t=0.25 Figure 6: Comparison of BH, AP, localFDR and LIS in the dominant model and the recessive model with different t values and different nominal FDR α values. Table (1) summarizes results. Subfigures (2a)-(2f) shows ROC and PR curves of LIS (solid red lines) and individual p-values (dashed green lines) in the dominant model. Subfigures (3a)-(3f) shows ROC and PR curves of LIS and individual p-values in the recessive model. Li et al., 2007; Bernert et al., 2011). The other cluster includes rs11200014, rs2981579, rs1219648, and rs2420946 on chromosome 10. They are exactly the 4 SNPs reported by Hunter et al. (2007). Their associated gene FGFR2 is also well known to be associated with breast cancer. SNP rs4866929 on chromosome 5 is also very likely to be associated because it is highly correlated (r2 =0.957) with SNP rs981782 (not included in our data) which was identified from a much larger dataset (4, 398 cases and 4, 316 controls and a follow-up confirmation stage on 21, 860 cases and 22, 578 controls) by Easton et al. (2007). 6 Conclusion In this paper, we use an MRF-coupled mixture model to leverage the dependence in multiple testing problems, and show the improved numerical performance on a variety of simulations and its applicability in a real-world GWAS problem. A theoretical question of interest is whether this graphical model based procedure is optimal in the sense that it has the smallest FNR among all the valid procedures. The optimality of the oracle procedure can be proved under the compound decision framework (Sun & Cai, 2007, 2009), as long as an exact inference algorithm exists or an approximate inference algorithm can be guaranteed to converge to the correct marginal probabilities. The asymptotic optimality of the data-driven procedure (the FNR yielded by the data-driven procedure approaches the FNR yielded by the oracle procedure as the number of tests m → ∞) requires consistent estimates of the unknown parameters in the graphical models. Parameter learning in undirected models is more complicated than in directed models due to the normalization constant. To the best of our knowledge, asymptotic properties of parameter learning for hidden MRFs and MRF-coupled mixture models have not been investigated. Therefore, we cannot prove the asymptotic optimality of the data-driven procedure so far, although we can observe its close-to-oracle performance in the basic simulations. Acknowledgements The authors acknowledge the support of the Wisconsin Genomics Initiative, NCI grant R01CA127379-01 and its ARRA supplement 3R01CA127379-03S1, NIGMS grant R01GM097618-01, NLM grant R01LM01102801, NIEHS grant 5R01ES017400-03, eMERGE grant 1U01HG004608-01, NSF grant DMS-1106586 and the UW Carbone Cancer Center. References Armitage, P. (1955). Tests for linear trends in proportions and frequencies. BIOMETRICS, 11, 375C386. Asuncion, A. U., Liu, Q., Ihler, A. T., & Smyth, P. (2010). 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1 No Blind Spots: Full-Surround Multi-Object Tracking for Autonomous Vehicles using Cameras & LiDARs arXiv:1802.08755v1 [] 23 Feb 2018 Akshay Rangesh, Member, IEEE, and Mohan M. Trivedi, Fellow, IEEE Abstract—Online multi-object tracking (MOT) is extremely important for high-level spatial reasoning and path planning for autonomous and highly-automated vehicles. In this paper, we present a modular framework for tracking multiple objects (vehicles), capable of accepting object proposals from different sensor modalities (vision and range) and a variable number of sensors, to produce continuous object tracks. This work is inspired by traditional tracking-by-detection approaches in computer vision, with some key differences - First, we track objects across multiple cameras and across different sensor modalities. This is done by fusing object proposals across sensors accurately and efficiently. Second, the objects of interest (targets) are tracked directly in the real world. This is a departure from traditional techniques where objects are simply tracked in the image plane. Doing so allows the tracks to be readily used by an autonomous agent for navigation and related tasks. To verify the effectiveness of our approach, we test it on real world highway data collected from a heavily sensorized testbed capable of capturing full-surround information. We demonstrate that our framework is well-suited to track objects through entire maneuvers around the ego-vehicle, some of which take more than a few minutes to complete. We also leverage the modularity of our approach by comparing the effects of including/excluding different sensors, changing the total number of sensors, and the quality of object proposals on the final tracking result. Index Terms—Multi-object tracking (MOT), panoramic surround behavior analysis, highly autonomous vehicles, computer vision, sensor fusion, collision avoidance, path planning. Y X (a) I. I NTRODUCTION T RACKING for autonomous vehicles involves accurately identifying and localizing dynamic objects in the environment surrounding the vehicle. Tracking of surround vehicles is essential for many tasks crucial to truly autonomous driving, such as obstacle avoidance, path planning, and intent recognition. To be useful for such high-level reasoning, the generated tracks should be accurate, long and robust to sensor noise. In this study, we propose a full-surround MOT framework to create such desirable tracks. Traditional MOT techniques for autonomous vehicles can roughly be categorized into 3 groups based on the sensory inputs they use - 1) dense point clouds from range sensors, 2) stereo vision, and 3) a fusion of range and vision sensors. Studies like [1]–[4] make use of dense point clouds created by 3D LiDARs like the Velodyne HDL-64E. Such sensors, although bulky and expensive, are capable of capturing finer The authors are with the Laboratory for Intelligent and Safe Automobiles, University of California, San Diego, CA 92092, USA. email - arangesh, [email protected] (b) Fig. 1: (a) Illustration of online MOT for autonomous vehicles. The surround vehicles (in red) are tracked in a coordinate system centered on the ego-vehicle (center). The ego-vehicle has full-surround coverage from vision and range sensors, and must fuse proposals from each of them to generate continuous tracks (dotted lines) in the real world. (b) An example of images captured from a full-surround camera array mounted on our testbed, along with color coded vehicle annotations. 2 details of the surroundings owing to its high vertical resolution. Trackers can therefore create suitable mid-level representations like 2.5D grids, voxels etc. that retain unique statistics of the volume they enclose, and group such units together to form coherent objects that can be tracked. It must be noted however, that these approaches are reliant on having dense point representations of the scene, and would not scale well to LiDAR sensors that have much fewer scan layers. On the other hand, studies such as [5]–[8] make use of stereo vision alone to perform tracking. The pipeline usually involves estimating the disparity image and optionally creating a 3D point cloud, followed by similar mid-level representations like stixels, voxels etc. which are then tracked from frame to frame. These sensors are limited by the quality of disparity estimates and the field of view (FoV) of the stereo pair. Unlike 3D LiDAR based systems, they are unable to track objects in full-surround. There are other single camera approaches to surround vehicle behavior analysis [9], [10], but they too are limited in their FoVs and localization capabilities. Finally, there are fusion based approaches like [11]–[13], that make use of LiDARs, stereo pairs, monocular cameras, and Radars in a variety of configurations. These techniques either perform early or late fusion based on their sensor setup and algorithmic needs. However, none of them seem to offer full-surround solutions for vision sensors, and are ultimately limited to fusion only in the FoV of the vision sensors. In this study, we take a different approach to full-surround MOT and try to overcome some of the limitations in previous approaches. Specifically, we propose a framework to perform full-surround MOT using calibrated camera arrays, with varying degrees of overlapping FoVs, and with an option to include low resolution range sensors for accurate localization of objects in 3D. We term this the M3 OT framework, which stands for multi-perspective, multi-modal, multi-object tracking framework. To train and validate the M3 OT framework, we make use of naturalistic driving data collected from our testbed (illustrated in Figure 1) that has full-surround coverage from vision and range modalities. Since we use vision as our primary perception modality, we leverage recent approaches from the 2D MOT community which studies tracking of multiple objects in the 2D image plane. Recent progress in 2D MOT has focused on the tracking-by-detection strategy, where object detections from a category detector are linked to form trajectories of the targets. To perform tracking-by-detection online (i.e. in a causal fashion), the major challenge is to correctly associate noisy object detections in the current video frame with previously tracked objects. The basis for any data association algorithm is a similarity function between object detections and targets. To handle ambiguities in association, it is useful to combine different cues in computing the similarity, and learn an association based on these cues. Many recent 2D MOT methods such as [14]–[18] use some form of learning (online or offline) to accomplish data association. Similar to these studies, we formulate the online multi-object tracking problem using Markov Decision Processes (MDPs) proposed in [19], where the lifetime of an object is modeled with a MDP (see Figure 2), and multiple MDPs are assembled for multi-object tracking. In this method, learning a similarity function for data association is equivalent to learning a policy for the MDP. The policy learning is approached in a reinforcement learning fashion which benefits from advantages of both offline-learning and online-learning in data association. The M3 OT framework is also capable to naturally handle the birth/death and appearance/disappearance of targets by treating them as state transitions in the MDP, and also benefits from the strengths of online learning approaches in single object tracking ( [20]–[23]). Our main contributions in this work can be summarized as follows - 1) We extend and improve the MDP formulation originally proposed for 2D MOT, and modify it to track objects in 3D (real world). 2) We make the M3 OT framework capable of tracking objects across multiple vision sensors in calibrated camera arrays by carrying out efficient and accurate fusion of object proposals. 3) The M3 OT framework is made highly modular, capable of working with any number of cameras, with varying degrees of overlapping FoVs, and with the option to include range sensors for improved localization and fusion in 3D. 4) Finally, we carry out experiments using naturalistic driving data collected on highways using full-surround sensory modalities, and validate the accuracy, robustness and modularity of our framework. II. R ELATED R ESEARCH We highlight some representative works in 2D and 3D MOT below. We also summarize the some key aspects of related 3D MOT studies in Table I. For a recent survey on detection and tracking for intelligent vehicles, we refer the reader to [25]. 2D MOT: Recent research in MOT has focused on trackingby-detection, where the main challenge is data association for linking object detections to targets. Majority of batch (offline) methods ( [17], [26]–[31]) formulate MOT as a global optimization problem in a graph-based representation, while online methods solve the data association problem either probabilistically [32]–[34] or deterministically (e.g., Hungarian algorithm [35] in [14], [15] or greedy association [36]). A core component in any data association algorithm is a similarity function between objects. Both batch methods [16], [17] and online methods [14], [15], [18] have explored the idea of learning to track, where the goal is to learn a similarity function for data association from training data. In this work, we extend and improve the MDP framework for 2D MOT proposed in [19], which is an online method that uses reinforcement learning to learn associations. 3D MOT for autonomous vehicles: In [5], the authors use a stereo rig to first calculate the disparity using SGM, followed by height based segmentation and free-space calculation to create a mid-level representation using stixels that encode the height within a cell. Each stixel is then represented by a 6D state vector, which is tracked using the Extended Kalman Filter (EKF). In [6], the authors use a voxel based representation instead, and cluster neighboring voxels based on color to create objects that are then tracked using a greedy association model. The authors in [7] use a grid based representation of the scene, where cells are grouped to create objects, each 3 TABLE I: Relevant research in 3D MOT for intelligent vehicles. Sensors used Study Tracker Type Monocular camera Stereo pair Full-surround camera array LiDAR Choi et al. [1] - - - Broggi et al. [6] - 3 Song et al. [2] - - Experimental Analysis Radar Single object tracker Multi-object tracker Online (causal) Dataset Evaluation metrics 3 - - 3 3 Proposed Distance and velocity errors - - - - 3 3 Proposed True positives, false positives, false negatives - 3 - 3 7 3 KITTI Position error, intersection ratio Cho et al. [11] 3 - - 3 3 - 3 3 Proposed Correctly tracked, falsely tracked, true and false positive rates Asvadi et al. [3] Vatavu et al. [7] - 3 - 3 - - - 3 3 3 3 KITTI Proposed - Asvadi et al. [4] - - - 3 - - 3 3 KITTI Number of missed and false obstacles Asvadi et al. [12] 3 - - 3 - 3 7 3 KITTI Average center location errors in 2D and 3D, orientation errors Ošep et al. [8] - 3 - - - - 3 3 KITTI Class accuracy, GOP recall, tracking recall Allodi et al. [13] - 3 - 3 - - 3 3 KITTI MOTP, IDS, Frag, average localization error Dueholm et al. [24] - - 3 - - - 3 7 VIVA Surround MOTA, MOTP, IDS, Frag, Precision, Recall This work1 (M3 OT) - - 3 3 - - 3 3 Proposed MOTA, MOTP, MT, ML, IDS for a variety of sensor configurations 1 this framework can work with or without LiDAR sensors, and with any subset of camera sensors. of which is represented by a set of control points on the object surface. This creates a high dimensional state-space representation, which is accounted for by a Rao-Blackwellized particle filter. More recently, the authors of [8] propose to carry out semantic segmentation on the disparity image, which is then used to generate generic object proposals by creating a scale-space representation of the density, followed by multiscale clustering. The proposed clusters are then tracked using a QPBO framework. The work in [24] use a camera setup similar to ours, but the authors propose an offline framework for tracking, hence limiting their use to surveillance related applications. Alternatively, there are approaches that make use of dense point clouds generated by LiDARs as opposed to creating point clouds from a disparity image. In [1], the authors first carry out ground classification based on the variance in the radius of each scan layer, followed by a 2.5D occupancy grid representation of the scene. The grid is then segmented, and regions of interest (RoIs) identified within, each of which is tracked by a standard Kalman Filter (KF). Data association is achieved by simple global nearest neighbor. Similar to this, the authors in [3] use a 2.5D occupancy grid based representation, but augment this with an occupancy grid map which accumulates past grids to create a coherent global map of occupancy by accounting for ego-motion. Using these two representations, a 2.5D motion grid is created by comparing the map with the latest occupancy grid, which isolates and identifies dynamic objects in the scene. Although the work in [4] follows the same general idea, the authors propose a piecewise ground plane estimation scheme capable of handling nonplanar surfaces. In a departure from grid based methods, the authors in [2] project the 3D point cloud onto a virtual image plane, creating an object appearance model based on 4 image- based cues for each template of the desired target. A particle filtering framework is implemented, where the particle with least reconstruction error with respect to the stored template is chosen to update the tracker. Background filtering and occlusion detection are implemented to improve performance. Finally, we list recent methods that rely on fusion of different sensor modalities to function. In [11], the authors propose an EKF based fusion scheme, where measurements from each modality are fed sequentially. Vision is used to classify the object category, which is then used to choose appropriate motion and observation models for the object. Once again, observations are associated based on a global nearest neighbor policy. This is somewhat similar to the work in [12], where given an initial 3D detection box, the authors propose to project 3D points within the box to the image plane and calculate the convex hull of projected points. In this case, a KF is used to perform both fusion and tracking, where fusion is carried out by projecting the 2D hull to a sparse 3D cloud, and using both 3D cues to perform the update. In contrast, the authors of [13] propose using the Hungarian algorithm (for bipartite matching) for both data association and fusion of object proposals from different sensors. The scores for association are obtained from Adaboost classifiers trained on high-level features. The objects are then tracked using an Unscented Kalman Filter (UKF). III. F USION OF O BJECT P ROPOSALS In this study, we make use of full-surround camera arrays comprising of sensors with varying FoVs. The M3 OT framework, however, is capable of working with any type and number of cameras, as long as they are calibrated. In addition to this, we also propose a variant of the framework for cases where LiDAR point clouds are available. To effectively utilize 4 a3 fused proposals a1 Object Proposal (u, v) w a4 Tracked h a6 Active Lost P a5 a2 Inactive a7 (a) Flattened LiDAR Point Cloud Object Proposal l1 Fig. 2: The Markov Decision Process (MDP) framework proposed in [19]. In this work, we retain the structure of the MDP, and modify the actions, rewards and inputs to enable multi-sensory tracking. all available sensors, we propose an early fusion of object proposals obtained from each of them. At the very start of each time step during tracking, we identify and fuse all proposals belonging to the same object. These proposals are then utilized by the M3 OT framework to carry out tracking. a) Projection & Back-projection: It is essential to have a way of associating measurements from different sensors to track objects across different camera views, and to carry out efficient fusion across sensor modalities. This is achieved by defining a set of projection mappings, one from each sensor’s unique coordinate system to the global coordinate system, and a set of back-projection mappings that take measurements in the global coordinate system to individual coordinate systems. In our case, the global coordinate system is centered at the mid-point of the rear axle of the ego-vehicle. The axes are oriented as shown in Figure 1. The LiDAR sensors output a 3D point cloud in a common coordinate system at every instant. This coordinate frame may either be centered about a single LiDAR sensor, or elsewhere depending on the configuration. In this case, the projection and back-projection mappings are simple 3D coordinate transformations: Prange→G (xrange ) = Rrange · xrange + trange l, (1) and, Prange←G (xG ) = RTrange · xG − RTrange trange , (2) where Prange→G (·) and Prange←G (·) are the projection and back-projection mappings from the LiDAR (range) coordinate system to the global (G) coordinate system and vice-versa. The vectors xrange and xG are the corresponding coordinates in the LiDAR and global coordinate frames. The 3 × 3 l2 (b) Fig. 3: Projection of object proposals using: (a) IPM: The bottom center of the bounding boxes are projected into the global coordinate frame (right), (b) LiDAR point clouds: LiDAR points that fall within a detection window are flattened and lines are fit to identify the vehicle center (right). orthonormal rotation matrix Rrange and translation vector trange are obtained through calibration. Similarly, the back-projection mappings for each camera k ∈ {1, · · · , K} can be defined as: Pcamk ←G (xG ) = (u, v)T ,   u s.t v  = C k · [Rk |tk ] · [(xG )T |1]T , 1 (3) (4) where the set of camera calibration matrices {C k }K k=1 are obtained after the intrinsic calibration of cameras, and the set of tuples {(Rk , tk )}K k=1 obtained after extrinsic calibration. Unlike the back-projection mappings, the projection mappings for camera sensors are not well defined. In fact, the mappings are one-to-many due to the depth ambiguity of single camera images. To find a good estimate of the projection, we use two different approaches. In case of a vision-only system, we use the inverse perspective mapping (IPM) approach: Pcamk →G (xk ) = (x, y)T ,   x s.t. y  = H k · [(xk )T |1]T , 1 (5) (6) where {H k }K k=1 are the set of homographies obtained after IPM calibration. Since we are only concerned with lateral and 5 longitudinal displacements of vehicles in the global coordinate system, we only require the (x, y)T coordinates, and set the altitude coordinate to a fixed number. b) Sensor Measurements & Object Proposals: As we adopt a tracking-by-detection approach, each sensor is used to produce object proposals to track and associate. In case of vision sensors, a vehicle detector is run on each individual camera’s image to obtain multiple detections d, each of which is defined by a bounding box in the corresponding image. Let (u, v) denote the top left corner and (w, h) denote the width and height of a detection d respectively. In case of a vision-only system, the corresponding location of d in the global coordinate system is obtained using the mapping Pcamk →G ((u + w2 , v + h)T ), where k denotes the camera from which the proposal was generated. This procedure is illustrated in Figure 3a. In cases where LiDAR sensors are available, an alternative is considered (shown in Figure 3b). First, the back-projected LiDAR points that fall within a detection box d are identified using a look-up table with pixel coordinates as the key, and the corresponding global coordinates as the value. These points are then flattened by ignoring the altitude component of their global coordinates. Next, a line l1 is fitted to these points using RANSAC with a small inlier ratio (0.3). This line aligns with the dominant side of the detected vehicle. The other side of the vehicle corresponding to line l2 is then identified by removing all inliers from the previous step and repeating a RANSAC line fit with a slightly higher inlier ratio (0.4). Finally, the intersection of lines l1 and l2 along with the vehicle dimensions yield the global coordinates of the vehicle center. Depending on the type of LiDAR sensors used, object proposals along with their dimensions in the real world can be obtained. However, we decide not to make use of LiDAR proposals, but rather use vision-only proposals with high recall by trading off some of the precision. This was seen to provide sufficient proposals to track all surrounding vehicles, at the expense of more false positives which the tracker is capable of handling. c) Early Fusion of Proposals: Since we operate with camera arrays with overlapping FoVs, the same vehicle may be detected in two adjacent views. It is important to identify and fuse such proposals to track objects across camera views. Once again, we propose two different approaches to carry out this fusion. For vision-only systems, the fusion of proposals is carried out in 4 steps: i) Project proposals from all cameras to the global coordinate system using proposed mappings, ii) Sort all proposals in descending order based on their confidence scores (obtained from the vehicle detector), iii) Starting with the highest scoring proposal, find the subset of proposals whose euclidean distance in the global coordinate system falls within a predefined threshold. These proposals are considered to belong to the same object and removed from the original set of proposals. iv) The projection of each proposal within this subset is set to the mean of projections of all proposals within the subset. This process is repeated for the remaining proposals until no proposals remain. Alternatively, for a system consisting of LiDAR sensors, Fig. 4: Fusion of object proposals using LiDAR point clouds: Points common to both detections are drawn in green, and the rest are drawn red. we project the 3D point cloud onto each individual camera image. Next, for each pair of proposals, we make a decision as to whether or not they belong to the same object. This is done by considering the back-projected LiDAR points that fall within the bounding box of each proposal (see Figure 4). Let P1 and P2 denote the index set of LiDAR points falling within each bounding box. Then, two proposals are said to belong to the same object if:   P1 ∩ P2 P1 ∩ P2 , ≥ 0.8, (7) max |P1 | |P2 | where |P1 | and |P2 | denote the cardinalities of sets P1 and P2 respectively. It should be noted that after fusion is completed, the union of LiDAR point sets that are backprojected into fused proposals can be used to obtain better projections. IV. M3 OT F RAMEWORK Once we have a set of fused object proposals, we feed them into the MDP as illustrated in Figure 2. This forms a crucial component of the M3 OT framework, details of which are presented in this section. A. Markov Decision Process As detailed in [19], we model the lifetime of a target with a Markov Decision Process (MDP). The MDP consists of the tuple (S, A, T (·, ·), R(·, ·)), where: • States s ∈ S encode the status of the target. • Actions a ∈ A define the actions that can be taken. • The state transition function T : S ×A 7→ S dictates how the target transitions from one state to another, given an action. • The real-valued reward function R : S × A 7→ R assigns the immediate reward received after executing action a in state s. In this study, we retain the states S, actions A and the state transition function T (·, ·) from the MDP framework for 2D MOT [19], while changing only the reward function R(·, ·). States: The state space is partitioned into four subspaces, i.e., S = SActive ∪ ST racked ∪ SLost ∪ SInactive , where each subspace contains infinite number of states which encode the information of the target depending on the feature representation, such as appearance, location, size and history of the target. Figure 2 illustrates the transitions between the four subspaces. Active is the initial state for any target. Whenever 6 an object is detected by the object detector, it enters an Active state. An active target can transition to Tracked or Inactive. Ideally, a true positive from the object detector should transition to a Tracked state, while a false alarm should enter an Inactive state. A tracked target can stay tracked, or transition to a Lost state if the target is not visible due to some reason, e.g. occlusion, or disappearance from sensor range. Likewise, a lost target can stay Lost, or go back to a Tracked state if it appears again, or transition to an Inactive state if it has been lost for a sufficiently long time. Finally, Inactive is the terminal state for any target, i.e., an inactive target stays inactive forever. Actions and Transition Function: Seven possible transitions are designed between the state subspaces, which correspond to seven actions in our target MDP. Figure 2 illustrates these transitions and actions. In the MDP, all the actions are deterministic, i.e., given the current state and an action, we specify a new state for the target. For example, executing action a6 on a Lost target would transfer the target into a Tracked state, i.e., T (sLost , a6 ) = sT racked . Reward Function: As in the original study [19], we learn the reward function from training data, i.e., an inverse reinforcement learning problem, where we use ground truth trajectories of the targets as supervision. B. Policy In the context of our MDP, a policy π is a mapping from the state space S to the action space A, i.e., π : S 7→ A. Given the current state of the target, a policy determines which action to take. Ideally, the policy is chosen to maximize the total reward obtained. In this subsection, we list the policies chosen for the Active and Tracked subspaces, after which we describe the reinforcement learning algorithm used to generate a good policy for data association in the Lost subspace. a) Policy in Active States: In an Active state s, the MDP makes the decision between transferring an object proposal into a Tracked or Inactive state based on whether the detection is true or noisy. To do this, we train a set of binary Support Vector Machines (SVM) offline, one for each camera view, to classify a detection belonging to that view into Tracked or Inactive states using a normalized 5D feature vector φActive (s), i.e., 2D image plane coordinates, width, height and score of the detection, where training examples are collected from training video sequences. This is equivalent to learning the reward function: k RActive (s, a) = y(a)((wActive )T · φActive (s) + bkActive ), (8) for an object proposal belonging to camera k ∈ {1, · · · , K}. k (wActive , bkActive ) defines the learned weights and bias of the SVM for camera k, y(a) = +1 if action a = a1 , and y(a) = −1 if a = a2 (see Figure 2). Training a separate SVM for each camera view allows weights to be learned based on object dimensions and locations in that particular view, and thus works better than training a single SVM for all views. Since a single object can result in multiple proposals, we initialize a tracker for that object if any of the fused proposals result in a positive reward. Note that a false positive from the object detector can still be misclassified and transferred to a Tracked state, which we then leave to be handled by the MDP in Tracked and Lost states. b) Policy in Tracked States: In a Tracked state, the MDP needs to decide whether to keep tracking the target or to transfer it to a Lost state. As long as the target is visible, we should keep tracking it. Else, it should be marked ”lost”. We build an appearance model for the target online and use it to track the target. If the appearance model is able to successfully track the target in the next video frame, the MDP leaves the target in a Tracked state. Otherwise, the target is transferred to a Lost state. Template Representation: The appearance of the target is simply represented by a template that is an image patch of the target in a video frame. Whenever an object detection is transferred to a Tracked state, we initialize the target template with the detection bounding box. If the target is initialized with multiple fused proposals, then each detection is stored as a template. We make note of detections obtained from different camera views, and use these to model the appearance of the target in that view. This is crucial to track objects across camera views under varying perspective changes. When the target is being tracked, the MDP collects its templates in the tracked frames to represent the history of the target, which will be used in the Lost state for decision making. Template Tracking: Tracking of templates is carried out by performing dense optical flow as described in [19]. The stability of the tracking is measured using the median of the FB n errors of all sampled points: emedF B = median(e(ui )i=1 ), where n is the number of points. If emedF B is larger than some threshold, the tracking is considered to be unstable. Moreover, after filtering out unstable matches whose FB error is larger than the threshold, a new bounding box of the target is predicted using the remaining matches by measuring scale change between points before and after. This process is carried out for all camera views in which a target template has been initialized and tracking is in progress. Similar to the original MDP framework, we use the optical flow information in addition to the object proposals history to prevent drifting of the tracker. To do this, we compute the bounding box overlap between the target box for l past frames, and the corresponding detections in each of those frames. Then we compute the mean bounding box overlap for the past L tracked frames omean as another cue to make the decision. Once again, this process is repeated for each camera view the target is being tracked in. In addition to the above features, we also gate the target track. This involves introducing a check to see if the current global position of the tracked target falls within a window (gate) of it’s last known global position. This forbids the target track from latching onto objects that appear close on the image plane, yet are much farther away in the global coordinate frame. We denote the last know global position and the currently tracked global position of the target as xG (t − 1) and x̂G (t) respectively. Finally, we define the reward function in a Tracked state s using the feature set φT racked (s) = 7 0 0 0 0 k K G G ({ekmedF B }K k0 =1 , {omean }k0 =1 , x (t − 1), x̂ (t)) as:  y(a), if ∃k 0 ∈ {1, · · · , K 0 } s.t.    0 0  (ekmedF B < e0 ) ∧ (okmean > o0 ) RT racked (s, a) =  ∧(|xG (t − 1) − x̂G (t)| ≤ tgate ),    −y(a), otherwise, (9) where e0 and o0 are fixed thresholds, y(a) = +1 if action a = a3 , and y(a) = −1 if a = a4 (see Figure 2). k 0 above indexes camera views in which the target is currently being tracked and tgate denotes the gating threshold. So the MDP keeps the target in a Tracked state if emedF B is smaller and omean is larger than their respective thresholds for any one of K 0 camera views in addition to satisfying the gating check. Otherwise, the target is transferred to a Lost state. Template Updating: The appearance model of the target needs to be regularly updated in order to accommodate appearance changes. As in the original work, we adopt a ”lazy” updating rule and resort to the object detector in preventing tracking drift. This is done so that we don’t accumulate tracking errors, but rather rely on data association to handle appearance changes and continue tracking. In addition to this, templates are initialized in views where the target is yet to be tracked by using proposals that are fused with detections corresponding to the tracked location in an adjacent camera view. This helps track objects that move across adjacent camera views, by creating target templates in the new view as soon as they are made available. c) Policy in Lost States: In a Lost state, the MDP needs to decide whether to keep the target in a Lost state, or transition it to a Tracked state, or mark it as Inactive. We simply mark a lost target as Inactive and terminate the tracking if the target has been lost for more than LLost frames. The more challenging task is to make the decision between tracking the target and keeping it as lost. This is treated as a data association problem where, in order to transfer a lost target into a Tracked state, the target needs to be associated with an object proposal, else, the target retains its Lost state. Data Association: Let t denote a lost target, and d be an object detection. The goal of data association is to predict the label y ∈ {+1, −1} of the pair (t, d) indicating that the target is linked (y = +1) or not linked (y = −1) to the detection. Assuming that the detection d belongs to camera view k, this binary classification is performed using the real-valued k linear function f k (t, d) = (wLost )T ·φLost (t, d)+bkLost , where k k (wLost , bLost ) are the parameters that control the function (for camera view k), and φLost (t, d) is the feature vector which captures the similarity between the target and the detection. The decision rule is given by y = +1 if f k (t, d) ≥ 0, else y = −1. Consequently, the reward function for data association in a lost state s given the feature set {φLost (t, dj )}M m=1 is defined as   M km T km RLost (s, a) = y(a) max ((wLost ) · φLost (t, dm ) + bLost ) , m=1 (10) where y(a) = +1 if action a = a6 , y(a) = −1 if a = a5 (see Figure 2), and m indexes M potential detections for input : Set of multi-video sequences V = {(vi1 , · · · , viK )}N i=1 , Ni ground truth trajectories Ti = {tij }j=1 , object proposals Mi Di = {dim }m=1 and their corresponding projections for each multi-video sequence (vi1 , · · · , viK ) k output: Binary classifiers {(wLost , bkLost )}K k=1 for data association 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 repeat foreach multi-video sequence (vi1 , · · · , viK ) in V do foreach target tij do Initialize the MDP in an Active state; l ← index of the first frame in which tij is correctly detected; Transfer the MDP to a Tracked state and initialize the target template for each camera view in which target is observed; while l ≤ index of last frame of tij do Fuse object proposals as described in III; Follow the current policy and choose an action a; Compute the action agt according to the ground truth; if current state is lost and a 6= agt then foreach camera view k in which the target has been seen do k Decide the label ym of the pair k (tkij , dkimk ); S Sk ← Sk {(φ(tkij , dkimk ), ymk )}; k (wLost , bkLost ) ← solution of Eq.11 on S k ; end break; else Execute action a; l ← l + 1; end if l > index of last frame of tij then Mark target tij as successfully tracked; end end end end until all targets are successfully tracked; Algorithm 1: Reinforcement learning of the binary classifier for data association. association. Potential detections for association with a target are simply obtained by applying a gating function around the last known location of the target in the global coordinate system. Note that based on which camera view each detection km m dm originates from, the appropriate weights (wLost , bkLost ) associated with that view are used. As a result, the task of policy learning in the Lost state reduces to learning the set k of parameters {(wLost , bkLost )}K k=1 for the decision functions k K {f (t, d)}k=1 . Reinforcement Learning: We train the binary classifiers described above using the reinforcement learning paradigm. Let V = {(vi1 , · · · , viK )}N i=1 denote a set of multi-video sequences for training, where N is the number of sequences and K is the total number of camera views. Suppose there th i are Ni ground truth targets Ti = {tij }N multij=1 in the i 1 K video sequence (vi , · · · , vi ). Our goal is to train the MDP to successfully track all these targets across all camera views they appear in. We start training with initial weights (w0k , bk0 ) and an empty training set S0k = ∅ for the binary classifier corresponding to each camera view k. Note that when the 8 TABLE II: Features used for data association [19]. We introduce two new features (highlighted in bold) based on the global coordinate positions of targets and detections. Type Notation FB error φ1 , · · · , φ 5 φ6 Feature Description Mean of the median forward-backward errors from the entire, left half, right half, upper half and lower half of the templates obtained from optical flow Mean of the median Normalized Correlation Coefficients (NCC) between image patches around the matched points in optical flow input : A multi-video sequence (v 1 , · · · , v K ), corresponding object proposals D = {dm }M m=1 and their projections, learned binary k classifier weights {(wActive , bkActive )}K k=1 and k {(wLost , bkLost )}K k=1 output: Trajectories of targets T = {tj }N j=1 in the sequence 1 2 3 4 5 6 7 NCC φ7 φ8 Height ratio φ9 Overlap φ10 Score φ11 φ12 Distance φ13 φ14 Mean of the median Normalized Correlation Coefficients (NCC) between image patches around the matched points obtained from optical flow Mean of ratios of the bounding box height of the detection to that of the predicted bounding boxes obtained from optical flow Ratio of the bounding box height of the target to that of the detection Mean of the bounding box overlaps between the detection and the predicted bounding boxes from optical flow Normalized detection score Euclidean distance between the centers of the target and the detection after motion prediction of the target with a linear velocity model Lateral offset between last known global coordinate position of the target and that of the detection Longitudinal offset between last known global coordinate position of the target and that of the detection 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 foreach frame l in (v 1 , · · · , v K ) do Fuse object proposals as described in III; /* process targets in tracked states */ foreach tracked target tj in T do Follow the policy, move the MDP of tj to the next state; end /* process targets in lost states */ foreach lost target tj in T do foreach proposal dm not covered by any tracked target do Compute km f km (tj , dm ) = (wkm )T Lost · φ(tj , dm ) + bLost ; end end Data association with Hungarian algorithm for the lost targets; Initialize target templates for uninitialized camera views using matched (fused) proposals; foreach lost target tj in T do Follow the assignment, move the MDP of tj to the next state; end /* initialize new targets */ foreach proposal dm not covered by any tracked target in T do Initialize a MDP for a new target t with proposal dm ; if action a1 is taken following the policy then Transfer t to the tracked state and initialize the target template for each camera view in which target is observed; S T ←T {t}; else Transfer t to the Inactive state; end end end Algorithm 2: Multi-object tracking with MDPs. weights of the binary classifiers are specified, we have a complete policy for the MDP to follow. So the training algorithm loops over all the multi-video sequences and all the targets, follows the current policy of the MDP to track the targets. The binary classifier or the policy is updated only when the MDP makes a mistake in data association. In this case, the MDP takes a different action than what is indicated by the ground truth trajectory. Suppose the MDP is tracking the j th target tij in the video vik , and on the lth frame of the video, the MDP is in a lost state. Consider the two types of mistakes that can happen: i) The MDP associates the target tkij (l) to an object detection dkm which disagrees with the ground truth, i.e., the target is incorrectly associated to a detection. Then φ(tkij (l), dkm ) is added to the training set S k of the binary classifier for camera k as a negative example. ii) The MDP decides to not associate the target to any detection, but the target is visible and correctly detected by a detection dkm based on the ground truth, i.e., the MDP missed the correct association. Then φ(tkij (l), dkm ) is added to the training set as a positive example. After the training set has been augmented, we update the binary classifier by re-training it on the new training set. Specifically, given the k current training set S k = {(φ(tkm , dkm ), ym )}M m=1 , we solve the following soft-margin optimization problem to obtain a max-margin classifier for data association in camera view k: M X 1 k ||wLost ||2 + C ξm k 2 wLost ,bk Lost ,ξ m=1
k k s.t. ym (wLost )T ·φ(tkm , dkm )+bkLost ≥ 1−ξm , ξm ≥ 0, ∀m, (11) min where ξm , m = 1, · · · , M are the slack variables, and C is a regularization parameter. Once the classifier has been updated, we obtain a new policy which is used in the next iteration of the training process. Note that based on which view the data association is carried out in, the weights of the classifier in that view are updated in each iteration. We keep iterating and updating the policy until all the targets are successfully tracked. Algorithm 1 summarizes the policy learning algorithm. Feature Representation: We retain the same feature representation described in [19], but add two features based on the lateral and longitudinal displacements of the last known target location and the object proposal location in the global coordinate system. This leverages 3D information that is otherwise unavailable in 2D MOT. Table II summarizes our feature representation. 9 TABLE III: Quantitative results showing ablative analysis of our proposed tracker. Criteria for Comparison Tracker Variant Sensor Configuration MOT Metrics [38], [39] # of Cameras Range Sensors MOTA (↑) MOTP (↓) MT (↑) ML (↓) IDS (↓) - 2 3 4 4† 6 8 3 3 3 3 3 3 73.38 77.26 72.81 74.18 79.06 75.10 0.03 0.03 0.05 0.05 0.04 0.04 71.36% 77.34% 72.48% 74.10% 79.66% 70.37% 16.13% 14.49% 20.76% 18.18% 11.93% 14.07% 16 38 49 45 51 59 Projection Scheme (Section V-B) Point cloud based projection IPM projection 8 8 3 3(for fusion) 75.10 47.45 0.04 0.39 70.37% 53.70% 14.07% 19.26% 59 152 Fusion Scheme (Section V-C) Point cloud based fusion Distance based fusion 8 8 3 3(for projection) 75.10 72.20 0.04 0.04 70.37% 68.23% 14.07% 12.23% 59 65 Sensor Modality (Sections V-B,V-C) Cameras+LiDAR Cameras 8 8 3 7 75.10 40.98 0.04 0.40 70.37% 50.00% 14.07% 27.40% 59 171 Vehicle Detector (Section V-D) RefineNet [40] RetinaNet [41] SubCat [42] 8 8 8 3 3 3 75.10 73.89 69.93 0.04 0.05 0.04 70.37% 68.37% 66.67% 14.07% 17.07% 22.22% 59 72 81 Global Position based Features (Section V-E) with {φ13 , φ14 } without {φ13 , φ14 } 8 8 3 3 75.10 71.32 0.04 0.05 70.37% 64.81% 14.07% 17.78% 59 88 Number of Cameras Used (Section V-A) C. Multi-Object Tracking with MDPs After learning the policy/reward of the MDP, we apply it to the multi-object tracking problem. We dedicate one MDP for each target, and the MDP follows the learned policy to track the object. Given a new input video frame, targets in tracked states are processed first to determine whether they should stay as tracked or transfer to lost states. Then we compute pairwise similarity between lost targets and object detections which are not covered by the tracked targets, where non-maximum suppression based on bounding box overlap is employed to suppress covered detections, and the similarity score is computed by the binary classifier for data association. After that, the similarity scores are used in the Hungarian algorithm [35] to obtain the assignment between detections and lost targets. According to the assignment, lost targets which are linked to some object detections are transferred to tracked states. Otherwise, they stay as lost. Finally, we initialize a MDP for each object detection which is not covered by any tracked target. Algorithm 2 describes our 3D MOT algorithm using MDPs in detail. Note that, tracked targets have higher priority than lost targets in tracking, and detections covered by tracked targets are suppressed to reduce ambiguities in data association. V. E XPERIMENTAL A NALYSIS Testbed: Since we propose full-surround MOT using vision sensors, we use a testbed comprising of 8 outside looking RGB cameras (seen in Figure 1). This setup ensures full surround coverage of the scene around the vehicle, while retaining a sufficient overlap between adjacent camera views. Frames captured from these cameras along with annotated surround vehicles are shown in Figure 1. In addition to full vision coverage, the testbed has full-surround Radar and LiDAR FoVs. Despite the final goal of this study being full-surround MOT, we additionally consider cases where only a subset of the vision sensors are used to illustrate the modularity of the approach. More details on the sensors, their synchronization and calibration, and the testbed can be found in [43]. Dataset: To train and test our 3D MOT system, we collect a set of four sequences, each 3-4 minutes long, comprising of multi-camera videos and LiDAR point clouds using our testbed described above. The sequences are chosen much longer than traditional MOT sequences so that long range maneuvers of surround vehicles can be tracked. This is very crucial to autonomous driving. We also annotate all vehicles in the 8 camera videos for each sequence with their bounding box, as well as track IDs. It should be noted that each unique vehicle in the scene is assigned the same ID in all camera views. With these sequences set up, we use one sequence for training our tracker, and reserve the rest for testing. All our results are reported on the entire test set. Evaluation Metrics: We use multiple metrics to evaluate the multiple object tracking performance as suggested by the MOT Benchmark [38]. Specifically, we use the 3D MOT metrics described in [39]. These include Multiple Object Tracking Accuracy (MOTA), Multiple Object Tracking Precision (MOTP), Mostly Track targets (MT, percentage of ground truth objects who trajectories are covered by the tracking output for at least 80%), Mostly Lost targets (ML, percentage of ground truth objects who trajectories are covered by the tracking output less than 20%), and the total number of ID Switches (IDS). In addition to listing the metrics in Table III, we also draw arrows next to each of them indicating if a high (↑) or low (↓) value is desirable. Finally, we provide top-down visualizations of the tracking results in a global coordinate system centered on the ego-vehicle for qualitative evaluation. A. Experimenting with Number of Cameras As our approach to tracking is designed to be extremely modular, we test our tracker with different camera configurations. We experiment with 2, 3, 4, 6 and 8 cameras 10 (a) Ground truth (b) 8 cameras (c) 6 cameras (d) 4 cameras (e) 4† cameras (f) 3 cameras (g) 2 cameras Fig. 5: Tracking results with different number of cameras. The camera configuration used is depicted above each result. respectively. Top-down visualizations of the generated tracks for a test sequence are depicted in Figure 5. The ground truth tracks are provided for visual comparison. As can be seen, the tracker provides consistent results in its FoV irrespective of the camera configuration used, even if the cameras have no overlap between them. The quantitative results on the test set for each camera configuration are listed in Table III. It must be noted that the tracker for each configuration is scored only based on the ground truth tracks visible in that camera configuration. The tracker is seen to score very well on each metric, irrespective of the number of cameras used. This illustrates the robustness of the M3 OT framework. More importantly, it is seen that our tracker performs exceptionally well in the MT and ML metrics, especially in camera configurations with overalapping FoVs. Even though our test sequences are about 3 minutes long in duration, the tracker mostly tracks more than 70% of the targets, while mostly losing only a few. This demonstrates that our M3 OT framework is capable of long-term target tracking. B. Effect of Projection Scheme Figure 6 depicts the tracking results for a test sequence using the two projection schemes proposed. It is obvious that LiDAR based projection results in much better localization in 3D, which leads to more stable tracks and fewer fragments. The IPM based projection scheme is very sensitive to small changes in the input domain, and this leads to considerable errors during gating and data association. This phenomenon is verified by the high MOTP value obtained for IPM based projection as listed in Table III. C. Effect of Fusion Scheme Once again, we see that the LiDAR point cloud based fusion scheme is more reliable in comparison to the distance based approach, albeit this difference is much less noticeable when proposals are projected using LiDAR point clouds. The LiDAR based fusion scheme results in objects being tracked longer (across camera views), and more accurately. The distance based fusion approach on the other hand fails to associate (a) Ground truth (b) LiDAR based projection (c) IPM based projection Fig. 6: Tracking results on a test sequence with different projection schemes. certain proposals, which results in templates not being stored for new camera views, thereby cutting short the track as soon as the target exits the current view. This superiority is reflected in the quantitative results shown in Table III. The drawbacks of the distance based fusion scheme are exacerbated when using IPM to project proposals, reflected by the large drop in MOTA for a purely vision based system. This drop in performance is to be expected in the absence of LiDAR sensors. However, it must be noted that half the targets are still tracked for most of their lifetime, while only a quarter of the targets are mostly lost. D. Effect of using Different Vehicle Detectors Ideally, a tracking-by-detection approach should be detector agnostic. To observe how the tracking results change for different vehicle detectors, we ran the proposed tracker on vehicle detections obtained from three commonly used object detectors [40]–[42]. All three detectors were trained on the KITTI dataset [44] and have not seen examples from the proposed multi-camera dataset. The ROC curves for the detectors on the proposed dataset are shown in Figure 7. The corresponding tracking results for each detector are listed in Table III. Despite the sub-optimal performance of all three 11 our colleagues Kevan Yuen, Nachiket Deo, Ishan Gupta and Borhan Vasli at the Laboratory for Intelligent and Safe Automobiles (LISA), UC San Diego for their useful inputs and help in collecting and annotating the dataset. R EFERENCES Fig. 7: ROC curves for different vehicle detectors on the 4 test sequences. detectors in addition to significant differences in their ROC curves, the tracking results are seen to be relatively unaffected. This indicates that the tracker is less sensitive to errors made by the detector, and consistently manages to correct for it. E. Effect of Global Position based Features Table III indicates a clear benefit in incorporating features {φ13 , φ14 } for data association in Lost states. These features express how near/far a proposal is from the last know location of a target. This helps the tracker disregard proposals that are unreasonably far away from the latest target location. Introduction of these features leads to an improvement in all metrics and therefore justifies their inclusion. VI. C ONCLUDING R EMARKS In this work, we have described a full-surround camera and LiDAR based approach to multi-object tracking for autonomous vehicles. To do so, we extend a 2D MOT approach based on the tracking-by-detection framework, and make it capable of tracking objects in the real world. The proposed M3 OT framework is also made highly modular so that it is capable of working with any camera configuration with varying FoVs, and also with or without LiDAR sensors. An efficient and fast early fusion scheme is adopted to handle object proposals from different sensors within a calibrated camera array. We conduct extensive testing on naturalistic full-surround vision and LiDAR data collected on highways, and illustrate the effects of different camera setups, fusion schemes and 2D-to-3D projection schemes, both qualitatively and quantitatively. Results obtained on the dataset support the modular nature of our framework, as well as its ability to track objects for a long duration. In addition to this, we believe that the M3 OT framework can be used to test the utility of any camera setup, and make suitable modifications thereof to ensure optimum coverage from vision and range sensors. VII. 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Girshick, K. He, and P. Dollár, “Focal loss for dense object detection,” arXiv preprint arXiv:1708.02002, 2017. [42] E. Ohn-Bar and M. M. Trivedi, “Fast and robust object detection using visual subcategories,” in Computer Vision and Pattern Recognition Workshops (CVPRW), 2014 IEEE Conference on. IEEE, 2014, pp. 179– 184. [43] A. Rangesh, K. Yuen, R. K. Satzoda, R. N. Rajaram, P. Gunaratne, and M. M. Trivedi, “A multimodal, full-surround vehicular testbed for naturalistic studies and benchmarking: Design, calibration and deployment,” arXiv preprint arXiv:1709.07502, 2017. [44] A. Geiger, P. Lenz, C. Stiller, and R. Urtasun, “Vision meets robotics: The kitti dataset,” The International Journal of Robotics Research, vol. 32, no. 11, pp. 1231–1237, 2013. Akshay Rangesh is currently working towards his PhD in electrical engineering from the University of California at San Diego (UCSD), with a focus on intelligent systems, robotics, and control. His research interests span computer vision and machine learning, with a focus on object detection and tracking, human activity recognition, and driver safety systems in general. He is also particularly interested in sensor fusion and multi-modal approaches for real time algorithms. Mohan Manubhai Trivedi is a Distinguished Professor at University of California, San Diego (UCSD) and the founding director of the UCSD LISA: Laboratory for Intelligent and Safe Automobiles, winner of the IEEE ITSS Lead Institution Award (2015). Currently, Trivedi and his team are pursuing research in intelligent vehicles, machine perception, machine learning, human-robot interactivity, driver assistance, active safety systems. Three of his students have received ”best dissertation” recognitions. Trivedi is a Fellow of IEEE, ICPR and SPIE. He received the IEEE ITS Society’s highest accolade ”Outstanding Research Award” in 2013. 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arXiv:1410.0640v3 [] 6 Oct 2014 Term-Weighting Learning via Genetic Programming for Text Classification Hugo Jair Escalantea,∗, Mauricio A. Garcı́a-Limóna , Alicia Morales-Reyesa , Mario Graffb , Manuel Montes-y-Gómeza , Eduardo F. Moralesa a Computer Science Department, Instituto Nacional de Astrofı́sica, Óptica y Electrónica, Luis Enrique Erro 1, Puebla, 72840, Mexico b División de Estudios de Postgrado Facultad de Ingenierı́a Eléctrica Universidad Michoacana de San Nicolás de Hidalgo, México Abstract This paper describes a novel approach to learning term-weighting schemes (TWSs) in the context of text classification. In text mining a TWS determines the way in which documents will be represented in a vector space model, before applying a classifier. Whereas acceptable performance has been obtained with standard TWSs (e.g., Boolean and term-frequency schemes), the definition of TWSs has been traditionally an art. Further, it is still a difficult task to determine what is the best TWS for a particular problem and it is not clear yet, whether better schemes, than those currently available, can be generated by combining known TWS. We propose in this article a genetic program that aims at learning effective TWSs that can improve the performance of current schemes in text classification. The genetic program learns how to combine a set of basic units to give rise to discriminative TWSs. We report an extensive experimental study comprising data sets from thematic and non-thematic text classification as well as from image classification. Our study shows the validity of the proposed method; in fact, we show that TWSs learned with the genetic program outperform traditional schemes and other Corresponding author. Email addresses: [email protected] (Hugo Jair Escalante), [email protected] (Mauricio A. Garcı́a-Limón), [email protected] (Alicia Morales-Reyes), [email protected] (Mario Graff), [email protected] (Manuel Montes-y-Gómez), [email protected] (Eduardo F. Morales) ∗ Preprint submitted to Elsevier October 8, 2014 TWSs proposed in recent works. Further, we show that TWSs learned from a specific domain can be effectively used for other tasks. Keywords: Term-weighting Learning, Genetic Programming, Text Mining, Representation Learning, Bag of words. 2000 MSC: code 68T10, 2000 MSC: 68T20 1. Introduction Text classification (TC) is the task of associating documents with predefined categories that are related to their content. TC is an important and active research field because of the large number of digital documents available and the consequent need to organize them. The TC problem has been approached with pattern classification methods, where documents are represented as numerical vectors and standard classifiers (e.g., naı̈ve Bayes and support vector machines) are applied (Sebastiani, 2008). This type of representation is known as the vector space model (VSM) (Salton and Buckley, 1988). Under the VSM one assumes a document is a point in a N-dimensional space and documents that are closer in that space are similar to each other (Turney and Pantel, 2010). Among the different instances of the VSM, perhaps the most used model is the bag-of-words (BOW) representation. In the BOW it is assumed that the content of a document can be determined by the (orderless) set of terms1 it contains. Documents are represented as points in the vocabulary space, that is, a document is represented by a numerical vector of length equal to the number of different terms in the vocabulary (the set of all different terms in the document collection). The elements of the vector specify how important the corresponding terms are for describing the semantics or the content of the document. BOW is the most used document representation in both TC and information retrieval. In fact, the BOW representation has been successfully adopted for processing other media besides text, including, images (Csurka et al., 2004), videos (Sivic and Zisserman, 2003), speech signals (S.Manchala et al., 2014), and time series (Wanga et al., 2013) among others. A crucial component of TC systems using the BOW representation is the 1 A term is any basic unit by which documents are formed, for instance, terms could be words, phrases, and sequences (n-grams) of words or characters. 2 term-weighting scheme (TWS), which is in charge of determining how relevant a term is for describing the content of a document (Feldman and Sanger, 2006; Altyncay and Erenel, 2010; Lan et al., 2009; Debole and Sebastiani, 2003). Traditional TWS are term-frequency (TF ), where the importance of a term in a document is given by its frequency of occurrence in the document; Boolean (B ), where the importance of a term in document is either 1, when the term appear in the document or 0, when the term does not appear in the document; and term-frequency inverse-document-frequency (TF-IDF ), where the importance of a term for a document is determined by its occurrence frequency times the inverse frequency of the term across the corpus (i.e., frequent terms in the corpus, as prepositions and articles, receive a low weight). Although, TC is a widely studied topic with very important developments in the last two decades (Sebastiani, 2008; Feldman and Sanger, 2006), it is somewhat surprising that little attention has been paid to the development of new TWSs to better represent the content of documents for TC. In fact, it is quite common in TC systems that researchers use one or two common TWSs (e.g., B, TF or TF-IDF ) and put more effort in other processes, like feature selection (Forman, 2003; Yang and Pedersen, 1997), or the learning process itself (Agarwal and Mittal, 2014; Aggarwal, 2012; Escalante et al., 2009). Although all of the TC phases are equally important, we think that by putting more emphasis on defining or learning effective TWSs we can achieve substantial improvements in TC performance. This paper introduces a novel approach to learning TWS for TC tasks. A genetic program is proposed in which a set of primitives and basic TWSs are combined through arithmetic operators in order to generate alternative schemes that can improve the performance of a classifier. Genetic programming is a type of evolutionary algorithm in which a population of programs is evolved (Langdon and Poli, 2001), where programs encode solutions to complex problems (mostly modeling problems), in this work programs encode TWSs. The underlying hypothesis of our proposed method is that an evolutionary algorithm can learn TWSs of comparable or even better performance than those proposed so far in the literature. Traditional TWSs combine term-importance and term-document-importance factors to generate TWSs. For instance in TF-IDF, TF and IDF are termdocument-importance and term-importance factors, respectively. Term-document weights are referred as local factors, because they account for the occurrence of a term in a document (locally). On the other hand, term-relevance weights are considered global factors, as they account for the importance of a term 3 across the corpus (globally). It is noteworthy that the actual factors that define a TWS and the combination strategy itself have been determined manually. Herein we explore the suitability of learning these TWSs automatically, by providing a genetic program with a pool of TWSs’ building blocks with the goal of evolving a TWS that maximizes the classification performance for a TC classifier. We report experimental results in many TC collections that comprise both: thematic and non-thematic TC problems. Throughout extensive experimentation we show that the proposed approach is very competitive, learning very effective TWSs that outperform most of the schemes proposed so far. We evaluate the performance of the proposed approach under different settings and analyze the characteristics of the learned TWSs. Additionally, we evaluate the generalization capabilities of the learned TWSs and even show that a TWS learned from text can be used to effectively represent images under the BOW formulation. The rest of this document is organized as follows. Next section formally introduces the TC task and describes common TWSs. Section 3 reviews related work on TWSs. Section 4 introduces the proposed method. Section 5 describes the experimental settings adopted in this work and reports results of experiments that aim at evaluating different aspects of the proposed approach. Section 6 presents the conclusions derived from this paper and outlines future research directions. 2. Text classification with the Bag of Words The most studied TC problem is the so called thematic TC (or simply text categorization) (Sebastiani, 2008), which means that classes are associated to different themes or topics (e.g., classifying news into “Sports” vs. “Politics” categories). In this problem, the sole occurrence of certain terms may be enough to determine the topic of a document; for example, the occurrence of words/terms “Basketball”, “Goal”, “Ball”, and “Football” in a document is strong evidence that the document is about “Sports”. Of course, there are more complex scenarios for thematic TC, for example, distinguishing documents about sports news into the categories: “Soccer” vs. “NFL”. Non-thematic TC, on the other hand, deals with the problem of associating documents with labels that are not (completely) related to their topics. Nonthematic TC includes the problems of authorship attribution (Stamatatos, 2009), opinion mining and sentiment analysis (Pang et al., 2002), authorship verification (Koppel and Schler, 2004), author profiling (Koppel et al., 2002), 4 among several others (Reyes and Rosso, 2014; Kiddon and Brun, 2011). In all of these problems, the thematic content is of no interest, nevertheless, it is common to adopt standard TWSs for representing documents in nonthematic TC as well (e.g., BOW using character n-grams or part-of-speech tags (Stamatatos, 2009)). It is noteworthy that the BOW representation has even trespassed the boundaries of the text media. Nowadays, images (Csurka et al., 2004), videos (Sivic and Zisserman, 2003), audio (S.Manchala et al., 2014), and other types of data (Wanga et al., 2013) are represented throughout analogies to the BOW. In non-textual data, a codebook is first defined/learned and then the straight BOW formulation is adopted. In image classification, for example, visual descriptors extracted from images are clustered and the centers of the clusters are considered as visual words (Csurka et al., 2004; Zhang et al., 2007). Images are then represented by numerical vectors (i.e., a VSM) that indicate the relevance of visual words for representing the images. Interestingly, in other media than text (e.g., video, images) it is standard to use only the TF TWS, hence motivating the study on the effectiveness of alternative TWSs in non-textual tasks. Accordingly, in this work we also perform experiments on learning TWSs for a standard computer vision problem (Fei-Fei et al., 2004). TC is a problem that has been approached mostly as a supervised learning task, where the goal is to learn a model capable of associating documents to categories (Sebastiani, 2008; Feldman and Sanger, 2006; Agarwal and Mittal, 2014). Consider a data set of labeled documents D = (xi , yi ){1,...,N } with N pairs of documents (xi ) and their classes (yi ) associated to a TC problem; where we assume xi ∈ Rp (i.e., a VSM) and yi ∈ C = {1, . . . K}, for a problem with K−classes. The goal of TC is to learn a function f : Rp → C from D that can be used to make predictions for documents with unknown labels, the so called test set: T = {xT1 , . . . , xTM }. Under the BOW formulation, the dimensionality of documents’ representation, p, is defined as p = |V |, where V is the vocabulary (i.e., the set all the different terms/words that appear in a corpus). Hence, each document di is represented by a numerical vector xi = hxi,1 . . . , xi,|V | i, where each element xi,j , j = 1, . . . , |V |, of xi indicates how relevant word tj is for describing the content of di , and where the value of xi,j is determined by the TWS. Many TWSs have been proposed so far, including unsupervised (Sebastiani, 2008; Salton and Buckley, 1988; Feldman and Sanger, 2006) and supervised schemes (Debole and Sebastiani, 2003; Lan et al., 2009), see Section 3. Unsupervised TWSs are the most used ones, they were firstly proposed for 5 information retrieval tasks and latter adopted for TC (Sebastiani, 2008; Salton and Buckley, 1988). Unsupervised schemes rely on term frequency statistics and measurements that do not take into account any label information. For instance, under the Boolean (B) scheme xi,j = 1 if f term tj appears in document i and 0 otherwise; while in the term-frequency (TF ) scheme, xi,j = #(di , tj ), where #(di , tj ) accounts for the times term tj appears in document di . On the other hand, supervised TWSs aim at incorporating discriminative information into the representation of documents (Debole and Sebastiani, 2003). For example in the TF-IG scheme, xi,j = #(di , tj ) × IG(tj ), is the product of the TF TWS for term tj and document di (a local factor) with the information gain of term tj (IG(tj ), global factor). In this way, the discrimination power of each term is taken into account for the document representation; in this case through the information gain value (Yang and Pedersen, 1997). It is important to emphasize that most TWSs combine information from both term-importance (global) and term-document-importance (local) factors (see Section 3), for instance, in the TF-IG scheme, IG is a term-importance factor, whereas TF is a termdocument-importance factor. Although acceptable performance has been reported with existing TWS, it is still an art determining the adequate TWS for a particular data set; as a result, mostly unsupervised TWSs (e.g., B, TF and TF-IDF ) have been adopted for TC systems (Feldman and Sanger, 2006; Aggarwal, 2012). A first hypothesis of this work is that different TWSs can achieve better performance on different TC tasks (e.g., thematic TC vs. non-thematic TC); in fact, we claim that within a same domain (e.g., news classification) different TWSs are required to obtain better classification performance on different data sets. On the other hand, we notice that TWSs have been defined as combinations of term-document weighting factors (which can be seen as other TWSs, e.g., TF ) and term-relevance measurements (e.g., IDF or IG), where the definition of TWSs has been done by relying on the expertise of users/researchers. Our second hypothesis is that the definition of new TWSs can be automated. With the aim of verifying both hypotheses, this paper introduces a genetic program that learns how to combine term-document-importance and termrelevance factors to generate effective TWSs for diverse TC tasks. 6 3. Related work As previously mentioned, in TC it is rather common to use unsupervised TWSs to represent documents, specifically B, TF and TF-IDF schemes are very popular (see Table 1). Their popularity derives from the fact that these schemes have proved to be very effective in information retrieval (Salton and Buckley, 1988; Baeza-Yates and Ribeiro-Neto, 1999; Turney and Pantel, 2010) and in many TC problems as well as (Sebastiani, 2008; Feldman and Sanger, 2006; Agarwal and Mittal, 2014; Aggarwal, 2012; Aggarwal and Zhai, 2012). Unsupervised TWSs mainly capture term-document occurrence (e.g., term occurrence frequency, TF ) and term-relevance (e.g., inverse document frequency, IDF ) information. While acceptable performance has been obtained with such TWSs in many applications, in TC one has available labeled documents, and hence, document-label information can also be exploited to obtain more discriminative TWSs. This observation was noticed by Debole & Sebastiani and other authors that have introduced supervised TWSs (Debole and Sebastiani, 2003; Lan et al., 2009). Supervised TWSs take advantage of labeled data by incorporating a discriminative term-weighting factor into the TWSs. In (Debole and Sebastiani, 2003) TWSs were defined by combining the unsupervised TF scheme with the following term-relevance criteria: information gain (TF-IG), which measures the reduction of entropy when using a term as classifier (Yang and Pedersen, 1997); χ2 (TF-CHI ), makes an independence test regarding a term and the classes (Sebastiani, 2008); and gain-ratio (TF-GR) measuring the gain-ratio when using the term as classifier (Debole and Sebastiani, 2003). The conclusions from (Debole and Sebastiani, 2003) were that small improvements can be obtained with supervised TWSs over unsupervised ones. Although somewhat disappointing, it is interesting that for some scenarios supervised TWSs were beneficial. More recently, Lan et al. proposed an alternative supervised TWS (Lan et al., 2009), the so called TF-RF scheme. TF-RF combines TF with a criterion that takes into account the true positives and true negative rates when using the occurrence of the term as classifier. In (Lan et al., 2009) the proposed TF-RF scheme obtained better performance than unsupervised TWSs and even outperformed the schemes proposed in (Debole and Sebastiani, 2003). In (Altyncay and Erenel, 2010) the RF term-relevance factor was compared with alternative weights, including mutual information, odds ratio and χ2 ; in that workRF outperformed the other term-importance criteria. 7 Table 1 shows most of the TWSs proposed so far for TC. It can be observed that TWSs are formed by combining term-document (TDR) and term (TR) relevance weights. The selection of what TDR and TR weights to use rely on researchers choices (and hence on their biases). It is quite common to use TF as TDR, because undoubtedly the term-occurrence frequency carries on very important information: we need a way to know what terms a document is associated with. However, it is not that clear what TR weight to use, as there is a wide variety of TR factors that have been proposed. The goal of TRs is to determine the importance of a given term, with respect to the documents in a corpus (in the unsupervised case) or to the classes of the problem (in the supervised case). Unsupervised TRs include: global term-frequency, and inverse document frequency (IDF) TRs. These weights can capture word importance depending on its global usage across a corpus, however, for TC seems more appealing to use discriminative TRs as one can take advantage of training labeled data. In this aspect, there is a wide variety of supervised TRs that have been proposed, including: mutual information, information gain, odds ratio, etcetera (Aggarwal and Zhai, 2012). Table 1: Common term weighting schemes for TC. In every TWS, xi,j indicates how relevant term tj is for describing the content of document di , under the corresponding TWS. N is the number of documents in training data set, #(di , tj ) indicates the frequency of term tj in document di , df (tj ) is the number of documents in which term tj occurs, IG(tj ) is the information gain of term tj , CHI(tj ) is the χ2 statistic for term tj , and T P , T N are the true positive and true negative rates for term tj (i.e., the number of positive, resp. negative, documents that contain term tj ). Acronym Name B Boolean Formula xi,j = 1{#(t ,d )>0} i j TermFrequency TF - Inverse Document Frequency TF - Information Gain xi,j = #(ti , dj ) TFCHI TF square xi,j = #(ti , dj ) × CHI(tj ) TF-RF TF - Relevance Frequency TF TFIDF TF-IG Chi- xi,j = #(ti , dj ) × ) log( df N (tj ) xi,j = IG(tj ) xi,j = #(ti , dj ) × #(ti , dj ) × TP ) log(2 + max(1,T N) Description Indicates the prescense/abscense of terms Accounts for the frequency of occurrence of terms An TF scheme that penalizes the frequency of terms across the collection TF scheme that weights term occurrence by its information gain across the corpus. TF scheme that weights term occurrence by its χ2 statistic TF scheme that weights term occurrence by its χ2 statistic Ref. (Salton and Buckley, 1988) (Salton and Buckley, 1988) (Salton and Buckley, 1988) (Debole and Sebastiani, 2003) (Debole and Sebastiani, 2003) (Lan et al., 2009) The goal of a supervised TR weight is to determine the importance of a given term with respect to the classes. The simplest, TR would be to estimate the correlation of term frequencies and the classes, although any other 8 criterion that accounts for the association of terms and classes can be helpful as well. It is interesting that although many TRs are available out there, they have been mostly used for feature selection rather than for building TWSs for TC. Comprehensive and extensive comparative studies using supervised TRs for feature selection have been reported (Altyncay and Erenel, 2010; Forman, 2003; Yang and Pedersen, 1997; Mladenic and Grobelnik, 1999). Although not being conclusive, these studies serve to identify the most effective TRs weights, such weights are considered in this study. To the best of our knowledge, the way we approach the problem of learning TWSs for TC is novel. Similar approaches based on genetic programming to learn TWSs have been proposed in (Cummins and O’Riordan, 2006, 2007, 2005; Trotman, 2005; Oren, 2002; Fan et al., 2004a), however, these researchers have focused on the information retrieval problem, which differs significantly from TC. Early approaches using genetic programming to improve the TF-IDF scheme for information retrieval include those from (Trotman, 2005; Oren, 2002; Fan et al., 2004a,b). More recently, Cummins et al. proposed improved genetic programs to learn TWSs also for information retrieval (Cummins and O’Riordan, 2006, 2007, 2005). Although the work by Cummins et al. is very related to ours, there are major differences (besides the problem being approached): Cummins et al. approached the information retrieval task and defined a TWS as a combination of three factors: local, global weighting schemes and a normalization factor2 . The authors designed a genetic program that aimed at learning a TWS by evolving the local and global schemes separately. Only 11 terminals, including constants, were considered. Since information retrieval is an unsupervised task, the authors have to use a whole corpus with relevance judgements (i.e., a collection of documents with queries and the set of relevant documents to each query) to learn the TWS, which, once learned, could be used for other information retrieval tasks. Hence they require a whole collection of documents to learn a TWS. On the other hand, the authors learned a TWS separately, first a global TWS was evolved fixing a binary local scheme, then a local scheme was learned by fixing the learned global weight. Hence, they restrict the search space for the genetic program, which 2 Recall a local factor incorporates term information (locally) available in a document, whereas a global term factor takes into account term statistics estimated across the whole corpus. In information retrieval it is also common to normalize the vectors representing a document to reduce the impact of the length of a document. 9 may limit the TWSs that can be obtained. Also, it is worth noticing that the focus of the authors of (Cummins and O’Riordan, 2006, 2007, 2005) was on learning a single, and generic TWS to be used for other information retrieval problems, hence the authors performed many experiments and reported the single best solution they found after extensive experimentation. Herein, we provide an extensive evaluation of the proposed approach, reporting average performance over many runs and many data sets. Finally, one should note that the approach from (Cummins and O’Riordan, 2006, 2007, 2005) required of large populations and numbers of generations (1000 individuals and 500 generations were used), whereas in this work competitive performance is obtained with only 50 individuals and 50 generations. 4. Learning term-weighting schemes via GP As previously mentioned, the traditional approach for defining TWSs has been somewhat successful so far. Nevertheless, it is still unknown whether we can automatize the TWS definition process and obtain TWSs of better classification performance in TC tasks. In this context, we propose a genetic programming solution that aims at learning effective TWSs automatically. We provide the genetic program with a pool of TDR and TR weights as well as other TWSs and let a program search for the TWS that maximizes an estimate of classification performance. Thus, instead of defining TWSs based on our own experiences on text mining, we let a computer itself to build an effective TWS. The advantages of this approach are that it may allow to learn a specific TWS for each TC problem, or to learn TWSs from one data set (e.g., a small one) and implement it in a different collection (e.g., a huge one). Furthermore, the method reduces the dependency on users/dataanalysts and their degree of expertise and biases for defining TWSs. The rest of this section describes the proposed approach. We start by providing a brief overview of genetic programming, then we explain in detail the proposal, finally, we close this section with a discussion on the benefits and limitations of our approach. 4.1. Genetic programming Genetic programming (GP) (Langdon and Poli, 2001) is an evolutionary technique which follows the reproductive cycle of other evolutionary algorithms such as genetic algorithms (see Figure 1): an initial population is created (randomly or by a pre-defined criterion), after that, individuals are 10 selected, recombined, mutated and then placed back into the solutions pool. The distinctive feature of GP, when compared to other evolutionary algorithms, is in that complex data structures are used to represent solutions (individuals), for example, trees or graphs. As a result, GP can be used for solving complex learning/modeling problems. In the following we describe the GP approach to learn TWSs for TC. Figure 1: A generic diagram of an evolutionary algorithm. 4.2. TWS learning with genetic programming We face the problem of learning TWSs as an optimization one, in which we want to find a TWSs that maximizes the classification performance of a classifier trained with the TWS. We define a valid TWS as the combination of: (1) other TWSs, (2) TR and (3) TDR factors, and restrict the way in which such components can be combined by a set of arithmetic operators. We use GP as optimization strategy, where each individual corresponds to a tree-encoded TWS. The proposed genetic program explores the search space of TWSs that can be generated by combining TWSs, TRs and TDRs with 11 a predefined set of operators. The rest of this section details the components of the proposed genetic program, namely, representation, terminals and function set, genetic operators and fitness function. 4.2.1. Representation Solutions to our problem are encoded as trees, where we define terminal nodes to be the building blocks of TWSs. On the other hand, we let internal nodes of trees to be instantiated by arithmetic operators that combine the building blocks to generate new TWSs. The representation is graphically described in Figure 2. Figure 2: Representation adopted for TWS learning. 4.2.2. Terminals and function set As previously mentioned, traditional TWSs are usually formed by two factors: a term-document relevance (TDR) weight and a term-relevance (TR) factor. The most used TDR is term frequency (TF ), as allows one to relate documents with the vocabulary. We consider TF as TDR indicator, but also we consider standard TWSs (e.g., Boolean, TD, RF ) as TDR weights. The decision to include other TWSs as building blocks is in order to determine whether standard TWSs can be enhanced with GP. Regarding TR, there are many alternatives available. In this work we analyzed the most common and effective TR weights as reported in the literature (Sebastiani, 2008; 12 Altyncay and Erenel, 2010; Lan et al., 2009; Debole and Sebastiani, 2003; Forman, 2003) and considered them as building blocks for generating TWSs. Finally we also considered some constants as building blocks. The full set of building blocks (terminals in the tree representation) considered is shown in Table 1, whereas the set of operators considered in the proposed method √ 2 (i.e., the function set) is the following: F = {+, −, ∗, /, log2 x, x, x }, where F includes operators of arities one and two. Table 2: Terminal set. Variable W1 W2 W3 W4 W5 W6 W7 W8 W9 W10 W11 W12 W13 W14 W15 W16 W17 W18 W19 W20 W21 W22 Meaning N, Constant matrix, the total number of training documents. kV k, Constant matrix, the number of terms. CHI, Matrix containing in each row the vector of χ2 weights for the terms. IG, Matrix containing in each row the vector of information gain weights for the terms. T F − IDF , Matrix with the TF-IDF term weighting scheme. T F , Matrix containing the TF term-weighting scheme. F GT , Matrix containing in each row the global term-frequency for all terms. T P , Matrix containing in each row the vector of true positives for all terms. F P , Matrix containing in each row the vector of false positives. T N, Matrix containing in each row the vector of true negatives. F N, Matrix containing in each row the vector of false negatives. Accuracy, Matrix in which each row contains the accuracy obtained when using the term as classifier. Accuracy Balance, Matrix containing the AC Balance each (term, class). BNS, An array that contains the value for each BNS per (term, class). DF req, Document frequency matrix containing the value for each (term, class). F M easure, F-Measure matrix containing the value for each (term, class). OddsRatio, An array containing the OddsRatio term-weighting. P ower, Matrix containing the Power value for each (term, class). P robabilityRatio, Matrix containing the ProbabilityRatio each (term, class). M ax T erm, Matrix containing the vector with the highest repetition for each term. RF , Matrix containing the RF vector. T F × RF , Matrix containing T F × RF . In the proposed approach, a TWS is seen as a combination of building blocks by means of arithmetic operators. One should note, however, that three types of building blocks are considered: TDR, TR and constants. Hence we must define a way to combine matrices (TDR weights), vectors (TR scores) and scalars (the constants), in such a way that the combination leads to a TWS (i.e., a form of TDR). Accordingly, and for easiness of implementation, each building block shown in Table 1 is processed as a matrix of the same length as the TWS (i.e., N × |V |) and operations are performed element-wise. In this way a tree can be directly evaluated, and the operators are applied between each element of the matrices, leading to a TWS. TDRs are already matrices of the same size as the TWSs: N × |V |. In the case of TRs, we have a vector of length |V |, thus for each TR we generate a matrix of size N × |V | where each of its rows is the TR; that is, we repeat N times the TR weight. In this way, for example, a TWS like TF-IDF can 13 be obtained as T F × IDF , where the × operator means that each element tfi,j of TF is multiplied by each element of the IDF matrix idfi,j and where idfi,j = log( dfN(tj ) ) for i = 1, . . . , N, all TRs were treated similarly. In the case of constants we use a scalar-matrix operator, which means that the constant is operated with each element of the matrix under analysis. Estimating the matrices each time a tree is evaluated can be a time consuming process, therefore, at the beginning of the search process we compute the necessary matrices for every terminal from Table 1. Hence, when evaluating an individual we only have to use the values of the precomputed matrices and apply the operators specified by a tree. 4.2.3. Genetic operators As explained above, in GP a population of individuals is initialized and evolved according to some operators that aim at improving the quality of the population. For initialization we used the standard ramped-half-and-half strategy (Eiben and Smith, 2010), which generates half of the population with (balanced) trees of maximum depth, and the other half with trees of variable depth. As genetic operators we also used standard mechanisms: we considered the subtree crossover and point mutation. The role of crossover is to take two promising solutions and combine their information to give rise to two offspring, with the goal that the offspring have better performance than the parents. The subtree crossover works by selecting two parent solutions/trees (in our case, via tournament) and randomly select an internal node in each of the parent trees. Two offspring are created by interchanging the subtrees below the identified nodes in the parent solutions. The function of the mutation operator is to produce random variations in the population, facilitating the exploration capabilities of GP. The considered mutation operator first selects an individual to be mutated. Next an internal node of the individual is identified, and if the internal node is an operator (i.e., a member of F ) it is replaced by another operator of the same arity. If the chosen node is a terminal, it is replaced by another terminal. Where in both cases the replacing node is selected with uniform probability. 4.2.4. Fitness function As previously mentioned, the aim of the proposed GP approach is to generate a TWS that obtains competitive classification performance. In this direction, the goodness of an individual is assessed via the classification performance of a predictive model that uses the representation generated by the 14 TWS. Specifically, given a solution to the problem, we first evaluate the tree to generate a TWS using the training set. Once training documents are represented by the corresponding TWS, we perform a k−fold cross-validation procedure to assess the effectiveness of the solution. In k−fold cross validation, the training set is split into k disjoint subsets, and k rounds of training and testing are performed; in each round k − 1 subsets are used as training set and 1 subset is used for testing, the process is repeated k times using a different subset for testing each time. The average classification performance is directly used as fitness function. Specifically, we evaluate the performance of classification models with the f1 measure. Let T P , F P and F N to denote the true positives, false positives and false negative rates for a particular class, P P precision (P rec) is defined as T PT+F and recall (Rec) as T PT+F . f1 -measure P N rec×Rec . is simply the harmonic average between precision and recall: f1 = 2×P P rec+Rec The average across classes is reported (also called, macro-average f1 ), this way of estimating the f1 -measure is known to be particularly useful when tackling unbalanced data sets (Sebastiani, 2008). Since under the fitness function k models have to be trained and tested for the evaluation of a single TWS, we need to look for an efficient classification model that, additionally, can deal naturally with the high-dimensionality of data. Support vector machines (SVM) comprise a type of models that have proved to be very effective for TC (Sebastiani, 2008; Joachims, 2008). SVMs can deal naturally with the sparseness and high dimensionality of data, however, training and testing an SVM can be a time consuming process. Therefore, we opted for efficient implementations of SVMs that have been proposed recently (Zhang et al., 2012; Djuric et al., 2013). That methods are trained online and under the scheme of learning with a budget. We use the predictions of an SVM as the fitness function for learning TWSs. Among the methods available in (Djuric et al., 2013) we used the low-rank linearized SVM (LLSMV) (Zhang et al., 2012). LLSVM is a linearized version of non-linear SVMs, which can be trained efficiently with the so called block minimization framework (Chang and Roth, 2011). We selected LLSVM instead of alterative methods, because this method has outperformed several other efficient implementations of SVMs, see e.g., (Djuric et al., 2013; Zhang et al., 2012). 4.3. Summary We have described the proposed approach to learn TWSs via GP. When facing a TC problem we start by estimating all of the terminals described in Table 1 for the training set. The terminals are feed into the genetic program, 15 together with the function set. We used the GPLAB toolbox for implementing the genetic program with default parameters (Silva and Almeida, 2003). The genetic program searches for the tree that maximizes the k−fold cross validation performance of an efficient SVM using training data only. After a fixed number of generations, the genetic program returns the best solution found so far, the best TWS. Training and test (which was not used during the search process) data sets are represented according to such TWS. One should note that all of the supervised term-weights in Table 1 are estimated from the training set only (e.g., the information gain for terms is estimated using only the labeled training data); for representing test data we use the precomputed term-weights. Next, the LLSVM is trained in training data and the trained model makes predictions for test samples. We evaluate the performance of the proposed method by comparing the predictions of the model and the actual labels for test samples. The next section reports results of experiments that aim at evaluating the validity of the proposed approach. 5. Experiments and results This section presents an empirical evaluation of the proposed TWL approach. The goal of the experimental study is to assess the effectiveness of the learned TWSs and compare their performance to existing schemes. Additionally, we evaluate the generalization performance of learned schemes, and their effectiveness under different settings. 5.1. Experimental settings For experimentation we considered a suite of benchmark data sets associated to three types of tasks: thematic TC, authorship attribution (AA, a non-thematic TC task) and image classification (IC). Table 3 shows the characteristics of the data sets. We considered three types of tasks because we wanted to assess the generality of the proposed approach. Seven thematic TC data sets were considered, in these data sets the goal is to learn a model for thematic categories (e.g., sports news vs. religion news). The considered data sets are the most used ones for the evaluation of TC systems (Sebastiani, 2008). For TC data sets, indexing terms are the words (unigrams). Likewise, seven data sets for AA were used, the goal in these data sets is to learn a model capable of associating documents with authors. Opposed to thematic collections, the goal in AA is to model the 16 Table 3: Data sets considered for experimentation Text categorization Data set Classes Terms Train Test Reuters-8 8 23583 5339 2333 Reuters-10 10 25283 6287 2811 20-Newsgroup 20 61188 11269 7505 TDT-2 30 36771 6576 2818 WebKB 4 7770 2458 1709 Classic-4 4 5896 4257 2838 CADE-12 12 193731 26360 14618 Authorship attribution Data set Classes Terms Train Test CCA-10 10 15587 500 500 Poetas 5 8970 71 28 Football 3 8620 52 45 Business 6 10550 85 90 Poetry 6 8016 145 55 Travel 4 11581 112 60 Cricket 4 10044 98 60 Image Classification Data set Classes Terms Train Test Caltech-101 101 12000 1530 1530 Caltech-tiny 5 12000 75 75 writing style of authors, hence, it has been shown that different representations and attributes are necessary for facing this task (Stamatatos, 2009). Accordingly, indexing terms in AA data sets were 3-grams of characters, that is, sequences of 3-characters found in documents, these terms have proved to be the most effective ones in AA (Stamatatos, 2009; Escalante et al., 2011; Luyckx and Daelemans, 2010). Finally, two data sets for image classification, taken from the CALTECH-101 collection, were used. We considered the collection under the standard experimental settings (15 images per class for training and 15 images for testing), two subsets of the CALTECH-101 data set were used: a small one with only 5 categories and the whole data set with 102 classes (101 object categories plus background) (Fei-Fei et al., 2004). Images were represented under the Bag-of-Visual-Words formulation using dense sift descriptors (PHOW features): descriptors extracted from images were clustered using k−means, the centers of the clusters are the visual words (indexing terms), images are then represented by accounting the occurrence of visual words, the VLFEAT toolbox was used for processing images (Vedaldi and Fulkerson, 2008). The considered data sets have been partitioned into training and test subsets (the number of documents for each partition and each data set are shown in Table 3). For some data sets there were predefined categories, while 17 for others we randomly generated them using 70% of documents for training and the rest for testing. All of the preprocessed data sets in Matlab format are publicly available3 . For each experiment, the training partition was used to learn the TWS, as explained in Section 4. The learned TWS is then evaluated in the corresponding test subset. We report two performance measures: accuracy, which is the percentage of correctly classified instances, and f1 measure, which assesses the tradeoff between precision and recall across classes (macro-average f1 ), recall that f1 was used as fitness function (see Section 4). The genetic program was run for 50 generations using populations of 50 individuals, we would like to point out that in each run of the proposed method we have used default parameters. It is expected that by optimizing parameters and running the genetic program for more generations and larger populations we could obtain even better results. The goal of our study, however, was to show the potential of our method even with default parameters. 5.2. Evaluation of TWS Learning via Genetic Programming This section reports experimental results on learning TWSs with the genetic program described in Section 4. The goal of this experiment is to assess how TWSs learned via GP compare with traditional TWSs. The GP method was run on each of the 16 data sets from Table 3, since the vocabulary size for some data sets is huge we decided to reduce the number of terms by using term-frequency as criterion. Thus, for each data set we considered the top 2000 more frequent terms during the search process. In this way, the search process is accelerated at no significant loss of accuracy. In Section 5.3 we analyze the robustness of our method when using the whole vocabulary size for some data sets. For each data set we performed 5 runs with the GP-based approach, we evaluated the performance of each learned TWS and report the average and standard deviation of performance across the five runs. Tables 4, 5, and 6 show the performance obtained by TWSs learned for thematic TC, AA and IC data sets, respectively. In the mentioned tables we also show the result obtained by the best baseline in each collection. Best baseline is the best TWS we found (from the set of TWSs reviewed in related work and the TWSs in Table 1) for each data set (using the test-set performance). Please note that 3 http://ccc.inaoep.mx/ hugojair/TWL/ 18 under these circumstances best baseline is in fact, a quite strong baseline for our GP method. Also, we would like to emphasize that no parameter of the GP has been optimized, we used the same default parameters for every execution of the genetic program. Table 4: Classification performance on thematic TC obtained with learned TWSs and the best baseline. PG- Avg. Best baseline Data set f1 Acc. f1 Acc. Baseline + Reuters-8 90.56+ 1.43 91.35 1.99 86.94 88.63 TF − − + Reuters-10 88.21+ 2.69 91.84 1.01 85.24 93.25 TFIDF − − 20-Newsgroup 66.23+ 67.97+ 59.21 61.99 TF − 3.84 − 4.16 TDT-2 96.95+ 96.95+ 95.20 95.21 TFIDF − 0.41 − 0.57 WebKB 88.79+ 89.12+ 87.49 88.62 B − 1.26 − 1.30 + Classic-4 94.75− 1.08 95.42+ 94.68 94.86 TF − 0.67 CADE-12 41.03+ 53.80+ 39.30 41.89 TF − 4.45 − 4.0 From Table 4 it can be seen that, regarding the best baseline, different TWSs obtained better performance for different data sets. Hence evidencing the fact that different TWSs are required for different problems. On the other hand, it can be seen that the average performance of TWSs learned with our GP outperformed significantly the best baseline in all but one result (accuracy for Reuters-10 data set). The differences in performance are large, mainly for the f1 measure, which is somewhat expected as this was the measure used as fitness function (recall f1 measure is appropriate to account for the class imbalance across classes); hence showing the competitiveness of our proposed approach for learning effective TWSs for thematic TC tasks. Table 5: Classification performance on AA obtained with learned TWSs and the best baseline. PG- Avg. Best baseline Data set f1 Acc. f1 Acc. Baseline + CCA-10 70.32+ 2.73 73.72 2.14 65.90 73.15 TF-IG − − + Poetas 72.23+ 1.49 72.63 1.34 70.61 71.84 TF-IG − − + Football 76.37+ 9.99 83.76 4.27 76.45 83.78 TF-CHI − − Business 78.08+ 83.58+ 73.77 81.49 TF-CHI − 4.87 − 1.57 + Poetry 70.03+ 7.66 74.05 7.38 59.93 76.71 B − − + Travel 73.92+ 10.26 78.45 6.72 71.75 75.32 TF-CHI − − Cricket 88.10+ 92.06+ 89.81 91.89 TF-CHI − 7.12 − 3.29 From Table 5 it can be seen that for AA data sets the best baseline 19 performs similarly as the proposed approach. In terms of f1 measure, our method outperforms the best baseline in 5 out of 7 data sets, while in accuracy our method beats the best baseline in 4 out of 7 data sets. Therefore, our method still obtains comparable (slightly better) performance to the best baselines, which for AA tasks were much more competitive than in thematic TC problems. One should note that for PG we are reporting the average performance across 5 runs, among the 5 runs we found TWSs that consistently outperformed the best baseline. Table 6: Classification performance on IC obtained with learned TWSs and the best baseline. PG- Avg. Best baseline Data set f1 Acc. f1 Acc. Baseline + Caltech-101 61.91+ 1.41 64.02 1.42 58.43 60.28 B − − + Caltech-tiny 89.70+ 2.44 91.11 2.36 85.65 86.67 TF − − It is quite interesting that, comparing the best baselines from Tables 4 and 5, for AA tasks supervised TWSs obtained the best results (in particular TF-CHI in 4 out of 7 data sets), whereas for thematic TC unsupervised TWSs performed better. Again, these results show that different TWSs are required for different data sets and different types of problems. In fact, our results confirm the fact that AA and thematic TC tasks are quite different, and, more importantly, our study provides evidence on the suitability of supervised TWSs for AA; to the best of our knowledge, supervised TWSs have not been used in AA problems. Table 6 shows the results obtained for the image categorization data sets. Again, the proposed method obtained TWSs that outperformed the best baselines. This result is quite interesting because we are showing that the TWS plays a key role in the classification of images under the BOVWs approach. In computer vision most of the efforts so far have been devoted to the development of novel/better low-level image-descriptors, using a BOW with predefined TWS. Therefore, our results pave the way for research on learning TWSs for image categorization and other tasks that rely in the BOW representation (e.g. speech recognition and video classification). Figure 3 and Table 7 complement the results presented so far. Figure 3 indicates the difference in performance between the (average of) learned TWSs and the best baseline for each of the considered data sets. We can clearly appreciate from this figure the magnitude of improvement offered by the learned TWSs, which in some cases is too large. 20 12 f 1 Difference in performance 10 Accuracy 8 6 4 2 0 Catech−tiny Cricket Travel Poetry Business Caltech−101 Data set Poetas CCA Football CADE12 Classic4 Web−KB TDT−2 20−Newsgroup Reuters−10 −4 Reuters−8 −2 Figure 3: Difference in performance between learned TWSs and best baseline per each data set, values above zero indicate better performance obtained by the TWSs. Table 7, on the other hand, shows a more fair comparison between our method and the reference TWSs: it shows the average performance obtained by reference schemes and the average performance of our method for thematic TC, AA and IC data sets. It is clear from this table that in average our method performs consistently better than any of the reference methods in terms of both accuracy and f1 measure for the three types of tasks. Thus, from the results of this table and those from Tables 4, 5, and 6, it is evident that standard TWSs are competitive, but one can take advantage of them only when the right TWS is selected for each data set. Also, the performance of TWSs learned with our approach are a better option than standard TWSs, as in average we were able to obtain much better representations. Summarizing the results from this section, we can conclude that: • The proposed GP obtained TWSs that outperformed the best baselines in the three types of tasks: thematic TC, AA and IC. Evidencing the generality of our proposal across different data types and modalities. Larger improvements were observed for thematic TC and IC data sets. In average, learned TWSs outperformed standard ones in the three types of tasks. 21 Table 7: Average performance on thematic TC obtained with learned TWSs and the baselines. Thematic TC AA IC TWS f1 Acc. f1 Acc. f1 Acc. TF 76.60 79.53 62.17 72.43 68.86 71.54 B 77.42 79.73 66.07 76.76 71.22 72.78 TFIDF 61.69 76.17 40.88 55.26 62.27 67.56 TF-CHI 71.56 75.63 68.75 73.69 65.38 67.45 TF-IG 64.22 69.00 68.96 74.91 66.02 67.93 PG-worst 77.81 81.19 66.47 74.84 74.30 75.67 PG-Avg. 81.01 83.63 75.58 79.75 75.81 77.07 PG-best 82.88 85.81 81.37 83.98 76.97 78.18 • Our results confirm our hypothesis that different TWSs are required for facing different tasks, and within a same task (e.g., AA) a different TWS may be required for a different data set. Hence, motivating further research on how to select TWS for a particular TC problem. • We show evidence that the proposed TWS learning approach is a promising solution for enhancing the classification performance in other tasks than TC, e.g., IC. • Our results show that for AA supervised TWS seem to be more appropriate, whereas unsupervised TWS performed better on thematic TC and IC. This is a quite interesting result that may have an impact in non-thematic TC and supervised term-weighting learning. 5.3. Varying vocabulary size For the experiments from Section 5.2 each TWS was learned by using only the top 2000 most frequent terms during the search process. This reduction in the vocabulary allowed us to speed up the search process significantly, however, it is worth asking ourselves what the performance of the TWSs would be when using an increasing number of terms. We aim to answer such question in this section. For this experiment we considered three data sets, one from each type of task: thematic TC, AA, and IC. The considered data sets were the Reuters8 (R8) for thematic TC, the CCA benchmark for AA, and Caltech-101 for IC. These data sets are the representative ones from each task: Reuters-8 is among the most used TC data sets, CCA has been widely used for AA as well, and Caltech-101 is the benchmark in image categorization For each 22 of the considered data sets we use a specific TWS learned using the top2000 most frequent terms (see Section 5.2), and evaluate the performance of such TWSs when increasing the vocabulary size: terms were sorted in ascending order of their frequency. Figures 4, 5, and 6 show the results of this experiment in terms of f1 measure and accuracy (the selected TWS is shown in the caption of each image). R8 R8 1 1 0.9 0.9 0.8 0.8 0.7 Accuracy 0.6 0.5 1 f measure 0.7 0.4 0.3 0.2 0.1 0 10 TF B TFIDF TF−RF TF−CHI TF−IG GP 20 0.6 TF B TFIDF TF−RF TF−CHI TF−IG GP 0.5 0.4 0.3 30 40 50 60 70 Percentage of features 80 90 0.2 10 100 20 30 40 50 60 70 Percentage of features 80 90 100 √ Figure 4: Classification performance on the Reuters-8 data set for the TWS: W5 − √ log W19 when increasing the number of considered terms. The left plot shows results in W21 terms of f1 measure while the right plot shows accuracy performance. f CCA− Sthatis 0.8 0.78 0.78 0.76 TF B TFIDF TF−RF TF−CHI TF−IG GP 0.74 0.72 0.7 0.68 10 Accuracy 0.76 1 measure CCA (Sthatis) 0.8 20 30 40 50 60 70 Percentage of features TF B TFIDF TF−RF TF−CHI TF−IG GP 0.74 0.72 0.7 80 90 0.68 10 100 20 30 40 50 60 70 Percentage of features 80 90 100 Figure 5: Classification performance on the CCA data set of the TWS: W4 − (W22 + W5 ) when increasing the number of considered terms. Different performance behavior can be observed in the different data sets. Regarding Figure 4, which shows the performance for a thematic TC data set, it can be seen that the TWS learned by our method outperformed all other TWSs for any data set size. Hence confirming the suitability of the proposed method for thematic TC. Figure 5, on the other hand, behaves differently: the proposed method outperforms all the other TWSs only for a single data set size (when 20% 23 Caltech−101 0.7 0.6 0.6 0.5 0.5 Accuracy f1 measure Caltech−101 0.7 0.4 TF B TFIDF TF−RF TF−CHI TF−IG GP 0.3 0.2 0.1 0 10 20 30 40 50 60 70 Percentage of features 80 90 0.4 TF B TFIDF TF−RF TF−CHI TF−IG GP 0.3 0.2 0.1 0 10 100 20 30 40 50 60 70 Percentage of features 80 90 100 Figure 6: Classification performance on the Caltech-101 data set of the TWS: rq p√ W17 − W22 when increasing the number of considered terms. of the terms were used). In general, our method consistently outperformed TF-CHI and TF-IG TWSs, and performs similarly as TF-IDF, but it was outperformed by the TF-RF TWS. This result can be due to the fact that for this AA data set, the genetic program learned a TWS that was suitable only for the vocabulary size that was used during the optimization. Although, interesting, this result is not that surprising: in fact, it is well known in AA that the number of terms considered in the vocabulary plays a key role on the performance of AA systems. AA studies suggest using a small amount of the most-frequent terms when approaching an AA problem (Stamatatos, 2009; Escalante et al., 2011; Luyckx and Daelemans, 2010). Results from Figure 5 corroborate the latter and seem to indicate that when approaching an AA problem, one should first determine an appropriate vocabulary size and then apply our method. One should note, however, that our method outperforms the other TWSs for the data set size that was used during the optimization, and this is, in fact, the highest performance that can be obtained with any other TWS and data set size combination. Finally, Figure 6 reports the performance of TWSs for the Caltech-101 data set under different data set sizes. In this case, the learned TWS outperforms all other TWSs when using more than 20% and 30% in terms of f1 measure and accuracy, respectively. The improvement is consistent and monotonically increases as more terms are considered. Hence showing the robustness of the learned TWS when increasing the vocabulary size for IC tasks. Among the other TWSs, TFIDF obtains competitive performance when using a small vocabulary, this could be due to the fact that when considering a small number of frequent terms the IDF component is important 24 for weighting the contribution of each of the terms. Summarizing the results from this section we can conclude the following: • TWSs learned with our method are robust to variations in the vocabulary size for thematic TC and IC tasks. This result suggests, we can learn TWSs using a small number of terms (making more efficient the search process) and evaluating the learned TWSs with larger vocabularies. • Learned TWSs outperform standard TWSs in thematic TC and IC tasks when varying the vocabulary size. • For AA, TWSs learned with our proposed approach seem to be more dependent on the number of terms used during training. Hence, when facing this type of problems it is a better option to fix the number of terms beforehand and then running our method. 5.4. Generalization of the learned term-weights In this section we evaluate the inter-data set generalization capabilities of the learned TWSs. Although results presented so far show the generality of our method across three types of tasks, we have reported results obtained with TWSs that were learned for each specific data set. It remains unclear whether the TWSs learned for a collection can perform similarly in other collections, we aim to answer to this question in this section. To assess the inter-data set generalization of TWSs learned with our method we performed an experiment in which we considered for each data set a single TWS and evaluated its performance across all the 16 considered data sets. The considered TWSs are shown in Table 8, we named the variables with meaningful acronyms for clarity but also show the mathematical expression using variables as defined in Table 2. Before presenting the results of the experiments it is worth analyzing the type of solutions (TWSs) learned with the proposed approach. First of all, it can be seen that the learned TWSs are not too complex: the depth of the trees is small and solutions have few terminals as components. This is a positive result because it allows us to better analyze the solutions and, more importantly, it is an indirect indicator of the absence of the over-fitting phenomenon. Secondly, as in other applications of genetic programming, it is unavoidable to have unnecessary terms in the solutions, for instance, the subtree: div(pow2(TF-RF),pow2(TF-RF))), (from TWS 2) is unnecessary 25 Table 8: Considered TWSs for the inter-data set generalization experiment for each data set. In column 2 each TWS is shown as a prefix expression, the names of the variables are self-explanatory. Column 3 shows the mathematical expression of each TWS using the terminal set from Table 2. Text categorization ID Data set Learned TWS 1 Reuters-8 -(sqrt(TFIDF),div(log2(sqrt(ProbR)),RF)) 2 Reuters-10 sqrt(div(pow2(sqrt(TFIDF)),div(pow2(TFRF),pow2(TF-RF)))) 3 20-Newsg. Formula √ √ log W19 W5 − W21 v √ u ( W )2 u 5 u W2 t 22 sr sqrt(sqrt(div(TF,GLOBTF))) 4 TDT-2 sqrt(×(sqrt(sqrt(TFIDF)), sqrt(×(sqrt(TFIDF),IG)))) 5 WebKB div(TF-RF,+(+(+(RF,TF-RF),FMEAS),FMEAS)) 6 7 Classic-4 CADE-12 ×(ProbR,TFIDF) div(TF,sqrt(log2(ACCU))) qp 2 W22 W6 W7 √ W5 × p√ W5 × W4 W22 (((W21 +W22 )+W16 )+W16 ) W5 × W19 √ W6 log W12 Authorship attribution ID 8 9 10 Data set CCA-10 Poetas Football Learned TWS -(IG,plus(TF-RF,TFIDF)) -(-(RF,TF-RF),TF-IDF) div(TF-RF,pow2(ODDSR)) Formula W4 − (W22 + W5 ) (W21 − W22 ) − W5 11 12 Business Poetry minus(TF-RF,PROBR) div(TF,log2(div(TF,log(TF-RF)))) W22 − W19 W22 W2 17 log 13 Travel 14 Cricket ID Data set 15 Caltech101 Caltechtiny 16 +(-(TF-RF,-(-(TF-RF,-(TFRF,POWER)),POWER)),TF) ×(IG,TF-RF) Image Classification Learned TWS sqrt(sqrt(-(ODDSR,sqrt(sqrt(TF-RF))))) (W22 − ((W22 − (W22 − W18 )) − W18 )) + W6 W4 × W22 Formula rq p√ W17 − W22 √ sqrt(-(TF-RF,ACBAL)) W6 W6 log W22 W22 − W13 because it reduces to a constant matrix; the same happens with the term pow2(sqrt(TFIDF)). Nevertheless, it is important to emphasize that this type of terms do not harm the performance of learned TWSs, and there are not too many of these type of subtrees. On the other hand, it is interesting that all of the learned TWSs incorporate supervised information. The most used TR weight is RF, likewise the most used TDR is TFIDF. Also it is interesting that simple operations over standard TWSs, TR and TDR weights results in significant performance improvements. For instance, compare the performance of TF-RF and the learned weight for Caltech-101 in Figure 6. By simply subtracting an odds-ratio from the TF-RF TWS and applying scaling operations, the resultant TWS outperforms significantly TF-RF. The 16 TWSs shown in Table 8 were evaluated in the 16 data sets in order to determine the inter-data set generality of the learned TWSs. Figure 7 shows the results of this experiment. We show the results with boxplots, 26 where each boxplot indicates the normalized4 performance of each TWSs across the 16 data sets, for completion we also show the performance of the reference TWSs on the 16 data sets. 1 f1 measure 0.9 0.8 0.7 0.6 Catech−tiny Cricket Travel Poetry Business CCA Poetas Caltech−101 Learned TWSs Football Classic4 CADE12 TDT−2 Web−KB 20−NewsG. Reuters−8 Reuters−10 TF−GI TF−RF TF−CHI TF−IDF B TF 0.5 Figure 7: Heatmap that shows the performance of TWSs (rows 7-22) from Table 8 in the 16 data sets (x axis) considered in the study. For completion, we also show the performance of standard TWSs (rows 1-6). It can be seen from Figure 7 that the generalization performance of learned TWSs is mixed. On the one hand, it is clear that TWSs learned for thematic TC (boxplots 7-13) achieve the highest generalization performance. Clearly, the generalization performance of these TWSs is higher than that of traditional TWSs (boxplots 1-6). It is interesting that TWSs learned for a particular data set/problem/modality perform well across different data sets/problems/modalities. In particular, TWSs learned for Reuters-10 and TDT-2 obtained the highest performance and the lowest variance among all 4 Before generating the boxplots we normalized the performance on a per-data set basis: for each data set, the performance of the 16 TWSs was normalized to the range [0, 1], in this way, the variation in f1 -values across data sets is eliminated, i.e, all f1 values are in the same scale and are comparable to each other. 27 of the TWSs. On the other hand, TWSs learned for AA and IC tasks obtained lower generalization performance: the worst in terms of variance is the TWS learned for the Poetry data set, while the worst average performance was obtained by the TWS learned for the Football data set. TWSs learned for IC are competitive (in generalization performance) with traditional TWSs. Because of the nature of the tasks, the generalization performance of TWSs learned from TC is better than that of TWSs learned for AA and IC. One should note that these results confirm our findings from previous sections: (i) the proposed approach is very effective mainly for thematic TC and IC tasks; and, (ii) AA data sets are difficult to model with TWSs. Finally, we evaluate the generality of learned TWSs across different classifiers. The goal of this experiment is to assess the extend to which the learned TWSs are tailored for the classifier they were learn for. For this experiment, we selected two TWSs corresponding to Caltech-tiny and Caltech-101 (15 and 16 in Table 8) and evaluated their performance of different classifiers across the 16 data sets. Figure 8 shows the results of this experiment. It can be seen from Figure 8 that the considered TWSs behaved quite differently depending on the classifier. On the one hand, the classification performance when using naı̈ve Bayes (Naive), kernel-logistic regression (KLogistic), and 1−nearest neighbors (KNN ) classifiers degraded significantly. On the other hand, the performance SVM and the neural network (NN) was very similar. These results show that TWSs are somewhat robust across classifiers of similar nature as SVM and NN are very similar classifiers: both are linear models in the parameters. The other classifiers are quite different to the reference SVM and, therefore, the performance is poor5 . It is interesting that in some cases the NN classifier outperformed the SVM, although in average the SVM performed better. This is a somewhat expected result as the performance of the SVM was used as fitness function. According to the experimental results from this section we can draw the following conclusions: • TWSs learned with the proposed approach are not too complex despite their effectiveness. Most of the learned TWSs included a supervised component, evidencing the importance of taking advantage of labeled 5 One should note that among the three worse classifiers, KNN, Naive and KLogistic, the latter obtained better performance than the former two, this is due to the fact that KLogistic is closer, in nature, to an SVM. 28 1 0.6 0.4 Catech−tiny Data set Caltech−101 Cricket Travel Poetry Poetas Poetas Business CCA Football CADE12 TDT−2 Web−KB 20−Newsgroup Reuters−10 0 Reuters−8 0.2 CCA KNN Naive SVM KLogistic Neural Classic4 f1 − measure 0.8 1 0.6 0.4 Catech−tiny Cricket Travel Poetry Caltech−101 Data set Business Football Web−KB TDT−2 20−Newsgroup Reuters−10 0 Reuters−8 0.2 CADE12 KNN Naive SVM KLogistic Neural Classic4 f1 − measure 0.8 Figure 8: Classification performance of selected TWSs across different classifiers, f1 measure is reported. The plot at the top is a TWS learned for Caltech-tiny, while the bottom plot shows the performance for a TWS learned for Caltech-101. documents. • TWSs offer acceptable inter-data set generalization performance, in particular, TWSs learned for TC generalize pretty well across data sets. • We showed evidence that TWSs learned for a modality (e.g., text / images) can be very competitive when evaluated on other modality. • TWSs are somewhat robust to the classifier choice. It is preferable 29 to use the classifier used to estimate the fitness function, although classifiers of similar nature perform similarly. 6. Conclusions We have described a novel approach to term-weighting scheme (TWS) learning in text classification (TC). TWSs specify the way in which documents are represented under a vector space model. We proposed a genetic programming solution in which standard TWSs, term-document, and term relevance weights are combined to give rise to effective TWSs. We reported experimental results in 16 well-known data sets comprising thematic TC, authorship attribution and image classification tasks. The performance of the proposed method is evaluated under different scenarios. Experimental results show that the proposed approach learns very effective TWSs that outperform standard TWSs. The main findings of this work can be summarized as follows: • TWSs learned with the proposed approach outperformed significantly to standard TWSs and those proposed in related work. • Defining the appropriate TWS is crucial for image classification tasks, an ignored issue in the field of computer vision. • In authorship attribution, supervised TWSs are beneficial, in comparison with standard TWSs. • The performance of learned TWSs do not degrades when varying the vocabulary size for thematic TC and IC. 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AN ANALYSIS OF GENE EXPRESSION DATA USING PENALIZED FUZZY C-MEANS APPROACH P.K. Nizar Banu1 H. Hannah Inbarani2 1 Department of Computer Applications, B. S. Abdur Rahman University, Chennai, Tamilnadu, India Email: [email protected] 2 Department of Computer Science, Periyar University, Salem, Tamilnadu, India Email: [email protected] ABSTRACT With the rapid advances of microarray technologies, large amounts of high-dimensional gene expression data are being generated, which poses significant computational challenges. A first step towards addressing this challenge is the use of clustering techniques, which is essential in the data mining process to reveal natural structures and identify interesting patterns in the underlying data. A robust gene expression clustering approach to minimize undesirable clustering is proposed. In this paper, Penalized Fuzzy C-Means (PFCM) Clustering algorithm is described and compared with the most representative off-line clustering techniques: K-Means Clustering, Rough K-Means Clustering and Fuzzy C-Means clustering. These techniques are implemented and tested for a Brain Tumor gene expression Dataset. Analysis of the performance of the proposed approach is presented through qualitative validation experiments. From experimental results, it can be observed that Penalized Fuzzy C-Means algorithm shows a much higher usability than the other projected clustering algorithms used in our comparison study. Significant and promising clustering results are presented using Brain Tumor Gene expression dataset. Thus patterns seen in genome-wide expression experiments can be interpreted as indications of the status of cellular processes. In these clustering results, we find that Penalized Fuzzy C-Means algorithm provides useful information as an aid to diagnosis in oncology. Keywords: Clustering, Microarray, Gene Expression, Brain Tumor, Fuzzy Clustering, FCM, PFCM produced. Searching for meaningful information patterns and dependencies among genes, in order to provide a basis for hypothesis testing, typically includes the initial step of a natural basis for organizing gene expression data to group genes together with similar patterns of expression. The field of gene expression data analysis has grown in the past few years from being purely datacentric to integrative, aiming at complementing microarray analysis with data and knowledge from diverse available sources. Advances in microarray technologies have made it possible to measure the expression profiles of thousands 1.INTRODUCTION Gene expression is the fundamental link between genotype and phenotype in a species, with microarray technologies facilitating the measurement of thousands of Gene expression values under tightly controlled conditions, e.g. (i) from a particular point in the cell cycle, (ii) after an interval response to some environmental change, (iii) from RNA, isolated from a tissue exhibiting certain phenotypic characteristics and so on (Kerr, et. al., 2008). A problem inherent in the use of microarray technologies is the huge amount of data 1 of genes in parallel under varying experimental conditions. Due to the large number of genes and complex gene regulation networks, clustering is a useful exploratory technique for analyzing these data. It divides data of interest into a small number of relatively homogeneous groups or clusters. Clustering is a popular data mining technique for various applications. One of the reasons for its popularity is the ability to work on datasets with minimum or on a priori knowledge. This makes clustering practical for real world applications. We can view the expression levels of different genes as attributes of the samples, or the samples as the attributes of different genes. Clustering can be performed on genes or samples (Michael et. al. 1998). This paper introduces the application of Penalized Fuzzy C-Means algorithm to cluster Brain Tumor genes. expression profiles are hierarchical clustering algorithms (Michael et. al. 1998), self-organizing maps (Paul, 1999) and K-Means clustering algorithms (Tavazoie, et. al., 1999). Hierarchical algorithms merge genes with the most similar expression profiles iteratively in a bottom-up manner. Self-organizing maps and K-means algorithms partition genes into userspecified k optimal clusters. Other full space clustering algorithms applied on gene expression data include Bayesian network (Friedman et. al. 2000) and neural network. A robust image segmentation method that combines the watershed segmentation and penalized fuzzy Hopfield neural network algorithms to minimize over-segmentation is described in (Kuo et. al. 2006). (Brehelin et. al., 2008) evaluates the stability of clusters derived from hierarchical clustering by taking repeated measurements. A wide variety of clustering algorithms are available for clustering gene expression data (Bezdek, 1981). They are mainly classified as Partitioning methods, hierarchical methods, Density based methods, Model based methods, Graph Theoretic methods, soft computing methods etc. This paper is organized as follows. In Section 2, Research background for clustering Gene Expression patterns is discussed. In Section 3, Methodology for preparing Gene Expression patterns is presented. Section 4 presents a method for extracting highly suppressed and expressed genes based on Penalized Fuzzy C-Means algorithm. The Experimental results are discussed in Section 5. Finally, section 6 concludes this paper by enumerating the merits of the proposed approaches. Multiple expression measurements are commonly recorded as a real-valued matrix, with row objects corresponding to gene expression measurements over a number of experiments and columns corresponding to the pattern of expression of all genes for a given microarray experiment. Each entry xij, is the measured expression of gene i in experiment j. Dimensionality of a gene refers to the number of expression values recorded for it. A gene/gene-profile x is a single data item (row) consisting of d measurements, x = (x1, x2, . . . , xd). An experiment/sample y is a single microarray experiment corresponding to a single column in the gene expression matrix, y = (x1, x2, …, xn)T where n is the number of 2. RESEARCH BACKGROUND 2.1. Clustering Clustering genes, groups similar genes into the same cluster based on a proximity measure. Genes in the same cluster have similar expression patterns. One of the characteristics of gene expression data is that it is meaningful to cluster both genes and samples. The most commonly applied full space clustering algorithms on gene 2 genes in the dataset. Clustering is considered an interesting approach for finding similarities in data and putting similar data into groups. Initial step in the analysis of gene expression data is the detection of groups of genes that exhibit similar expression patterns. In gene expression, elements are usually genes and the vector of each gene is its expression pattern. Patterns that are similar are allocated in the same cluster, while the patterns that differ significantly are put in different clusters. Gene expression data are usually of high dimensions and relatively small samples, which results in the main difficulty for the application of clustering algorithms. Clustering the microarray matrix can be achieved in two ways: (i) Genes can form a group which show similar expression across conditions, (ii) Samples can form a group which shows similar gene expression across all genes. This gives rise to global clustering, where a gene or sample is grouped across all dimensions. Additionally, the clustering can be complete or partial. A complete clustering assigns each gene to a cluster, whereas a partial clustering does not. Partial clustering tends to be more suited to gene expression, as the dataset often contains irrelevant genes or samples. Clearly this allows: (i) Noisy genes to be left out, with correspondingly less impact on the outcome and (ii) Genes belonging to no cluster omitting a large number of irrelevant contributions. Microarrays measure expression for the entire genome in one experiment, but genes may change expression, independent of the experimental condition. Forced inclusion in well-defined but inappropriate groups may impact the final structures found for the data. Partial clustering avoids the situation where an interesting sub-group in a cluster is hidden through forcing membership of unrelated genes (Kerr, et. al., 2008). 2.2. Categories of Gene Expression data clustering Methods of clustering can be categorized as Hard Clustering or Soft Clustering. Hard Clustering requires each gene to belong to a single cluster, whereas Soft Clustering permit genes to simultaneously be members of numerous clusters. Hard Clustering tells whether a gene belongs to a cluster or not. Whereas in Soft Clustering, with membership values, every gene belongs to each cluster with a membership weight between 0 (doesn’t belong) and 1 (belongs). Clustering algorithms which permit genes to belong to more than one cluster are more applicable to Gene expression. Gene expression data has certain special characteristics and is a challenging research problem A modern working definition of a gene is "a locatable region of genomic sequence, corresponding to a unit of inheritance, which is associated with regulatory regions, transcribed regions, and or other functional sequence regions". Currently, a typical microarray experiment contains 103 to 104 genes, and this number is expected to reach to the order of 106. However, the number of samples involved in a microarray experiment is generally less than 100. One of the characteristics of gene expression data is that it is meaningful to cluster both genes and samples. On one hand, co-expressed genes can be grouped into clusters based on their expression patterns (Ben-Dor, et. al., 1999&Michael et. al. 1998). In such gene-based clustering, the genes are treated as the objects, while the samples are the features. On the other hand, the samples can be partitioned into homogeneous groups. Each group may 3 correspond to some particular macroscopic phenotype, such as clinical syndromes or cancer types (Golub et. al., 1999). Such sample-based clustering considers the samples as the objects and the genes as the features. The distinction of gene-based clustering and sample-based clustering is based on different characteristics of clustering tasks for gene expression data. Some clustering algorithms, such as KMeans and hierarchical approaches, can be used both to group genes and to partition samples. Hard clustering algorithms like K-Means and k-medoid place a restriction that a data object can belong precisely to only one cluster during clustering process. This can be too restrictive while clustering high dimensional data like Gene Expression data because genes have a property of getting expressed in multiple conditions. Fuzzy set clustering like Fuzzy C-Means, Penalized Fuzzy C-Means allows data objects to belong to multiple clusters based on the degree of membership. to remember that when a gene expression profile is analyzed in a given sample, it is just a snapshot in time and space. 2.4 Clustering Techniques K-Means Algorithm The K-means method aims to minimize the sum of squared distances between all points and the cluster centre. This procedure includes the steps, as described by Tou and Gonzalez (Tou et. al. 1974). Rough Set Theory Rough set theory introduced by Pawlak (Pawlak, 1982) deals with uncertainty and vagueness. Rough set theory became popular among scientists around the world due to its fundamental importance in the field of artificial intelligence and cognitive sciences. Similar to fuzzy set theory it is not an alternative to classical set theory but it is embedded in it. Rough set theory can be viewed as a specific implementation of Frege’s idea of vagueness, i.e., imprecision in this approach is expressed by a boundary region of a set, and not by a partial membership, like in fuzzy set. 2.3. Analysis of Gene Expression data Gene expression data is usually represented by a matrix, with rows corresponding to genes, and columns corresponding to conditions, experiments or time points. The content of the matrix is the expression levels of each gene under each condition. Those levels may be absolute, relative or otherwise normalized. Each column contains the results obtained from a single array in a particular condition, and is called the profile of that condition. Each row vector is the expression pattern of a particular gene across all the conditions. Analyzing gene expression data is a process by which a gene's information is converted into the structures and functions of a cell. Thousands of different mRNAs are present in a given cell; together they make up the transcriptional profile. It is important Rough Clustering A rough cluster is defined in a similar manner to a rough set that is with lower and upper approximation. The lower approximation of a rough cluster contains genes that only belong to that cluster. The upper approximation of a rough cluster contains genes in the cluster which are also members of other clusters (SushmitaMitra, 2004 & SushmitaMitra, 2006). To use the theory of rough sets in clustering, the value set (Va) need to be ordered. This allows a measure of the distance between each object to be defined. Distance is a form of similarity, which is a relaxing of the strict 4 requirement of indiscernibility outlined in canonical rough sets theory, and allows the inclusion of genes that are similar rather than identical. Clusters of genes are then formed on the basis of their distance from each other. An important distinction between rough clustering and traditional clustering approaches is that, with rough clustering, an object can belong to more than one cluster. 3. METHODOLOGY Cluster analysis, is an important tool in gene expression data analysis. For experimentation, we used a set of gene expression data that contains a series of gene expression measurements of the transcript (mRNA) levels of brain tumor gene. In clustering gene expression data, the genes are treated as objects and the samples are treated as attributes. Gene pattern extraction consists of 4 steps. i) Data Preparation ii) Data Normalization iii) Clustering iv) Pattern analysis Fuzzy Clustering Cluster analysis is a method of grouping data with similar characteristics into larger units of analysis. First in (Zadeh, 1965) fuzzy set theory that gave rise to the concept of partial membership, based on a membership function, fuzziness was articulated and has received increasing attention. Fuzzy clustering which produces overlapping cluster partitions has been widely studied and applied in various areas. In fuzzy clustering, the Fuzzy C-Means (FCM) clustering algorithm is the best known and most powerful methods used in cluster analysis (Bezdek, 1981). In (Yu et. al., 2007), a general theoretical method to evaluate the performance of fuzzy clustering algorithm is proposed. The Fuzzy integrated model is accurate than rough integrated model and conventional integrated model (Banu et. al., 2011). Fuzzy clustering approach captures the uncertainty that prevails in gene expression and becomes more suitable for tumor prediction. One of the important parameters in the FCM is the weighting exponent m. When m is close to one, the FCM approaches the hard C-Means algorithm. When m approaches infinity, the only solution of the FCM will be the mass center of the data set. Therefore, the weighting exponent m plays an important role in the FCM algorithm. 3.1. Data Preparation We represent the gene expression ˆ {m |i 1,2,...,n , j 1,2,...,n}, data as ng by ns matrix: M i, j g s There are ns columns, one for each sample and ng rows, one for each gene. One row of genes is also called a gene vector, denoted  . Thus a gene as g i  m , m ,..., m vector contains the values of a particular attribute for all samples. i ,1 i,2 i ,n s 3.2. Data Normalization Data sometimes need to be transformed before being used. For example; attributes may be measured using different scales, such as centimeters and kilograms. In instances where the range of values differ widely from attribute to attribute, these differing attribute scales can dominate the results of the cluster analysis. It is therefore common to normalize the data so that all attributes are on the same scale. The following are two common approaches for data normalization of each gene vector: m i, j  m i, j  m i m i, j  5 , mi m i, j  m i i or , where mi   ns j 1 ns m i, j  m ns , i  j 1 The membership weighting system reduces noise effects, as a low membership grade is less important in centroid calculation. PFCM algorithm helps in identifying hidden pattern and providing enhanced understanding of the functional genomics in a better way.  mi  2 i, j ns  1 m i, j denotes the normalized value and for gene vector i of sample j, mi,j represents the original value for gene i of sample j, ns is the number of samples, m i is the mean of the values for gene vector i over all samples, and  i is the standard deviation of the ith gene vector. Fuzzy Clustering permit genes to belong to more than one cluster, is more applicable to Gene Expression. Noisy genes are unlikely to be members of several clusters and genes with similar change in expression for a set of samples are involved in several biological functions and groups should not be co-active under all conditions. This gives rise to high inconsistency in the gene groups and some overlap between them. The boundary of a cluster is usually fuzzy for three reasons: The Brain Tumor gene expression data is used for our experiments. This data set is publically available in Broad Institute web site. The various cluster validation techniques namely Root Mean Square Error (RMSE), Mean Absolute Error (MAE) and Xie-Beni (XB) validity index are used to validate the clusters obtained after applying the clustering algorithms. i. 4. PROPOSED PPROACH: PENALIZED FUZZY C-MEANS ii. Penalized Fuzzy C-Means (PFCM) algorithm for clustering gene expression data is introduced in this paper, which modified Fuzzy C-Means (FCM) algorithm to produce more meaningful fuzzy clusters. Genes are assigned a membership degree to a cluster indicating its percentage association with that cluster. The two algorithms differ in the weighting scheme used for the contribution of a gene to the mean of the cluster. FCM membership values for a gene are divided among clusters in proportion to similarity with that clusters mean. The contribution of each gene to the mean of a cluster is weighted, based on its membership grade. Membership values are adjusted iteratively until the variance of the system falls below a threshold. These calculations require the specification of a degree of fuzziness parameter which is problem specific (Dembele et. al., 2003). iii. The gene expression dataset might be noisy and incomplete The similarity measurement between genes is continuous and there is no clear cutoff value for group membership A gene might behave similarly to gene1 under a set of samples and behave similarly to another gene2 under another set of samples. Therefore, there is a great need for a fuzzy clustering method, which produces clusters in which genes can belong to a cluster partially and to multiple clusters at the same time with different membership degrees. The main objective of using this method is to minimize the objective function value so that the highly suppressed genes and highly expressed genes are clustered separately and also it helps to diagnose at an early stage of tumor formation. 6 Based on the numerical results PFCM is more accurate than FCM. The steps of the PFCM algorithm are given as follows: Step 1: Initialize the cluster centroids wj (2  j  c),v(v  0), fuzzification parameter, m (1  m   ), and the value   0. Gives a fuzzy C-partition  ( 0 ) and t=1. 4.1. Penalized Fuzzy C-Means Algorithm Another strategy for fuzzy clustering, called the penalized Fuzzy CMeans (PFCM) algorithm, with the addition of a penalty term was proposed by Yang (Yang, 1993 &Yang, 1994). Yang made the fuzzy extension of the Classification Maximum Likelihood (CML) procedure in conjunction with fuzzy C-partitions and called it a class of fuzzy CML procedures. The idea of penalization is also important in statistical learning. Combining the CML procedure and penalty idea, Yang (Yang, 1993) added a penalty term to the FCM objective function JFCM and then extended the FCM to the so-called Penalized FCM (PFCM) which produces more meaningful and effective results than the FCM algorithm. Thus, the PFCM objective function is defined as follows: Step 2: Calculate  (j t ) , w(jt ) with ( t 1) using Eqs. (1) and (2). Step 3: Calculate the membership matrix  ( t )  [ui , j ]with  (jt ) , w(jt ) using Eq. (3)  c n 1c n m 2 1 JPFCM ui, j || xi wj ||  vuim, j lnj 2 j1 i1 2 j1 i1 1 c n m  JFCM vui, j lnj 2 j1 i1 ( 5. EXPERIMENTAL RESULTS where  j is a proportional constant of class j and v (≥ 0) is a constant. The penalty term  1 v   u ln  is added to the c n j1 i1 m i, j 2 The effectiveness of the algorithms based on cluster validity measure is demonstrated in this section. j objective function, when v=0, JPFCM is equal to JFCM. αj, wj and ui,j are defined as n u im, j   j  c i 1 n , j  1, 2,..., c m ui, j  (1) j 1 i 1 wj  n 1 n  u   u  m i, j xi 5.1. Data Source A set of gene expression data that contains a series of gene expression measurements of the transcript (mRNA) levels of brain Tumor gene is used in this paper to analyze the efficiency of the proposed approach. The brain tumor dataset is taken from the Broad-Institute website (http://www.broadinstitute.org/cgibin/cancer/datasets.cgi). The dataset is titled as “Gene Expression-Based Classification and Outcome Prediction of Central Nervous (2) m i 1 i, j i 1 ui, j    Step 4: Compute   max |  (t 1)   ( t ) | . If    , and , t  t  1 go to Step 2; otherwise go to Step 5. ( Step 5: Find the results1for the final class centroids. )  c || x  w || 2 v ln  i j j    l 1 || x  w || 2 v ln  i l l    1 /( m 1) 1 /( m 1) 1  (3)    7 System Embryonal Tumors”. The Datasets Consists of three types of Brain Tumors data namely Medulloblastoma classic and desmoplastic, Multiple Brain tumors, Medulloblastoma treatment outcome. Each dataset contains nearly 7000 genes with 42 samples. In order to evaluate the proposed algorithm, we applied it to the Brain Tumor gene expression data taken from Broad Institute by taking 7129, 5000, 3000, and 1000 genes for all samples with various numbers of clusters. above mentioned validity measures and the effectiveness of the proposed algorithm is well understood. We tested our method for the Brain Tumour gene expression dataset to cluster the highly suppressed and highly expressed genes and are depicted for various dataset sizes. It is observed that for each set of genes taken, the value of validity measures for the proposed algorithm is lower than the value of validity measures for other algorithms and it is graphically illustrated from Figure 1 to Figure 12. Among these clustering algorithms Penalized Fuzzy C-Means produces better results in identification of differences between data sets. This helps to correlate the samples according to the level of gene expression. 5.2. Comparative Analysis A comparative study of the performance of K-Means (Tou et. al. 1974), Rough K-Means (Peters, 2006) and Fuzzy C-Means (Peters, 2006) is made with Penalized Fuzzy C-Means algorithm (Shen et. al, 2006). In terms of MAE, Penalized Fuzzy C-Means shows superior performance and K-Means and rough K-Means exhibits better performance than Fuzzy C-Means. Cluster Validation In this paper, Root Mean Square Error, Mean Absolute Error and Xie-Beni validity index are used to validate the clusters obtained after applying the clustering algorithms. To assess the quality of our method, we need an objective external criterion. In order to validate our clustering results, we employed Root Mean Square Error (RMSE), Mean Absolute Error (MAE) (Pablo de Castro et. al., 2007). Xie-Beni validity index has also been chosen as the cluster validity measure because it has been shown to be able to detect the correct number of clusters in several experiments (Pal et. al. 1995). Also, With respect to RMSE and Xie-Beni Index Penalized Fuzzy C-Means produces greater performance than the other algorithms. The comparative results based on Root Mean Square Error, Mean Absolute Error and Xie-Beni validity measure for all the Gene expression data clustering algorithms for various Brain Tumor data sets taken with different number of clusters are enumerated in Table 1. Figure 1 to Figure 3 shows the comparative analysis of various approaches for 7129 genes taking K as 7, 5 and 3 respectively. It can be observed from the figures, Penalized Fuzzy C-Means outperforms other approaches Fuzzy CMeans, Rough K-Means and K-Means. K Means, Rough K-Means, Fuzzy C-Means and Penalized Fuzzy C-Means clustering algorithm are applied and analysed for Brain Tumour genes. Table.1 shows the experimental results that are obtained by applying the 8 Data Set Size=7129 k=7 Value of Valididty Measures Value of Valididty Measures 1.2 1 0.8 0.6 0.4 0.2 0 FCMeans RKMeans KMeans MAE RMSE XB PFCMeans 1.2 1 0.8 0.6 0.4 0.2 0 Value of Valididty Measures Value of Valididty Measures FCMeans RKMeans KMeans XB PFCMeans 1.2 1 0.8 0.6 0.4 0.2 0 FCMeans 0.4 RKMeans Value of Valididty Measures Value of Valididty Measures 0.6 KMeans MAE RMSE XB PFCMeans FCMeans RKMeans KMeans RMSE XB PFCMeans Figure5: Validity Measure for Data set size = 5000, k=5 Data Set Size=7129 k=3 0.2 0 XB Validity Measures Figure 2: Validity Measure for Data set size =7129, k=5 0.8 RMSE Data Set Size=5000 k=5 MAE Validity Measures 1 KMeans Figure4: Validity Measure for Data set size =5000, k=7 Data Set Size=7129 k=5 RMSE RKMeans Validity Measures Figure 1: Validity Measure for Data set size = 7129, k=7 MAE FCMeans MAE Validity Measures 1.2 1 0.8 0.6 0.4 0.2 0 Data Set Size=5000 k=7 PFCMeans 1 0.8 Data Set Size=5000 k=3 0.6 0.4 FCMeans 0.2 KMeans RKMeans 0 MAE RMSE XB PFCMeans Validity Measures Validity Measures Figure 3: Validity Measure for Data set size = 7129, k=3 Figure6: Validity Measure for Data set size =5000, k=3 Figure 4 to Figure 6 shows the comparative analysis of various approaches for 5000 genes taking K as 7, 5 and 3 respectively. The experimental result shows that the Penalized Fuzzy C-Means outperforms other approaches Fuzzy CMeans, Rough K-Means and K-Means. Figure 7 to Figure 9 shows the comparative analysis of various approaches for 3000 genes taking K as 7, 5 and 3 respectively. The experimental result shows that the Penalized Fuzzy C-Means gives better results than the other algorithms. 9 Data Set Size=3000 k=7 3 Value of Valididty Measures Value of Valididty Measures 4 FCMeans 2 RKMeans 1 KMeans 0 MAE RMSE XB PFCMeans 3 2.5 2 1.5 1 0.5 0 Value of Valididty Measures Value of Valididty Measures FCMeans RKMeans 0.5 KMeans 0 MAE RMSE XB PFCMeans 4 3 PFCMeans Data Set Size=1000 k=5 FCMeans RKMeans 1 KMeans 0 MAE RMSE XB PFCMeans Validity Measures Figure8: Validity Measure for Data set size =3000, k=5 Figure 11: Validity Measure for Dataset Size =1000, k=5 Data Set Size=3000 k=3 Value of Valididty Measures Value of Valididty Measures XB 2 Validity Measures 1.2 1 0.8 0.6 0.4 0.2 0 KMeans Figure10: Validity Measure for Data set size =1000, k=7 Data Set Size=3000 k=5 1 RKMeans Validity Measures Figure 7: Validity Measure for Data set size = 3000, k=7 1.5 FCMeans MAE RMSE Validity Measures 2 Data Set Size=1000 k=7 FCMeans RKMeans KMeans PFCMeans Validity Measures 1.5 Data Set Size=1000 k=3 1 FCMeans RKMeans 0.5 KMeans 0 MAE RMSE XB PFCMeans Validity Measures Figure12: Validity Measure for Data set size =1000, k=3 Figure 9: Validity Measure for Data set size = 3000, k=3 Figure 10 to Figure 12 shows the comparative analysis of various approaches for 1000 genes taking K as 7, 5 and 3 respectively. The experimental result shows that the Penalized Fuzzy C-Means performs better than the other algorithms 10 Table 1: Performance Analysis of K-Means, Rough K-Means, Fuzzy C-Means and Penalized Fuzzy C-Means No. of No. of Clusters Genes 7 5 3 7 7129 5000 3000 1000 Root Mean Mean Square Absolute Error Error K-Means 0.0019 0.0044 0.2804 Rough K-Means 0.0511 0.0874 0.0020 Fuzzy C-Means 1.0438 0.8627 0.9548 Penalized Fuzzy C-Means 0.0043 0.0009 0.0001 K-Means 0.0034 0.0074 0.4798 Rough K-Means 0.0624 0.4654 0.0581 Fuzzy C-Means 1.0336 0.7504 0.0184 Penalized Fuzzy C-Means 0.0255 0.0015 0.0002 K-Means 0.0065 0.0150 0.1240 Rough K-Means 0.0336 0.0082 0.0203 Fuzzy C-Means 0.9925 0.6331 0.0624 Penalized Fuzzy C-Means 0.0079 0.0002 0.0001 K-Means 0.0225 0.0470 0.4994 Rough K-Means 0.1204 1.0548 0.1079 Fuzzy C-Means 1.8542 2.7108 0.1107 Penalized Fuzzy C-Means 0.0056 0.0015 0.0001 Clustering Algorithms Xie-Beni Index Table .2 Parameter setting and other issues Clustering Algorithm Cluster Membership Input Proximity Measure K-Means Binary Starting Prototype, Stopping Threshold, K Pair wise Distance Rough K-Means Rough Membership Starting Prototype, Stopping Threshold, K Pair wise Distance Fuzzy C-Means Fuzzy Membership Degree of fuzziness, Starting Prototypes, Stopping Threshold, K Pair wise Distance Penalized Fuzzy C-Means Improved Fuzzy Membership Fuzzification Parameter, K Pair wise Distance 11 Other Issues Very Sensitive to Input parameters and order of Input Imprecision in Gene Expression data can be captured Careful interpretations of membership values. Sensitive to Input parameters and order of Input Quality of membership is increased by introducing Penalty term Pattern Analysis The parameter setting and other issues of the clustering approaches which are discussed are given in Table.2. For each gene, red indicates a high level of expression (highly expressed Genes) relative to the mean; green indicates a low level of expression (highly suppressed Genes) relative to the mean. Transcriptional Initiation is the most important mode for control of gene expression level. Suppressed gene expression level may be stimulated by gene therapy (i.e.) promoter insertion or up regulation of suppressed gene and radiation therapy (i.e.) more amount of radiation may cause sudden mutation in gene, due to this sudden change the gene expression may be in high level. This helps to correlate the samples according to the level of gene expression. When precise functions for over or under expressed genes are determined, new avenues for intervention strategies may emerge. These studies are in their infancy; however, the improved technology employed here shows reasonable promise as our understanding of these deadly tumours increases. Therefore, future treatment decisions based on the expression profile of a primary tumour is a rational approach towards preventing the outgrowth of metastases. 6. CONCLUSION Cluster analysis applied to microarray measurements aims to highlight meaningful patterns for gene expression. The goal of gene clustering is to identify the important gene markers. Gene expression data are the representation of nonlinear interactions among genes and environmental factors. Brain Tumor is so deadly because, it is not usually diagnosed until it has reached an advance stage. Early detection can help prolong or save lives, but clinicians currently have no specific and sensitive method. Computing analysis of these data is expected to gain knowledge of gene functions and disease mechanisms. We used a set of gene expression data that contains a series of gene expression measurements of the transcript (mRNA) levels of Brain Tumor gene. Highly expressed genes and suppressed genes are identified and clustered using various clustering techniques. The following methods such as Post-Translational Modification, Small RNAs and Control of Transcript Levels, Translational Initiation, Transcript Stability, RNA Transport, Transcript Processing and Modification, Epigenetic Control and Transcriptional Initiation can be used to reduce the tumor level. The empirical results also reveal the importance of using Penalized Fuzzy CMeans (PFCM) clustering methods for clustering more meaningful highly suppressed and highly expressed genes from the gene expression data set. 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arXiv:1708.01751v2 [q-bio.QM] 8 Aug 2017 DNA Sequence Complexity Reveals Structure Beyond GC Content in Nucleosome Occupancy Hector Zenil 1,2 ∗ Peter Minary 1 August 9, 2017 1 Department of Computer Science, University of Oxford, Oxford, U.K. Algorithmic Dynamics Lab, Unit of Computational Medicine, SciLife Lab, Centre for Molecular Medicine, Department of Medicine Solna, Karolinska Institute, Stockholm, Sweden. 2 Abstract We introduce methods that rapidly evaluate a battery of informationtheoretic and algorithmic complexity measures on DNA sequences in application to potential binding sites for nucleosomes. The first application of this new tool demonstrates structure beyond GC content on DNA sequences in the context of nucleosome binding. We tested the measures on well-studied genomic sequences of size 20K and 100K bps. The measures reveal the known in vivo versus in vitro predictive discrepancies, but they also uncover the internal structure of G and C within the nucleosome length, thus disclosing more than simply GC content when one examines alphabet transformations that separate and scramble the GC content signal and the DNA sequence. Most current prediction methods are based upon training (e.g. k-mer discovery), the one here advanced, however, is a training-free approach to investigating informative measures of DNA information content in connection with structural nucleosomic packing. Keywords: Nucleosome positioning/occupancy; DNA sequence complexity; DNA structure; genomic information content 1 The challenge of Predicting Nucleosome Organization DNA in the cell is organized into a compact form, called chromatin. Nucleosome organization in the cell is referred to as the primary chromatin structure and can depend on the ‘suitability’ of a sequence for accommodating a nucleosome, which may in turn be influenced by the packing of neighbouring nucleosomes. Depending on the context, nucleosomes can inhibit or facilitate transcription ∗ To whom correspondence should be addressed: [email protected] 1 factor binding and are thus a very active area of research. The location of low nucleosomic occupancy is key to understanding active regulatory elements and genetic regulation that is not directly encoded in the genome but rather in a structural layer of information. Structural organization of DNA in the chromosomes is widely known to be heavily driven by GC content, involving a simple count of G and C occurrences in the DNA sequence. Despite its simplicity, uncovering exactly how (and to what extent) GC content drives/affects nucleosome organization is among the central questions in modern molecular biology. GC content, local and short-range signals carried by DNA sequence ‘motifs’ or fingertips (k-mer statistical regularities), have been found (Refs. [1] and [2]) to be able to determine a good fraction of the structural (and thus functional) properties of DNA, such as nucleosome occupancy, but the explanatory (and predictive) power of GC content (the G or C count in a sequence) alone and sequence motifs display very significant differences in vivo versus in vitro [3]. Despite intensive analysis of the statistical correspondence between in vitro and in vivo positioning, there is a lack of consensus as to the degree to which the nucleosome landscape is intrinsically specified by the DNA sequence [4], as well as in regards to the apparently profound difference in dependence in vitro versus in vivo. Because the nucleosome landscape is known to be significantly dependent on the DNA sequence, it encodes the structural information of the DNA (particularly demonstrated in vitro). We consider this an opportunity to compare the performance of complexity measures, in order to discover how much of the information encoded in a sequence in the context of the nucleosome landscape can be recovered from information-content versus algorithmic complexity measures. Nucleosome location is thus an ideal test case to probe how informative sequence-based indices of complexity can be in determining a structural (and ultimately functional) property of genomic DNA. 1.1 Algorithmic Information Theory in Genomic Sequence Profiling Previous applications of measures based upon algorithmic complexity include experiments on the evaluation of lossless compression lengths of sets of genomes [5, 7] and more recently [13] with interesting results. For example, in a landmark paper in the area, a measure of algorithmic mutual information was introduced to distinguish sequence similarities by way of minimal encodings and lossless compression algorithms in which a mitochondrial phylogenetic tree that conformed to the evolutionary history of known species was reconstructed [6, 7]. These approaches have, however, either been purely theoretical or have effectively been reduced to applications of Shannon entropy rather than of algorithmic complexity because, implementations of lossless compression are actually entropy estimators [8, 38]. In some other cases, control tests have been missing. For example, in the comparison of the compressibility indices of different genomes [6, 7], GC content (counting every G and C in the sequence) can reconstruct the same if not a more accurate phylogenetic tree. This is because 2 two species that are close to each other evolutionarily will have similar GC content (see e.g. [9]). Species close to each other will have similar DNA sequence entropy values, allowing lossless compression algorithms to compress statistical regularities of genomes of related species with similar compression rates. Here we intend to go beyond previous attempts, in breadth as well as depth, using better-grounded algorithmic measures and more biologically relevant test cases. 1.2 Current Sequence-based Prediction Methods While the calculation of GC content is extremely simple, the reasons behind its ability to predict the structural properties of DNA are largely unknown [10, 11]. For example, it has been shown that low GC content can explain low occupancy, but high GC content can mean either high or low occupancy [12]. Current algorithms that build upon while probing beyond GC content have been largely influenced by sequence motif ([14, 2]) and dinucleotide models [15]—and to a lesser degree by k-mers [1], DNA sequence motifs that are experimentally isolated and used for their informative value in determining the structural properties of DNA. Table 1 (SI) shows the in vitro nucleosome occupancy dependence on GC content, with a correlation of 0.684 (similar to that reported by Kaplan [3]) for the well-studied 20K bp genomic region (187K – 207K) of Yeast Chromosome 14 [16]. Knowledge-based methods dependent on observed sequence motifs [17, 18] are computationally cost-efficient alternatives for predicting genome-wide nucleosome occupancy. However, they are trained on experimental statistical data and are not able to predict anything that has not been observed before. They also require context, as it may not be sufficient to consider only short sequence motifs such as dinucleotides [19, 3]. More recently, deep machine learning techniques have been applied to DNA accessibility related to chromatin and nucleosome occupancy [17]. However, these techniques require a huge volume of data for training if they are to predict just a small fraction of data with marginally improved accuracy as compared to more traditional approaches based on k-mers, and they have not shed new light on the sequence dependence of occupancy. Here we test the ability of a general set of measures, statistical and algorithmic, to be informative about nucleosome occupancy and/or about the relationship between the affinity of nucleosomes with certain sequences and their complexities. 2 2.1 Methods The Dinucleotide Wedge Model The formulation of models of DNA bending was initially prompted by a recognition that DNA must be bent for packaging into nucleosomes, and that bending would be an informative index of nucleosome occupancy. Various dinucleotide 3 models can account reasonably well for the intrinsic bending observed in different sets of sequences, especially those containing A-tracts [19]. The Wedge model [20] suggests that bending is the result of driving a wedge between adjacent base pairs at various positions in the DNA. The model assumes that bending can be explained by wedge properties attributed solely to an AA dinucleotide (8.7 degrees for each AA). No current model provides a completely accurate explanation of the physical properties of DNA such as bending [21], but the Wedge model (like the more basic Junction model which is less suitable for short sequences and less general [22]) reasonably predicts the bending of many DNA sequences [23]. Although it has been suggested that trinucleotide models may make for greater accuracy in explaining DNA curvature in some of the sequences, dinucleotide models remain the most effective [19]. 2.2 The Segal Model Segal et al. established a probabilistic model to characterize the possibility that one DNA sequence is occupied by a nucleosome [16]. They constructed a nucleosome-DNA interaction model and used a hidden Markov model (HMM) to obtain a probability score. The model is based mainly on a 10-bp sequence periodicity that indicates the probability of any base pair being covered by a nucleosome. All k-nucleotide models, including that of Segal et al., are based upon knowledge-based sequence motifs and are thereby dependent on certain previously learned patterns. They can only account for local curvature and local predictions, not longer range correlations. Perhaps the fact that k-nucleotide models for k > 2 have not been proven to provide a significant advantage over k = 2 has led researchers to disregard longer range signals across DNA sequences involved in both DNA curvature and nucleosome occupancy [24]. To date, these models, including that of Kaplan [3] (which considers up to k = 5) and of Segal et al., are considered the gold standard for comparison purposes. To study the extent of different signals in the determination of nucleosome occupancy, we applied some basic transformations to the original genomic DNA sequence. The SW transformation substitutes G and C for S (Strong interaction), and A and T for W (for Weak interaction). The RY transformation substitutes A and G for R (for puRines), and C and T for Y (pYrimidines). 2.3 Complexity-based Genomic Profiling In what follows, we generate a function score fc for every complexity measure c (detailed descriptions in the S.I.) by applying each measure to a sliding window of length 147 nucleotides (nts) across a 20K and 100K base pair (bps) DNA sequence from Yeast chromosome 14. At every position of the sliding window, we get a sequence score for every c that is used to compare against in vivo and in vitro experimental occupancy. The following measures (followed by the name we refer to in parenthesis throughout the text) are here introduced. Among the measures considered are 4 Figure 1: Top: Correlation values of nucleosome occupancy (measured experimentally from chromosomic Yeast) on a sliding window of length 4K nt for both in vitro and in vivo against different measures/signals: the occupancy predictive Segal model (clearly better for in vitro). Middle: Calculated correlation values between the SW DNA transformation, carrying the GC content signal, found highly correlated to the Segal model but poorly explaining in vivo occupancy data. Bottom: The RY DNA transformation, an orthogonal signal to SW (and thus to GC content) whose values report a non-negligible max-min correlation, suggesting that the mixing of AT and GC carries some information about nucleosome occupancy (even if weaker than GC content), with in vivo values showing greatest correlation values unlike SW/GC and thus possibly neglected in predictive models (such as Segal’s). 5 entropy-based ones (see Supplementary Material for exact definitions): • Shannon entropy (Entropy) with uniform probability distribution. • Entropy rate with uniform probability distribution. • Lossless compression (Compress) A set of measures of algorithmic complexity (see Supplementary Material for exact definitions): • Coding Theorem Method (CTM) as an estimator of algorithmic randomness by way of algorithmic probability via the algorithmic Coding theorem (see Supplementary Material) relating causal content and classical probability [33, 34]. • Logical Depth (LD) as a BDM-based (see below) estimation of logical depth [25], a measure of sophistication that assigns both algorithmically simple and algorithmically random sequences shallow depth, and everything else higher complexity, believed to be related to biological evolution [26, 27]. And a hybrid measure of complexity combining local approximations of algorithmic complexity by CTM and global estimations of (block) Shannon entropy (see Supplementary Material for exact definitions): • The Block Decomposition Method (BDM) that approximates Shannon entropy—up to a logarithmic term—for long sequences, but KolmogorovChaitin complexity otherwise, as in the case of short nucleotides [28]. We list lossless compression under information-theoretic measures and not under algorithmic complexity measures, because implementations of lossless compression algorithms such as Compress and all those based on LempelZivWelch (LZ or LZW) as well as derived algorithms (ZIP, GZIP, PNG, etc.) are actually entropy estimators [38, 8, 28]. BDM allows us to expand the range of application of both CTM and LD to longer sequences by using Shannon entropy. However, if sequences are divided into short enough subsequences (of 12 nucleotides) we can apply CTM and avoid any trivial connection to Shannon entropy and thus to GC content. Briefly, to estimate the algorithmic probability [29, 30]—on which the measure BDM is based—of a DNA sequence (e.g. the sliding window of length 147 nucleoides or nt), we produce an empirical distribution [33, 34] to compare with by running a sample of up to 325 433 427 739 Turing machines with 2 states and 4 symbols (the number of nucleotide types in a DNA sequence) with empty input. If a DNA sequence is algorithmically random, then very few computer programs (Turing machines) will produce it, but if it has a regularity, either statistical or algorithmic, then there is a high probability of its being produced. Producing approximations to algorithmic probability provides approximations to algorithmic complexity by way of the so-called algorithmic 6 Coding Theorem [30, 33, 34]. Because the procedure is computationally expensive (and ultimately uncomputable) only the full set of strings of up to 12 bits was produced, and thus direct values can be given only to DNA sequences of up to 12 digits (binary for RY and SW and quaternary for full-alphabet DNA sequences). The tool is available at http://complexitycalculator.com/ where the user can calculate the information content and algorithmic complexity using the methods here introduced on any DNA segment for the purpose of similar of any other investigation of the structure of DNA and beyond. 3 3.1 Results Complexity-based Indices Fig. 1 shows the correlations between in vivo, in vitro, and the Segal model. In contrast, the SW transformation captures GC content, which clearly drives most of the nucleosome occupancy, but the correlation with the RY transformation that loses all GC content is very interesting. While significantly lower, it is existent and indicates a signal not contained in the GC content alone, as verified in Fig. 4. In Table 1 (SI), we report the correlation values found between experimental nucleosome occupancy data and ab initio training-free complexity measures. BDM alone explains more than any other index, including GC content in vivo, and unlike all other measures LD is negatively correlated, as theoretically expected [35] and numerically achieved [28], it being a measure that assigns low logical depth to high algorithmic randomness, with high algorithmic randomness implying high entropy (but not the converse). Surprisingly, entropy alone does not capture all the GC signals, which means that there is more structure in the distributions of Gs and Cs beyond the GC content alone. However, entropy does capture GC content in vivo, suggesting that local nucleotide arrangements (for example, sequence motifs) have a greater impact on in vivo prediction. Compared to entropy, BDM displays a higher correlation with in vivo nucleosome occupancy, thereby suggesting more internal structure than is captured by GC content and Shannon entropy alone. 3.2 Model Curvature versus Complexity Indices The dinucleotide model incorporates knowledge regarding sequence motifs that are known to have specific natural curvature properties and adds to the knowledge and predictive power that GC content alone offers. Using the Wedge dinucleotide model we first estimated the predicted curvature on a set of 20 artificially generated sequences (Table 4 (SI)) with different statistical properties, in order to identify possibly informative informationtheoretic and algorithmic indices. As shown in Table 2 (SI), we found all measures negatively correlated to the curvature modelled, except for LD, which 7 displays a positive correlation–and the highest in absolute value–compared to all the others. Since BDM negatively correlates with curvature, it is expected that the minima may identify nucleosome positions (see next subsection). An interesting observation in Table 2 (SI) concerns the correlation values between artificially generated DNA sequences and DNA structural curvature according to the Wedge nucleotide model: all values are negatively correlated, but curvature as predicted by the model positively correlates with LD, in exact inverse fashion vis-a-vis the correlation values reported in Table 1 (SI). This is consonant with the theoretically predicted relation between algorithmic complexity and logical depth [28]. All other measures (except for LD) behave similarly to BDM. The results in Tables 1 and 2 (SI) imply that for all measures both extrema (min and max for BDM and max and min for LD) may be indicative of high nucleosome occupancy. In the next section we explore whether extrema of these measures are also informative about nucleosome locations. 3.3 Nucleosome Dyad and Centre Location Test Positioning and occupancy of nucleosomes are closely related. Nucleosome positioning is the distribution of individual nucleosomes along the DNA sequence and can be described by the location of a single reference point on the nucleosome, such as its dyad of symmetry [36]. Nucleosome occupancy, on the other hand, is a measure of the probability that a certain DNA region is wrapped onto a histone octamer. Fig. 2 shows the predictive capabilities of algorithmic indices for nucleosome dyad and centre location when nucleosomic regions are placed against a background of (pseudo-) randomly generated DNA sequences with the same average GC content as themselves (∼ 0.5). BDM outperforms all methods in accuracy and strength. When taking the local min/max as potential indicators of nucleosome centres, we find that GC content fails (by design, as the surrounding sequences have the same GC content as the nucleosomic region of interest); lossless compression (Compress) performs well on the second half of the nucleosomes (left panel) but fails for the first half (right panel). Entropy performs as poorly as Compress—not surprisingly, as lossless compression algorithms are Entropy estimators. The results for BDM and LD suggest that the first 4 nucleosomal DNA sequences, of which 3 are clones, display greater algorithmic randomness (BDM) than the statistically pseudo-randomly generated background (surrounding sequences), while all other nucleosomes are of significantly lower algorithmic randomness (BDM) and mixed (both high and low) structural complexity (LD). The same robust results were obtained after several replications with different pseudo-random backgrounds. Moreover, the signal produced by similar nucleosomes with strong properties [37], such as clones 601, 603 and 605, had similar shapes and convexity. Fig. 3 shows the strength of the BDM signal at indicating the nucleosome centres based on the min/max value of the corresponding functions. The signal8 Figure 2: Nucleosome centre prediction (red dots) of 14 nucleosomes on an intercalated background of pseudo-random DNA segments of 147 nts with the same average GC content as the surrounding nucleosomic regions. Values are normalized between 0 and 1 and they were smoothed by taking one data point per 40 and using and interpolating order 2. Experimentally known nucleosome centres (called dyads) are marked with dashed lines. Panels on the right have their nucleotide centre estimated by the centre of the nucleosomic sequence (also dashed lines). Predictions are based on the local min/max values (up to 75 nts to each side) from the actual/estimated dyad/centre. Some red dots may appear to be placed slightly off but this is because there was a local min or max that vanished after the main curve was made smoother for visualization purposes. 9 Figure 3: Signal-to-noise ratio histogram. Distributions of centre predicted values demonstrating how BDM and LD are removed from a normal distribution thus picking a signal, unlike GC content that distributes values normally and performs no better than chance on a pseudo-random background with similar GC content. BDM carries the strongest signal followed by LD skewed in the opposite direction both peaking closer to the nucleosome centres than GC content. On the x-axis are complexity values arranged in bins of 1000 as reported in Fig. 2. to-noise ratio is much stronger for BDM, and is also shifted by LD in the opposite direction (to BDM), as would be consistent with the relation between algorithmic complexity and logical depth (see S.I.). Both BDM and LD spike at nucleosome positions stronger than GC content on a random DNA background with the same GC content, and perform better than entropy and compression. BDM is informative about every dyad or centre of nucleosomes, with 10 out of the 14 predicted within 1 to 3 bps distance and the rest within a 20 bps range. Unlike all other measures, LD performed better for the first half (left panel) of nucleosome centre locations than for the second half (right panel), suggesting that the nucleosomes of the first half may have greater structural organization. Table 3 (SI) compares distances to the nucleosome centres and error percentages as predicted without any training with BDM, to GC content prediction. The average distance between the predicted and the actual nucleosome centre is calculated to the closest local extreme (minima or maxima) within a window of 41 base pairs or bps (20 bps to each side plus the centre) from the actual centre (the experimentally known dyad or the centre nucleotide when the dyad was not known). In accordance with the results in Table 1 (SI) the maxima of BDM (minima of LD) could be informative about nucleosome positions and for those sequences, whose natural curvature is a fit to the superhelix, the minima BDM (maxima of LD) could also indicate nucleosome locations. This latter finding is supported by results in Table 2 (SI). Our results suggest that if some measures of complexity peak where GC 10 content is (purposely) tricked, there must be some structure different to GC content along the DNA sequence, either a distribution of GC content within the nucleosome length that is not related simply to the G and C count, or some other signal. 3.4 Informative Measures of High and Low Occupancy To find the most informative measures of complexity c we maximized the potential separation by taking only the sequences with highest (X% high) and lowest (Y% low) nucleosome occupancy. To this end we took as cutoff values 2 and 0.2 respectively, generating 300 disjoint sequences each from a 100K DNA segment for highest and lowest nucleosome occupancy values. The 100K segment starting and ending points are 187K − 40K and 207K + 40K nts in the 14th Yeast chromosome, so 40K nts surrounding the original shorter 20K sequence first explored. In Fig. 4 it was puzzling to find that the Segal model correlates less strongly than GC content alone for in vivo, suggesting that the model assigns greater weight to k-mer information than to GC content for these extreme cases given that we already knew that the Segal model is mostly driven by GC content (Fig. 1 middle). The box plot for the Segal model indicates that the model does not work as well for extreme sequences of high occupancy, with an average of 0.6 where the maximum over the segments on which these nucleosome regions are contained reaches an average correlation of ∼ 0.85 (in terms of occupancy), as shown in Fig. 1 for in vitro data. This means that these high occupancy sequences are on the outer border of the standard deviation in terms of accuracy in the Segal model. While the best model is the one that best separates the highest from the lowest occupancy, and therefore is clearly Segal’s model. Except for informationtheoretic indices (entropy and Compress), all algorithmic complexity indices were found to be informative of high and low occupancy. Moreover, all algorithmic complexity measures display a slight reduction in accuracy in vivo versus in vitro, as is consistent with the limitations of current models such as Segal’s. All but the Segal model are, however, training-free measures, in the sense that they do not contain any k-mer information related to high or low occupancy and thus are naive indices, yet all algorithmic complexity measures were informative to different extents, with CTM and BDM performing best and LD performing worst, LD displaying inverted values for high and low occupancy as theoretically expected (because LD assigns low LD to high algorithmic complexity) [35]. Also of note is the fact that CTM and BDM applied to the RY transformation were informative of high versus low occupancy, thereby revealing a signal different to GC content that models such as Segal’s partially capture in their encoded k-mer information. Interestingly, GC content alone outperforms the Segal model for high occupancy both in vitro and in vivo, but the Segal model outperforms GC content for low occupancy. Lossless compression was the worst behaved, showing how CTM and BDM outperform what is usually used as an estimator of algorithmic complexity [38, 11 Figure 4: Box plots reporting the informative value of complexity indices in vivo and in vitro for segments of lowest and highest occupancy, providing an overview of the informative value of sequence-dependent complexity measures. The occupancy score is given by the re-scaling of the complexity value fc (yaxis) so that the highest value is 1 and the lowest 0. In the case of the Segal model, fc is the direct score sequence based on the probability assigned by the model [16], with no re-scaling (because it is already scaled from origin with probability values between 0 and 1). Other cases not shown (e.g. entropy rate or Compress on RY or SW) had no significant results. Magenta and pink (bright colours) signify measures of algorithmic complexity; in dark gray colour are information-theoretic based measures. 12 8, 28]. Unlike entropy alone, however, lossless compression does take into consideration sequence repetitions, averaging over all k-mers up to the compression algorithm sliding window length. The results thus indicate that averaging over all sequence motifs—both informative and not—deletes all advantages, thereby justifying specific knowledge-driven k-mer approaches introduced in models such as Segal’s. 4 Conclusions The Kaplan and Segal models are considered to be the most accurate and the gold standard for predicting in vitro nucleosome occupancy. However, previous approaches including Segal and Kaplan requires extensive (pre-)training. In contrast, all measures considered by our approach are training free. These training-free measures revealed that there is more structure to nucleosome occupancy than GC content, and potentially to k-mer structure as well (e.g. non-AT-based mers), based on the correlations found in RY transformations indicative of low versus high occupancy. The nucleotide location test suggests a complexity hierarchy in which natural non-nucleosomic regions are less algorithmically random than nucleosomic regions, which in turn are less algorithmically random than pseudo-randomly generated DNA sequences with GC content equal to the nucleosomic regions. When pseudo-random regions are placed between nucleosomes, we showed that BDM tends to identify nucleosomic regions with a preference for lower algorithmic randomness with relative accuracy, and more consistently than GC content, which showed no pattern and mostly failed, as was expected, when fooled with a background of similar GC content. We have thus gone beyond previous attempts to connect and apply measures of complexity to structural and functional properties of genomic DNA, specifically in the highly active and open challenge of nucleosome occupancy in molecular biology. A direction for future research suggested by our work is the exploration of the use of these complexity indices to complement current machine learning approaches for reducing the feature space, by, e.g., determining which kmers are more and less informative, and thereby ensuring better prediction results. Another direction is a more extensive investigation of the possible use of genomic profiling for other types of structural and functional properties of DNA, with a view to contributing to, e.g., HiC techniques or protein encoding/promoter/enhancer region detection, and to furthering our understanding of the effect of extending the alphabet transformation of a sequence to epigenetics. 13 References [1] Gu C et al. (2015) DNA structural correlation in short and long ranges. The Journal of Physical Chemistry B 119(44):13980–13990. [2] Schep AN et al. (2015) Structured nucleosome fingerprints enable highresolution mapping of chromatin architecture within regulatory regions. 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[39] Collier JD (1998) Information increase in biological systems: how does adaptation fit? in Evolutionary systems. (Springer), pp. 129–139. [40] Rado T (1962) On non-computable functions. Bell System Technical Journal 41(3):877–884. 16 Supplementary Material 5 Indices of Information and of Algorithmic Complexity Here we describe alternative measures to explore correlations from an informationtheoretic and algorithmic (hence causal) complexity perspective. 5.1 Shannon Entropy Central to information theory is the concept of Shannon’s information entropy, which quantifies the average number of bits needed to store or communicate a message. Shannon’s entropy determines that one cannot store (and therefore communicate) a symbol with n different symbols in less than log(n) bits. In this sense, Shannon’s entropy determines a lower limit below which no message can be further compressed, not even in principle. Another application (or interpretation) of Shannon’s information theory is as a measure for quantifying the uncertainty involved in predicting the value of a random variable. For an ensemble X(R, p(xi )), the Shannon information content or entropy of X is then given by H(X) = − n X p(xi ) log2 p(xi ) i=1 where R is the set of possible outcomes (the random variable), n = |R| and p(xi ) is the probability of an outcome in R. 5.1.1 Entropy Rate The function R gives what is variously denominated as rate or block entropy and is Shannon entropy over blocks or subsequences of X of length b. That is, b=|X| R(X) = min H(Xb ) b=1 If the sequence is not statistically random, then R(X) will reach a low value for some b, and if random, then it will be maximally entropic for all blocks b. R(X) is computationally intractable as a function of sequence size, and typically upper bounds are realistically calculated for a fixed value of b (e.g. a window length). Notice that, as discussed in the main text, having maximal entropy does not by any means imply algorithmic randomness (c.f. 5.3). 5.2 Compression algorithms Two widely used lossless compression algorithms were employed. 17 5.2.1 Bzip2 Bzip2 is a lossless compression method that uses several layers of compression techniques stacked one on top of the other, including Run-length encoding (RLE), BurrowsWheeler transform (BWT), Move to Front (MTF) transform, and Huffman coding, among other sequential transformations. Bzip2 compresses more effectively than LZW, LZ77 and Deflate, but is considerably slower. 5.2.2 Compress Compress is a lossless compression algorithm based on the LZW compression algorithm. LempelZivWelch (LZW) is a lossless data compression algorithm created by Abraham Lempel, Jacob Ziv, and Terry Welch, and is considered universal for an infinite sliding window (in practice the sliding window is bounded by memory or choice). It is considered universal in the sense of Shannon entropy, meaning that it approximates the entropy rate of the source (an input in the form of a file/sequence). It is the algorithm of the widely used Unix file compression utility ‘Compress’, and is currently in the international public domain. 5.3 Measures of Algorithmic Complexity A binary sequence s is said to be random if its Kolmogorov complexity C(s) is at least twice its length. It is a measure of the computational resources needed to specify the object. Formally, C(s) = min{|p| : T (p) = s} where p is a program that outputs s running on a universal Turing machine T . A technical inconvenience of C as a function taking s to the length of the shortest program that produces s is its uncomputability. This is usually considered a major problem. The measure was first conceived to define randomness and is today the accepted objective mathematical measure of complexity, among other reasons because it has been proven to be mathematically robust (by virtue of the fact that several independent definitions converge to it). The invariance theorem guarantees that complexity values will only diverge by a constant (e.g. the length of a compiler, a translation program between T1 and T2 ) and will converge at the limit. Formally, |C(s)T1 − C(s)T2 | < c 5.3.1 Lossless Compression as Approximation to C Lossless compression is traditionally the method of choice when a measure of algorithmic content related to Kolmogorov-Chaitin complexity C is needed. The Kolmogorov-Chaitin complexity of a sequence s is defined as the length of the 18 shortest computer program p that outputs s running on a reference universal Turing machine T . While lossless compression is equivalent to algorithmic complexity, actual implementations of lossless compression (e.g. Compress) are heavily based upon entropy rate estimations [38, 8, 28] that mostly deal with statistical repetitions or k-mers of up to a window length size L, such that k ≤ L. 5.3.2 Algorithmic Probability as Approximation to C Another approach consists in making estimations by way of a related measure, Algorithmic Probability [33, 34]. The Algorithmic Probability of a sequence s is the probability that s is produced by a random computer program p when running on a reference Turing machine T . Both algorithmic complexity and Algorithmic Probability rely on T , but invariance theorems for both guarantee that the choice of T is asymptotically negligible. One way to minimize the impact of the choice of T is to average across a large set of different Turing machines all of the same size. The chief advantage of algorithmic indices is that causal signals in a sequence may escape entropic measures if they do not produce statistical regularities. And it has been the case that increasing the length of k in k-nucleotide models of structural properties of DNA have not returned more than a marginal advantage. The Algorithmic Probability [29] (also known as Levin’s semi-measure [30]) of a sequence s is a measure that describes the expected probability of a random program p running on a universal prefix-free Turing machine T producing s. Formally, X m(s) = 1/2|p| p:T (p)=s The Coding theorem beautifully connects C(s) and m(s): C(s) ∼ − log m(s) 5.3.3 Bennett’s Logical Depth Another measure of great interest is logical depth [25]. The logical depth (LD) of a sequence s is the shortest time logged by the shortest programs pi that produce s when running on a universal reference Turing machine. In other words, just as algorithmic complexity is associated with lossless compression, LD can be associated with the shortest time that a Turing machine takes to decompress the sequence s from its shortest computer description. A multiplicative invariance theorem for LD has also been proven [25]. Estimations of Algorithmic Probability and logical depth of DNA sequences were performed as determined in [33, 34]. Unlike algorithmic (Kolmogorov-Chaitin) complexity C, logical depth is a measure related to ‘structure’ rather than randomness. LD can be identified 19 with biological complexity [26, 39] and is therefore of great interest when comparing different genomic regions. 5.4 Measures Based on Algorithmic Probability and on Logical Depth The Coding theorem method (or simply CTM) is a method [33, 34] rooted in the relation between C(s) and m(s) specified by Algorithmic Probability, that is, between frequency of production of a sequence from a random program and its Kolmogorov complexity as described by Algorithmic Probability. Essentially, it uses the fact that the more frequent a sequence the lower its Kolmogorov complexity, and sequences of lower frequency have higher Kolmogorov complexity. Unlike algorithms for lossless compression, the Algorithmic Probability approach not only produces estimations of C for sequences with statistical regularities, but it is deeply rooted in a computational model of Algorithmic Probability, and therefore, unlike lossless compression, has the potential to identify regularities that are not statistical (e.g. a sequence such as 1234...), that is, sequences with high entropy or no statistical regularities but low algorithmic complexity [38, 8]. Let (n, m) be the space of all n-state m-symbol Turing machines, n, m > 1 and s a sequence, then: D(n, m)(s) = |{T ∈ (n, m) : T produces s}| |{T ∈ (n, m)}| (1) T is a standard Turing machine as defined in the Busy Beaver problem by Radó [40] with 4 symbols (in preparation for the calculation of the DNA alphabet size). Then using the relation established by the Coding theorem, we have: CT M (s) = − log2 (D(n, m)(s)) (2) That is, the more frequently a sequence is produced the lower its Kolmogorov complexity, and vice versa. CTM is an upper bound estimation of KologorovChaitin complexity. From CTM, a measure of Logical Depth can also be estimated–as the computing time that the shortest Turing machine (i.e. the first in the quasilexicographic order) takes to produce its output s upon halting. CTM thus produces both an empirical distribution of sequences up to a certain size, and an LD estimation based on the same computational model. Because CTM is computationally very expensive (equivalent to the Busy Beaver problem [40]), only short sequences (currently only up to length k = 12) have associated estimations of their algorithmic complexity. To approximate the complexity of genomic DNA sequences up to length k = 12, we calculated D(5, 4)(s), from which CT M (s) was approximated. To calculate the Algorithmic Probability of a DNA sequence (e.g. the sliding window of length 147 nt) we produced an empirical Algorithmic Probability 20 Table 1: Spearman correlations between complexity indices with in vivo and in vitro experimental nucleosome occupancy data from position 187 001 bp to 207 000 bp on the 14th Yeast chromosome in vitro in vivo GC content LD Entropy BDM Compress in vitro 1 0.5 0.684 -0.29 0.588 0.483 0.215 in vivo 0.5 1 0.26 -0.23 0.291 0.322 0.178 distribution from (5, 4) to compare with by running a sample of 325 433 427 739 Turing machines with up to 5 states and 4 symbols (the number of nucleotides in a DNA sequence) with empty input (as required by Algorithmic Probability). The resulting distribution came from 325 378 582 327 non-unique sequences (after removal of those sequences only produced by 5 or fewer machines/programs). 5.5 Relation of BDM to Shannon Entropy and GC Content The Block Decomposition Method (BDM) is a divide-and-conquer method that can be applied to longer sequences on which local approximations of C(s) using CTM can be averaged, thereby extending the range of application of CTM. Formally, X BDM (s, k) = log(n) + CTM(r). (3) sk The set of subsequences sk is composed of the pairs (r, n), where r is an element of the decomposition of sequence s of size k, and n the multiplicity of each subsequence of length k. BDM (s) is a computable approximation from below to the algorithmic information complexity of s, C(s). BDM approximations to K improve with smaller departures (i.e. longer k-mers) from the Coding Theorem method. When k decreases in size, however, we have shown [28] that BDM approximates the Shannon entropy of s for the chosen k-mer distribution. In this sense, BDM is a hybrid complexity measure that in the ‘worst case’ behaves like Shannon entropy and in the best approximates K. We have also shown that BDM is robust when instead of partitioning a sequence, overlapping subsequences are used, but this latter method tends to over-fit the value of the resultant complexity of the original sequence that was broken into k-mers. 21 Table 2: Spearman correlation values of complexity score functions versus the Wedge dinucleotide model prediction of DNA curvature on 20 synthetically generated DNA sequences p rho GC content 0.047 -0.45 Entropy 0.051 -0.44 Entropy rate (4) 0.0094 -0.57 Compress BZip2 BDM LD 0.0079 -0.58 0.048 -0.45 0.0083 -0.57 0.0019 0.65 Table 3: Distance in nucleotides to local min/max within a window of 2 tests, around 40 and 140 nts around the centre on a pseudo-randomly generated DNA background with the same GC content as the mean of the GC content of the next contiguous nucleosomic region. Clearly the experiment is designed for GC content to fail, yet BDM predicts the nucleosome position (by its centre) in a high number of cases and with great accuracy, with 10 out of the 14 centres predicted to within a 1 to 3 nt distance, thereby suggesting that there is more structure than GC content. Contrast this to GC content performing no better than chance, with an average fractional distance of 0.538 versus 0.105 for BDM from the predicted centre. Likewise for windows around the centre of 40 nt and 140 nt. All other methods (not included) reported intermediate values between GC content and BDM BDM 601 603 605 5Sr DNA pGub chicken β− globulin -48 -2 -1 -1 19 25 msat CAG TATA CA NoSecs TGGA TGA BadSecs 2 1 1 1 14 1 1 1 Table 4: The 20 short artificial DNA sequences generated covering a wide range of patterns and regularities used to find informative measures of DNA curvature. AAAAAAAAAAAA AAAAAAAAATAA ATAGAACGCTCC TCGTTCGCGAAT CTCTCAGGTCGT GGCGGGGGGTGG GCGCGCGCGCGC ATATATATATAT AAAAAAAACAAT ACCTATGAAAGC TGCACGTGTGGA CTCGTGGATATC GGGGGGGCGGGC GGGGGGGGGGGG 22 AAAAAATTTTTT AAGATCTACACT TAGGCGGCGGGC CTAAACACAATA CCACGATCCCGT GGGGGGCCCCCC Table 5: 14 Experimental nucleosome sequences [24]. Only the first 6 have known dyads name 601 dyad position 74 sequence ACAGGATGTATATATCTGACACGTGCCTGGAGACTAGGGAGTA ATCCCCTTGGCGGTTAAAACGCGGGGGACAGCGCGTACGTGCG TTTAAGCGGTGCTAGAGCTGTCTACGACCAATTGAGCGGCCTCG GCACCGGGATTCTCCAG 603 154 CGAGACATACACGAATATGGCGTTTTCCTAGTACAAATCACCCCA GCGTGACGCGTAAAATAATCGACACTCTCGGGTGCCCAGTTCGC GCGCCCACCTACCGTGTGAAGTCGTCACTCGGGCTTCTAAGTACG CTTAGGCCACGGTAGAGGGCAATCCAAGGCTAACCACCGTGCAT CGATGTTGAAAGAGGCCCTCCGTCCTTATTACTTCAAGTCCCTGG GGTACCGTTTC 605 132 TACTGGTTGGTGTGACAGATGCTCTAGATGGCGATACTGACAGG TCAAGGTTCGGACGACGCGGGATATGGGGTGCCTATCGCACATT GAGTGCGAGACCGGTCTAGATACGCTTAAACGACGTTACAACCC TAGCCCCGTCGTTTTAGCCGCCCAAGGGTATTCAAGCTCGACGCT AATCACCTATTGAGCCGGTATCCACCGTCACGACCATATTAATAG GACACGCCG 5Sr DNA 74, 92 AACGAATAACTTCCAGGGATTTATAAGCCGATGACGTCATAACAT CCCTGACCCTTTAAATAGCTTAACTTTCATCAAGCAAGAGCCTAC GACCATACCATGCTGAATATACCGGTTCTCGTCCGATCACCGAAG TCAAGCAGCATAGGGCTCGGTTAGTACTTGGATGGGAGACCGCC TGGGAATACCG pGub 84, 104 GATCCTCTAGACGGAGGACAGTCCTCCGGTTACCTTCGAACCACGT GGCCGTCTAGATGCTGACTCATTGTCGACACGCGTAGATCTGCTAG CATCGATCCATGGACTAGTCTCGAGTTTAAAGATATCCAGCTGCCC GGGAGGCCTTCGCGAAATATTGGTACCCCATGGAATCGAGGGATC chicken β-globulin 125 CTGGTGTGCTGGGAGGAAGGACCCAACAGACCCAAGCTGTGGTC TCCTGCCTCACAGCAATGCAGAGTGCTGTGGTTTGGAATGTGTGA GGGGCACCCAGCCTGGCGCGCGCTGTGCTCACAGCACTGGGGTG AGCACAGGGTGCCATGCCCACACCGTGCATGGGGATGTATGGCGC ACTCCGGTATAGAGCTGCAGAGCTGGGAATCGGGGGG mouse minor satellite ATTTGTAGAACAGTGTATATCAATGAGCTACAATGAAAATCATGGA CAG AGCAGCAGCAGCAACAGTAGTAGAAGCAGCAGCACTAACGACAG AAATGATAAAAACCACACTGTAGAACATATTAGATGAGTGAGTTA CACTGAAAAACACATCCGTTGGAAACCGGCAT CACAGCAGTAGCAGTAATAGAAGCAGCAGCAGCAGCAGTAGCAG TAGCAGCAGCAGCAGCAGCAATTTCAACAACAGCAGCAGCAGCT TATA AGGTCTATAAGCGTCTATAAGCGTCTATGAACGTCTATAAACGTCT ATAAACGCCTATAAACGCCTATAAACGCCTATACAAGCCTATAAAC GCCTATACACGTCTATGCACGACTATACACGTCT CA GAGAGTAACACAGGCACAGGTGTGGAGAGTAACACAGGCACAG GTGTGGGAGAGTGACACACAGGCACAGGTGAGGAGAGTACACA CAGGCACAGGTGTGGAGAGCACACACAGGTGCGGAGAG NoSecs GGGCTGTAGAATCTGATGGAGGTGTAGGATGGATGGACAGTATGA CAAAAGGGTACTAGCCTGGGACAGCAGGATTGGTGGAAAGGTTA CAGGCAGGCCCAGCAGGCTCGGACGCTGTATAGAG TGGA AGATGGATGGATGATGGATGGATGATGGATAGATGGATGATGGAT GGATGGATGATGATGGATGAATAGATGGATGGATGGATGATGGAT GGATGGACGATGGATGGATAGATGGATGGATGG TGA ATAGATGGATGAGTGGATGGATGGGTGGATGGATAGATGGGTGG ATGGGTGGATGGGTGGATGGATGATGGATGGATGAGTGGATGGA TGGATGGATGGGTGGATGGGTGGACGG BadSecs TCTAGAGTGTACAACTATCTACCCTGTAGGCATCAAGTCTATTTCGG 23 TAATCACTGCAGTTGCATCATTTCGATACGTTGCTCTTGCTTCGCTAG CAACGGACGATCGTACAAGCAC 

1 arXiv:1711.10662v1 [] 29 Nov 2017 Undefined 1 (2010) 1–5 IOS Press An Adaptive Fuzzy-Based System to Simulate, Quantify and Compensate Color Blindness Jinmi Lee a and Wellington Pinheiro dos Santos b,∗ a Escola Politécnica de Pernambuco, Universidade de Pernambuco, Rua Benfica, 455, Madalena, Recife, Pernambuco, 50720-001, Brazil E-mail: [email protected] b Núcleo de Engenharia Biomédica, Universidade Federal de Pernambuco, Av. Prof. Morais Rego, 1235, Cidade Universitária, Recife, Pernambuco, 50670-901, Brazil E-mail: [email protected] Abstract. About 8% of the male population of the world are affected by a determined type of color vision disturbance, which varies from the partial to complete reduction of the ability to distinguish certain colors. A considerable amount of color blind people are able to live all life long without knowing they have color vision disabilities and abnormalities. Nowadays the evolution of information technology and computer science, specifically image processing techniques and computer graphics, can be fundamental to aid at the development of adaptive color blindness correction tools. This paper presents a software tool based on Fuzzy Logic to evaluate the type and the degree of color blindness a person suffer from. In order to model several degrees of color blindness, herein this work we modified the classical linear transform-based simulation method by the use of fuzzy parameters. We also proposed four new methods to correct color blindness based on a fuzzy approach: Methods A and B, with and without histogram equalization. All the methods are based on combinations of linear transforms and histogram operations. In order to evaluate the results we implemented a web-based survey to get the best results according to optimize to distinguish different elements in an image. Results obtained from 40 volunteers proved that the Method B with histogram equalization got the best results for about 47% of volunteers. Keywords: Color blindness correction, color blindness simulation, linear color systems, color transformations, digital image processing 1. Introduction Millions of years of evolution have made the human visual system one of the most important sensory faculties. Nevertheless, about 8% of the male population has some sort of color vision disturbance. This condition is characterized by the partial or complete reduction of the ability to distinguish some colors [1]. It is a relatively common fact that people with color blindness use to live several years without realizing that they have a color vision deficiency. The reason for this * Corresponding author. E-mail: [email protected] fact is that the disorder can appear in different intensities. The increasing user interaction with graphical interfaces has been evidencing several problems related to color discrimination, which often restrics the use of these web-based applications [8]. The increasing evolution of information technology and computer science, as well as digital image processing on the improvement of visual information for human interpretation, have improved the visual quality of digital images for people with certain degrees of disturbance in color perception. However, most applications intended to lessen the color blindness effects do 0000-0000/10/$00.00 c 2010 – IOS Press and the authors. All rights reserved 2 J. Lee and W. P. dos Santos / An Adaptive Fuzzy-Based System to Simulate, Quantify and Compensate Color Blindness not take in account that such disorders could occur in many degrees, depending on each person [10]. The classical methods to simulate color blindness are based on mathematical color models to represent extreme cases of color blindness, i.e. the absence of one of the three photoreceptors of the human eye (red, green, and blue, i.e. low, medium, and high frequencies) [32]. Real cases of color blindness are characterized by some degree of anomaly at the absorbance spectrum of red, green, and blue spectral bands. Such a degree of chromatic deviation can be measured by software tools designed to collect parameters that can simulate real color blindness using fuzzy membership functions. One of the targets of this paper is to perform a study on the chromatic abnormalities of the human visual system and to develop computational tools for adaptability of human-machine interfaces, providing the inclusion of individuals with color blindness and creating more accessible solutions. In order to reach this target, we developed a simulation tool of color blindness from the application of linear transformation matrices that model the nonexistence or disability of the cone cells responsible for sensitivity to low, medium and high frequency spectral bands. We also developed a software tool to test color blindness and assess the degree of color blindness, using a fuzzy-based approach. The last tool was developed to improve the visual quality of digital images for people with color blindness. This integrated solution includes tests for the diagnosis of the color disturbance type to the image preview with the adjustments that aims to reduce the color blindness effects. In order to contribute for the evaluation of future color blindness correction tools, another purpose of this paper is to present a fuzzy-based method to simulate real color blindness. The development of color blindness simulators is very important both for the development and design of GUIs (Graphical User Interfaces) and for supporting the development of new mathematical methods of correction of color blindness. We developed a tool which simulates color blindness from linear transformation matrices which model either the nonexistence or the disability of the cone cells responsible for the sensitivities to low, medium and high frequencies. Those matrices are adapted according to the fuzzy parameters obtained by a previous tool, briefly cited above, which evaluates color blindness and assesses its severity degree, using an approach based on Fuzzy Logic. 2. Materials and Methods In the eyes, specifically in the retina, there is a zone where we can find the sensory cells specialized to capture the light stimuli: the photoreceptors known as rods and cones [8]. The rods have a scotopic characteristic, i.e they have a high sensitivity to achromatic light. Since the cones have the photopic characteristic, they are less sensitive to light, but are able to discriminate between different wavelengths [8]. There are three different types of cones in human eyes, each containing one type of photosensitive pigment. One type detects the spectrum of low frequencies of light (red color), while another one detects the medium frequencies spectrum (green color). The third type of cones detects the high frequencies spectral band (blue color). All these cones working together allow color vision [8,32]. The eye cones can be classified according to their sensitivity to different wavelengths. Those which are sensitive to the red spectral band are stimulated by long wavelengths; those ones sensitive to green are stimulated by wavelengths considered average, while cones that are sensitive to blue color are stimulated by short wavelengths. The vision of different types of colors becomes possible when those three types of cones are stimulated at the same time [8]. Several statistical studies show that about 8% of males and 0.4% of females have some form of disability related to the perception of colors [8]. Deficiencies for the colors are also called dyschromatopsias, or simply color blindness [32,8]. Most people have trichromatic vision. However, regarding to color vision disorders such as color blindness, it is possible to classify them in [8]: Anomalous trichromacy: anomaly in the proportions of red, green and blue. There are three types of anomalous trichomatic classified: 1. Protanomaly: less sensitive to red. 2. Deuteranomaly: less sensitive to green. 3. Tritanomaly: less sensitive to colors in the range of blue-yellow. Dichromatism: it can be considered as special absolute cases of anomalous dichromatism, i.e. the total lack of sensitivity to red, green or blue. Due to the absence of one kind of cone, it is presented in the form of: 1. Protanopia: absence of red retinal photoreceptors. J. Lee and W. P. dos Santos / An Adaptive Fuzzy-Based System to Simulate, Quantify and Compensate Color Blindness 2. Deuteranopia: absence of green retinal photoreceptors. 3. Tritanopia: absence of blue retinal photoreceptors. Monochromatism: Total inability to perceive color. This type of color blindness makes one see the world in gray levels. It is a very rare deficiency called achromatic vision. Considering monochromatism as monospectral vision, it is not possible to realce image features depending on color. Furthermore, dichromatism can be perceived as extremes cases of anomalous trichromacy. Therefore, herein this paper we focused only on anomalous trichromacy and dichromatism. In order to simplify notation and avoid transcribing the full name of the deficiencies of the chromatic visual system, throughout this article we decided to adopt the terms protan, deuteran and tritan to the deficiency of sensitivity to the frequencies of red (low frequencies), green (medium frequencies) and blue (high frequencies), respectively. About twenty methods of diagnosis and classification of color disorders are most often used. Among them are: pseudoisochromatic plates or color discrimination tests, color arrangement tests or color hue test, equalization, appointment or designation, etc. Pseudoisochromatic plates are used in the discrimination tests. These various types of available plates are combined in various tests, among which the Ishihara test, that is quite popular, being the most known and used worldwide [8]. Although this test does not provide a quantitative assessment of the problem and does not identify deficiencies of the tritan type, studies show that the Ishihara test remains the most effective test for rapid identification of congenital deficiencies in color vision [8]. Once visual information depends on image color distribution, it also depends on the frequency response of the visual system. Consequently, the lack of information can constitute a considerable problem for people with color blindness. Thus, it is possible to say that color blindness constitutes an obstacle to the effective use of computers, which nowadays uses more and more graphics in its interface and its visual communication. There is a growing commitment to create computational tools focused on the accessibility of people with color visual disturbances. The color blindness simulators are already quite common and avoid serious problems of accessibility, aiding to comprehend the percep- 3 tual limitations of a color blind individual [32]. Nevertheless, these simulators are designed just for dichromatism [32]. There exist also applications designed to improve the visual quality of images. However, in most applications, it is assumed that the user already knows his type of disorder and does not consider that the disorder may occur in varying degrees. The difference of using an adaptive filter is related to the diagnosis and the use of an approach that considers the uncertainty associated with the problem, where Fuzzy Logic appears as a natural candidate to solve the adaptation to the user according to his degree of color blindness. It is quite common for people with color blindness not to realize that they have visual disturbances, and many - when they discover the problem - do not know how to classify it. However, it is very important to improve efficiently the quality of life of those people, the knowledge of information as the type of color blindness and in what degree it is. In order to fill this information gap, a test tool called DaltonTest was developed. The goal of this tool is to classify the color blindness, showing the degree of the disability and its possible forms of presentation. In DaltonTest tool, the user is submitted to the Ishihara test (see figure 1), that has been customized through the use of weights. Different weights were assigned to the Ishihara test questions, so that the inaccurate responses received fewer points than the precise answers. This simple change makes it possible to evaluate approximately the user’s blindness degree. Figure 2 shows an example of the data structure adopted to describe and store a question of the Ishihara test used in the DaltonTest tool. The test, when completed, presents an estimated diagnosis of the user color blindness, containing three factors: the degree of color blindness, the degree of protanomaly, and the degree of deuteranomaly. These degrees are in fact degrees of membership to fuzzy sets associated to types of color blindness, protanomaly, and deuteranomaly, calculated from the fuzzyfication of determined scores. Such result is then used by the correction tool, providing a fuzzy characteristic for the application. The developed tool DaltonSim allows the simulation of the most common cases of anomalous trichomatic: protanomaly and the deuteranomaly. The simulation algorithm of colorblindness is based on the color model LMS (Longwave, Middlewave, Shortwave), once this color model is the most adequate to model the behaviour of light reception: notice eye cones are organized in receptor groups of short, mid- 4 J. Lee and W. P. dos Santos / An Adaptive Fuzzy-Based System to Simulate, Quantify and Compensate Color Blindness follows [32]:      Lp 0 2.0234 −2.5258 L  Mp  =  0 1  M  , 0 Sp 0 0 1 S (3) and for deuteranopia [32]: (a) (c)      Ld 1 0 0 L  Md  =  0.4942 0 1.2483   M  . Sd 0 0 1 S (4) Finally, there must be a transformation of the LMS color model to RGB. This transformation is obtained using the inverse matrix of the first step matrix [32]. (b)  (d) Fig. 1. Examples of Ishihara plates: (a) demonstration plate; (b) hidden plate; (c) masked plate; and (d) diagnosis plate.    R L  G  = TLM S−RGB  M  , B S (5)    L R M  ,  G  = T −1 RGB−LM S S B (6)      0.0809 −0.1305 0.1167 L R  G  =  −0.0102 0.0540 −0.1136   M  . (7) B −0.0004 −0.0041 0.6935 S  Fig. 2. Example of XML code describing the data structure of the DaltonTest tool dle, and high frequencies [32]. Conversion of the RGB components into components of the LMS model is the first step of the algorithm. The conversion is achieved by the application of a matrix, and, therefore, a linear conversion [8]:     L R  M  = TRGB−LM S  G  , S B (1)      L 17.8824 43.5161 4.1194 R  M  =  3.4557 27.1554 3.8671   G  . (2) S 0.0300 0.1843 1.4671 B The second step is reducing the normal domain of colors to the domain of a color blind individual. The linear transformation for protanopia is expressed as In order to modify these dichromatism models, we generate other matrices by including several fuzzy parameters, in order to get a mathematical model useful to deal with anomalous trichromatism. Therefore, our fuzzy-based model considers not only the extreme cases of protanopia and deuteranopia, but also hybrid cases where each case of color reception, including color “normality”, is represented by a given degree of membership to fuzzy sets designed to model these cases [15,35,6,33,11,5,34,20,31,23,30,18,24,21, 1,28,27,4,26,16,29,12,2,3,13,22,17,25,7,7,14]. The fuzzy parameter were designed to generate a linear dichromatism model to get LMS models able to vary from nondichromatic matrices represented by identity matrices to the absolute dichromatic simulation matrices given by expressions 3 and 4. Consequently, for the fuzzy degree of protanomaly αp , if αp = 0 we have the identity matrix, whilst in case we have αp = 1, we get the transform matrix described by expression 3. The analysis is similar for the deuteranopia simulation matrix. Our proposal deals with the J. Lee and W. P. dos Santos / An Adaptive Fuzzy-Based System to Simulate, Quantify and Compensate Color Blindness (a) (a) (a) Fig. 3. Result of dichromate simulation. (a) Original image, normal vision. (b) Image simulating protan type color blindness. (c) Image simulating type deuteran color blindness. generation of intermediate simulation matrices able to model nonabsolute cases of dichromatism. Therefore, considering the components already converted into the LMS model, the linear transformation for protanomaly is proposed as following:      Lp (1 − αp ) 2.0234αp −2.5258αp L  Mp  =   M  , 0 1 0 Sp S 0 0 1 (8) where αp is equal to the degree of protan; and for deuteranomaly: 5 individuals. Figure 3 shows some results from the simulation tool. Notice the presence of green color, since the absence of cones that detect green, in this case the type deuteran color blindness, does not prevent the perception of this spectral range, as there is a compensation due to the presence of other cones, as evidenced in the mathematical model of the anomaly deuteran. However, there is no perception of the real green [32]. Herein this work we also implemented a software tool called DaltonCor, intended to improve the visual quality of the color blind individuals, whereas its deficiency may be presented in different degrees. Two methods (A and B) were developed for the DaltonCor tool. Method A is based on the generation of a compensated image generated by the linear combination of the original image and the corrected versions for protan and deuteran cases. The weights are based on fuzzy rules. Method B is based on a linear transform matrix based on the fuzzy degrees of protan and deuteran color blindness, without using additional fuzzy rules. Both methods try to compensate for the lack of sensitivity to a particular color with new values for the normal color perception. 2.1. Method A      Ld 1 0 0 L  Md  =  0.4942αd (1 − αd ) 1.2483αd   M  , Sd 0 0 1 S (9) where αd is equal to the degree of deuteran. Based on both linear transformations, we propose a hybrid model of protanomaly and deuteranomaly:      Lh (1 − αp ) 2.0234αp −2.5258αp L  Mh  =  0.4942αd (1 − αd ) 1.2483αd   M  . Sh 0 0 1 S (10) The degrees of protanomaly αp and deuteranomaly αd are parameters obtained from the testing tool DaltonTest. Therefore, it is possible to simulate real cases of color blindness. The DaltonSim has a very simple interface, and the result from the simulation is the visualization of the original images alongside the simulated images. The simulation has shown to be essential to understand the problems of accessibility of the color blind The Method A was divided into three modules: Filter, Fuzzy and Control. This last one, in addition to link the other two modules, communicates with the graphical interface and feeds it with information. The solution developed in the filter module is based on color perception in cases of absolute color blindness. First, let us focus on the protan color blindness: Consider f = (fr , fg , fb ) the original image followed by its three bands of color r, g and b. Its correction is performed by following two steps. The first one (equation 11) is to assign new values to the bands not affected by color blindness. To protans, these bands of color are fg and fb , as following: f 0 = (fr , fg0 , fb0 ),  fg0 = 12 (fr + fg ) . fb0 = 12 (fr + fb ) (11) In order to increase the visual quality of the image, the second step (equation 12) is intended to improve the contrast. The chosen technique was the standard histogram equalization presented by Gonzalez and Woods (2002) [10]. Considering γ the contrast optimizer (histogram equalizer), we have the following 6 J. Lee and W. P. dos Santos / An Adaptive Fuzzy-Based System to Simulate, Quantify and Compensate Color Blindness expression: fp = (fr , fg00 , fb00 ),  fg00 = γ(fg0 ) . fb00 = γ(fb0 ) (12) The correction equations for deuterans (13 and 14) are similar to those presented above: 0 f = (fr0 , fg , fb0 ),  fr0 = 12 (fr + fg ) . fb0 = 12 (fg + fb ) (13) xn = x0p . x0p + x0d + x0n (20) The corrected image f ∗ is a weighted average of the corrected images for protan and deuteran type color blindness, and the original image, as represented in the following expression: f ∗ = xp fp + xd fd + xn f. fd = (fr00 , fg , fb00 ),  fr00 = γ(fr0 ) . fb00 = γ(fb0 ) (14) The Filter Module returns two corrected images, fp and fd . The Fuzzy Module is responsible for the filter customization and, based on the outcome from the test tool, assigns a fuzzy character of this correction. From an experimental approach, the following fuzzy rules were proposed, based on the output data we collect from DaltonTest color blindness software test [20,34,9,19]: x0p = β ∧ αp , (15) where x0p is equal to the degree of color blindness β with conjunction the degree of protan αp . x0d = β ∧ αd , (16) where x0d is equal to the degree of color blindness β with conjunction the degree of deuteran αd . x0n = αn ∧ (¬β), (17) where x0n is equal to the degree of normality αn and in conjunction with not color blindness. From the measurements obtained from the rules above, it is possible to make a fuzzification process on these magnitudes to obtain the weights expressed by the following equations: xp = x0p , x0p + x0d + x0n x0p xd = 0 , xp + x0d + x0n (18) (19) (21) 2.2. Method B The Method B is based on linear transformations designed to adaptatively correct the effects of chromatic disturbance. Differently from method A, this method is divided into two modules: Correction and Control. It is important to notice that, in this method, there is no previous correction for absolute color blindness, as it occurs in the module Filter of Method A. The module Control furnishes the graphic interface important information to be described in the following paragraphs. The solution implemented in module Correction is inspired by the perception of colors in absolute color blindness. However, instead of using the absolute degrees of color blindness, we take into account the fuzzy degrees of protanomaly and deuteranomaly. Therefore, this method is an adaptative correction method. Similarly to the method A, the proposal of correction of method B also tries to compensate the lack of sensitivity for determined colors (in fact, the deficiency to receive determined spectral bands), increasing their occurrence in the digital image and modifying the colors acquired with normal perception. Let f = (fr , fg , fb ) be the original image with its three color bands: r, g and b. This image is compensated by the use of a transform matrix to correct the pixel values by changing them using linear combinations. The expressions 22 and 23 represent the corrections for the absolute cases of protan and deuteran using fuzzy features. These empirical expressions are based on the following idea: compensating the lack of color band in color blind vision by equally distributing the information of this band to the other two bands. For protans, the corrected image is proposed as following: fp = (fr , fg0 , fb0 ),  fg0 = fb0 = αp 2 fr αp 2 fr 2−α + 2 p fg ; (22) 2−α + 2 p fb J. Lee and W. P. dos Santos / An Adaptive Fuzzy-Based System to Simulate, Quantify and Compensate Color Blindness for deuterans, the customization is performed in a similar manner, as can be seen in expression 23: fd = (fr0 , fg , fb0 ),  fr0 = fb0 = αd 2 fg αd 2 fg + + 2−αd 2 fr 2−αd 2 fb . (23) In order to deal with hybrid cases of color blindness where the individual has anomalous trichromatism for the red and green spectral bands at the same time, the new values of bands fr , fg and fb are calculated as following: f ∗ = (fr0 , fg0 , fb0 ), (24) fr0 = 2 − αd αd fg + fr 2 2 (25) fg0 = αp 2 − αp fr + fg 2 2 (26) fb0 = αp αd 4 − αp − αd fr + fg + fb 4 4 4 (27) Based on the new values of the three spectral bands, r, g and b, it was possible to generate an unique transform matrix to deal with the adaptative correction of anomalous trichromatism cases, as seen in expression 28:  0    1 − α2d α2d 0 R R  G0  =  α2p 1 − α2p   G  . (28) 0 αp α +α αd B0 B 1 − p4 d 4 4 The result of the application of the transform matrix is an image modified based on the given degrees of protan and deuteran. 3. Results 3.1. Simulation Results The fuzzy-based simulation tool presented in this paper receives as inputs the results obtained by the customized Ishihara testing tool, DaltonTest. These parameters allow our simulation tool to generate results in accordance to what is expected from real color 7 blindness cases, which is essential to understand how people with different degrees of color blindness perceive color variation. Notice in figure 4 a simulation for different degrees of the color vision anomalies and the gradual loss of ability to distinguish different colors. It is important to notice the similarity of images figure 4(c) is a protanomaly in 50% of cones and figure 4(j) is a hybrid case: 25% protanomaly and 25% deuteranomaly, which together reduce the compensation due to the presence of other cones, as can be perceived in the new mathematical model we presented herein this work. The simulation results with different levels of abnormality proved to be very satisfactory, with exception of hybrid cases with more than 50% of protan and deuteran. In order to analyze the results of the correction tool proposed, 10 images in bitmap 32 bits format were used. The use of the simulation tool was essential to understand how the corrections would be perceived by people with color blindness. We also analyzed all the correction variations in RGB, LMS, and with and without histogram equalization. In figure 5, it can be seen the correction result for an individual 100% color blind, 0% protan, 100% deuteran and 0% normal. Notice that in figure 5(b) the red tomatoes and the green peppers appear nearly equal. However, in figure 5(d) the shades of tomatoes and peppers are very different. Thus it is easy to see the gain of visual information after correcting the image. 3.2. Results for Color Blind People Although the analysis done by the simulator display is satisfactory, a group of four people with color blindness volunteered to test the tool. The volunteers were evaluated with the customized Ishihara test using the test tool. The test set was composed by a set of 32 bits bitmap images with size of 300×300 distributed as following: 10 original images without correction and, for each of these images, 4 corrected images using Method A with and without histogram equalization, taking into account RGB and LMS images, composing a total of 44 images. Through the obtained results, the four combinations of correction were analyzed. The results are presented in Table 1. Althoug these results were not conclusive due to the small number of volunteers, it was important as a first feedback from color blind people. 8 J. Lee and W. P. dos Santos / An Adaptive Fuzzy-Based System to Simulate, Quantify and Compensate Color Blindness ORIGINAL PROTAN (b) (a) DEUTERAN (f) (a) (b) (c) (d) (e) (f) (j) (g) (k) (d) (h) (l) (i) SIMULATED HYBRID (c) (e) ORIGINALS (m) Fig. 4. Results of dichromate simulation: (a) original image, normal vision; (b), (c), (d) and (e): protan type color blindness, for degrees of 25%, 50%, 75%, and 100%, respectively; (f), (g), (h) and (i): deuteran type color blindness, for degrees of 25%, 50%, 75%, and 100%, respectively; (j), (k), (l) and (m): hybrid (protan and deuteran) type color blindness, for equal degrees of protan and deuteran of 25%, 50%, 75%, and 100%, respectively. Each corrected image was subjectively evaluated and judged as much better, better, indifferent, worse or much worse. Distortion and greater ability to distinguish elements in relation to the original image were considered as evaluation criteria. It was observed that, by the correction in RGB domain with histogram equalization, the images became more understandable, because elements that were perceived with the same color (due to color blindness) re- Fig. 5. Results from image correction using the methods A and B in RGB with histogram equalization: (a) original image; (b) simulation for the absolute protan type; (c) corrected image with Method A; (d) corrected image with Method A simulated for protan color blindness; (e) corrected image with Method B; (f) corrected image with Method B simulated for protan color blindness. ceived different colors. Some images after the correction showed less saturation in their colors and some of these cases were considered worse. With the correction in RGB without histogram equalization there was not a big gain in the visual improvement of images, although this type of correction has been found to cause less negative impact on the distortion of the original colors. The correction in LMS, in general, changed overmuch the original colors of the images, although it performed efficiently on images that purposely have hidden elements, like the images used in pseudoisochromatic Ishihara plates. J. Lee and W. P. dos Santos / An Adaptive Fuzzy-Based System to Simulate, Quantify and Compensate Color Blindness With equalization RGB LMS 9 Without equalization Much better 20% Much better 0% Better Indifferent Worse Much worse 46% 20% 14% 0% Better Indifferent Worse Much worse 43% 43% 14% 0% Much better Better Indifferent Worse Much worse 20% 17% 20% 7% 36% Table 1 Much better Better Indifferent Worse Much worse 7% 33% 3% 7% 50% Preliminar evaluation by a set of 4 volunteers of 40 images corrected using Method A based on RGB and LMS, with and without histogram equalization 3.3. Results for People without Color Blindness Considering that: 1) the amount of people with color blindness represents a considerably lower percentage in the world’s population [32,8], 2) the occurrence of color blindness is not related to factors that might allow a clear separation between color blind individuals and the rest of the population, it is quite difficult to assemble a good set of test tools to validate the correction of color blindness proposed in this paper. Thus, we build a test based on the web to collect information of a subjective nature that could be useful to distinguish the methods proposed in this paper according to a qualitative evaluation. In order to build the platform test site we used 10 basic images. For each of these 10 images we generated images for protan and deuteran color blindness using the simulation method proposed in this work, with degrees of color blindness of 0%, 25%, 50%, 75% and 100%. We also used images with 0% of blindness as a way of maintaining quality control of results. Thus, we generated a total amount of 450 images. The web survey consists of 90 questions with five randomly arranged choices. Each issue consists in presenting an image, converted to the vision of a colorblind using four simulation methods proposed, with a certain degree of colorblindness. For each presented image there are five options, where four of them are obtained from the application of the correction methods, while the other image is simply a copy of the presented image. The user must choose the option among five options that best highlights different elements in the presented image, i.e. the option that optimizes the contrast between different elements, even if these ele- Fig. 6. Percentages of responses versus color blindness correction methods, considering a set of 40 volunteers without color blindness ments are not clearly distinguished in the original image. The web survey was administered to a universe of 40 users without color blindness. The absolute results are present in figure 6. From the results presented, we can notice that subjective aesthetic criteria influenced the results: 20.8% of volunteers chose the option “No improvements”, when the expected number was 11.1% (10 of the 90 presented images with 0% of color blindness). Figure 7 shows the results without the option “No improvements”. The results for protan and deuteran colorblindness distributed according to the degree of color blindness are shown in figures 8 and 10, respectively. Figure 8 shows that Method B with histogram equalization received the greatest number of positive responses from volunteers. Method B without histogram equalization also had a reasonable amount of positive responses for all degrees of color blindness, yet its performance was inferior to that obtained by Method B with histogram equalization. Method A, with and without histogram equalization, is shown below the Method B, for the degrees of color blindness of 25%, 50% and 75%. However, for protans with 100% of color blindness, Method A with histogram equalization can achieve a similar performance to the performance of Method B without histogram equalization. For protans with 100% color blindness, Method A without histogram equalization reaches almost twice 10 J. Lee and W. P. dos Santos / An Adaptive Fuzzy-Based System to Simulate, Quantify and Compensate Color Blindness Fig. 7. Percentages of positive responses (responses different from “no improvements”) versus color blind correction methods, considering a set of 40 volunteers without color blindness Fig. 8. Percentages of positive responses (responses different from “no improvements”) versus the degree of protan color blindness, considering a set of 40 volunteers without color blindness the performance obtained for 25% and 50% protans. However, despite the improvement acquired for protan cases with 100% of color blindness, Method A without histogram equalization is still much lower than other methods. Figure 9 shows the overall results for protan colorblindness. The results show that, for protan colorblindness, Method B with histogram equalization had the best performance: 47.3% of positive responses, compared with Method B without histogram equalization, Fig. 9. Percentages of positive responses (responses different from “no improvements”) versus color blind correction methods, considering a set of 40 volunteers without color blindness and protan simulated images with 31.4% of positive responses. Method A was very inferior to Method B, with 14.1% and 7.2% of positive responses, with and without histogram equalization, respectively. The results also show that the histogram equalization is also an important factor in improved distinction of different elements in the images. Figure 10 shows the results for deuteran color blindness, considering all degrees of color blindness. Here again the results obtained with Method B with histogram equalization were superior to those obtained by other methods. However, the distance to Method B without histogram equalization increased. The performance of Method A, with and without histogram equalization, improved slightly to 25%, 50% and 75% of blindness in relation to protan cases, although it remains well below the Method B with histogram equalization, but the behavior for 100% of color blindness was similar. Unlike the protan cases, Method B without histogram equalization had the worst performance among the four methods for 75% of blindness. Figure 11 illustrates the general results for the cases of deuteran color blindness. Comparing the results for deuterans with the results for protans, it is clear to notice that the performance of Method B with histogram equalization remained virtually the same, at approximately 47%. However, Method A with and without histogram equalization had much better results for deuterans than for protans. Already Method B with- J. Lee and W. P. dos Santos / An Adaptive Fuzzy-Based System to Simulate, Quantify and Compensate Color Blindness Fig. 10. Percentages of positive responses (responses different from “no improvements”) versus the degree of deuteran color blindness, considering a set of 40 volunteers without color blindness Fig. 11. Percentages of positive responses (responses different from “no improvements”) versus color blind correction methods, considering a set of 40 volunteers without color blindness and deuteran simulated images out histogram equalization proved to be far worse for deuterans than for protans. Therefore it is important to notice that there is a clear improvement of results due to the addition of histogram equalization. 4. Discussion and Conclusion Despite the rapid technological developments and advancements in the area of digital image processing, 11 it is not easy to find tools that simulate milder, intermediate forms of color vision impairment and reduce the effects of chromatic visual disturbances. As cited before, the extreme forms of color blindness which are characterized by the complete functional absence of one type of cones - are not as common as the milder forms with partial or shifted sensitivity. However, many studies assume that if a color scheme is legible for someone with extreme color vision impairment, it will also be easily legible for those with a minor affliction, and ignore the possibility of adaptive and customized correction methods. This work presents an adaptive tool to simulate color blindness. The fuzzy parameters designed to make the simulation flexible are obtained from the testing tool previously presented. Therefore, this paper proposed the development of a set of computational tools to improve the accessibility and visual quality of life for color blind people. However, one of the greatest difficulties encountered in developing this work is the relative lack in the literature about other mathematical methods for adaptive compensation of color blindness as the proposed. As mentioned, color blindness affects only a low percentage of population. The absence of color blindness cases in a statistically significant amount can be an obstacle to the development of adaptive tools for correction of the anomaly, especially when the focus is on real cases, rather than in extreme cases (dichromatism). Thus good color blindness simulators are of great importance on generating of synthetic cases of color vision disturbance in a statistically significant amount, so that they can be used in the development of correction tools. The fuzzy-based simulation proved to be essential to understand the accessibility problems of people with red and green color disorders in different degrees. Tests were conducted with a group of people with color blindness. This experience was important to evaluate the results obtained from the correction filters, and to more accurately gauge the difficulties encountered by the volunteers. The results were very positive, confirming that the proposed correction is able to extract a higher amount of information from an image. Moreover, all the tools proved to be intuitive and easy to use, providing a better user experience, and consequently their habitual use. Extensive tests were also conducted with people without color blindness using color blindness simulators and a web tool based on a survey. These tests were performed by a group of 40 volunteers. Results indi- 12 J. Lee and W. P. dos Santos / An Adaptive Fuzzy-Based System to Simulate, Quantify and Compensate Color Blindness cated that the Method B with histogram equalization got better results for both protan and deuteran color blindness. [18] [19] References [1] H. Adeli and S. L. Hung. An Adaptive Conjugate Gradient Learning Algorithm for Effective Training of Multilayer Neural Networks. Applied Mathematics and Computation, 62(1):81–102, 1994. [2] H. Adeli and X. Jiang. Neuro-Fuzzy Logic Model for Freeway Work Zone Capacity Estimation. Journal of Transportation Engineering, 129(5):484–493, 2003. [3] H. Adeli and X. Jiang. Dynamic Fuzzy Wavelet Neural Network Model for Structural System Identification. Journal of Structural Engineering, 132(1):102–111, 2006. [4] H. Adeli and A. 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MODULES AND LIE SEMIALGEBRAS OVER SEMIRINGS WITH A NEGATION MAP arXiv:1705.01075v1 [math.RA] 2 May 2017 GUY BLACHAR Abstract. In this article, we present the basic definitions of modules and Lie semialgebras over semirings with a negation map. Our main example of a semiring with a negation map is ELT algebras, and some of the results in this article are formulated and proved only in the ELT theory. When dealing with modules, we focus on linearly independent sets and spanning sets. We define a notion of lifting a module with a negation map, similarly to the tropicalization process, and use it to prove several theorems about semirings with a negation map which possess a lift. In the context of Lie semialgebras over semirings with a negation map, we first give basic definitions, and provide parallel constructions to the classical Lie algebras. We prove an ELT version of Cartan’s criterion for semisimplicity, and provide a counterexample for the naive version of the PBW Theorem. Contents 0. Introduction 0.1. Semirings with a Negation Map 0.2. Modules Over Semirings with a Negation Map 0.3. Supertropical Algebras 0.4. Exploded Layered Tropical Algebras 1. Modules over Semirings with a Negation Map 1.1. The Surpassing Relation for Modules 1.2. Basic Definitions for Modules 1.3. -morphisms 1.4. Lifting a Module Over a Semiring with a Negation Map 1.5. Linearly Independent Sets 1.6. d-bases and s-bases 1.7. Free Modules over Semirings with a Negation Map 2. Symmetrized Versions of Important Algebraic Structures 2.1. Semialgebras over Symmetrized Semirings 2.2. Lie Semialgebras with a Negation Map 2.3. Free Lie Algebras with a Negation Map 2.4. Symmetrized Versions of the Classical Lie Algebras 3. Solvable and Nilpotent Symmetrized Lie Algebras 3.1. Basic Definitions 3.2. Lifts of Lie Semialgebras with a Negation Map 3.3. Cartan’s Criterion for Lie Algebras Over ELT Algebras 4. PBW Theorem 4.1. Tensor Power and Tensor Algebra of Modules over a Semirings 4.2. The Universal Enveloping Algebra of a Lie Algebra with a Negation Map 4.3. Counterexample to the Naive PBW Theorem Page 2 2 3 4 4 5 6 7 9 10 13 14 18 20 20 21 24 28 30 30 33 35 37 37 37 38 Date: 14th March 2018. This article contains work from the author’s M.Sc. Thesis, which was submitted to the Math Department at BarIlan University. The work was carried under the supervision of Prof. Louis Rowen from Bar-Ilan University, to whom the author thanks deeply for his help and guidance. 1 2 GUY BLACHAR References 38 0. Introduction This paper’s objective is to form an algebraic basis in the context of semirings with negation maps. Semirings need not have additive inverses to all of the elements. While some of the theory of rings can be copied “as-is” to semirings, there are many facts about rings which use the additive inverses of the elements. The idea of negation maps on semirings is to imitate the additive inverse map. In order to do that, we assume that we are given a map a 7→ (−) a, which satisfies all of the properties of the usual inverse, except for the fact that a + (−) a = 0. This allows us to use the concept of additive inverses, even if they do not exist in the usual definition. Semirings with a negation map are discussed in [1, 8, 9, 2, 3, 24]. During this paper, we will first deal with modules which have a negation map over a semiring, and then study Lie semialgebras with a negation map over semirings. In the context of modules, our main interest would be to study linearly independent sets and spanning sets. We will study maximal linearly independent sets, minimal spanning sets and the connection between them. As the introduction demonstrates, the tropical world provides us with several nontrivial examples of semirings with a negation map – supertropical algebras and Exploded Layered Tropical (ELT) algebras. In the study of tropical algebra and tropical geometry, one tool is using Puiseux series – which allow us to “lift” a tropical problem to the classical algebra, and using the classical tools to solve it. We will give a general definition of what a lift is, and study some of its properties. We will move to considering Lie semialgebras with a negation map. We will study at first the basic definitions related to Lie algebras when working over semirings. We will consider on Lie semialgebras which are free as modules, such that their basis contains 1, 2 or 3 elements, and also consider nilpotent, solvable and semisimple Lie semialgebras. Cartan’s criterion regarding semisimple Lie algebras states that a Lie algebra is semisimple if and only if its Killing form is nondegenerate. In this paper, we prove an ELT version of this theorem for Lie semialgebras over ELT algebras. Another point of interest is Poincaré-Birkhoff-Witt (PBW) Theorem. In the classical theory of Lie algebras, the PBW Theorem states that every Lie algebra can be embedded into an associative algebra (its universal enveloping algebra), where the Lie bracket on this algebra is the commutator. We will construct the universal enveloping algebra of a Lie semialgebra, and give a counterexample to the naive version of the PBW theorem. Throughout this paper, addition and multiplication will always be denoted as + and ·, in order to emphasize the algebraic structure. Negation maps will always be denoted by (−). N stands for the set of natural numbers, whereas N0 = N ∪ {0}. Z stands for the ring of integers, and Z /nZ denotes the quotient ring of Z by nZ. 0.1. Semirings with a Negation Map. Definition 0.1. Let R be a semiring. A map (−) : R → R is a negation map (or a symmetry) on R if the following properties hold: (1) ∀a, b ∈ R : (−) (a + b) = (−) a + (−) b. (2) (−) 0R = 0R . MODULES AND LIE SEMIALGEBRAS OVER SEMIRINGS WITH A NEGATION MAP 3 (3) ∀a, b ∈ R : (−) (a · b) = a · ((−) b) = ((−) a) · b. (4) ∀a ∈ R : (−) ((−) a) = a. We say that (R, (−)) is a semiring with a negation map. If (−) is clear from the context, we will not mention it. For example, the map (−) a = a defines a trivial negation map on every semiring R. If R is a ring, it has a negation map given by (−) a = −a. Throughout this article, we use the following notations: • a (−) a is denoted a◦ . • R◦ = {a◦ |a ∈ R}. • R∨ = R \ R◦ ∪ {0R }. • We define two partial orders on R: – The relation  (which we call the surpassing relation) defined by a  b ⇔ ∃c ∈ R◦ : a = b + c – The relation ∇ defined by a∇b ⇔ a (−) b ∈ R◦ 0.2. Modules Over Semirings with a Negation Map. We now consider modules over semirings with a negation map. Definition 0.2. Let R be a semiring with a negation map. A (left) R-module is a commutative monoid (M, +, 0M ), equipped with an operation of “scalar multiplication”, R × M → M , where we denote (α, x) 7→ αx, such that the following properties hold: (1) ∀α ∈ R ∀x, y ∈ M : α (x + y) = αx + αy. (2) ∀α, β ∈ R ∀x ∈ M : (α + β) x = αx + βx. (3) ∀α, β ∈ R ∀x ∈ M : α (βx) = (α · β) x. (4) ∀α ∈ R : α0M = 0M . (5) ∀x ∈ M : 0R x = 0M . (6) ∀x ∈ M : 1R x = x. A right R-module is defined similarly. Example 0.3. The following are examples of R-modules, where R is a semiring: (1) R is an R-module, where the scalar multiplication is the multiplication of R. The submodules of R are its left ideals, meaning sets I ⊆ R that are closed under addition and satisfy RI ⊆ I. (2) Taking the direct sum of Mi = R for i ∈ I, we obtain the free R-module: M RI = R i∈I Two important special cases are: (3) Mm×n (R) is an R-module with the standard scalar multiplication, (αA)ij = α · (A)ij . Y M (4) Let {Mi }i∈I be R-modules. Then Mi and Mi are R-modules with the usual comi∈I i∈I ponentwise sum and scalar multiplication. As in [24], we consider negation maps on modules. Definition 0.4. Let R be a semiring, and let M be an R-module. A map (−) : M → M is a negation map (or a symmetry) on M if the following properties hold: (1) ∀x, y ∈ M : (−) (x + y) = (−) x + (−) y. (2) (−) 0M = 0M . (3) ∀α ∈ R ∀x ∈ M : (−) (αx) = α ((−) x). (4) ∀x ∈ M : (−) ((−) x) = x. 4 GUY BLACHAR If the underlying semiring has a negation map (−), every R-module M has an induced negation map given by (−) x = ((−) 1R ) x. Unless otherwise written, when working with a module over a semiring with a negation map, the negation map will be the induced one. We note that if it is not mentioned that the semiring has a negation map, the negation map on the module can be arbitrary (although the main interest is with the induced negation map). Definition 0.5. Let R be a semiring, and let M be an R-module with a negation map. A submodule with a negation map of M is a submodule N of M which is closed under the negation map. Since we are in the context of negation maps, and every module in this paper has a negation map, we will write “submodule” for a “submodule with a negation map”, where we understand that every submodule should be closed under the negation map. A main example of semirings and modules with a negation map uses the process of symmetrization defined in [24]. Briefly, If R is an arbitrary semiring, we may define R̂ = R × R with componentwise addition, and with multiplication given by (r1 , r2 ) · (r1′ , r2′ ) = (r1 r1′ + r2 r2′ , r1 r2′ + r2 r1′ ) . This is indeed a semiring with a negation map given by (r1 , r2 ) 7→ (r2 , r1 ). Of course, there is a natural injection R ֒→ R̂ by r 7→ (r, 0). The idea of this construction is to imitate the way Z is constructed from N. One should think of the element (r1 , r2 ) as r1 + (−) r2 . Now, if M is any R-module, we may define M̂ = M × M . We want to define it as an R̂-module; so we use componentwise addition, and define multiplication by (r1 , r2 ) (x1 , x2 ) = (r1 x1 + r2 x2 , r1 x2 + r2 x1 ) . This multiplication is called the twist action on M̂ over R̂. It endows M̂ with a R̂-module structure, and the induced negation map is (−) (x1 , x2 ) = (0, 1) (x1 , x2 ) = (x2 , x1 ) For the remainder of the introduction, we will present our two main examples of semirings with a negation map – supertropical algebras and ELT algebras. 0.3. Supertropical Algebras. Supertropical algebras are a refinement of the usual max-plus algebra. This is a refinement of the max-plus algebra, in which one adds “ghost elements” to replace the role of the classical zero. Supertropical algebras are discussed in several articles, including [13, 16, 17, 14, 18, 15]. Definition 0.6. A supertropical semiring is a quadruple R := (R, T , G, ν), where R is a semiring, T ⊆ R is a multiplicative submonoid, and G0 = G ∪ {0} ⊆ R is an ordered semiring ideal, together with a map ν : R → G0 , satisfying ν 2 = ν as well as the conditions:  a, ν (a) > ν (b) a+b= ν (a) ν (a) = ν (b) The monoid T is called the monoid of tangible elements, while the elements of G are called ghost elements, and ν : R → G0 is called the ghost map. Intuitively, the tangible elements correspond to the original max-plus algebra, although now a + a = ν (a) instead of a + a = a. Any supertropical semiring R is a semiring with a negation map, when the negation map is (−) a = a. Endowed with this negation map, R◦ = G0 . 0.4. Exploded Layered Tropical Algebras. ELT algebras are a more refined degeneration of the classical algebra than the classical max-plus algebra. The main idea which stands behind this structure is not only to remember the element, but rather remember also another information – its layer – which tells us “how many times we added this element to itself”. ELT algebras originate in [22], were formally defined in [25] and are discussed in [5, 6]. MODULES AND LIE SEMIALGEBRAS OVER SEMIRINGS WITH A NEGATION MAP 5 Definition 0.7. Let L be a semiring, and F a totally ordered semigroup. An Exploded Layered Tropical algebra (or, in short, an ELT algebra) is the pair R = R (L , F ), whose elements are denoted [ℓ]a for a ∈ F and ℓ ∈ L , together with the semiring (without zero) structure:  [ℓ ]  a1 > a2  1 a1 [ℓ ] [ℓ ] [ℓ ] 1 2 2 (1) a1 + a2 := a2 a1 < a2 .  [ℓ1 +L ℓ2 ] a1 a1 = a2 (2) [ℓ1 ]a1 · [ℓ2 ]a2 := [ℓ1 ·L ℓ2 ] (a1 +F a2 ). We write . For [ℓ]a, ℓ is called the layer, whereas a is called the tangible value. Let R be an ELT algebra. We write s : R → L for the projection on the first component (the sorting map):
s [ℓ]a = ℓ We also write τ : R → F for the projection on the second component:
τ [ℓ]a = a We denote the zero-layer subset R◦ = {α ∈ R|s (α) = 0} and R∗ = {α ∈ R|s (α) 6= 0} = R \ R◦ Definition 0.8. An ELT algebra R = R (F , L ) in which F is a totally ordered group and L is a ring (with 1) is called an ELT ring. Definition 0.9. For any ELT ring R, we define (−) [ℓ]a = [−ℓ]a. Lemma 0.10. a 7→ (−) a is a negation map on any ELT ring R. Proof. Straightforward from the definition.  Remark 0.11. When dealing with ELT algebra, the relation  is denoted . We point out some important elements in any ELT ring R: (1) [1]0, which is the multiplicative identity of R. (2) [0]0, which is idempotent to both operations of R. (3) [−1]0, which has the role of “−1” in our theory. Since ELT algebras lack an additive identity element, we adjoin one, denoted 0R (denoted −∞ in [5, 6]), and we denote R = R ∪ {0R }. This endows R with an antiring structure. 1. Modules over Semirings with a Negation Map Modules over rings have an important role in the classical theory. In the classical theory, a module M over a ring R is an abelian group, which has a scalar multiplication operation by elements from R. If one takes R to be a semiring rather than a ring, then a semimodule M is a commutative monoid, which has a scalar multiplication operation. Since semirings with a negation map are not rings, modules over them lack some basic properties of modules over rings. The study of modules can be found in [10, Chapters 14-18]. However, knowing that we are working over a semiring with a negation map instead of a general semiring, we note some unique properties of the module. For example, over a semiring with a negation map we have the notion of quasi-zeros, i.e. elements of the form x + (−) x, which play the role of the classical zero. In this part, we look into modules over semirings with a negation map. The main issue which we will deal with is different definitions of base. Following [14], we will give four definitions: (1) d-base – a maximal linearly independent set, Definition 1.45. 6 GUY BLACHAR (2) s-base – a minimal spanning set, Definition 1.54. (3) d,s-base – an s-base which is also a d-base, Definition 1.62. (4) base – defined as the classical base, Definition 1.66. We will point out the properties satisfied by modules possessing these bases, and we will examine the differences between these definitions. 1.1. The Surpassing Relation for Modules. We will now give analogous definitions of some concepts defined previously for semirings with a negation map. Definition 1.1. Let R be a semiring, and let M be a module with a negation map. An element of the form x + (−) x for some x ∈ M is called a quasi-zero (or a balanced element, as in [2]). We denote x◦ = x + (−) x. The submodule of quasi-zeros is M ◦ = {x◦ | x ∈ M } If R has a negation map, and the negation map of M coincides with the induced negation map, then M ◦ = 1◦R M . Definition 1.2. Let R be a semiring, and let M be an R-module with a negation map. We define a relation  on M in the following way: x  y ⇐⇒ ∃z ∈ M ◦ : x = y + z If x  y, we say that x surpasses y. Example 1.3. We refer to some of the examples of modules given here, and demonstrate the meaning of the surpassing relation in these modules. (1) In R, the surpassing relation of the module coincides with the surpassing relation of the semiring with a negation map. (2) In RI , the surpassing relation means surpassing componentwise. (3) As a special case of the free module, in Mm×n (R) the surpassing relation is equivalent to surpassing componentwise. (4) As another special case of the free module, in R [λ], the surpassing relation means surpassing of each coefficient of the polynomials. Lemma 1.4.  is a reflexive and transitive relation on any R-module with a negation map. However, it may not be antisymmetric, as demonstrated in [2, Example 4.12]. Proof. Let M be an R-module with a negation map. (1) If x ∈ M , then x = x + 0M , and 0M ∈ M ◦ ; so x  x. (2) If x, y, z ∈ M satisfy x  y  z, then x = y + z1 , y = z + z2 , where z1 , z2 ∈ M ◦ . So x = y + z1 = z + z1 + z2 ◦ and z1 + z2 ∈ M , implying x  z.  Although in general  may not be a partial order relation, we give a necessary and sufficient condition for it to be one, and we point out a specific case in which it is a partial order relation. Lemma 1.5. The following are equivalent for an R-module M with a negation map: (1) ∀x ∈ M ∀z1 , z2 ∈ M ◦ : x = x + z1 + z2 ⇒ x = x + z1 ∧ x = x + z2 . (2) ∀x ∈ M ∀z1 , z2 ∈ M ◦ : x = x + z1 + z2 ⇒ x = x + z1 ∨ x = x + z2 . (3)  is a partial order relation on M . Proof. 1 ⇒ 2 is trivial. 2 ⇒ 3: Assume that condition 2 holds. By Lemma 1.4, it is enough to show that  is antisymmetric. Let x, y ∈ M such that x  y and y  x. Then there are z1 , z2 ∈ M ◦ such that x = y + z1 and y = x + z2 . Therefore, x = x + z1 + z2 . By condition 2, either x = x + z2 = y (in which case we are done) or x = x + z1 . In the latter case, x = x + z1 + z2 = x + z2 = y MODULES AND LIE SEMIALGEBRAS OVER SEMIRINGS WITH A NEGATION MAP 7 which proves that  is a partial order relation on M . 3 ⇒ 1: Assume that  is a partial order relation on M , and let x ∈ M and z1 , z2 ∈ M ◦ such that x = x + z1 + z2 . Write y = x + z1 . By definition, y  x. On the other hand, x = x + z1 + z2 = y + z2  y Since  is antisymmetric, x = y = x + z1 , and thus also x = x + z1 + z2 = x + z2 .  Corollary 1.6. Let R be a semiring with a negation map, and let M be an R-module (with the induced negation map). The following are equivalent: (1) 1◦R is idempotent. (2) R◦ is idempotent. (3) M ◦ is idempotent for every R-module M . In this case,  is a partial order relation on every R-module M . Proof. 1 ⇒ 2: Let α◦ ∈ R◦ . Then α◦ + α◦ = 1◦R · α + 1◦R · α = (1◦R + 1◦R ) α = 1◦R α = α◦ 2 ⇒ 3: Assume that R◦ is idempotent, let M be an R-module, and let x◦ ∈ M ◦ . Then x◦ + x◦ = 1◦R x + 1◦R x = (1◦R + 1◦R ) x = 1◦R x = x◦ 3 ⇒ 1: Trivial. It is left to prove that in this case, if M is an R-module then  is a partial order relation on M . We prove condition 2 in Lemma 1.5. Indeed, if x ∈ M and z1 , z2 ∈ M ◦ satisfy x = x + z1 + z2 , then x = x + z1 + z2 = x + z1 + z2 + z2 = x + z2  Example 1.7. If R is a ring with (−) a = −a, a supertropical algebra with (−) a = a or an ELT ring with (−) a = [−1]0a, then  is a partial order relation on every R-module. Example 1.8. If  is a partial order relation on R, the induced surpassing relation on an R-module M might not be a partial order relation on M . For example, consider R = N0 with the usual addition and multiplication and the negation map (−) a = a. Then N◦0 = 2N0 , and thus for m, n ∈ N0 , m  n ⇔ ∃k ∈ N0 : m = n + 2k This is clearly a partial order relation. However, consider the R-module M = Z. The induced negation map is (−) a = a, and thus Z◦ = 2Z. Thus, for m, n ∈ Z, m  n ⇔ ∃k ∈ Z : m = n + 2k which is not only not a partial order relation, but an equivalence relation. We will later see that if  is not a partial order, we can “enforce” it to be one, by taking the module modulo the congruence it generates (see Definition 1.21). We note another property of M ◦ : Lemma 1.9. If x ∈ M ◦ and y  x, then y ∈ M ◦ . Proof. Since y  x, there exists z ∈ M ◦ such that y = x + z. The result follows, since M ◦ is a submodule of M .  1.2. Basic Definitions for Modules. 8 GUY BLACHAR 1.2.1. Spanning Sets. Definition 1.10. Let R be a semiring with a negation map, let M be an R-module, and let S ⊆ M be a subset of M . The R-module spanned by S, denoted Span (S), is ) ( k X αi xi ∈ M k ∈ N, α1 , . . . , αk ∈ R, x1 , . . . , xk ∈ S Span (S) = i=1 It is easy to see that S ⊆ Span (S), and that Span (S) ⊆ M is also an R-module with respect to the induced operations. Definition 1.11. Let R be a semiring with a negation map, and let M be an R-module. We say that a subset S ⊆ M is a spanning set of M , if Span (S) = M . Definition 1.12. A module with a finite spanning set is called finitely generated. 1.2.2. Congruences and the First Isomorphism Theorem. Since the usual construction of quotient modules using ideals fails over semirings, we use the language of congruences. In this subsubsection, we study congruences of modules over semirings with a negation map. Since this is a special case of congruences in the context of universal algebra, as written in [10], we mainly cite some known facts. Definition 1.13. Let R be a semiring, and let M1 , M2 be two R-modules with a negation map. An R-module homomorphism is a function ϕ : M1 → M2 , which satisfies: (1) ∀x, y ∈ M1 : ϕ (x + y) = ϕ (x) + ϕ (y). (2) ∀α ∈ R, ∀x ∈ M1 : ϕ (αx) = αϕ (x). (3) ∀x ∈ M1 : ϕ ((−) x) = (−) ϕ (x). We note that if R has a negation map, and the negation maps on M1 and M2 are the induced negation maps, then condition 3 follows from condition 2, since ϕ ((−) x) = ϕ (((−) 1R ) x) = ((−) 1R ) ϕ (x) = (−) ϕ (x) . Remark 1.14. ϕ (M1 ) ⊆ M2 ◦ , since ◦ ϕ (x◦ ) = ϕ (x + (−) x) = ϕ (x) + (−) ϕ (x) ∈ M2 ◦ Definition 1.15. Let R be a semiring, and let M be an R-module with a negation map. An equivalence relation ∼ on M is called a module congruence, if: (1) ∀x1 , x2 , y1 , y2 ∈ M : x1 ∼ x2 ∧ y1 ∼ y2 ⇒ x1 + y1 ∼ x2 + y2 . (2) ∀α ∈ R ∀x, y ∈ M : x ∼ y ⇒ αx ∼ αy. (3) ∀x, y ∈ M : x ∼ y ⇒ (−) x ∼ (−) y. The kernel of ∼ is M∼ = {x ∈ M | ∃y ∈ M ◦ : x ∼ y} Again, if the negation map on M is induced from a negation map from R, condition 3 follows from condition 2. Therefore, the negation map (−) of M a negation map (−) on M /∼ by (−) [x] = [(−) x]. It is easy to see that this negation map and the induced negation map from R coincide, since (−) [x] = [(−) x] = [((−) 1R ) x] = ((−) 1R ) [x] Lemma 1.16. M /∼ is an R-module. Lemma 1.17. M∼ is a submodule of M . Proof. If x1 , x2 ∈ M∼ , take y1 , y2 ∈ M ◦ such that x1 ∼ y1 and x2 ∼ y2 ; then x1 + x2 ∼ y1 + y2 , and y1 + y2 ∈ M ◦ . Thus, x1 + x2 ∈ M∼ . Now, if α ∈ R, x ∈ M∼ , take y ∈ M ◦ such that x ∼ y; then αx ∼ αy, and αy ∈ M ◦ . Thus, αx ∈ M∼ .  Lemma 1.18. Let ρ : M → M /∼ be the canonical map. Then
M /∼ ◦ = ρ (M∼ ) MODULES AND LIE SEMIALGEBRAS OVER SEMIRINGS WITH A NEGATION MAP Proof. ρ (x) ∈ M /∼
◦ 9 ⇐⇒ ρ (x) = 1◦R ρ (x) = ρ (1◦R x) ⇐⇒ x ∼ 1◦R x ⇐⇒ x ∈ M∼  We recall the first isomorphism theorem: Lemma 1.19. Let ϕ : M1 → M2 be an R-module homomorphism. Then x ∼ y ⇔ ϕ (x) = ϕ (y) is a module congruence on M1 . Theorem 1.20 (The First Isomorphism Theorem). Let R be a semiring with a negation map, and let M1 , M2 be R-modules. If ϕ : M1 → M2 is an R-module homomorphism, then there exists a module congruence ∼ on M1 such that M1 /∼ ∼ = Imϕ We return to  to demonstrate how we can “enforce”  to be a partial order on M . Definition 1.21. Let R be a semiring, and let M be an R-module with a negation map. Define a relation ≡◦ on M as a ≡◦ b ⇔ a  b ∧ b  a  Remark 1.22. ≡◦ is a congruence on M , and the R-module M ≡◦ is partially ordered by the induced negation map (−) [x] = [(−) x]. 1.3. -morphisms. As we will see more when dealing with Lie algebras with a negation map, we cannot always construct functions that will preserve every operation of the Lie algebra. We will now define the notion of -morphisms, as also defined in [24, Section 8.2]. Definition 1.23. Let R be a semiring, and let M1 , M2 be two R-modules with a negation map. A -morphism is a function ϕ : M1 → M2 , which satisfies: (1) ∀x, y ∈ M1 : ϕ (x + y)  ϕ (x) + ϕ (y). (2) ∀α ∈ R, ∀x ∈ M1 : ϕ (αx)  αϕ (x). (3) ∀x ∈ M1 : ϕ ((−) x) = (−) ϕ (x). (4) ϕ (M1 ◦ ) ⊆ M2 ◦ . Our purpose now will be to formulate a version of the First Isomorphism Theorem for morphisms. Assume that ϕ : M1 → M2 is a surjective -morphism. We define the equivalence relation ∼ (which is not necessarily a congruence) on M1 as x ∼ y ⇐⇒ ϕ (x) = ϕ (y) We now wish to define addition and scalar multiplication on M1 /∼ . The usual definition (adding or scalar multiplying the representatives) will no longer work, because ∼ may not be a congruence; so we define [x] + [y] = α · [x] = {z ∈ M1 |ϕ (z) = ϕ (x) + ϕ (y)} = ϕ−1 (ϕ (x) + ϕ (y)) {z ∈ M1 |ϕ (z) = α · ϕ (x)} = ϕ−1 (α · ϕ (x)) and the natural negation map (−) [x] = [(−) x]. The usual verifications show that: Lemma 1.24. M1 /∼ is an R-module with the above operations. The quasi-zeros are equivalence classes of the form [x], where x ∈ ker ϕ. We define the usual projection map ρ : M1 → M1 /∼ as ρ (x) = [x]. Lemma 1.25. ρ is a -morphism. Proof. 10 GUY BLACHAR (1) Let x, y ∈ M1 . Since ϕ (x + y)  ϕ (x) + ϕ (y), and since ϕ is injective, there is some z ∈ M1 such that ϕ (x + y) + ϕ (z) = ϕ (x) + ϕ (y) ◦ and ϕ (z) ∈ M2 , that is, z ∈ ker ϕ. Hence ρ (x + y) = [x + y]  [x + y] + [z] = [x] + [y] = ρ (x) + ρ (y) (2) Similarly as 1. (3) Given x ∈ M1 , it is easily seen that ρ ((−) x) = [(−) x] = (−) [x] = (−) ρ (x) (4) This is obvious.  We now get a version of the First Isomorphism Theorem: Theorem 1.26 (The First Isomorphism Theorem for -morphisms). Let R be a semiring, let M1 , M2 be two R-modules with a negation map, and let ϕ : M1 → M2 be a -morphism. Let ∼ be the equivalence relation defined above, and rho : M1 → M1 /∼ the projection map. Then there exists a unique R-isomorphism ϕ̂ : M1 /∼ → M2 such that ϕ = ϕ̂ ◦ ρ. Proof. We define ϕ̂ : M1 /∼ as ϕ̂ ([x]) = ϕ (x). The usual verifications as in the proof of the First Isomorphism Theorem prove that ϕ̂ is an R-isomorphism, and that ϕ̂ is uniquely defined.  We define a similar concept for semirings with a negation map: Definition 1.27. Let R1 and R2 be semirings with a negation map. A -morphism is a function ϕ : R1 → R2 such that the following properties hold: (1) ∀α, β ∈ R1 : ϕ (α + β)  ϕ (α) + ϕ (β). (2) ∀α, β ∈ R1 : ϕ (α · β)  ϕ (α) · ϕ (β). (3) ∀α ∈ R1 : (−) ϕ (α) = ϕ ((−) α). (4) ϕ (0R1 ) = 0R2 . (5) ϕ (1R1 ) = 1R2 . 1.4. Lifting a Module Over a Semiring with a Negation Map. 1.4.1. Lifting a semiring with a negation map. When dealing with tropical algebra, we had a powerful tool – Puiseux series. This tool enables one to use known results in the classical theory to prove tropical results. In this section, we attempt to give a similar construction for an arbitrary semiring with a negation map. Definition 1.28. Let R be a semiring with a negation map. For any subset A ⊆ R, define A = {β ∈ R | ∃α ∈ A : β  α} If A = R, we say that A is -dense in R. For example, R∨ = R. We note that this defines a topology on R; however, this topology is usually not even T1 , since, for example, {0R } = R◦ . b and a map Definition 1.29. Let R be a semiring with a negation map. A lift of R is a ring R b → R, such that the following properties hold: ϕ b:R b is (−) α (1) ϕ b is a -morphism, where the negation map on R b = −b α. ∨ (2) Imϕ b ⊆ R is -dense in R. (3) ϕ b (b α) = 0R ⇐⇒ α b = 0Rb Example 1.30. We give several examples of lifts. (1) If R is a ring with the negation map (−) a = −a, then the identity map R → R is a lift of R. b = L {{t}} with the EL tropicalization map is a (2) Given an ELT algebra R = R (F , L ), R lift of R, as defined and proved in [5, Lemma 0.4]. MODULES AND LIE SEMIALGEBRAS OVER SEMIRINGS WITH A NEGATION MAP 11 (3) Although the intuition is that lifts should be “very big”, this example proves that this is not always the case. Consider the semiring (N0 , +, ·) with the negation map (−) a = a. Then Z /2Z is a lift of N0 , with the map defined by ϕ b (0̄) = 0 and ϕ b (1̄) = 1. We now prove the following theorem: Theorem 1.31. Let R be an antiring with a negation map. Then R possesses a lift. Proof. We shall first construct the lift. We denote by A the following set: A = {aα | α ∈ R∨ } (We use the notation aα rather than α to distinguish the elements of A and those of R∨ ). We define a multiplication on the elements of A as follows:  aαβ , αβ ∈ R∨ aα · aβ = a0R , Otherwise This multiplication endows A with a monoid structure. b = Z [A], meaning We also denote R ) ( k X ∨ b= ni aαi ni ∈ Z, αi ∈ R R i=1 b is a ring. Since A is a monoid, R b → R. We define an action of {−1, 0, 1} on R by We are left with defining the lifting map ϕ b:R 1 · α = α, 0 · α = 0R , (−1) · α = (−) α
b → R is defined by ϕ The map ϕ b:R b 0Rb = 0R and ! k k X X |ni | (sign (ni ) · αi ) ni aαi = ϕ b i=1 i=1 where k X ni aαi is in its reduced representation, i.e. αi 6= αj for i 6= j, and sign (ni ) · αi is calculated i=1 by the above action.   b ϕ We shall now prove that R, b is a lift of R. We first prove that ϕ b is a -morphism. (1) It is easy to see that if αi 6= αj whenever i 6= j, ! k k X X ϕ b ϕ b (ni aαi ) ni aαi = i=1 i=1 Therefore we only have to prove that ϕ b (maα ) + ϕ b (naα )  ϕ b (maα + naα ) where m, n 6= 0. If sign (m) = sign (n), then ϕ b (maα ) + ϕ b (naα ) = |m| (sign (m) · α) + |n| (sign (n) · α) = |m + n| (sign (m + n) · α) = = ϕ b ((m + n) aα ) = ϕ b (maα + naα ) Now, suppose sign (m) 6= sign (n). Without loss of generality, we assume n < 0 < m and −n < m (the other cases are proved similarly). Thus, ϕ b (maα ) + ϕ b (naα ) = mα + (−n) ((−) α) = (m + n) α + (−n) α + (−n) ((−) α) = = (m + n) α + (−n) α◦  (m + n) α = ϕ b ((m + n) aα ) = = ϕ b (maα + naα ) as we needed to prove. 12 GUY BLACHAR (2) Let us begin by observing that ϕ b (aα ) ϕ b (aβ )  ϕ b (aαβ ) ∨ (Since if αβ ∈ R , there is equality; otherwise, the LHS is αβ ∈ R◦ , whereas the RHS is 0R ). Now,    !  ℓ ! k ℓ k X X X X n j βj  = mi αi ·  b nj aβ j  = mi aαi · ϕ ϕ b j=1 i=1 j=1 i=1 X = (mi nj ) αi βj + αi βj ∈R∨ X  αi βj = (3) ϕ b − k X i=1 ni aαi ! = k X ∈R∨ X (mi nj ) αi βj  αi βj ∈R / ∨  (mi nj ) αi βj = ϕ b X αi βj ∈R∨  !  ℓ k X X nj aβj  mi aαi ·  ϕ b  (mi nj ) aαi βj  j=1 i=1 |−ni | (sign (−ni ) · αi ) = (−) k X i=1 i=1  |ni | (sign (ni ) · αi ) = (−) ϕ b k X i=1 ni aαi !
(4) ϕ b 0Rb = 0R by definition.
b 1Rb = 1R . (5) Noting that 1Rb = 1 · a1R , ϕ This proves that ϕ b is a -morphism. Since Imϕ b = R∨ , we are left to prove that ϕ b (b α) = 0R if and only if α b = 0Rb . But this follows from the fact that R is an antiring.   b ϕ Hence R, b is a lift of R.  1.4.2. Lifting a module. We move towards lifting a module. We use again the concept of -density: Definition 1.32. Let R be a semiring, let M be an R-module with a negation map. For any subset S ⊆ M , define S = {y ∈ M | ∃x ∈ S : y  x} If S = M , we say that S is -dense in M .   b ϕ Definition 1.33. Let R be a semiring with a negation map with a lift R, b , and let M be an b c and a map ψb : M c → M such that the following properties R-module. A lift of M is an R-module M hold: c : ψb (x1 ) + ψb (x2 )  ψb (x1 + x2 ). (1) ∀x1 , x2 ∈ M b ∀x ∈ M c:ϕ (2) ∀b α∈R b (b α) ψb (x)  ψb (b αx b).
b (3) ψ 0M c = 0M . (4) Imψb is -dense in M . Theorem 1.34. If R has a lift, then every R-module has a (free) lift. Furthermore, if the module is generated by µ elements (for some cardinal number µ), it possesses a lift which is a free module generated by µ elements.   b ϕ c=R bS , and define a Proof. Let R, b be a lift of R. Let S ⊆ M be a generating set. Define M c → M by function ψb : M
X ψb (b rs ) = ϕ b (b rs ) s s∈S   c, ψb is a lift of M . We now prove that M s∈S MODULES AND LIE SEMIALGEBRAS OVER SEMIRINGS WITH A NEGATION MAP 13 c. Then (1) Let (b xs )s∈S , (b ys )s∈S ∈ M X X X

ψb (b xs )s∈S + ψb (b ys )s∈S = ϕ b (b xs ) s + ϕ b (b ys ) s = (ϕ b (b xs ) + ϕ b (b ys )) s  s∈S  X s∈S s∈S s∈S ϕ b (b xs + ybs ) s = ψb (b xs )s∈S + (b ys )s∈S
b and (b c. Then (2) Let α b∈R xs )s∈S ∈ M X X X

ϕ b (b α) ψb (b xs )s∈S = ϕ b (b α) ϕ b (b xs ) s = (ϕ b (b α) ϕ b (b xs )) s  ϕ b (b αx bs ) s = ψb α b (b xs )s∈S s∈S s∈S s∈S (3) Follows immediately. P b b is -dense in R, for each αs there is α bs ∈ R (4) Let x ∈ M . Write x = s∈S αs s. Since Imϕ such that αs  ϕ b (b αs ). If αs = 0R , we may choose α bs = 0Rb . Therefore, X X
x= αs s  ϕ b (b αs ) s = ψb (b αs )s∈S s∈S s∈S    n  b ϕ b , ψb is a lift of Rn , where Remark 1.35. If R, b is a lift of R, then R   b ((b x)i ) ψb (b x) = ϕ  i n I.e., ψb applies ϕ b on each entry of the given vector. In this case, Imψb = (R∨ ) . We will later see theorems which hold over modules which have a lift, such as Corollary 1.49.   b ⊆M c be a submodule. Then ψb N b is a submodule of M . Lemma 1.36. Let N   b and let α, β ∈ R. Take x c such that x  ψb (b b, yb ∈ M x) and y  ψb (b y ), and Proof. Let x, y ∈ ψb N   b such that α  ϕ take α b, βb ∈ R b (b α) and β  ϕ b βb . Then       by  ψb α by αx + βy  ϕ b (b α) ψb (b x) + ϕ b βb ψb (b y )  ψb (b αx b) + ψb βb bx b + βb by ∈ N , we are finished. Since α bx b + βb  1.5. Linearly Independent Sets. Up until now, most of our results were formulated and proved for a module with a negation map, where we didn’t assume that the underlying semiring has a negation map. However, now that we are going to deal with linear dependency, we need the notion of quasi-zero scalars; hence, we assume for the rest of this section that our semiring has a negation map. We first specialize our underlying semiring, to avoid the problem of “quasi-zero divisors”: Definition 1.37. A semiring with a negation map R is called entire, if R is commutative in both of the operations and if ∀α, β ∈ R : αβ ∈ R◦ ⇒ α ∈ R◦ ∨ β ∈ R◦ Definition 1.38. Let R be an entire semiring with a negation map, and let M be an R-module. A set S ⊆ M is called ◦-linearly dependent, if ∃k ∈ N, ∃x1 , . . . , xk ∈ S, ∃α1 , . . . , αk ∈ R∨ \ {0R } : α1 x1 + · · · + αk vk ∈ M ◦ S is called ◦-linearly independent, if it is not linearly dependent. Since we are dealing with negation maps, we will omit the ◦ in ◦-linearly dependent and ◦-linearly independent. Lemma 1.39. Let R be an entire semiring with a negation map, let M be an R-module, and let S ⊆ M be a linearly independent set. Then S ∩ M◦ = ∅ 14 GUY BLACHAR Proof. Assume there exists some x ∈ S such that x ∈ M ◦ . Then the linear combination 1R x is quasi-zero, contradicting the fact that S is linearly independent.  Lemma 1.40. Let R be an entire semiring with a negation map, let M be an R-module, and let S = {x1 , . . . , xm } be a linearly dependent subset of M . Assume that S ′ = {y1 , . . . , ym } ⊆ M satisfies ∀i : yi  xi . Then S ′ is also linearly dependent. X αi xi ∈ M ◦ , where ∀i : αi ∈ R∨ ; then Proof. Assume that i implying X X αi yi  αi xi ∈ M ◦ i i ◦ X αi yi ∈ M , by Lemma 1.9.  i Lemma 1.41. Let M be an R-module. If x  y, then x + (−) y ∈ M ◦ . Proof. By definition, there is some z ∈ M ◦ such that x = y + z. Therefore, x + (−) y = y + z + (−) y = 1◦R y + z ∈ M ◦  Lemma 1.42. Let R be an entire semiring with a negation map, and let M be an R-module. Assume k X αi xi . Then the set {x1 , . . . , xk , y} is linearly dependent. that y  i=1 Proof. Write I = {i = 1, . . . , n|αi ∈ / R◦ } ◦ We note that if i ∈ / I, then αi ∈ R , and thus αi xi ∈ M ◦ . If I = ∅, then y ∈ M ◦ , and thus {x1 , . . . , xk , y} is linearly dependent. Therefore, we may assume that I 6= ∅, and thus y k X αi xi  i=1 By Lemma 1.41, y+ X X αi xi i∈I ((−) αi ) xi = y + (−) i∈I X αi xi ∈ M ◦ . i∈I We found a linear combination of some of the vectors in the set {x1 , . . . , xk , x}, which is quasi-zero, and the coefficients are not quasi-zero, implying {x1 , . . . , xk , x} is linearly dependent.  Lemma 1.43. Let S ⊆ M be a linearly independent set. Then for all x ∈ S, S\ {x} is not a spanning set of M . Proof. Assume that S\ {x} is a spanning set of M . Then ∃k ∈ N, ∃α1 , . . . , αk ∈ R, ∃x1 , . . . , xk ∈ S : α1 x1 + · · · + αk xk = x = 1R x By Lemma 1.42, (S\ {x}) ∪ {x} = S is linearly dependent, a contradiction. Thus, S\ {x} is not a spanning set of M .  Corollary 1.44. Suppose A ⊆ M is a linearly independent set, and that B ⊆ M is a spanning set of M . If B ⊆ A, then A = B. Proof. If A 6= B, there exists x ∈ A \ B. Therefore, there exists a linear combination X x= αb b b∈B By Lemma 1.42, B ∪ {x} is linearly dependent. But B ∪ {x} ⊆ A, contradicting the assumption that A is linearly independent. Thus A = B.  1.6. d-bases and s-bases. MODULES AND LIE SEMIALGEBRAS OVER SEMIRINGS WITH A NEGATION MAP 15 1.6.1. d-bases. Definition 1.45. Let R be an entire semiring with a negation map. A d-base (for dependence base) of an R-module M is a maximal linearly independent subset of M . Definition 1.46. Let R be an entire semiring with a negation map, and let M be an R-module. The rank of M is rk (M ) = max {|B||B is a d-base of M } Lemma 1.47. Every linearly independent set is contained in some d-base. Proof. Let M be an R-module, and let S ⊆ M be a linearly independent set. Consider {S ′ ⊆ M |S ⊆ S ′ is linearly independent} This set satisfies Zorn’s condition, and thus, has a maximal element S ′′ , which is a d-base of M .  1.6.2. d-bases of modules  which possess a lift. For this part, we fix an entire semiring with a negation b map R with a lift R, ϕ b .   c, ψb , and let x c be vectors. If Lemma 1.48. Let M be an R-module with a lift M b1 , . . . , x bm ∈ M x b1 , . . . , x bm are linearly dependent, then ψb (b x1 ) , . . . , ψb (b xm ) are also linearly dependent. b not all are 0 b , such that Proof. x b1 , . . . , x bm are linearly dependent, so there are α b1 , . . . , α bm ∈ R, R m X i=1 Applying ψb on the equation, we get m X i=1 Using Lemma 1.9, we get ϕ b (b αi ) ψb (b xi )  ψb m X i=1 α bi x bi = 0M c m X i=1 α bi x bi !
= ψb 0M c = 0M ϕ b (b αi ) ψb (b xi ) ∈ M ◦ Since Imϕ b ⊆ R∨ , we get that for each i, ϕ b (b αi ) ∈ R∨ . But we know that not all of the α bi ’s are 0Rb , and thus not all of the ϕ b (b αi ) are 0R , as required.  Corollary 1.49. If R has a lift, than any n + 1 vectors in Rn are linearly dependent.  n  b , ψb . Let Proof. Let x1 , . . . , xn+1 ∈ Rn be any vectors. By Remark 1.35), Rn has a lift R  n b x′1 , . . . , x′n+1 ∈ Imψb such that xi  x′i for each i. Now, let y1 , . . . , yn+1 ∈ R be some vectors ′ b such that for each i, ψ (yi ) = x . i  n b is a ring, hence any n + 1 vectors in R b R are linearly dependent; in particular, y1 , . . . , yn+1 are linearly dependent. By Lemma 1.48, x′1 , . . . , x′n+1 are linearly dependent. By Lemma 1.40, we are finished.  Theorem 1.50. If R has a lift, then rk (Rn ) = n. Proof. By Corollary 1.49, any n + 1 vectors in Rn are linearly dependent; so rk (Rn ) ≤ n. However, since {e1 , . . . , en } are linearly dependent, rk (Rn ) ≥ n, which together yield the desired equality.  Recall the relation ≡◦ from Definition 1.21, defined as  a ≡◦ b ⇔ a  b ∧ b  a Theorem 1.51. Assume that R ≡◦ has a lift (rather than R). Then any n + 1 vectors in Rn are linearly dependent. 16 GUY BLACHAR n Proof. Let x1 , . . . , xn+1 ∈ Rn , and let [x1 ] , . . . , [xn+1 ] ∈ R ≡◦ be their equivalence classes under ≡◦ . Noting that  
n Rn ≡ ∼ = R ≡ ◦ ◦ by Corollary 1.49, there are α1 , . . . , αn+1 ∈ R∨ , not all are 0R , such that Hence,
◦ n [α1 x1 + · · · + αn+1 xn+1 ] = α1 [x1 ] + · · · + αn+1 [xn+1 ] ∈ R ≡◦ [α1 x1 + · · · + αn+1 xn+1 ] = 1◦R [α1 x1 + · · · + αn+1 xn+1 ] = [1◦R (α1 x1 + · · · + αn+1 xn+1 )] By the definition of ≡◦ , ◦ α1 x1 + · · · + αn+1 xn+1  1◦R (α1 x1 + · · · + αn+1 xn+1 ) ∈ (Rn ) By Lemma 1.9, ◦ α1 x1 + · · · + αn+1 xn+1 ∈ (Rn ) as required.  n Corollary 1.52. Let R be a ring with some negation map. Then any n+1 vectors in R are linearly dependent.  Proof. In this case, R ≡◦ is a ring, and the induced negation map is (−) [a] = − [a]; therefore,  R ≡ has a lift, and Theorem 1.51 can be applied.  ◦ In the above cases, we have another corollary: Corollary 1.53. If M ⊆ Rn is a submodule, then rk (M ) ≤ n. Proof. Any d-base of M is contained in a d-base of Rn , due to Lemma 1.47, whose order is at most n.  1.6.3. s-bases. Definition 1.54. Let R be an entire semiring with a negation map. An s-base (for spanning base) of an R-module M is a minimal spanning subset of M . An easy corollary from the First Isomorphism Theorem is the following: Corollary 1.55. Let M be an R-module with a finite spanning set S, |S| = n. Then there exists a congruence ∼ on Rn , such that n M∼ = R /∼ Corollary 1.56. Any spanning set of an R-module M must have at least rk (M ) elements. n Proof. By Corollary 1.55, if M = Span {x1 , . . . , xn }, then M ∼ = R /∼ for some congruence ∼. n ′ ]} of R /∼ . Thus, any d-base {y1 , . . . , ym } of M maps to a linearly independent set {[y1′ ] , . . . , [ym ′ ′ n  Then also y1 , . . . , ym are linearly independent in R , and thus m ≤ n. 1.6.4. Critical elements versus s-bases. We define a similar concept to the concept of critical elements presented in [14]: Definition 1.57. Let R be an entire semiring with a negation map, and let M be an R-module. (1) We define an equivalence relation ∼ on M , called projective equivalence, as the transitive closure of the following relation: we say that x ∼ y if there is an invertible α ∈ R∨ such that x = αy. (2) We define the equivalence class of x to be [x]∼ = {y ∈ M |y ∼ x} ◦ (3) We say that x ∈ M \M is critical, if there is no linear combination x = x1 + x2 where x1 , x2 ∈ M \ [x]∼ . (4) A critical set is a set of representatives of all the equivalence classes of ∼. Remark 1.58. Any critical set is projectively unique. MODULES AND LIE SEMIALGEBRAS OVER SEMIRINGS WITH A NEGATION MAP 17 Lemma 1.59. Suppose x ∈ M is critical. Then it is not spanned by M \ [x]∼ . Proof. Assume that x= n X αi yi i=1 for yi ∈ M \ [x]∼ . x is critical, thus n > 1. Write x1 = α1 y1 and x2 = of criticality, x2 ∈ [x]∼ . We get a contradiction by induction on n. Pn i=2 αi yi . By the definition  Lemma 1.60. Suppose S is an s-base of M . Then S contains a critical set of M . Proof. Suppose x ∈ M is critical. S is a s-base, thus x is spanned by S. So, by Lemma 1.59, it must be an element of S (up to projective equivalence).  In the classical theory (meaning when working with modules over rings), this definition is meaningless; there are no critical vectors. However, In [14, Theorem 5.24] it is proven that in the supertropical theory, any s-base (if it exists) is a critical set. One could hope that over any entire antiring with a negation map these definition will coincide – but even in the ELT theory there is a counterexample. Example 1.61. Consider R = R (C, R), and       [1]0 −∞  [1]0  M = Span [1]0 , −∞ ,  [1]0   [0] [1] (−1)  −∞ 0 {z } | S The set is a spanning set of M . It is straightforward to prove that any vector in S cannot be presented as a linear combination of the others. However, [1]  [1]      [1]    [1]0 −∞ −∞ 0 0 0 [1]0 = [0]0 [1]0 + −∞ +  [1]0  = [0]0 +  [1]0  [1] (−1) [1] (−1) [0]0 [0]0 [0]0 −∞ [1]  0 implying [1]0 is not critical. Since this vector is not critical, we get an example of an ELT module [0]0 with two s-bases which are not projectively equivalent:             [1]0 [1]0 −∞ −∞  [1]0   [1]0  M = Span [1]0 , −∞ ,  [1]0  = Span [0]0 , −∞ ,  [1]0   [0]  [0] [1] (−1)  [1] (−1)  −∞ −∞ 0 0 1.6.5. d,s-bases. Definition 1.62. Let R be an entire semiring with a negation map, and let M be an R-module. A d,s-base of M is an s-base of M , which is also a d-base. Lemma 1.63. If S is a d,s-base of M , with M finitely generated, then |S| = rk (M ). Proof. By Corollary 1.56, |S| ≥ rk (M ); but, since S is also a d-base, we get equality.  Lemma 1.64. Any linearly independent spanning set is a d,s-base. Proof. Assume that S is a linearly independent spanning set of M . If S ′ ⊆ S is an s-base, we get that S ′ = S, by Corollary 1.44. Now, assume that S ⊆ S ′′ is a d-base, using the same corollary we get S = S ′′ . So S is a d-base. To sum up, S is an s-base and a d-base, and thus S is a d,s-base.  Note that not every s-base is a d,s-base. Example 1.65. Any submodule spanned by zero-layered elements has no d,s-base, since the only linearly independent subset of this submodule is the empty set, which is not a spanning set. 18 GUY BLACHAR 1.7. Free Modules over Semirings with a Negation Map. 1.7.1. Definitions and examples. Definition 1.66. Let R be an entire semiring with a negation map. An R-module M is called free, if there exists a set B ⊆ M such that for all x ∈ M \ {0M } there exists a unique choice of n ∈ N, α1 , . . . , αn ∈ R and x1 , . . . , xn ∈ B such that x = α1 x1 + · · · + αn xn Such a set B is called a base of M . Example 1.67. Let R be an entire semiring with a negation map. Then Rn is a free R-module, with the base {e1 , . . . , en }. Indeed, the linear combination is determined uniquely, each component separately. Definition 1.68. Let R be an entire semiring with a negation map, and let M be a free Rmodule with a base B (|B| = n). For every x ∈ M there exists a unique representation x = αi1 vi1 + · · · + αik vik . We define the coordinate vector of x by the base B, [x]B ∈ Rn , to be: ( 0R j ∈ / {i1 , . . . , ik } ([x]B )j = , j = 1, . . . , n αij j ∈ {i1 , . . . , ik } It is easy to verify that [x + y]B = [x]B + [y]B , and [αx]B = α [x]B . Lemma 1.69. Let R be an entire semiring with a negation map, and let M be an R-module. M is free with a base B of size n if and only if M ∼ = Rn . Proof. If M ∼ = Rn , then M is free with a base of size n. Now, assume that M is free with a base B of size n. Denote B = {x1 , . . . , xn }, and define a function ϕ : M → Rn by ϕ (x) = [x]B . (1) ϕ is an R-module homomorphism: since [x + y]B = [x]B + [y]B and [αx]B = α [x]B . (2) ϕ is injective: If ϕ (x) = ϕ (y), then the representations of x and y with respect to B are the same. But we also know that these representations are unique, and thus x = y. ! n X αi xi = (αi ). (3) ϕ is surjective: If (αi ) ∈ Rn , then ϕ i=1 Thus, M ∼ = Rn .  We now prove that any base is a d,s-base. Lemma 1.70. If B is a base of M , and if n X αi bi ∈ M ◦ i=1 for α1 , . . . , αn ∈ R, b1 , . . . , bn ∈ B, then α1 , . . . , αn ∈ R◦ . Proof. Since n X αi bi ∈ M ◦ , there is some x ∈ M such that αi bi = x◦ . Writing x = i=1 i=1 have n X n X i=1 αi bi = x◦ = X b∈B βb b ! = X X βb b, we b∈B βb◦ b b∈B But B is a base, and thus α1 , . . . , αn ∈ R◦ (since all of the coefficients in the right side are quasizeros).  Lemma 1.71. If B is a base of M , then it is a linearly independent set. MODULES AND LIE SEMIALGEBRAS OVER SEMIRINGS WITH A NEGATION MAP Proof. Assume that X 19 αb b ∈ M ◦ b∈B ◦ By Lemma 1.70, ∀b ∈ B : αb ∈ R , and thus B is linearly independent.  Corollary 1.72. Any base is a d,s-base. Proof. Let M be a free R-module with base B. By definition, B is a spanning set of M . In addition, by Lemma 1.71, B is linearly dependent. We are finished, by Lemma 1.64.  Corollary 1.73. The cardinality of a base of a free module is uniquely determined. Example 1.74. We give an example of a submodule of a free module, which has a d,s-base which is not a base. Take R = R (C, R), and consider    [1]   [1]  0 1  [1]1  M = Span  [1]0  ,  [1]1  ,  [−1]0   [1]  [1]1 [1]0 0    [1]   [1]  0 1  [1]1  The set  [1]0  ,  [1]1  ,  [−1]0  is a d,s-base of M (to prove this set is linearly inde [1]  [1]1 [1]0 0 pendent, we use [25, Theorem 3.3.5]; their determinant is [2]2, which is not zero-layered, and thus this set is linearly dependent). However, it is not a base, because there is a vector which can be written as a linear combination of its elements in two different ways:  [1]   [1]   [1]   [1]   [1]  1 0 1 0 1  [1]0  +  [1]1  =  [1]1  =  [1]1  +  [−1]0  [1]0 [1]1 [1]1 [1]1 [1]0 However, if we know that a module is free, we also know all of its d,s-bases: Lemma 1.75. Let M be a free R-module, and let S be a d,s-base of M . Then S is a base of M . Proof. By Lemma 1.69, we can assume M = Rn . Since S is an s-base of M , by Lemma 1.60, S contains some critical set, A. Since the critical elements in Rn are precisely αi ei for some αi ∈ R∨ , we may write A = {α1 e1 , . . . , αn en } We now prove that each αi is invertible. Otherwise, assume without loss of generality that α1 is not invertible. Since S is an s-base of Rn , and since e1 is critical, there are β1 e1 , . . . , βk e1 ∈ S such that ∃γ1 , . . . , γk ∈ R : γ1 β1 e1 + · · · + γk βk e1 = e1 Since S is linearly independent, we must have k = 1, and thus βe1 ∈ S such that β is invertible. Again, since S is linearly independent, α1 = β is invertible, as required.  1.7.2. Free modules over entire antirings with a negation map. For the remainder of this part, we consider free modules over entire antirings with a negation map. Definition 1.76. Let R be an entire semiring with a negation map, and let M be a free R-module with two bases B = {v1 , . . . , vn } and C = {w1 , . . . , wn }. We define the transformation matrix B from B to C, [I]C , by: B [I]C = ([v1 ]C , . . . , [vn ]C ) Notice that [I]B C ∈ Mn (R). B Lemma 1.77. Under the conditions of Definition 1.76, ∀x ∈ M : [I]C [x]B = [x]C . 20 GUY BLACHAR Proof. Denote x= B [I]C n X k=1 = (αi,j ), [x]B = (βk ). That means x = βk vk = n X k=1 βk n X ℓ=1 αℓ,k wℓ ! n X βk vk , and vk = k=1 = n n X X ℓ=1 k=1 αℓ,k βk ! n X αℓ,k wℓ . We get ℓ=1 wℓ = n  X   And by the definition, ∀ℓ = 1, . . . , n : [I]B [x] = ([x]C )ℓ , as needed. C B B [I]C [x]B ℓ=1  ℓ wℓ  ℓ Lemma 1.78. Under the conditions of Definition 1.76, B [I]C B · [I]C = In Proof. We will show equality of columns. For all i = 1, . . . , n,   Lemma 1.77 B C B C B · [I] = [I]C Ci [I]C B C = [I]B · [I]C · ei = [I]B · [I]C · [vi ]B B · [vi ]C Lemma = 1.77 as needed. [vi ]B = ei  Corollary 1.79. A base of a free module over an entire antiring with a negation map is unique, up to invertible scalar multiplication of each element and permutation. In other words, any two bases of a free ELT module are projectively equivalent. B Proof. If B and C are two bases of a free R-module M , then [I]C is invertible. We also know (by B Corollary 1.73) that [I]C is a square matrix. By [7, Corollary 3], it is a generalized permutation matrix, namely C is B up to scalar multiplication of each element (ci ’s) and change of order (σ).  2. Symmetrized Versions of Important Algebraic Structures We now turn to study Lie semialgebras over semirings with a negation map. In the classical theory, a Lie algebra is a vector space endowed with a bilinear alternating multiplication, which satisfies Jacobi’s identity. In the context of negation maps, our definition will be quite similar to the classical one. We will mostly follow the approach of [12]. 2.1. Semialgebras over Symmetrized Semirings. Definition 2.1. A nonassociative semialgebra with a negation map over a semiring R is an R-module with a negation map A, together with an R-bilinear multiplication, A × A → A, that is distributive over addition and satisfies the axioms: (1) ∀α ∈ R ∀x, y ∈ A : α (xy) = (αx) y = x (αy). (2) ∀x, y ∈ A : (−) (xy) = ((−) x) y = x ((−) y). Note that “nonassociative” means “not necessarily associative”. As in the case of homomorphisms and congruences, if the negation map on A is induced from a negation map on R, the last condition is follows from the other. Example 2.2. Let A, B be nonassociative semialgebras with a negation map over a semiring R. Then A × B is also a nonassociative semialgebra with a negation map (a module as in Example 0.3), where the multiplication and the negation map are defined componentwise. Definition 2.3. An associative semialgebra is a nonassociative semialgebra, whose multiplication is associative. Definition 2.4. Let A, B be nonassociative semialgebras with a negation map over a semiring. A function ϕ : A → B is called an R-homomorphism, if it satisfies the following properties: (1) ϕ is an R-module homomorphism (as defined in Definition 1.13). (2) ∀x, y ∈ A : ϕ (xy) = ϕ (x) ϕ (y). The set of all R-homomorphisms ϕ : A → B is denoted Hom (A, B). We also say that: • ϕ is an R-monomorphism, if ϕ is injective. • ϕ is an R-epimorphism, if ϕ is surjective. • ϕ is an R-isomorphism, if ϕ is bijective. In this case we denote A ∼ = B. MODULES AND LIE SEMIALGEBRAS OVER SEMIRINGS WITH A NEGATION MAP 21 In addition, if A is a semialgebra with negation over a semiring R, then End (A) = Hom (A, A) is also semialgebra with a negation map over R, with the usual addition and scalar multiplication, composition as multiplication and negation map ((−) ϕ) (x) = (−) ϕ (x) One may also define -morphism of algebras with a negation map similarly. Definition 2.5. Let A, B be nonassociative semialgebras with a negation map over a semiring R, and let ϕ : A → B be an R-homomorphism. The kernel of ϕ is ker ϕ = {x ∈ A | ϕ (x) ∈ B ◦ } = ϕ−1 (B ◦ ) By Remark 1.14, A◦ ⊆ ker ϕ. We recall the definition of an ideal: Definition 2.6. Let A be a nonassociative semialgebra over a semiring R. A subalgebra I ⊆ A which satisfies IA, AI ⊆ I is called an ideal of A, denoted I ⊳ A. Lemma 2.7. ker ϕ is an ideal of A. Proof. We prove it directly: (1) If x, y ∈ ker ϕ, then ϕ (x + y) = ϕ (x) + ϕ (y) ∈ B ◦ . Hence, x + y ∈ ker ϕ. (2) If α ∈ R and x ∈ ker ϕ, then ϕ (αx) = αϕ (x) ∈ B ◦ , and thus αx ∈ ker ϕ. (3) If x ∈ ker ϕ, then ϕ ((−) x) = (−) ϕ (x) ∈ B ◦ , so (−) x ∈ ker ϕ. (4) If x ∈ ker ϕ and y ∈ A, then ϕ (x) ∈ B ◦ . Hence, ϕ (xy) = ϕ (x) ϕ (y) = (1◦R ϕ (x)) ϕ (y) = 1◦R (ϕ (x) ϕ (y)) Therefore, ϕ (xy) ∈ B ◦ , implying xy ∈ ker ϕ. Similarly, yx ∈ ker ϕ.  2.2. Lie Semialgebras with a Negation Map. 2.2.1. Basic definitions. We first restrict our base semiring to be a semifield with a negation map, i.e. a commutative semiring with a negation map in which every element which is not quasizero is invertible. Definition 2.8. Let R be a semifield. A Lie semialgebra with a negation map over R is a nonassociative semialgebra with a negation map over R, whose multiplication [·, ·] : L × L → L, called a negated Lie bracket, satisfies the following axioms: (1) Commutes with the negation map: ∀x, y ∈ L : [(−) x, y] = (−) [x, y]. (2) Alternating on L: ∀x ∈ L : [x, x] ∈ L◦ . (3) Anticommutativity: ∀x, y ∈ L : [y, x] = (−) [x, y]. (4) Jacobi’s identity: ∀x, y, z ∈ L : [x, [y, z]] + [z, [x, y]] + [y, [z, x]] ∈ L◦ . Remark 2.9. (1) If charR 6= 2, then 2 ⇒ 1. (2) If R has a negation map and the negation map of L is the induced negation map from R, then condition 1 follows from the bilinearity of the negated Lie bracket, since [(−) x, y] = [((−) 1R ) x, y] = ((−) 1R ) [x, y] = (−) [x, y] . Our definition is a bit different than Rowen’s definition [24]. The difference is reflected in Jacobi’s identity; Rowen’s definition requires that ∀x, y, z ∈ L : [x, [y, z]] (−) [y, [x, z]]  [[x, y] , z] We call this axiom the strong Jacobi’s identity. Note that while the strong Jacobi’s identity implies our version of Jacobi’s identity, the converse does not hold. An example is given in Example 4.6. Lemma 2.10. ∀x, y ∈ L : [x, y] + [y, x] ∈ L◦ . 22 GUY BLACHAR Proof. For all x, y ∈ L, [x, y] + [y, x] = [x, y] + (−) [x, y] ∈ L◦  Definition 2.11. A Lie semialgebra with a negation map L is called abelian, if ∀x, y ∈ L : [x, y] ∈ L◦ Example 2.12. Given a semifield R, let A be an associative semialgebra with a negation map over R. We define an operation as follows: [x, y] = xy + (−) yx, which is called the negated commutator. We will now show that it forms a structure of a Lie semialgebra with a negation map over R, which satisfied the strong Jacobi’s identity: (1) Bilinear: Let x, y, z ∈ A and α, β ∈ R. Then, [αx + βy, z] = (αx + βy) z + (−) (z (αx + βy)) = = α (xz + (−) zx) + β (yz + (−) zy) = α [x, z] + β [y, z] Linearity in the second component is proved similarly. (2) [·, ·] commutes with (−): For all x, y ∈ L, [(−) x, y] = ((−) x) y + (−) (y ((−) x)) = (−) (xy + (−) yx) = (−) [x, y] (3) Alternating on A: For all x ∈ A, [x, x] = x2 + (−) x2 ∈ L◦ (4) Anticommutativity: For all x, y ∈ A, [y, x] = yx + (−) xy = (−) (xy + (−) yx) = (−) [x, y] (5) Strong Jacobi’s identity: a proof can be found in [24], which uses the strong transfer principle. Lemma 2.13. ∀x, y ∈ L : x ∈ L◦ ∨ y ∈ L◦ =⇒ [x, y] ∈ L◦ Proof. Assume x ∈ L◦ ; write x = z + (−) z for some z ∈ L. Then [x, y] = [z + (−) z, y] = [z, y] + (−) [z, y] ∈ L◦  Definition 2.14. Let R be a semifield with a negation map, and let L be a Lie semialgebra with a negation map over R. A subset L1 ⊆ L is called a subalgebra of L, if it is a Lie semialgebra with a negation map over R with the restrictions of the operations of L. Definition 2.15. Let R be a semifield with a negation map, and let L be a Lie semialgebra with a negation map over R. The center of L is Z (L) = {x ∈ L | ∀y ∈ L : [x, y] ∈ L◦ } Lemma 2.16. L◦ ⊆ Z (L). Proof. By Lemma 2.13.  Definition 2.17. Let R be a semifield with a negation map, let L be a Lie semialgebra with a negation map over R, and let L1 , L2 ⊆ L be subalgebras of L. Define [L1 , L2 ] = Span {[x1 , x2 ]|x1 ∈ L1 , x2 ∈ L2 } Lemma 2.18. [Z (L) , L] ⊆ L◦ ⊆ Z (L) Proof. ∀x ∈ Z (L) ∀y ∈ L : [x, y] ∈ L◦ ⊆ Z (L)  MODULES AND LIE SEMIALGEBRAS OVER SEMIRINGS WITH A NEGATION MAP 23 2.2.2. Homomorphisms and Ideals. We use the general definitions for homomorphisms and ideals of nonassociative algebras given in section 2.1. Lemma 2.19. L◦ ⊳ L and Z (L) ⊳ L. Proof. L◦ is immediate by Lemma 2.13, and Z (L) by Lemma 2.18.  Lemma 2.20. If I, J ⊳ L, then I ∩ J ⊳ L. Proof. Let x ∈ I ∩ J, y ∈ L. x ∈ I, so [x, y] ∈ I. Also x ∈ J, so [x, y] ∈ J. In conclusion, [x, y] ∈ I ∩ J.  Definition 2.21. Let R be a semifield, let L be a Lie semialgebra with a negation map over R, and let I, J ⊳ L be two ideals of L. Then their sum is defined as I + J = {x + y|x ∈ I, y ∈ J} Lemma 2.22. If I, J ⊳ L, then I + J ⊳ L. Proof. Obviously, I + J is a submodule of L. Now, let x ∈ I + J, y ∈ L. x ∈ I + J, so there exists x′ ∈ I and x′′ ∈ J such that x = x′ + x′′ . We get [x, y] = [x′ + x′′ , y] = [x′ , y] + [x′′ , y] ∈ I + J  2.2.3. The Adjoint Algebra. Definition 2.23. Let R be a semifield, let L be a Lie semialgebra with a negation map over R, and let x ∈ L. We define a homomorphism adx : L → L by adx (y) = [x, y] Definition 2.24. Let R be a semifield, and let L be a Lie semialgebra with a negation map over R. The adjoint algebra of L is the set AdL = {adx |x ∈ L} endowed with the following negated Lie bracket: ∀adx , ady ∈ AdL : [adx , ady ] = ad[x,y] and the induced negation map from End (L): (−) adx = ad(−)x . Jacobi’s identity can be rewritten now as ∀x, y, z ∈ L : adx (ady (z)) + adz (adx (y)) + ady (adz (x)) ∈ L◦ and the strong Jacobi’s identity can be rewritten now as adx ady + (−) ady adx  ad[x,y] Lemma 2.25. AdL is a Lie semialgebra with a negation map over R. Proof. We first check that AdL is a submodule of End (L). But this is obvious, since adx + ady = adx+y ∈ AdL, αadx = adαx ∈ AdL and (−) adx = ad(−)x ∈ AdL. Now, we need to check that it is a Lie semialgebra with a negation map. (1) ∀α, β ∈ R ∀adx , ady , adz ∈ AdL : [αadx + βady , adz ] = [adαx+βy , adz ] = ad[αx+βy,z] = adα[x,z]+β[y,z] = αad[x,z] + βad[y,z] =  α [adx , adz] + β [ady , adz ] (2) ∀adx , ady ∈ AdL : [(−) adx , ady ] = ad(−)x , ady = ad[(−)x,y] = (−) ad[x,y] = (−) [adx , ady ]. (3) ∀adx ∈ AdL : [adx , adx ] = ad[x,x] , which is a quasi-zero. (4) ∀adx , ady ∈ AdL : [ady , adx ] = ad[y,x] = ad(−)[x,y] = (−) ad[x,y] = (−) [adx , ady ]. (5) ∀adx , ady , adz ∈ AdL : [adx [ady , adz ]]+[adz [adx , ady ]]+[ady [adz , adx ]] = ad[x[y,z]]+[z[x,y]]+[y[z,x]] , which is a quasi-zero.  24 GUY BLACHAR Lemma 2.26. There is a homomorphism of Lie semialgebras with a negation map L → AdL given by x 7→ adx Therefore, given the congruence ≡ on L defined by x ≡ y ⇔ adx = ady , one has L/ ≡ ∼ = AdL Proof. Using the definition of the operation on AdL, this is a homomorphism. By Theorem 1.20, we get the conclusion.  A word of caution: with the definition given above to the negated Lie bracket, AdL is not a subalgebra of End (L). Generally, there is no obvious way of fixing this problem; however, one can use the strong Jacobi’s identity. If L satisfies the strong Jacobi’s identity, then one can define adL = {adx | x ∈ L} ⊆ End (L) as in [24], and this is also a Lie semialgebra with a negation map over R (under the negated commutator). Nevertheless, we now have only a -morphism L → adL given by x 7→ adx . 2.3. Free Lie Algebras with a Negation Map. In this subsection we turn to study Lie semialgebras with a negation map which are free as modules and have a base consisting of one, two or three elements. 2.3.1. 1-dimensional Lie Algebras with a negation map. Lemma 2.27. Any Lie semialgebra L over a semifield with a negation map R with a base B = {x} is abelian. Proof. Given y, z ∈ L, there exist α, β ∈ R such that y = αx, z = βx. Therefore, [y, z] = [αx, βx] = αβ [x, x] ∈ L◦ Thus, L is abelian.  2.3.2. 2-dimensional Lie Algebras with a negation map. Lemma 2.28. Let R be a semifield with a negation map, and let L be a free R-module with base B = {x1 , x2 }. Define [x1 , x1 ] = α1 x1 + α2 x2 , [x2 , x2 ] = β1 x1 + β2 x2 , [x1 , x2 ] = (−) [x2 , x1 ] = γ1 x1 + γ2 x2 where α1 , α2 , β1 , β2 , γ1 , γ2 ∈ R and α1 , α2 , β1 , β2 ∈ R◦ . Now, extend [·, ·] to [·, ·] : L × L → L by bilinearity. Then L equipped with [·, ·] is a Lie semialgebra over R. Proof. By bilinearity, it is enough to check Jacobi’s identity in two cases: (1) [x1 , [x1 , x2 ]] + [x1 , [x2 , x1 ]] + [x2 , [x1 , x1 ]] ∈ L◦ . (2) [x2 , [x2 , x1 ]] + [x2 , [x1 , x2 ]] + [x1 , [x2 , x2 ]] ∈ L◦ . We shall prove the first part; the second is proved similarly. Indeed,  ◦ ◦ [x1 , [x1 , x2 ]] + [x1 , [x2 , x1 ]] + [x2 , [x1 , x1 ]] = x1 , [x1 , x2 ] + [x2 , [x1 , x1 ]] = [x1 , [x1 , x2 ]] + [x2 , [x1 , x1 ]] We note that [x1 , x1 ] ∈ L◦ , and thus [x1 , [x1 , x2 ]] + [x1 , [x2 , x1 ]] + [x2 , [x1 , x1 ]] ∈ L◦ , as required.  Lemma 2.29. Let L be a Lie semialgebra over a semifield with a negation map R with base B = {x1 , x2 }. Then there are α1 , α2 , β1 , β2 , γ1 , γ2 ∈ R such that [x1 , x1 ] = α1 x1 + α2 x2 , [x2 , x2 ] = β1 x1 + β2 x2 , [x1 , x2 ] = (−) [x2 , x1 ] = γ1 x1 + γ2 x2 and α1 , α2 , β1 , β2 ∈ R◦ . Therefore, each such Lie semialgebra is obtained from Lemma 2.28. Proof. The existence of α1 , α2 , β1 , β2 , γ1 , γ2 ∈ R follows from the fact that L is free, and by the antisymmetry of [·, ·]. Since [·, ·] is alternating, combined with Lemma 1.70, α1 , α2 , β1 , β2 ∈ R◦ .  MODULES AND LIE SEMIALGEBRAS OVER SEMIRINGS WITH A NEGATION MAP 25 2.3.3. 3-dimensional Lie Algebras with a negation map. We turn to studying free Lie semialgebras with a base consisting of three elements. The purpose of this case is to define a Lie semialgebra that will be parallel to sl (2, F ) in the classical theory. We are now going to give a necessary and sufficient condition for a bilinear multiplication defined on a module over a semifield with a negation map to be a negated Lie bracket. This is formulated in Corollary 2.32 and in Lemma 2.33. Throughout this part, R is a semifield with a negation map. Lemma 2.30. Let L be a free R-module with base B = {x1 , x2 , x3 }. Define ∀i ≤ j : [xi , xj ] = (−) [xj , xi ] = 3 X αi,j,ℓ xℓ ℓ=1 where ∀i, j, ℓ : αi,j,ℓ ∈ R, and ∀i, ℓ : αi,i,ℓ ∈ R◦ . If i > j, we write αi,j,ℓ = (−) αj,i,ℓ . Assume also that 3 XX ∀i, j, k, m : αi,ℓ,m αj,k,ℓ ∈ R◦ where X i,j,k ℓ=1 is the cyclic sum over i, j, k. Extend [·, ·] to [·, ·] : L × L → L by bilinearity. i,j,k Then L equipped with [·, ·] is a Lie semialgebra over R. Proof. Again, it is enough to check Jacobi’s identity. We need to ensure that ∀i, j, k : [xi , [xj , xk ]] + [xk , [xi , xj ]] + [xj , [xk , xi ]] ∈ L◦ Take some i, j, k. By calculating [xi , [xj , xk ]], we get: ! # " 3 3 3 3 3 X X X X X αj,k,ℓ [xi , xℓ ] = αj,k,ℓ αi,ℓ,m xm = [xi , [xj , xk ]] = xi , αj,k,ℓ xℓ = ℓ=1 ℓ=1 By permuting the indices, [xk , [xi , xj ]] = 3 X m=1 [xj , [xk , xi ]] = 3 X m=1 By summing all of the above, we get 3 X ℓ=1 3 X αi,j,ℓ αk,ℓ,m αk,i,ℓ αj,ℓ,m ℓ=1 [xi , [xj , xk ]] + [xk , [xi , xj ]] + [xj , [xk , xi ]] = 3 X m=1 Thus, the condition ∀m : 3 XX m=1 m=1 ℓ=1 ! ! 3 X αj,k,ℓ αi,ℓ,m ℓ=1 xm xm 3 XX αj,k,ℓ αi,ℓ,m cyc ℓ=1 ! xm αi,ℓ,m αj,k,ℓ ∈ R◦ i,j,k ℓ=1 is equivalent to Jacobi’s identity (by Lemma 1.70), and the assertion follows.  The above condition involving the cyclic sum may seem rather strong, since it has to hold for each choice of 1 ≤ i, j, k, m ≤ 3. However, it turns out that it is enough to check that it holds for a specific choice of i, j, k in which they are all different – for example, (i, j, k) = (1, 2, 3). This is formulated in the following lemma: Lemma 2.31. Assume that ∀i, ℓ : αi,i,ℓ ∈ R◦ and ∀i, j, ℓ : αi,j,ℓ = (−) αj,i,ℓ . If ∀m : 3 X ℓ=1 (α1,ℓ,m α2,3,ℓ + α3,ℓ,m α1,2,ℓ + α2,ℓ,m α3,1,ℓ ) ∈ R◦ ! xm 26 GUY BLACHAR holds, then 3 XX ∀i, j, k, m : αi,ℓ,m αj,k,ℓ ∈ R◦ i,j,k ℓ=1 Proof. We prove the lemma in two cases. Case 1. We first assume that two of i, j, k are equal. Without loss of generality, we assume that j = k, and thus ∀ℓ : αj,k,ℓ ∈ R◦ , and ∀ℓ : αi,j,ℓ = (−) αk,i,ℓ . We expand the cyclic sum: 3 XX αi,ℓ,m αj,k,ℓ = i,j,k ℓ=1 3 X (αi,ℓ,m αj,k,ℓ + αk,ℓ,m αi,j,ℓ + αj,ℓ,m αk,i,ℓ ) = ℓ=1 = = 3 X ℓ=1 3 X αi,ℓ,m αj,k,ℓ + 3 X 1◦R αi,ℓ,m αj,k,ℓ + ℓ=1 = 1◦R 3 X 1◦R αk,ℓ,m αi,j,ℓ = ℓ=1 3 X αi,ℓ,m αj,k,ℓ + ℓ=1 3 X αk,ℓ,m αi,j,ℓ ℓ=1 and thus the sum is quasi-zero. Case 2. (αk,ℓ,m αi,j,ℓ + (−) αj,ℓ,m αi,j,ℓ ) = ℓ=1 ! Now we may assume that i, j, k are different indices. Since 1 ≤ i, j, k ≤ 3, and by the cyclic symmetry of i, j, k, there are two options: either (i, j, k) = (1, 2, 3), and the sum is S(1,2,3) = 3 XX αi,ℓ,m αj,k,ℓ = 3 XX αi,ℓ,m αj,k,ℓ = i,j,k ℓ=1 3 X (α1,ℓ,m α2,3,ℓ + α3,ℓ,m α1,2,ℓ + α2,ℓ,m α3,1,ℓ ) 3 X (α1,ℓ,m α3,2,ℓ + α2,ℓ,m α1,3,ℓ + α3,ℓ,m α2,1,ℓ ) ℓ=1 or (i, j, k) = (1, 3, 2), and the sum is S(1,3,2) = ℓ=1 i,j,k ℓ=1 We notice that S(1,3,2) = (−) S(1,2,3) , since S(1,3,2) = = 3 X ℓ=1 3 X (α1,ℓ,m α3,2,ℓ + α2,ℓ,m α1,3,ℓ + α3,ℓ,m α2,1,ℓ ) = ((−) α1,ℓ,m α2,3,ℓ + (−) α2,ℓ,m α3,1,ℓ + (−) α3,ℓ,m α1,2,ℓ ) = ℓ=1 = (−) 3 X (α1,ℓ,m α2,3,ℓ + α2,ℓ,m α3,1,ℓ + α3,ℓ,m α1,2,ℓ ) = (−) S(1,2,3) ℓ=1 But we assumed that S(1,2,3) ∈ R◦ , and thus S(1,3,2) ∈ R◦ , as required.  Corollary 2.32. Let L be a free R-module with base B = {x1 , x2 , x3 }. Define ∀i ≤ j : [xi , xj ] = (−) [xj , xi ] = 3 X αi,j,ℓ xℓ ℓ=1 where ∀i, j, ℓ : αi,j,ℓ ∈ R, and ∀i, ℓ : αi,i,ℓ ∈ R◦ . If i > j, we write αi,j,ℓ = (−) αj,i,ℓ . Assume that also 3 X ∀m : (α1,ℓ,m α2,3,ℓ + α3,ℓ,m α1,2,ℓ + α2,ℓ,m α3,1,ℓ ) ∈ R◦ ℓ=1 Extend [·, ·] to [·, ·] : L × L → L by bilinearity. MODULES AND LIE SEMIALGEBRAS OVER SEMIRINGS WITH A NEGATION MAP 27 Then L equipped with [·, ·] is a Lie semialgebra over R. Proof. By Lemma 2.31, the conditions of Lemma 2.30 hold, and thus the assertion follows.  Lemma 2.33. Let L be a Lie semialgebra over R with base B = {x1 , x2 , x3 }. Then there are αi,j,ℓ ∈ R such that 3 X αi,j,ℓ xℓ ∀i ≤ j : [xi , xj ] = (−) [xj , xi ] = ℓ=1 and ∀i, j, ℓ : αi,j,ℓ ∈ R, and ∀i, ℓ : αi,i,ℓ ∈ R◦ . If i > j, we denote αi,j,ℓ = (−) αj,i,ℓ . Moreover, ∀m : 3 X (α1,ℓ,m α2,3,ℓ + α3,ℓ,m α1,2,ℓ + α2,ℓ,m α3,1,ℓ ) ∈ R◦ ℓ=1 Therefore, each such Lie semialgebra is obtained from Corollary 2.32. Proof. The existence of αi,j,ℓ follows from the fact that L is free, and by the antisymmetry of [·, ·]. Since [·, ·] is also alternating, and by Lemma 1.70, ∀i, ℓ : αi,i,ℓ ∈ R◦ . In addition, in the proof of Lemma 2.30, we showed that Jacobi’s identity is equivalent to ∀i, j, k, m : 3 XX αj,k,ℓ αi,ℓ,m ∈ R◦ i,j,k ℓ=1 In particular, substituting (i, j, k) = (1, 2, 3), ∀m : 3 X (α1,ℓ,m α2,3,ℓ + α3,ℓ,m α1,2,ℓ + α2,ℓ,m α3,1,ℓ ) ∈ R◦ ℓ=1 and thus we are finished.  We use the above way to construct a Lie semialgebra which has the same relations as sl (2, F ). In section 2.4 we will see the naive way of constructing sl (n, R), which will be different from the following Lie semialgebra: Example 2.34. We take a free R-module L with base {e, f, h}, and define [e, e] = [f, f ] = [h, h] = 0L [e, f ] = (−) [f, e] = [h, e] = (−) [e, h] = h 2e [h, f ] = (−) [f, h] = 2f where 2e and 2f mean e + e and f + f . In the notations of Lemma 2.30, if x1 = e, x2 = f and x3 = h, then α1,2,3 = (−) α2,1,3 = 1R , α3,1,1 = (−) α1,3,1 = 2, α3,2,2 = (−) α2,3,2 = (−) 2 and all other coefficients are 0R . We want to prove that these coefficients satisfy the conditions of Corollary 2.32, and thus L equipped with the bilinear extension of [·, ·] is a Lie semialgebra over R. We need to ensure that 3 X ∀m : (α1,ℓ,m α2,3,ℓ + α3,ℓ,m α1,2,ℓ + α2,ℓ,m α3,1,ℓ ) ∈ R◦ ℓ=1 (1) If m = 1, αi,ℓ,1 = 0R unless (i, ℓ) ∈ {(1, 3) , (3, 1)}. We are left with two summands 6= 0R : α1,3,1 α2,3,3 + α3,1,1 α1,2,1 = 0R (2) If m = 2, αi,ℓ,2 = 0R unless (i, ℓ) ∈ {(2, 3) , (3, 2)}. We are left with two summands 6= 0R : α2,3,2 α3,1,3 + α3,2,2 α1,2,2 = 0R (3) If m = 3, αi,ℓ,3 = 0R unless (i, ℓ) ∈ {(1, 2) , (2, 1)}. We are left with two summands 6= 0R : α1,2,3 α2,3,2 + α2,1,3 α3,1,1 = 1 · 2 (−) 1 · 2 = 2◦ 28 GUY BLACHAR Since the sum is quasi-zero for each choice of m, such L exists. 2.4. Symmetrized Versions of the Classical Lie Algebras. In this subsection we construct negated versions of the classical Lie algebras: An , Bn , Cn and Dn . We assume again that our underlying semiring R is a semifield with a negation map. Definition 2.35. The general linear algebra, gl (n, R), is the Lie semialgebra of all matrices of size n × n, where the Lie bracket is the negated commutator. Definition 2.36. We define matrices ei,j ∈ gl (n, R) by ( 1R (i, j) = (k, ℓ) (ei,j )k,ℓ = 0R (i, j) 6= (k, ℓ) these matrices form a base of gl (n, R). Remark 2.37. In gl (n, R), we have the formula [ei,j , ek,ℓ ] = δj,k ei,ℓ + (−) δi,ℓ ek,j where δi,j = ( 1R 0R i=j i 6= j Definition 2.38. An , or the negated special linear algebra, is An = {A ∈ gl (n + 1, R) | s (tr (A)) = 0} Lemma 2.39. An is a subalgebra of gl (n + 1, R). Proof. Obviously, An is a submodule of gl (n + 1, R). Since tr (AB) = tr (BA), we have s (tr ([A, B])) = s (tr (AB + (−) BA)) = 0 so An is a subalgebra of gl (n + 1, R).  Lemma 2.40. An has the following s-base: B = {ei,j | i 6= j} ∪ {ei,i + (−) ej,j | 1 ≤ i < j ≤ n} ∪ {1◦R ei,i | 1 ≤ i ≤ n} Proof. Assume that A = (ai,j ) ∈ An . Therefore, s (tr (A)) = 0. There are two options: (1) If tr (A) is achieved from a dominant zero-layered element in the diagonal of A, without loss of generality a1,1 , then A= n X ai,j ei,j + n X ai,i (ei,i + (−) e1,1 ) i=1 i,j=1 i6=j (2) Otherwise, tr (A) is achieved from (at least) two elements, not of layer zero, such that they “cancel” each other. Assume without loss of generality that these elements are a1,1 , . . . , ak,k ; that means that τ (a1,1 ) = · · · = t (ak,k ) = τ (tr (A)), s (a1,1 ) , . . . , s (ak,k ) 6= 0 and for each ℓ > k, either τ (aℓ,ℓ ) < τ (tr (A)) or s (aℓ,ℓ ) = 0. Then A= n X ai,j ei,j + i,j=1 i6=j n X ai,i (ei,i + (−) e1,1 ) i=2 Therefore, B spans An . It is easy to see that no element of B is spanned by the others, and thus it is an s-base.  Lemma 2.41. Considering A1 , we see that its s-base consists of the elements e = e1,2 , f = e2,1 , h = e1,1 + (−) e2,2 they satisfy the relations [e, f ] = h; [e, h] = [−2]0e; [f, h] = [2]0f MODULES AND LIE SEMIALGEBRAS OVER SEMIRINGS WITH A NEGATION MAP 29 Proof. Considering the formula mentioned in Remark 2.37 we get: [e, f ] = [e1,2 , e2,1 ] = e1,1 + (−) e2,2 = h [e, h] = [e1,2 , e1,1 + (−) e2,2 ] = (−) e1,2 + (−) e1,2 = [−2]0e [f, h] = [e2,1 , e1,1 + (−) e2,2 ] = e2,1 + (−) (−) e2,1 = [2]0f  Let L be the Lie semialgebra constructed in Example 2.34. Note that A1 is similar to L; however, A1 has no d,s-base, whereas L is free. Unlike the definition of An , the Lie semialgebras Bn , Cn and Dn are all defined using involutions. In this part, we follow [23]. Definition 2.42. Let R be a semifield with a negation map, and let A1 , A2 be nonassociative semialgebras. An antihomomorphism is a function ϕ : A1 → A2 that satisfies: (1) ∀x, y ∈ A1 : ϕ (x + y) = ϕ (x) + ϕ (y). (2) ∀α ∈ R, ∀x ∈ A1 : ϕ (αx) = αϕ (x). (3) ∀x, y ∈ A1 : ϕ (xy) = ϕ (y) ϕ (x). Definition 2.43. Let R be a semifield with a negation map, and let A be a nonassociative semialgebra. An involution is an antihomomorphism ϕ : A → A for which ∀x ∈ A : ϕ (ϕ (x)) = x In other words, ϕ is its own inverse. We also denote involutions as ∗ : A → A, x 7→ x∗ . Example 2.44. Let R be a semifield with a negation map. Then the transpose on Mn×n (R) is an involution. Definition 2.45. Let R be a semifield with a negation map, and let A be a nonassociative semialgebra with an involution ∗. An element x ∈ A is called symmetric, if x∗ = x x is called skew-symmetric, if x∗ = (−) x Lemma 2.46. Let A be a nonassociative semialgebra over R. Denote by [·, ·] the negated commutator on A, and let ∗ be an involution on A. Then the set of all skew-symmetric elements in A, à = {x ∈ A | x∗ = (−) x} is a Lie subalgebra of A. Proof. ∗ is an involution, and in particular an antihomomorphism, so à is a submodule of A. Assume x, y ∈ A. Then ∗ ∗ [x, y] = (xy + (−) yx) = y ∗ x∗ + (−) x∗ y ∗ = [y ∗ , x∗ ] If x, y ∈ Ã, then [x, y]∗ = [y ∗ , x∗ ] = [(−) y, (−) x] = [y, x] = (−) [x, y] so [x, y] ∈ Ã, as needed.  Now we can define Bn , Cn and Dn by defining their involutions. Definition 2.47. Let R be a semifield with a negation map. We define an involution ∗ on gl (k, R) by the transpose (A∗ = At ). As stated in Lemma 2.46, we get a Lie subalgebra consisting all of the skew-symmetric elements. (1) When k = 2n + 1 is odd, this Lie semialgebra is called Bn , the negated odd-dimensional orthogonal algebra. (2) When k = 2n is even, it is called Dn , the negated even-dimensional orthogonal algebra. 30 GUY BLACHAR Remark 2.48. Bn has the following s-base: {ei,j + (−) ej,i | 1 ≤ i < j ≤ 2n + 1} ∪ {1◦R ei,i | 1 ≤ i ≤ 2n + 1} whereas Dn has the following s-base: {ei,j + (−) ej,i | 1 ≤ i < j ≤ 2n} ∪ {1◦R ei,i | 1 ≤ i ≤ 2n} Definition 2.49. Let R be a semifield with a negation map. We define an involution ∗ on gl (2n, R) by  ∗   A B Dt (−) B t = C D (−) C t At By taking the skew-symmetric elements, we get the Lie subalgebra Cn , the negated symplectic algebra. Lemma 2.50. Cn has the following base: {ei,j + (−) en+j,n+i | 1 ≤ i, j ≤ n} ∪ {ei,n+j + ej,n+i | 1 ≤ i, j ≤ n} ∪ {en+i,j + en+j,i | 1 ≤ i, j ≤ n}   A1 A2 Proof. Denote the above set by B , and assume ∈ Cn . Then A3 A4 ∗      At4 (−) At2 A1 A2 A1 A2 = = (−) A3 A4 A3 A4 (−) At3 At1 we get the following conditions: At4 = (−) A1 ; At2 = A2 ; At3 = A3 If (Ak )i,j = ak,i,j , then   n n n X X X A1 A2 a3,i,j (en+i,j + en+j,i ) a2,i,j (ei,n+j + ej,n+i )+ a1,i,j (ei,j + (−) en+j,n+i )+ = A3 A4 i,j=1 i,j=1 i,j=1 and this combination is unique, implying B is a base of Cn .  3. Solvable and Nilpotent Symmetrized Lie Algebras 3.1. Basic Definitions. 3.1.1. Solvability. Definition 3.1. Let R be a semifield with a negation map, and let L be a Lie semialgebra with a negation map over R. The derived series of L is the series defined by L(0) = L, L(1) = L′ = [L, L],  (n−1)  (n) (n−1) and in general: L = L ,L . Definition 3.2. Let R be a semifield with a negation map, and let L be a Lie semialgebra with a negation map over R. We say that L is solvable, if ∃n ∈ N : L(n) ⊆ L◦ Example 3.3. All abelian Lie semialgebras with a negation map are solvable. Lemma 3.4. Any Lie semialgebra with a negation map L, for which L′ ⊆ L◦ , is abelian. Proof. The condition is equivalent to ∀x, y ∈ L : [x, y] ∈ L◦ , which shows that L is abelian.  Lemma 3.5. Let R be a semifield with a negation map, and let L be a Lie semialgebra with a negation map over R. If L is solvable, then so are all of its subalgebras and homomorphic images. Proof. If K is a subalgebra of L, then it is easy to see by induction that ∀n ∈ N : K (n) ⊆ L(n) . Also, if ϕ : L → L1 is an R-epimorphism, then   ∀n ∈ N : (L1 )(n) = ϕ L(n) Therefore, if L is solvable, there is some n ∈ N such that L(n) = L◦ ; by the above formula, (n)  (L1 ) ⊆ L1 ◦ . MODULES AND LIE SEMIALGEBRAS OVER SEMIRINGS WITH A NEGATION MAP 31 Lemma 3.6. Let R be a semifield with a negation map, and let L be a Lie semialgebra with a negation map over R. L is solvable if and only if there exists a chain of subalgebras Lk ⊳ Lk−1 ⊳ · · · ⊳ L0 = L, such that Lk ⊆ L◦ and L′i ⊆ Li+1 . Proof. It is easy to check that in such a chain, L(i) ⊆ Li ; so, if such a chain exists, L(k) ⊆ L◦ , implying L is solvable. If L is solvable with L(k) ⊆ L◦ , than the chain L(k) ⊳ · · · ⊳ L(0) = L works.  Lemma 3.7. If I, J ⊳ L are two ideals of L, and if I is solvable, then I ∩ J is also solvable. Proof. I ∩ J ⊳ I, so this is a direct corollary of Lemma 3.5.  Lemma 3.8. If I, J ⊳ L are two solvable ideals of L, then I + J is also solvable. Proof. We shall prove by induction that (k) ∀k ∈ N ∪ {0} : (I + J) ⊆ I (k) + J (k) + I ∩ J If k = 0 the assertion is clear. Assume it is true for k, i.e. (k) (I + J) ⊆ I (k) + J (k) + I ∩ J We get i h i h (k) (k) ⊆ I (k) + J (k) + I ∩ J, I (k) + J (k) + I ∩ J = (I + J) , (I + J) h i h i h i h i = I (k) , I (k) + I (k) , J (k) + I ∩ J + J (k) , J (k) + J (k) , I (k) + I ∩ J + h i + I ∩ J, I (k) + J (k) + I ∩ J ⊆ I (k+1) + J (k+1) + I ∩ J       Note that I (k) , J (k) + I ∩ J , J (k) , I (k) + I ∩ J , I ∩ J, I (k) + J (k) + I ∩ J ⊆ I ∩ J. (I + J) (k+1) = I and J are solvable, so ∃m, n ∈ N : I (m) , J (n) ⊆ L◦ . Taking k = max {m, n}, we get (k) (I + J) (k) We also know that (I ∩ J) ⊆ L◦ + I ∩ J ⊆ L◦ , and thus, (I + J) (2k) ⊆ (L◦ ) (k) + (I ∩ J) (k) + L◦ ∩ I ∩ J ⊆ L◦ implying I + J is solvable.  3.1.2. Nilpotency. Definition 3.9. Let R be a semifield with a negation map, and let L be a Lie semialgebra with a negation map over R. The descending central series, or the lower  central series, is the series defined by L0 = L, L′ = L1 = [L, L], and in general: Ln = L, Ln−1 . Definition 3.10. Let R be a semifield with a negation map, and let L be a Lie semialgebra with a negation map over R. L is called nilpotent, if ∃n ∈ N : Ln ⊆ L◦ Example 3.11. All abelian Lie semialgebras with a negation map are nilpotent. Lemma 3.12. ∀n ∈ N ∪ {0} : L(n) ⊆ Ln . Proof. By induction: for n = 0, it is clear, because L(0) = L = L0 . Assume the statement is true for n ∈ N ∪ {0}, and let x ∈ L(n+1) . Then there exist y1 , y2 ∈ L(n) such that x = [y1 , y2 ] So, y1 ∈ L, and by the induction hypothesis, y2 ∈ Ln . Thus, x = [y1 , y2 ] ∈ [L, Ln ] = Ln+1 and we get L(n) ⊆ Ln .  32 GUY BLACHAR Corollary 3.13. Any nilpotent Lie semialgebra with a negation map is solvable. Lemma 3.14. Let R be a semifield with a negation map, and let L be a nilpotent Lie semialgebra with a negation map over R. (1) All of the subalgebras and homomorphic images of L are nilpotent. (2) If L 6= L◦ , then L◦ 6= Z (L) (in fact, L◦ ( Z (L)). Proof. (1) If K is a subalgebra of L, then it is easy to see by induction that ∀n ∈ N : K n ⊆ Ln . Also, n if ϕ : L → L′ is an R-epimorphism, then ∀n ∈ N : (L′ ) = ϕ (Ln ). (2) Assume n is the maximal index for which Ln * L◦ . Then [L, Ln ] ⊆ L◦ , so Ln ⊆ Z (L). ∃x0 ∈ Ln \L◦ , so x0 ∈ Z (L) \L◦ , as required.  Lemma 3.15. If L′ is nilpotent, then L is solvable. Proof. We shall prove that n ∀n ∈ N ∪ {0} : L(n+1) ⊆ (L′ ) We use induction on n, the case n = 0 being trivial. We get h i h i h i n−1 n−1 n−1 n L(n+1) = L(n) , L(n) ⊆ (L′ ) , (L′ ) ⊆ L′ , (L′ ) = (L′ ) Thus, if L′ is nilpotent, for some n we have (L′ )n ⊆ (L′ )◦ ⊆ L◦ , implying L(n+1) ⊆ L◦ , so L is solvable.  Similarly to Lemma 3.7 and Lemma 3.8, we have the following lemmas: Lemma 3.16. If I, J ⊳ L are two ideals of L, and if I is nilpotent, then I ∩ J is also nilpotent. Lemma 3.17. If I, J ⊳ L are two nilpotent ideals of L, then I + J is also nilpotent. 3.1.3. Solvability and Nilpotency Modulo Ideals. In the semiring, the quotient algebra structure fails, and the replacement is congruences. However, we would define equivalent definitions to solvability and nilpotency of quotient algebras. Definition 3.18. Let R be a semifield with a negation map, let L be a Lie semialgebra with a negation map over R, and let I ⊳ L. We say that L is solvable modulo I, if ∃n ∈ N : L(n) ⊆ I Remark 3.19. Solvability modulo L◦ is equivalent to solvability. Lemma 3.20. If L is solvable modulo I, and if I is solvable (as a Lie semialgebra with a negation map), then L is also solvable. Proof. L is solvable modulo I, so there exists n ∈ N for which L(n) ⊆ I. But I is also solvable, so there exists m ∈ N for which I (m) ⊆ I ◦ ⊆ L◦ . To conclude, (m)  ⊆ I (m) ⊆ L◦ L(m+n) = L(n) implying L is solvable.  Definition 3.21. Let R be a semifield with a negation map, let L be a Lie semialgebra with a negation map over R, and let I ⊳ L. We say that L is nilpotent modulo I, if ∃n ∈ N : Ln ⊆ I Remark 3.22. Nilpotency modulo L◦ is equivalent to nilpotency. Definition 3.23. Let R be a semifield with a negation map, let L be a Lie semialgebra with a negation map over R, and let I ⊳ L. We say that L is abelian modulo I, if [L, L] ⊆ I. Remark 3.24. Lemma 3.6 can be rephrased as follows: L is solvable if and only if there exists a chain of subalgebras Lk ⊳ Lk−1 ⊳ · · · ⊳ L0 = L such that each Li+1 is abelian modulo Li and Lk is abelian. MODULES AND LIE SEMIALGEBRAS OVER SEMIRINGS WITH A NEGATION MAP 33 3.1.4. The Symmetrized Radical and Semisimplicity. Definition 3.25. Let R be a semifield with a negation map, let L be a Lie semialgebra with a negation map over R. The (negated) radical of L is X RadL = I I⊳L is solvable That is, the negated radical is the sum of all solvable ideals of L. Remark 3.26. In the classical theory, the radical of a finitely generated Lie algebra is its unique maximal solvable ideal. However, in our theory, there is no guarantee that there would be one maximal solvable ideal, since being finitely generated as a module with a negation map usually does not mean being Noetherian (and thus, there may be infinite ascending chains of solvable ideals). Definition 3.27. Let R be a semifield with a negation map, let L be a Lie semialgebra over R. We say that L is locally solvable, if any finitely generated subalgebra of L is solvable. Lemma 3.28. RadL is locally solvable. Proof. Let L1 ⊆ RadL be a finitely generated subalgebra of L. Write L1 = Span {x1 , . . . , xn } By the definition of RadL, for each i there are solvable ideals Ii,1 , . . . , Ii,mi such that xi ∈ mi X Ii,j j=1 Therefore, L1 ⊆ mi n X X Ii,j i=1 j=1 Since the sum is a finite sum of solvable ideals, by Lemma 3.8, it is also solvable. Therefore, by Lemma 3.5, also L1 is solvable, as required.  Definition 3.29. Let R be a semifield with a negation map, let L be a Lie semialgebra over R. If RadL = L◦ , then L is called semisimple. Lemma 3.30. L is semisimple if and only if any abelian ideal of L is contained in L◦ . Proof. Let L be semisimple, and let I be an abelian ideal of L. Then I is solvable, implying I ⊆ RadL = L◦ . Now suppose that any abelian ideal of L is contained in L◦ . If L is not semisimple, it has a solvable ideal I * L◦ . Take a maximal n such that I (n) * L◦ ; then I (n) is an abelian ideal, a contradiction.  3.2. Lifts of Lie Semialgebras with a Negation Map. As a continuation of subsection 1.4, we give a definition of a lift of a Lie semialgebra with a negation map.   b ϕ Definition 3.31. Let R be a semiring with a negation map with a lift R, b , and let L be a Lie b L, b and a map ψb : L b → L, such that the semialgebra over R. A lift of L is a Lie algebra over R, following properties hold:   b ψb is a lift of L as modules. (1) L, h i b : ψb ([x1 , x2 ])  ψb (x1 ) , ψb (x2 ) . (2) ∀x1 , x2 ∈ L We also have two parallel lemmas to Lemma 1.36:   b ⊆L b be a subalgebra. Then ψb K b is a subalgebra of L. Lemma 3.32. Let K 34 GUY BLACHAR   b is a submodule of L. Therefore, we need to check that it is closed Proof. By Lemma 1.36, ψb K with respect to the negated Lie bracket.   b . Take x b such that x  ψb (b Let x, y ∈ ψb K b, yb ∈ K x) and y  ψb (b y). Then h i [x, y]  ψb (b x) , ψb (b y)  ψb ([b x, yb]) b we are done. Since [b x, yb] ∈ K,    b be an ideal. Then ψb Ib is an ideal of L. Lemma 3.33. Let Ib ⊳ L   Proof. By Lemma 3.32, ψb Ib is a subalgebra of L.   b and y ∈ L. Take x b such that y  ψb (b Let x ∈ ψb K b ∈ Ib such that x  ψb (b x), and take yb ∈ L y). Then h i [x, y]  ψb (b x) , ψb (b y)  ψb ([b x, yb]) b we are done. Since [b x, yb] ∈ I,  Although in general there is no obvious way to find a lift of a given Lie semialgebra (even if there is a lift as modules), there is some consolation:   b ϕ Theorem 3.34. Let R be a semiring with a negation map with a lift R, b . Consider the function   b → gl (n, R) ψb : gl n, R     b , ψb is a lift of gl (n, R) as Lie which applies ϕ b on each entry of a given matrix. Then gl n, R semialgebras.   bB b ∈ gl n, R b , Proof. Since this is already a lift of modules, we will prove that given A,       b ψb B b  ψb A bB b ψb A Indeed,      b ψb B b ψb A i,j = n X k=1 n   X b ϕ b (b ai,k ) ϕ b bbk,j  ϕ b ai,kbbk,j k=1 !    bB b = ψb A i,j Now the assertion follows, since h    i             b , ψb B b b ψb B b + (−) ψb B b ψb A b  ψb A bB b + (−) ψb B bA b  ψb A = ψb A i  h  b B b bB b + (−) B bA b = ψb A,  ψb A    b ϕ 3.2.1. Solvability, Nilpotency and Semisimplicity of the Lift. Throughout this subsection, R, b   b ψb will be a lift of a semifield with a negation map R, L will be a Lie semialgebra over R, and L, will be a lift of L. b1, L b 2 be subalgebras of L. b Then Lemma 3.35. Let L      i h b 1 , ψb L b 2 ⊆ ψb L b 1, L b2 ψb L MODULES AND LIE SEMIALGEBRAS OVER SEMIRINGS WITH A NEGATION MAP 35 i     h b 1 and b 1 and y ∈ ψb L b 2 . We will prove that [x, y] ∈ ψb L b 1, L b 2 . Take x Proof. Let x ∈ ψb L b∈L b 2 such that x  ψb (b yb ∈ L x) and y  ψb (b y). Then h i [x, y]  ψb (b x) , ψb (b y)  ψb ([b x, yb]) i h b1, L b 2 , we are finished.  Since [b x, yb] ∈ L Corollary 3.36. and h i bn ∀n ∈ N ∪ {0} : Ln ⊆ ψb L i h b (n) ∀n ∈ N ∪ {0} : L(n) ⊆ ψb L Proof. We will prove the first assertion; the second is proven similarly. We use induction on n, the case n = 0 being trivial. If the assertion holds for n ∈ N ∪ {0}, then, by Lemma 3.35 and the induction hypothesis,      i h b , ψb L b n ⊆ ψb L, b L b n = ψb (Ln+1 ) Ln+1 = [L, Ln ] ⊆ ψb L as required.  b is nilpotent (resp. solvable), then L is nilpotent (resp. solvable). Corollary 3.37. If L Proof. We will prove the assertion for nilpotency; the assertion for solvability is proved similarly. b is nilpotent, there is n such that L b n = 0. By Corollary 3.36, Since L   b n = ψb (0) = L◦ Ln ⊆ ψb L and thus L is nilpotent. Corollary 3.38.     b ⊆ Rad (L) ψb Rad L      b ⊳ L, b and thus, by Lemma 3.33, ψb Rad L b Proof. Rad L ⊳ L. In addition, by Corollary 3.37,    b ψb Rad L is solvable. Therefore, the assertion follows.  b is semisimple. Corollary 3.39. If L is semisimple, then L Proof. L is semisimple, hence Rad (L) ⊆ L◦ . By Corollary 3.38,    b ψb Rad L ⊆ Rad (L) ⊆ L◦   b = 0, as required. Hence Rad L  3.3. Cartan’s Criterion for Lie Algebras Over ELT Algebras. For this subsection, we work only with ELT algebras. Specifically, we are interested in the following type of ELT algebra: Definition 3.40. Let R = R (F , L ) be an ELT algebra. We say that R is a divisible ELT field, if L is a field, and F is a divisible group. In our version of Cartan’s criterion, we will use the essential trace defined in [6]. Let us recall its definition: 36 GUY BLACHAR
Definition 3.41. Let R be a divisible ELT field, and let A ∈ Mn R . Let n X
αi λn−i pA (λ) = det λI + [−1]0A = λn + i=1 be its ELT characteristic polynomial. We define   τ (αk ) τ (αℓ ) ≥ L (A) = ℓ ≥ 1 | ∀k ≥ n : ℓ k µ (A) = min L (A) In words, αµ λn−µ is the first monomial after λn which is not inessential in pA (λ). The essential trace of A is ( etr (A) = Recall: [0] tr (A) ,  τ (αµ ) µ [−1]0tr (A) , is essential in pA (λ) Otherwise Corollary ([6, Corollary 2.22]). If A is ELT nilpotent, then s (etr (A)) = 0. Returning to Lie semialgebras, we specialize our symmetrized Lie semialgebra to be free; therefore, given a homomorphism ϕ : L → L, its trace and essential trace are well-defined. Definition 3.42. Let R be a divisible ELT field, and let L be a free symmetrized Lie semialgebra over R. The Killing form of L is κ (x, y) = tr (adx ◦ ady ) The essential Killing form of L is κes (x, y) = etr (adx ◦ ady ) Definition 3.43. Let R be a divisible ELT field, let L be a free symmetrized Lie semialgebra over R, and let κ be the Killing form of L. We say that κ is non-degenerate, if Radκ = {x ∈ L|∀y ∈ L : s (κ (x, y)) = 0} ⊆ L◦ We say that κes is non-degenerate, if SRadκ = {x ∈ L|∀y ∈ L : s (κes (x, y)) = 0} ⊆ L◦ Remark 3.44. Since Radκ ⊆ SRadκ (by [6, Lemma 2.11]), if κes is non-degenerate, then κ is nondegenerate. Theorem 3.45 (Cartan’s criterion). If L has a non-degenerate essential Killing form, then L is semisimple. Proof. By Lemma 3.30, it is enough to prove that any abelian ideal I of L is of layer zero. Let I be an abelian ideal, and let x ∈ I. Take an arbitrary y ∈ L. Since adx ◦ ady maps L to I, (adx ◦ ady )2 maps L to [I, I], which is of layer zero. So adx ◦ ady is ELT nilpotent. Thus, by [6, Corollary 2.22], s (etr (adx ◦ ady )) = 0. This argument implies x ∈ SRadκ; so I ⊆ SRadκ. But κ is strongly non-degenerate, and thus I ⊆ L◦ , as required. So L is semisimple.  MODULES AND LIE SEMIALGEBRAS OVER SEMIRINGS WITH A NEGATION MAP 37 4. PBW Theorem In this subsection we will construct the universal enveloping algebra of an arbitrary Lie semialgebra with a negation map. Then, we will give a counterexample to a naive version of PBW theorem. The tensor product is a well-known process in general categories (for example, [4], [11], [19] and [26]). Regarding tensor product of modules over semirings, there are two (nonisomorphic) notions of tensor product of modules over semirings in the literature, both denoted by ⊗R . For our purposes, we use the tensor product discussed in [20], [21] and [4]. The other notion of tensor product is discussed in [26] and [10]. Their construction satisfies a different universal property, and the outcome is a cancellative monoid. 4.1. Tensor Power and Tensor Algebra of Modules over a Semirings. Definition 4.1. Let R be a commutative semiring, and let M be an R-module. The k-th tensor product of M is defined as M ⊗k = M ⊗ · · · ⊗ M {z } | k times If k = 0, we define M ⊗0 = R. We note that this definition is well-defined, because the tensor product is associative. Definition 4.2. Let R be a commutative semiring, and let M be an R-module. The tensor algebra of M , denoted T (M ), is ∞ M T (M ) = M ⊗k k=0 Remark 4.3. The natural isomorphism M ⊗k ⊗ M ⊗ℓ ∼ = M ⊗(k+ℓ) yields a multiplication M ⊗k × M ⊗ℓ → M ⊗(k+ℓ) given by (a, b) 7→ a ⊗ b. Endowed with this multiplication, T (M ) is an R-semialgebra. Lemma 4.4. Let A be an R-semialgebra, and assume that ϕ : M → A is a homomorphism of R-modules. Then there exists a unique homomorphism of R-semialgebras ϕ̂ : T (M ) → A, such that ϕ̂|M = ϕ. Proof. Uniqueness forces that ϕ̂ (x1 ⊗ · · · ⊗ xk ) = ϕ̂ (x1 ) ⊗ · · · ⊗ ϕ̂ (xk ) = ϕ (x1 ) ⊗ · · · ⊗ ϕ (xk ) This defines a homomorphism of R-semialgebras.  4.2. The Universal Enveloping Algebra of a Lie Algebra with a Negation Map. We now return to our motivation, where we work with a Lie semialgebra L over a semifield with a negation map R. Definition 4.5. Let R be a semifield with a negation map, and let L be a Lie semialgebra over R. We define a congruence ∼ on T (L) as the congruence generated by ∀x, y ∈ L : [x, y] ∼ x ⊗ y + (−) y ⊗ x The universal enveloping algebra of L is U (L) = T (L) /∼ If ρ : T (L) → U (L) is the canonical map, we have the PBW homomorphism of L, which we denote ϕL : L → U (L), defined by ϕL = ρ|L . However, the PBW homomorphism of L need not be injective, as the next subsection demonstrates. 38 GUY BLACHAR 4.3. Counterexample to the Naive PBW Theorem. We now present a Lie semialgebra with a negation map, which cannot be embedded into a Lie semialgebra with a negation map obtained from an associative algebra together with the negated commutator operator. In particular, ϕL will not be injective. Example 4.6. The example will use the ELT structre. For convenience, we assume that our underlying ELT field is R = R (C, R). Recall that in the ELT theory, the relation  is denoted . We take a free R-module L with base B = {x1 , x2 }, and define: [x1 , x1 ] = [0]1x 1 + [0]1x2 [x2 , x2 ] = [0]1x 1 + [0]1x2 [x1 , x2 ] = [x2 , x1 ] = x1 + x2 (−) [x1 , x2 ] By Lemma 2.28, L equipped with [·, ·] is a Lie semialgebra with a negation map over R. Consider y1 = [x1 , [x1 , x2 ]] + [x1 , [x2 , x1 ]] and y2 = [x2 , [x1 , x1 ]] . From Jacobi’s identity, we know that y1 + y2 ∈ L◦ . Let us expand them, using the definition of the Lie bracket:     y1 = [x1 , [x1 , x2 ]] + [x1 , [x2 , x1 ]] = x1 , [0]0 [x1 , x2 ] = x1 , [0]0x1 + [0]0x2 = = [0]0 [x1 , x1 ] + [0]0 [x1 , x2 ] = [0]1x1 + [0]1x2 + [0]0x1 + [0]0x2 = [0]1x1 + [0]1x2 whereas   y2 = [x2 , [x1 , x1 ]] = x2 , [0]1x1 + [0]1x2 = [0]1 [x2 , x1 ] + [0]1 [x2 , x2 ] =

= [0]1 [−1]0x1 + [−1]0x2 + [0]1 [0]1x1 + [0]1x2 = [0]2x1 + [0]2x2 Therefore, y2  y1 , but y1 6= y2 . Now, assume that there is an ELT associative algebra A, which is a Lie semialgebra with a negation map together with the ELT commutator, such that there exists an R-monomorphism ϕ : L → A of Lie semialgebra with a negation maps. Denote a1 = ϕ (x1 ), a2 = ϕ (x2 ). We will show that ϕ (y1 ) = ϕ (y2 ), which will show that ϕ is not injective. We note that b1 = ϕ (y1 ) = [a1 , [a1 , a2 ]] + [a1 , [a2 , a1 ]] and b2 = ϕ (y2 ) = [a2 , [a1 , a1 ]] Since any associative algebra with the symmetrized commutator satisfies the strong Jacobi’s identity (Example 2.12), b1  b2 . However, since y2  y1 , also b1 = ϕ (y2 )  ϕ (y1 ) = b2 . By Corollary 1.6, since R◦ is idempotent,  is a partial order on L. In particular, ϕ (y1 ) = b1 = b2 = ϕ (y2 ). But ϕ is a monomorphism, implying y1 = y2 , a contradiction. References [1] Marianne Akian, Guy Cohen, Stephane Gaubert, R Nikoukhah, and Jean Pierre Quadrat. Linear systems in (max,+) algebra. In Decision and Control, 1990., Proceedings of the 29th IEEE Conference on, pages 151–156. IEEE, 1990. [2] Marianne Akian, Stéphane Gaubert, and Alexander Guterman. Linear independence over tropical semirings and beyond. Contemporary Mathematics, 495:1, 2009. [3] Marianne Akian, Stéphane Gaubert, and Alexander Guterman. Tropical cramer determinants revisited. Tropical and Idempotent Mathematics and Applications, 616:45, 2014. [4] Markus Banagl. The tensor product of function semimodules. Algebra universalis, 70(3):213–226, 2013. MODULES AND LIE SEMIALGEBRAS OVER SEMIRINGS WITH A NEGATION MAP 39 [5] Guy Blachar and Erez Sheiner. ELT linear algebra. to be published, 2016. [6] Guy Blachar and Erez Sheiner. ELT linear algebra II. to be published, 2016. [7] David Dolžan and Polona Oblak. Invertible and nilpotent matrices over antirings. arXiv preprint arXiv:0806.2996, 2008. [8] Stéphane Gaubert. Théorie des systèmes linéaires dans les dioı̈des. PhD thesis, 1992. [9] Stephane Gaubert. Methods and applications of (max,+) linear algebra. In STACS 97, pages 261–282. Springer, 1997. [10] Jonathan S Golan. Semirings and Their Applications. Springer Science & Business Media, 1999. [11] Alexandre Grothendieck. Produits tensoriels topologiques et espaces nucléaires. Séminaire Bourbaki, 2:193–200, 1955. [12] James E Humphreys. Introduction to Lie Algebras and Representation Theory, volume 9. Springer Science & Business Media, 1972. [13] Zur Izhakian. Tropical arithmetic and tropical matrix algebra. arXiv preprint math/0505458, 2005. [14] Zur Izhakian, Manfred Knebusch, and Louis Rowen. Supertropical linear algebra. Pacific Journal of Mathematics, 266(1):43–75, 2013. [15] Zur Izhakian, Adi Niv, and Louis Rowen. Supertropical SLn . arXiv preprint arXiv:1508.04483, 2015. [16] Zur Izhakian and Louis Rowen. Supertropical matrix algebra. Israel Journal of Mathematics, 182(1):383–424, 2011. [17] Zur Izhakian and Louis Rowen. Supertropical matrix algebra II: Solving tropical equations. Israel Journal of Mathematics, 186(1):69–96, 2011. [18] Zur Izhakian and Louis Rowen. Supertropical matrix algebra III: Powers of matrices and their supertropical eigenvalues. Journal of Algebra, 341(1):125–149, 2011. [19] EB Katsov. Tensor product of functors. Siberian Mathematical Journal, 19(2):222–229, 1978. [20] Yefim Katsov. Tensor products and injective envelopes of semimodules over additively regular semirings. In Algebra Colloquium, volume 4, pages 121–131. SPRINGER VERLAG 175 FIFTH AVE, NEW YORK, NY 10010, 1997. [21] Yefim Katsov. Toward homological characterization of semirings: Serres conjecture and basss perfectness in a semiring context. algebra universalis, 52(2-3):197–214, 2005. [22] Brett Parker. Exploded manifolds. Advances in Mathematics, 229(6):3256–3319, 2012. [23] Louis Halle Rowen. Graduate Algebra: Noncommutative View, volume 91. American Mathematical Society, 2008. [24] Louis Halle Rowen. Symmetries in tropical algebra. arXiv preprint arXiv:1602.00353, 2016. [25] Erez Sheiner. Exploded Layered Tropical Algebra. PhD thesis, Bar-Ilan University, 2015. [26] Michihiro Takahashi. On the bordism categories III. In Mathematics seminar notes, volume 10, pages 211–236. , 1982. Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel. E-mail address: [email protected] 

Ultimate Intelligence Part III: Measures of Intelligence, Perception and Intelligent Agents Eray Özkural arXiv:1709.03879v1 [] 8 Sep 2017 Gök Us Sibernetik Ar&Ge Ltd. Şti. Abstract. We propose that operator induction serves as an adequate model of perception. We explain how to reduce universal agent models to operator induction. We propose a universal measure of operator induction fitness, and show how it can be used in a reinforcement learning model and a homeostasis (self-preserving) agent based on the free energy principle. We show that the action of the homeostasis agent can be explained by the operator induction model. “Wir müssen wissen – wir werden wissen!” — David Hilbert 1 Introduction The ultimate intelligence research program is inspired by Seth Lloyd’s work on the ultimate physical limits to computation [15]. We investigate the ultimate physical limits and conditions of intelligence. This is the third installation of the paper series, the first two parts proposed new physical complexity measures, priors and limits of inductive inference [18,17]. We frame the question of ultimate limits of intelligence in a general physical setting, for this we provide a general definition of an intelligent system and a physical performance criterion, which as anticipated turns out to be a relation of physical quantities and information, the latter of which we had conceptually reduced to physics with minimum machine volume complexity in [18]. 2 Notation and Background 2.1 Universal Induction Let us recall Solomonoff’s universal distribution [21]. Let U be a universal computer which runs programs with a prefix-free encoding like LISP; y = U (x) denotes that the output of program x on U is y where x and y are bit strings. 1 Any unspecified variable or function is assumed to be represented as a bit string. 1 A prefix-free code is a set of codes in which no code is a prefix of another. A computer file uses a prefix-free code, ending with an EOF symbol, thus, most reasonable programming languages are prefix-free. 2 Eray Özkural |x| denotes the length of a bit-string x. f (·) refers to function f rather than its application. The algorithmic probability that a bit string x ∈ {0, 1}+ is generated by a random program π ∈ {0, 1}+ of U is: X 2−|π| (1) PU (x) = U(π)∈x(0|1)∗ ∧π∈{0,1}+ which conforms to Kolmogorov’s axioms [13]. PU (x) considers any continuation of x, taking into account non-terminating programs.2 PU is also called the universal prior for it may be used as the prior in Bayesian inference, for any data can be encoded as a bit string. We also give the basic definitions of Algorithmic Information Theory (AIT) [14], where the algorithmic entropy, or complexity of a bit string x ∈ {0, 1}+ is HU∗ (x) = − log2 PU (x) HU (x) = min({|π| | U (π) = x}) (2) We use some variables in overloaded fashion in the paper, e.g., π might be a program, a policy, or a physical mechanism depending on the context. 2.2 Operator induction Operator induction is a general form of supervised machine learning where we learn a stochastic map from n question and answer pairs D = {(qi , ai )} sampled from a (computable) stochastic source µ. Operator induction can be solved by finding in available time a set of operators Oj (·|·), each a conditional probability density function (cpdf), such that the following goodness of fit is maximized X ψnj (3) Ψ= j for a stochastic source µ where each term in the summation is the contribution of a model: n Y j Oj (ai |qi ). (4) ψnj = 2−|(O (·|·)| i=1 qi and ai are question/answer pairs in the input dataset drawn from µ, and Oj is a computable cpdf in Equation 4. We can use the found m operators to predict unseen data with a mixture model [24] PU (an+1 |qn+1 ) = m X ψnj Oj (an+1 |qn+1 ) (5) j=1 The goodness of fit in this case strikes a balance between high a priori probability and reproduction of data like in minimum message length (MML) method [27,26], yet uses a universal mixture like in sequence induction. The convergence theorem for operator induction was proven in [23] using Hutter’s extension to arbitrary alphabet, and it bounds total error by HU (µ) ln 2 similarly to sequence induction. 2 We used the regular expression notation in language theory. Measuring Intelligence 2.3 3 Set induction Set induction generalizes unsupervised machine learning where we learn a probability density function (pdf) from a set of n bitstrings D = {d1 , d2 , ..., dn } sampled from a stochastic source µ. We can then inductively infer new members to be added to the set with: P (dn+1 ) = PU (D ∪ dn+1 ) PU (D) (6) Set induction is clearly a restricted case of operator induction where we set Qi ’s to null string. Set induction is a universal form of clustering, and it perfectly models perception. If we apply set induction over a large set of 2D pictures of a room, it will give us a 3D representation of it necessarily. If we apply it to physical sensor data, it will infer the physical theory – perfectly general, with infinite domains – that explains the data, perception is merely a specific case of scientific theory inference in this case, though set induction works both with deterministic and non-deterministic problems. 2.4 Universal measures of intelligence There is much literature on the subject of defining a measure of intelligence. Hutter has defined an intelligence order relation in the context of his universal reinforcement learning (RL) model AIXI [8], which suggests that intelligence corresponds to the set of problems an agent can solve. Also notable is the universal intelligence measure [10,11], which is again based on the AIXI model. Their universal intelligence measure is based on the following philosophical definition compiled from their review of definitions of intelligence in the AI literature. Definition 1 (Legg & Hutter). Intelligence measures an agent’s ability to achieve goals in a wide range of environments. It implies that intelligence requires an autonomous goal-following agent. The intelligence measure of [10] is defined as X Υ (π) = 2−HU (µ) Vµπ (7) µ∈E where µ is a computable reward bounded environment, And Vµπ is the expected sum P of future rewards in the total interaction sequence of agent π. ∞ Vµπ = Eµ,π [ t=1 γ t rt ], where rt is the instantaneous reward at time t generated from the interaction between the agent π and the environment µ, and γ t is the time discount factor. 2.5 The free energy principle. In Asimov’s story titled “The Last Question”, the task of life is identified as overcoming the second law of thermodynamics, however futile. Variational free energy essentially measures predictive error, and it was introduced by Feynmann to 4 Eray Özkural address difficult path integral problems in quantum physics. In thermodynamic free energy, energies are negative log probabilities like entropy. The free energy principle states that any system must minimize its free energy to maintain its order. An adaptive system that tends to minimize average surprise (entropy) will tend to survive longer. A biological organism can be modelled as an adaptive system that has an implicit probabilistic model of the environment, and the variational free energy puts an upper bound on the surprise, thus minimizing free energy will improve the chances of survival. The divergence between the pdf of environment and an arbitrary pdf encoded by its own mechanism is minimized in Friston’s model [9]. It has been shown in detail that the free energy principle adequately models a self-preserving agent in a stochastic dynamical system [6,9], which we can interpret as an environment with computable pdf. An active agent may be defined in the formalism of stochastic dynamical systems, by partitioning the physical states X of the environment into X = E × S × A × Λ where e ∈ E is an external state, s ∈ S is a sensory state, a ∈ A an active state, and λ ∈ Λ is an internal state. Self-preservation is defined by the Markov blanket S × A, the removal of which partitions X into external states E and internal states Λ that influence each other only through sensory and action states. E influences sensations S, which in turn influence internal states Λ, resulting in the choice of action signals S, which impact E, forming the feedback loop of the adaptive system. The system states x ∈ X evolve according to the stochastic equation:   fe (e, s, a)  f (e, s, a)    s (10) f (x) =   ẋ(t) = f (x) + ω (8) fa (s, a, λ) x(0) = x0 (9) fλ (s, a, λ) where f (x) is the flow of system states and it is decomposed into flows over the sets in the system partition, explicitly showing the dependencies among state sets; ω models fluctuations. Friston formalizes the self-preservation (homeostasis) problem as finding an internal dynamics that minimizes the uncertainty (Shannon entropy) of the external states, and shows a solution based on the principle of least action [9] wherein minimizing free energy is synonymous with minimizing the entropy of the external states (principle of least action), which subsequently corresponds to active inference. We have space for only some key results from the rather involved mathematical theory. p(s, f |m) is the generative pdf that generates sensorium s and fictive (hidden) states f ∈ F from probabilistic model m, and q(f |λ) is the recognition pdf that predicts hidden states F in the world given internal state. Generative pdf factorizes as p(s, f |m) = p(s|f, m)p(f |m). Free energy is defined as energy minus entropy F (s, λ) = Eq [− ln p(s, f |m)] − H(q(f |λ)) (11) which can be subjectively computed by the system. Free energy is also equal to surprise plus divergence between recognition and generative pdf’s. F (s, λ) = Eq [− ln p(s, f |m)] + DKL (q(f |λ)||p(f |s, m)) (12) Measuring Intelligence 5 Minimizing divergence minimizes free energy, internal states λ may be optimized to minimize predictive error using Equation 12, and surprise is invariant with respect to λ. Free energy may be formulated as complexity plus accuracy of recognition, as well. F (s, λ) = Eq [− ln p(s, a|f, m)] + DKL (q(f |λ)||p(f, m)) (13) In this case, we may choose an action that changes sensations to reduce predictive error. Only the first term is a function of action signals. Minimization of free energy turns out to be equivalent to the information bottleneck principle of Tishby [9,25]. The information bottleneck method is equivalent to the pioneering work of Ashby, which is simple enough to state here [3,2]: SB = I(λ; F ) − I(S; λ) (14) where the first term is the mutual information between internal and hidden states, and the second term is the mutual information between sensory states and internal states. Both terms are expanded using conditional entropy, and then two terms in the middle are eliminated because they are not relevant to the optimization problem – we do not know the hidden variables in H(λ|F ) and H(S) is constant. SB = H(λ) − H(λ|F ) − H(S) + H(S|λ) ∗ SB = H(λ) + H(S|λ) (15) (16) ∗ Minimizing SB Equation 16 thus minimizes the sum of the entropy of internal states and the entropy required to encode sensory states given internal states. In other words, it strikes an optimal balance between model complexity H(λ), and model accuracy H(S|λ). Friston further shows that Equation 16 directly derives from the free energy principle, closing potential loopholes in the theory. Please see [5] for a comprehensive application of the free energy principle to agents and learning. Note also that the bulk of the theory assumes the ergodic hypothesis. 3 Perception as General Intelligence Since we are chiefly interested in stochastic problems in the physical world, we propose a straightforward informal definition of intelligence: Definition 2. Intelligence measures the capability of a mechanism to solve prediction problems. Mechanism is any physical machine as usual, see [4] which suggests likewise. Therefore, a general formulation of Solomonoff induction, operator induction, might serve as a model of general intelligence, as well [24]. Recall that operator induction can infer any physically plausible cpdf, thus its approximation can solve any classical supervised machine learning problem. The only slight issue with Equation 7 might be that it seems to exclude classical AI systems that are not agents, e.g., expert systems, machine learning tools, knowledge representation systems, search and planning algorithms, and so forth, which are somewhat more naturally encompassed by our informal definition. 6 3.1 Eray Özkural Is operator induction adequate? A question naturally arises as to whether operator induction can adequately solve every prediction problem we require in AI. There are two strong objections to operator induction that we know of. It is argued that in a dynamic environment, as in a physical environment, we must use an active agent model so that we can account for changes in the environment, as in the space-time embedded agent [16] which also provides an agent-based intelligence measure. This objection may be answered by the simple solution that each decision of an active intelligent system may be considered a separate induction problem. The second objection is that the basic Solomonoff induction can only predict the next bit, but not the expected cumulative reward, which its extensions can solve. We counter this objection by stating that we can reduce an agent model to a perception and action-planning problem as in OOPS-RL [20]. In OOPS-RL, the perception module searches for the best world-model given the history of sensory input and actions in allotted time using OOPS, and the planning module searches for the best control program using the world-model of the perception module to determine the action sequence that maximizes cumulative reward likewise. OOPS has a generalized Levin Search [12] which may be tweaked to solve either prediction or optimization problems. Hutter has also observed that standard sequence induction does not readily address optimization problems [8]. However, Solomonoff induction is still complete in the sense of Turing, and can infer any computable cpdf; and when the extension to Solomonoff induction is applied to sequence prediction, it does not yield a better error bound, which seems like a conundrum. On the other hand, Levin Search with a proper universal probability density function (pdf) of programs can be modified to solve induction problems (sequence, set, operator, and sequence prediction with arbitrary loss), inversion problems (computer science problems in P and NP), and optimization problems [23]. The planning module of OOPS-RL likewise requires us to write such an optimization program. In that sense, AIXI implies yet another variation of Levin Search for solving a particular universal optimization problem, however, it also has the unique advantage that formal transformations between AIXI problem and many important problems including function minimization and strategic games have been shown [8]. Nevertheless, the discussion in [23] is rather brief. Also see [1] for a discussion of universal optimization. Proposition 1. A discrete-time universal RL model may be reduced to operator induction. More formally, the perceptual task of an RL agent would be inferring from a history the cumulative rewards in the future, without loss of generality. Let the chronology C be a sequence of sensory, reward, and action data C = [(s1 , r1 , a1 ), (s2 , r2 , a2 ), . . . , (sn , rn , an )] where Ci accesses ith element, and Ci:j accesses the subsequence [Ci , Ci+1 , . . . , Cj ]. Let rc be the cumulative reward function where Pk≤j rc (C, i, j) = k=i rk . After observing (sn , rn , an ), we construct dataset Dc as follows. For every unique (i, j) pair such that 1 < i ≤ j ≤ n, we concatenate history tuples C1:(i−1) , and we form a question string that also includes the next Measuring Intelligence 7 action, i and j, q = [(s1 , r1 , a1 ), (s2 , r2 , a2 ), . . . , (s(i−1) , r(i−1) , a(i−1) )], ai , i, j, and an answer string which is the cumulative reward a = rc (C, i, j). Solving the operator induction problem for this dataset DC will yield a cpdf which predicts cumulative rewards in the future. After that, choosing the next action is a simple matter of maximizing r(C1:n , ai , n + 1, λ) where λ is the planning horizon. The reduction causes quadratic blow-up in the number of data items. Our somewhat cumbersome reduction suggests that all of the intelligence here comes from operator induction, surely an argmax function, or a summation of rewards does not provide it, but rather it builds constraints into the task. In other words, we interpret that the intelligence in an agent model is provided by inductive inference, rather than an additional application of decision theory. 4 Physical Quantification of Intelligence Definition 1 corresponds to any kind of reinforcement-learning or goal-following agent in AI literature quite well, and can be adapted to solve other kinds of problems. The unsupervised, active inference agent approach is proposed instead of reinforcement learning approach in [7], and the authors argue that they did not need to invoke the notion of reward, value or utility. The authors in particular claim that they could solve the mountain-car problem by the free-energy formulation of perception. We thus propose a perceptual intelligence measure. 4.1 Universal measure of perception fitness Note that operator induction is considered to be insufficient to describe universal agents such as AIXI, because basic sequence induction is inappropriate for modelling optimization problems [8]. However, a modified Levin search procedure can solve such optimization problems as in finding an optimal control program [20]. In OOPS-RL, the perception module searches for the best world-model given the history of sensory input and actions in allotted time using OOPS, and the planning module searches for the best control program using the world-model of the perception module to determine the control program that maximizes cumulative reward likewise. In this paper, we consider the perception module of such a generic agent which must produce a world-model, given sensory input. We can use the intelligence measure Equation 7 in a physical theory of intelligence, however it contains terms like utility that do not have physical units (i.e., we would be preferring a more reductive definition). We therefore attempt to obtain such a measure using the more benign goodness-of-fit (Equation 3). Let the universal measure of the fitness of operator induction be defined as X ΥO (π) = 2−HU (µ) Ψ (µ, π) (17) µ∈S where S is the set of possible stochastic sources in the observable universe U and π is a physical mechanism, and Ψ is relative to a stochastic source µ and a 8 Eray Özkural physical mechanism (computer) π. This would be maximum if we assume that operator induction were solved exactly by an oracle machine. Note that HU (µ) is finite; Ψ (µ, π) is likewise bounded by the amount of computation π will spend on approximating operator induction. 4.2 Application to homeostasis agent In a presentation to Friston’s group in January 2015, we noted that the mini∗ mization of SB is identical to Minimum Message Length principle, which can be further refined as ′ SB = H ∗ (Λ) + H ∗ (S|Λ) (18) using Solomonoff’s entropy formulation that takes the negative logarithm of algorithmic probability [22]. In the unsupervised agent context, solving this minimization problem corresponds to inferring an optimal behavioral policy as Λ constitutes internal dynamics which may be modeled as a non-terminating program. We could directly apply induction to minimize KL divergence, as well. Note the correspondence to operator induction. Theorem 1. Minimizing the free energy is equivalent to solving the operator induction problem for (λ, s) pairs where qi ∈ Λ and ai ∈ S. Proof. Observe that minimizing Equation 16 corresponds to picking maximum ψnj since in entropy form, − log2 (ψnj ) = − log2 (2 −|Oj (·|·)| ) − log2 ( n Y Oj (si |λi )) i=1 = |Oj (·|·)| − n X log2 (Oj (ai |qi )) = |Oj (·|·)| + H(Oj (ai |qi )). i=1 We define a non-redundant selection of ψnj ’s, |Oj (·|·)| = HU (Oj (·|·)), e.g., we pick only the shortest programs that produce the same cpdf, otherwise the entropy form would diverge. Minimizing Equation 18 is exactly operator induction, even though the questions are programs, the ensemble is of all programs P here j and all sensory state, program pairs in space-time. |O (·|·)| = H ∗ (Λ) and P j ∗ H(O (ai |qi )) = H (S|Λ). Note that this merely establishes model equivalence, we have not yet explained how it is to be computed in detail. Proposition 2. By the above theorem, Equation 17 measures the goodness of fit for a given homeostasis agent mechanism, for all possible environments. The mechanism π that maximizes Ψ (µ, π) achieves less error with respect to a source (which may be taken to correspond to the whole random dynamical system in the framework of free energy principle), while ΥO (π) normalizes Ψ (µ, π) with respect to a random dynamical system. It holds for the same reasons Legg’s Measuring Intelligence 9 measure holds, which are not discussed due to space limits in the present paper. We prefer the unsupervised homeostasis agent among the two agent models we discussed because it provides an exceptionally elegant and reductionist model of autonomous behavior, that has been rigorously formulated physically. Note that this agent is conceptually related to the survival property of RL agents discussed in [19]. 4.3 Discussion The unsupervised model still achieves exploration and curiosity, because it would stochastically sample and navigate the environment to reduce predictive errors. While we either optimize perceptual models or choose an action that would befit expectations, it might be possible to express the optimal adaptive agent policy in a general optimization framework. A more in-depth analysis of the unsupervised agent will be presented in a subsequent publication. A more general reductive definition of intelligence should also be researched. These developments could eventually help unify AGI theory. References 1. 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arXiv:1404.1957v3 [math.PR] 29 Oct 2015 The Annals of Applied Probability 2015, Vol. 25, No. 6, 3511–3570 DOI: 10.1214/14-AAP1081 c Institute of Mathematical Statistics, 2015 ERGODIC CONTROL OF MULTI-CLASS M/M/N + M QUEUES IN THE HALFIN–WHITT REGIME By Ari Arapostathis∗,1, Anup Biswas∗,2 and Guodong Pang†,3 The University of Texas at Austin∗ and Pennsylvania State University† We study a dynamic scheduling problem for a multi-class queueing network with a large pool of statistically identical servers. The arrival processes are Poisson, and service times and patience times are assumed to be exponentially distributed and class dependent. The optimization criterion is the expected long time average (ergodic) of a general (nonlinear) running cost function of the queue lengths. We consider this control problem in the Halfin–Whitt (QED) regime, that is, the number of servers n and the total offered load r scale √ like n ≈ r + ρ̂ r for some constant ρ̂. This problem was proposed in [Ann. Appl. Probab. 14 (2004) 1084–1134, Section 5.2]. The optimal solution of this control problem can be approximated by that of the corresponding ergodic diffusion control problem in the limit. We introduce a broad class of ergodic control problems for controlled diffusions, which includes a large class of queueing models in the diffusion approximation, and establish a complete characterization of optimality via the study of the associated HJB equation. We also prove the asymptotic convergence of the values for the multi-class queueing control problem to the value of the associated ergodic diffusion control problem. The proof relies on an approximation method by spatial truncation for the ergodic control of diffusion processes, where the Markov policies follow a fixed priority policy outside a fixed compact set. Received April 2014; revised November 2014. Supported in part by the Office of Naval Research Grant N00014-14-1-0196. 2 Supported in part by an award from the Simons Foundation (# 197982) to The University of Texas at Austin and in part by the Office of Naval Research through the Electric Ship Research and Development Consortium. 3 Supported in part by the Marcus Endowment Grant at the Harold and Inge Marcus Department of Industrial and Manufacturing Engineering at Penn State. AMS 2000 subject classifications. Primary 60K25; secondary 68M20, 90B22, 90B36. Key words and phrases. Multi-class Markovian queues, reneging/abandonment, Halfin–Whitt (QED) regime, diffusion scaling, long time-average control, ergodic control, stable Markov optimal control, spatial truncation, asymptotic optimality. 1 This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in The Annals of Applied Probability, 2015, Vol. 25, No. 6, 3511–3570. This reprint differs from the original in pagination and typographic detail. 1 2 A. ARAPOSTATHIS, A. BISWAS AND G. PANG CONTENTS 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Contributions and comparisons . . . . . . . . . . . . . . . 1.2. Organization . . . . . . . . . . . . . . . . . . . . . . . . . 1.3. Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. The controlled system in the Halfin–Whitt regime . . . . . . 2.1. The multi-class Markovian many-server model . . . . . . 2.2. The ergodic control problem in the Halfin–Whitt regime 3. A broad class of ergodic control problems for diffusions . . . 3.1. The controlled diffusion model . . . . . . . . . . . . . . . 3.2. Structural assumptions . . . . . . . . . . . . . . . . . . . 3.3. Piecewise linear controlled diffusions . . . . . . . . . . . . 3.4. Existence of optimal controls . . . . . . . . . . . . . . . . 3.5. The HJB equation . . . . . . . . . . . . . . . . . . . . . . 3.6. Technical proofs . . . . . . . . . . . . . . . . . . . . . . . 4. Approximation via spatial truncations . . . . . . . . . . . . . 5. Asymptotic convergence . . . . . . . . . . . . . . . . . . . . . . 5.1. The lower bound . . . . . . . . . . . . . . . . . . . . . . . 5.2. The upper bound . . . . . . . . . . . . . . . . . . . . . . . 6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 4 6 6 8 8 10 14 14 17 18 22 26 31 42 48 49 53 61 61 61 1. Introduction. One of the classical problems in queueing theory is to schedule the customers/jobs in a network in an optimal way. These problems are known as the scheduling problems which arise in a wide variety of applications, in particular, whenever there are different customer classes present in the network and competing for the same resources. The optimal scheduling problem has a long history in the literature. One of the appealing scheduling rules is the well-known cµ rule. This is a static priority policy in which it is assumed that each class-i customer has a marginal delay cost ci and an average service time 1/µi , and the classes are prioritized in the decreasing order of ci µi . This static priority rule has proven asymptotically optimal in many settings [4, 28, 32]. In [11], a single-server Markov modulated queueing network is considered and an averaged cµ-rule is shown asymptotically optimal for the discounted control problem. An important aspect of queueing networks is abandonment/reneging, that is, customers/jobs may choose to leave the system while being in the queue before their service. Therefore, it is important to include customer abandonment in modeling queueing systems. In [5, 6], Atar et al. considered a multi-class M/M/N +M queueing network with customer abandonment and proved that a modified priority policy, referred to as cµ/θ rule, is asymptotically optimal for the long run average cost in the fluid scale. Dai and Tezcan [13] showed the asymptotic optimality of a static priority policy on a finite ERGODIC CONTROL IN THE HALFIN–WHITT REGIME 3 time interval for a parallel server model under the assumed conditions on the ordering of the abandonment rates and running costs. Although static priority policies are easy to implement, it may not be optimal for control problems of many multi-server queueing systems. For the same multi-class M/M/N + M queueing network, discounted cost control problems are studied in [3, 7, 22], and asymptotically optimal controls for these problems are constructed from the minimizer of a Hamilton–Jacobi–Bellman (HJB) equation associated with the controlled diffusions in the Halfin–Whitt regime. In this article, we are interested in an ergodic control problem for a multi-class M/M/N + M queueing network in the Halfin–Whitt regime. The network consists of a single pool of n statistically identical servers and a buffer of infinite capacity. There are d customer classes and arrivals of jobs/customers are d independent Poisson processes with parameters λni , i = 1, . . . , d. The service rate for class-i customers is µni , i = 1, . . . , d. Customers may renege from the queue if they have not started to receive service before their patience times. Class-i customers renege from the queue at rates γin > 0, i = 1, . . . , d. The scheduling policies are work-conserving, that is, no server stays idle if any of the queues is nonempty. We assume the system operates in the Halfin–Whitt regime, where the arrival rates and the number of servers are scaled appropriately in a manner that the traffic intensity of the system satisfies ! d X √ λni −→ ρ̂ ∈ R. n 1− nµni n→∞ i=1 In this regime, the system operations achieve both high quality (high server levels) and high efficiency (high servers’ utilization), and hence it is also referred to as the Quality-and-Efficiency-Driven (QED) regime; see, for example, [7, 16, 17, 19, 21] on the many-server regimes. We consider an ergodic cost function given by  Z T 1 n r(Q̂ (s)) ds , lim sup E T →∞ T 0 where the running cost r is a nonnegative, convex function with polynomial growth and Q̂n = (Q̂n1 , . . . , Q̂nd )T is the diffusion-scaled queue length process. It is worth mentioning that in addition to the running cost above which is based on the queue-length, we can add an idle-server cost provided that it has at most polynomial growth. For such, a running cost structure the same analysis goes through. The control is the allocation of servers to different classes of customers at the service completion times. The value function is defined to be the infimum of the above cost over all admissible controls (among all work-conserving scheduling policies). In this article, we are interested in the existence and uniqueness of asymptotically optimal stable 4 A. ARAPOSTATHIS, A. BISWAS AND G. PANG stationary Markov controls for the ergodic control problem, and the asymptotic behavior of the value functions as n tends to infinity. In [7], Section 5.2, it is stated that analysis of this type of problems is important for modeling call centers. 1.1. Contributions and comparisons. The usual methodology for studying these problems is to consider the associated continuum model, which is the controlled diffusion limit in a heavy-traffic regime, and to study the ergodic control problem for the controlled diffusion. Ergodic control problems governed by controlled diffusions have been well studied in literature [1, 9] for models that fall in these two categories: (a) the running cost is near-monotone, which is defined by the requirement that its value outside a compact set exceeds the optimal average cost, thus penalizing unstable behavior (see Assumption 3.4.2 in [1] for details), or (b) the controlled diffusion is uniformly stable, that is, every stationary Markov control is stable and the collection of invariant probability measures corresponding to the stationary Markov controls is tight. However, the ergodic control problem at hand does not fall under any of these frameworks. First, the running cost we consider here is not near-monotone because the total queue length can be 0 when the total number of customers in the system are O(n). On the other hand, it is not at all clear that the controlled diffusion is uniformly stable (unless one imposes nontrivial hypotheses on the parameters), and this remains an open problem. One of our main contributions in this article is that we solve the ergodic control problem for a broad class of nondegenerate controlled diffusions, that in a certain way can be viewed as a mixture of the two categories mentioned above. As we show in Section 3, stability of the diffusion under any optimal stationary Markov control occurs due to certain interplay between the drift and the running cost. The model studied in Section 3 is far more general than the queueing problem described, and thus it is of separate interest for ergodic control. We present a comprehensive study of this broad class of ergodic control problems that includes existence of a solution to the ergodic HJB equation, its stochastic representation and verification of optimality (Theorem 3.4), uniqueness of the solution in a certain class (Theorem 3.5), and convergence of the vanishing discount method (Theorem 3.6). These results extend the well-known results for near-monotone running costs. The assumptions in these theorems are verified for the multiclass queueing model and the corresponding characterization of optimality is obtained (Corollary 3.1), which includes growth estimates for the solution of the HJB. We also introduce a new approximation technique, spatial truncation, for the controlled diffusion processes; see Section 4. It is shown that if we freeze the Markov controls to a fixed stable Markov control outside a compact set, then we can still obtain nearly optimal controls in this class of Markov ERGODIC CONTROL IN THE HALFIN–WHITT REGIME 5 controls for large compact sets. We should keep in mind that this property is not true in general. This method can also be thought of as an approximation by a class of controlled diffusions that are uniformly stable. We remark that for a fixed control, the controlled diffusions for the queueing model can be regarded as a special case of the piecewise linear diffusions considered in [14]. It is shown in [14] that these diffusions are stable under constant Markov controls. The proof is via a suitable Lyapunov function. We conjecture that uniform stability holds for the controlled diffusions associated with the queueing model. For the same multi-class Markovian model, Gamarnik and Stolyar show that the stationary distributions of the queue lengths are tight under any work-conserving policy [15], Theorem 2. We also wish to remark here that we allow ρ̂ to be negative, assuming abandonment rates are strictly positive, while in [15], ρ̂ > 0 and abandonment rates can be zero. Another important contribution of this work is the convergence of the value functions associated with the sequence of multi-class queueing models to the value of the ergodic control problem, say ̺∗ , corresponding to the controlled diffusion model. It is not obvious that one can have asymptotic optimality from the existence of optimal stable controls for the HJB equations of controlled diffusions. This fact is relatively straightforward when the cost under consideration is discounted. In that situation, the tightness of paths on a finite time horizon is sufficient to prove asymptotic optimality [7]. But we are in a situation where any finite time behavior of the stochastic process plays no role in the cost. In particular, we need to establish the convergence of the controlled steady states. Although uniform stability of stationary distributions for this multi-class queueing model in the case where ρ̂ > 0 and abandonment rates can be zero is established in [15], it is not obvious that the stochastic model considered here has the property of uniform stability. Therefore, we use a different method to establish the asymptotic optimality. First, we show that the value functions are asymptotically bounded below by ̺∗ . To study the upper bound, we construct a sequence of Markov scheduling policies that are uniformly stable (see Lemma 5.1). The key idea used in establishing such stability results is a spatial truncation technique, under which the Markov policies follow a fixed priority policy outside a given compact set. We believe these techniques can also be used to study ergodic control problems for other many-server queueing models. The scheduling policies we consider in this paper allow preemption, that is, a customer in service can be interrupted for the server to serve a customer of a different class and her service will be resumed later. In fact, the asymptotic optimality is shown within the class of the work-conserving preemptive policies. In [7], both preemptive and nonpreemptive policies are studied, where a nonpreemptive scheduling control policy is constructed from the HJB equation associated with preemptive policies and thus is shown to be 6 A. ARAPOSTATHIS, A. BISWAS AND G. PANG asymptotically optimal. However, as far as we know, the optimal nonpreemptive scheduling problem under the ergodic cost remains open. For a similar line of work in uncontrolled settings, we refer the reader to [16, 19]. Admission control of the single class M/M/N + M model with an ergodic cost criterion in the Halfin–Whitt regime is studied in [26]. For controlled problems and for finite server models, asymptotic optimality is obtained in [12] in the conventional heavy-traffic regime. The main advantage in [12] is the uniform exponential stability of the stochastic processes, which is obtained by using properties of the Skorohod reflection map. A recent work studying ergodic control of a multi-class single-server queueing network is [25]. To summarize our main contributions in this paper: – We introduce a new class of ergodic control problems and a framework to solve them. – We establish an approximation technique by spatial truncation. – We provide, to the best of our knowledge, the first treatment of ergodic control problems at the diffusion scale for many server models. – We establish asymptotic optimality results. 1.2. Organization. In Section 1.3, we summarize the notation used in the paper. In Section 2, we introduce the multi-class many server queueing model and describe the Halfin–Whitt regime. The ergodic control problem under the heavy-traffic setting is introduced in Section 2.2, and the main results on asymptotic convergence are stated as Theorems 2.1 and 2.2. Section 3 introduces a class of controlled diffusions and associated ergodic control problems, which contains the queueing models in the diffusion scale. The key structural assumptions are in Section 3.2 and these are verified for a generic class of queueing models in Section 3.3, which are characterized by piecewise linear controlled diffusions. Section 3.4 concerns the existence of optimal controls under the general hypotheses, while Section 3.5 contains a comprehensive study of the HJB equation. Section 3.6 is devoted to the proofs of the results in Section 3.5. The spatial truncation technique is introduced and studied in Section 4. Finally, in Section 5 we prove the results of asymptotic optimality. 1.3. Notation. The standard Euclidean norm in Rd is denoted by |·|. The set of nonnegative real numbers is denoted by R+ , N stands for the set of natural numbers, and I denotes the indicator function. By Zd+ we denote the set of d-vectors of nonnegative integers. The closure, the boundary and the complement of a set A ⊂ Rd are denoted by A, ∂A and Ac , respectively. The open ball of radius R around 0 is denoted by BR . Given two real numbers a and b, the minimum (maximum) is denoted by a ∧ b (a ∨ b), respectively. ERGODIC CONTROL IN THE HALFIN–WHITT REGIME 7 Define a+ := a ∨ 0 and a− := −(a ∧ 0). The integer part of a real number a is denoted by ⌊a⌋. We use the notation ei , i = 1, . . . , d, to denote the vector with ith entry equal to 1 and all other entries equal to 0. We also let e := (1, . . . , 1)T . Given any two vectors x, y ∈ Rd the inner product is denoted by x · y. By δx we denote the Dirac mass at x. For any function f : Rd → R and domain D ⊂ R we define the oscillation of f on D as follows: osc(f ) := sup{f (x) − f (y) : x, y ∈ D}. D For a nonnegative function g ∈ C(Rd ), we let O(g) denote the space of |f (x)| < ∞. This is a Banach space functions f ∈ C(Rd ) satisfying supx∈Rd 1+g(x) under the norm |f (x)| kf kg := sup . x∈Rd 1 + g(x) We also let o(g) denote the subspace of O(g) consisting of those functions f satisfying lim sup |x|→∞ |f (x)| = 0. 1 + g(x) By a slight abuse of notation, we also denote by O(g) and o(g) a generic member of these spaces. For two nonnegative functions f and g, we use the notation f ∼ g to indicate that f ∈ O(g) and g ∈ O(f ). We denote by Lploc (Rd ), p ≥ 1, the set of real-valued functions that are p d d locally p-integrable and by Wk,p loc (R ) the set of functions in Lloc (R ) whose p ith weak derivatives, i = 1, . . . , k, are in Lloc (Rd ). The set of all bounded k,α continuous functions is denoted by Cb (Rd ). By Cloc (Rd ) we denote the set of functions that are k-times continuously differentiable and whose kth derivatives are locally Hölder continuous with exponent α. We define Cbk (Rd ), k ≥ 0, as the set of functions whose ith derivatives, i = 1, . . . , k, are continuous and bounded in Rd and denote by Cck (Rd ) the subset of Cbk (Rd ) with compact support. For any path X(·), we use the notation ∆X(t) to denote the jump at time t. Given any Polish space X , we denote by P(X ) the set of probability measures on X and we endow P(X ) with the Prokhorov metric. For ν ∈ P(X ) and a Borel measurable map f : X → R, we often use the abbreviated notation Z f dν. ν(f ) := X The quadratic variation of a square integrable martingale is denoted by h·, ·i and the optional quadratic variation by [·, ·]. For presentation purposes we use the time variable as the subscript for the diffusion processes. Also κ1 , κ2 , . . . and C1 , C2 , . . . are used as generic constants whose values might vary from place to place. 8 A. ARAPOSTATHIS, A. BISWAS AND G. PANG Fig. 1. A schematic model of the system. 2. The controlled system in the Halfin–Whitt regime. 2.1. The multi-class Markovian many-server model. Let (Ω, F, P) be a given complete probability space and all the stochastic variables introduced below are defined on it. The expectation w.r.t. P is denoted by E. We consider a multi-class Markovian many-server queueing system which consists of d customer classes and n parallel servers capable of serving all customers (see Figure 1). The system buffer is assumed to have infinite capacity. Customers of class i ∈ {1, . . . , d} arrive according to a Poisson process with rate λni > 0. Customers enter the queue of their respective classes upon arrival if not being processed. Customers of each class are served in the first-come-first-serve (FCFS) service discipline. While waiting in queue, customers can abandon the system. The service times and patience times of customers are classdependent and both are assumed to be exponentially distributed, that is, class i customers are served at rate µni and renege at rate γin . We assume that customer arrivals, service and abandonment of all classes are mutually independent. The Halfin–Whitt regime. We study this queueing model in the Halfin– Whitt regime [or the Quality-and-Efficiency-Driven (QED) regime]. Consider a sequence of such systems indexed by n, in which the arrival rates λni and the number of servers n both increase appropriately. Let rni := λni /µni be the mean offered load of classPi customers. The traffic intensity of the nth system is given by ρn = n−1 di=1 rni . In the Halfin–Whitt regime, the ERGODIC CONTROL IN THE HALFIN–WHITT REGIME 9 parameters are assumed to satisfy the following: as n → ∞, (2.1) λni → λi > 0, µni → µi > 0, n √ λni − nλi √ n(µni − µi ) → µ̂i , → λ̂i , n λi rni → ρi := < 1, n µi This implies that √ n(1 − ρn ) → ρ̂ := d X γin → γi > 0, ρi = 1. i=1 d X ρi µ̂i − λ̂i i=1 µi ∈ R. The above scaling is common in multi-class multi-server models [7, 22]. Note that we do not make any assumption on the sign of ρ̂. State descriptors. Let Xin = {Xin (t) : t ≥ 0} be the total number of class i customers in the system, Qni = {Qni (t) : t ≥ 0} the number of class i customers in the queue and Zin = {Zin (t) : t ≥ 0} the number of class i customers in service. The following basic relationships hold for these processes: for each t ≥ 0 and i = 1, . . . , d, (2.2) Xin (t) = Qni (t) + Zin (t), Qni (t) ≥ 0, Zin (t) ≥ 0 and e · Z n (t) ≤ n. We can describe these processes using a collection {Ani , Sin , Rin , i = 1, . . . , d} of independent rate-1 Poisson processes. Define Ãni (t) := Ani (λni t),  Z t  S̃in (t) := Sin µni Zin (s) ds , 0  Z t  n n n n R̃i (t) := Ri γi Qi (s) ds . 0 Then the dynamics take the form (2.3) Xin (t) = Xin (0) + Ãni (t) − S̃in (t) − R̃in (t), t ≥ 0, i = 1, . . . , d. Scheduling control. Following [7, 22], we only consider work-conserving policies that are nonanticipative and allow preemption. When a server becomes free and there are no customers waiting in any queue, the server stays idle, but if there are customers of multiple classes waiting in the queue, the 10 A. ARAPOSTATHIS, A. BISWAS AND G. PANG server has to make a decision on the customer class to serve. Service preemption is allowed, that is, service of a customer class can be interrupted at any time to serve some other class of customers and the original service is resumed at a later time. A scheduling control policy determines the processes Z n , which must satisfy the constraints in (2.2) and the work-conserving constraint, that is, e · Z n (t) = (e · X n (t)) ∧ n, t ≥ 0. Define the action set An (x) as An (x) := {a ∈ Zd+ : a ≤ x and e · a = (e · x) ∧ n}. Thus, we can write Z n (t) ∈ An (X n (t)) for each t ≥ 0. We also assume that all controls are nonanticipative. Define the σ-fields Ftn := σ{X n (0), Ãni (t), S̃in (t), R̃in (t) : i = 1, . . . , d, 0 ≤ s ≤ t} ∨ N and Gtn := σ{δÃni (t, r), δS̃in (t, r), δR̃in (t, r) : i = 1, . . . , d, r ≥ 0}, where δÃni (t, r) := Ãni (t + r) − Ãni (t),  Z t  n n n n n δS̃i (t, r) := Si µi Zi (s) ds + µi r − S̃in (t), 0 δR̃in (t, r) := Rin  γin Z 0 t Qni (s) ds + γin r  − R̃in (t), and N is the collection of all P-null sets. The filtration {Ftn , t ≥ 0} represents the information available up to time t while Gtn contains the information about future increments of the processes. We say that a working-conserving control policy is admissible if: (i) Z n (t) is adapted to Ftn , (ii) Ftn is independent of Gtn at each time t ≥ 0, (iii) for each i = 1, . . . , d, and t ≥ 0, the process δS̃in (t, ·) agrees in law with Sin (µni ·), and the process δR̃in (t, ·) agrees in law with Rin (γin ·). We denote the set of all admissible control policies (Z n , F n , G n ) by Un . 2.2. The ergodic control problem in the Halfin–Whitt regime. Define the diffusion-scaled processes X̂ n = (X̂1n , . . . , X̂dn )T , Q̂n = (Q̂n1 , . . . , Q̂nd )T and Ẑ n = (Ẑ1n , . . . , Ẑdn )T , ERGODIC CONTROL IN THE HALFIN–WHITT REGIME 11 by (2.4) 1 X̂in (t) := √ (Xin (t) − ρi nt), n 1 Q̂ni (t) := √ Qni (t), n 1 Ẑin (t) := √ (Zin (t) − ρi nt) n for t ≥ 0. By (2.3), we can express X̂in as Z t Z t n n n n n n X̂i (t) = X̂i (0) + ℓi t − µi Ẑi (s) ds − γi Q̂ni (s) ds 0 0 (2.5) n n n + M̂A,i (t) − M̂S,i (t) − M̂R,i (t), where ℓn = (ℓn1 , . . . , ℓnd )T is defined as 1 ℓni := √ (λni − µni ρi n), n and (2.6) 1 n M̂A,i (t) := √ (Ani (λni t) − λni t), n   Z t   Z t 1 n n n n n n M̂S,i (t) := √ Si µi Zi (s) ds − µi Zi (s) ds , n 0 0   Z t   Z t 1 n n n n n n M̂R,i (t) := √ Ri γi Qi (s) ds − γi Qi (s) ds n 0 0 are square integrable martingales w.r.t. the filtration {Ftn }. Note that √ 1 (λ̂i − ρi µ̂i ) . ℓni = √ (λni − λi n) − ρi n(µni − µi ) −→ ℓi := n→∞ n µi Define S := {u ∈ Rd+ : e · u = 1}. For Z n ∈ Un we define, for t ≥ 0 and for adapted Û n (t) ∈ S, (2.7) Q̂n (t) := (e · X̂ n (t))+ Û n (t), Ẑ n (t) := X̂ n (t) − (e · X̂ n (t))+ Û n (t). If Q̂n (t) = 0, we define Û n (t) := ed = (0, . . . , 0, 1)T . Thus, Ûin represents the fraction of class-i customers in the queue when the total queue size is positive. As we show later, it is convenient to view Û n (t) as the control. Note that the controls are nonanticipative and preemption is allowed. 12 A. ARAPOSTATHIS, A. BISWAS AND G. PANG 2.2.1. The cost minimization problem. We next introduce the running cost function for the control problem. Let r : Rd+ → R+ be a given function satisfying (2.8) c1 |x|m ≤ r(x) ≤ c2 (1 + |x|m ) for some m ≥ 1, and some positive constants ci , i = 1, 2. We also assume that r is locally Lipschitz. This assumption includes linear and convex running cost functions. For example, if we let hi be the holding cost rate for class i customers, then some of the typical running cost functions are the following: r(x) = d X hi xm i , m ≥ 1. i=1 These running cost functions evidently satisfy the condition in (2.8). Given the initial state X n (0) and a work-conserving scheduling policy n Z ∈ Un , we define the diffusion-scaled cost function as  Z T 1 n n n (2.9) r(Q̂ (s)) ds , J(X̂ (0), Ẑ ) := lim sup E T →∞ T 0 where the running cost function r satisfies (2.8). Note that the running cost is defined using the scaled version of Z n . Then the associated cost minimization problem becomes (2.10) V̂ n (X̂ n (0)) := ninf n J(X̂ n (0), Ẑ n ). Z ∈U We refer to V̂ n (X̂ n (0)) as the diffusion-scaled value function given the initial state X̂ n (0) in the nth system. From (2.7), it is easy to see that by redefining r as r(x, u) = r((e · x)+ u) we can rewrite the control problem as ˜ X̂ n (0), Û n ), V̂ n (X̂ n (0)) = inf J( where (2.11) ˜ X̂ n (0), Û n ) := lim sup 1 E J( T →∞ T Z T 0  r(X̂ n (s), Û n (s)) ds , and the infimum is taken over all admissible pairs (X̂ n , Û n ) satisfying (2.7). For simplicity, we assume that the initial condition X̂ n (0) is deterministic and X̂ n (0) → x as n → ∞ for some x ∈ Rd . ERGODIC CONTROL IN THE HALFIN–WHITT REGIME 13 2.2.2. The limiting controlled diffusion process. As in [7, 22], one formally deduces that, provided X̂ n (0) → x, there exists a limit X for X̂ n on every finite time interval, and the limit process X is a d-dimensional diffusion process with independent components, that is, (2.12) dXt = b(Xt , Ut ) dt + Σ dWt, with initial condition X0 = x. In (2.12), the drift b(x, u) : Rd × S → Rd takes the form (2.13) with b(x, u) = ℓ − R(x − (e · x)+ u) − (e · x)+ Γu, ℓ := (ℓ1 , . . . , ℓd )T , R := diag(µ1 , . . . , µd ), Γ := diag(γ1 , . . . , γd ). The control Ut lives in S and is nonanticipative, W (t) is a d-dimensional standard Wiener process independent of the initial condition X0 = x, and the covariance matrix is given by ΣΣT = diag(2λ1 , . . . , 2λd ). A formal derivation of the drift in (2.13) can be obtained from (2.5) and (2.7). A detailed description of equation (2.12) and related results are given in Section 3. Let U be the set of all admissible controls for the diffusion model (for a definition see Section 3). 2.2.3. The ergodic control problem in the diffusion scale. Define r̃ : Rd+ × Rd+ → R+ by r̃(x, u) := r((e · x)+ u), where r is the same function as in (2.9). In analogy with (2.11) we define the ergodic cost associated with the controlled diffusion process X and the running cost function r̃(x, u) as  Z T 1 U J(x, U ) := lim sup Ex r̃(Xt , Ut ) dt , U ∈ U. T →∞ T 0 We consider the ergodic control problem (2.14) ̺∗ (x) = inf J(x, U ). U ∈U We call ̺∗ (x) the optimal value at the initial state x for the controlled diffusion process X. It is shown later that ̺∗ (x) is independent of x. A detailed treatment and related results corresponding to the ergodic control problem are given in Section 3. We next state the main results of this section, the proof of which can be found in Section 5. 14 A. ARAPOSTATHIS, A. BISWAS AND G. PANG Theorem 2.1. Let X̂ n (0) → x ∈ Rd as n → ∞. Also assume that (2.1) and (2.8) hold. Then lim inf V̂ n (X̂ n (0)) ≥ ̺∗ (x), n→∞ where ̺∗ (x) is given by (2.14). Theorem 2.2. Suppose the assumptions of Theorem 2.1 hold. In addition, assume that r in (2.9) is convex. Then lim sup V̂ n (X̂ n (0)) ≤ ̺∗ (x). n→∞ Thus, we conclude that for any convex running cost function r, Theorems 2.1 and 2.2 establish the asymptotic convergence of the ergodic control problem for the queueing model. 3. A broad class of ergodic control problems for diffusions. 3.1. The controlled diffusion model. The dynamics are modeled by a controlled diffusion process X = {Xt , t ≥ 0} taking values in the d-dimensional Euclidean space Rd , and governed by the Itô stochastic differential equation (3.1) dXt = b(Xt , Ut ) dt + σ(Xt ) dWt . All random processes in (3.1) live in a complete probability space (Ω, F, P). The process W is a d-dimensional standard Wiener process independent of the initial condition X0 . The control process U takes values in a compact, metrizable set U, and Ut (ω) is jointly measurable in (t, ω) ∈ [0, ∞) × Ω. Moreover, it is nonanticipative: for s < t, Wt − Ws is independent of Fs := the completion of σ{X0 , Ur , Wr , r ≤ s} relative to (F, P). Such a process U is called an admissible control, and we let U denote the set of all admissible controls. We impose the following standard assumptions on the drift b and the diffusion matrix σ to guarantee existence and uniqueness of solutions to equation (3.1). (A1) Local Lipschitz continuity: The functions b = [b1 , . . . , bd ]T : Rd × U → Rd and σ = [σ ij ] : Rd → Rd×d are locally Lipschitz in x with a Lipschitz constant CR > 0 depending on R > 0. In other words, for all x, y ∈ BR and u ∈ U, |b(x, u) − b(y, u)| + kσ(x) − σ(y)k ≤ CR |x − y|. We also assume that b is continuous in (x, u). ERGODIC CONTROL IN THE HALFIN–WHITT REGIME 15 (A2) Affine growth condition: b and σ satisfy a global growth condition of the form |b(x, u)|2 + kσ(x)k2 ≤ C1 (1 + |x|2 ) ∀(x, u) ∈ Rd × U, where kσk2 := trace(σσ T ). (A3) Local nondegeneracy: For each R > 0, it holds that d X i,j=1 −1 2 |ξ| aij (x)ξi ξj ≥ CR ∀x ∈ BR , for all ξ = (ξ1 , . . . , ξd )T ∈ Rd , where a := σσ T . In integral form, (3.1) is written as Z t Z t (3.2) σ(Xs ) dWs . b(Xs , Us ) ds + Xt = X0 + 0 0 The third term on the right-hand side of (3.2) is an Itô stochastic integral. We say that a process X = {Xt (ω)} is a solution of (3.1), if it is Ft -adapted, continuous in t, defined for all ω ∈ Ω and t ∈ [0, ∞), and satisfies (3.2) for all t ∈ [0, ∞) a.s. It is well known that under (A1)–(A3), for any admissible control there exists a unique solution of (3.1) [1], Theorem 2.2.4. We define the family of operators Lu : C 2 (Rd ) → C(Rd ), where u ∈ U plays the role of a parameter, by (3.3) Lu f (x) := 12 aij (x)∂ij f (x) + bi (x, u)∂i f (x), u ∈ U. We refer to Lu as the controlled extended generator of the diffusion. In ∂ (3.3) and elsewhere in this paper, we have adopted the notation ∂i := ∂x i 2 and ∂ij := ∂x∂i ∂xj . We also use the standard summation rule that repeated subscripts and superscripts are summed from 1 through d. In other words, the right-hand side of (3.3) stands for d d X 1 X ij ∂f ∂2f bi (x, u) a (x) (x) + (x). 2 ∂xi ∂xj ∂xi i=1 i,j=1 Of fundamental importance in the study of functionals of X is Itô’s formula. For f ∈ C 2 (Rd ) and with Lu as defined in (3.3), it holds that Z t LUs f (Xs ) ds + Mt , (3.4) a.s., f (Xt ) = f (X0 ) + 0 where Mt := Z 0 t h∇f (Xs ), σ(Xs ) dWs i 16 A. ARAPOSTATHIS, A. BISWAS AND G. PANG is a local martingale. Krylov’s extension of Itô’s formula [27], page 122, d extends (3.4) to functions f in the local Sobolev space W2,p loc (R ), p ≥ d. Recall that a control is called Markov if Ut = v(t, Xt ) for a measurable map v : R+ × Rd → U, and it is called stationary Markov if v does not depend on t, that is, v : Rd → U. Correspondingly, (3.1) is said to have a strong solution if given a Wiener process (Wt , Ft ) on a complete probability space (Ω, F, P), there exists a process X on (Ω, F, P), with X0 = x0 ∈ Rd , which is continuous, Ft -adapted, and satisfies (3.2) for all t a.s. A strong solution is called unique, if any two such solutions X and X ′ agree P-a.s., when viewed as elements of C([0, ∞), Rd ). It is well known that under assumptions (A1)–(A3), for any Markov control v, (3.1) has a unique strong solution [20]. Let USM denote the set of stationary Markov controls. Under v ∈ USM , the process X is strong Markov, and we denote its transition function by Pvt (x, ·). It also follows from the work of [8, 31] that under v ∈ USM , the transition probabilities of X have densities which are locally Hölder continuous. Thus, Lv defined by Lv f (x) := 12 aij (x)∂ij f (x) + bi (x, v(x))∂i f (x), v ∈ USM , for f ∈ C 2 (Rd ), is the generator of a strongly-continuous semi-group on Cb (Rd ), which is strong Feller. We let Pvx denote the probability measure and Evx the expectation operator on the canonical space of the process under the control v ∈ USM , conditioned on the process X starting from x ∈ Rd at t = 0. We need the following definition. Definition 3.1. A function h : Rd × U → R is called inf-compact on a set A ⊂ Rd if the set Ā ∩ {x : minu∈U h(x, u) ≤ β} is compact (or empty) in Rd for all β ∈ R. When this property holds for A ≡ Rd , then we simply say that h is inf-compact. Recall that control v ∈ USM is called stable if the associated diffusion is positive recurrent. We denote the set of such controls by USSM , and let µv denote the unique invariant probability measure on Rd for the diffusion under the control v ∈ USSM . We also let M := {µv : v ∈ USSM }. Recall that v ∈ USSM if and only if there exists an inf-compact function V ∈ C 2 (Rd ), a bounded domain D ⊂ Rd , and a constant ε > 0 satisfying Lv V(x) ≤ −ε ∀x ∈ D c . We denote by τ (A) the first exit time of a process {Xt , t ∈ R+ } from a set A ⊂ Rd , defined by τ (A) := inf{t > 0 : Xt ∈ / A}. ERGODIC CONTROL IN THE HALFIN–WHITT REGIME 17 The open ball of radius R in Rd , centered at the origin, is denoted by BR , c ). and we let τR := τ (BR ), and τ̆R := τ (BR We assume that the running cost function r(x, u) is nonnegative, continuous and locally Lipschitz in its first argument uniformly in u ∈ U. Without loss of generality, we let κR be a Lipschitz constant of r(·, u) over BR . In summary, we assume that (A4) r : Rd ×U → R+ is continuous and satisfies, for some constant CR > 0 |r(x, u) − r(y, u)| ≤ CR |x − y| ∀x, y ∈ BR , ∀u ∈ U, and all R > 0. In general, U may not be a convex set. It is therefore often useful to enlarge the control set to P(U). For any v(du) ∈ P(U) we can redefine the drift and the running cost as Z Z (3.5) b̄(x, v) := b(x, u)v(du) and r̄(x, v) := r(x, u)v(du). U U It is easy to see that the drift and running cost defined in (3.5) satisfy all the aforementioned conditions (A1)–(A4). In what follows, we assume that all the controls take values in P(U). These controls are generally referred to as relaxed controls. We endow the set of relaxed stationary Markov controls with the following topology: vn → v in USM if and only if Z Z Z Z f (x) g(x, u)v(du|x) dx f (x) g(x, u)vn (du|x) dx −→ Rd n→∞ Rd U L1 (Rd ) L2 (Rd ) U (Rd for all f ∈ ∩ and g ∈ Cb × U). Then USM is a compact metric space under this topology [1], Section 2.4. We refer to this topology as the topology of Markov controls. A control is said to be precise if it takes value in U. It is easy to see that any precise control Ut can also be understood as a relaxed control by Ut (du) = δUt . Abusing the notation, we denote the drift and running cost by b and r, respectively, and the action of a relaxed control on them is understood as in (3.5). 3.2. Structural assumptions. Assumptions 3.1 and 3.2, described below, are in effect throughout the analysis, unless otherwise stated. Assumption 3.1. For some open set K ⊂ Rd , the following hold: (i) The running cost r is inf-compact on K. (ii) There exist inf-compact functions V ∈ C 2 (Rd ) and h ∈ C(Rd × U), such that (3.6) Lu V(x) ≤ 1 − h(x, u) ∀(x, u) ∈ Kc × U, Lu V(x) ≤ 1 + r(x, u) ∀(x, u) ∈ K × U. 18 A. ARAPOSTATHIS, A. BISWAS AND G. PANG Without loss of generality, we assume that V and h are nonnegative. Remark 3.1. In the statement of Assumption 3.1, we refrain from using any constants in the interest of notational economy. There is no loss of generality in doing so, since the functions V and h can always be scaled to eliminate unnecessary constants. The next assumption is not a structural one, but rather the necessary requirement that the value of the ergodic control problem is finite. Otherwise, the problem is vacuous. For U ∈ U, define  Z T 1 U ̺U (x) := lim sup Ex (3.7) r(Xs , Us ) ds . T →∞ T 0 Assumption 3.2. x ∈ Rd . There exists U ∈ U such that ̺U (x) < ∞ for some Assumption 3.2 alone does not imply that ̺v < ∞ for some v ∈ USSM . However, when combined with Assumption 3.1, this is the case as the following lemma asserts. Lemma 3.1. Let Assumptions 3.1 and 3.2 hold. Then there exists u0 ∈ USSM such that ̺u0 < ∞. Moreover, there exists a nonnegative inf-compact function V0 ∈ C 2 (Rd ), and a positive constant η such that (3.8) Lu0 V0 (x) ≤ η − r(x, u0 (x)) ∀x ∈ Rd . Conversely, if (3.8) holds, then Assumption 3.2 holds. Proof. The first part of the result follows from Theorem 3.1(e) and (3.23) whereas the converse part follows from Lemma 3.2. These proofs are stated later in the paper.  Remark 3.2. There is no loss of generality in using only the constant η in Assumption 3.2, since V0 can always be scaled to achieve this. We also observe that for K = Rd the problem reduces to an ergodic control problem with near-monotone cost, and for K = ∅ we obtain an ergodic control problem under a uniformly stable controlled diffusion. 3.3. Piecewise linear controlled diffusions. The controlled diffusion process in (2.12) belongs to a large class of controlled diffusion processes, called piecewise linear controlled diffusions [14]. We describe this class of controlled diffusions and show that it satisfies the assumptions in Section 3.2. ERGODIC CONTROL IN THE HALFIN–WHITT REGIME 19 Definition 3.2. A square matrix R is said to be an M -matrix if it can be written as R = sI − N for some s > 0 and nonnegative matrix N with property that ρ(N ) ≤ s, where ρ(N ) denotes the spectral radius of N . Let Γ = [γ ij ] be a given matrix whose diagonal elements are positive, = 0 for i = 1, . . . , d − 1, and the remaining elements are in R. (Note that for the queueing model, Γ is a positive diagonal matrix. Our results below hold for the more general Γ.) Let ℓ ∈ Rd and R be a nonsingular M -matrix. Define γ id (3.9) b(x, u) := ℓ − R(x − (e · x)+ u) − (e · x)+ Γu, with u ∈ S := {u ∈ Rd+ : e · u = 1}. Assume that eT R ≥ 0T . We consider the following controlled diffusion in Rd : (3.10) dXt = b(Xt , Ut ) dt + Σ dWt, where Σ is a constant matrix such that ΣΣT is invertible. It is easy to see that (3.10) satisfies conditions (A1)–(A3). Analysis of these types of diffusion approximations is an established tradition in queueing systems. It is often easy to deal with the limiting object and it also helps to obtain information on the behavior of the actual queueing model. We next introduce the running cost function. Let r : Rd × S → [0, ∞) be locally Lipschitz with polynomial growth and (3.11) c1 [(e · x)+ ]m ≤ r(x, u) ≤ c2 (1 + [(e · x)+ ]m ), for some m ≥ 1 and positive constants c1 and c2 that do not depend on u. Some typical examples of such running costs are + m r(x, u) = [(e · x) ] d X i=1 hi um i with m ≥ 1, for some positive vector (h1 , . . . , hd )T . Remark 3.3. The controlled dynamics in (3.9) and running cost in (3.11) are clearly more general than the model described in Section 2.2. In (3.10), X denotes the diffusion approximation for the number customers in the system in the Halfin–Whitt regime and its ith component X i denotes the diffusion approximation of the number of class i customers. Therefore, (e · X)+ denotes the total number of customers in the queue. For R and Γ diagonal as in (2.13), the diagonal entries of R and Γ denote the service 20 A. ARAPOSTATHIS, A. BISWAS AND G. PANG and abandonment rates, respectively, of the customer classes. The ith coordinate of U denotes the fraction of class-i customers waiting in the queue. Therefore, the vector-valued process Xt − (e · Xt )+ Ut denotes the diffusion approximation of the numbers of customers in service from different customer classes. Proposition 3.1. Let b and r be given by (3.9) and (3.11), respectively. Then (3.10) satisfies Assumptions 3.1 and 3.2, with h(x) = c0 |x|m and (3.12) K := {x : δ|x| < (e · x)+ } for appropriate positive constants c0 and δ. Proof. We recall that if R is a nonsingular M -matrix, then there exists a positive definite matrix Q such that QR + RT Q is strictly positive definite [14]. Therefore, for some positive constant κ0 it holds that 2 κ0 |y|2 ≤ y T [QR + RT Q]y ≤ κ−1 0 |y| ∀y ∈ Rd . The set K in (3.12), where δ > 0 is chosen later, is an open convex cone, and the running cost function r is inf-compact on K. Let V be a nonnegative function in C 2 (Rd ) such that V(x) = [xT Qx]m/2 for |x| ≥ 1, where the constant m is as in (3.11). Let |x| ≥ 1 and u ∈ S. Then m[xT Qx]m/2−1 T x [QR + RT Q]x 2 + m[xT Qx]m/2−1 Qx · (R − Γ)(e · x)+ u   m/2−1 κ0 T 2 + ≤ ℓ · ∇V(x) − m[x Qx] |x| − C|x|(e · x) 2 ∇V(x) · b(x, u) = ℓ · ∇V(x) − κ0 , then on Kc ∩ {|x| ≥ 1} for some positive constant C. If we choose δ = 4C we have the estimate mκ0 T (3.13) [x Qx]m/2−1 |x|2 . ∇V(x) · b(x, u) ≤ ℓ · ∇V(x) − 4 Note that ℓ · V is globally bounded for m = 1. For any m ∈ (1, ∞), it follows by (3.13) that mκ0 T ∇V(x) · b(x, u) ≤ m(ℓT Qx)[xT Qx]m/2−1 − [x Qx]m/2−1 |x|2 4 (3.14) m/2 T mκ0 (λ(Q))m/2 m m|ℓ Q|(λ(Q)) |x|m−1 − |x| ≤ λ(Q) 4λ(Q) for x ∈ Kc ∩ {|x| ≥ 1}, where λ(Q) and λ(Q) are the smallest and largest eigenvalues of Q, respectively. We use Young’s inequality |a|m m − 1 m/(m−1) + |b| , a, b ≥ 0, |ab| ≤ m m ERGODIC CONTROL IN THE HALFIN–WHITT REGIME 21 in (3.14) to obtain the bound (3.15) ∇V(x) · b(x, u) ≤ κ1 − mκ0 (λ(Q))m/2 |x|m 8λ(Q) for some constant κ1 > 0. A similar calculation shows for some constant κ2 > 0 it holds that (3.16) ∇V(x) · b(x, u) ≤ κ2 (1 + [(e · x)+ ]m ) ∀x ∈ K ∩ {|x| ≥ 1}. Also note that we can select κ3 > 0 large enough such that (3.17) 1 mκ0 |trace(ΣΣT ∇2 V(x))| ≤ κ3 + (λ(Q))m/2 |x|m . 2 16λ(Q) Hence, by (3.13)–(3.17) there exists κ4 > 0 such that mκ0 (λ(Q))m/2 |x|m IKc (x) + κ2 [(e · x)+ ]m IK (x) (3.18) Lu V(x) ≤ κ4 − 16λ(Q) for all x ∈ Rd . It is evident that we can scale V, by multiplying it with a constant, so that (3.18) takes the form (3.19) Lu V(x) ≤ 1 − c0 |x|m IKc (x) + c1 [(e · x)+ ]m IK (x) ∀x ∈ Rd . By (3.11), the running cost r is inf-compact on K. It then follows from (3.11) and (3.19) that (3.6) is satisfied with h(x) := c0 |x|m . We next show that (3.10) satisfies Assumption 3.2. Let u0 (·) ≡ ed = (0, . . . , 0, 1)T . Then we can write (3.10) as (3.20) dXt = (ℓ − R(Xt − (e · Xt )+ u0 ) − (e · x)+ Γu0 ) dt + Σ dWt. It is shown in [14] that the solution Xt in (3.20) is positive recurrent and, therefore, u0 is a stable Markov control. This is done by finding a suitable Lyapunov function. In particular, in [14], Theorem 3, it is shown that there exists a positive definite matrix Q̃ such that if we define (3.21) ψ(x) := (e · x)2 + κ̃[x − ed φ(e · x)]T Q̃[x − ed φ(e · x)], for some suitably chosen constant κ̃ and   y, φ(y) = − 21 δ̃,  smooth, a function φ ∈ C 2 (R), given by if y ≥ 0, if y ≤ −δ̃, if − δ̃ < y < 0, where δ̃ > 0 is a suitable constant and 0 ≤ φ′ (y) ≤ 1, then it holds that (3.22) Lu0 ψ(x) ≤ −κ̃1 |x|2 , 22 A. ARAPOSTATHIS, A. BISWAS AND G. PANG for |x| large enough and positive constant κ̃1 . We define V0 := eaψ where a is to be determined later. Note that |∇ψ(x)| ≤ κ̃2 (1 + |x|) for some constant κ̃2 > 0. Hence, a straightforward calculation shows that if we choose a small enough, then for some constant κ̃3 > 0 it holds that Lu0 V0 (x) ≤ (−κ̃1 a|x|2 + a2 kΣk2 κ̃2 (1 + |x|)2 )V0 (x) ≤ −κ̃3 |x|2 V0 (x), for all |x| large enough. Since V0 (x) > [(e · x)+ ]m , m ≥ 1, for all large enough |x| we see that V0 satisfies (3.8) with control u0 . Hence, Assumption 3.2 holds by Lemma 3.1.  3.4. Existence of optimal controls. Definition 3.3. Recall the definition of ̺U in (3.7). For β > 0, we define Uβ := {U ∈ U : ̺U (x) ≤ β for some x ∈ Rd }. We also let UβSM := Uβ ∩ USM , and ̺ˆ∗ := inf{β > 0 : Uβ 6= ∅}, ̺∗ := inf{β > 0 : UβSM 6= ∅}, ̺˜∗ := inf{π(r) : π ∈ G}, where  Z d G := π ∈ P(R × U) : Rd ×U u L f (x)π(dx, du) = 0 ∀f ∈ Cc∞ (Rd )  , and Lu f (x) is given by (3.3). It is well known that G is the set of ergodic occupation measures of the controlled process in (3.1), and that G is a closed and convex subset of P(Rd × U) [1], Lemmas 3.2.2 and 3.2.3. We use the notation πv when we want to indicate the ergodic occupation measure associated with the control v ∈ USSM . In other words, πv (dx, du) := µv (dx)v(du|x). Lemma 3.2. If (3.8) holds for some V0 and u0 , then we have πu0 (r) ≤ η. Therefore, ̺ˆ∗ < ∞. Proof. Let (Xt , u0 (Xt )) be the solution of (3.1). Recall that τR is the first exit time from BR for R > 0. Then by Itô’s formula  Z T ∧τR u0 u0 Ex [V0 (XT ∧τR )] − V0 (x) ≤ ηT − Ex r(Xs , u0 (Xs )) ds . 0 ERGODIC CONTROL IN THE HALFIN–WHITT REGIME 23 Therefore, letting R → ∞ and using Fatou’s lemma, we obtain the bound  Z T u0 Ex r(Xs , u0 (Xs )) ds ≤ ηT + V0 (x) − min V0 , Rd 0 and thus 1 lim sup Eux0 T →∞ T Z T 0  r(Xs , u0 (Xs )) ds ≤ η.  In the analysis, we use a function h̃ ∈ C(Rd × U) which, roughly speaking, is of the same order as r in K × U and lies between r and a multiple of r + h on Kc × U, with K as in Assumption 3.1. The existence of such a function is guaranteed by Assumption 3.1 as the following lemma shows. Lemma 3.3. Define H := (K × U) ∪ {(x, u) ∈ Rd × U : r(x, u) > h(x, u)}, where K is the open set in Assumption 3.1. Then there exists an inf-compact function h̃ ∈ C(Rd × U) which is locally Lipschitz in its first argument uniformly w.r.t. its second argument, and satisfies (3.23) r(x, u) ≤ h̃(x, u) ≤ k0 (1 + h(x, u)IHc (x, u) + r(x, u)IH (x, u)) 2 for all (x, u) ∈ Rd × U, and for some positive constant k0 ≥ 2. Moreover, (3.24) Lu V(x) ≤ 1 − h(x, u)IHc (x, u) + r(x, u)IH (x, u) for all (x, u) ∈ Rd × U, where V is the function in Assumption 3.1. Proof. If f (x, u) denotes the right-hand side of (3.23), with k0 = 4, then f (x, u) − r(x, u) > h(x, u)IHc (x, u) + r(x, u)IH (x, u) ≥ h(x, u)IKc (x) + r(x, u)IK (x), since r(x, u) > h(x, u) on H \ (K × U). Therefore, by Assumption 3.1, the set {(x, u) : f (x, u) − r(x, u) ≤ n} is bounded in Rd × U for every n ∈ N. Hence, there exists an increasing sequence of open balls Dn , n = 1, 2, . . . , centered at 0 in Rd such that f (x, u) − r(x, u) ≥ n for all (x, u) ∈ Dnc × U. Let g : Rd → R be any nonnegative smooth function such that n − 1 ≤ g(x) ≤ n for x ∈ Dn+1 \ Dn , n = 1, 2, . . . , and g(x) = 0 on D1 . Clearly, h̃ := r + g is continuous, inf-compact, locally Lipschitz in its first argument, and satisfies (3.23). That (3.24) holds is clear from (3.6) and the fact that H ⊇ K × U.  24 A. ARAPOSTATHIS, A. BISWAS AND G. PANG Remark 3.4. It is clear from the proof of Lemma 3.3 that we could fix the value of the constant k0 in (3.23), say k0 = 4. However, we keep the variable k0 because this provides some flexibility in the choice of h̃, and also in order to be able to trace it along the different calculations. Remark 3.5. Note that if h ≥ r and r is inf-compact, then H = K × U and h̃ := r satisfies (3.23). Note also, that in view of (3.11) and Proposition 3.1, for the multi-class queueing model we have r(x, u) ≤ c2 (1 + [(e · x)+ ]m ) c2 dm−1 (1 + (1 ∧ c0 )|x|m ) 1 ∧ c0   c2 dm−1 1 + m m c ≤ 1 + c0 |x| IK (x) + m [(e · x) ] IK (x) 1 ∧ c0 δ   m−1 1 c2 d r(x, u)IK (x) 1 + h(x)IKc (x) + ≤ 1 ∧ c0 c1 δm ≤ ≤ ≤ c2 dm−1 (1 + h(x)IKc (x) + r(x, u)IK (x)) 1 ∧ c0 ∧ c1 δm c2 dm−1 (1 + h(x)IHc (x, u) + r(x, u)IH (x, u)). 1 ∧ c0 ∧ c1 δm Therefore, h̃(x, u) := c2 + c2 dm−1 |x|m satisfies (3.23). Remark 3.6. We often use the fact that if g ∈ C(Rd × U) is bounded below, then the map P(Rd × U) ∋ ν 7→ ν(g) is lower semi-continuous. This easily follows from two facts: (a) g can be expressed as an increasing limit of bounded continuous functions, and (b) if g is bounded and continuous, then π 7→ π(g) is continuous. Theorem 3.1. Let β ∈ (ˆ ̺∗ , ∞). Then: (a) For all U ∈ Uβ and x ∈ Rd such that ̺U (x) ≤ β, then Z T  1 U lim sup Ex (3.25) h̃(Xs , Us ) ds ≤ k0 (1 + β). t→∞ T 0 (b) ̺ˆ∗ = ̺∗ = ̺˜∗ . (c) For any β ∈ (̺∗ , ∞), we have UβSM ⊂ USSM . (d) The set of invariant probability measures Mβ corresponding to controls in UβSM satisfies Z h̃(x, v(x))µv (dx) ≤ k0 (1 + β) ∀µv ∈ Mβ . Rd ERGODIC CONTROL IN THE HALFIN–WHITT REGIME 25 In particular, UβSM is tight in P(Rd ). (e) There exists (Ṽ , ̺˜) ∈ C 2 (Rd ) × R+ , with Ṽ inf-compact, such that (3.26) min[Lu Ṽ (x) + h̃(x, u)] = ̺˜. u∈U Proof. Using Itô’s formula, it follows by (3.24) that (3.27) 1 U (E [V(XT ∧τR )] − V(x)) T x Z T ∧τR  1 U h(Xs , Us )IHc (Xs , Us ) ds ≤ 1 − Ex T 0  Z T ∧τR 1 U + Ex r(Xs , Us )IH (Xs , Us ) ds . T 0 Since V is inf-compact, (3.27) together with (3.23) implies that  Z T 1 U 2 lim sup E h̃(Xs , Us ) ds k0 T →∞ T x 0 (3.28)  Z T 1 U ≤ 2 + 2 lim sup Ex r(Xs , Us ) ds . T →∞ T 0 Part (a) then follows from (3.28). Now fix U ∈ Uβ and x ∈ Rd such that ̺U (x) ≤ β. The inequality in (3.25) U : t ≥ 1}, defined by implies that the set of mean empirical measures {ζx,t  Z t 1 U U IA×B (Xs , Us ) ds ζx,t(A × B) := Ex t 0 for any Borel sets A ⊂ Rd and B ⊂ U, is tight. It is the case that any limit point of the mean empirical measures in P(Rd × U) is an ergodic occupation measure [1], Lemma 3.4.6. Then in view of Remark 3.6 we obtain (3.29) U π(r) ≤ lim sup ζx,t (r) ≤ β t→∞ for some ergodic occupation measure π. Therefore, ̺˜∗ ≤ ̺ˆ∗ . Disintegrating the measure π as π(dx, du) = v(du|x)µv (dx), we obtain the associated control v ∈ USSM . From ergodic theory [33], we also know that  Z T 1 lim sup Evx r(Xs , v(Xs )) ds = πv (r) for almost every x. T →∞ T 0 It follows that ̺∗ ≤ ̺˜∗ , and since it is clear that ̺ˆ∗ ≤ ̺∗ , equality must hold among the three quantities. If v ∈ UβSM , then (3.28) implies that (3.29) holds with U ≡ v and π ≡ πv . Therefore, parts (c) and (d) follow. 26 A. ARAPOSTATHIS, A. BISWAS AND G. PANG Existence of (Ṽ , ̺˜), satisfying (3.26), follows from Assumption 3.2 and [1], Theorem 3.6.6. The inf-compactness of Ṽ follows from the stochastic representation of Ṽ in [1], Lemma 3.6.9. This proves (e).  Existence of a stationary Markov control that is optimal is asserted by the following theorem. Theorem 3.2. Let G denote the set of ergodic occupation measures corresponding to controls in USSM , and Gβ those corresponding to controls in UβSM , for β > ̺∗ . Then: (a) The set Gβ is compact in P(Rd ) for any β > ̺∗ . (b) There exists v ∈ USM such that ̺v = ̺∗ . Proof. By Theorem 3.1(d), the set Gβ is tight for any β > ̺∗ . Let {πn } ⊂ Gβ , for some β > ̺∗ , be any convergent sequence in P(Rd ) such that πn (r) → ̺∗ as n → ∞ and denote its limit by π∗ . Since G is closed, π∗ ∈ G, and since the map π → π(r) is lower semi-continuous, it follows that π∗ (r) ≤ ̺∗ . Therefore, Gβ is closed, and hence compact. Since π(r) ≥ ̺∗ for all π ∈ G, the equality π∗ (r) = ̺∗ follows. Also v is obtained by disintegrating π∗ .  Remark 3.7. The reader might have noticed at this point that Assumption 3.1 may be weakened significantly. What is really required is the existence of an open set Ĥ ⊂ Rd × U and inf-compact functions V ∈ C 2 (Rd ) and h ∈ C(Rd × U), satisfying (H1) inf {u : (x,u)∈Ĥ} r(x, u) −→ ∞. (H2) |x|→∞ u L V(x) ≤ 1 − h(x, u)IĤc (x, u) + r(x, u)IĤ (x, u) ∀(x, u) ∈ Rd × U. In (H1), we use the convention that the ‘inf’ of the empty set is +∞. Also note that (H1) is equivalent to the statement that {(x, u) : r(x, u) ≤ c} ∩ Ĥ is bounded in Rd × U for all c ∈ R+ . If (H1)–(H2) are met, we define H := Ĥ ∪ {(x, u) ∈ Rd × U : r(x, u) > h(x, u)}, and following the proof of Lemma 3.3, we assert the existence of an inf-compact h̃ ∈ C(Rd × U) satisfying (3.23). In fact, throughout the rest of the paper, Assumption 3.1 is not really invoked. We only use (3.24), the inf-compact function h̃ satisfying (3.23), and, naturally, Assumption 3.2. 3.5. The HJB equation. For ε > 0, let rε (x, u) := r(x, u) + εh̃(x, u). By Theorem 3.1(d), for any π ∈ Gβ , β > ̺∗ , we have the bound (3.30) π(rε ) ≤ β + εk0 (1 + β). ERGODIC CONTROL IN THE HALFIN–WHITT REGIME 27 Therefore, since rε is near-monotone, that is, lim inf min rε (x, u) > inf π(rε ), |x|→∞ u∈U π∈G there exists πε ∈ arg minπ∈G π(rε ). Let π∗ ∈ G be as in the proof of Theorem 3.2. The sub-optimality of π∗ relative to the running cost rε and (3.30) imply that πε (r) ≤ πε (rε ) (3.31) ≤ π∗ (rε ) ≤ ̺∗ + εk0 (1 + ̺∗ ) ∀ε > 0. It follows from (3.31) and Theorem 3.1(d) that {πε : ε ∈ (0, 1)} is tight. Since πε 7→ πε (r) is lower semi-continuous, if π̄ is any limit point of πε as ε ց 0, then taking limits in (3.31), we obtain (3.32) π̄(r) ≤ lim sup πε (r) ≤ ̺∗ . εց0 Since G is closed, π̄ ∈ G, which implies that π̄(r) ≥ ̺∗ . Therefore, equality must hold in (3.32), or in other words, π̄ is an optimal ergodic occupation measure. Theorem 3.3. There exists a unique function V ε ∈ C 2 (Rd ) with V ε (0) = 0, which is bounded below in Rd , and solves the HJB (3.33) min[Lu V ε (x) + rε (x, u)] = ̺ε , u∈U where ̺ε := inf π∈G π(rε ), or in other words, ̺ε is the optimal value of the ergodic control problem with running cost rε . Also a stationary Markov control vε is optimal for the ergodic control problem relative to rε if and only if it satisfies (3.34) Hε (x, ∇V ε (x)) = b(x, vε (x)) · ∇V ε (x) + rε (x, vε (x)) where (3.35) Hε (x, p) := min[b(x, u) · p + rε (x, u)]. u∈U Moreover: (a) for every R > 0, there exists kR such that (3.36) osc V ε ≤ kR ; BR a.e. in Rd , 28 A. ARAPOSTATHIS, A. BISWAS AND G. PANG (b) if vε is a measurable a.e. selector from the minimizer of the Hamiltonian in (3.35), that is, if it satisfies (3.33), then for any δ > 0,  Z τ̆δ V ε (x) ≥ Evxε (rε (Xs , vε (Xs )) − ̺ε ) ds + inf V ε ; Bδ 0 (c) for any stationary control v ∈ USSM and for any δ > 0,  Z τ̆δ ε v ε V (x) ≤ Ex (rε (Xs , v(Xs )) − ̺ε ) ds + V (Xτ̆δ ) , 0 where τ̆δ is hitting time to the ball Bδ . Theorem 3.4. following hold: Let V ε , ̺ε , and vε , for ε > 0, be as in Theorem 3.3. The (a) The function V ε converges to some V∗ ∈ C 2 (Rd ), uniformly on compact sets, and ̺ε → ̺∗ , as ε ց 0, and V∗ satisfies min[Lu V∗ (x) + r(x, u)] = ̺∗ . (3.37) u∈U Also, any limit point v∗ (in the topology of Markov controls) as ε ց 0 of the set {vε } satisfies Lv∗ V∗ (x) + r(x, v∗ (x)) = ̺∗ a.e. in Rd . (b) A stationary Markov control v is optimal for the ergodic control problem relative to r if and only if it satisfies a.e. in Rd , (3.38) H(x, ∇V∗ (x)) = b(x, v(x)) · ∇V∗ (x) + r(x, v(x)) where H(x, p) := min[b(x, u) · p + r(x, u)]. u∈U Moreover, for an optimal v ∈ USM , we have  Z T 1 v r(Xs , v(Xs )) ds = ̺∗ lim Ex T →∞ T 0 ∀x ∈ Rd . (c) The function V∗ has the stochastic representation Z τ̆δ  v Ex V∗ (x) = lim S inf (r(Xs , v(Xs )) − ̺∗ ) ds δց0 v∈ (3.39) = lim δց0 Ev̄x β β>0 USM Z 0 τ̆δ 0 (r(Xs , v∗ (Xs )) − ̺∗ ) ds for any v̄ ∈ USM that satisfies (3.38).  ERGODIC CONTROL IN THE HALFIN–WHITT REGIME 29 (d) If U is a convex set, u 7→ {b(x, u) · p + r(x, u)} is strictly convex whenever it is not constant, and u 7→ h̃(x, u) is strictly convex for all x, then any measurable minimizer of (3.33) converges pointwise, and thus in USM , to the minimizer of (3.37). Theorem 3.4 guarantees the existence of an optimal stable control, which is made precise by (3.38), for the ergodic diffusion control problem with the running cost function r. Moreover, under the convexity property in part (d), the optimal stable control can be obtained as a pointwise limit from the minimizing selector of (3.33). For instance, if we let + r(x, u) = (e · x) d X hi um i , m > 1, i=1 then by choosing h and h̃ + |u|2 as in Proposition 3.1, we see that the approximate value function V ε and approximate control vε converge to the desired optimal value function V∗ and optimal control v∗ , respectively. Concerning the uniqueness of the solution to the HJB equation in (3.37), recall that in the near-monotone case the existing uniqueness results are as follows: there exists a unique solution pair (V, ̺) of (3.37) with V in the class of functions C 2 (Rd ) which are bounded below in Rd . Moreover, it satisfies V (0) = 0 and ̺ ≤ ̺∗ . If the restriction ̺ ≤ ̺∗ is removed, then in general, there are multiple solutions. Since in our model r is not near-monotone in Rd , the function V∗ is not, in general, bounded below. However, as we show later in Lemma 3.10 the negative part of V∗ grows slower than V, that is, it holds that V∗− ∈ o(V), with o(·) as defined in Section 1.3. Therefore, the second part of the theorem that follows may be viewed as an extension of the well-known uniqueness results that apply to ergodic control problems with near-monotone running cost. The third part of the theorem resembles the hypotheses of uniqueness that apply to problems under a blanket stability hypothesis. Theorem 3.5. (3.40) Let (V̂ , ̺ˆ) be a solution of min[Lu V̂ (x) + r(x, u)] = ̺ˆ, u∈U such that V̂ − ∈ o(V) and V̂ (0) = 0. Then the following hold: (a) Any measurable selector v̂ from the minimizer of the associated Hamiltonian in (3.38) is in USSM and ̺v̂ < ∞. (b) If ̺ˆ ≤ ̺∗ then necessarily ̺ˆ = ̺∗ and V̂ = V∗ . (c) If V̂ ∈ O(minu∈U h̃(·, u)), then ̺ˆ = ̺∗ and V̂ = V∗ . 30 A. ARAPOSTATHIS, A. BISWAS AND G. PANG Applying these results to the multi-class queueing diffusion model, we have the following corollary. Corollary 3.1. For the queueing diffusion model with controlled dynamics given by (3.10), drift given by (3.9), and running cost as in (3.11), there exists a unique solution V , satisfying V (0) = 0, to the associated HJB in the class of functions C 2 (Rd ) ∩ O(|x|m ), whose negative part is in o(|x|m ). This solution agrees with V∗ in Theorem 3.4. Proof. Existence of a solution V follows by Theorem 3.4. Select V ∼ |x|m as in the proof of Proposition 3.1. That the solution V is in the stated class then follows by Lemma 3.10 and Corollary 4.1 that appear later in Sections 3.6 and 4, respectively. With h ∼ |x|m as in the proof of Proposition 3.1, it follows that minu∈U h̃(x, u) ∈ O(|x|m ). Therefore, uniqueness follows by Theorem 3.5.  We can also obtain the HJB equation in (3.37) via the traditional vanishing discount approach as the following theorem asserts. Similar results are shown for a one-dimensional degenerate ergodic diffusion control problem in [29] and certain multi-dimensional ergodic diffusion control problems (allowing degeneracy and spatial periodicity) in [2]. Theorem 3.6. Let V∗ and ̺∗ be as in Theorem 3.4. For α > 0, we define  Z ∞ −αt U e r(Xt , Ut ) dt . Vα (x) := inf Ex U ∈U 0 The function Vα − Vα (0) converges, as α ց 0, to V∗ , uniformly on compact subsets of Rd . Moreover, αVα (0) → ̺∗ , as α ց 0. The proofs of the Theorems 3.3–3.6 are given in Section 3.6. The following result, which follows directly from (3.31), provides a way to find ε-optimal controls. Proposition 3.2. Let {vε } be the minimizing selector from Theorem 3.3 and {µvε } be the corresponding invariant probability measures. Then almost surely for all x ∈ Rd ,  Z Z T 1 vε lim Ex r(x, vε (x))µvε (dx) r(Xs , vε (Xs )) ds = T →∞ T Rd 0 ≤ ̺∗ + εk0 (1 + ̺∗ ). ERGODIC CONTROL IN THE HALFIN–WHITT REGIME 31 3.6. Technical proofs. Recall that rε (x, u) = r(x, u) + εh̃(x, u), with h̃ as in Lemma 3.3. We need the following lemma. For α > 0 and ε ≥ 0, we define  Z ∞ ε U −αt Vα (x) := inf Ex (3.41) e rε (Xt , Ut ) dt , U ∈U 0 where we set r0 ≡ r. Clearly, when ε = 0, we have Vα0 ≡ Vα . We quote the following result from [1], Theorem 3.5.6, Remark 3.5.8. Lemma 3.4. Provided ε > 0, then Vαε defined above is in C 2 (Rd ) and is the minimal nonnegative solution of min[Lu Vαε (x) + rε (x, u)] = αVαε (x). u∈U The HJB in Lemma 3.4 is similar to the equation in [7], Theorem 3, which concerns the characterization of the discounted control problem. Lemma 3.5. Let u be any precise Markov control and Lu be the corresponding generator. Let ϕ ∈ C 2 (Rd ) be a nonnegative solution of Lu ϕ − αϕ = g, d where g ∈ L∞ loc (R ). Let κ : R+ → R+ be any nondecreasing function such that kgkL∞ (BR ) ≤ κ(R) for all R > 0. Then for any R > 0 there exists a constant D(R) which depends on κ(4R), but not on u, or ϕ, such that   osc ϕ ≤ D(R) 1 + α inf ϕ . BR B4R Proof. Define g̃ := α(g − 2κ(4R)) and ϕ̃ := 2κ(4R) + αϕ. Then g̃ ≤ 0 in B4R and ϕ̃ solves Lu ϕ̃ − αϕ̃ = g̃ in B4R . Also kg̃kL∞ (B4R ) ≤ α(2κ(4R) + kgkL∞ (B4R ) ) ≤ 3α(2κ(4R) − kgkL∞ (B4R ) ) = 3 inf |g̃| B4R ≤ 3|B4R |−1 kg̃kL1 (B4R ) . Hence by [1], Theorem A.2.13, there exists a positive constant C̃H such that sup ϕ̃(x) ≤ C̃H inf ϕ̃(x), x∈B3R x∈B3R 32 A. ARAPOSTATHIS, A. BISWAS AND G. PANG implying that   α sup ϕ(x) ≤ C̃H 2κ(4R) + inf αϕ(x) . (3.42) x∈B3R x∈B3R We next consider the solution of Lu ψ = 0 in B3R , ψ=ϕ on ∂B3R . Then Lu (ϕ − ψ) = αϕ + g in B3R . If ϕ(x̂) = inf x∈B3R ϕ(x), then applying the maximum principle ([1], Theorem A.2.1, [18]) it follows from (3.42) that sup |ϕ − ψ| ≤ Ĉ(1 + αϕ(x̂)). (3.43) x∈B3R Again ψ attains its minimum at the boundary ([1], Theorem A.2.3, [18]). Therefore, ψ − ϕ(x̂) is a nonnegative function, and hence by the Harnack inequality, there exists a constant CH > 0 such that ψ(x) − ϕ(x̂) ≤ CH (ψ(x̂) − ϕ(x̂)) ≤ CH Ĉ(1 + αϕ(x̂)) ∀x ∈ B2R . Thus, combining the above display with (3.43) we obtain osc ϕ ≤ sup(ϕ − ψ) + sup ψ − ϕ(x̂) ≤ Ĉ(1 + CH )(1 + αϕ(x̂)). B2R B2R B2R This completes the proof.  Lemma 3.6. Let Vαε be as in Lemma 3.4. Then for any R > 0, there exists a constant kR > 0 such that osc Vαε ≤ kR for all α ∈ (0, 1] and ε ∈ [0, 1]. BR Proof. Recall that µu0 is the stationary probability distribution for the process under the control u0 ∈ USSM in Lemma 3.1. Since u0 is sub-optimal for the α-discounted criterion in (3.41), and Vαε is nonnegative, then for any ball BR , using Fubini’s theorem, we obtain Z ε Vαε (x)µu0 (dx) µu0 (BR ) inf Vα ≤ BR Rd ≤ Z Rd Eux0 Z ∞ 0 −αt e  rε (Xt , u0 (Xt )) dt µu0 (dx) 1 µu (rε ) α 0 1 ≤ (η + εk0 (1 + η)), α = ERGODIC CONTROL IN THE HALFIN–WHITT REGIME 33 where for the last inequality we used Lemma 3.2 and Theorem 3.1(a). Therefore, we have the estimate α inf Vαε ≤ BR η + εk0 (1 + η) . µu0 (BR ) The result then follows by Lemma 3.5.  We continue with the proof of Theorem 3.3. Proof of Theorem 3.3. Consider the function V̄αε := Vαε − Vαε (0). In view of Lemma 3.5 and Lemma 3.6, we see that V̄αε is locally bounded uniformly in α ∈ (0, 1] and ε ∈ (0, 1]. Therefore, by standard elliptic theory, V̄αε and its first- and second-order partial derivatives are uniformly bounded in Lp (B), for any p > 1, in any bounded ball B ⊂ Rd , that is, for some constant CB depending on B and p, kV̄αε kW2,p (B) ≤ CB ([18], Theorem 9.11, page 117). Therefore, we can extract a subsequence along which V̄αε converges. Then the result follows from Theorems 3.6.6, Lemma 3.6.9 and Theorem 3.6.10 in [1]. The proof of (3.36) follows from Lemma 3.5 and Lemma 3.6.  Remark 3.8. In the proof of the following lemma, and elsewhere in the paper, we use the fact that if U ⊂ USSM is a any set of controls such that the corresponding set {µv : v ∈ U } ⊂ M of invariant probability measures is tight then the map v 7→ πv from the closure of U to P(Rd × U) is continuous, and so is the map v 7→ µv . In fact, the latter is continuous under the total variation norm topology [1], Lemma 3.2.6. We also recall that G and M are closed and convex subsets of P(Rd × U) and P(Rd ). Therefore,{πv : v ∈ Ū } is compact in G. Note also that since U is compact, tightness of a set of invariant probability measures is equivalent to tightness of the corresponding set of ergodic occupation measures. Lemma 3.7. If {vε : ε ∈ (0, 1]} is a collection of measurable selectors from the minimizer of (3.33), then the corresponding invariant probability measures {µε : ε ∈ (0, 1]} are tight. Moreover, if vεn → v∗ along some subsequence εn ց 0, then the following hold: (a) µεn → µv∗ as εn ց 0, (b) Rv∗ is a stable Markov control, (c) Rd r(x, v∗ (x))µv∗ (dx) = limεց0 ̺ε = ̺∗ . Proof. By (3.25) and (3.31), the set of ergodic occupation measures corresponding to {vε : ε ∈ (0, 1]} is tight. By Remark 3.8, the same applies to the set {µε : ε ∈ (0, 1]}, and also part (a) holds. Part (b) follows from the equivalence of the existence of an invariant probability measure for a 34 A. ARAPOSTATHIS, A. BISWAS AND G. PANG controlled diffusion and the stability of the associated stationary Markov control (see [1], Theorem 2.6.10). Part (c) then follows since equality holds in (3.32).  We continue with the following lemma that asserts the continuity of the mean hitting time of a ball with respect to the stable Markov controls. Lemma 3.8. Let {vn : n ∈ N} ⊂ UβSM , for some β > 0, be a collection of Markov controls such that vn → v̂ in the topology of Markov controls as n → ∞. Let µn , µ̂ be the invariant probability measures corresponding to the controls vn , v̂, respectively. Then for any δ > 0, it holds that Evxn [τ̆δ ] −→ Ev̂x [τ̆δ ] ∀x ∈ Bδc . n→∞ Proof. Define H(x) := minu∈U h̃(x, u). It is easy to see that H is infcompact and locally Lipschitz. Therefore, by Theorem 3.1(d) we have sup µn (H) ≤ k0 (1 + β), n∈N and since µn → µ̂, we also have µ̂(H) ≤ k0 (1 + β). Then by [1], Lemma 3.3.4, we obtain  Z τ̆δ  Z τ̆δ sup Evxn H(Xs ) ds + Ev̂x (3.44) H(Xs ) ds < ∞. n∈N 0 0 Let R be a positive number greater than |x|. Then by (3.44), there exists a positive k such that  Z τ̆δ  Z τ̆δ 1 k Evx I{H>R} (Xs ) ds ≤ Evx H(Xs )I{H>R} (Xs ) ds ≤ R R 0 0 for v ∈ {{vn }, v̂}. From this assertion and (3.44), we see that  Z τ̆δ v sup Ex I{H>R} (Xs ) ds −→ 0. v∈{{vn },v̂} R→∞ 0 Therefore, in order to prove the lemma it is enough to show that, for any R > 0, we have Z τ̆δ  Z τ̆δ  vn v̂ Ex I{H≤R} (Xs ) ds −→ Ex I{H≤R} (Xs ) ds . 0 n→∞ 0 But this follows from [1], Lemma 2.6.13(iii).  ERGODIC CONTROL IN THE HALFIN–WHITT REGIME 35 Lemma 3.9. Let (V ε , ̺ε ) be as in Theorem 3.3, and vε satisfy (3.35). There exists a subsequence εn ց 0, such that V εn converges to some V∗ ∈ C 2 (Rd ), uniformly on compact sets, and V∗ satisfies min[Lu V∗ (x) + r(x, u)] = ̺∗ . (3.45) u∈U Also, any limit point v∗ (in the topology of Markov controls) of the set {vε }, as ε ց 0, satisfies (3.46) Lv∗ V∗ (x) + r(x, v∗ (x)) = ̺∗ a.e. in Rd . Moreover, V∗ admits the stochastic representation  Z τ̆δ v V∗ (x) = S inf Ex (r(Xs , v(Xs )) − ̺∗ ) ds + V∗ (Xτ̆δ ) v∈ (3.47) = Evx∗ β β>0 USM Z 0 τ̆δ 0  (r(Xs , v∗ (Xs )) − ̺∗ ) ds + V∗ (Xτ̆δ ) . It follows that V∗ is the unique limit point of V ε as ε ց 0. Proof. From (3.36), we see that the family {V ε : ε ∈ (0, 1]} is uniformly locally bounded. Hence, applying the theory of elliptic PDE, it follows that d {V ε : ε ∈ (0, 1]} is uniformly bounded in W2,p loc (R ) for p > d. Consequently, 1,γ ε {V : ε ∈ (0, 1]} is uniformly bounded in Cloc for some γ > 0. Therefore, along some subsequence εn ց 0, V εn → V∗ ∈ W2,p ∩ C 1,γ , as n → ∞, uniformly on compact sets. Also, limεց0 ̺ε = ̺∗ by Lemma 3.6(c). Therefore, passing to the limit we obtain the HJB equation in (3.45). It is straightforward to verify that (3.46) holds [1], Lemma 2.4.3. By Theorem 3.3(c), taking limits as ε ց 0, we obtain  Z τ̆δ Evx (3.48) V∗ (x) ≤ S inf (r(Xs , v(Xs )) − ̺∗ ) ds + V∗ (Xτ̆δ ) . v∈ β β>0 USM 0 Also by Theorem 3.3(b) we have the bound V ε (x) ≥ −̺ε Evxε [τ̆δ ] + inf V ε . Bδ Using Lemma 3.8 and taking limits as εn ց 0, we obtain the lower bound (3.49) V∗ (x) ≥ −̺∗ Evx∗ [τ̆δ ] + inf V∗ . Bδ By Lemma 3.7(c) and Theorem 3.1(d), v∗ ∈ USSM , and πv∗ (h̃) ≤ k0 (1 + ̺∗ ). Define Z τ̆δ  v∗ ϕ(x) := Ex (3.50) h̃(Xs , v∗ (Xs )) ds . 0 36 A. ARAPOSTATHIS, A. BISWAS AND G. PANG For |x| > δ, we have Evx∗ [I{τR <τ̆δ } ϕ(XτR )] = Evx∗  I{τR <τ̆δ } Z τ̆δ τR ∧τ̆δ  h̃(Xs , v∗ (Xs )) ds . Therefore, by the dominated convergence theorem and the fact that ϕ(x) < ∞ we obtain Evx∗ [ϕ(XτR )I{τR <τ̆δ } ] −→ 0. Rր∞ By (3.48) and (3.49), we have |V∗ | ∈ O(ϕ). Thus (3.49) and (3.50) imply that lim inf Evx∗ [V∗ (XτR )I{τR <τ̆δ } ] = 0, Rր∞ and thus lim inf Evx∗ [V∗ (XτR ∧τ̆δ )] = Evx∗ [V∗ (Xτ̆δ )]. (3.51) Rր∞ Applying Itô’s formula to (3.46), we obtain Z τ̆δ ∧τR  v∗ (3.52) V∗ (x) = Ex (r(Xs , v∗ (Xs )) − ̺∗ ) ds + V∗ (Xτ̆δ ∧τR ) . 0 Taking limits as R → ∞, and using the dominated convergence theorem, we obtain (3.47) from (3.48).  Recall the definition of o(·) from Section 1.3. We need the following lemma. Lemma 3.10. Let V∗ be as in Lemma 3.9. It holds that V∗− ∈ o(V). Proof. Let v∗ be as in Lemma 3.9. Applying Itô’s formula to (3.24) with u ≡ v∗ we obtain  Z τ̆δ v∗ Ex h(Xs , v∗ (Xs ))IHc (Xs , v∗ (Xs )) ds (3.53) 0 ≤ Evx∗ Z τ̆δ 0  r(Xs , v∗ (Xs ))IH (Xs , v∗ (Xs )) ds + Evx∗ [τ̆δ ] + V(x). Therefore, adding the term  Z τ̆δ v∗ Ex r(Xs , v∗ (Xs ))IH (Xs , v∗ (Xs )) ds − (1 + 2̺∗ )Evx∗ [τ̆δ ] 0 ERGODIC CONTROL IN THE HALFIN–WHITT REGIME 37 to both sides of (3.53) and using the stochastic representation of V∗ we obtain Z τ̆δ  −1 v∗ F (x) := 2k0 Ex h̃(Xs , v∗ (Xs )) ds − 2(1 + ̺∗ )Evx∗ [τ̆δ ] (3.54) 0 ≤ 2V∗ (x) + V(x) − 2 inf V∗ . Bδ From the stochastic representation of V∗ we have V∗− (x) ≤ ̺∗ Evx∗ [τ̆δ ]−inf Bδ V∗ . For any R > δ, we have Z τ̆δ    v∗ c (3.55) Ex h̃ Ex [τ̆R ] ∀x ∈ BR . h̃(Xs , v∗ (Xs )) ds ≥ inf c BR ×U 0 It is also straightforward to show that lim|x|→∞ EExx[τ̆[τ̆Rδ ]] = 1. Therefore, since h̃ is inf-compact, it follows by (3.54) and (3.55) that the map x 7→ Evx∗ [τ̆δ ] is in o(F ), which implies that V∗− ∈ o(F ). On the other hand, by (3.54) we obtain F (x) ≤ V(x) − 2 supBδ V∗ for all x such that V∗ (x) ≤ 0, which implies that the restriction of F to the support of V∗− is in O(V). It follows that V∗− ∈ o(V).  We next prove Theorem 3.4. Proof of Theorem 3.4. Part (a) is contained in Lemma 3.9. To prove part (b), let v̄ be any control satisfying (3.38). By Lemma 3.10 the map V + 2V∗ is inf-compact and by Theorem 3.4 and (3.24) it satisfies Lv̄ (V + 2V∗ )(x) ≤ 1 + 2̺∗ − r(x, v̄(x)) − h(x, v̄(x))IHc (x, v̄(x)) ≤ 2 + 2̺∗ − 2k0−1 h̃(x, v̄(x)) ∀x ∈ Rd . This implies that v̄ ∈ USSM . Applying Itô’s formula, we obtain  Z T 1 v̄ lim sup Ex (3.56) h̃(Xs , v̄(Xs )) ds ≤ k0 (1 + ̺∗ ). T →∞ T 0 Therefore, πv̄ (h̃) < ∞. By (3.24), we have  Z t v̄ v̄ Ex [V(Xt )] ≤ V(x) + t + Ex r(Xs , v̄(Xs )) ds , 0 and since r ≤ h̃, this implies by (3.56) that 1 (3.57) lim sup Ev̄x [V(XT )] ≤ 1 + k0 (1 + ̺∗ ). T →∞ T Since V∗− ∈ o(V), it follows by (3.57) that 1 lim sup Ev̄x [V∗− (XT )] = 0. T →∞ T 38 A. ARAPOSTATHIS, A. BISWAS AND G. PANG Therefore, by Itô’s formula, we deduce from (3.37) that  Z T 1 v̄ (3.58) r(Xs , v̄(Xs )) ds ≤ ̺∗ . lim sup Ex T →∞ T 0 On the other hand, since the only limit point of the mean empirical measures v̄ , as t → ∞, is π , and π (r) = ̺ , then in view of Remark 3.6, we obtain ζx,t v̄ v̄ ∗ v̄ (r) ≥ ̺ . This proves that equality holds in (3.58) and that lim inf t→∞ ζx,t ∗ the “lim sup” may be replaced with “lim.” Conversely, suppose v ∈ USM is optimal but does not satisfy (3.38). Then there exists R > 0 and a nontrivial nonnegative f ∈ L∞ (BR ) such that fε (x) := IBR (x)(Lv V ε (x) + rε (x, v(x)) − ̺ε ) converges to f , weakly in L1 (BR ), along some subsequence ε ց 0. By applying Itô’s formula to (3.33), we obtain  Z T ∧τR 1 v 1 v ε ε (E [V (XT ∧τR )] − V (x)) + Ex rε (Xs , v(Xs )) ds T x T 0 (3.59) Z T ∧τR  1 v fε (Xs , v(Xs )) ds . ≥ ̺ε + Ex T 0 Define, for some δ > 0, G(x) := Evx Z 0 τ̆δ  rε (Xs , v(Xs )) ds . Since V is bounded from below, by Theorem 3.3(c) we have V ε ∈ O(G). Invoking [1], Corollary 3.7.3, we obtain ε 1 v ε Ex [V (XT )] = 0, T →∞ T lim and lim Evx [V ε (XT ∧τR )] = Evx [V ε (XT )]. R→∞ Therefore, taking limits in (3.59), first as R ր ∞, and then as T → ∞, we obtain (3.60) πv (rε ) ≥ ̺ε + πv (fε ). Taking limits as ε ց 0 in (3.60), since µv has a strictly positive density in BR , we obtain πv (r) ≥ ̺∗ + πv (f ) > ̺∗ , which is a contradiction. This completes the proof of part (b). ERGODIC CONTROL IN THE HALFIN–WHITT REGIME 39 The first equality (3.39) follows by Lemma 3.9, taking limits as δ ց 0. To show that the second equality holds for any optimal control, suppose v̄ satisfies (3.38). By (3.24) we have, for δ > 0 and |x| > δ,  Z τR ∧τ̆δ (1 + r(Xs , v̄(Xs ))) ds . Ev̄x [V(XτR )I{τR <τ̆δ } ] ≤ V(x) + sup V − + Ev̄x Bδ 0 It follows that (see [1], Lemma 3.3.4) lim sup Ev̄x [V(XτR )I{τR <τ̆δ } ] < ∞, R→∞ and since V∗− ∈ o(V) we must have lim sup Ev̄x [V∗− (XτR )I{τR <τ̆δ } ] = 0. R→∞ By the first equality in (3.47), we obtain V∗+ ∈ O(ϕ), with ϕ as defined in (3.50) with v∗ replaced by v̄. Thus, in analogy to (3.51), we obtain lim inf Ev̄x [V∗ (XτR ∧τ̆δ )] = Ev̄x [V∗ (Xτ̆δ )]. Rր∞ The rest follows as in the proof of Lemma 3.9 via (3.52). We next prove part (d). We assume that U is a convex set and that c(x, u, p) := {b(x, u) · p + r(x, u)} is strictly convex in u if it is not identically a constant for fixed x and p. We fix some point ū ∈ U. Define B := {x ∈ Rd : c(x, ·, p) = c(x, ū, p) for all p}. It is easy to see that on B both b and r do not depend on u. It is also easy to check that B is a closed set. Let (V∗ , v∗ ) be the limit of (V ε , vε ), where V∗ is the solution to (3.37) and v∗ is the corresponding limit of vε . We have already shown that v∗ is a stable Markov control. We next show that it is, in fact, a precise Markov control. By our assumption, vε is the unique minimizing selector in (3.34) and, moreover, vε is continuous in x. By the definition of rε it is clear that the restriction of vε to B does not depend on ε. Let vε (x) = v ′ (x) on B. Using the strict convexity property of c(x, ·, ∇V∗ ) it is easy to verify that vε converges to the unique minimizer of (3.37) on B c . In fact, since B c is open, then for any sequence xε → x ∈ B c it holds that vε (xε ) → v∗ (x). This follows from the definition of the minimizer and the uniform convergence of ∇V ε to ∇V∗ . Therefore, we see that v∗ is a precise Markov control, v∗ = v ′ on B, and vε → v∗ pointwise as ε → 0. It is also easy to check that pointwise convergence implies convergence in the topology of Markov controls.  We now embark on the proof of Theorem 3.5. 40 A. ARAPOSTATHIS, A. BISWAS AND G. PANG Proof of Theorem 3.5. The hypothesis that V̂ − ∈ o(V) implies that the map V + 2V̂ is inf-compact. Also by (3.24) and (3.40), it satisfies Lv̂ (V + 2V̂ )(x) ≤ 1 + 2ˆ ̺ − r(x, v̂(x)) − h(x, v̂(x))IHc (x, v̂(x)) ≤ 2 + 2ˆ ̺ − 2k0−1 h̃(x, v̂(x)) ∀x ∈ Rd . R Therefore, h̃(x, v̂(x)) dπv̂ < ∞ from which it follows that ̺v̂ < ∞. This proves part (a). By (3.24), we have  Z t v̂ v̂ r(Xs , v̂(Xs )) ds , Ex [V(Xt )] ≤ V(x) + t + Ex 0 and since ̺v̂ < ∞, this implies that 1 lim sup Ev̂x [V(XT )] ≤ 1 + ̺v̂ . (3.61) T →∞ T Since V̂ − ∈ o(V), it follows by (3.61) that lim sup T →∞ 1 v̂ − E [V̂ (XT )] = 0. T x Therefore, by Itô’s formula, we deduce from (3.40) that  Z T  1 v̂ 1 v̂ + E [V̂ (XT )] + Ex r(Xs , v̂(Xs )) ds = ̺ˆ. lim sup T x T T →∞ 0 This implies that ̺v̂ ≤ ̺ˆ and since by hypothesis ̺ˆ ≤ ̺∗ we must have ̺ˆ = ̺∗ . Again by (3.24), we have Z τR ∧τ̆δ  v̂ − v̂ Ex [V(XτR )I{τR <τ̆δ } ] ≤ V(x) + sup V + Ex (1 + r(Xs , v̂(Xs ))) ds . Bδ 0 It follows by [1], Lemma 3.3.4, that lim sup Ev̂x [V(XτR )I{τR <τ̆δ } ] < ∞, R→∞ and since V̂ − ∈ o(V) we must have (3.62) lim sup Ev̂x [V̂ − (XτR )I{τR <τ̆δ } ] = 0. R→∞ Using (3.62) and following the steps in the proof of the second equality in (3.47), we obtain Z τ̆δ  v̂ V̂ (x) ≥ Ex (r(Xs , v̂(Xs )) − ̺∗ ) ds + inf V̂ Bδ 0 ≥ V∗ (x) − sup V∗ + inf V̂ . Bδ Bδ 41 ERGODIC CONTROL IN THE HALFIN–WHITT REGIME Taking limits as δ ց 0, we have V∗ ≤ V̂ . Since Lv̂ (V∗ − V̂ ) ≥ 0 and V∗ (0) = V̂ (0), we must have V̂ = V∗ on Rd , and the proof of part (b) is complete. R To prove part (c) note that by part (a) we haveR ̺v̂ < ∞. Therefore, h̃ dπv̂ ≤ ∞ by Theorem 3.1(a), which implies that |V̂ | dµv̂ ≤ ∞ by the hypothesis. Therefore, Ev̂x (|V̂ (Xt )|) converges as t → ∞ by [23], Proposition 2.6, which of course implies that 1t Ev̂x (|V̂ (Xt )|) tends to 0 as t → ∞. Similarly, we deduce that 1t Evx∗ (|V̂ (Xt )|) as t → ∞. Applying Itô’s formula to (3.40), with u ≡ v∗ , we obtain ̺ˆ ≤ ̺∗ . Another application with u ≡ v̂ results in ̺ˆ = ̺v̂ . Therefore, ̺ˆ = ̺∗ . The result then follows by part (b).  We finish this section with the proof of Theorem 3.6. Proof of Theorem 3.6. We first show that limαց0 αVα (0) = ̺∗ . Let Ṽ(t, x) := e−αt V(x), and τn (t) := τn ∧ t. Applying Itô’s formula to (3.24), we obtain  Z τn (t) U U Ex [Ṽ(τn (t), Xτn (t) )] ≤ V(x) − Ex αṼ(s, Xs ) ds 0 It follows that Z U Ex (3.63) τn (t) −αs e 0 ≤ + EU x Z + EU x Z τn (t) −αs e 0 τn (t) −αs e 0 (1 − h(Xs , Us ))IHc (Xs , Us ) ds  (1 + r(Xs , Us ))IH (Xs , Us ) ds . h(Xs , Us )IHc (Xs , Us ) ds 1 + V(x) + EU x α Z τn (t) 0    e−αs r(Xs , Us )IH (Xs , Us ) ds . Taking limits first as n ր ∞ and then as t ր ∞ in (3.63), and evaluating U at an optimal α-discounted control vα∗ , relative to r we obtain the estimate, using also (3.23),  Z ∞ 2 ∗ e−αs h̃(Xs , vα∗ (Xs )) ds ≤ + V(x) + 2Vα (x). (3.64) 2k0−1 Exvα α 0 By (3.23) and (3.64), it follows that Z ∗ ε vα Vα (x) ≤ Vα (x) ≤ Ex 0 ∞ −αs e rε (Xs , vα∗ (Xs )) ds ≤ Vα (x) + εk0 (α−1 + V(x) + Vα (x)).  42 A. ARAPOSTATHIS, A. BISWAS AND G. PANG Multiplying by α and taking limits as α ց 0 we obtain lim sup αVα (0) ≤ ̺ε ≤ (1 + εk0 ) lim sup αVα (0) + εk0 . αց0 αց0 The same inequalities hold for the “lim inf.” Therefore, limαց0 αVα (0) = ̺∗ . Let Ṽ := lim (Vα − Vα (0)). αց0 (Note that a similar result as Lemma 3.5 holds.) Then Ṽ satisfies  Z τ̆δ [ β v Ṽ (x) ≤ lim Ex (r(Xs , v(Xs )) − ̺∗ ) ds ∀v ∈ USM . δց0 0 β>0 This can be obtained without the near-monotone assumption on the running cost; see, for example, [1], Lemma 3.6.9 or Lemma 3.7.8. It follows from (3.39) that Ṽ ≤ V∗ . On the other hand, since Lv∗ (Ṽ − V∗ ) ≥ 0, and Ṽ (0) = V∗ (0), we must have Ṽ = V∗ by the strong maximum principle.  4. Approximation via spatial truncations. We introduce an approximation technique which is in turn used to prove the asymptotic convergence results in Section 5. Let v0 ∈ USSM be any control such that πv0 (r) < ∞. We fix the control v0 on the complement of the ball B̄l and leave the parameter u free inside. In other words, for each l ∈ N we define  b(x, u), if (x, u) ∈ B̄l × U, bl (x, u) := b(x, v0 (x)), otherwise,  r(x, u), if (x, u) ∈ B̄l × U, rl (x, u) := r(x, v0 (x)), otherwise. We consider the family of controlled diffusions, parameterized by l ∈ N, given by (4.1) dXt = bl (Xt , Ut ) dt + σ(Xt ) dWt , with associated running costs rl (x, u). We denote by USM (l, v0 ) the subset of USM consisting of those controls v which agree with v0 on B̄lc . Let η0 := πv0 (r). It is well known that there exists a nonnegative solution ϕ0 ∈ d W2,p loc (R ), for any p > d, to the Poisson equation (see [1], Lemma 3.7.8(ii)) Lv0 ϕ0 (x) = η0 − h̃(x, v0 (x)) x ∈ Rd , which is inf-compact, and satisfies, for all δ > 0,  Z τ̆δ v0 ϕ0 (x) = Ex (h̃(Xs , v0 (Xs )) − η0 ) ds + ϕ0 (Xτ̆δ ) 0 ∀x ∈ Rd . ERGODIC CONTROL IN THE HALFIN–WHITT REGIME 43 We recall the Lyapunov function V from Assumption 3.1. We have the following theorem. Theorem 4.1. Let Assumptions 3.1 and 3.2 hold. Then for each l ∈ N d l there exists a solution V l in W2,p loc (R ), for any p > d, with V (0) = 0, of the HJB equation min[Lul V l (x) + rl (x, u)] = ̺l , (4.2) u∈U Lul where is the elliptic differential operator corresponding to the diffusion in (4.1). Moreover, the following hold: (i) (ii) 2ϕ0 (x) (iii) (iv) ̺l is nonincreasing in l; there exists a constant C0 , independent of l, such that V l (x) ≤ C0 + for all l ∈ N; (V l )− ∈ o(V + ϕ0 ) uniformly over l ∈ N; the restriction of V l on Bl is in C 2 . Proof. As earlier, we can show that  Z ∞ l U −αs Vα (x) := inf Ex e rl (Xs , Us ) ds U ∈U 0 is the minimal nonnegative solution to (4.3) min[Lul Vαl (x) + rl (x, u)] = αVαl (x), u∈U d and Vαl ∈ W2,p loc (R ), p > d. Moreover, any measurable selector from the minimizer in (4.3) is an optimal control. A similar estimate as in Lemma 3.5 holds and, therefore, there exists a subsequence {αn }, along which Vαl n (x) − Vαl n (0) d l converges to V l in W2,p loc (R ), p > d, and αn Vαn (0) → ̺l as αn ց 0, and (V l , ̺l ) satisfies (4.2) (see also [1], Lemma 3.7.8). To show that πvl (r) = ̺l , v l is a minimizing selector in (4.2), we use the following argument. Since πv0 (r) < ∞, we claim that there exists a nonnegative, inf-compact function g ∈ C(Rd ) such that πv0 (g · (1 + r)) < ∞. Indeed, this is true since integrability and uniform integrability of a function under any given measure are equivalent (see also the proof of [1], Lemma 3.7.2). Since every control in USM (l, v0 ) agrees with v0 on Blc , then for any x0 ∈ B̄lc the map  Z τ̆l v v 7→ Ex0 g(Xs )(1 + r(Xs , v(Xs ))) ds 0 is constant on USM (l, v0 ). By the equivalence of (i) and (iii) in Lemma 3.3.4 of [1], this implies that sup v∈USM (l,v0 ) πv (g · (1 + r)) < ∞ ∀l ∈ N, 44 A. ARAPOSTATHIS, A. BISWAS AND G. PANG and thus r is uniformly integrable with respect to the family {πv : v ∈ USM (l, v0 )} for any l ∈ N. It then follows by [1], Theorem 3.7.11, that (4.4) ̺l = inf v∈USM (l,v0 ) πv (r), l ∈ N. This yields part (i). Moreover, in view of Lemmas 3.5 and 3.6, we deduce that for any δ > 0 it holds that supBδ |V l | ≤ κδ , where κδ is a constant independent of l ∈ N. It is also evident by (4.4) that ̺l is decreasing in l and ̺l ≤ η0 for all l ∈ N. Fix δ such that minu∈U h̃(x, u) ≥ 2η0 on Bδc . Since ϕ0 is nonnegative, we obtain  Z τ̆δ v0 (4.5) ∀x ∈ Rd . (h̃(Xs , v0 (Xs )) − η0 ) ds ≤ ϕ0 (x) Ex 0 Using an analogous argument as the one used in the proof of [1], Lemma 3.7.8, we have Z τ̆δ  l v (4.6) V (x) ≤ Ex (rl (Xs , v(Xs )) − ̺l ) ds + κδ ∀v ∈ USM (l, v0 ). 0 Thus, by (4.5) and (4.6), and since by the choice of δ > 0, it holds that r ≤ h̃ ≤ 2(h̃ − η0 ) on Bδc , we obtain Z τ̆δ  V l (x) ≤ Evx0 2(h̃(Xs , v0 (Xs )) − η0 ) ds + κδ 0 (4.7) ≤ κδ + 2ϕ0 (x) ∀x ∈ Rd . This proves part (ii). Now fix l ∈ N. Let vαl be a minimizing selector of (4.3). Note then that l vα ∈ USM (l, v0 ). Therefore, vαl is a stable Markov control. Let vαl n → v l in the topology of Markov controls along the same subsequence as above. Then it is evident that v l ∈ USM (l, v0 ). Also from Lemma 3.8, we have vl l Exαn [τ̆δ ] −→ Evx [τ̆δ ] ∀x ∈ Bδc , ∀δ > 0. αn ց0 Using [1], Lemma 3.7.8, we obtain the lower bound l V l (x) ≥ −̺l Evx [τ̆δ ] − κδ . (4.8) By [1], Theorem 3.7.12(i) (see also (3.7.50) in [1]), it holds that  Z τ̆δ l vl l l V (x) = Ex (rl (Xs , v (Xs )) − ̺l ) ds + V (Xτ̆δ ) 0 (4.9) ≥ l Evx Z 0 τ̆δ l  l rl (Xs , v (Xs )) ds − ̺l Evx [τ̆δ ] − κδ ∀x ∈ Blc . ERGODIC CONTROL IN THE HALFIN–WHITT REGIME 45 By (3.23), we have 2k0−1 h̃(x, u)IH (x, u) ≤ 1 + r(x, u)IH (x, u). Therefore, using the preceding inequality and (4.9), we obtain l (4.10) V l (x) + (1 + ̺l )Evx [τ̆δ ] + κδ Z τ̆δ  2 vl l l ≥ Ex h̃(Xs , v (Xs ))IH (Xs , v (Xs )) ds . k0 0 By (3.24), (4.9) and the fact that V is nonnegative, we have Z τ̆δ  2 vl l Ex h̃(Xs , v l (Xs ))IHc (Xs , v l (Xs )) ds − V(x) − Evx [τ̆δ ] k0 0 Z τ̆δ  vl l l ≤ Ex (4.11) r(Xs , v (Xs ))IH (Xs , v (Xs )) ds 0 l l ≤ V (x) + ̺l Evx [τ̆δ ] + κδ . Combining (4.7), (4.10) and (4.11), we obtain  Z τ̆δ l l h̃(Xs , v l (Xs )) ds ≤ k0 (1 + ̺l )Evx [τ̆δ ] Evx 0 + k0 V(x) + 2k0 (ϕ0 (x) + κδ ) 2 for all l ∈ N. As earlier, using the inf-compact property of h̃ and the fact that ̺l ≤ η0 is bounded, we can choose δ large enough such that Z τ̆δ  vl vl l (4.12) η0 Ex [τ̆δ ] ≤ Ex h̃(Xs , v (Xs )) ds ≤ k0 V(x) + 4k0 (ϕ0 (x) + κδ ) 0 for all l ∈ N. Since h̃ is inf-compact, part (iii) follows by (4.8) and (4.12). Part (iv) is clear from regularity theory of elliptic PDE [18], Theorem 9.19, page 243.  Similar to Theorem 3.3, we can show that oscillations of {V l } are uniformly bounded on compacts. Therefore, if we let l → ∞ we obtain a HJB equation (4.13) min[Lu V̂ (x) + r(x, u)] = ̺ˆ, u∈U with V̂ ∈ C 2 (Rd ) and liml→∞ ̺l = ̺ˆ. By Theorem 4.1, we have the bound (4.14) V̂ (x) ≤ C0 + 2ϕ0 (x), 46 A. ARAPOSTATHIS, A. BISWAS AND G. PANG for some positive constant C0 . This of course, implies that V̂ + (x) ≤ C0 + 2ϕ0 (x). Moreover, it is straightforward to show that for any v ∈ USSM with ̺v < ∞, we have 1 lim sup Evx [V(Xt )] < ∞. t→∞ t Therefore, if in addition, we have 1 lim sup Evx [ϕ0 (Xt )] < ∞, t→∞ t then it follows by Theorem 4.1(iii) that (4.15) 1 lim sup V̂ − (Xt ) −→ 0. t→∞ t→∞ t Theorem 4.2. Suppose that ϕ0 ∈ O(minu∈U h̃(·, u)). Then, under the assumptions of Theorem 4.1, we have liml→∞ ̺l = ̺ˆ = ̺∗ , and V̂ = V∗ . Moreover, V∗ ∈ O(ϕ0 ). Proof. Let {v̂l } be any sequence of measurable selectors from the minimizer of (4.2) and {πl } the corresponding sequence of ergodic occupation measures. Since by Theorem 3.1 {πl } is tight, then by Remark 3.8 if v̂ is a limit point of a subsequence {v̂l }, which we also denote by {v̂l }, then π̂ = πv̂ is the corresponding limit point of {πl }. Therefore, by the lower semi-continuity of π → π(r) we have ̺ˆ = lim πl (r) ≥ π̂(r) = ̺v̂ . l→∞ It also holds that (4.16) Lv̂ V̂ (x) + r(x, v̂(x)) = ̺ˆ, a.s. By (4.15), we have lim inf T →∞ 1 v̂ E [V̂ (XT )] = 0, T x and hence applying Itô’s rule on (4.16) we obtain ̺v̂ ≤ ̺ˆ. On the other hand, if v∗ is an optimal stationary Markov control, then by the hypothesis ϕ0 ∈ O(h̃), the fact that πv∗ (h̃) < ∞, (4.14) and [23], Proposition 2.6, we deduce that Evx∗ [V̂ + (Xt )] converges as t → ∞, which of course together with (4.15) implies that 1t Ev̂x [V̂ (Xt )] tends to 0 as t → ∞. Therefore, evaluating (4.13) at v∗ and applying Itô’s rule we obtain ̺v∗ ≥ ̺ˆ. Combining the two estimates, we have ̺v̂ ≤ ̺ˆ ≤ ̺∗ , and thus equality must hold. Here, we have used the fact that there exists an optimal Markov control for r by Theorem 3.4. ERGODIC CONTROL IN THE HALFIN–WHITT REGIME Next, we use the stochastic representation in (4.9)  Z τ̆δ l v̂l l (4.17) V (x) = Ex (r(Xs , v̂l (Xs )) − ̺l ) ds + V (Xτ̆δ ) , 0 47 x ∈ Bδc . ̺v Fix any x ∈ Bδc . Since USM0 is compact, it follows that for each δ and R with ̺v 0 < δ < R, the map Fδ,R (v) : USM0 → R+ defined by  Z τ̆δ ∧τR v Fδ,R (v) := Ex r(Xs , v(Xs )) ds 0 is continuous. Therefore, the map F̄δ := limRր∞ Fδ,R is lower semi-continuous. It follows that  Z τ̆δ  Z τ̆δ v̂ v̂l r(Xs , v̂(Xs )) ds ≤ lim Ex (4.18) Ex r(Xs , v̂l (Xs )) ds . l→∞ 0 0 On the other hand, since h̃ is inf-compact, it follows by (4.12) that τ̆δ is uniformly integrable with respect to the measures {Pv̂xl }. Therefore, as also shown in Lemma 3.8, we have lim Ev̂xl [τ̆δ ] = Ev̂x [τ̆δ ]. (4.19) l→∞ Since V l → V̂ , uniformly on compact sets, and ̺l → ̺∗ , as l → ∞, it follows by (4.17)–(4.19) that Z τ̆δ  V̂ (x) ≥ Ev̂x (r(Xs , v̂(Xs )) − ̺∗ ) ds + V̂ (Xτ̆δ ) , x ∈ Bδc . 0 Therefore, by Theorem 3.4(b), for any δ > 0 and x ∈ Bδc we obtain Z τ̆δ  v̂ V∗ (x) ≤ Ex (r(Xs , v̂(Xs )) − ̺∗ ) ds + V∗ (Xτ̆δ ) 0 ≤ V̂ (x) + Ev̂x [V ∗ (Xτ̆δ )] − Ev̂x [V̂ (Xτ̆δ )], and taking limits as δ ց 0, using the fact that V̂ (0) = V∗ (0) = 0, we obtain V∗ ≤ V̂ on Rd . Since Lv̂ (V∗ − V̂ ) ≥ 0, we must have V∗ = V̂ . By Theorem 4.1(ii), we have V∗ ∈ O(ϕ0 ).  Remark 4.1. It can be seen from the proof of Theorem 4.2 that the assumption ϕ0 ∈ O(h̃) can be replaced by the weaker hypothesis that 1 v∗ T Ex [ϕ0 (XT )] → 0 as T → ∞. It is easy to see that if one replaces rl by  1   r(x, u) + f (u), for x ∈ B̄l , l rl (x, u) =   r(x, v0 (x)) + 1 f (v0 (x)), otherwise, l Remark 4.2. 48 A. ARAPOSTATHIS, A. BISWAS AND G. PANG for some positive valued continuous function f , the same conclusion of Theorem 4.2 holds. If we consider the controlled dynamics given by (3.20), with running cost as in (3.11), then there exists a function V ∼ |x|m satisfying (3.6). This fact is proved in Proposition 3.1. There also exists a Lyapunov function V0 ∈ O(|x|m ), satisfying the assumption in Theorem 4.2, relative to any control v0 with πv0 (h̃) < ∞, where h̃ is selected as in Remark 3.5. Indeed, in order to construct V0 we recall the function ψ in (3.21). Let V0 ∈ C 2 (Rd ) be any function such that V0 = ψ m/2 on the complement of the unit ball centered at the origin. Observe that for some positive constants κ1 and κ2 it holds that κ1 |x|2 ≤ ψ(x) ≤ κ2 |x|2 . Then a straightforward calculation from (3.22) shows that (3.8) holds with the above choice of V0 . By the stochastic representation of ϕ0 , it follows that ϕ0 ∈ O(V0 ). We have proved the following corollary. Corollary 4.1. For the queueing diffusion model with controlled dynamics given by (3.20), and running cost given by (3.11), there exists a solution (up to an additive constant) to the associated HJB in the class of functions in C 2 (Rd ) whose positive part grows no faster than |x|m and whose negative part is in o(|x|m ). We conclude this section with the following remark. Remark 4.3. Comparing the approximation technique introduced in this section with that in Section 3, we see that the spatial truncation technique relies on more restrictive assumption on the Lyapunov function V0 and the running cost function (Theorem 4.2). In fact, the growth of h̃ also restricts the growth of r by (3.23). Therefore, the class of ergodic diffusion control problems considered in this section is more restrictive. For example, if the running cost r satisfies (3.11) and h̃ ∼ |x|m , then it is not obvious that one can obtain a Lyapunov function V0 with growth at most of order |x|m . For instance, if the drift has strictly sub-linear growth, then it is expected that the Lyapunov function should have growth larger than |x|m . Therefore, the class of problems considered in Section 3 is larger than those considered in this section. 5. Asymptotic convergence. In this section, we prove that the value of the ergodic control problem corresponding to the multi-class M/M/N + M queueing network asymptotically converges to ̺∗ , the value of the ergodic control for the controlled diffusion. ERGODIC CONTROL IN THE HALFIN–WHITT REGIME 49 Recall the diffusion-scaled processes X̂ n , Q̂n and Ẑ n defined in (2.4), and from (2.5) and (2.6) that Z t Z t n n n n n n X̂i (t) = X̂i (0) + ℓi t − µi Ẑi (s) ds − γi Q̂ni (s) ds (5.1) 0 0 n n n + M̂A,i (t) − M̂S,i (t) − M̂R,i (t), n (t), M̂ n (t) and M̂ n (t), i = 1, . . . , d, as defined in (2.6), are where M̂A,i S,i R,i square integrable martingales w.r.t. the filtration {Ftn } with quadratic variations λni t, n Z µni t n n hM̂S,i i(t) = Z (s) ds, n 0 i Z γin t n n Q (s) ds. hM̂R,i i(t) = n 0 i n hM̂A,i i(t) = 5.1. The lower bound. In this section, we prove Theorem 2.1. Proof of Theorem 2.1. Recall the definition of V̂ n in (2.10), and consider a sequence such that supn V̂ n (X̂ n (0)) < ∞. Let ϕ ∈ C 2 (Rd ) be any function satisfying ϕ(x) := |x|m for |x| ≥ 1. As defined in Section 1.3, ∆X(t) denotes the jump of the process X at time t. Applying Itô’s formula on ϕ (see, e.g., [24], Theorem 26.7), we obtain from (5.1) that  Z t Θn1 (X̂1n (s), Ẑ1n (s))ϕ′ (X̂1n (s)) ds E[ϕ(X̂1n (t))] = E[ϕ(X̂1n (0))] + E 0 +E Z 0 +E t Θn2 (X̂1n (s), Ẑ1n (s))ϕ′′ (X̂1n (s)) ds X s≤t  ∆ϕ(X̂1n (s)) − ϕ′ (X̂1n (s−)) · ∆X̂1n (s)  1 ′′ n n n − ϕ (X̂ (s−))∆X̂1 (s)∆X̂1 (s) , 2 where Θn1 (x, z) := ℓn1 − µn1 z − γ1n (x − z),   µn1 z + γ1n (x − z) λn1 1 n n √ + µ ρ1 + . Θ2 (x, z) := 2 1 n n 50 A. ARAPOSTATHIS, A. BISWAS AND G. PANG Since {ℓn1 } is a bounded sequence, it is easy to show that for all n there exist positive constants κi , i = 1, 2, independent of n, such that Θn1 (x, z)ϕ′ (x) ≤ κ1 (1 + |(e · x)+ |m ) − κ2 |x|m , κ2 Θn2 (x, z)ϕ′′ (x) ≤ κ1 (1 + |(e · x)+ |m ) + |x|m , 4 provided that x − z ≤ (e · x)+ and √zn ≤ 1. We next compute the terms corresponding to the jumps. For that, first we see that the jump size is of order √1n . We can also find a positive constant κ3 such that sup |ϕ′′ (y)| ≤ κ3 (1 + |x|m−2 ) ∀x ∈ Rd . |y−x|≤1 Using Taylor’s approximation, we obtain the inequality ∆ϕ(X̂1n (s)) − ϕ′ (X̂1n (s−)) · ∆X̂1n (s) ≤ 1 sup |ϕ′′ (y)|[∆(X̂1n (s))]2 . 2 |y−X̂ n (s−)|≤1 1 Hence, combining the above facts we obtain X E ∆ϕ(X̂1n (s)) − ϕ′ (X̂1n (s−)) · ∆X̂1n (s) s≤t (5.2)  1 ′′ n n n − ϕ (X̂1 (s−))∆X̂1 (s)∆X̂1 (s) 2 X ≤E κ3 (1 + |X̂1n (s−)|m−2 )(∆(X̂1n (s)))2 s≤t t   λn1 µn1 Z1n (s) γ1n Qn1 (s) = κ3 E + + ds n n n 0 Z t    κ2 ≤E κ4 + |X̂1n (s)|m + κ5 ((e · X̂ n (s))+ )m ds , 4 0 Z  (1 + |X̂1n (s)|m−2 ) for some suitable positive constants κ4 and κ5 , independent of n, where in the second inequality we use the fact that the optional martingale [X̂1n ] is the sum of the squares of the jumps, and that [X̂1n ] − hX̂1n i is a martingale. Therefore, for some positive constants C1 and C2 it holds that 0 ≤ E[ϕ(X̂1n (t))] (5.3) ≤ E[ϕ(X̂1n (0))] + C1 t − + C2 E Z t 0 κ2 E 2 Z 0 t |X̂1n (s)|m ds  ((e · X̂ (s)) ) ds . n + m  ERGODIC CONTROL IN THE HALFIN–WHITT REGIME 51 By (2.8), we have r(Q̂n (s)) ≥ c1 ((e · X̂ n (s))+ )m , dm which, combined with the assumption that supn V̂ n (X̂ n (0)) < ∞, implies that  Z T 1 + m n ((e · X̂ (s)) ) ds < ∞. sup lim sup E n T →∞ T 0 In turn, from (5.3) we obtain sup lim sup n T →∞ 1 E T Z T 0  |X̂1n (s)|m ds < ∞. Repeating the same argument for coordinates i = 2, . . . , d, we obtain  Z T 1 (5.4) |X̂ n (s)|m ds < ∞. sup lim sup E n T →∞ T 0 We introduce the process  n  X̂i (t) − Ẑin (t) , n n + Ui (t) :=  (e · X̂ (t)) ed , i = 1, . . . , d, if (e · X̂ n (t))+ > 0, otherwise. Since Z n is work-conserving, it follows that U n takes values in S, and Uin (t) represents the fraction of class i customers in queue. Define the mean empirical measures  Z T 1 n n n ΦT (A × B) := E IA×B (X̂ (s), U (s)) ds T 0 for Borel sets A ⊂ Rd and B ⊂ S. From (5.4), we see that the family {ΦnT : T > 0, n ≥ 1} is tight. Hence, for any sequence Tk → ∞, there exists a subsequence, also denoted by Tk , such that ΦnTk → π n , as k → ∞. It is evident that {π n : n ≥ 1} is tight. Let π n → π along some subsequence, with π ∈ P(Rd × S). Therefore, it is not hard to show that Z n n r̃(x, u)π(dx, du), lim V̂ (X̂ (0)) ≥ n→∞ Rd ×U where, as defined earlier, r̃(x, u) = r((e · x)+ u). To complete the proof of the theorem, we only need to show that π is an ergodic occupation measure for the diffusion. For that, consider f ∈ Cc∞ (Rd ). Recall that [X̂in , X̂jn ] = 0 52 A. ARAPOSTATHIS, A. BISWAS AND G. PANG for i 6= j [30], Lemmas 9.2 and 9.3. Therefore, using Itô’s formula and the definition of ΦnT , we obtain 1 E[f (X̂ n (T ))] T 1 = E[f (X̂ n (0))] T ! Z d X (5.5) Ani (x, u) · fxi (x) + Bin (x, u)fxi xi (x) ΦnT (dx, du) + Rd ×U + i=1 d " X 1 X fxi (X̂ n (s−)) · ∆X̂in (s) E ∆f (X̂ n (s)) − T s≤T i=1 # d 1 X n n n fxi xj (X̂ (s−))∆X̂i (s)∆X̂j (s) , − 2 i,j=1 where Ani (x, u) := ℓni − µni (xi − (e · x)+ ui ) − γin (e · x)+ ui ,   λn µn xi + (γin − µni )(e · x)+ ui 1 n √ µi ρi + i + i . Bin (x, u) := 2 n n We first bound the last term in (5.5). Using Taylor’s formula, we see that n ∆f (X̂ (s)) − − d X i=1 ∇f (X̂ n (s−)) · ∆X̂ n (s) d 1 X fxi xj (X̂ n (s−))∆X̂in (s)∆X̂jn (s) 2 i,j=1 = d kkf kC 3 X √ |∆X̂in (s)∆X̂jn (s)| n i,j=1 for some positive constant k, where we use the fact that the jump size is √1 . Hence, using the fact that independent Poisson processes do not have n simultaneous jumps w.p.1, using the identity Q̂ni = X̂in − Ẑin , we obtain " d X 1 X ∇f (X̂ n (s−)) · ∆X̂ n (s) E ∆f (X̂ n (s)) − T i=1 s≤T (5.6) # d 1 X − fxi xj (X̂ n (s−))∆X̂in (s)∆X̂jn (s) 2 i,j=1 ERGODIC CONTROL IN THE HALFIN–WHITT REGIME kkf k 3 ≤ √C E T n "Z T 0 53 d  n X λ i=1  # n Z n (s) n Qn (s) µ γ i + i i + i i ds . n n n Therefore, first letting T → ∞ and using (5.2) and (5.4) we see that the expectation on the right-hand side of (5.6) is bounded above. Therefore, as n → ∞, the left-hand side of (5.6) tends to 0. Thus, by (5.5) and the fact that f is compactly supported, we obtain Z Lu f (x)π(dx, du) = 0, Rd ×U where Lu f (x) = λi ∂ii f (x) + (ℓi − µi (xi − (e · x)+ ui ) − γi (e · x)+ ui )∂i f (x). Therefore, π ∈ G.  5.2. The upper bound. The proof of the upper bound in Theorem 2.2 is a little more involved than that of the lower bound. Generally, it is very helpful if one has uniform stability across n ∈ N (see, e.g., [12]). In [12], uniform stability is obtained from the reflected dynamics with the Skorohod mapping. However, here we establish the asymptotic upper bound by using the technique of spatial truncation that we have introduced in Section 4. Let vδ be any precise continuous control in USSM satisfying vδ (x) = u0 = (0, . . . , 0, 1) for |x| > K > 1. First, we construct a work-conserving admissible policy for each n ∈ N (see [7]). Define a measurable map ̟ : {z ∈ Rd+ : e · z ∈ Z} → Zd+ as follows: for z = (z1 , . . . , zd ) ∈ Rd , let ! d X (zi − ⌊zi ⌋) . ̟(z) := ⌊z1 ⌋, . . . , ⌊zd−1 ⌋, ⌊zd ⌋ + i=1 Note that |̟(z) − z| ≤ 2d. Define uh (x) := ̟((e · x − n)+ vδ (x̂n )), x ∈ Rd ,   x1 − ρ1 n xd − ρd n n √ x̂ := ,..., √ , n n n √ o An := x ∈ Rd+ : sup |xi − ρi n| ≤ K n . i We define a state-dependent, work-conserving policy as follows:  n X − uh (X n ), if X n ∈ An ,    i ! + i−1 X Zin [X n ] := (5.7) n n  , otherwise. Xj X ∧ n−   i j=1 54 A. ARAPOSTATHIS, A. BISWAS AND G. PANG Therefore, whenever the state of the system is in Acn , the system works under the fixed priority policy with the least priority given to class-d jobs. First, we show that this is a well-defined policy for all large n. It is enough to show that Xin − uh (X n ) ≥ 0 for all i when X n ∈ An . If not, then for some i, 1 ≤ i ≤ d, we must have Xin − uh (X n ) < 0 and so Xin < (e · X n − n)+ + d. Since X n ∈ An , we obtain √ −K n + ρi n ≤ Xin < (e · X n − n)+ + d !+ d X n (Xi − ρi n) = +d i=1 √ ≤ dK n + d. But this cannot hold for large n. Hence, this policy is well defined for all large n. Under the policy defined in (5.7), X n is a Markov process and its generator given by Ln f (x) = d X i=1 + λni (f (x + ei ) − f (x)) + d X i=1 d X i=1 µni Zin [x](f (x − ei ) − f (x)) γin Qni [x](f (x − ei ) − f (x)), x ∈ Zd+ , where Z n [x] is as above and Qn [x] := x − Z n [x]. It is easy to see that, for x∈ / An , ! + #+ " i−1 X n xj . Qi [x] = xi − n − j=1 Lemma 5.1. Let X n be the Markov process corresponding to the above control. Let q be an even positive integer. Then there exists n0 ∈ N such that  Z T 1 q n sup lim sup E |X̂ (s)| ds < ∞, n≥n0 T →∞ T 0 where X̂ n = (X̂1n , . . . , X̂dn )T is the diffusion-scaled process corresponding to the process X n , as defined in (2.4). Proof. The proof technique is inspired by [6], Lemma 3.1. Define fn (x) := d X i=1 βi (xi − ρi n)q , ERGODIC CONTROL IN THE HALFIN–WHITT REGIME 55 where βi , i = 1, . . . , d, are positive constants to be determined later. We first show that for a suitable choice of βi , i = 1, . . . , d, there exist constants Ci , i = 1, 2, independent of n ≥ n0 , such that Ln fn (x) ≤ C1 nq/2 − C2 fn (x), (5.8) x ∈ Zd+ . Choose n large enough so that the policy is well defined. We define Yin := xi − ρi n. Note that Also, (a ± 1)q − aq = ±qa · aq−2 + O(aq−2 ), µni Zin [x] = µni xi Ln fn (x) = d X i=1 − − ≤ (5.9) i=1 d X i=1 i=1 d X d X i=1 + βi µni xi [qYin |Yin |q−2 + O(|Yin |q−2 )] βi (γin − µni )Qni [x][qYin |Yin |q−2 + O(|Yin |q−2 )] βi (λni + µni xi + |γin − µni |Qin [x])O(|Yin |q−2 ) i=1 ≤ Then a ∈ R. βi λni [qYin |Yin |q−2 + O(|Yin |q−2 )] d X d X + − µni Qni [x]. βi qYin |Yin |q−2 (λni − µni xi − (γin − µni )Qni [x]) βi (λni + (µni + |γin − µni |)(Yin + ρi n))O(|Yin |q−2 ) d X i=1 βi qYin |Yin |q−2 (λni − µni xi − (γin − µni )Qni [x]), where in the last inequality we use the fact that Qni [x] ≤ xi for x ∈ Zd+ . Let √ δin := λni − µni ρi n = O( n). The last estimate is due to the assumptions in (2.1) concerning the parameters in the Halfin–Whitt regime. Then (5.10) d X i=1 βi qYin |Yin |q−2 (λni − µni xi − (γin − µni )Qni [x]) = −q d X i=1 βi µni |Yin |q + d X i=1 βi qYin |Yin |q−2 (δin − (γin − µni )Qni [x]). 56 A. ARAPOSTATHIS, A. BISWAS AND G. PANG If x ∈ An and n is large, then Qni [x] = uh (x) = ̟((e · x − n)+ vδ (x̂n )) √ ≤ (e · x − n)+ + d ≤ 2dK n. Let x ∈ Acn . We use the fact that for any a, b ∈ R it holds that a+ − b+ = ξ[a − b] for some ξ ∈ [0, 1]. Also, !+ #+ " i−1 X nρj = 0, i = 1, . . . , d. nρi − n − j=1 Thus, we obtain maps ξ, ξ̃ : Rd → [0, 1]d such that !+ #+ " i−1 X nρj − Qni [x] −Qni [x] = nρi − n − j=1 = ξi (x)(nρi − xi ) − ξ˜i (x) i−1 X (xj − nρj ), x ∈ Acn . j=1 Hence, from (5.10) we obtain d X i=1 βi qYin |Yin |q−2 (λni − µni xi − (γin − µni )Qni [x]) d d X X √ βi ((1 − ξi (x))µni + ξi (x)γin )|Yin |q βi |Yin |q−1 − q ≤ O( n)q i=1 i=1 +q d X i=1 βi Yin |Yin |q−2 δin − (γin − µni )ξ̃i (x) where we used the fact that on An we have !+ #+ " i−1 X √ = O( n) xj xi − n − j=1 i−1 X Yjn j=1 ! , ∀i. Observe that there exists ϑ > 0, independent of n due to (2.1), such that (1 − ξi (x))µni + ξi (x)γin ≥ min(µni , γin ) ≥ ϑ for all n ∈ N, all x ∈ Rd , and all i = 1, . . . , d. As a result, we obtain d X i=1 βi qYin |Yin |q−2 (λni − µni xi − (γin − µni )Qni [x]) 57 ERGODIC CONTROL IN THE HALFIN–WHITT REGIME d d X X √ n q−1 βi |Yin |q βi |Yi | − qϑ ≤ O( n)q (5.11) i=1 i=1 +q d X i=1 βi Yin |Yin |q−2 δin − (γin − µni )ξ̃i (x) i−1 X j=1 ! Ynj . We next estimate the last term on the right-hand side of (5.11). Let κ := ϑ . Using Young’s inequality, we obtain the estisupn,i |γin − µni |, and ε1 := 8κ mate q i−1 i−1 X 1 X n n q−1 j n q Yj . |Yi | Yn ≤ ε1 |Yi | + q−1 ε 1 j=1 j=1 Therefore, q d X i=1 βi Yin |Yin |q−2 −(γin − µni )ξ̃i (x) ≤ qκ d X ε1 βi |Yin |q ≤ qκ d X ε1 βi |Yin |q i=1 i=1 + + i−1 βi X ε1q−1 j=1 βi q−1 ε1q−1 i−1 X Yjn j=1 Yjn d ! q! i−1 X |Yjn |q j=1 ! ! d i−1 qϑ X βi q−1 X n q n q = βi |Yi | + q d |Yj | . 8 ε1 i=1 j=1 We choose β1 = 1 and for i ≥ 2, we define βi by εq1 βi := q min βj . d j≤i−1 With this choice of βi it follows from above that q d X i=1 βi Yin |Yin |q−2 −(γin − µni )ξ̃i (x) i−1 X j=1 Ynj ! Using the preceding inequality in (5.11), we obtain (5.12) d X i=1 d qϑ X ≤ βi |Yin |q . 4 i=1 βi qYin |Yin |q−2 (λni − µni xi − (γin − µni )Qni [x]) d d X √ 3 X βi |Yin |q . ≤ O( n)q βi |Yin |q−1 − qϑ 4 i=1 i=1 58 A. ARAPOSTATHIS, A. BISWAS AND G. PANG Combining (5.9) and (5.12), we obtain Ln fn (x) ≤ (5.13) d X i=1 d X √ O(n)O(|Yin |q−2 ) O( n)O(|Yin |q−1 ) + i=1 d 3 X βi |Yin |q . − qϑ 4 i=1 By Young’s inequality, for any ε > 0, we have the bounds √ √ O( n)O(|Yin |q−1 ) ≤ ε[O(|Yin |q−1 )]q/(q−1) + ε(1−q) [O( n)]q , O(n)O(|Yin |q−2 ) ≤ ε[O(|Yin |q−2 )]q/(q−2) + ε(1−q/2) [O(n)]q/2 . Thus, choosing ε properly in (5.13) we obtain (5.8). We proceed to complete the proof of the lemma by applying (5.8). First, we observe that E[sups∈[0,T ] |X n (s)|p ] is finite for any p ≥ 1 as this quantity is dominated by the Poisson arrival process. Therefore, from (5.8) we see that  Z T n n n Ln fn (X (s)) ds E[fn (X (T ))] − fn (X (0)) = E 0 ≤ C1 n q/2 T − C2 E Z T 0  fn (X (s)) ds , n which implies that # "Z d d T X X βi (X̂in (0))q . C2 E βi (X̂in (s))q ds ≤ C1 T + 0 i=1 i=1 Hence, the proof follows by dividing both sides by T and letting T → ∞.  Proof of Theorem 2.2. Let r be the given running cost with polynomial growth with exponent m in (2.8). Let q = 2(m + 1). Recall that r̃(x, u) = r((e · x)+ u) for (x, u) ∈ Rd × S. Then r̃ is convex in u and satisfies (3.11) with the same exponent m. For any δ > 0, we choose vδ ∈ USSM such that vδ is a continuous precise control with invariant probability measure µδ and Z (5.14) r̃(x, vδ (x))µδ (dx) ≤ ̺∗ + δ. Rd We also want the control vδ to have the property that vδ (x) = (0, . . . , 0, 1) outside a large ball. To obtain such vδ , we see that by Theorems 4.1, 4.2 and ERGODIC CONTROL IN THE HALFIN–WHITT REGIME 59 Remark 4.2 we can find vδ′ and a ball Bl for l large, such that vδ′ ∈ USSM , vδ′ (x) = ed for |x| > l, vδ′ is continuous in Bl , and Z δ r̃(x, vδ′ (x))µ′δ (dx) − ̺∗ < , 2 d R where µ′δ is the invariant probability measure corresponding to vδ′ . We note that vδ′ might not be continuous on ∂Bl . Let {χn : n ∈ N} be a sequence c of cut-off functions such that χn ∈ [0, 1], it vanishes on Bl−(1/n) , and it n n takes the value 1 on Bl−(2/n) . Define the sequence vδ (x) := χ (x)vδ′ (x) + (1 − χn (x))ed . Then vδn → vδ′ , as n → ∞, and the convergence is uniform on the complement of any neighborhood of ∂Bl . Also by Proposition 3.1 the corresponding invariant probability measures µnδ are exponentially tight. Thus, Z Z ′ ′ r̃(x, vδn (x))µnδ (dx) −→ 0. r̃(x, vδ (x))µδ (dx) − Rd Rd n→∞ Combining the above two expressions, we can easily find vδ which satisfies (5.14). We construct a scheduling policy as in Lemma 5.1. By Lemma 5.1, we see that for some constant K1 it holds that  Z T 1 q n (5.15) |X̂ (s)| ds < K1 , q = 2(m + 1). sup lim sup E n≥n0 T →∞ T 0 Define vh (x) := ̟((e · x − n)+ vδ (x̂n )), √ v̂h (x̂n ) := ̟( n(e · x̂n )+ vδ (x̂n )). Since vδ (x̂n ) = (0, . . . , 0, 1) when |x̂n | ≥ K, it follows that Qn [X n ] = X n − Z n [X n ] = vh (X n ) P n for large n, provided that d−1 i=1 Xi ≤ n. Define ( d−1 ) X √ x̂ni > ρd n . Dn := x : i=1 Then   1 n r(Q̂ (t)) = r √ v̂h (X̂ (t)) + r(X̂ n (t) − Ẑ n (t))I{X̂ n (t)∈Dn } n   1 n − r √ v̂h (X̂ (t)) I{X̂ n (t)∈Dn } . n n 60 A. ARAPOSTATHIS, A. BISWAS AND G. PANG Define, for each n, the mean empirical measure ΨnT by  Z T 1 IA (X̂ n (t)) dt . ΨnT (A) := E T 0 By (5.15), the family {ΨnT : T > 0, n ≥ 1} is tight. We next show that  Z Z T 1 n (5.16) lim lim sup E r((e · x)+ vδ (x))µδ (dx). r(Q̂ (t)) dt = n→∞ T →∞ T Rd 0 For each n, select a sequence {Tkn : k ∈ N} along which the “lim sup” in (5.16) is attained. By tightness, there exists a limit point Ψn of ΨnT n . Since Ψn has k support on a discrete lattice, we have     Z Z 1 1 n r √ v̂h (x) Ψn (dx). r √ v̂h (x) ΨT n (dx) −→ k k→∞ d n n d R R Therefore, 1 lim sup E T →∞ T Z T 0    1 r √ v̂h (x) Ψn (dx) ± E n , r(Q̂ (t)) dt ≶ n d R n Z where E n = lim sup T →∞ 1 E T Z 0 T    1 r(Q̂n (t)) + r √ v̂h (X̂ n (t)) I{X̂ n (t)∈Dn } dt . n By (5.15), the family {Ψn : n ≥ 1} is tight. Hence, it has a limit Ψ. By definition, we have 1 2d √ v̂h (x) − (e · x)+ vδ (x) ≤ √ . n n Thus, using the continuity property of r and (2.8) it follows that   Z Z 1 n r √ v̂h (x) Ψ (dx) −→ r((e · x)+ vδ (x))Ψ(dx), n→∞ n d d R R along some subsequence. Therefore, in order to complete the proof of (5.16) we need to show that lim sup E n = 0. n→∞ Since the policies are work-conserving, we observe that 0 ≤ X̂ n − Ẑ n ≤ (e · X̂ n )+ , and therefore for some positive constants κ1 and κ2 , we have   1 n r √ v̂h (X̂ (t)) ∨ r(X̂ n (t) − Ẑ n (t)) ≤ κ1 + κ2 [(e · X̂ n )+ ]m . n ERGODIC CONTROL IN THE HALFIN–WHITT REGIME 61 Given ε > 0 we can choose n1 so that for all n ≥ n1 ,  Z T 1 + m n √ √ [(e · X̂ (s)) ] I{|X̂ n (s)|>(ρd / d) n} ds ≤ ε, lim sup E T →∞ T 0 p where we use (5.15). We observe that Dn ⊂ {|x̂n | > ρd d/n}. Thus, (5.16) holds. In order to complete the proof, we only need to show that Ψ is the invariant probability measure corresponding to vδ . This can be shown using the convergence of generators as in the proof of Theorem 2.1.  6. Conclusion. We have answered some of the most interesting questions for the ergodic control problem of the Markovian multi-class many-server queueing model. This current study has raised some more questions for future research. One of the interesting questions is to consider nonpreemptive policies and try to establish asymptotic optimality in the class of nonpreemptive admissible polices [7]. It will also be interesting to study a similar control problem when the system has multiple heterogeneous agent pools with skill-based routing. It has been observed that customers’ service requirements and patience times are nonexponential [10] in some situations. It is therefore important and interesting to address similar control problems under general assumptions on the service and patience time distributions. Acknowledgements. We thank the anonymous referee for many helpful comments that have led to significant improvements in our paper. Ari Arapostathis acknowledges the hospitality the Department of Industrial and Manufacturing Engineering in Penn State while he was visiting at the early stages of this work. Guodong Pang acknowledges the hospitality of the Department of Electrical and Computer Engineering at University of Texas at Austin while he was visiting for this work. Part of this work was done while Anup Biswas was visiting the Department of Industrial and Manufacturing Engineering in Penn State. Hospitality of the department is acknowledged. REFERENCES [1] Arapostathis, A., Borkar, V. S. and Ghosh, M. K. (2012). Ergodic Control of Diffusion Processes. Encyclopedia of Mathematics and Its Applications 143. Cambridge Univ. Press, Cambridge. MR2884272 [2] Arisawa, M. and Lions, P.-L. (1998). On ergodic stochastic control. Comm. Partial Differential Equations 23 2187–2217. MR1662180 [3] Atar, R. (2005). Scheduling control for queueing systems with many servers: Asymptotic optimality in heavy traffic. Ann. Appl. Probab. 15 2606–2650. MR2187306 [4] Atar, R. and Biswas, A. (2014). Control of the multiclass G/G/1 queue in the moderate deviation regime. Ann. Appl. Probab. 24 2033–2069. 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(1980). Functional Analysis, 6th ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 123. Springer, Berlin. MR0617913 A. Arapostathis A. Biswas Department of Electrical and Computer Engineering The University of Texas at Austin 1616 Guadalupe St., UTA 7.508 Austin, Texas 78701 USA E-mail: [email protected] [email protected] G. Pang The Harold and Inge Marcus Department of Industrial and Manufacturing Engineering College of Engineering Pennsylvania State University University Park, Pennsylvania 16802 USA E-mail: [email protected] 

arXiv:1607.07837v4 [math.OC] 17 Apr 2017 First Efficient Convergence for Streaming k-PCA: a Global, Gap-Free, and Near-Optimal Rate∗ Zeyuan Allen-Zhu [email protected] Institute for Advanced Study Yuanzhi Li [email protected] Princeton University July 26, 2016† Abstract We study streaming principal component analysis (PCA), that is to find, in O(dk) space, the top k eigenvectors of a d × d hidden matrix Σ with online vectors drawn from covariance matrix Σ. We provide global convergence for Oja’s algorithm which is popularly used in practice but lacks theoretical understanding for k > 1. We also provide a modified variant Oja++ that runs even faster than Oja’s. Our results match the information theoretic lower bound in terms of dependency on error, on eigengap, on rank k, and on dimension d, up to poly-log factors. In addition, our convergence rate can be made gap-free, that is proportional to the approximation error and independent of the eigengap. In contrast, for general rank k, before our work (1) it was open to design any algorithm with efficient global convergence rate; and (2) it was open to design any algorithm with (even local) gap-free convergence rate in O(dk) space. 1 Introduction Principle component analysis (PCA) is the problem of finding the subspace of largest variance in a dataset consisting of vectors, and is a fundamental tool used to analyze and visualize data in machine learning, computer vision, statistics, and operations research. In the big-data scenario, since it can be unrealistic to store the entire dataset, it is interesting and more challenging to study the streaming model (a.k.a. the stochastic online model) of PCA. Suppose the data vectors x ∈ Rd are drawn i.i.d. from an unknown distribution with covariance matrix Σ = E[xx> ] ∈ Rd×d , and the vectors are presented to the algorithm in an online fashion. Following [10, 12], we assume the Euclidean norm kxk2 ≤ 1 with probability 1 (therefore Tr(Σ) ≤ 1) and we are interested in approximately computing the top k eigenvectors of Σ. We are interested in algorithms with memory storage O(dk), the same as the memory needed to store any k vectors in d dimensions. We call this the streaming k-PCA problem. For streaming k-PCA, the popular and natural extension of Oja’s algorithm originally designed for the k = 1 case works as follows. Beginning with a random Gaussian matrix Q0 ∈ Rd×k (each ∗ We thank Jieming Mao for discussing our lower bound Theorem 6, and thank Dan Garber and Elad Hazan for useful conversations. Z. Allen-Zhu is partially supported by a Microsoft research award, no. 0518584, and an NSF grant, no. CCF-1412958. † An earlier version of this paper appeared at https://arxiv.org/abs/1607.07837. This newer version contains a stronger Theorem 2, a new lower bound Theorem 6, as well as the new Oja++ results Theorem 4 and Theorem 5. 1 entry i.i.d ∼ N (0, 1)), it repeatedly applies rank-k Oja’s algorithm: Qt ← (I + ηt xt x> t )Qt−1 , Qt = QR(Qt ) (1.1) where ηt > 0 is some learning rate that may depend on t, vector xt is the random vector in iteration t, and QR(Qt ) is the Gram-Schmidt decomposition that orthonormalizes the columns of Qt . Although Oja’s algorithm works reasonably well in practice, very limited theoretical results are known for its convergence in the k > 1 case. Even worse, little is known for any algorithm that solves streaming PCA in the k > 1. Specifically, there are three major challenges for this problem: 1. Provide an efficient convergence rate that only logarithmically depends on the dimension d. 2. Provide a gap-free convergence rate that is independent of the eigengap. 3. Provide a global convergence rate so the algorithm can start from a random initial point. In the case of k > 1, to the best of our knowledge, only Shamir [18] successfully analyzed the original Oja’s algorithm. His convergence result is only local and not gap-free.1 Other groups of researchers [3, 9, 12] studied a block variant of Oja’s, that is to sample multiple vectors x in each round t, and then use their empirical covariance to replace the use of xt x> t . This algorithm is more stable and easier to analyze, but only leads to suboptimal convergence. We discuss them more formally below (and see Table 1): • Shamir [18] implicitly provided a local but efficient convergence result for Oja’s algorithm,2 which requires a very accurate starting matrix Q0 : his theorem relies on Q0 being correlated with the top k eigenvectors by a correlation value at least k−1/2. If using random initialization, this event happens with probability at most 2−Ω(d) . • Hardt and Price [9] analyzed the block variant of Oja’s,3 and obtained a global convergence that linearly scales with the dimension d. Their result also has a cubic dependency on the gap between the k-th and (k + 1)-th eigenvalue which is not optimal. They raised an open question regarding how to provide any convergence result that is gap-free. • Balcan et al. [3] analyzed the block variant of Oja’s. Their results are also not efficient and cubically scale with the eigengap. In the gap-free setting, their algorithm runs in space more than O(kd), and also outputs more than k vectors.4 For such reason, we do not include their gap-free result in Table 1, and shall discuss it more in Section 4. • Li et al. [12] also analyzed the block variant of Oja’s. Their result also cubically scales with the eigengap, and their global convergence is not efficient. • In practice, researchers observed that it is advantageous to choose the learning rate ηt to be high at the beginning, and then gradually decreasing (c.f. [22]). To the best of our knowledge, there is no theoretical support behind this learning rate scheme for general k. In sum, it remains open before our work to obtain (1) any gap-free convergence rate in space O(kd), (2) any global convergence rate that is efficient, or (3) any global convergence rate that has the optimal quadratic dependence on eigengap. 1 A local convergence rate means that the algorithm needs a warm start that is sufficiently close to the solution. However, the complexity to reach such a warm start is not clear. 2 The original method of Shamir [18] is an offline one. One can translate his result into a streaming setting and this requires a lot of extra work including the martingale techniques we introduce in this paper. 3 They are in fact only able to output 2k vectors, guaranteed to approximately include the top k eigenvectors. 4 They require space O((k + m)d) where k + m is the number of eigenvalues in the interval [λk − ρ, 1] for some “virtual gap” parameter ρ. See our Theorem 2 for a definition. This may be as large as O(d2 ). Also, they output k + m vectors which are only guaranteed to approximately “contain” the top k eigenvectors. 2 Shamir [17] Sa et al. [16] k=1 gapdependent Li et al. [11] a Jain et al. [10] Theorem 1 (Oja) k=1 gap-free Shamir [17] (Remark 1.3) Theorem 2 (Oja) Hardt-Price [9] Li et al. [12] k≥1 gapdependent b d gap2 · 1 ε e O d gap2 · 1 ε e O dλ1 gap2 · 1 ε e O λ1 gap2 · 1 ε e O λ1 gap2 · 1 ε e O d ρ2 1 ε2 e O λ1∼(1+m) ρ2 e O dλk gap3 · kλk gap3 · kd + ·




1 ε
[ · 1 ε unknown Shamir [18] Balcan et al. [3]
e O e O b b Theorem 4 (Oja++ ) Theorem 6 (LB) 2 e O d(λ1∼k ) λk gap3 λ1∼k gap2 · e O λ1∼k gap2 · e O Theorem 1 (Oja) 1 ε 1 ε

e O Theorem 5 (Oja++ ) Theorem 6 (LB) λ1∼k+m ρ2 · [

1 [ ε [
1 ·ε [ (when λ1∼k ≥ k/d) c

+k Ω 1 ε no yes kλk gap2 · 1 ε

min{1, (λ1∼k +k·λ(k+1)∼(k+m) )} ·k ρ2
e λ1∼k+m +O · 1ε ρ2 e O Theorem 2 (Oja) k≥1 gap-free Is It Local Convergence “Efficient”?
e 12 · 1 O [ no [ gap ε
e d2 · 1 O [ no [ gap ε
e dλ12 · 1 [ no O [ gap ε
e λ12 · 1 yes O gap ε
e λ12 · 1 yes O gap ε Global Convergence Paper
Ω def kλk ρ2 · 1 ε
no no no no yes yes e O e O yes · 1 ε2
dλk gap3 · 1 ε kλk gap3 · ε O 1 gap2 · e O e O e O e O e O e O (lower bound) [ λ1∼(1+m) ρ2 e O (lower bound) yes 1 ρ2
· 1 ε

1 [ [
1 [ ε
1 2 d(λ1∼k ) λk gap3 ·ε [ (when λ1∼k ≥ k/d)
λ1∼k 1 gap2 · ε
λ1∼k 1 gap2 · ε λ1∼k+m ρ2 · 1 ε λ1∼k+m ρ2 · 1 ε

Table 1: Comparison of known results. For gap = λk − λk+1 , every ε ∈ (0, 1) and ρ ∈ (0, 1): 2 “gap-dependent convergence” means kQ> T ZkF ≤ ε where Z consists of the last d − k eigenvectors. 2 “gap-free convergence” means kQ> T WkF ≤ ε where W consists of all eigenvectors with eigenvalues ≤ λk − ρ. a global convergence is “efficient” if it only (poly-)logarithmically depend on the dimension d. k is the target rank; in gap-free settings m be the largest index so that λk+m > λk − ρ. def P • we denote by λa∼b = bi=a λi in this table. Since kxk2 ≤ 1 for each sample vector, we have • • • • gap ∈ [0, 1/k], λi ∈ [0, 1/i], kgap ≤ kλk ≤ λ1∼k ≤ λ1∼k+m ≤ 1 . • we use [ to indicate the result is outperformed. • some results in this table (both ours and prior work) depend on λ1∼k . In principle, this requires the algorithm to know a constant approximation of λ1∼k upfront. In practice, however, since one always tunes the learning rate η (for any algorithm in the table), we do not need additional knowledge on λ1∼k . 2 e dλ12 · 1 ) by under a stronger 4-th moment assumption. It slows down at least by a The result of [11] is in fact O( ε gap factor 1/λ1 if the 4-th moment assumption is removed. b 2 These results give guarantees on spectral norm kQ> T Wk2 , so we increased them by a factor k for a fair comparison. c If kxt k2 is always 1 then λ1∼k ≥ k/d always holds. Otherwise, even in the rare case of λ1∼k < k/d, their
e k2 λk3 and is still worse than ours. complexity becomes O d·gap a 3 Over Sampling. Let us emphasize that it is often desirable to directly output a d × k matrix QT . Some of the previous results, such as Hardt and Price [9], or the gap-free case of Balcan et al. [3], are only capable of finding an over-sampled matrix d × k 0 for some k 0 > k, with the guarantee that these k 0 columns approximately contain the top k eigenvectors of Σ. However, it is not clear how to find “the best k vectors” out of this k 0 -dimensional subspace. Special Case of k = 1. Jain [10] obtained the first convergence result that is both efficient and global for streaming 1-PCA. Shamir [17] obtained the first gap-free result for streaming 1-PCA, but his result is not efficient. Both these results are based on Oja’s algorithm, and it remains open before our work to obtain a gap-free result that is also efficient even when k = 1. Other Related Results. Mitliagkas et al. [13] obtained a streaming PCA result but in the restricted spiked covariance model. Balsubramani et al. [4] analyzed a modified variant of Oja’s algorithm and needed an extra O(d5 ) factor in the complexity. The offline problem of PCA (and SVD) can be solved via iterative algorithms that are based on variance-reduction techniques on top of stochastic gradient methods [2, 18] (see also [6, 7] for the k = 1 case); these methods do multiple passes on the input data so are not relevant to the streaming model. Offline PCA can also be solved via power method or block Krylov method [14], but since each iteration of these methods relies on one full pass on the dataset, they are not suitable for streaming setting either. Other offline problems and efficient algorithms relevant to PCA include canonical correlation analysis and generalized eigenvector decomposition [1, 8, 21]. Offline PCA is significantly easier to solve because one can (although non-trivially) reduce a general k-PCA problem to k times of 1-PCA using the techniques of [2]. However, this is not the case in streaming PCA because one can lose a large polynomial factor in the sampling complexity. 1.1 Results on Oja’s Algorithm We denote by λ1 ≥ · · · ≥ λd ≥ 0 the eigenvalues of Σ, and it satisfies λ1 + · · · + λd = Tr(Σ) ≤ 1. We present convergence results on Oja’s algorithm that are global, efficient and gap-free. Our first theorem works when there is a eigengap between λk and λk+1 : def Theorem 1 (Oja, gap-dependent). Letting gap = λk − λk+1 ∈ 0, k1 for every ε, p ∈ (0, 1) define learning rates 
1 e Θ       gap·T0
kΛ Λ 1 e e e Θ T0 = Θ , T1 = Θ , ηt = gap·T1
 gap2 p2 gap2  Θ 1 e gap·(t−T0 )  def and Λ = Pk i=1 λi 1 ≤ t ≤ T0 ; T0 < t ≤ T0 + T1 ; t > T0 + T1 .  ∈ 0, 1 , 5 Let Z be the column orthonormal matrix consisting of all eigenvectors of Σ with values no more than λk+1 . Then, the output QT ∈ Rd×k of Oja’s algorithm satisfies with probability at least 1 − p:
e T1 it satisfies kZ> QT k2 ≤ ε . for every6 T = T0 + T1 + Θ ε e hides poly-log factors in Above, Θ 1 1 p , gap F and d. k In other words, after a warm up phase of length T0 , we obtain a λ1 +···+λ · gap2 > 2 the quantity kZ QT kF . We make several observations (see also Table 1): 1 T convergence rate for • In the k = 1 case, Theorem 1 matches the best known result of Jain et al. [10]. 5 6 The intermediate stage [T0 , T0 + T1 ] is in fact unnecessary, but we add this phase to simplify proofs.
e T1 /ε by making ηt poly-logarithmically dependent on T . Theorem also applies to every T ≥ T0 + T1 + Ω 4 • In the k > 1 case, Theorem 1 gives the first efficient global convergence rate. • In the k > 1 case, even in terms of local convergence rate, Theorem 1 is faster than the best known result of Shamir [18] by a factor λ1 + · · · + λk ∈ (0, 1). Remark 1.1. The quantity kZ> QT k2F captures the correlation between the resulting matrix QT ∈ Rd×k and the smallest d − k eigenvectors of Σ. It is a natural generalization of the sin-square quantity widely used in the k = 1 case, because if k = 1 then kZ> QT k2F = sin2 (q, ν1 ) where q is the only column of Q and ν1 is the leading eigenvector of Σ. Some literatures instead adopt the spectral-norm guarantee (i.e., bounds on kZ> QT k22 ) as opposed to the Frobenius-norm one. The two guarantees are only up to a factor k away. We choose to prove Frobenius-norm results because: (1) it makes the analysis significantly simpler, and (2) k is usually small comparing to d, so if one can design an efficient (i.e., dimension free) convergence rate for the Frobenius norm that also implies an efficient convergence rate for the spectral norm. Remark 1.2. Our lower bound later (i.e. Theorem 6) implies, at least when λ1 and λk are within a constant factor of each other, the local convergence rate in Theorem 1 is optimal up to log factors. Gap-Free Streaming k-PCA. When the eigengap is small which is usually true in practice, it is desirable to obtain gap-free convergence [14, 17]. We have the following theorem which answers the open question of Hardt and Price [9] regarding gap-free convergence rate for streaming k-PCA. Theorem 2 (Oja, gap-free). For every ρ, ε, p ∈ (0, 1), let λ1 , . . . , λm be all eigenvalues of Σ   def Pk def Pk+m that are > λk − ρ, let Λ1 = i=1 λi ∈ 0, 1 , Λ2 = j=k+1 λj ∈ 0, 1 , define learning rates 
e 1 ! Θ t ≤ T0 ;    kΛ2  ρ·T 0 k · min{1, Λ1 + p2 }
Λ1 + Λ2 1 e e e Θ ρ·T1 t ∈ (T0 , T0 + T1 ]; T0 = Θ , T1 = Θ , ηt =
 ρ2 · p2 ρ2  Θ 1 e t > T0 + T1 . ρ·(t−T0 ) Let W be the column orthonormal matrix consisting of all eigenvectors of Σ with values no more than λk − ρ. Then, the output QT ∈ Rd×k of Oja’s algorithm satisfies with prob. at least 1 − p:
e T1 for every7 T = T0 + T1 + Θ it satisfies kW> QT k2 ≤ ε . F ε e hides poly-log factors in 1 , 1 and d. Above, Θ p ρ Note that the above theorem is a double approximation. The number of iterations depend both on ρ and ε, where ε is an upper bound on the correlation between QT and all eigenvectors in W (which depends on ρ). This is the first known gap-free result for the k > 1 case using O(kd) space. One may also be interested in single-approximation guarantees, such as the rayleigh-quotient guarantee. Note that a single-approximation guarantee by definition loses information about the ε-ρ tradeoff; furthermore, (good) single-approximation guarantees are not easy to obtain.8 We show in this paper the following theorem regarding the rayleigh-quotient guarantee: Theorem 3
(Oja, rayleigh quotient, informal). There exist learning rate choices so that, for every e 2k 2 , letting qi be the i-th column of the output matrix QT , then T =Θ ρ ·p   e Pr ∀i ∈ [k], qi> Σqi ≥ λi − Θ(ρ) ≥1−p . e hides poly-log factors in 1 , 1 and d. Again, Θ p ρ
e T0 /ε by making ηt poly-logarithmically dependent on T . Theorem also applies to every T ≥ T0 + Ω Pointed out by [10], a direct translation from double approximation to a rayleigh-quotient type of convergence loses a factor on the approximation error. They raised it as an open question regarding how to design a direct proof without sacrificing this loss. Our next theorem answers this open question (at least in the gap-free case). 7 8 5 Remark 1.3. Before our work, the only gap-free result with space O(kd) is Shamir [17] — but it is not efficient and only for k = 1. His result is in Rayleigh quotient but not double-approximation. If the initialization phase is ignored, Shamir’s local convergence rate matches our global one in Theorem 3. However, if one translates his result into double approximation, the running time loses a factor ε. This is why in Table 1 Shamir’s result is in terms of 1/ε2 as opposed to 1/ε. 1.2 Results on Our New Oja++ Algorithm Oja’s algorithm has a slow initialization phase (which is also observed

in practice [22]). For example, λ1 +···+λk 1 e in the gap-dependent case, Oja’s running time O · k+ ε is dominated by its initialization ρ2 when ε > 1/k. We propose in this paper a modified variant of Oja’s that initializes gradually. Our Oja++ Algorithm. At iteration 0, instead putting all the dk random Gaussians into Q0 like Oja’s, our Oja++ only fills the first k/2 columns of Q0 with random Gaussians, and sets the remaining columns be zeros. It applies the same iterative rule as Oja’s to go from Qt to Qt+1 , but after every T0 iterations for some T0 ∈ N∗ , it replaces the zeros in the next k/4, k/8, . . . columns with random Gaussians and continues.9 This gradual initialization ends when all the k columns become nonzero, and the remaining algorithm of Oja++ works exactly the same as Oja’s. We provide pseudocode of Oja++ in Algorithm 2 on page 58, and state below its main theorems:  def 1 ++ Theorem 4 (Oja++ , gap-dependent, informal). Letting gap = λk − λk+1  ∈ 0, k , our Oja k e λ1 +···+λ outputs a column-orthonormal QT ∈ Rd×k with kZ> QT k2 ≤ ε in T = Θ iterations. 2 F gap ε ++ outputs a column-orthonormal Theorem 5 (Oja++ , gap-free, informal). Given  our Oja  ρ ∈ (0, 1), λ +···+λ e 1 2 k+m iterations. QT ∈ Rd×k with kW> QT k2F ≤ ε in T = Θ ρ ε 1.3 Result on Lower Bound We have the following information-theoretical lower bound for any (possibly offline) algorithm: Theorem 6 (lower bound, informal). For every integer k ≥ 1, integer m ≥ 0, every 0 < ρ < λ < 1/k, every (possibly randomized) algorithm A, we can construct a distribution µ over unit vectors with λk+m+1 (Eµ [xx> ]) ≤ λ − ρ and λk (Eµ [xx> ]) ≥ λ. The output QT of A with samples x1 , ..., xT i.i.d. drawn from µ satisfies     kλ > 2 E kW QT kF = Ω . x1 ,...,xT ,A ρ2 · T (W consists of the last d − (k + m) eigenvectors of Eµ [xx> ].) Our Theorem 6 (with m = 0 and ρ = gap) implies that, in the gap-dependent setting, the global convergence rate of Oja++ is optimal up to log factors, at least when λ1 = O(λk ). Our gap-free result does not match this lower bound. We explain in Section 4 that if one increases the space from O(kd) to O((k + m)d) in the gap-free case, our Oja++ can also match this lower bound. 9 Zeros columns will remain zero according to the usage of Gram-Schmidt in Oja’s algorithm. 6 2 Preliminaries We denote by 1 ≥ λ1 ≥ · · · ≥ λd ≥ 0 the eigenvalues of the positive semidefinite (PSD) matrix Σ, and ν1 , ν2 , . . . , νd the corresponding normalized eigenvectors. Since we assumed kxk2 ≤ 1 for each   def data vector it satisfies λ1 + · · · + λd = Tr(Σ) ≤ 1. We define gap = λk − λk+1 ∈ 0, k1 . Slightly abusing notations, we also use λk (M) to denote the k-th largest eigenvalue of an arbitrary M. def def Unless otherwise noted, we denote by V = [ν1 , . . . , νk ] ∈ Rd×k and Z = [νk+1 , . . . , νd ] ∈ Rd×(d−k) . For a given parameter ρ > 0 in our gap-free results, we also define W = [νk+m+1 , . . . , νd ] ∈ Rd×(d−k−m) where m is the largest index so that λk+m > λk −ρ. We write Σ≤k = VDiag{λ1 , . . . , λk }V> def and Σ>k = ZDiag{λk+1 , . . . , λd }Z> so Σ = Σ≤k + Σ>k . For a vector y, we sometimes denote by y[i] or y (i) the i-th coordinate of y. We may use different notations in different lemmas in order to obtain the cleanest representations; when we do so, we shall clearly point out inQthe statement of the lemmas. def We denote by Pt = ts=1 (I + ηs xs x> s ) where xs is the s-th data sample and ηs is the learning def rate of iteration s. We denote by Q ∈ Rd×k (or Q0 ) the random initial matrix, and by Qt = 10 We use the QR((I + ηt xt x> t )Qt−1 ) = QR(Pt Q0 ) the output of Oja’s algorithm for every t ≥ 1. notation Ft to denote the sigma-algebra generated by xt . We denote F≤t to be the sigma-algebra generated by x1 , ..., xt , i.e. F≤t = ∨ts=1 Fs . In other words, whenever we condition on F≤t it means we have fixed x1 , . . . , xt . For a vector x we denote by kxk or kxk2 the Euclidean norm of x. We write A  B if A, B are symmetric matrices and A − B is PSD. We denote by kAkS1 the Schatten-1 norm which is the summation of the (nonnegative) singular values of A. It satisfies the following simple properties: Proposition 2.1. For not necessarily symmetric matrices A, B ∈ Rd×d we have (1): Tr(A) ≤ kAkS1 (2): Tr(AB) ≤ kABkS1 ≤ kAkS1 kBk2 .
1/2 (3): Tr(AB) ≤ kAkF kBkF = Tr(A> A)Tr(B> B) .
Proof. (1) is because Tr(A) = 21 Tr(A + A> ) ≤ 12 kA + A> kS1 ≤ 12 kAkS1 + kA> kS1 = kAkS1 . (2) is because of (1) and the matrix Holder’s inequality. P (3) is owing to von Neumann’s trace inequality (together with Cauchy’s) which says Tr(AB) ≤ i σA,i · σB,i ≤ kAkF kBkF . (Here, we have noted by σA,i the i-th largest eigenvalue of A and similarly for B.  2.1 A Matrix View of Oja’s Algorithm The following lemma tells us that, for analysis purpose only, we can push the QR orthogonalization step in Oja’s algorithm to the end: Lemma 2.2 (Oja’s algorithm). For every s ∈ [d], every X ∈ Rd×s , every t ≥ 1, every initialization matrix Q ∈ Rd×k , it satisfies kX> Qt kF ≤ kX> Pt Q(V> Pt Q)−1 kF . e t = Pt Q, we first observe that for every t ≥ 0, Qt = Q e t Rt for Proof of Lemma 2.2. Denoting by Q k×k some (upper triangular) invertible matrix Rt ∈ R (if Rt is not invertible, then the right hand side of the statement is +∞ so we already done). The claim is true for t = 0. Suppose it holds for t by induction, then > Qt+1 = QR[(I + ηt+1 xt+1 x> t+1 )Qt ] = (I + ηt+1 xt+1 xt+1 )Qt St for some St ∈ Rk×k by the definition of Gram-Schmidt. This implies that e t Rt St = Pt+1 QRt St = Q e t+1 Rt St = Q e t+1 Rt+1 Qt+1 = (I + ηt+1 xt+1 x> )Q t+1 10 The second equality is a simple fact but anyways proved in Lemma 2.2 later. 7 e t Rt . As a result, since if we define Rt+1 = Rt St . This completes the proof that Qt = Q > each Qt is column orthogonal for t ≥ 1, we have kQt Vk2 ≤ 1 and therefore kX> Qt kF ≤ kX> Qt (V> Qt )−1 kF kV> Qt k2 ≤ kX> Qt (V> Qt )−1 kF . Finally, e t Rt (V> Q e t Rt )−1 kF ≤ kX> Q e t (V> Q e t )−1 kF .  kX> Qt kF ≤ kX> Qt (V> Qt )−1 kF = kX> Q Observation. Due to Lemma 2.2, in order to prove our upper bound theorems, it suffices to upper bound the quantity kX> Pt Q(V> Pt Q)−1 kF for X = W (gap-free) or X = Z (gap-dependent). 3 Overview of Our Proofs and Techniques Oja’s Algorithm. To illustrate the idea, let us simply focus on the gap-dependent case. Denoting def in this section by st = kZ> Pt Q(V> Pt Q)−1 kF , owing to Lemma 2.2, we want to bound st in terms of xt and st−1 . A simple calculation using the Sherman-Morrison formula gives h η a 2 i t t > −1 E[s2t ] ≤ (1 − ηt gap) E[s2t−1 ] + E where at = kx> (3.1) t Pt−1 Q(V Pt−1 Q) k2 1 − η t at At a first look, E[s2t ] is decaying by a multiplicative factor (1 − ηt gap) at every iteration; however, this bound could be problematic when ηt at is close to 1 and thus we need to ensure ηt ≤ a1t with high probability for every step. > −1 √One can naively bound at ≤ kPt−1 Q(V Pt−1 Q) √ k2 ≤ st−1 + 1. However, since st−1 can be Ω( d) even at t = 1, we must choose ηt ≤ O(1/ d) and the resulting convergence rate will be not efficient (i.e., proportional to d). This is why most known global convergence results are not efficient (see Table 1). On the other hand, if one ignores initialization and starts from a point t0 when st0 ≤ 1 is already satisfied, then we can prove a local convergence rate that is efficient (c.f. [18]). Note that this local rate is still slower than ours by a factor λ1 + · · · + λk . Our first contribution is the following crucial observation: for a random initial matrix Q, a1 = > −1 kx> 1 Q(V Q) k2 is actually quite small. √ A simple fact on the singular value distribution of inverseWishart distribution √ implies a1 = O( k) with high probability. Thus, at least in the first iteration, we can set η1 = Ω(1/ k) independent of the dimension d. Unfortunately, in subsequent iterations, it is not clear whether at remains small or increases. Our second contribution is to control at using the fact that at itself “forms another random > −1 process.” More precisely, denoting by at,s = kx> t Ps Q(V Ps Q) k2 for 0 ≤ s ≤ t − 1, we wish to bound at,s in terms of at,s−1 and show that it does √ not increase by much. (If we could achieve so, combining it with the initialization at,0 ≤ O( k), we would know that all at,s are small for s ≤ t − 1.) Unfortunately, since xt is not an eigenvector of Σ, the recursion looks like h
2 i η a E[a2t,s ] ≤ (1 − ηs λk ) E[a2t,s−1 ] + ηs λk E[b2t,s−1 ] + E 1−ηs ss,s−1 (3.2) as,s−1 > −1 where bt,s = kx> t ΣPs Q(V Ps Q) k2 . Now three difficulties arise from formula (3.2): • bt,s can be very different from at,s — in worse case, the ratio between them can be unbounded. • the problematic term now becomes as,s−1 (rather than the original at = at,t−1 in (3.1)) which t−1 is not present in the chain {at,s }s=1 . • since bt,s differs from at,s by an additional factor Σ in the middle, to analyze bt,s , we need to 2 > −1 further study kx> t Σ Ps Q(V Ps Q) k2 and so on. (i) def We solve these issues by carefully considering a multi-dimensional random process ct,s with ct,s = i > −1 kx> t Σ Ps Q(V Ps Q) k2 . Ignoring the last term, we can derive that  (i)
2   (i)
2   (i+1)
2  ∀t, ∀s ≤ t − 1, E ct,s / (1 − ηs λk ) E ct,s−1 + ηs λk E ct,s−1 . (3.3) 8 Our third contribution is a new random process concentration bound to control the change in this multi-dimensional chain (3.3). To achieve this, we adapt the prove of standard Chernoff bound to multi dimensions (which is not the same as matrix concentration bound). After establishing (0) this non-trivial concentration result, all terms of at = ct,t−1 can be simultaneously bounded by a constant. This ensures that the problematic term in (3.1) is well-controlled. The overall plan looks promising; however, there are holes in the above thought experiment. • In order to apply a random-process concentration bound (e.g., Azuma concentration), we need the process to not depend on the future. However, the random vector ct,s is not F≤s measurable but F≤s ∨ Ft measurable (i.e., it depends on xt for a future t > s). • Furthermore, the expectation bounds such as (3.1), (3.2), (3.3) only hold if E[xt xt ] = Σ; however, if we take away a failure event C —C may correspond to the event when at is large— the conditional expectation E[xt xt | C] becomes Σ + ∆ where ∆ is some error matrix. This can amplify the failure probability in next iteration. Our fourth contribution is a “decoupling” framework to deal with the above issues (Appendix i.D). At a high level, to deal with the first issue we fix xt and study {ct,s }s=0,1,...,t−1 conditioning on xt ; in this way the process decouples and each ct,s becomes F≤s measurable. We can do so because we can carefully ensure that the failure events only depend on xs for s ≤ t − 1 but not on xt . To deal with the second issue, we convert the random process into an unconditional random process (see (i.D.2)); this is a generalization of using stopping time on martingales. Using these tools, we manage to show that the failure probability only grows linearly with respect to T and henceforth (i) bound the value of ct,s for all t, s and i. we are able to show that Oja’s algorithm achieves convergence rate Putting them together,  λ1 +...+λk 1 e O ( ε + k) . The rate matches our lower bound asymptotically when λ1 and λk are gap2 within a constant factor of each other, however, if we only care about crude approximation of the eigenvectors (e.g. for constant ε), then the Oja’s algorithm is off by a factor k. Remark 3.1. The ideas above are insufficient for our gap-free results. To prove Theorem 2 and 3, we def also need to bound quantities s0t = kW> Pt Q(V> Pt Q)−1 kF where W consists of all eigenvectors of Σ with values no more than λk − ρ. This is so because the interesting quantity in a gap-free case changes from st to s0t according to Lemma 2.2. Now, to bound s0t one has to bound ct,s ; however, the ct,s process still depends on the original st as opposed to s0t . In sum, we unavoidably have to bound st , s0t , and ct,s all together, making the proofs even more sophisticated. Our Oja++ Algorithm. The factor k in Oja’s √ algorithm comes from the fact that √ the earlier > Q)−1 k is at least Ω( k) at t = 1, so we must set η ≤ O(1/ k) and this quantity a1 = kx> Q(V 2 1 1 incurs a factor k in the running time. After warm start, at drops to O(1) and we can choose ηt ≤ 1. A similar issue was also observed by Hardt and Price [9] and they solved it using “oversampling”. Namely, to put it into our setting, we can use a d × 2k random starting matrix Q0 and run Oja’s to produce QT ∈ Rd×2k . In this way, the quantity a1 becomes O(1) even at the beginning due to some property of the inverse-Wishart distribution. However, the output QT is now a 2k dimensional space that is only guaranteed to “approximately contain” the top k eigenvectors. It is not clear how to find this k-subspace (recall the algorithm does not see Σ). Our key observation behind Oja++ is that a similar effect also occurs via “under-sampling”. If we initialize Q0 randomly with dimension d × k/2, we can also obtain a speed-up factor of k. Unlike Hardt and Price, this time the output QT0 is a k/2-dimensional subspace that approximately lies entirely in the column span of V ∈ Rd×k . 9 After getting QT0 , one could hope to get the rest by running the same algorithm again, but restricted to the orthogonal complement of QT0 . This approach would work if QT0 were exactly the eigenvectors of Σ; however, due to approximation error, this approach would eventually lose a factor 1/gap in the sample complexity which is even bigger than the factor k that we could gain. Instead, our Oja++ algorithm is divided into log k epochs. At each epoch i = 1, 2, . . . , log k, we attach k/2i new random columns to the working matrix Qt in Oja’s algorithm, and then run Oja’s for a fixed number of iterations. Note that every time (except the last time) we attach new random columns, we are in an “under-sampling” mode because if we add k/2i columns there must be k/2i remaining dimensions. This ensures that the quantity at only increases by a constant so we have at = O(1) throughout execution of Oja++ . Finally, there are only log k epochs so the total running   k 1 e λ1 +...+λ e time is still O ε and this O notion hides a log k factor. ρ2 Roadmap. Our proofs are highly technical so we carefully choose what to present in this main body. In Section 5 we state properties of the initial matrix Q which corresponds to our first contribution. In Section 6 we provide expected guarantees on st , s0t and at,s and they correspond to our second contribution. The third (martingale lemmas) and fourth contributions (decoupling lemma) are deferred to the appendix. Most importantly, in Section 7 we present (although in weaker forms) two Main Lemmas to deal with the convergence one for t ≤ T0 (before warm start) and one for t > T0 (after warm start). These sections, when put together, directly imply two weaker variants of Theorem 1 and 2. We state these weaker variants in Appendix i and include all the mathematical details there. Appendix ii includes our Rayleigh quotient Theorem 3 and lower bound Theorem 6. Appendix iii strengthens the main lemmas into their stronger forms, and prove Theorem 1 and 2 formally. Our Oja++ results, namely Theorem 4 and 5, are also proved in Appendix iii. In Figure 1 on page 14, we present a dependency graph of all of our main theorems and lemmas. We hope that the readers could appreciate our organization of this paper. 4 Discussions, Extensions and Future Directions In this paper we give global convergence analysis of the Oja’s algorithm, and a twisted version Oja++ which has better complexity. We also give an information-theoretic bound that  showing  lower
any kλk > 2 algorithm, offline or online, must have final accuracy Ex1 ,...,xT ,A kW QT kF = Ω gap2 ·T . This matches our gap-dependent result on Oja++ when λ1 + · · · + λk = O(kλk ); that is, when there is an eigengap and when “the spectrum is flat.” When the spectrum is not flat, our algorithm can be improved to have better accuracy. However, this requires good prior knowledge of λ1 , · · · λk , and may not be realistic.
λ +···λ In the gap-free case, Oja++ only achieves accuracy O 1 ρ2 ·Tk+m , which appears worse than

λ1 +···λk k the lower bound O ρkλ if we allow more space, namely, 2 ·T . In fact, we can also achieve O ρ2 ·T space up to O((k + m)d) as opposed to O(kd). More generally, we have a space-accuracy tradeoff. If we run Oja++ on k 0 initial random vectors, and thus using space O(dk 0 ) for k 0 ∈ [k, k + m], we can randomly pick k columns from the output and have the following accuracy: Theorem 4.1 (tradeoff). For every k 0 ∈ [k, k + m] with λk0 − λk+m ≥ 0 ρ log d , 0 let Q ∈ Rd×k be a random gaussian matrix and QT ∈ Rd×k be the output of Oja++ with random input Q. Then, letting Q0T ∈ Rd×k be k random columns of QT (chosen uniformly at random), we have     k λ1 + · · · λk+m0 > 0 2 e E kW QT kF = O k0 ρ2 T 10 where m0 ≤ m is any index satisfying λk0 − λk+m0 ≥ ρ log d . Proof of Theorem 4.1. Observethat Oja++ guarantees (using ρ/ log d instead of ρ as the gap-free k+m0 e λ1 +···λ parameter) E[kW> QT k2F ] = O . Then, k random columns of Q0T decreases the squared ρ2 T  Frobenius norm by a factor of k 0 /k. We also have the following crucial observation: Corollary 4.2. There exists k 0 ∈ [k, k + m] such that λk0 − λk+m ≥ ρ log d and k (λ1 + · · · λk+m0 ) = O (λ1 + · · · λk ) . k0 Proof of Corollary 4.2. The is by
counting. Divide [λk
− ρ, λk] into log d intervals
of
equal  proof  ρ ρ span in descending order λk − log d , λk , λk − 2 log d , λk − logρ d , · · · , λk − ρ, λk − 1 − log1 d ρ , and let Si ⊆ {k + 1, .P . . , k + m} be the indices of λj is in the i-th interval above,P for i = 1, 2, . . . , log d. Define Λi = j∈Si λj and Λ0 = Λ = λ1 + · · · λk . Also define ki = k + 1≤j<i |Sj |. We then P have, for every k 0 = ki , we can choose m0 such that λ1 + · · · λk+m0 = Λ + ij=1 Λj . By Λ0 ≤ dΛlog d , we know that there must exist some i such that Λi ≤ 100Λi−1 . We compute    Pi Pi−1  k k k Λ ≤ 100 Λ + Λ + j j=1 j=1 Λi = 100 ki (λ1 + · · · + λki ) ≤ 100Λ ki ki  so we have found such k 0 satisfying the statement. ++ to at most O((k + m)d), we can In sum. In the gap-free case,
if we increase the space of Oja λ1 +···λk achieve accuracy O ρ2 ·T , and thus match the lower bound when the spectrum is flat.11 It was also studied by Balcan et al. [3] that increasing the space could enhance the performance. However, their algorithm always uses space Ω((k + m)d), and furthermore, in the Frobenius-norm accuracy, their performance is always worse than Oja++ , not
to say Oja++ only uses space O(kd).12 k It is an important future direction to directly get O λ1ρ+···λ without increasing the space. 2 ·T Matrix Bernstein. Using matrix Bernstein and a gap-free variant of the Wedin theorem [2], one can QT be the top-k
eigenvectors of the empirical covariance matrix PT show>that, if we simply let > e λ21 . If one directly translates this to a Frobenius x x , then we have E[kW QT k22 ] = O t=1 t t ρ T
λ1 k > 2 e norm bound, it gives E[kW QT kF ] = O ρ2 T and is worse than ours. However, our result, if naively translated to spectral norm, also loses a factor k. It is a future direction to directly get a spectral-norm guarantee for streaming PCA. 5 Random Initialization We state our main lemma for initialization. Let Q ∈ Rd×k be a matrix with each entry i.i.d drawn from N (0, 1), the standard gaussian. Lemma 5.1 (initialization). For every p, q ∈ (0, 1), every T ∈ N∗ , every distribution on vector set {xt }Tt=1 with kxt k2 ≤ 1, with probability at least 1 − p − 2q over the random choice of Q:  2 ln dp and  (Z> Q)(V> Q)−1 F ≤ 576dk p2   1/2  i−1 18 T > (Σ/λ > Q)−1  Prx1 ,...,xT ∃i ∈ [T ], ∃t ∈ [T ], x> 2k ln ≤q ZZ ) Q(V ≥ k+1 t p q 2 0 Of course, this requires the algorithm to know k which can be done by trying k0 = k + 2, k + 4, k + 8, etc. 2

12 e λ1∼k+m e dk(λ1∼k+m3 ) λk , but ours is only O . For instance, when λ1∼k+m ≥ k+m , their global convergence is O 2 11 d (k+m)ρ T 11 ρ T (i) The two statements of the above lemma correspond to s0 and ct,0 that we defined in Section 3. The second statement is of the form “Pr[event] ≤ q” instead of “for every fixed xt , event holds with probability ≤ q” because we cannot afford taking union bound on xt . 6 Expected Results We now formalize inequalities (3.1), (3.2), (3.3) which characterize to the behavior of our target random processes. Let X ∈ Rd×r be a generic matrix that shall later be chosen as either X = W (corresponding to s0t ), X = Z (corresponding to st ), or X = [w] for some vector w (corresponding (i) to ct,s ). We introduce the following notions that shall be used extensively: Lt = Pt Q(V> Pt Q)−1 ∈ Rd×k r×k R0t = X> xt x> t Lt−1 ∈ R St = X> Lt ∈ Rr×k k×k H0t = V> xt x> t Lt−1 ∈ R Lemma 6.1 (Appendix i.B). For every t ∈ [T ], suppose C≤t is an event on random x1 , . . . , xt and 1 > > −1 C≤t implies kx> , t Lt−1 k2 = kxt Pt−1 Q(V Pt−1 Q) k2 ≤ φt where ηt φt ≤ 2   > and suppose Ext xt xt | F≤t−1 , C≤t = Σ + ∆. Then, we have: (a) When X = Z, h i 2 2 E Tr(S> S ) | F , C ≤ (1 − 2ηt gap + 14ηt2 φ2t )Tr(St−1 S> t ≤t−1 ≤t t t−1 ) + 10ηt φt  3/2  1/2  > + 2ηt k∆k2 Tr(S> + 2Tr(S> t−1 St−1 ) t−1 St−1 ) + Tr(St−1 St−1 ) (b) When X = W, h i 2 2 > 2 2 E Tr(S> t St ) | F≤t−1 , C≤t ≤ (1 − 2ηt ρ + 14ηt φt )Tr(St−1 St−1 ) + 10ηt φt   1/2  1/2  + 2ηt k∆k2 Tr(S> + Tr(S> 1 + Tr(Z> Lt−1 L> t−1 St−1 ) t−1 St−1 ) t−1 Z) (c) When X = [w] ∈ Rd×1 where w is a vector with Euclidean norm at most 1, h i
ηt 2 2 > 2 2 kw> ΣLt−1 k22 E Tr(S> t St ) | F≤t−1 , C≤t ≤ 1 − ηt λk + 14ηt φt Tr(St−1 St−1 ) + 10ηt φt + λk   1/2  1/2  > > > + 2ηt k∆k2 Tr(S> S ) + Tr(S S ) 1 + Tr(Z L L Z) t−1 t−1 t−1 t−1 t−1 t−1 The above three expectation results will be used to provide upper bounds on the quantities we (i) care about (i.e., st , s0t , ct,s ). In the appendix, to enable proper use of martingale concentration,  > > we also bound their absolute changes |Tr(S> t St ) − Tr(St−1 St−1 )| and variance E |Tr(St St ) − > 2 13 Tr(St−1 St−1 )| in changes. 7 Main Lemmas Our main lemmas in this section can be proved by combining (1) the expectation results in Section 6, (2) the martingale concentrations in Appendix i.C, and (3) our decoupling lemma in Appendix i.D. 13 Recall that even in the simplest martingale concentration, one needs upper bounds on the absolute difference between consecutive variables; furthermore, the concentration can be tightened if one also has an (expected) variance upper bound between variables. 12 Our first lemma describes the behavior of quantities st = kZ> Pt Q(V> Pt Q)−1 kF and s0t = kW> Pt Q(V> Pt Q)−1 kF before warm start. At a high level, it shows if the st sequence starts from s20 ≤ ΞZ , under mild conditions, s2t never increases to more than 2ΞZ . Note that ΞZ = O(d) according to Lemma 5.1. The other sequence (s0t )2 also never increases to more than 2ΞZ because s0t ≤ st , but most importantly, (s0t )2 drops below 2 after t ≥ T0 . Therefore, at point t = T0 we need to adjust the learning rate so the algorithm achieves best convergence rate, and this is the goal of our Lemma Main 2. (We emphasize that although we are only interested in st and s0t , our proof of the lemma also needs to bound the multi-dimensional ct,s sequence discussed in Section 3.)  Lemma Main 1 (before warm start). For every ρ ∈ (0, 1), q ∈ 0, 12 , ΞZ ≥ 2, Ξx ≥ 2, and fixed matrix Q ∈ Rd×k , suppose it satisfies • kZ> Q(V> Q)−1 k2F ≤ ΞZ , and h j−1 > Q(V> Q)−1 • Prxt ∀j ∈ [T ], x> t ZZ (Σ/λk+1 ) Suppose also the learning rates {ηs }s∈[T ] satisfy (1): 3/2 ∀s ∈ [T ], qΞZ ≤ ηs ≤ ρ 4000Ξ2x ln (3): 24T q2 2 i ≤ Ξx ≥ 1 − q 2 /2 for every t ∈ [T ]. (2): PT ∃T0 ∈ [T ] such that 2 2 t=1 ηt Ξx PT0 t=1 ηt ≤ 1 100 ln ≥ ln(3ΞZ ) ρ 32T q2 . Then, for every t ∈ [T − 1], with probability at least 1 − 2qT (over the randomness of x1 , . . . , xt ): • kZ> Pt Q(V> Pt Q)−1 k2F ≤ 2ΞZ , and • if t ≥ T0 then kW> Pt Q(V> Pt Q)−1 2 F ≤ 2. Our second lemma asks for a stronger assumption on the learning rates and shows that after warm start (i.e., for t ≥ T0 ), the quantity (s0t )2 scales as 1/t. √ Lemma Main 2 (after warm start). In the same setting as Lemma Main 1, if there exists δ ≤ 1/ 8 s.t. T0 1 1 9 ln(8/q 2 ) , ∀s ∈ {T0 +1, . . . , T } : 2ηs ρ−56ηs2 Ξ2x ≥ and ηs ≤ ≥ , 2 2 δ s−1 20(s − 1)δΞx ln T0 then, with probability at least 1 − 2qT (over the randomness of x1 , . . . , xT ): • kZ> Pt Q(V> Pt Q)−1 k2F ≤ 2ΞZ for every t ∈ {T0 , . . . , T }, and • kW> Pt Q(V> Pt Q)−1 k2F ≤ 5T0 / ln2 (T0 ) t/ ln2 t for every t ∈ {T0 , . . . , T }. Parameter 7.1. There exist constants C1 , C2 , C3 > 0 such that for every q > 0 that is sufficiently small (meaning q < 1/poly(T, ΞZ , Ξx , 1/ρ)), the following parameters satisfy both Lemma Main 1 and Lemma Main 2: ( ln ΞZ Ξ2x ln Tq ln2 ΞZ t ≤ T0 ; ρ T0 T0 ·ρ , and δ = C3 · = C · , η = C · . 1 t 2 1 2 2 t > T0 . ρ Ξx ln (T0 ) t·ρ Using such learning rates for our main lemmas, one can prove in one page (see Appendix i.F) • a weaker version of Theorem 2 where (Λ1 , Λ2 ) are replaced by (1, 0), and • a weaker version of Theorem 1 where Λ = λ1 + · · · + λk is replaced by 1. 13 Appendix Overview Main Lemma Appendix ii.G Lower Bound Theorem 6 Rayleigh Quotient Theorem 3 (before warm start) Lemma Main 3 Appendix ii upgrade Appendix i Expectation Lemma Lemma 6.1 and Appendix i.B upgra Main Lemma Appendix i.E de Oja (GF, weak) Theorem 2’ Decoupling Lemma Appendix i.D (after warm start) Lemma Main 2 Oja (GD, weak) Theorem 1’ Initialization Lemma Lemma 5.1 Appendix i.A rade (before warm start) Lemma Main 1 upg Martingale Lemma Appendix i.C upgrade u p grade Appendix iii Main Lemma Appendix iii.L (before warm start) Lemma Main 4 (after warm start) Lemma Main 5 Expectation Lemma Appendix iii.K (under sampling) Lemma Main 6 Oja (GF) Theorem 2 Oja (GD) Theorem 1 Initialization Lemma Lemma iii.J.2 Appendix iii.J Oja++ (GF) Theorem 5 Oja++ (GD) Theorem 4 Figure 1: Overall structure of this paper. GF and GD stand for gap-free and gap-dependent respectively. We divide our appendix sections into three parts, Appendix i, ii, and iii. • Appendix i (page 16) provides complete proof but for two weaker versions of our Theorem 1 and 2. – Appendix i.A and i.B give missing proofs for Section 5 and 6; – Appendix i.C and i.D provide details for our martingale and decoupling lemmas; – Appendix i.E proves main lemmas in Section 7 and Appendix i.F puts everything together. • Appendix ii (page 35) includes proofs for Theorem 6 and Theorem 3. – Appendix ii.G extends our main lemmas to better serve for the rayleigh quotient setting; 14 – Appendix ii.H provides the final proof for our Rayleigh Quotient Theorem 3; – Appendix ii.I includes a three-paged proof of our lower bound Theorem 6. • Appendix iii (page 42) provide full proofs not only to the stronger Theorem 1 and Theorem 2 for Oja’s algorithm, but also to Theorem 4 and Theorem 5 for Oja++ . – – – – – Appendix Appendix Appendix Appendix Appendix iii.J extends our initialization lemma in Appendix i.A to stronger settings; iii.K extends our expectation lemmas in Appendix i.B to stronger settings; iii.L extends our main lemmas in Appendix i.E to stronger settings; iii.M provides the final proofs for our Theorem 1 and Theorem 2; iii.N provides the final proofs for our Theorem 4 and Theorem 5. We include the dependency graphs of all of our main sections, lemmas and theorems in Figure 1 for a quick reference. 15 Appendix (Part I) In this Part I of the appendix, we provide complete proof but two weaker versions of our Theorem 1 and 2. We state these weaker versions Theorem 1’ and 2’ here, and meanwhile: • Appendix i.A and i.B give missing proofs for Section 5 and 6; • Appendix i.C and i.D provide details for our martingale and decoupling lemmas; • Appendix i.E proves main lemmas in Section 7; and • Appendix i.F puts everything together and proves Theorem 1’ and 2’.  def Theorem 1’ (gap-dependent streaming k-PCA). Letting gap = λk − λk+1 ∈ 0, k1 , for every ε, p ∈ (0, 1) define learning rates      1 e  Θ gap·T0 1 ≤ t ≤ T0 ; k e   T0 = Θ , ηt = 2 2 1 e  Θ gap · p t > T0 . gap·t Let Z be the column orthonormal matrix consisting of all eigenvectors of Σ with values no more than λk+1 . Then, the output QT ∈ Rd×k of Oja’s algorithm satisfies with prob. at least 1 − p:   T0 e for every T = T0 + Θ it satisfies kZ> QT k2F ≤ ε . ε e hides poly-log factors in 1 , 1 and d. Above, Θ p gap Theorem 2’ (gap-free streaming k-PCA). For every ρ, ε, p ∈ (0, 1), define learning rates      1 e  Θ t ≤ T0 ; k 0 e  ρ·T T0 = Θ , ηt = 2 2 e 1  Θ ρ ·p t > T0 . ρ·t Let W be the column orthonormal matrix consisting of all eigenvectors of Σ with values no more than λk − ρ. Then, the output QT ∈ Rd×k of Oja’s algorithm satisfies with prob. at least 1 − p:   T0 e for every T = T0 + Θ it satisfies kW> QT k2F ≤ ε . ε e hides poly-log factors in 1 , 1 and d. Above, Θ p ρ 16 i.A Random Initialization (for Section 5) Recall that Q ∈ Rd×k is a matrix with each entry i.i.d drawn from N (0, 1), the standard gaussian. i.A.1 Preparation Lemmas Lemma i.A.1. For every x ∈ Rd that has Euclidean norm kxk2 ≤ 1, every PSD matrix A ∈ Rk×k , and every λ ≥ 1, we have   − λ Pr x> ZZ> QAQ> ZZ> x ≥ Tr(A) + λ ≤ e 8Tr(A) . Q Proof of Lemma i.A.1. Let A = UΣA U> be the eigendecomposition of A, and we denote by Qz = Z> QU ∈ R(d−k)×d . Since a random Gaussian matrix is rotation invariant, and since U is unitary and Z is column orthonormal, we know that each entry of Qz draw i.i.d. from N (0, 1). Next, since we have kZ> xk2 ≤ 1, it satisfies that y = x> ZZ> QU is a vector with each coordinate i independently drawn from distribution N (0, σi ) for σi ≤ 1. This implies x> ZZ> QAQ> ZZ> x = y > ΣA y = P )2 k X [ΣA ]i,i (yi )2 . i=1 distribution14 is a subexponential with parameter (σ 2 , b) where σ 2 , b ≤ Now, i∈[k] [ΣA ]i,i (yi Pk 4 i=1 [ΣA ]i,i . Using the subexponential concentration bound [20], we have for every λ ≥ 1, " k # ( ) k X X λ 2 Pr [ΣA ]i,i (yi ) ≥ [ΣA ]i,i + λ ≤ exp − Pk . 8 i=1 [ΣA ]i,i i=1 i=1 After rearranging, we have λ − 8Tr(A) Pr[x> ZZ> QAQ> ZZ> x ≥ Tr(A) + λ] ≤ e  . The following lemma is on the singular value distribution of a random Gaussian matrix: Lemma i.A.2 (Theorem 1.2 of [19]). Let Q ∈ Rk×k be a random matrix with each entry i.i.d. drawn from N (0, 1), and σ1 ≤ σ2 ≤ · · · ≤ σk be its singular values. We have for every j ∈ [k] and α ≥ 0:    j 2 αj Pr σj ≤ √ ≤ (2e)1/2 α . k Lemma i.A.3. Let Q be our initial matrix, then for every p ∈ (0, 1):  h  √
−1 i π 2 ek p > > > ≥ ≤ . Pr Tr (V Q) (V Q) Q 3p 1−p Proof of Lemma i.A.3. Using Lemma i.A.2, we know that (using the famous equation π2 6 )  Pr Tr 14  P∞ 1 j=1 j 2 =   −1  π 2 ek  2ek −2 > ≥ (V Q) (V Q) ≤ Pr ∃j ∈ [k], σj (V Q) ≥ 2 3p j p  √ √  X k j p p 2 = Pr ∃j ∈ [k], σj (V> Q) ≤ √ ≤ pj /2 ≤ . 1−p 2ek j=1 > > > Recall that a random variable X is (σ 2 , b)-subexponential if log E exp(λ(X − E X)) ≤ λ2 σ 2 /2 for all λ ∈ [0, 1/b]. The squared standard Gaussian variable is (4, 4)-subexponential. 17 i.A.2 Proof of Lemma 5.1 2 Proof of Lemma 5.1. Applying Lemma i.A.3 with the choice of probability = p4 , we know that    −1 36k def . Pr Tr(A) ≥ 2 ≤ p where A = (V> Q)> (V> Q) Q p n o Conditioning on event C = Tr(A) ≤ 36k , and setting r = 36k , we have for every fixed x1 , ..., xT p2 p2 and fixed i ∈ [T ], it satisfies    T 1/2 i−1 > > −1 ≥ 18r ln Pr x> ZZ (Σ/λ ) Q(V Q) C, x t k+1 t Q q 2    ¬ T 1/2 ≤ Pr yt ZZ> Q(V> Q)−1 ≥ 18r ln C, x1 , .., xt q 2   ® q2 ­ T2 > > > > ≤ Pr yt ZZ QAQ ZZ yt ≥ 9r ln 2 | C, x1 , .., xt ≤ 2 . q T i−1 > Above, ¬ uses the definition yt = x> ; ­ is from the definition of A; and ® is t ZZ (Σ/λk+1 ) > Σ
i−1 owing to Lemma i.A.1 together with the fact that kyt k2 ≤ kxt k2 · ZZ ≤ 1 and the fact λk+1 2 def that Z> Q is independent of V> Q.15 Next, define event  i−1 > C2 = ∃i ∈ [T ], ∃t ∈ [T ], x> Q(V> Q)−1 t ZZ (Σ/λk+1 )  T 1/2 ≥ 18r ln . q 2 The above derivation, after taking union bound, implies that for every fixed x1 , ..., xT , it satisfies PrQ [C2 | C, x1 , ..., xT ] ≤ q 2 . Therefore, denoting by 1C2 the indicator function of event C2 ,     1 Pr [C2 | Q] C Pr Pr [C2 | Q] ≥ q C ≤ E Q x1 ,...,xT q Q x1 ,...,xT   1 = E E [1C | Q] C q Q x1 ,...,xT 2   1 = E E[1C | C, x1 , . . . , xT ] q x1 ,...,xT Q 2   1 = E Pr[C2 | C, x1 , . . . , xT ] ≤ q . q x1 ,...,xT Q Above, the first inequality uses Markov’s bound. In an analogous manner, we define event n  d 1/2 o C3 = ∃j ∈ [d], j ≥ k + 1, kνj> Q(V> Q)−1 k2 ≥ 18r ln p where νj is the j-th eigenvector of Σ corresponding to eigenvalue λj . A completely analogous proof as the lines above also shows PrQ [C3 | C] ≤ q. Finally, using union bound h ^ i h i Pr C3 Pr [C2 | Q] ≥ q ≤ Pr[C3 | C] + Pr Pr [C2 | Q] ≥ q C + Pr[C] ≤ q + q + p , Q x1 ,...,xT Q Q x1 ,...,xT  Q we conclude that with probability at least 1 − p − 2q over the random choice of Q, it satisfies • Prx1 ,...,xT [C2 | Q] < q, and • C3 holds (which implies kZ> Q(V> Q)−1 k2F < 18rd ln dp as desired). 15  In principle, we only proved Lemma i.A.1 when Q is a random matrix, independent of A. Here, A also depends on Q but only on V> Q. Therefore, A is independent from Z> Q, so we can still safely apply Lemma i.A.1. 18 i.B Expectation Lemmas (for Section 6) Let X ∈ Rd×r be a generic matrix that shall later be chosen as either X = W, X = Z, or X = [w] for some vector w. We recall the following notions from Section 6 Lt = Pt Q(V> Pt Q)−1 ∈ Rd×k r×k R0t = X> xt x> t Lt−1 ∈ R St = X> Lt ∈ Rr×k k×k H0t = V> xt x> t Lt−1 ∈ R Lemma i.B.1. For every Q ∈ Rd×k and every t ∈ [T ], suppose for φt ≥ 0, xt satisfies: 1 > > −1 kx> . t Lt−1 k2 = kxt Pt−1 Q(V Pt−1 Q) k2 ≤ φt and ηt φt ≤ 2 Then the following holds: > > 0 > 0 (a) Tr(S> t St ) ≤ Tr(St−1 St−1 ) − 2ηt Tr(St−1 St−1 Ht ) + 2ηt Tr(St−1 Rt ) 2 0 2 2 0 0 + (12ηt2 kH0t k22 + 2ηt2 kR0t k2 kH0t k2 )Tr(S> t−1 St−1 ) + 8ηt kRt k2 + 2ηt kRt k2 kHt k2 > 2 2 0 2 > 2 2 kR0 k2 Tr(S> S 4 2 0 2 (b) |Tr(S> t St )−Tr(St−1 St−1 )| ≤ 243ηt kHt k2 Tr(St−1 St−1 ) +12η t 2 t−1 t−1 )+300ηt φt kRt k2 q t > > 2 2 > (c) |Tr(S> t St ) − Tr(St−1 St−1 )| ≤ 9ηt φt Tr(St−1 St−1 ) + 2ηt φt Tr(St−1 St−1 ) + 10ηt φt Proof of Lemma i.B.1. We first notice that X> Pt Q = X> Pt−1 Q + ηt X> xt x> t Pt−1 Q and V> Pt Q = V> Pt−1 Q + ηt V> xt x> t Pt−1 Q , where the second equality further implies (using the Sherman-Morrison formula) that (V> Pt Q)−1 = (V> Pt−1 Q)−1 − > −1 ηt (V> Pt−1 Q)−1 V> xt x> t Pt−1 Q(V Pt−1 Q) > −1 > 1 + η t x> t Pt−1 Q(V Pt−1 Q) V xt = (V> Pt−1 Q)−1 − (ηt − αt ηt2 )(V> Pt−1 Q)−1 H0t , def and above we denote by αt = ψt 1+ηt ψt def > where ψt = x> t Lt−1 V xt . Therefore, we can write ¬ St = X> Pt Q(V> Pt Q)−1 ­ = St−1 − (ηt − αt ηt2 )St−1 H0t + ηt R0t − (ηt2 − αt ηt3 )R0t H0t ® = St−1 − (ηt − αt ηt2 )St−1 H0t + (ηt − ψt ηt2 + αt ψt ηt3 )R0t ¯ = St−1 − ηt St−1 Ht + ηt Rt . Above, equality ¬ uses the definition of St and Lt ; equality ­ uses our derived equations for X> Pt Q and (V> Pt Q)−1 ; equality ® uses R0t H0t = ψt R0t ; and in quality ¯ we have denoted by Ht = (1 − αt ηt )H0t and Rt = (1 − ψt ηt + αt ψt ηt2 )R0t to simplify the notations. Note that H0t , R0t are rank one matrices so kH0t kF = kH0t k2 and kR0t kF = 19 kR0t k2 . We now proceed and compute > > > Tr(S> t St ) = Tr(St−1 St−1 ) − 2ηt Tr(St−1 St−1 Ht ) + 2ηt Tr(St−1 Rt ) ¬ > 2 > 2 > +ηt2 Tr(H> t St−1 St−1 Ht ) + ηt Tr(Rt Rt ) − 2ηt Tr(Rt St−1 Ht ) > > ≤ Tr(St−1 S> t−1 ) − 2ηt Tr(St−1 St−1 Ht ) + 2ηt Tr(St−1 Rt ) > 2 > +2ηt2 Tr(H> t St−1 St−1 Ht ) + 2ηt Tr(Rt Rt ) ­ > > ≤ Tr(St−1 S> t−1 ) − 2ηt Tr(St−1 St−1 Ht ) + 2ηt Tr(St−1 Rt ) ® 2 2 2 0 2 +2ηt2 (1 − αt ηt )2 kH0t k22 Tr(St−1 S> t−1 ) + 2ηt (1 − ψt ηt + αt ψt ηt ) kRt k2 > 0 > 0 ≤ Tr(St−1 S> t−1 ) − 2ηt Tr(St−1 St−1 Ht ) + 2ηt Tr(St−1 Rt ) 0 2 > 0 +2ηt2 |αt | Tr(S> t−1 St−1 Ht ) + 2ηt (ηt |ψt | + ηt |αt ||ψt |) Tr(St−1 Rt ) ¯ 2 2 2 2 0 2 +2ηt2 (1 + 2φt ηt )2 kH0t k22 Tr(St−1 S> t−1 ) + 2ηt (1 + φt ηt + 2φt ηt ) kRt k2 > 0 > 0 ≤ Tr(St−1 S> t−1 ) − 2ηt Tr(St−1 St−1 Ht ) + 2ηt Tr(St−1 Rt ) 0 2 0 > 0 +4ηt2 kH0t k2 Tr(S> t−1 St−1 Ht ) + 4ηt kHt k2 Tr(St−1 Rt ) ° 2 0 2 +8ηt2 kH0t k22 Tr(St−1 S> t−1 ) + 8ηt kRt k2 > 0 > 0 ≤ Tr(St−1 S> t−1 ) − 2ηt Tr(St−1 St−1 Ht ) + 2ηt Tr(St−1 Rt ) 0 2 0 2 > 2 0 2 +4ηt2 kH0t k2 Tr(S> t−1 Rt ) + 12ηt kHt k2 Tr(St−1 St−1 ) + 8ηt kRt k2 . (i.B.1) Above, ¬ is because 2Tr(A> B) ≤ Tr(A> A) + Tr(B> B) which is Young’s inequality in the matrix case; ­ and ® are both because Ht = (1 − αt ηt )H0t and Rt = (1 − ψt ηt + αt ψt ηt2 )R0t ; ¯ follow from the parameter properties |ψt | ≤ kH0t k2 ≤ φt , |αt | ≤ 2kH0t k2 ≤ 2φt , and 0 ≤ ηt φt ≤ 21 ; ° follows 0 > 0 from |Tr(S> t−1 St−1 Ht )| ≤ Tr(St−1 St−1 )kHt k2 which uses Proposition 2.1. Next, Proposition 2.1 tells us q  kR0t k2  0 0 0 > > S )| ≤ kR k kS k ≤ kR k |Tr(S> R Tr(S ) ≤ Tr(S S ) + 1 , (i.B.2) t−1 2 t−1 t t S1 t 2 t−1 t−1 t−1 t−1 2 (the second inequality is because R0t is rank 1, and the spectral norm of a matrix is no greater than its Frobenius norm.) we can further simplify the upper bound in (i.B.1) as > > 0 > 0 Tr(S> t St ) ≤ Tr(St−1 St−1 ) − 2ηt Tr(St−1 St−1 Ht ) + 2ηt Tr(St−1 Rt )   2 0 2 > 2 0 2 +2ηt2 kR0t k2 kH0t k2 Tr(S> t−1 St−1 ) + 1 + 12ηt kHt k2 Tr(St−1 St−1 ) + 8ηt kRt k2 > 0 > 0 = Tr(St−1 S> t−1 ) − 2ηt Tr(St−1 St−1 Ht ) + 2ηt Tr(St−1 Rt ) 2 0 2 2 0 0 +(12ηt2 kH0t k22 + 2ηt2 kR0t k2 kH0t k2 )Tr(S> t−1 St−1 ) + 8ηt kRt k2 + 2ηt kRt k2 kHt k2 . This finishes the proof of Lemma i.B.1-(a). A completely symmetric analysis of the above derivation also gives > > 0 > 0 Tr(S> t St ) ≥ Tr(St−1 St−1 ) − 2ηt Tr(St−1 St−1 Ht ) + 2ηt Tr(St−1 Rt ) 2 0 2 2 0 0 −(12ηt2 kH0t k22 + 2ηt2 kR0t k2 kH0t k2 )Tr(S> t−1 St−1 ) − 8ηt kRt k2 − 2ηt kRt k2 kHt k2 , 20 and thus combining the upper and lower bounds we have > > 0 > 0 |Tr(S> t St ) − Tr(St−1 St−1 )| ≤ 2ηt |Tr(St−1 St−1 Ht )| + 2ηt |Tr(St−1 Rt )| +(12ηt2 kH0t k22 + 2ηt2 kR0t k2 kH0t k2 )Tr(S> t−1 St−1 ) + 8ηt2 kR0t k22 + (i.B.3) 2ηt2 kR0t k2 kH0t k2 0 ≤ (2ηt kH0t k2 + 12ηt2 kH0t k22 + 2ηt2 kR0t k2 kH0t k2 )Tr(S> t−1 St−1 ) + 2ηt kRt k2 q Tr(S> (i.B.4) ) t−1 St−1 q 2 0 Tr(S> t−1 St−1 ) + 10ηt φt kRt k2 . (i.B.5) ¬ +8ηt2 kR0t k22 + 2ηt2 kR0t k2 kH0t k2 ­ 0 ≤ 9ηt kH0t k2 Tr(S> t−1 St−1 ) + 2ηt kRt k2 Above, ¬ again uses Proposition 2.1 and (i.B.2); ­ uses ηt φt ≤ 1/2 and kH0t k2 , kR0t k2 ≤ φt . Finally, if we take square on both sides of (i.B.5), we have (using again ηt kR0t k2 ≤ 12 ): > 2 2 0 2 > 2 2 0 2 > 4 2 0 2 |Tr(S> t St ) − Tr(St−1 St−1 )| ≤ 243ηt kHt k2 Tr(St−1 St−1 ) + 12ηt kRt k2 Tr(St−1 St−1 ) + 300ηt φt kRt k2 and this finishes the proof of Lemma i.B.1-(b). If we continue to use kH0t k2 , kR0t k2 ≤ φt to upper bound the right hand side of (i.B.5), we finish the proof of Lemma i.B.1-(c).  Proof of Lemma 6.1 from Lemma i.B.1. According to the expectation we have E[H0t | F≤t−1 , C≤t ] = V> (Σ + ∆)Lt−1 and E[R0t | F≤t−1 , C≤t ] = X> (Σ + ∆)Lt−1 . Now we consider the subcases separately: (a) By Lemma i.B.1-(a), h i¬ 2 2 > 2 2 E Tr(S> t St ) | F≤t−1 , C≤t ≤ (1 + 14ηt φt )Tr(St−1 St−1 ) + 10ηt φt h i 0 > 0 + E −2ηt Tr(S> . t−1 St−1 Ht ) + 2ηt Tr(St−1 Rt ) | F≤t−1 , C≤t (i.B.6) kR0t k2 , kH0t k2 ≤ φt . Next, we compute the expectation Above, ¬ uses h i 0 > 0 E −2ηt Tr(S> S H ) + 2η Tr(S R ) | F , C t ≤t−1 ≤t t−1 t−1 t t−1 t > > > = −2ηt Tr(S> t−1 St−1 V (Σ + ∆)Lt−1 ) + 2ηt Tr(St−1 Z (Σ + ∆)Lt−1 ) ­ > > > > ≤ −2ηt gap · Tr(St−1 S> t−1 ) − 2ηt Tr(St−1 St−1 V ∆Lt−1 ) + 2ηt Tr(St−1 Z ∆Lt−1 ) .(i.B.7) > > > > > Above, ­ is because Tr(S> t−1 Z ΣLt−1 ) = Tr(St−1 Σ>k Z Lt−1 ) = Tr(St−1 Σ>k St−1 ) ≤ λk+1 Tr(St−1 St−1 ), > > > > > > as well as Tr(St−1 St−1 V ΣLt−1 ) = Tr(St−1 St−1 Σ≤k V Lt−1 ) = Tr(St−1 St−1 Σ≤k ) ≥ λk Tr(St−1 St−1 ). Using the decomposition I = VV> + ZZ> , kVk2 ≤ 1, kZk2 ≤ 1, and Proposition 2.1 multiple times, we have > > > > > Tr(S> t−1 St−1 V ∆Lt−1 ) = Tr(St−1 St−1 V ∆(VV + ZZ )Lt−1 ) > > > ≤ Tr(S> t−1 St−1 V ∆V) + Tr(St−1 St−1 V ∆ZSt−1 )  ¬  3/2  > ≤ k∆k2 Tr(S> t−1 St−1 ) + Tr(St−1 St−1 ) > > > > > Tr(S> t−1 Z ∆Lt−1 ) = Tr(St−1 Z ∆(VV + ZZ )Lt−1 ) > > > ≤ Tr(S> t−1 Z ∆V) + Tr(St−1 Z ∆ZSt−1 )   > 1/2 ≤ k∆k2 Tr(S> S ) + Tr(S S ) . t−1 t−1 t−1 t−1  3/2 > > Above, ¬ uses the fact that kSt−1 S> t−1 St−1 kS1 ≤ kSt−1 St−1 kS1 kSt−1 k2 ≤ Tr(St−1 St−1 ) 21 Plugging them into (i.B.7) gives h i 0 > 0 > E −2ηt Tr(S> S H ) + 2η Tr(S R ) | F , C t ≤t−1 ≤t ≤ −2ηt gap · Tr(St−1 St−1 ) t−1 t−1 t t−1 t  3/2  1/2  > > +2ηt k∆k2 Tr(S> S ) + 2Tr(S S ) + Tr(S S ) . t−1 t−1 t−1 t−1 t−1 t−1 (i.B.8) Putting this back to (i.B.6) finishes the proof of Corollary 6.1-(a). (b) In this case (i.B.7) also holds but one needs to replace gap with ρ because of the definitional difference between W and Z. We compute the following upper bounds similar to case (a): > > > > > Tr(S> t−1 St−1 V ∆Lt−1 ) = Tr(St−1 St−1 V ∆(VV + ZZ )Lt−1 ) > > > > ≤ Tr(S> t−1 St−1 V ∆V) + Tr(St−1 St−1 V ∆ZZ Lt−1 )  ¬  1/2  > > ≤ k∆k2 Tr(S> t−1 St−1 ) 1 + Tr(Z Lt−1 Lt−1 Z) > > > > > Tr(S> t−1 Z ∆Lt−1 ) = Tr(St−1 Z ∆(VV + ZZ )Lt−1 ) > > > > ≤ Tr(S> t−1 Z ∆V) + Tr(St−1 Z ∆ZZ Lt−1 )   ­ > > 1/2 1/2 (i.B.9) 1 + Tr(Z L L Z) ≤ k∆k2 Tr(S> S ) t−1 t−1 t−1 t−1 Above, ¬ is because (using Proposition 2.1)
1/2  1/2 > > · Tr(V> ∆ZZ> Lt−1 L> t−1 ZZ ∆ V)  1/2 > > ≤ k∆k2 Tr(S> t−1 St−1 ) · Tr(Z Lt−1 Lt−1 Z) > > > 2 Tr(S> t−1 St−1 V ∆ZZ Lt−1 ) ≤ Tr (St−1 St−1 ) and ­ holds for a similar reason. Putting these upper bounds into (i.B.7) finishes the proof of Corollary 6.1-(b). (c) When X = [w], a slightly different derivation of (i.B.7) gives h i 2 2 > 2 2 E Tr(S> t St ) | Ft−1 , C≤t ≤ (1 − 2ηt λk + 14ηt φt )Tr(St−1 St−1 ) + 10ηt φt > > > > > − 2ηt Tr(S> t−1 St−1 V ∆Lt−1 ) + 2ηt Tr(St−1 w ∆Lt−1 ) + 2ηt Tr(St−1 w ΣLt−1 ) . (i.B.10) Note that the third and fourth terms can be upper bounded similarly using (i.B.9). As for the fifth term, we have 1 λk > Tr(S> Tr(w> ΣLt−1 L> Tr(S> t−1 w ΣLt−1 ) ≤ t−1 St−1 ) + t−1 Σw) 2 2λk Putting these together, we have: h i
ηt 2 2 > 2 2 E Tr(S> S ) | F , C kw> ΣLt−1 k22 ≤t−1 ≤t ≤ 1 − ηt λk + 14ηt φt Tr(St−1 St−1 ) + 10ηt φt + t t λk   1/2  1/2  > > > + 2ηt k∆k2 Tr(S> S ) + Tr(S S ) 1 + Tr(Z L L Z)  t−1 t−1 t−1 t−1 t−1 t−1 i.C Martingale Concentrations We prove in the appendix the following two martingale concentration lemmas. Both of them are stated in their most general form for the purpose of this paper. The first lemma is for 1-d martingales and the second is for multi-d martingales. At a high level, Lemma i.C.1 will only be used to analyze the sequences st or s0t (see Section 3) after warm start — that is, after t ≥ T0 . Our Lemma i.C.2 can be used to analyze ct,s as well as st and s0t before warm start. 22 Lemma i.C.1 (1-d martingale). Let {zt }∞ t=t0 be a non-negative random process with starting time 1 ∗ t0 ∈ N . Suppose there exists δ > 0, κ ≥ 2, and τt = δt such that   E[zt+1 | F≤t ] ≤ (1 − δτt )zt + τt2   E[(zt+1 − zt )2 | F≤t ] ≤ τt2 zt + κ2 τt4 ∀t ≥ t0 : (i.C.1) √   |zt+1 − zt | ≤ κτt zt + κ2 τt2 2 If there exists φ ≥ 36 satisfying lnt20t ≥ 7.5κ2 (φ + 1) with zt0 ≤ φ2δln2 t0t0 , we have: 0 h i exp{− φ −1
ln t } 0 (φ+1) ln2 t 36 Pr ∃t ≥ t0 , zt > ≤ . φ δ2 t 36 −1 Lemma i.C.2 (multi-dimensional martingale). Let {zt }Tt=0 be a random process where each zt ∈ T −1 RD ≥0 is F≤t -measurable. Suppose there exist nonnegative parameters {βt , δt , τt }t=0 satisfying κ ≥ 0 and κτt ≤ 1/6 such that, ∀i ∈ [D], ∀t ∈ {0, 1, . . . , T − 1}, (denoting by [zt ]i is the i-th coordinate of zt and [zt ]D+1 = 0)    2 )[z ] + δ [z ] 2 , E [z ] | F ≤ (1 − β − δ + τ + τ  t+1 i ≤t t t t i t t i+1 t t   
E |[zt+1 ]i − [zt ]i |2 | F≤t ≤ τt2 [zt ]2i + [zt ]i +κ2 τt4 , and (i.C.2) p   [zt+1 ]i − [zt ]i ≤ κτt [zt ]i + [zt ]i + κ2 τt2 .   Then, we have: for every λ > 0, every p ∈ 1, mins∈[t] { 6κτ1s−1 } :  nP o   t−1 2 τ 2 − pβ Pr [zt ]1 ≥ λ ≤ λ−p maxj∈[t+1] {[z0 ]pj } exp 5p s s s=0 nP o Pt−1 t−1 2 2 . + 1.4 s=0 exp u=s+1 5p τu − pβu The above two lemmas are stated in the most general way in order to be used towards all of our three theorems each requiring different parameter choices of βt , δt , τt , κ. For instance, to prove Theorem 2 it suffices to use κ = O(1). i.C.1 Martingale Corollaries We provide below four instantiations of these lemmas, each of them can be verified by plugging in the specific parameters. Corollary i.C.3 (1-d martingale). Consider the same setting as Lemma i.C.1. Suppose p ∈ (0, e12 ),   1 δ ≤ √18 , τt = δt , κ ∈ 2, √12δ , lnt20t ≥ 9 ln(1/p) , and zt0 ≤ 2 we have: δ2 0 h i 2 0 / ln t0 ) ≤p . Pr ∃t ≥ t0 , zt > 5(tt/ 2 ln t Corollary i.C.4 (multi-d martingale). Consider the same setting as Lemma i.C.2. Suppose q ∈ Pt−1 −1 4t 1 2 (0, 1), mins∈[t] { 6κτ1s−1 } ≥ 4 ln 4t and s=0 τs ≤ 100 ln q q , then h i  Pr [zt ]1 ≥ 2 max 1, max {[z0 ]j } ≤ q . j∈[t+1] Corollary i.C.5 (multi-d martingale). Consider the same setting as Lemma i.C.2. Given q ∈ def (0, 1), suppose there exists parameter γ ≥ 1 such that, denoting by l = 10γ ln 3t q,   t−1 X ^ 1 2 βs − lτs ≥ ln max {[z0 ]j } and ∀s ∈ {0, 1, . . . , t − 1} : βs ≥ lτs2 κτs ≤ . j∈[t+1] 12 ln 3t q s=0   Then, we have Pr [zt ]1 ≥ 2/γ ≤ q . 23 i.C.2 Proofs for One-Dimensional Martingale Proof of Lemma i.C.1. Define yt = δ 2 tzt ln t − ln t, then we have: δ 2 (t + 1) E[zt+1 | F≤t ] − ln(t + 1) ln(t + 1) δ 2 (t + 1)(1 − δτt )zt δ 2 (t + 1)τt2 + − ln(t + 1) ≤ ln(t + 1) ln(t + 1)
¬ δ 2 tzt δ 2 (t + 1) 1 − 1t t+1 ≤ zt + 2 − ln(t + 1) ≤ − ln t = yt , ln(t + 1) t ln(t + 1) ln t
t2 1 t+1 where ¬ is because for every t ≥ 4 it satisfies (t+1)(t−1) ln(t+1) ≤ ln t and t2 ln(t+1) ≤ ln 1 + t . At the same time, we have ­ δ2t δ2 1 |yt+1 − yt | ≤ |zt+1 − zt | + zt+1 + , (i.C.3) ln t ln t t t+1 where ­ is because for every t ≥ 3 it satisfies 0 ≤ ln(t+1) − lnt t ≤ ln1 t and ln(t + 1) − ln(t) ≤ 1/t. Taking square on both sides, we have  2 2  2 2 δ t δ 3 2 2 2 |yt+1 − yt | ≤ 3 |zt+1 − zt | + 3 zt+1 + 2 . ln t ln t t Taking expectation on both sides, we have  2 2 (yt + ln t)2 3 δ t 2 (τt2 zt + κ2 τt4 ) + 3 + 2 E[|yt+1 − yt | | F≤t ] ≤ 3 2 ln t t t 2 2 3(yt + ln t) 3(yt + ln t) 3(1 + κ ) < + + t ln t t2 t2 2 2 2 ® ¯ 4(φt + 1) 3(φ + 1) 3(φ + 1) ln t 15κ ≤ + + ≤ . t t2 4t2 t  Above, ® uses yt ≤ φ ln t and κ ≥ 2; ¯ uses lnt2 t ≥ lnt20t ≥ max 7.5κ2 , 6(φ + 1) and ln t ≥ 1. 0 Therefore, if yt ≤ φ ln t holds true for t = t0 , ..., T and t0 ≥ 8 (which implies lnt2 t ≥ lnt20t ), then 0 Z T T T X X dt 4(φ + 1) ≤ 4(φ + 1) ≤ 4(φ + 1) ln(T ) . E[|yt+1 − yt |2 | F≤t ] ≤ t t=t0 −1 t t=t t=t E[yt+1 | F≤t ] = 0 0 Now we can check about the absolute difference. We continue from (i.C.3) and derive that, if yt ≤ φ ln t, then
√ δ2t δ2 1 δ2t δ2 1 |yt+1 − yt | ≤ |zt+1 − zt | + zt+1 + ≤ κτt zt + κ2 τt2 + zt+1 + ln t ln t t ln t! ln t t ! r r ° yt + ln t κ (yt + ln t) 1 yt + ln t yt + ln t + κ ≤ κ + + + ≤ κ + t ln t t ln t t t t ln t t ! r r ± ² (φ + 1) (φ + 1) ln t + κ (φ + 1) ≤ κ + ≤ 2κ t t t where ° uses ln t ≥ 2 and κ ≥ 2, ± uses yt ≤ φ ln t, and ² uses lnt2 t ≥ lnt20t ≥ 4 max{φ + 1, κ}. 0 From the above inequality, we have that if t0 ≥ 4κ2 (φ + 1) and yt ≤ φ ln t holds true for t = t0 , ..., T − 1 then |yt+1 − yt | ≤ 1 for all t = t0 , . . . , T − 1. 2 Finally, since we have assumed φ > 36 and zt0 ≤ φ2δln2 t0t0 which implies yt0 ≤ φ ln2 t0 , we can apply 24 martingale concentration inequality (c.f. [5, Theorem 18]): ∞ X   Pr [∃t ≥ t0 , yt > φ ln t] ≤ Pr yT > φ ln T ; ∀t ∈ {t0 , ..., T − 1}, yt ≤ φ ln t ≤ T =t0 +1 ∞ X T =t0 +1 ∞ X   Pr yT − yt0 > φ ln T /2; ∀t ∈ {t0 , ..., T − 1}, yt ≤ φ ln t ) −(φ ln T /2)2 ≤ exp 2 · 4(φ + 1) ln(T − 1) + 23 (φ ln T /2) T =t0 +1   ∞ X φ2 /4 ≤ exp − ln T 8(φ + 1) + φ/3 T =t0 +1
  Z ∞ φ exp{− 36 − 1 ln t0 } φ exp − ln T dT ≤ ≤ . φ 36 T =t0 36 − 1 def 4δ 2 t0 ln2 t0 φ ln2 t0 2δ 2 t0 Proof of Corollary i.C.3. Define φ = √ 1) (because κ ≤ 1/( 2δ)) and zt0 ≤ ( ≥ 36 ln p1 ≥ 72. It is easy to verify that t0 ln2 t0 ≥ 7.5κ2 (φ+ = 2, so we can apply Lemma i.C.1: n o
φ      2  exp − − 1 ln t 0 36 φ (φ + 1) ln t ≤ exp − Pr ∃t ≥ t0 , zt > ≤ − 1 ln t 0 ≤p , φ δ2t 36 − 1 36
φ φ − 1 ln t0 ≥ 36 . Therefore, we conclude that where the last inequality uses ln t0 ≥ 2 and 36     2 5(t0 / ln t0 ) (φ + 1) ln2 t Pr ∃t ≥ t0 , zt > ≤ Pr ∃t ≥ t0 , zt > ≤p . δ2t t/ ln2 t i.C.3   Proofs for Multi-Dimensional Martingale  Proof of Corollary i.C.4. We apply Lemma i.C.2 with λ = 2 max 1, maxj∈[t+1] {[z0 ]j } ≥ 2. Using the fact that βt ≥ 0, we have ( )   t−1 X   Pr [zt ]1 ≥ λ = Pr [zt ]1 ≥ 2( max {[z0 ]j } + 1) ≤ (1 + 1.4t) exp − ln(2p ) + 5p2 τs2 . j∈[t+1] We can take p = 4 ln 4t q fore, denoting by α = ≤ mins∈[t] { 6κτ1s−1 } Pt−1 2 s=0 τs , we have s=0 which satisfies the assumption of Lemma i.C.2. There- n o   Pr [zt ]1 ≥ λ ≤ 4t exp − p ln 2 + 5p2 α ≤ q .  2 4t Above, the last inequality is due to −p ln 2 + 5p2 α2 ≤ −2 ln 4t + 5 4 ln α ≤ − ln 4t q q q which holds for every α ≤ 1 100 ln−1  4t q. Proof of Corollary i.C.5. We consider fixed p = with (using the fact that γ ≥ 1) l 5γ βt0 = βt , = 2 ln 3t q . Let yt = γ · zt , then yt satisfies (i.C.2) δt0 = δt , (τt0 )2 = γτt2 , κ0 = κ . def Pt−1 def P 2 We denote by b = s=0 βs = b and a = t−1 s=0 τs , and apply Lemma i.C.2 on yt with λ = 2. Using 2 2 0 the fact that βs ≥ lτs = 5zpτs we know pβt ≥ 5p2 (τt0 )2 . Therefore, for all s ∈ {0, 1, . . . , T − 1} we 25 have  Pr [[yt ]1 ≥ 2] ≤ exp −pb + 5p2 γa + p ln Ξ − p ln 2 + 1.4t exp {−p ln 2} , (i.C.4) def where we have denoted by Ξ = maxj∈[t+1] {[z0 ]j } for notational simplicity. Now, the choice p = 1 2 ln 3t q satisfies the presumption of Lemma i.C.2 because we have assumed κτs ≤ 12 ln 3t Therefore, q we have ^ 3t q ⇐= b − la ≥ ln Ξ p ≥ 2 ln 2 q q 3t −p ln 2 ≤ ln ⇐= p ≥ 2 ln . 3t q i h    Plugging them into (i.C.4) gives Pr [zt ]1 ≥ γ2 = Pr [yt ]1 ≥ 2 ≤ 2q + 2q = q . −pb + 5p2 γa + p ln Ξ − p ln 2 = p(−b + la + ln Ξ − ln 2) ≤ ln Proof of Lemma i.C.2. Define vector st for every t ∈ {0, 1, . . . , T − 1} and i ∈ [D], it satisfies def ]i [st ]i = [z[zt+1 − 1. We have t ]i   τ2 [zt ]i+1 + t . E [st ]i | F≤t ≤ −(δt + βt − τt2 ) + δt [zt ]i [zt ]i In particular,   κ2 τt4 τ2 ≤ (2 + (τt κ)2 )τt2 ≤ 3τt2 , E [st ]2i | F≤t ≤ τt2 + t + [zt ]i [zt ]2i κτt κ2 τt2 [st ]i ≤ κτt + p + ≤ κτt (2 + κτt ) ≤ 3κτt . [zt ]i [zt ]i if [zt ]i ≥ 1, then (i.C.5) (i.C.6) (i.C.7) We consider [zt+1 ]pi for some fixed value p ≥ 1 and derive that (using (i.C.7)) X  p   p 1 [st ]qi if (κτt )p ≤ and [zt ]i ≥ 1, then [zt+1 ]pi = [zt ]pi (1 + [st ]i )p = [zt ]pi 6 q q=0
≤ [zt ]pi 1 + p[st ]i + p2 [st ]2i . After taking expectation, we have if (κτt )p ≤ ¬ E [[zt+1 ]pi | F≤t ] ≤ ­ ≤ = ® ≤ = ¯ ≤ 1 6 and [zt ]i ≥ 1, then
[zt ]pi 1 + p E [[st ]i | F≤t ] + 3p2 τt2   τt2 p [zt ]i+1 p 2 2 2 [zt ]i 1 − p(δt + βt − τt ) + δt p + + 3p τt [zt ]i [zt ]i
[zt ]pi 1 − p(δt + βt − τt2 ) + 3p2 τt2 + δt p[zt ]p−1 [zt ]i+1 + pτt2 [zt ]ip−1 i  
p−1 p 1 p p 2 2 2 2 [zt ]i + [zt ]i+1 [zt ]i 1 − p(δt + βt − τt ) + 3p τt + pτt + δt p p p
p p 2 2 2 2 [zt ]i 1 − δt − pβt + pτt + 3p τt + pτt + δt [zt ]i+1
[zt ]pi 1 − δt − pβt + 5p2 τt2 + δt [zt ]pi+1 . Above, ¬ uses (i.C.6); ­ uses (i.C.5); ® uses [zt ]i ≥ 1 and Young’s inequality ab ≤ ap /p + bq /q for 1/p + 1/q = 1; and ¯ uses p ≥ 1. On the other hand, if (κτt )p ≤ 61 but [zt ]i < 1, we have the following simple bound (using κτt ≤ 1/6): p [zt+1 ]i ≤ (1 + κτt )[zt ]i + κτt [zt ]i + κ2 τt2 ≤ (1 + κτt ) + (κτt ) + κ2 τt2 < 1.4 . Therefore, as long as (κτt )p ≤ 16 we always have  
E [zt+1 ]pi | F≤t ≤ [zt ]pi 1 − δt − pβt + 5p2 τt2 + δt [zt ]pi+1 + 1.4 =: (1 − αt )[zt ]pi + δt [zt ]pi+1 + 1.4 , 26 def and in the last inequality we have denoted by αt = δt + pβt − 5p2 τt2 . Telescoping this expectation, and choosing i = 1, we have whenever p ∈ [1, mins∈[t] { 6κτ1s−1 }], it satisfies !   t t t Y X Y p p E [[zt+1 ]1 ] ≤ (1 − αs + δs ) max {[z0 ]j } + 1.4 (1 − αu + δu ) ≤ ≤ j∈[t+2] s=1 t Y (1 − pβs + 5p2 τs2 ) s=0 max j∈[t+2] {[z0 ]pj } exp (  −p s=0 u=s+1  t X p max {[z0 ]j } + 1.4 j∈[t+2] t X s=0 βs ! s=0 + 5p 2 t X s=0 τs2 ) t Y (1 − pβu + 5p2 τu2 ) u=s+1 t X + 1.4 s=0 ( exp −p u=s+1 Finally, using Markov’s inequality, we have for every λ > 0:  nP o   t 2 τ 2 − pβ Pr [zt+1 ]1 ≥ λ ≤ λ−p maxj∈[t+2] {[z0 ]pj } exp 5p s s s=0 nP o Pt t 2 2 + 1.4 s=0 exp . u=s+1 5p τu − pβu i.D t X ! βu ! + 5p 2 u=s+1  Decoupling Lemmas We prove the following general lemma. Let x1 , ..., xT ∈ Ω be random variables each i.i.d. drawn from some distribution D. Let Ft be the sigma-algebra generated by xt , and denote by F≤t = ∨ts=1 Ft .16 Lemma i.D.1 (decoupling lemma). Consider a fixed value q ∈ [0, 1). For every t ∈ [T ] and s ∈ {0, 1, ..., t − 1}, let yt,s ∈ RD be an Ft ∨ F≤s measurable random vector and let φt,s ∈ RD be a fixed vector. Let D0 ∈ [D]. Define events (we denote by (i) the i-th coordinate)   h i (i) (i) 0 0 def Ct = (x1 , ..., xt−1 ) satisfies Pr ∃i ∈ [D ] : yt,t−1 > φt,t−1 Ft−1 ≤ q xt n o def (i) (i) Ct00 = (x1 , ..., xt ) satisfies ∀i ∈ [D0 ] : yt,t−1 ≤ φt,t−1 def def V and denote by Ct = Ct0 ∧ Ct00 and C≤t = ts=1 Cs . Suppose the following three assumptions hold: (A1) The random process {yt,s }t,s satisfy that for every i ∈ [D], t ∈ [T − 1], s ∈ {0, 1, . . . , t − 2}  (i) 
(i) (a) E yt,s+1 | Ft , F≤s , C≤s ≤ fs yt,s , q ,  (i) 
(i) (i) (b) E |yt,s+1 − yt,s |2 | Ft , F≤s , C≤s ≤ hs yt,s , q , and
(i) (i) (i) (c) yt,s+1 − yt,s ≤ gs yt,s whenever C≤s holds. d Above, for each i ∈ [D] and s ∈ {0, 1, . . . , T − 2}, we have fs , hs : Rd × [0, 1] → RD ≥0 , gs : R → D d R≥0 are functions satisfying for every x ∈ R , (i) (i) (d) fs (x, p), hs (x, p) are monotone increasing in p, and 2 (i) (i) (i) (i) (i) (e) x(i) − fs (x, 0) ≤ hs (x, 0) and x(i) − fs (x, 0) ≤ gs (x) whenever fs (x, 0) ≤ x(i) . (A2) Each t ∈ [T ] satisfies Prxt [Et ] ≤ q 2 /2 where event def  (i) (i) Et = xt satisfies ∀i ∈ [D] : yt,0 ≤ φt,0 16 d . For the purpose of this paper, one can feel free view Ω as R , each xt as the t-th sample vector, and D as the distribution with covariance matrix Σ. 27 t X τu2 ) . (A3) For every t ∈ [T ], letting xt be any vector satisfying Et , consider any random process {zs }t−1 s=0 where each zs ∈ RD ≥0 is F≤s measurable with z0 = yt,0 as the starting vector. Suppose that t−1 whenever {zs }s=0 satisfies    (i)  (i)  E zs+1 | F≤s ≤ fs (zs , q)      (i) (i) (i) ∀i ∈ [D], ∀s ∈ {0, 1, . . . , t − 2} : (i.D.1) E |zs+1 − zs |2 | F≤s ≤ hs (zs , q)     (i) (i) (i) zs+1 − zs ≤ gs (zs ) (i) (i) then it holds Prx1 ,...,xt−1 [∃i ∈ [D0 ] : zt−1 > φt,t−1 ] ≤ q 2 /2. Under the above two assumptions, we have for every t ∈ [T ], it satisfies Pr[Ct ] ≤ 2tq . Proof of Lemma i.D.1. We prove the lemma by induction. For the base case, by applying assump (i) (i)  tion (A2) we know that Prx1 ∃i ∈ [D0 ] : y1,0 > φ1,0 ≤ Pr[E1 ] ≤ q 2 /2 ≤ q so event C1 holds with probability at least 1 − q. In other words, Pr[C≤1 ] = Pr[C1 ] ≤ q < 2q. Suppose Pr[C≤t−1 ] ≤ 2(t − 1)q is true for some t ≥ 2, we will prove Pr[C≤t ] ≤ 2tq. Since it satisfies Pr[C≤t ] ≤ Pr[C≤t−1 ] + Pr[Ct ], it suffices to prove that Pr[ Ct ] ≤ 2q. Note also Pr[ Ct ] ≤ Pr[ Ct0 ] + Pr[ Ct00 | Ct0 ] but the second quantity Pr[ Ct00 | Ct0 ] is no more than q according to our definition of Ct0 and Ct00 . Therefore, in the rest of the proof, it suffices to show Pr[ Ct0 ] ≤ q. We use yt,s (xt , x≤s ) to emphasize that yt,s is an Ft × F≤s measurable random vector. Let us D now fix xt to be a vector satisfying Et . Define {zs }t−1 s=0 to be a random process where each zs ∈ R is F≤s measurable: ( (i)
yt,s n xt , x≤s
def (i) (i) o if x≤s satisfies C≤s ;
(i) zs = zs x≤s = (i.D.2) (i) min fs−1 zs−1 (x≤s−1 ), 0 , zs−1 (x≤s−1 ) if x≤s satisfies C≤s . (i) Then zs satisfies for every i ∈ [D], s ≤ {0, 1, . . . , t − 2},  (i)   (i)    (i)   E zs+1 | F≤s = Pr[C≤s+1 | F≤s ] · E zs+1 | C≤s+1 , F≤s + Pr C≤s+1 | F≤s · E zs+1 | C≤s+1 , F≤s ¬  (i)    ≤ Pr[C≤s+1 | F≤s ] · E yt,s+1 | C≤s+1 , F≤s + Pr C≤s+1 | F≤s · fs(i) (zs , 0) ­ 

 ≤ Pr[C≤s+1 | F≤s ] · fs(i) yt,s , q + Pr C≤s+1 | F≤s · fs(i) zs , q ®
 
≤ Pr[C≤s+1 | F≤s ] · fs(i) yt,s , q + Pr C≤s+1 | F≤s · fs(i) yt,s , q (i.D.3) = fs(i) (zs , q) (i.D.4) Above, ¬ is because whenever C≤s+1 holds it satisfies (i) zs+1 (i) fs (zs , 0); (i) zs+1 = (i) yt,s+1 , as well as whenever C≤s+1 holds it satisfies ≤ ­ uses assumptions  (A1a) and (A1d) as well as the fact that we have fixed xt ; ® uses the fact that whenever Pr C≤s+1 | F≤s > 0 it must hold that C≤s is satisfied, and therefore it satisfies yt,s = zs . Similarly, we can also show for every i ∈ [D], s ≤ {0, 1, . . . , t − 2},  (i)  E |zs+1 − zs(i) |2 | F≤s   (i)   (i)   = Pr[C≤s+1 | F≤s ] · E |zs+1 − zs(i) |2 | C≤s+1 , F≤s + Pr C≤s+1 | F≤s · E |zs+1 − zs(i) |2 | C≤s+1 , F≤s ¬  (i)    (i) ≤ Pr[C≤s+1 | F≤s ] · E |yt,s+1 − yt,s |2 | C≤s+1 , F≤s + Pr C≤s+1 | F≤s · h(i) s (zs , 0) ­   (i) ≤ Pr[C≤s+1 | F≤s ] · h(i) s yt,s , q) + Pr C≤s+1 | F≤s · hs (zs , q) ® ≤ h(i) s (zs , q) . (i.D.5) 28 (i) (i) (i) Above, ¬ is because whenever C≤s+1 holds it satisfies zs+1 = yt,s+1 and zs (i) (i) (i) = yt,s , together (i) (i) 2 with whenever C≤s+1 holds it satisfies |zs+1 − ys |2 either equal zero or equal fs (zs , 0) − zs (i) fs (zs , 0) , (i) zs but in the latter case we must have < (owing to (i.D.2)) and therefore it holds (i) (i) 2 (i) fs (zs , 0) − zs ≤ hs (zs , 0) using assumption (A1e). ­ uses assumptions  (A1b) and (A1d) as well as the fact that we have fixed xt . ® uses the fact that whenever Pr C≤s+1 | F≤s > 0 then C≤s must hold, and therefore it satisfies yt,s = zs . Finally, we also have (i) |zs+1 − zs(i) | ≤ gs(i) (zs(i) ) . (i.D.6) (i) |zs+1 (i) (i) (i) This is so because whenever C≤s+1 holds it satisfies − zs | = |yt,s+1 − yt,s | so we can apply (i) (i) assumption (A1c). Otherwise, C≤s+1 holds we either have |zs+1 − zs | = 0 (so (i.D.6) trivially
(i) (i) (i) (i) (i) (i) holds) or |zs+1 − zs | = fs zs , 0 − zs , but in the latter case we must have fs (zs , 0) < zs
(i) (i) (i) (owing to (i.D.2)) so it must satisfy fs zs , 0 − zs ≤ gs (zs ) using assumption (A1e). We are now ready to apply assumption (A3), which together with (i.D.4), (i.D.5), (i.D.6), implies that (recalling we have fixed xt to be any vector satisfying Et )   (i) (i) Pr ∃i ∈ [D0 ] : zt−1 > φt,t−1 | Et ≤ q 2 /2 . x1 ,...,xt−1 This implies, after translating back to the random process {yt,s }, we have    (i) (i)  (i) (i) Pr ∃i ∈ [D0 ] : yt,t−1 > φt,t−1 ≤ Pr ∃i ∈ [D0 ] : yt,t−1 > φt,t−1 | Et + Pr[Et ] x1 ,...,xt x1 ,...,xt   (i) (i) ≤ Pr ∃i ∈ [D0 ] : zt−1 > φt,t−1 | Et + q 2 /2 x1 ,...,xt−1 ≤ q 2 /2 + q 2 /2 = q 2 . where the last inequality uses (A2). Finally, using Markov’s inequality,      0 (i) (i) 0 Ct = Pr Pr ∃i ∈ [D ] : yt,t−1 > φt,t−1 | F≤t−1 > q Pr x1 ,...,xt−1 xt x1 ,...,xt−1   1 (i) (i) 0 ≤ · E Pr[∃i ∈ [D ] : yt,t−1 > φt,t−1 | F≤t−1 ] q x1 ,...,xt−1 xt h i 1 (i) (i) = · Pr ∃i ∈ [D0 ] : yt,t−1 > φt,t−1 ≤ q . q x1 ,...,xt Therefore, we finish proving Pr[Ct0 ] ≤ q which implies Pr[C≤t ] ≤ 2tq as desired. This finishes the proof of Lemma i.D.1.  i.E i.E.1 Main Lemmas (for Section 7) Before Warm Start Proof of Lemma Main 1. For every t ∈ [T ] and s ∈ {0, 1, ..., t − 1}, consider random vectors yt,s ∈ RT +2 defined as: (1) def (2) def yt,s = kZ> Ps Q(V> Ps Q)−1 k2F , yt,s = kW> Ps Q(V> Ps Q)−1 k2F ,   x> ZZ> (Σ/λ )j P Q(V> P Q)−1 s s (3+j) def k+1 t yt,s = (3+j)  (1 − η λ ) · y , s k t,s−1 29 2 2 , for j ∈ {0, 1, . . . , t − s − 1}; for j ∈ {t − s, . . . , T − 1}. (3+j) (3+j) (In fact, we are only interested in yt,s for j ≤ t − s − 1, and can “almost” define yt,s = +∞ whenever j ≥ t − s. However, we still decide to give such out-of-boundary variables meaningful values in order to make all of our vectors yt,s (and functions f, g, h defined later) to be of the same dimension T + 2. This allows us to greatly simplify our notations.) We consider upper bounds  2ΞZ s < T0 ; (3+j) def (2) def (1) def , and φt,s = 2Ξ2x . φt,s = 2ΞZ , φt,s = 2 otherwise. For each t ∈ [T ], define event Ct0 and Ct00 in the same way as decoupling Lemma i.D.1 (with D0 = 3):   h i (i) (i) 0 def Ct = (x1 , ..., xt−1 ) satisfies Pr ∃i ∈ [3] : yt,t−1 > φt,t−1 Ft−1 ≤ q xt n o def (i) (i) Ct00 = (x1 , ..., xt ) satisfies ∀i ∈ [3] : yt,t−1 ≤ φt,t−1 def V def and denote by Ct = Ct0 ∧ Ct00 and C≤t = ts=1 Cs . As a result, if C≤s+1 holds, then we always have
2 > −1 2 > > > −1 > > > −1 kx> s+1 Ps Q(V Ps Q) k2 ≤ kxs+1 VV Ps Q(V Ps Q) k2 + kxs+1 ZZ Ps Q(V Ps Q) k2 √ (3) ≤ (1 + φs+1,s )2 = ( 2Ξx + 1)2 ≤ 4Ξ2x , where last inequality uses Ξx ≥ 2. This allows us to later apply Lemma i.B.1 and Lemma 6.1 with φt = 2Ξx . Verification of Assumption (A1) in Lemma i.D.1. def 00 Suppose E[xs x> s | C≤s , F≤s−1 ] = Σ + ∆, and we want to bound k∆k2 . Defining q1 = Pr[Cs | 00 0 0 Cs , C≤s−1 , F≤s−1 ], then we must have q1 ≤ q according to the definition of Cs and Cs . Using law of total expectation: 0 > > 00 0 E[xs x> s | Cs , C≤s−1 , F≤s−1 ] = E[xs xs | C≤s , F≤s−1 ] · (1 − q1 ) + E[xs xs | Cs , Cs , C≤s−1 , F≤s−1 ] · q1 , > 0 > and combining it with the fact that 0  xs x> s  I and E[xs xs | Cs , C≤s−1 , F≤s−1 ] = E[xs xs ] = Σ, 17 we have Σ  (Σ + ∆)(1 − q1 ) + q1 · I and Σ  (Σ + ∆)(1 − q1 ) . q1 q After rearranging, these two properties imply k∆k2 ≤ 1−q . ≤ 1−q 1 Now, we can apply Lemma 6.1 and obtain for every t ∈ [T ], s ∈ {0, 1, . . . , t − 2}, and every j ∈ {0, 1, . . . , T − 1}, it satisfies18 (1) (1) 2 2 E[yt,s+1 | Ft , F≤s , C≤s+1 ] ≤ (1 + 56ηs+1 Ξ2x )yt,s + 40ηs+1 Ξ2x + 20ηs+1 (2) (2) (3+j) (3+j) 3/2 qΞZ 1−q , 2 2 E[yt,s+1 | Ft , F≤s , C≤s+1 ] ≤ (1 − 2ηs+1 ρ + 56ηs+1 Ξ2x )yt,s + 40ηs+1 Ξ2x + 20ηs+1 2 E[yt,s+1 | Ft , F≤s , C≤s+1 ] ≤ (1 − ηs+1 λk + 56ηs+1 Ξ2x )yt,s (3+j+1) + ηs+1 λk yt,s 3/2 qΞZ 1−q , and 2 + 40ηs+1 Ξ2x + 20ηs+1 Moreover, for every i ∈ [T + 2], using Lemma i.B.1-(c) with φt = 2Ξx we have whenever C≤s+1 17 18 Here, we use notation A  B to indicate spectral dominance: that is, B − A is positive semidefinite. We make a few comments regarding how to derive these upper bounds. • Event C≤s+1 implies we can safely apply Lemma 6.1 with φt = 2Ξx . • To obtain the third inequality, one needs to use the fact that when w = xt ZZ> the quantity ληkt kw> ΣLt−1 k22 ηt λ2 k+1 kw> Lt−1 k22 ≤ ηt λk kw> Lt−1 k22 . λk (3+j) def (3+j) which means yt,s = (1 − ηs λk ) · yt,s−1 according that appeared in Corollary 6.1-(c) can be upper bounded by (3+j) • Whenever j ≥ t − s we have yt,s is “out of boundary” to our definition. In this case it easily satisfies the third inequality. (3+j+1) • When j = T − 1, in the third inequality we should replace yt,s with zero because it is out of bound. 30 3/2 qΞZ 1−q . holds it satisfies (i) (i) (i) yt,s+1 − yt,s ≤ 18ηs+1 Ξx · yt,s + 4ηs+1 Ξx · q (i) (i) 2 2 yt,s + 40ηs+1 Ξ2x ≤ 20ηs+1 Ξx · yt,s + 42ηs+1 Ξ2x . Putting the above bounds together, one can verify that the random process {yt,s }t∈[T ],s≤t−1 satisfy assumption (A1) of Lemma i.D.1 with19 2 2 Ξ2x )y (1) + 40ηs+1 Ξ2x + 20ηs+1 fs(1) (y, q) = (1 + 56ηs+1 3/2 qΞZ 1−q , 2 2 Ξ2x + 20ηs+1 fs(2) (y, q) = (1 − 2ηs+1 ρ + 56ηs+1 Ξ2x )y (2) + 40ηs+1 3/2 qΞZ 1−q , 2 2 Ξ2x )y (3+j) + ηs+1 λk y (3+j+1) + 40ηs+1 Ξ2x + 20ηs+1 fs(3+j) (y, q) = (1 − ηs+1 λk + 56ηs+1 3/2 qΞZ 1−q , 2 gs(i) (y) = 20ηs+1 Ξx · y (i) + 42ηs+1 Ξ2x , and
2 (i) h(i) s (y, q) = gs (y) Verification of Assumption (A2) of Lemma i.D.1. (i) For coordinates i = 1 and i = 2, our assumption kZ> Q(V> Q)−1 k2F ≤ ΞZ implies yt,0 ≤ ΞZ < h (i) j−1 > Q(V> Q)−1 2 ≤ φt,0 . For coordinates i ≥ 3, we have assumption Prxt ∀j ∈ [T ], x> t ZZ (Σ/λk+1 ) i def  (i) (i) Ξx ≥ 1 − q 2 /2. Together, event Et (recall Et = xt satisfies ∀i ∈ [D] : yt,0 ≤ φt,0 ) holds for all t ∈ [T ] with probability at least 1 − q 2 /2. In sum, assumption (A2) is satisfied in Lemma i.D.1. Verification of Assumption (A3) of Lemma i.D.1. For every t ∈ [T ], at a high level assumption (A3) is satisfied once we plug in the following three sets of parameter choices to Corollary i.C.4 and Corollary i.C.5: for every s ∈ [T − 1], define βs,1 = 0, δs,1 = 0, τs,1 = 20ηs+1 Ξx βs,2 = 2ηs+1 ρ, δs,2 = 0, τs,2 = 20ηs+1 Ξx βs,3 = 0, δs,3 = ηs+1 λk τs,3 = 20ηs+1 Ξx More specifically, for every t ∈ [T ], let Lemma i.D.1. Define q2 = q 2 /8. {zs }t−1 s=0 be the arbitrary random vector satisfying (i.D.1) of • For coordinate i = 1 of {zs }t−1 s=0 , – apply Corollary i.C.4 with {βs,1 , δs,1 , τs,1 }t−2 s=0 , q = q2 , D = 1, and κ = 1; • For coordinate i = 2 of {zs }t−1 s=0 , – if t < T0 , apply Corollary i.C.4 with {βs,2 , δs,2 , τs,2 }t−2 s=0 , q = q2 , D = 1, and κ = 1; t−2 – if t ≥ T0 , apply Corollary i.C.5 with {βs,2 , δs,2 , τs,2 }s=0 , q = q2 , D = 1, γ = 1, and κ = 1; • For coordinates i = 3, 4, . . . , T + 2 of {zs }t−1 s=0 , – apply Corollary i.C.4 with {βs,3 , δs,3 , τs,3 }t−2 s=0 , q = q2 , D = T , and κ = 1. 19
(i) (i) The only part of (A1) that is non-trivial to verify is (A1e) for gs . Whenever fs x, 0 ≤ x(i) , it satisfies  if i = 1;  0,
2ηs+1 ρ · x(2) , if i = 2; ≤ 2ηs+1 · x(i) ≤ gs(i) (x) , fs(i) x, 0 − x(i) ≤  ηs+1 λk · x(i) , if i ≥ 3. where the second inequality uses ρ, λk ≤ 1 and the last inequality uses Ξx ≥ 2. 31 One needs to verify that the assumptions of Corollary i.C.4 and i.C.5 are satisfied as follows. First of all, one can carefully check that our parameters β, δ, τ satisfy (i.C.2) with κ = 1 and this needs our assumption q ≤ ηs+1 3/2 . Next, we can apply Corollary i.C.4 because our assumptions on ηs ΞZ PT −1 2 1 1 imply s=0 τs,i ≤ 100 ln−1 4T q2 and τs,i ≤ 24 log(4T /q2 ) for i = 1, 2, 3. To verify the presumption of Corollary i.C.5 with γ = 1, we notice that • our assumption ηs ≤ ρ 4000·Ξ2x ln 3T q2 2 implies βs,2 ≥ 10 ln 3T q2 · τs,2 and κτs ≤ 1 12 ln 3T q2 for every s, P 0 −1 P 3t 2 • our assumption Ts=0 βs,2 ≥ 1 + ln ΞZ implies t−1 s=0 βs − 10 ln q2 τs ≥ ln ΞZ + 1 − 1 = ln ΞZ whenever t > T0 , Therefore, the conclusion of Corollary i.C.4 and Corollary i.C.5 imply that (i) (i) Pr[∃i ∈ [3] : zt−1 > φt,t−1 ] ≤ 3q2 < q 2 /2 so assumption (A3) of Lemma i.D.1 holds. Application of Lemma i.D.1. Applying Lemma i.D.1, we have Pr[CT ] ≤ 2qT which implies our desired bounds and this finishes the proof of Lemma Main 1.  i.E.2 After Warm Start Proof of Lemma Main 2. For every t ∈ [T ] and s ∈ {0, 1, ..., t − 1}, consider the same random vectors yt,s ∈ RT +2 defined in the proof of Lemma Main 1: (1) def (2) def yt,s = kZ> Ps Q(V> Ps Q)−1 k2F , yt,s = kW> Ps Q(V> Ps Q)−1 k2F ,   x> ZZ> (Σ/λ )j P Q(V> P Q)−1 s s (3+j) def k+1 t yt,s =  (1 − η λ ) · y (3+j) , s k 2 2 , for j ∈ {0, 1, . . . , t − s − 1}; t,s−1 This time, we consider slightly different upper bounds  2ΞZ if s < T0 ;   (1) def (2) def 2 if s = T0 ; , φt,s = 2ΞZ , φt,s =   5T0 / ln22(T0 ) if s > T0 . for j ∈ {t − s, . . . , T − 1}. (3+j) and φt,s def = 2Ξ2x . s/ ln s We stress that the only difference between the above upper bounds and the ones we used in the (2) proof of Lemma Main 1 is the choice of φt,s for s > T0 . Instead of setting it to be constant 2 for all such s, we make it decrease almost linearly with respect to index s. Again, define event   h i (i) (i) 0 def Ct = (x1 , ..., xt−1 ) satisfies Pr ∃i ∈ [3] : yt,t−1 > φt,t−1 Ft−1 ≤ q xt n o def (i) (i) Ct00 = (x1 , ..., xt ) satisfies ∀i ∈ [3] : yt,t−1 ≤ φt,t−1 def def V and denote by Ct = Ct0 ∧ Ct00 and C≤t = ts=1 Cs . We next want to apply the decoupling Lemma i.D.1. Verification of Assumption (A1) in Lemma i.D.1. (i) (i) (i) The same functions fs , gs , and hs used in the proof of Lemma Main 1 still apply here. (i) However, we want to make a minor change on gs whenever s ≥ T0 . 32 Applying Lemma i.B.1-(c) with φt = 2Ξx , we have whenever C≤s+1 holds for some s ≥ T0 (2) (which implies yt,s ≤ 5), q q (2) (2) (2) (2) (2) 2 2 |yt,s+1 − yt,s | ≤ 18ηs+1 Ξx yt,s + 4ηs+1 Ξx yt,s + 40ηs+1 Ξ2x ≤ 45ηs+1 Ξx yt,s + 40ηs+1 Ξ2x . Therefore, we can choose gs(2) (y) q 2 = 45ηs+1 Ξx y (2) + 40ηs+1 Ξ2x for all s ≥ T0 and this still satisfies assumption (A1) of Lemma i.D.1.20 Verification of Assumption (A2) of Lemma i.D.1. This is the same as the proof of Lemma Main 1. Verification of Assumption (A3) in Lemma i.D.1. Again, for every t ∈ [T ], let {zs }t−1 s=0 be the arbitrary random vector satisfying (i.D.1) of 2 Lemma i.D.1. Choosing q2 = q /8 again, the same proof of Lemma Main 1 shows that (i) (i) Pr[∃i ∈ {1, 3} : zt−1 > φt,t−1 ] ≤ 2q2 . (2) (2) Therefore, it suffices to prove that Pr[zt−1 > φt,t−1 ] ≤ 2q2 . (2) We only need to focus on the case t ≥ T0 + 2, because otherwise if t ≤ T0 + 1 then gs is (2) not changed for all s ∈ {0, . . . , t − 2} so the same proof of Lemma Main 1 also shows Pr[zt−1 > (2) φt,t−1 ] ≤ q2 . When t ≥ T0 + 2, we can first apply the same proof of Lemma Main 1 (for t = T0 + 1) to show (2) (2) (2) that Pr[zT0 > φT0 +1,T0 = 2] ≤ q2 . Next, conditioning on zT0 ≤ 2 which happens with probability 1 . at least 1 − q2 , we want to apply Corollary i.C.3 with κ = 2 and τs = δs More specifically, for every t ∈ {T0 + 2, . . . , T }, we have shown that the random sequence (2) {zs }t−1 s=T0 satisfies (i.D.1) with (2) 2 2 fs(2) (y, q) = (1 − 2ηs+1 ρ + 56ηs+1 Ξ2x )y (2) + 40ηs+1 Ξ2x + 20ηs+1 q (2) 2 gs (y) = 45ηs+1 Ξx y (2) + 40ηs+1 Ξ2x
2 (2) h(2) s (y, q) = gs (y) Therefore, {zs }t−1 s=T0 also satisfies (i.C.1) with κ = 2 and τs = our assumptions: 1 δτs 3/2 qΞZ 1−q because the following holds from 1 2 ≤ 2ηs+1 ρ − 56ηs+1 Ξ2x s 3/2 1 2 qΞZ 2 2 τs2 = 2 2 ≥ 60ηs+1 Ξ2x ≥ 40ηs+1 Ξ2x + 20ηs+1 1−q κτs = ≥ 40ηs+1 Ξx δ s δs Now, we are ready to apply Corollary i.C.3 with q = q2 , t0 = T0 , and κ = 2. Because q2 ≤ e−2 , √ (2) 2) zT0 ≤ 2, δ ≤ 1/ 8 and lnT2 0T ≥ 9 ln(1/q , the conclusion of Corollary i.C.3 tells us δ2 3/2 qΞZ ≤ ηs+1 δτs = 0 (2) (2) (2) Pr[zt−1 > φt,t−1 | zT0 ≤ 2] ≤ q2 . (2) (2) By union bound, we have Pr[zt−1 > φt,t−1 ] ≤ q2 + q2 = 2q2 as desired. Finally, we conclude (for every t ≥ T0 + 2) that (i) (i) Pr[∃i ∈ [3] : zt−1 > φt,t−1 ] ≤ 4q2 < q 2 /2

(i) (2) (2) Similar to Footnote 19, we also need to verify (A1e) for gs . Whenever fs x, 0 ≤ x(2) , it satisfies fs x, 0 − (2) x(2) ≤ 2ηs+1 · x(2) ≤ gs (x) , where the first inequality uses ρ ≤ 1 and the second uses Ξx ≥ 2. 20 33 so assumption (A3) of Lemma i.D.1 holds. Application of Lemma i.D.1. Applying Lemma i.D.1, we have Pr[CT ] ≤ 2qT which implies our desired bounds and this finishes the proof of Lemma Main 2.  i.F Proof of Theorem 1’ and 2’ Proof of Theorem apply Lemma 5.1 with p0 = p6  1 2’. pFirst for a sufficiently large constant C, we can and q = min CT 2 d2 , 4T and obtain: with probability at least 1−p0 −q 2 ≥ 1−p/2 over the random choice of Q, the following holds:  > > −1 2 ≤ 20736dk ln 6d , and   (Z Q)(V" Q) p p2 F # q i−1 > > > −1   Prx1 ,...,xT ∃i ∈ [T ], ∃t ∈ [T ], xt ZZ (Σ/λk+1 ) Q(V Q) 2 ≥ 216 k ln p 2T q ≤ q2 2 . Denote by C1 the union of the above two events, and we have PrQ [C1 ] ≥ 1 − p/2. Now, for every fixed Q, whenever C1 holds, we can let q 216 2k ln 2T p 20736dk 6d ln , Ξx = , ΞZ = 2 p p p so the initial conditions in Lemma Main 1 (and thus Lemma Main 2) is satisfied. Also, according to Parameter 7.1, our parameter choices satisfy the assumptions in Lemma Main 2. Finally, the conclusion of Lemma Main 2 immediately implies for every T ≥ T0     T0 p > > −1 2 e Pr ∀t = {T0 , . . . , T } : kW Pt Q(V Pt Q) kF ≤ O C1 ≥ 1 − 2qT ≥ 1 − . x1 ,...,xT t 2 Union bounding this with event C1 , we have    T0 > > −1 2 e Pr ∀t = {T0 , . . . , T } : kW Pt Q(V Pt Q) kF ≤ O ≥1−p . Q,x1 ,...,xT t Combining this with Lemma 2.2 completes the proof. Finally, Theorem 1’ is a direct corollary of Theorem 2’ by setting ρ ← gap. 34  Appendix (Part II) Theorem a. In this part II of the appendix, we provide our proofs for the lower bound Theorem 6, as well as that for the Rayleigh quotient Theorem 3. • Appendix ii.G extends our main lemmas to better serve for the rayleigh quotient setting; • Appendix ii.H provides the final proof for our Rayleigh Quotient Theorem 3; • Appendix ii.I includes a three-paged proof of our lower bound Theorem 6. Below we address, at a high level, the main ideas needed behind our main lemma extension as well as the proof of Theorem 3. Additional Ideas Needed for Theorem 3. In order to prove Theorem 3 which is the rayleighquotient guarantee in gap-free streaming PCA, we want to strengthen Lemma Main 1 so that it provides guarantee essentially of the form: for every γ ≥ 1 : Wγ> Pt Q(V> Pt Q)−1 2 F ≤ 2/γ , (ii.F.1) where Wγ is the column orthonormal matrix consisting of all eigenvectors of Σ with eigenvalues ≤ λk − γ · ρ. For obvious reason Lemma Main 1 is a special case of (ii.F.1) when restricting only to γ = 1. It is a simple exercise to show that (ii.F.1) implies our desired rayleigh-quotient guarantee (via an Abel transformation and an integral computation, see Appendix ii.H). Therefore, it suffices to prove (ii.F.1). If one were allowed to magically change learning rates and apply Lemma Main 1 multiple times, then (ii.F.1) would be trivial to prove: just replace W with Wγ and replacing ρ with γ · ρ and repeatedly apply Lemma Main 1. Unfortunately, the difficulty arises because want to prove (ii.F.1) for all γ ≥ 1 but with a fixed set of learning rates ηt . In Appendix ii.G, we showed that the same learning rates in Parameter 7.1, together with a more general martingale concentration lemma (i.e., Corollary i.C.5 with γ ≥ 1), one can obtain (ii.F.1) and we call it Lemma Main 3. This proof follows from the same structure as that of Lemma Main 1 except for the change in how we apply Corollary i.C.5. Finally, Theorem 3 follows from Lemma Main 3 for a one-paged reason, see Appendix ii.H. 35 ii.G Improved Main Lemma In this section we also sketch the proof to obtain Rayleigh quotient result. We will prove the following lemma which is a strengthened version of Lemma Main 1. Lemma Main 3 (before warm start). In the same setting as Lemma Main 1, suppose we redefine W = Wγ to be the column orthonormal matrix consisting of eigenvectors of Σ with values ≤ λk − γ · ρ. Then, for every γ ∈ [1, 1/ρ], with probability at least 1 − 2qT : 2 2 ∀t ∈ {T0 , . . . , T }, Wγ> Pt Q(V> Pt Q)−1 F ≤ . γ Proof of Lemma Main 3. The proof is a non-trivial adaption of the proof of Lemma Main 1. We redefine W = Wγ and consider random vectors yt,s ∈ RT +2 defined in the same way as the proof of Lemma Main 1. This time, we consider upper bounds  2ΞZ s < T0 ; (1) def (2) def (3+j) def φt,s = 2ΞZ , φt,s = , and φt,s = 2Ξ2x 2/γ s ≥ T0 . so the only difference we make here is on coordinate i = 2 for s ≥ T0 . For each t ∈ [T ], we also def def V consider events Ct0 , Ct00 , Ct = Ct0 ∧ Ct00 , and C≤t = ts=1 Cs defined in the same way as before. Verification of Assumption (A1) in Lemma i.D.1. We consider the same functions fs , gs , hs as defined in the proof of Lemma Main 1, except that we replace ρ with γ · ρ because this time we have redefined W = Wγ so that it consists of eigenvectors with values ≤ λk − γ · ρ. In other words, we redefine 2 2 fs(2) (y, q) = (1 − 2ηs+1 γρ + 56Ληs+1 Ξ2x )y (2) + 40ηs+1 ΛΞ2x + 20ηs+1 3/2 qΞZ 1−q . In the same way we can verify that these functions satisfy assumption (A1) of Lemma i.D.1. Verification of Assumption (A2) of Lemma i.D.1. This step is exactly the same as the proof of Lemma Main 1 so ignored here. Verification of Assumption (A3) of Lemma i.D.1. We consider the same parameters {βs , δs , τs }s as Lemma Main 1 except that at coordinate i = 2 we replace ρ with γ · ρ: βs,2 = 2ηs+1 γρ, δs,2 = 0, τs,2 = 20Ληs+1 Ξx . Now, for every t ∈ [T ], let {zs }t−1 s=0 be the arbitrary random vector satisfying (i.D.1) of Lemma i.D.1. 2 Letting q2 = q /8, we can handle coordinates i = 1 and i ≥ 3 in the same way as before. As for t−1 coordinate i = 2 of {zs }s=0 , • if t < T0 , apply Corollary i.C.4 with {βs,2 , δs,2 , τs,2 }t−2 s=0 , q = q2 , D = 1, and κ = √1 ; Λ • if t ≥ T0 , apply Corollary i.C.5 with {βs,2 , δs,2 , τs,2 }t−2 s=0 , q = q2 , D = 1, γ = γ, and κ = √1 ; Λ Note that the t < T0 case is exactly the same as before. When t ≥ T0 , we again apply Corollary i.C.5 but this time with value γ ≥ 1 rather than γ = 1. Since this is the only difference here, we only need to verify the the presumptions of Corollary i.C.5: • our assumption ηs ≤ ρ 4000Λ·Ξ2x ln 3T q2 2 implies βs,2 ≥ 20γ ln 3T q2 · τs,2 and κτs ≤ P 0 −1 P 3t 2 • our assumption Ts=0 βs,2 ≥ 1 + ln ΞZ implies t−1 s=0 βs,2 − 10γ ln q2 τs,2 ≥ whenever t > T0 . 36 1 12 ln 1 2 3T q2 Pt−1 for every s, s=0 βs,2 ≥ ln ΞZ Therefore, in the same way as the old proof in Lemma Main 1, we can conclude using Corollary i.C.4 and Corollary i.C.5 that (i) (i) Pr[∃i ∈ [3] : zt−1 > φt,t−1 ] ≤ 3q2 < q 2 /2 . This verifies assumption (A3) of Lemma i.D.1. Application of Lemma i.D.1. Applying Lemma i.D.1, we have Pr[CT ] ≤ 2qT which implies our desired bounds and this finishes the proof of Lemma Main 3.  ii.H Proof of Theorem 3: Rayleigh Quotient e Theorem 3 (restated). In the same setting as Theorem 2’, we have for every T = Θ letting qi be the i-th column of the output matrix QT , then h i e Pr ∀i ∈ [k], qi> Σqi ≥ λi − Θ(ρ) ≥1−p . k ρ2 ·p2
, e hides poly-log factors in 1 , 1 and d. Again, Θ p ρ Proof of Theorem 3. Since the statement of Theorem 3 uses the same learning rates21 as in Theorem 2’, the same proof of Theorem 2’ ensures that the initialization assumptions in Lemma Main 3 are satisfied and thus we can apply Lemma Main 3. We want to prove next the output matrix QT = [q1 , . . . , qk ] ∈ Rd×k satisfies 1 with probability at least 1 − (2kdT )q, ∀i ∈ [k] : qi> Σqi ≥ λi − 3ρ ln . (ii.H.1) ρ For every i ∈ [k], let QiT ∈ Rd×i denote the first i-columns of QT . By the property of Oja’s algorithm, the same QiT would have been the output if we started from an Rd×i random matrix Q0 for streaming i-PCA. In other words, we can write QiT = [q1 , . . . , qi ]. Letting Wγi be the column orthonormal matrix consisting of all eigenvectors of Σ with eigenvalue ≤ λi − γ · ρ, we applying Lemma Main 3 (with k = i) and obtain: w.p. at least 1 − 2qT : k(Wγi )> QiT k2F ≤ k(Wγi )> PT Qi (V> PT Qi )−1 k2F ≤ 2/γ . (Above, the first inequality uses Lemma 2.2.) This in particular implies k(Wγi )> qi k22 ≤ Let us define for each i ∈ [k], nλ − λ o  1 def def λi − λj i j λi − λj ≥ ρ ⊆ R≥1 and γi,j = . Γi = ∈ 1, ρ ρ ρ By union bound, w.p. at least 1 − 2qkdT , ∀i ∈ [k], ∀γ ∈ Γi : k(Wγi )> qi k22 ≤ 2/γ . 2 γ . (ii.H.2) We are now ready to bound Rayleigh quotient. For each i ∈ [k], let i0 be the index of the first def P (i.e., the largest) eigenvector with eigenvalue ≤ λi − ρ and define bi,j = ds=j hqi , νj i2 where νj is the j-th largest eigenvector of Σ. It satisfies bi,1 = 1. By Abel’s formula, qi> Σqi = d X j=1 21 2 λj hqi , νj i ≥ (λi − ρ) − d X j=i0 +1 bi,j (λj−1 − λj ) . More precisely, we can use the same Parameter 7.1 together with the same values of Ξx and ΞZ used in Section i.F. 37 Note that for every j ≥ i0 + 1, we have bi,j ≤ kWγi i,j qi k22 ≤ d X j=i0 +1 which implies ii.I bi,j (λj−1 − λj ) ≤ qi> Σqi ≥ λi − 3ρ ln 1 ρ d X j=i0 +1 2 ρ(γi,j γi,j 2 γi,j according to (ii.H.2). Therefore, Z 1 ρ 1 1 − γi,j−1 ) ≤ 2ρ dz ≤ 2ρ ln , ρ 1 γ  so (ii.H.1) holds. Proof of Theorem 6: Lower Bound In this section we prove the following lower bound. Without loss of generality, we only focus on the case when d = 2(k + m) and the case for d > 2(k + m) can be done by padding zeros. Denoting by S(2(k+m))×k the set of all (column) orthonormal matrix in R(2(k+m))×k , Theorem 6 (lower bound, restated). For every k ∈ N∗ , every m ≥ 0 that is an integral multiple of
1 λ λ ∗ k, for every λ ∈ (0, 4(k+m) ], every δ ∈ (0, 2 ], every T ∈ N satisfying T ≥ Ω δ2 , every algorithm A : (R2(k+m) )⊗T → S(2(k+m))×k , there exists a distribution µ over vectors in R2(k+m) such that     > > all x ∼ µ satisfies kxk2 ≤ 1, λk E [xx ] ≥ λ, λk+m+1 E [xx ] ≤ λ − δ . x∼µ x∼µ def Furthermore, let QT = A(x1 , . . . , xT ) be the output with respect to T i.i.d. random inputs x1 , ..., xT from µ, we have    >  kλ 2 , E kQT WkF = Ω 2 x1 ,...,xT ,A δ T where W consists of all the last d − (k + m) eigenvectors of Ex∼µ [xx> ]. We shall just prove this theorem for gap case, i.e. when m = 0. For the gap free case, the theorem follows by first obtaining µ for the m = 0 case and for λ0 = λ · m+k k , and then modifying it (m/k) independent copies from µ, and then concatenate into some µ0 as follows: first draw x(0) , . . . , x √ k them into x0 = (x(0) , . . . , x(m/k) ) · √m+k and output this x0 instead. Throughout this section, we use the phrase i-th eigenvalue to denote the i-th largest eigenvalue of a matrix (tie breaking arbitrarily) and the i-th eigenvector to denote that corresponding to the i-th eigenvector (with unit norm). We first state a lemma regarding 2 × 2 matrices: √ √
 p > Lemma ii.I.1 (2×2 matrix). For every β ∈ 22 , 23 and ε ∈ (0, 2β 2 −1], choose a = β, 1 − β 2 , b = p >  β, − 1 − β 2 ∈ R2 and define

1 1 1 1 A = aa> + bb> , B = + ε aa> + − ε bb> . 2 2 2 2 a b Then, letting ν1 and ν1 be the top eigenvectors of A and B respectively, letting λ1 , λ2 be the eigenvalues of A, and λb1 , λb2 be the eigenvalues of B, we have |hν1a , ν1b i|2 ≤ 1 − ε2 16(λ1 − λ2 )2 and λb1 > λ1 = β 2 > 1 − β 2 = λ2 > λb2 .   β2 0 Proof of Lemma ii.I.1. We can calculate that A = and therefore λ1 = β 2 and 0 1 − β2 ! p 2 2 β 2εβ 1 − β 2 p λ2 = 1 − β . We can also compute B = and the eigenvectors of B 2εβ 1 − β 2 1 − β2 p are νib = √ 12 (1, si )> with eigenvalues λbi = β 2 + 2εβ 1 − β 2 si for i = 1, 2. Here, s1 ≥ s2 are the si +1 38  2 2β √−1 2εβ 1−β 2 have that λb1 two roots of equation s2 +  s − 1 = 0. Since exactly one of the roots is positive (denoted p = β 2 + 2εβ 1 − β 2 s1 > λ1 . This also implies λb2 < λ2 since by s1 ), we immediately λb1 + λb2 = Tr(B) = 1 = λ1 + λ2 . 2 We now turn to the eigenvectors. Denote by α = 2β√−1 2 and it satisfies α ≤ 2(λ1ε−λ2 ) because 2εβ 1−β √ √  2 3 β ∈ 2 , 2 . Therefore, √ −α + α2 + 4 2 1 1 ε √ s1 = = ≥√ ≥q ≥√ . 2 4(λ1 −λ2 )2 8(λ − λ ) α + α2 + 4 α2 + 4 1 2 +4 ε2 Above, the last inequality uses our assumption that ε ≤ 2β 2 − 1 = λ1 − λ2 . On the other hand, s1 = α+√2α2 +4 ≤ 1. Therefore, we have: 1 s21 ε2 ≤ 1 − ≤ 1 − .  2 16(λ1 − λ2 )2 1 + s21 √ √
Corollary ii.I.2 (Corollary of Lemma ii.I.1). For every β ∈ 22 , 23 and every ε ∈ (0, gap] where def gap = 2β 2 − 1, choose a and b in R2 as defined as in Lemma ii.I.1. Define distributions over {a, b}: 1 1 1 µ1 : Pr[x = a] = Pr[x = b] = and µ2 : Pr[x = a] = + ε, Pr[x = b] = − ε . 2 2 2 2 ⊗T 2 Suppose we are given an arbitrary (possibly randomized) algorithm A : (R ) → R for T < O( ε12 ). Then, if we pick i ∈ {1, 2} each with probability 1/2 and sample T i.i.d. vectors x1 , ..., xT from distribution µi , it satisfies kqk22 ε2 E[hq, ν2+ i2 ] ≥ , 512gap2 where q = A(x1 , ..., xT ) is the output of A, ν2+ is the second eigenvector of Eµi [xx> ], and the expectation is taken over the randomness of i, x1 , ..., xT , and A. |hν1a , ν1b i|2 = Proof of Corollary ii.I.2. Without lose of generality, let us assume that kqk2 = 1. Let ν2a , ν2b be the second eigenvectors of Eµ1 [xx> ] and Eµ2 [xx> ] respectively. Let ν1+ , ν2+ be the two eigenvectors of Eµi [xx> ], and ν1− , ν2− be the two eigenvectors of Eµ3−i [xx> ]. ε2 Suppose by way of contradiction that E[hq, ν2+ i2 ] < 512gap 2 . Then, we shall construct a protocol that can, given samples x1 , . . . , xT , with success probability at least 3/4 to tell if the distribution µi is µ1 or µ2 . However, we cannot distinguish between a fair coin and a 21 ± ε biased coin with probability ≥ 3/4 with fewer than O(1/ε2 ) samples. This will give a contradiction and finish the ε2 proof that E[hq, ν2+ i2 ] ≥ 512gap 2. We define the following protocol: on input samples x1 , ..., xT , run algorithm A and get output ε2 q = A(x1 , . . . , xT ); then, declare distribution µ1 if hq, ν2a i2 < 128gap 2 , or distribution µ2 otherwise. Using Markov’s inequality, with probability at least 2 3 4, ε2 . In this 128gap2 + − 2 ε2 hν1 , ν2 i ≥ 16gap2 . Using it satisfies hq, ν2+ i2 < ε 1 case, we have hq, ν1+ i2 ≥ 1 − 128gap 2 ≥ 2 . By Lemma ii.I.1, we also have these two together we can derive that hq, ν2− i2 = (hq, ν1+ ihν1+ , ν2− i + hq, ν2+ ihν2+ , ν2− i)2 1 ε2 ε2 ε2 ≥ hq, ν1+ i2 hν1+ , ν2− i2 − hq, ν2+ i2 hν2+ , ν2− i2 ≥ − ≥ . 2 64gap2 128gap2 128gap2 This means, the protocol we just defined can declare the correct µi : indeed, among the two ε2 vectors {ν2a , ν2b } = {ν2+ , ν2− }, the incorrect one (namely, ν2− ) will not satisfy hq, ν2− i2 < 128gap  2. 39 q 1+δ/λ so 2β 2 − 1 = λδ . Let ε ∈ (0, λδ ] be defined such that Proof of Theorem 6 . Choose β = 2
2λT /β 2 = Θ( ε12 ). (We can do so because T ≥ Ω δλ2 .) We let a and b be the two vectors defined in Lemma ii.I.1 with parameters β, ε. We also denote by A0 and A1 respectively the two 2 × 2 matrices from Lemma ii.I.1: ! p  2  2 2 β 2εβ β 0 1 − β p A0 = and A1 = . 0 1 − β2 1 − β2 2εβ 1 − β 2 Now, consider the following procedure to generate T random vectors x1 , . . . , xT ∈ R2k . At the beginning, pick a vector z ∈ {0, 1}k uniformly at random from the 2k choices. Then, in each of the T rounds, 1. Pick y ∈ [0, 1] uniformly at random. 2. If y > kλ/β 2 , output xt = 0; otherwise continue to the next step. (Note kλ/β 2 ∈ [0, 1].) 3. Pick i ∈ [k] uniformly at random. 4. If zi = 0, then pick xt = 0⊕2(i−1) ⊕ a ⊕ 0⊕2(k−i) w.p. 1/2 and xt = 0⊕2(i−1) ⊕ b ⊕ 0⊕2(k−i) w.p. 1/2 If zi = 1, then pick xt = 0⊕(i−1) ⊕ a ⊕ 0⊕(k−i) w.p. 1/2 + ε and xt = 0⊕(i−1) ⊕ b ⊕ 0⊕(k−i) w.p. 1/2 − ε It is clear from the above definition that x1 , . . . , xT are generated i.i.d. from a distribution Dz which is characterized by vector z. Since kak2 = kbk2 = 1, we also have kxt k2 ≤ 1. By the construction, k   M λ > , A E [xx ] = z i x∼Dz β2 i=1 which implies, according to Lemma ii.I.1, for every possible z ∈ {0, 1}k     λ λ E [xx> ] ≥ 2 β 2 = λ and λk+1 E [xx> ] ≤ 2 (1 − β 2 ) ≤ λ − δ . λk x∼Dz x∼Dz β β Now, in the total number of T rounds, with probability ≥ 12 , there are less than 2T · kλ rounds β2 that xt 6= 0. Denote this event as E. If this happens, then at least 7/8 of i ∈ [k] are picked less than 16T · kλ · 1 = 16 λT times. Denote this set of indices as S1 ⊂ [k], and let QT = A(x1 , . . . , xT ) ∈ R2k×k β2 k β2 be the output of the algorithm on this random input sequence x1 , . . . , xT . (Recall that the algorithm does not know the vector z ∈ {0, 1}k ). Let [QT ]i be the i-th row of QT , it holds: 1. ∀i ∈ [2k], k[QT ]i k22 ≤ 1. P2k 2 2 2. i=1 k[QT ]i k2 = kQT kF = k. Therefore, at least 1/4 of j ∈ [k] satisfies k[QT ]2j−1 k22 + k[QT ]2j k22 ≥ 12 . We denote this set of j def as S2 ⊂ [k]. Let S = S1 ∩ S2 , then it holds that |S2 | ≥ k8 . Now we apply Corollary ii.I.2 to each index i ∈ S under event E. We can apply Corollary ii.I.2 because 16λT /β 2 ≤ O( ε12 ) and the algorithm A has received only 16λT /β 2 nonzero samples under event E. The conclusion of Corollary ii.I.2 implies: for every i ∈ S:
ε2 ε2 + > + 2 2 E[(νi,2 ) QT Q> ≥ . T νi,2 | E] ≥ k[QT ]2i−1 k2 + k[QT ]2i k2 · 512(2β 2 − 1)2 1024(2β 2 − 1)2 + Above, we have denoted by νi,2 = 0⊕2(i−1) ⊕ ν2+ ⊕ 0⊕2(k−i) where ν2+ is the second eigenvector + of Azi . It is clear that νi,2 QT = (ν2+ )> (q1 , ..., qk ) where each qj = ([QT ]2i−1,j , [QT ]2i,j )> ∈ R2 so we can apply Corollary ii.I.2 on each of them. Thus, using Pr[E] ≥ 21 we conclude: 40 E " X i∈S + > k(νi,2 ) QT k22 # Θ(k) ε2 1 k = =Θ ≥ · · 2 2 2 2 8 1024(2β − 1) λ(2β − 1)2 T /β 2  kλ δ2T  . Finally, let W be the column orthonormal matrix consisting of the (k + 1)-th till the 2k-th + eigenvectors of matrix Ex∼Dz [xx> ]. Since each νi,2 for i ∈ [k] is in one of the columns of W, the above lower bound implies   h i kλ > 2 E , kQT WkF ≥ Ω 2 z,x1 ,...,xT ,QT δ T where the expectation is over all possible choices z ∈ {0, 1}k , the random vectors x1 , . . . , xT generated from Dz , and the randomness of the algorithm. By an averaging argument, there exists some z ∈ {0, 1}k such that   h i kλ > 2 kQT WkF ≥ Ω 2 E .  x1 ,...,xT ∼Dz ,QT δ T 41 Appendix (Part III) In this part III of the appendix, we make non-trivial modifications to Appendix I and derive our final theorems. More specifically, • Appendix iii.J extends our initialization lemma in Appendix i.A to stronger settings; • Appendix iii.K extends our expectation lemmas in Appendix i.B to stronger settings; • Appendix iii.L extends our main lemmas in Appendix i.E to stronger settings; • Appendix iii.M provides the final proofs for our Theorem 1 and Theorem 2; • Appendix iii.N provides the final proofs for our Theorem 4 and Theorem 5. Theorem a. Main Ideas Behind Main Lemmas. Our New Main Lemmas in Appendix iii are extensions of Main Lemmas in Appendix i essentially with two additional ingredients. • Under Sampling This we have already discussed in Section 3 of the main body. • Variance Bound Recall that our weaker theorems (such as Theorem 1’) do not have the factor Λ = λ1 +· · ·+λk ∈ [0, 1] show up in the complexity. This speed-up factor 1/Λ could be large and has been argued as a very important factor in the total running time by [10]. To achieve this, we need tighter martingale concentrations on our random variables and below we discuss the main intuition. Recall that all martingale concentrations for a random process {zt }t require some upper bound between consecutive variables |zt − zt+1 |. If this upper bound holds with probability 1, that is, |zt − zt+1 |2 ≤ M , then an Azuma-type of concentration can be naturally used. However,   Azuma concentration is not tight: if one knows a better bound on E |zt+1 − zt |2 | zt , the M term can be replaced with this expected bound a tighter concentration bound can be implied. See for instance the survey [5]. This same issue also shows up in streaming PCA. Our Lemma Main 1 and Main 2 have adopted a probability-one absolute bound on |zt − zt+1 | because that gives the simplest proof. If one replaces it with a tighter (but very sophisticated) expected bound, the concentration result can be further improved and this improvement translates to a factor 1/Λ speed-up in the running time on Oja’s algorithm in the gap-dependent case (and similarly in the gap-free case). We present such expected bounds in Appendix iii.K. Additional Ideas for Theorem 2. While the extended main lemmas are sufficient to prove Theorem 1, they lead again to a weaker version of Theorem 2 where Λ1 , Λ2 are replaced by (kΛ, kΛ2 ).22 The reason that the factor k shows up is because in gap-free case, there is no “local convergence” (i.e. convergence when > −1 2 kPt Q(V> Pt Q)−1 k2 = O(1)) as opposite to the gap-case. Therefore, the quantity kx> t Pt Q(V Pt Q) k2 will stay at Ω(k) during the entire execution of Oja’s algorithm, and never decreased to O(1). 22 This new weaker version is still weaker than Theorem 2 but already stronger than Theorem 2’. We refrain from stating it formally in this paper. 42 To overcome this issue, we consider an auxiliary “under-sampled” objective that has a “local convergence” behavior. As illustrated by Figure 2 on page 57, we define W1 to consist of all the eigenvectors of eigenvalue ≤ λ − ρ/2, and V1 = W1⊥ contains all eigenvectors of eigenvalue > λ − ρ/2. Now, we define the auxiliary objective to be W1> Pt Q(V> Pt Q)−1 . Unlike Z, this new matrix W1 has an eigengap ρ/2 as compared with V, and thus after certain number of iterations, the quantity kW1> Pt Q(V> Pt Q)−1 k2F can drop to a constant, say ≤ 2. Now, let us suddenly “shift” our objective to “ kW> Pt Q(V1> Pt Q)−1 k2F ”.23 The crucial observation is that > −1 > > −1 > > −1 kx> t Pt Q(V1 Pt Q) k2 ≤ 1 + kW1 Pt Q(V1 Pt Q) k2 ≤ 1 + kW1 Pt Q(V Pt Q) k2 ≤ 3 , √ and this is k times smaller than the previous bound √ > −1 kx> t Pt Q(V Pt Q) k2 ≤ O( k) . Therefore, we can now focus on this random process “ kW> Pt Q(V1> Pt Q)−1 k2F ”, and it converges a factor Ω(k) faster than the old quantity kW> Pt Q(V> Pt Q)−1 k2F . Finally, Lemma 2.2 tells us that bounding the former (with respect to V1 ) as opposed to the old quantity (with respect to V) also implies the convergence of Oja’s algorithm, so we are done. Additional Ideas for Oja++ Algorithm. The analysis behind our Oja++ requires a more fine-grind argument regarding the alternation between “under-sampling” and “objective shift”. As illustrated by Figure 4 on page 60, we consider a logarithmic long sequence (V1 , W1 ), (V2 , W2 ), . . . and shift carefully and only when needed. Since our main goal of Oja++ is to remove the factor O(k) in the running time, the spirit behind these shiftings is the same as the “under-sampled” objective as we discussed above. However, in this case we need to shift our objective once per epoch, so we need in total logarithmic many of them. Also, we need to establish a stronger initialization lemma. Recall that at the beginning of each epoch of Oja++ , we insert random columns to the working matrix from the previous epoch. Therefore, the initial matrix Q0 for every epoch (except the first epoch) is not completely fresh e Q] where Q e is a (fixed) column orthonormal matrix and random, but consists of two parts Q0 = [Q, Q is a fresh random gaussian matrix. We therefore need to re-design our initialization lemma for this > −1 > e > e −1 particular case, especially to bound the quantity kx> t Q0 (V Q0 ) k2 = kxt [Q, Q](V [Q, Q]) k2 . This will be our main focus in the next sub-section. The matrix V1> Pt Q is not a square matrix anymore because of the shifting, so by inverse we actually mean the > −1/2 2 Moore-Penrose pseudo-inverse. kW> Pt Q(V1> Pt Q)−1 k2F is also equal to kW> Pt Q(Q> P> kF . t V1 V1 Pt Q) 23 43 iii.J Final Initialization Lemmas We state and prove the following lemma using random matrix theory: Lemma iii.J.1. There exists constants C such that the following holds: Let Q in RN ×n , N ≥ n be a random matrix with each entry i.i.d. N (0, 1), for every ε > 0, √ √ 1 Pr[λmin (Q> Q) ≤ ε2 ( N − n − 1)2 ] ≤ Cε log ε Proof of Lemma iii.J.1. Theorem 1.1 of [15] states that there exist constants C 0 , c > 0 such that √ √ Pr[λmin (Q> Q) ≤ ε2 ( N − n − 1)2 ] ≤ (C 0 ε)N −n+1 + e−cN . log 1 log 1 Therefore, if N > 2c ε then we are done. In the case when N ≤ 2c ε , we view Q as a submatrix e where the entries of Q e are also i.i.d. generated from N (0, 1). of an N × N random matrix [Q, Q] e and conclude that for every α ≤ 0, In such a case, we can apply Lemma i.A.2 on [Q, Q] h 2i e > [Q, Q]) e ≤ α ≤ C 00 · α . Pr λmin ([Q, Q] N > > e e e > [Q, Q]). e Since Q Q is a sub matrix of [Q, Q] [Q, Q], we know that λmin (Q> Q) ≥ λmin ([Q, Q] Choosing α = εN , we conclude that √ √ Pr[λmin (Q> Q) ≤ ε2 ( N − n − 1)2 ] ≤ Pr[λmin (Q> Q) ≤ ε2 N ] ≤ O(ε log(1/ε)) .  We prove the following the following initialization lemma that extends Lemma 5.1. Note that e Q] where Q is a random matrix but Q e is a fixed one. This the matrix we are interested now is [Q, will allow us to perform “under-sampling” and “objective shift” properly. e ∈ Rd×α , Q ∈ Rd×β , V ∈ Rd×k be three matrices such that Lemma iii.J.2 (initialization). Let Q • α, β, k ∈ N where α ≥ 0, β ≥ 1, and α + β ≤ k ≤ d; • each entry of Q is i.i.d. random from N (0, 1); e and V are (column) orthonormal; • Q e = [] has zero column or Q e > VV> Q e  1 I holds. • either Q 2 Denote by Z ∈ Rd×(d−k) the orthogonal complement of V. Then, for every p, q ∈ (0, 1), every T ∈ N∗ , every set of random vectors {xt }Tt=1 with kxt k2 ≤ 1, with probability at least 1 − p − q over the random choice of Q, the following holds:  −1/2 2 e Q] [Q, e Q]> VV> [Q, e Q] • Z> [Q, ≤d·♣ ; F    −1/2 2 e Q] [Q, e Q]> VV> [Q, e Q] ≥♣ ≤q ; • Pr ∃i, t ∈ [T ], x> ZZ> (Σ/λk+1 )i−1 [Q, t x1 ,...,xT • where νj> ZZ> Q(Q> VV> Q)−1/2 2 2 ≤ ♣ for every j ∈ [d] . β · log2 (1/p) · log(T d/q) e ♣=C· 2 √ =Θ √ p ( k − α − β − 1)2 def  β √ √ 2 p ( k − α − β − 1)2  . def e > VV> Q e  1 I ∈ Rα×α and by B> def e > VV> ∈ Rα×d . Proof of Lemma iii.J.2. Let A2 = Q = A−1 Q 2 We have   e > VV> Q] e −1 Q> V V . B> B = I and BB> = V> V> Q[Q 44 e > VV> Q] e −1/2 is a column orthonormal matrix, we can always find C such that Since V> Q[Q VV> − BB> = CC>  0, C> B = 0, Therefore,   e Q]> VV> [Q, e Q] [Q, = C ∈ Rd×(k−α) . ! e > VV> Q e Q e > VV> Q Q e Q> VV> Q Q> VV> Q     A I B> Q A = I I Q> B Q> (BB> + CC> )Q Let us denote Qb = B> Q ∈ Rα×β , Qc = C> Q ∈ R(k−α)×β , σ = σmin (Q> c Qc ), we then have:       I Qb A A > > e e [Q, Q] VV [Q, Q]  I I Q> Q> b b Qb + σI Which implies that  −1    −1   −1 A I + σ −1 Qb Q> −σ −1 Qb A b e Q]> VV> [Q, e Q] [Q,  I I −σ −1 Q> σ −1 I b   −σ −1 Qb I + σ −1 Qb Q> b  2 −1 > σ −1 I −σ Qb   I 0 > > + 2σ −1 [Q> (iii.J.1) = 2 b , −I] [Qb , −I] . 0 0  −1   I Qb −σ −1 Qb I + σ −1 Qb Q> b ; = Above, the first spectral dominance uses Q> Q> σ −1 I −σ −1 Q> b Qb + σI b b the second spectral dominance uses A2  12 I Now consider any fixed y ∈ Rd with kyk2 ≤ 1. We have  −1 e Q] [Q, e Q]> VV> [Q, e Q] e Q]> y y > [Q, [Q,  −1/2  −1/2 e Q] [Q, e Q]> VV> [Q, e Q] e Q] [Q, e Q]> VV> [Q, e Q] = y > VV> [Q, + y > ZZ> [Q, ≤ 2 y>V 2 2 −1/2 e Q] [Q, e Q]> VV> [Q, e Q] V> [Q, 2 > > > 2  2 −1/2 e Q] [Q, e Q]> VV> [Q, e Q] + 2 y > ZZ> [Q, −1 e Q] [Q, e Q] VV [Q, e Q] e Q]> ZZ> y . ≤ 2 + 2y ZZ [Q, [Q, >   2 (iii.J.2) Above, the last inequality uses ky > Vk2 ≤ 1 and the fact that kA(A> A)−1/2 k2 ≤ 1 for every matrix A that has full column rank. Next, using inequality (iii.J.1), we can bound  −1 1 > > e e Q]> VV> [Q, e Q] e Q]> ZZ> y y ZZ [Q, Q] [Q, [Q, 2
e 2 + σ −1 ky > ZZ> QQ e > − Q ]k2 ≤ ky > ZZ> Qk 2 2 b > > > 2 > > 2 e 2kQb Q ZZ yk2 + 2kQ ZZ yk2 ≤ 1+ . (iii.J.3) σ Note that Qb = B> Q ∈ Rα×β , Qc = C> Q ∈ R(k−α)×β , and Z> Q ∈ R(d−k)×β are independent of each other, because the entries of Q remain independent after rotation and the columns of def def > > β e> > B, C, Z are pairwise orthogonal. Therefore, letting u1 = Q> b Q ZZ y and u2 = Q ZZ y ∈ R , we have that e > ZZ> yk2 ); • u1 ∈ Rβ is a random vector with entries i.i.d. in N (0, kBQ 2 • u2 ∈ Rβ is a random vector with entries i.i.d. in N (0, kZZ> yk22 ); 45 2 2 • u1 , u2 and σ = σmin (Q> c Qc ) are independent variables. e > ZZ> yk2 ≤ 1 and kZZ> yk2 ≤ 1, it implies (using tail bound for chi-squared distriSince kBQ 2 bution) that ∀s ≥ 4 : Pr[ku1 k22 ≥ sβ], Pr[ku2 k22 ≥ sβ] ≤ (se1−s )β/2 ≤ e−s/6 . def Putting them into (iii.J.3) and then altogether to (iii.J.2), we claim that under event Cσ = {σmin (Q> CC> Q) ≥ σ}, we have    −1 16βs > e > > e > e e Pr y [Q, Q] [Q, Q] VV [Q, Q] [Q, Q] y ≥ 6 + (iii.J.4) Cσ ≤ 2e−s/6 . σ Final Probability Arguments. Let us now take σ = √ √ p2 ( k−α− β−1)2 c2 ·log2 (1/p) for a large enough constant
c2 , we have Pr[Cσ ] ≥ 1 − p/2 according to Lemma iii.J.1. We also take s = Θ log(T d/q) , and it satisfies   β · log2 (1/p) · log(T d/q) 16βs √ =♣ . =Θ 6+ √ σ p2 ( k − α − β − 1)2 We now apply (iii.J.4) multiple times in two different manners • We can apply (iii.J.4) with y = ν1 , ν2 , . . . , νd which are the eigenvector of Σ. By union bound, we have Pr[C1 |Cσ ] ≥ 1 − q where C1 is the following event    −1/2 2     > > > e Q] [Q, e Q] VV [Q, e Q]  Z [Q,  ≤ d · ♣ , and def C1 = F  −1/2 2    e Q] [Q, e Q]> VV> [Q, e Q]  ∀j ∈ [d] : νj ZZ> [Q,  ≤♣  2 • Define  event  −1/2 i−1 e > > > e e ZZ (Σ/λ ) [ Q, Q] [ Q, Q] VV [ Q, Q] C2 = ∃i ∈ [T ], ∃t ∈ [T ], x> k+1 t i−1 > x> t ZZ (Σ/λk+1 ) 2  ≥♣ . (and we can do so Then, we apply (iii.J.4) multiple times each with y = because kyk2 ≤ 1). By union bound, we have for every fixed x1 , ..., xT , it satisfies PrQ [C2 | Cσ , x1 , ..., xT ] ≤ q 2 . Denoting by 1C2 the indicator function of event C2 , then     1 Pr [C2 | Q] Cσ Pr Pr [C2 | Q] ≥ q Cσ ≤ E Q x1 ,...,xT q Q x1 ,...,xT   1 E [1C | Q] Cσ = E q Q x1 ,...,xT 2   1 E E[1C | Cσ , x1 , . . . , xT ] = q x1 ,...,xT Q 2   1 = E Pr[C2 | Cσ , x1 , . . . , xT ] ≤ q . q x1 ,...,xT Q Above, the first inequality uses Markov’s bound.   In sum, we just derived that PrQ [C1 |Cσ ] ≥ 1 − q and PrQ Prx1 ,...,xT [C2 | Q] ≥ q ≤ q. Applying union bound again we have h i Pr C1 ∧ Pr [C2 | Q] ≤ q ≥ 1 − p − 2q . Q x1 ,...,xT  This finishes the proof of Lemma iii.J.2. 46 iii.K Final Expectation Lemmas This section extends Appendix i.B to provide better bounds regarding the expected behaviors of the random process we are interested. Recall that if X ∈ Rd×r is a generic matrix (either X = W, X = Z, or X = [w] for some vector w), we have the following notions from Section 6. Lt = Pt Q(V> Pt Q)−1 ∈ Rd×k r×k R0t = X> xt x> t Lt−1 ∈ R St = X> Lt ∈ Rr×k k×k H0t = V> xt x> t Lt−1 ∈ R Our next lemma is a stronger version of Lemma 6.1. It provide tighter expected bounds byh introducing an additional factor Γ, and introduce new bounds regarding the variance terms i 2 > > E Tr(St St ) − Tr(St−1 St−1 ) . Lemma iii.K.1. For every t ∈ [T ], For every t ∈ [T ], let C≤t be any event that depends on random x1 , . . . , xt and implies 1 > > −1 ∀j ∈ [d], kνj> Lt−1 k2 ≤ φt , kx> , t Lt−1 k2 = kxt Pt−1 Q(V Pt−1 Q) k2 ≤ φt where ηt φt ≤ 2 and E[xt x> t | F≤t−1 , C≤t ] = Σ + ∆. Let m be the largest integer such that λk+m > λk − ρ, and   k k+m X  X def def Γ = min λi + φ2t λj + k∆k2 , 1 and χ = k + mφ2t   i=1 j=k+1 . We have: (a) If X = [w] ∈ Rd×1 where w is a vector with Euclidean norm at most 1, h i
ηt 2 2 > 2 2 kw> ΣLt−1 k22 E Tr(S> t St ) | F≤t−1 , C≤t ≤ 1−ηt λk +14Γηt φt Tr(St−1 St−1 )+10Γηt φt + λk   1/2  1/2  > > > + 2ηt k∆k2 Tr(S> S ) + Tr(S S ) 1 + Tr(Z L L Z) t−1 t−1 t−1 t−1 t−1 t−1 (b) If X = [w] ∈ Rd×1 where w is a vector with Euclidean norm at most 1, i h 2 > | F , C E Tr(S> S ) − Tr(S S ) t−1 ≤t t−1 t−1 t t 2 2 2 > 4 4 ≤ 243Γηt2 φ2t Tr(S> t−1 St−1 ) + 12Γηt φt Tr(St−1 St−1 ) + 300Γηt φt (c) If X = W, h i E Tr(S> S ) | F , C t−1 ≤t t t
2 ≤ 1 − 2ηt ρ + 12Γηt2 φ2t + ηt2 (6φt + 8)λk+1 Tr(St−1 S> t−1 ) + 10Γηt (2φt + 8)   3/2 > > + 2ηt k∆k2 ηt (4 + φt )χ + Tr(L> + (5 + 4ηt )Tr(L> t−1 ZZ Lt−1 ) t−1 ZZ Lt−1 )  1/2  > > + Tr(Lt−1 ZZ Lt−1 ) . (d) If X = W, > 2 E[|Tr(S> t St ) − Tr(St−1 St−1 )|2 | Ft−1 , C≤t ] 2 2 2 > ≤ 192Γηt2 φ2t Tr(St−1 S> t−1 ) + 4ηt (4φt + 8) λk+1 Tr(St−1 St−1 ) + 192Γηt4 φ2t + k∆k2 · 4ηt2 (4φt + 8)2 (χ + Tr(S> t−1 St−1 )) . 47 Proof. The proof of the first two cases rely on the following tighter upper bounds when X = [w]: n P    
o 2 k 2 E kH0t k22 | F≤t−1 , C≤t ≤ φ2t E kxt Vk22 | F≤t−1 , C≤t ≤ min λ + k∆k 2 , 1 φt ≤ Γφt i=1 i o n
    Pk 2 2 E kR0t k22 | F≤t−1 , C≤t ≤ φ2t E kxt Xk22 | F≤t−1 , C≤t ≤ min i=1 λi + k∆k2 , 1 φt ≤ Γφt (iii.K.1) φ2t as opposed to that we have used in the past. The proof of the last two cases rely on different upper bounds for X = W. We introduce def some notations that shall be only used in this proof. Let Y = [νk+1 , . . . , νk+m ] ∈ Rd×m be the matrix consisting of all the eigenvectors of Σ with eigenvalues λk+1 , . .P . , λk+m . In this Pnotation, k k+m > > > e ZZ = YY + WW . We also denote by V = [V, Y]. Let Λ1 = j=1 λj , Λ2 = j=k+1 λj , Λ = Λ1 + Λ2 φ2t ≤ Γ. We make three quick observations:  P P > 2 2 > e e>  = kj=1 λj + k+m  j=k+1 λj kνj Lt k2 ≤ Λ1 + Λ2 φt = Λ ,  Tr(Lt−1 VΣ≤k+m V Lt )P k+m 2 2 e e> (iii.K.2) Tr(L> t−1 VV Lt ) = k + j=k+1 kνj Lt k2 ≤ k + mφt = χ , and    eV e > + WW> )Lt ) ≤ χ + Tr(S> St−1 ) . Tr(L> Lt ) = Tr(L> (V t−1 t−1 t−1 Therefore, we have: 2 2 E[kH0t k22 | F≤t−1 , C≤t ] ≤ φ2t · E[kx> t VkF | F≤t−1 , C≤t ] ≤ Γφt 2 > > E[kR0t k22 | F≤t−1 , C≤t ] ≤ E[kx> t Lt−1 k2 | F≤t−1 , C≤t ] = Tr(Lt−1 ΣLt−1 ) + Tr(Lt−1 ∆Lt−1 ) e ≤k+m V e > + WΣ>k+m W> )Lt−1 ) + k∆k2 Tr(L> Lt−1 ) ≤ Tr(L> (VΣ t−1 ≤Λ+ =Λ+ t−1 > λk+1 kW Lt−1 k2F + k∆k2 · (χ + Tr(S> t−1 St−1 )) > λk+1 Tr(St−1 St−1 ) + k∆k2 · (χ + Tr(S> t−1 St−1 )) . (iii.K.3) We are now ready to prove our four cases individually. (a) This follows from almost the same proof of Corollary 6.1-(c), except that one can replace the use of (i.B.10) with the following (owing to (iii.K.1)) h i 2 2 E Tr(S> S ) | F , C ≤ (1 − 2ηt λk + 14Γηt2 φ2t )Tr(St−1 S> ≤t−1 ≤t t t t−1 ) + 10Γηt φt > > > > > −2ηt Tr(S> t−1 St−1 V ∆Lt−1 ) + 2ηt Tr(St−1 w ∆Lt−1 ) + 2ηt Tr(St−1 w ΣLt−1 ) . (b) This follows directly from Lemma i.B.1-(b) and (iii.K.1). (c) The exact same first half of the proof of Lemma i.B.1 gives (i.B.1) which using kH0t k2 ≤ φt gives > > 0 > 0 Tr(S> t St ) ≤ Tr(St−1 St−1 ) − 2ηt Tr(St−1 St−1 Ht ) + 2ηt Tr(St−1 Rt ) 0 2 0 2 > 2 0 2 +4ηt2 φt Tr(S> t−1 Rt ) + 12ηt kHt k2 Tr(St−1 St−1 ) + 8ηt kRt k2 . (iii.K.4) This time, we upper bound 0 > > > > > > e e> > |Tr(S> t−1 Rt )| = |Tr(St−1 W xt xt Lt−1 )| = |Tr(St−1 W xt xt (VV + WW )Lt−1 )| eV e > Lt−1 )| (iii.K.5) = |Tr(S> W> xt x> WSt−1 ) + Tr(S> W> xt x> V t−1 t t−1 ¬ t 3 1 >e e> > > 2 ≤ Tr(S> t−1 W xt xt WSt−1 ) + kxt VV Lt−1 k2 . 2 2 Above, inequality ¬ is because 2Tr(A> B) ≤ Tr(A> A)+Tr(B> B) which is Young’s inequality 48 in the matrix case. We take expectation and get:   0 E |Tr(S> t−1 Rt )| | Ft−1 , C≤t 3 1 > > e e> e e> ≤ Tr(S> t−1 W (Σ + ∆)WSt−1 ) + Tr(Lt−1 VV (Σ + ∆)VV Lt−1 ) 2 2   1 3 1 3 > > e > > > e e> e Tr(St−1 St−1 ) + Tr(Lt−1 VV Lt−1 ) ≤ λk+1 Tr(St−1 St−1 ) + Tr(Lt−1 VΣ≤k+m V Lt ) + k∆k2 · 2 2 2 2   ¬ 3 1 3 χ (iii.K.6) Tr(S> ≤ λk+1 Tr(S> t−1 St−1 ) + Λ + k∆k2 · t−1 St−1 ) + 2 2 2 2 Above, inequality ¬ has relied on our earlier observations (iii.K.2). At this point, plugging (iii.K.6), (iii.K.3) into (iii.K.4) and using the assumption ηt φt ≤ 1/2, we have
2 E[Tr(S> 1 + 12Γηt2 φ2t + ηt2 (6φt + 8)λk+1 Tr(St−1 S> t St ) | F≤t−1 , C≤t ] ≤ t−1 ) + ηt (2φt + 8)Λ   0 > 0 + E − 2ηt Tr(S> t−1 St−1 Ht ) + 2ηt Tr(St−1 Rt ) | F≤t−1 , C≤t   +2ηt k∆k2 · ηt (4 + φt )χ + ηt (4 + 3φt )Tr(S> . t−1 St−1 ) Finally, inequality (i.B.8) (from the proof of Corollary 6.1-(a)) gives an upper bound on the 0 > 0 expected value of −Tr(S> t−1 St−1 Ht ) + Tr(St−1 Rt ). Putting it in gives us the desired bound. (d) This time we compute a slightly different upper bound from (iii.K.5) 0 > > > > > > > > |Tr(S> t−1 Rt )| = |Tr(St−1 W xt xt Lt−1 )| = |Tr(St−1 W xt xt (VV + ZZ )Lt−1 )| > > > > > > = |Tr(S> t−1 W xt xt V) + Tr(St−1 W xt xt ZZ Lt−1 )| > > > > > ≤ |Tr(S> t−1 W xt xt ZZ Lt−1 )| + kSt−1 W xt k2 . Plugging this into (i.B.1), we obtain > > 0 > 0 2 0 > 0 |Tr(S> t St ) − Tr(St−1 St−1 )| ≤ 2ηt |Tr(St−1 St−1 Ht )| + 2ηt |Tr(St−1 Rt )| + 4ηt kHt k2 Tr(St−1 Rt ) ¬ 2 0 2 +12ηt2 kH0t k22 Tr(St−1 S> t−1 ) + 8ηt kRt k2 . > 0 2 0 2 ≤ 8ηt kH0t k2 Tr(St−1 S> t−1 ) + 4ηt |Tr(St−1 Rt )| + 8ηt kRt k2 > > > > ≤ 8ηt kH0t k2 Tr(St−1 S> t−1 ) + 4ηt |Tr(St−1 W xt xt ZZ Lt−1 )| ­ > 2 0 2 +4ηt kS> t−1 W xt k2 + 8ηt kRt k2 > > > 1/2 ≤ 8ηt kH0t k2 Tr(St−1 S> t−1 ) + ηt (4φt + 8)Tr(St−1 W xt xt WSt−1 ) +8ηt2 φt kR0t k2 Above, ¬ uses the fact that ηt kH0t k2 ≤ ηt φt ≤ 1/2, and ­ uses kR0t k2 ≤ φt as well as > > > > kx> t ZZ Lt−1 k2 ≤ kxt Lt−1 k2 + kxt VV Lt−1 k2 ≤ (φt + 1) Taking square on both sides, we have > 2 2 0 2 > 2 2 2 > > > |Tr(S> t St ) − Tr(St−1 St−1 )|2 ≤ 192ηt kHt k2 Tr(St−1 St−1 ) + 3ηt (4φt + 8) Tr(St−1 W xt xt WSt−1 ) +192ηt4 φ2t kR0t k22 49 Finally, taking expectation and using (iii.K.3), we have (noticing that ηt φt ≤ 1/2) > 2 E[|Tr(S> t St ) − Tr(St−1 St−1 )|2 | F≤t−1 , C≤t ] 2 2 2 > > ≤ 192Γηt2 φ2t Tr(St−1 S> t−1 ) + 3ηt (4φt + 8) Tr(St−1 W (Σ + ∆)WSt−1 )   > +192ηt4 φ2t Λ + λk+1 Tr(S> S ) + k∆k · (χ + Tr(S S )) . t−1 2 t−1 t−1 t−1  
2 2 2 > ≤ 192Γηt2 φ2t Tr(St−1 S> ) + 3η (4φ + 8) λ + k∆k Tr(S S ) t 2 t−1 k+1 t−1 t t−1   > +192ηt4 φ2t Λ + λk+1 Tr(S> t−1 St−1 ) + k∆k2 · (χ + Tr(St−1 St−1 )) . 2 2 2 > ≤ 192Γηt2 φ2t Tr(St−1 S> t−1 ) + 4ηt (4φt + 8) λk+1 Tr(St−1 St−1 ) +192Γηt4 φ2t + k∆k2 · 4ηt2 (4φt + 8)2 (χ + Tr(S> t−1 St−1 )) . iii.L  Final Main Lemmas In this section, we extend our main lemmas in Section 7 to their strong forms. Specifically, • Lemma Main 4 is an extension of Lemma Main 1; • Lemma Main 5 is an extension of Lemma Main 2; • Lemma Main 6 is a new main lemma that takes into account “under-sampling”. Recall that given parameter ρ ∈ (0, λk ), • V is a matrix consisting of all the eigenvectors of Σ with eigenvalue ≥ λk , • Z is a matrix consisting of all the eigenvectors of Σ with eigenvalue < λk , • W is a matrix consisting of all the eigenvectors of Σ with eigenvalue ≤ λk − ρ. When we apply these main lemmas in later sections, we may redefine the meaning of (V, Z, W) to be with respect to some other λ that is not necessarily λk . iii.L.1 Before Warm Start  Lemma Main 4 (before warm start). For every q ∈ 0, 12 , ΞZ ≥ 2, Ξx ≥ 2, and fixed matrix Q ∈ Rd×k , suppose the initial matrix Q satisfies • kZ> Q(V> Q)−1 k2F ≤ ΞZ , h j−1 > • Prxt ∀j ∈ [T ], x> Q(V> Q)−1 t ZZ (Σ/λk+1 ) • 24 νj> ZZ> Q(V> Q)−1 2 ≤ Ξx for every j ∈ [d].24 2 i ≤ Ξx ≥ 1 − q 2 /2 for every t ∈ [T ], This assumption is redundant for j ∈ [k] because νj> Z = 0 for j ∈ [k]. 50 Pk Let m be the number of eigenvalues of Σ in (λk − ρ, λk+1 ]. Let Λ = Suppose also the learning rates {ηs }s∈[T ] satisfy (1): ∀s ∈ [T ], 3/2   2q(ΞZ + dΞ2x ) ρ ≤ ηs ≤ O Λ Λ · Ξ2x ln dT q (3): (2): T X t=1 i=1 λi + Ξ2x Ληt2 Ξ2x ≤ O ∃T0 ∈ [T ] such that T0 X t=1 Pk+m j=k+1 λj .  1  . ln dT q  ln(Ξ )  Z ηt ≥ Ω ρ (iii.L.1) Then, for every t ∈ [T − 1], we have with probability at least 1 − 2qT (over the randomness of x1 , . . . , xt ): • if t ≥ T0 then kW> Pt Q(V> Pt Q)−1 2 F ≤ 2. Proof of Lemma Main 4. The proof is a non-trivial adaption of the proof of Lemma Main 1. This time we consider random vectors yt,s ∈ R2+d+T defined as: (1) def (2) def (2+T +j) def yt,s = kZ> Ps Q(V> Ps Q)−1 k2F , yt,s = kW> Ps Q(V> Ps Q)−1 k2F , yt,s (3+d+j) yt,s 2 = νj> ZZ> Ps Q(V> Ps Q)−1 , for j ∈ [d] , 2   x> ZZ> (Σ/λ )j P Q(V> P Q)−1 2 , for j ∈ {0, 1, . . . , t − s − 1}; def s s k+1 t = 2 (3+d+j)  (1 − η λ ) · y , for j ∈ {t − s, . . . , T − 1}. s k t,s−1 We again consider upper bounds  2ΞZ s < T0 ; (1) def (2) def φt,s = 2ΞZ , φt,s = , 2 otherwise. (3) (2+d+T ) and φt,s = · · · = φt,s def = 2Ξ2x . For each t ∈ [T ], define event Ct0 and Ct00 in the same way as before:   h i (i) (i) 0 def Ct = (x1 , ..., xt−1 ) satisfies Pr ∃i ∈ [3 + d] : yt,t−1 > φt,t−1 Ft−1 ≤ q xt n o def (i) (i) Ct00 = (x1 , ..., xt ) satisfies ∀i ∈ [3 + d] : yt,t−1 ≤ φt,t−1 def def V and denote by Ct = Ct0 ∧ Ct00 and C≤t = ts=1 Cs . Verification of Assumption (A1) in Lemma i.D.1. q1 q Suppose E[xs x> s | C≤s , F≤s−1 ] = Σ + ∆, then we have k∆k2 ≤ 1−q1 ≤ 1−q using the same proof as before. This time, we use Lemma iii.K.1 (instead of Lemma i.B.1) with φt = 2Ξx to obtain the following tighter bounds for i ∈ [T + d + 2]:  (i) 
 (i) 
(i) E yt,s+1 | Ft , F≤s , C≤s ≤ fs(i) yt,s , q and E |yt,s+1 − yt,s |2 | Ft , F≤s , C≤s ≤ h(i) s yt,s , q 51 where we define25  
def 2 2 fs(i) (y, q) = 1 + O(Ληs+1 Ξ2x ) y (i) + O Ληs+1 Ξx + Err for i = 1, 3, 4, . . . 2 + d  
def 2 2 fs(i) (y, q) = 1 − 2ηs+1 ρ + O(Ληs+1 Ξ2x ) y (i) + O Ληs+1 Ξx + Err for i = 2  
def 2 2 fs(i) (y, q) = 1 − ηs+1 λk + O(Ληs+1 Ξ2x ) y (i) + ηs+1 λk y (i+1) + O Ληs+1 Ξ2x + Err def  
(i) 2 for i = 3 + d, . . . , 2 + d + T 2 2 2 4 h(i) + Ληs+1 Ξ2x y (i) + Ληs+1 Ξ2x + Err for i = 1, 2, 3, . . . 2 + d s (y, q) = O Ληs+1 Ξx y  
def 2 2 (i) 2 2 4 h(i) + Ληs+1 Ξ2x y (i) + Ληs+1 Ξ4x for i = 3 + d, . . . , 2 + d + T . s (y, q) = O Ληs+1 Ξx y
def 3/2 q Above, we denote by Err = ηs+1 Ξx ΞZ + dΞ2x · 1−q the error term similar to the proof of 2q(Ξ 3/2 +dΞ2 ) x Z Lemma Main 1. Obviously if ≤ ηs is satisfied then the Err term can be absorbed into Λ the big-O notation. For every i ∈ [2 + d + T ], we consider the same gs as defined in the proof of Lemma Main 1: 2 gs(i) (y) = 20ηs+1 Ξx · y (i) + 42ηs+1 Ξ2x (i) (i) (i) and it satisfies whenever C≤s+1 holds then |yt,s+1 − yt,s | ≤ gs (yt,s ) . Putting the above bounds together, we finish verifying assumption (A1) of Lemma i.D.1. Verification of Assumption (A2) of Lemma i.D.1. This step is exactly the same as the proof of Lemma Main 1 so ignored here. Verification of Assumption (A3) of Lemma i.D.1. For every t ∈ [T ], at a high level assumption (A3) is satisfied once we plug in the following √ def three sets of parameter choices to Corollary i.C.4 and Corollary i.C.5: define κ = 1/ Λ > 1 and for every s ∈ [T − 1], √ βs,1 = 0, δs,1 = 0, τs,1 = O(ηs+1 Ξx · Λ) √ βs,2 = 2ηs+1 ρ, δs,2 = 0, τs,2 = O(ηs+1 Ξx · Λ) √ βs,3 = ηs+1 ρ, δs,3 = ηs+1 λk τs,3 = O(ηs+1 Ξx · Λ) More specifically, for every t ∈ [T ], let {zs }t−1 s=0 be the arbitrary random vector satisfying (i.D.1) of Lemma i.D.1. Define q2 = q 2 /(8 + 2d). (i) t−2 • For coordinate i = 1, 3, 4, . . . , 2 + d of {zs }t−1 s=0 , apply Corollary i.C.4 with {βs,1 , δs,1 , τs,1 }s=0 , q = q2 , D = 1, and κ; • For coordinate i = 2 of {zs }t−1 s=0 , – if t < T0 , apply Corollary i.C.4 with {βs,2 , δs,2 , τs,2 }t−2 s=0 , q = q2 , D = 1, and κ; t−2 – if t ≥ T0 , apply Corollary i.C.5 with {βs,2 , δs,2 , τs,2 }s=0 , q = q2 , D = 1, γ = 1, and κ; • For coordinates i = 2 + d + 1, . . . , 2 + d + T of {zs }t−1 s=0 , – apply Corollary i.C.4 with {βs,3 , δs,3 , τs,3 }t−2 s=0 , q = q2 , D = T , and κ. 25 We refer readers to Footnote 18 in the proof of Lemma Main 1 on page 30 for a detailed discussion as well as a careful treatment for out-of-bound indices. We remark here that to derive such bounds, one needs to use the fact that when w = xZZ> for some unit norm vector x (such as x = xt for some t ∈ [T ] or x = νj for some j ∈ [d]), the quantity ηt kw> ΣLt−1 k22 λk that appeared in Lemma iii.K.1-(a) can be upper bounded by 52 ηt λ2 k+1 kw> Lt−1 k22 λk ≤ ηt λk kw> Lt−1 k22 . Note that we can apply Corollary i.C.4 because for every i = 1, 2, 3, our assumptions on ηs √ PT −1 2 −1 4T 1 Λ imply s=0 τs,i ≤ 100 ln q2 and τs,i ≤ 24 log(4T /q2 ) .
We can apply Corollary i.C.5 with γ = 1 because our assumption ηs ≤ O Λ·Ξ2ρln dT implies x q PT0 −1 Pt−1 2 for every s, and our assumption βs,2 ≥ 10 ln 3(Tq+d) · τ β ≥ 1 + ln Ξ implies s,2 Z s,2 s=0 s=0 βs − 2 3t 2 10 ln q2 τs ≥ ln ΞZ + 1 − 1 = ln ΞZ whenever t > T0 . Therefore, the conclusion of Corollary i.C.4 and Corollary i.C.5 imply that (i) (i) Pr[∃i ∈ [3 + d] : zt−1 > φt,t−1 ] ≤ (3 + d)q2 < q 2 /2 so assumption (A3) of Lemma i.D.1 holds. Application of Lemma i.D.1. Applying Lemma i.D.1, we have Pr[CT ] ≤ 2qT which implies our desired bounds and this finishes the proof of Lemma Main 4.  iii.L.2 After Warm Start Lemma Main 5 √ (after warm start). In the same setting as Lemma Main 4, suppose in addition there exists δ ≤ 1/ 8 such that 9 ln((8 + 2d)/q 2 ) T0 ≥ , δ2 ln2 T0 ∀s ∈ {T0 +1, . . . , T } : 2ηs ρ−ηs2 Ξ2x ≥ Ω(1) s−1 and Then, with probability at least 1 − 2qT (over the randomness of x1 , . . . , xT ): • kW> Pt Q(V> Pt Q)−1 k2F ≤ 5T0 / ln2 (T0 ) t/ ln2 t ηs ≤ √ O(1) . Λ(s − 1)δΞx for every t ∈ {T0 , . . . , T }. Proof of Lemma Main 5. For every t ∈ [T ] and s ∈ {0, 1, ..., t − 1}, consider the same random vectors yt,s ∈ R2+d+T defined in the proof of Lemma Main 4. This time, we define upper bounds:  2ΞZ if s < T0 ;   (1) def (2) def 2 if s = T0 ; , and φ(3) = · · · = φ(2+d+T ) def φt,s = 2ΞZ , φt,s = = 2Ξ2x . t,s t,s 2   5T0 / ln 2(T0 ) if s > T0 . s/ ln s def def V Also consider the same events Ct0 , Ct00 , Ct = Ct0 ∧ Ct00 and C≤t = ts=1 Cs defined in the proof of Lemma Main 4. We again want to apply the decoupling Lemma i.D.1. Verification of Assumption (A1) in Lemma i.D.1. (i) (i) (i) The same functions fs , gs , and hs used in the proof of Lemma Main 4 still apply here. We make minor changes on the second coordinate (and this similar modification was also done in Lemma Main 2): whenever s ≥ T0 , define q   def def (2) 2 2 2 (2) 4 gs (y) = 45ηs+1 Ξx y (2) + 40ηs+1 Ξ2x and h(2) + Ληs+1 Ξ2x + Err . s (y, q) = O Ληs+1 Ξx y (2) Note that we can make this change for gs owing to exactly the same reason as the proof of (2) Lemma Main 2. We can do so for hs because whenever C≤s+1 holds for some s ≥ T0 (which (2) (2) implies yt,s ≤ 5), we have (y (2) )2 = O(y (2) ) so the formulation of hs can be simplified as above. (i) (i) (i) These choices of fs , gs , and hs satisfy assumption (A1) of Lemma i.D.1. Verification of Assumption (A2) of Lemma i.D.1. Same as before. Verification of Assumption (A3) in Lemma i.D.1. 53 Same as the proof of Lemma Main 2, for every t ∈ [T ], let {zs }t−1 s=0 be the arbitrary random 2 vector satisfying (i.D.1) of Lemma i.D.1. Choosing q2 = q /(8 + 2d) again, the same argument before indicates that it suffices to focus on t ≥ T0 + 2 and prove (2) (2) (2) Pr[zt−1 > φt,t−1 | zT0 ≤ 2] ≤ q2 . (iii.L.2) We want to apply Corollary i.C.3. Recall that for every t ∈ {T0 +2, . . . , T }, the random sequence satisfies (i.D.1) with   def 2 2 fs(2) (y, q) = (1 − 2ηs+1 ρ + O(Ληs+1 Ξ2x ))y (2) + O Ληs+1 Ξx + Err ,   def 2 2 (2) 4 2 h(2) (y, q) = O Λη Ξ y + Λη Ξ + Err , s s+1 x s+1 x q def 2 gs(2) (y) = 45ηs+1 Ξx y (2) + 40ηs+1 Ξ2x √ (2) t−1 1 Therefore, {zs }s=T satisfies (i.C.1) with κ = 2/ Λ and τs = δs because the following holds from 0 our assumptions: 1 3/2 2 Ξ2x ) qΞZ ≤ ηs+1 δτs = ≤ 2ηs+1 ρ − Ω(ηs+1 s 1 1 2 2 4 τs2 = 2 2 ≥ Ω(Ληs+1 Ξ2x ) κ2 τs4 = ≥ Ω(Ληs+1 Ξ2x ) κτs = √ ≥ Ω(ηs+1 Ξx ) 4 4 δ s Λδ s Λδs √ Finally, we are ready to apply Corollary i.C.3 with q = q2 , t0 = T0 , and κ = 2/ Λ. Because √ (2) 2) q2 ≤ e−2 , zT0 ≤ 2, δ ≤ 1/ 8 and lnT2 0T ≥ 9 ln(1/q , the conclusion of Corollary i.C.3 tells us δ2 (2) {zs }t−1 s=T0 (2) (2) (2) 0 Pr[zt−1 > φt,t−1 | zT0 ≤ 2] ≤ q2 , which is exactly (iii.L.2) so this finishes the verification of assumption (A3). Application of Lemma i.D.1. Applying Lemma i.D.1, we have Pr[CT ] ≤ 2qT which implies our desired bounds and this finishes the proof of Lemma Main 5.  Parameter iii.L.1. There exists constants C1 , C2 , C3 > 0 such that for every q > 0 that is sufficiently small (meaning q < 1/poly(T, ΞZ , Ξx , 1/gap)), the following parameters both satisfy Lemma Main 4 and Lemma Main 5: ( 2 ln ΞZ ΛΞ2x ln dT t ≤ T0 ; T0 ρ q ln ΞZ T0 ·ρ = C1 · , ηt = C2 · , and δ = C3 · √ . 1 2 2 t > T . ρ ln (T0 ) 0 ΛΞx t·ρ iii.L.3 Under-Sampling Lemma The previous two sections together (namely, Lemma Main 4 and Lemma Main 5 together), analyze the behavior of a rank-k Oja’s algorithm with an eigen-partition (λk , λk − ρ), meaning V consists of all eigenvectors with eigenvalues ≥ λk , Z consists of all the eigenvectors with eigenvalues < λk , and W consists of all the eigenvectors with eigenvalues ≤ λk − ρ. In this section, we consider a more general scenario Definition iii.L.2. Given λ ∈ [0, λk ] and ρ ∈ (0, λ), we say that V, Z, W form an (λ, λ − ρ) eigenpartition if V consists of all eigenvectors with eigenvalues ≥ λ, Z consist of all the eigenvectors with eigenvalues < λ, and W consist of all the eigenvectors with eigenvalues ≤ λ − ρ. The following lemma studies the behavior of a rank-k 0 Oja’s algorithm for an arbitrary k 0 ≤ r. Lemma Main 6 (under sampling). Let V, Z, W be an (λ, λ − ρ) eigen-partition where the V ∈ Rd×r , Z ∈ Rd×(d−r) , W∈ Rd×(d−m−r) . We study rank k 0 Oja’s algorithm for some k 0 ≤ r. 0 For every q ∈ 0, 21 , ΞZ ≥ 2, Ξx ≥ 2, and fixed matrix Q ∈ Rd×k , suppose it satisfies 54 • kZ> Q(Q> VV> Q)−1/2 k2F ≤ ΞZ , h j−1 > Q(Q> VV> Q)−1/2 • Prxt ∀j ∈ [T ], x> t ZZ (Σ/λr+1 ) • νj> ZZ> Q(Q> VV> Q)−1/2 2 2 i ≤ Ξx ≥ 1−q 2 /2 for every t ∈ [T ], ≤ Ξx for every j ∈ [d]. Suppose also the learning rates {ηs }s∈[T ] satisfy the all conditions in Lemma Main 4 and Lemma Main 5 P def P with Λ replaced by Λ0 = ri=1 λi + Ξ2x r+m j=r+1 λj . 0 d×k Then, letting Qt ∈ R be the output of the rank-k 0 Oja’s algorithm with respect to input Q, with probability at least 1 − 2qT (over the randomness of x1 , . . . , xT ), it satisfies • kW> Qt k2F ≤ 5T0 / ln2 (T0 ) t/ ln2 t for every t ∈ {T0 , . . . , T }. 0 Proof of Lemma Main 6. Since V> Q ∈ Rr×k for r ≥ k 0 , we can always find a (column) orthonor0 e = VS ∈ Rd×(r−k0 ) , we have mal matrix S ∈ Rr×(r−k ) such that S> V> Q = 0. Letting Q e > VV> Q e = I, Q e > VV> Q = 0, Z> Q e = 0, Z> Σj−1 Q e =0 . Q > j−1 , or X> = ν ZZ> , we always have Therefore, for every X> = Z> , X> = x> j t ZZ (Σ/λr+1 )   > e e −1 k2 = Tr X> [Q, Q]([Q, e e > VV> [Q, Q]) e −1 [Q, Q] e >X kX> [Q, Q](V [Q, Q]) Q] F   = Tr X> Q(Q> VV> Q)−1 Q> X = kX> Q(Q> VV> Q)−1/2 k2F ≤ Ξ2X . e ∈ Rd×r now satisfies This implies this matrix [Q, Q] > [Q, Q]) e e −1 k2 ≤ ΞZ , • kZ> [Q, Q](V F h j−1 > > [Q, Q]) e e −1 [Q, Q](V • Prxt ∀j ∈ [T ], x> t ZZ (Σ/λr+1 ) • > [Q, Q]) e e −1 νj> ZZ> [Q, Q](V 2 ≤ Ξx for every j ∈ [d]. 2 i ≤ Ξx ≥ 1 − q 2 /2 for all t ∈ [T ], e and k = r and conclude Therefore, we can apply Lemma Main 5 with initial matrix [Q, Q] with probability 1 − 2qT : > e −1 k2 ≤ e kW> Pt [Q, Q](V Pt [Q, Q]) F 5T0 / ln2 (T0 ) . t/ ln2 t e ∈ Rd×r be the output of the rank-r Oja’s algorithm on input [Q, Q] e Now, let Qt = QR(Pt [Q, Q]) after t steps. Owing to Lemma 2.2, we have 5T0 / ln2 (T0 ) . t/ ln2 t However, recall that QR decomposition orthonormalizes a column vectors only with respect to its previous columns. This implies, Qt , which is the output of the rank-k 0 Oja’s algorithm on input Q, is exactly identical to the first k 0 columns of Qt . Therefore, we have > e e −1 k2 ≤ kW> Qt k2F ≤ kW> Pt [Q, Q](V Pt [Q, Q]) F kW> Qt k2F ≤ kW> Qt k2F ≤ 55 5T0 / ln2 (T0 ) . t/ ln2 t  iii.M Proof of Theorems 1 and 2 (for Oja) We now prove the final Theorem of Oja’s algorithm in gap-free case: Theorem 2 (restated). For every ρ, ε, p ∈ (0, 1), let k + m be the number of eigenvalues of Σ in (λk − ρ, 1]. Let o nP P k k Pk+m Λ2 = k+m Λ1 = min i=1 λi . i=1 λi + p2 j=k+1 λj , 1 , Define learning rates e T0 = Θ  kΛ1 ρ2 p2  e T1 = Θ ,  Λ2 ρ2  ,    1 e  t ≤ T0 ; Θ     ρ·T0  1 e ηt = Θ t ∈ (T0 , T0 + T1 ]; ρ·T1      1 e  Θ t > T0 + T1 . ρ·(t−T0 ) Let W be the column orthonormal matrix consisting of all eigenvectors of Σ with values no more than λk − ρ. Then, the output QT ∈ Rd×k of Oja’s algorithm satisfies with prob. at least 1 − p:   e T1 it satisfies kW> QT k2F ≤ ε . for every T = T0 + T1 + Θ ε e hides poly-log factors in 1 , 1 and d. Above, Θ p ρ Note that Theorem 1 is a direct corollary of Theorem 2 by setting ρ ← gap and noticing that m = 0 so Λ1 = λ1 + · · · + λk and Λ2 = 0. Proof of Theorem 2. First for a sufficiently large constant C, we can apply Lemma 5.1 with p0 = p8 and some sufficiently small q = poly(1/T, 1/d, p), with probability at least 1 − p0 − q 2 ≥ 1 − p/4 over the random choice of Q, the following holds: 
2 d  , and (Z> Q)(V> Q)−1 F ≤ O dk 2 ln p  p # "    qk ln T    2 q i−1 > ≤ q2 Prx1 ,...,xT ∃i ∈ [T ], ∃t ∈ [T ], x> Q(V> Q)−1 ≥ Ω t ZZ (Σ/λk+1 ) p 2  q    T   k ln q   ν > ZZ> Q(V> Q)−1 ≤ O for every j ∈ [d] . j 2 p Denote by C1 the union of the above three events, and we have PrQ [C1 ] ≥ 1 − p/4. Let us introduce now some notations for this proof. As illustrated in Figure 2, besides the definitions of V, W, Z, let W1 be the (column) orthogonal matrix consisting of all eigenvectors of Σ with eigenvalue ≤ λk − ρ2 , and V1 be the (column) orthogonal matrix consisting of all eigenvectors of Σ with eigenvalue > λk − ρ2 . We now wish to apply our main lemma twice. Once with (V, Z, W) = (V, Z, W1 ), and once with (V, Z, W) = (V1 , W1 , W). Application One. For every fixed Q, whenever C1 holds, we can let q  dk d   k ln Tp  ΞZ = Θ 2 ln , Ξx = Θ , p p p so the initial conditions in Lemma Main 4 is satisfied. Also, according to Parameter iii.L.1, our parameter choices ηt for t ≤ T0 satisfy the assumptions in Lemma Main 4 with Λ replaced by Λ1 . We can therefore apply Lemma Main 4 with Q = Q, (V, Z, W) = (V, Z, W1 ), 56 λk = λk , ρ = ρ/2, ΞZ , Ξx Application One 𝑉 𝑍 Application Two 𝑊 𝑉 𝑽 𝑍 𝑊 𝒁 𝑾𝟏 𝑾 𝑽𝟏 eigenvalues 𝜆𝑘 𝜆𝑘 − 𝜌/2 𝜆𝑘 − 𝜌 eigenvalues Figure 2: Notations V1 , W1 for the proof of Theorem 2 and derive that Pr x1 ,...,xT0   kW1> PT0 Q(V> PT0 Q)−1 k2F ≥ 2 C1 ≤ 2qT0 ≤ p/4 . Now, let C2 be the event where kW1> PT0 Q(V> PT0 Q)−1 k2F ≤ 2 holds, and by union bound PrQ,x1 ,...,xT0 [C2 ] ≥ 1 − p/4 − p/4 = 1 − p/2. Application Two. If C2 is true then ¬ > −1/2 2 > −1/2 2 kW1> PT0 Q(Q> P> kF ≤ kW1> PT0 Q(Q> P> kF T0 V1 V1 PT0 Q) T0 VV PT0 Q) ­ V1 V1> = kW1> PT0 Q(V> PT0 Q)−1 k2F ≤ 2 . VV> (iii.M.1) > −1 (Q> P> T0 V1 V1 PT0 Q) Above, inequality ¬ is because  and this gives us  > > > −1 > (Q PT0 VV PT0 Q) ; equality ­ is because V PT0 Q is a square matrix and we therefore have
> −1 = (V> P Q)−1 (V> P Q)−1 > . (Q> P> T0 T0 T0 VV PT0 Q) Inequality (iii.M.1) also implies that, for every x ∈ Rd with kxk2 ≤ 1, it satisfies √ > > −1/2 kx> W1 W1> PT0 Q0 (Q> k2 ≤ 2 . 0 PT0 V1 V1 PT0 Q0 ) In sum, whenever C2 holds, we have that the initial conditions in Lemma Main 6 is satisfied. Also, according to Parameter iii.L.1, our parameter choices ηt for t = T0 + 1, . . . , T —once shifted left by T0 — satisfy the assumptions in Lemma Main 6 with Λ replaced by Λ2 . We can therefore apply Lemma Main 6 with Q = PT0 Q, (V, Z, W) = (V1 , W1 , W), and conclude that h Pr ∀T0 + T1 ≤ t ≤ T : kW> Qt k2F ≤ λ = λk − ρ/2, ρ = ρ/2, Ξx = ΞZ = 2 i 5T1 / ln2 (T1 ) C ≥ 1 − 2qT ≥ 1 − p/4 . 2 xT0 +1 ,...,xT (t − T0 )/ ln2 (t − T0 ) In sum, we have with probability at least Pr[C2 ](1 − p/4) ≥ 1 − p over the random choices of Q, e 1 /ε), it satisfies kW> Qt k2 ≤ ε. and x1 , . . . , xT , for every t = T0 + T1 + Θ(T  F 57 iii.N Proof of Theorem 4 and 5 (for Oja++ )
Algorithm 1 Oja {xt }Tt=1 , {ηt }Tt=1 , Q Input: vectors {xt ∈ Rd }Tt=1 , learning rates {ηt ∈ R>0 }Tt=1 , an initial matrix Q ∈ Rd×k . 1: Q0 ← Q. 2: for t ← 1 to T do
3: Qt ← QR (I + ηt xt x> t )Qt−1 4: end for 5: return QT Algorithm 2 Oja++ {xt }Tt=1 , {ηt }Tt=1 , {(T (i) , Q(i) )}si=1
Input: vectors {xt ∈ Rd }Tt=1 ; learning rates {ηt ∈ R>0 }Tt=1 ; initial matrices {Q(i) ∈ Rd×ri }si=1 ; lengths T (1) , . . . , T (s) satisfying T (1) + · · · + T (s) = T . Output: QT ∈ Rd×(r1 +···+rs ) . 1: Q0 ← []; t ← 0. 2: for i ← 1 to s do 3: Qt ← [Qt , Q(i) ] 4: for t0 ← 1 to T (i) do 5: t ← t + 1;
6: Qt ← QR (I + ηt xt x> t )Qt−1 7: end for 8: end for 9: return QT We formally write the pseudocode of Oja++ in Algorithm 2. We also write down the pseudocode of Oja because we shall use it in the analysis of this section. We emphasize here that Oja++ spends the same per-iteration running time and space complexity as Oja. 58 We prove the following main theorem: Theorem 5 (Oja++ , restated). For every ρ, ε, p ∈ (0, 1), let k + m be the number of eigenvalues of Σ in (λk − ρ, 1]. Define     k m k+m X X Λ2 1 X Λ1 e e Λ1 = , T1 = Θ λi + 2 λj , Λ2 = λi , s = dlog(k + 1)e, T0 = Θ p ρ2 p2 ρ2 i=1 i=1 j=k+1 and learning rates (where C     C · ρT1 0        1  C·  ρ(t−11iT  0) ηt = 1  C · ρT1        1  C· ρ(t−11sT0 ) e is some fixed value that is only Θ(1)): if t ∈ [11iT0 , 11iT0 + T0 ) for some i ∈ {0, 1, ..., s − 1}; if t ∈ [11iT0 + T0 , 11iT0 + 11T0 ) for some i ∈ {0, 1, ..., s − 1}26 ; if t ∈ (11sT0 , 11sT0 + T1 ); if t > 11sT0 + T1 . i−1 c−bk/2i c) For each i ∈ [s], let Q(i) ∈ Rd×(bk/2 Then, the output   T QT ← Oja++ xt t=1 , ηt satisfies with probability at least 1 − p for every be a random matrix with entries i.i.d. N (0, 1).  T s−1 , (T0 , Q(i) ) i=1 t=1 e T = 11sT0 + T1 + Θ  e hides poly-log factors in 1 , 1 and d. Above, Θ p ρ T1 ε    ∪ (T − 11(s − 1)T0 , Q(s) kW> QT k2F ≤ ε . it satisfies Note that Theorem 4 is a simple corollary of Theorem 5 by setting ρ ← gap and noticing that m = 0 so Λ1 = λ1 + · · · + λk and Λ2 = 0. 𝑘/2 - 𝑘/22 𝑘 − 𝑘/2 𝑑 𝑄 1 𝑄 𝑘/22 - 𝑘/23 2 𝑄 3 𝑘 𝑑 𝑄 𝑇 1 1 𝑄 apply Oja’s for = 11𝑇0 iterations 1 𝑄 𝑇 2 2 𝑄 apply Oja’s for = 11𝑇0 iterations 2 𝑄 𝑇 3 3 … 𝑄 𝑘 𝑠 apply Oja’s for = 11𝑇0 iterations 𝑄𝑇 𝑇 apply Oja’s for = 𝑇 − 11𝑠𝑇0 iterations s+1 Figure 3: Illustration of our Oja++ algorithm. e (0) = [] being empty matrix and for every j ∈ [s] by Proof of Theorem 5. Denote by Q     j e (j) def Q = Oja++ x1 , . . . , x11jT0 , η1 , . . . , η11jT0 , (11T0 , Q(j) ) i=1 def 26 In fact, the intermediate learning rates for t ∈ {1, 2, . . . , 11sT0 } can all be set to the same value them slightly decrease between epochs just for the sake of having a cleaner proof. 59 C . ρT0 We make the output of Oja++ if we run the outer loop only for j iterations. By definition, it satisfies   (j−1) (j)   e e (j) = Oja x11(j−1)T +b T0 , η11(j−1)T +b T0 , Q ,Q , ∀j ∈ [s] : Q 0 0 b=1 b=1 e (j) as the output of the old Oja’s algorithm when initialized on the previous so we can view each Q (j−1) e output Q appended with a new random matrix Q(j) . We illustrate this pictorially in Figure 3. We also introduce some notations (see Figure 4 for an illustration). For each i ∈ [s], we define Vi to be the (column) orthonormal matrix consisting of all eigenvectors of Σ with eigenvalue i > λk − s+1 ρ, and Wi to be the (column) orthonormal matrix consisting of all eigenvectors of Σ i with eigenvalue ≤ λk − s+1 ρ. Define V0 = V and W0 = Z. Let ki ≥ k be the column dimension of Vi . Application 1 𝑉 𝑍 Application 2 𝑊 𝑉 dim = 𝑘 𝑍 Application 𝑠 + 1 𝑉 𝑊 𝑽 = 𝑽0 𝑽1 𝒁 = 𝑾0 𝑾1 dim = 𝑘1 𝑽2 𝑽3 dim = 𝑑 − 𝑘1 𝑽𝑠 dim = 𝑑 − 𝑘2 𝑾3 dim = 𝑘3 dim = 𝑑 − 𝑘3 𝑾𝑠 dim = 𝑘s 𝑊 dim = 𝑑 − 𝑘 𝑾2 dim = 𝑘2 𝑍 dim = 𝑑 − 𝑘𝑠 𝑾 eigenvalues 𝜆𝑘 … 𝜆𝑘 − 𝜌 eigenvalues Figure 4: Notations V0 , . . . , Vs , W0 , . . . , Ws for the proof of Theorem 5. Below, we wish to apply our Lemma Main 6 a total of s + 1 times each corresponding to one outer loop of Oja++ . Each time we shall use our new initialization lemma (i.e., Lemma iii.J.2) in order to satisfy the preassumption of Lemma Main 6. The first s applications serve as a graduated warm-start phase and the last application provides the final convergence rate. def e (j) k2 ≤ 1 }. In this first Application 1 Through s. Define event Ci = {∀j ∈ N, j ≤ i : kWj> Q F 2 step we prove Pr[Ci ] ≥ 1 − 2ip for all i = 0, 1, . . . , s by induction. The base case Pr[C0 ] = 1 is e (0) is an empty matrix. obvious because Q Suppose Pr[Ci−1 ] ≥ 1 − 2(i − 1)p holds true for some i ∈ [s] and we wish to bound Pr[Ci ]. We e (i−1) is an empty matrix) first note that event Ci−1 implies (or if i = 1 then Q

e (i−1) > Vi−1 V> Q e (i−1) = I − Q e (i−1) > Wi−1 W> Q e (i−1)  1 I . Q i−1 i−1 2 (i−1) d×α (i) d×β i−1 e Next, we have Q ∈R and Q ∈ R where α = (k −bk/2 c) and β = bk/2i−1 c−bk/2i c. Since ki−1 − α ≥ k − α ≥ 2β − 1, we have β β β p √ ≤ √ ≤ √ <6 . √ √ 2 ( 2β − 1 − β − 1)2 ( k − α − β − 1) ( ki−1 − α − β − 1)2 Therefore, we can apply Lemma iii.J.2 on e =Q e (i−1) , Q Q = Q(i) , (V, Z) = (Vi−1 , Wi−1 ). and derive that, denoting by Q = [Q(i−1) , Q(i) ], then with probability 1 − p over the random choice 60 of Q(i) , the following event Bi holds (for some polynomially small q):     −1/2 2  d > Q > Q) Q> V e  ≤ O and V (W  i−1 i−1 i−1 p2   F  " #      −1/2 2 > def ∃`∈[T ] `−1 > Q > > e 12 ≥Ω Pr Q Q Vi−1 Vi−1 ≤q Bi = ∃t∈[T ] xt Wi−1 Wi−1 (Σ/λk+1 ) p x ,...,x  1 T  2   2        ν > W W> Q Q> V V> Q −1/2 ≤ O e 1 for every j ∈ [d].  j i−1 i−1 i−1 i−1 p2 2                  e (j) can be viewed as the output of the original Oja’s algorithm on Whenever Bi holds, because Q  T0 , we can apply Lemma Main 6 with input Q and applied with learning rates η11(i−1)T0 +b b=1   i−1 1 Q = Q, (V, Z, W) = (Vi−1 , Wi−1 , Wi ), (λ, ρ) = λk − s+1 ρ, s+1 ρ , 2 ), O(1/p 2 )) e e (ΞZ , Ξx ) = (O(d/p  T0 —once shifted left by 11(i − 1)T0 — are We emphasize that these learning rates, η11(i−1)T0 +b b=1 exactly    e 1  Θ t ≤ T0 0  ρT ηt = , T = 11T0 , Λ = Λ1 e 1  Θ t > T 0 ρt so they satisfy the assumption of Lemma Main 6 according to Parameter iii.L.1. Finally, the conclusion of Lemma Main 6 tells us  1 > e (i) 2 Pr kWi Q kF ≥ Ci−1 ∩ Bi ≤ p , 2 and by union bound, we have (recalling Pr[Bi ] ≥ 1 − p):   1 > e (i) 2 ≥ Pr[Ci−1 ∩ Bi ] × (1 − p) Pr[Ci ] = Pr Ci−1 ∧ kWi Q kF ≤ 2 = (1 − p) Pr[Ci−1 ] Pr[Bi | Ci−1 ] ≥ (1 − p)(1 − 2(i − 1)p)(1 − p) ≥ 1 − 2ip . This finishes the proof that Pr[Cs ] ≥ 1 − 2sp. Application s + 1. We now focus on the last outer loop of Oja++ , which satisfies    T T e (s) . QT = Oja xt t=11sT0 +1 , ηt t=11sT0 +1 , Q e (s) k2 ≤ 1 and thus (Q e (s) )> Vs V> Q e (s)  Similar to the first s outer loops, under event Cs we have kWs> Q s F 2 √ 1 > e (s) −1/2 k ≤ e (s) > 2 and therefore 2 2 I. This implies k((Q ) Vs Vs Q ) > e (s) e (s) > > e (s) −1/2 2 e (s) k2 k((Q e (s) )> Vs V> Q e (s) )−1/2 k2 ≤ 2. kW Q ((Q ) Vs V Q ) k ≤ kW> Q s s F s F s 2 Rd Thus, for every x ∈ with kxk2 ≤ 1: > > e (s) e (s) > e (s) )−1/2 k2 ≤ kW> Q e (s) k2 k((Q e (s) )> Vs V> Q e (s) )−1/2 k2 ≤ 2 . kx Ws Ws Q ((Q ) Vs Vs> Q s s Now we can apply Lemma Main 6 again with parameters  (s) e Q = Q , (V, Z, W) = (Vs , Ws , W), (λ, ρ) = λk −  s 1 ρ, ρ , s+1 s+1 ΞZ = Ξx = 2  T We emphasize that these learning rates, ηt t=11sT0 +1 —once shifted left by 11sT0 — are exactly    e 1  Θ t ≤ T1 1  ρT ηt = , T = T − 11sT0 , Λ = Λ2 e 1  Θ t > T1 ρt 61 so they satisfy the assumption of Lemma Main 6 according to Parameter iii.L.1. We can conclude from Lemma Main 6 that     T1 1 > 2 > e (s) 2 e ≤p . Pr kW QT kF = Ω kWs Q kF ≤ T − 11sT0 2 h i e (s) k2 ≥ 1 ≤ 2sp we complete the proof of Theorem 5 if we replace Taking into account Pr kWs> Q F 2  p with p/(2s + 1). 62 References [1] Zeyuan Allen-Zhu and Yuanzhi Li. 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On Integrating Fuzzy Knowledge Using a Novel Evolutionary Algorithm Nafisa Afrin Chowdhury, Murshida Khatun and M. M. A. Hashem Department of computer science and Engineering, Khulna University of Engineering & Technology, Khulna Khulna-920300, Bangladesh Ph: +88041-774318, Fax+88-0141774403 E-mail: [email protected]; [email protected]. Abstract Fuzzy systems may be considered as knowledge-based systems that incorporates human knowledge into their knowledge base through fuzzy rules and fuzzy membership functions. The intent of this study is to present a fuzzy knowledge integration framework using a Novel Evolutionary Strategy (NES), which can simultaneously integrate multiple fuzzy rule sets and their membership function sets. The proposed approach consists of two phases: fuzzy knowledge encoding and fuzzy knowledge integration Four application domains, the hepatitis diagnosis, the sugarcane breeding prediction, Iris plants classification, and Tic-tac-toe endgame were used to show the performance of the proposed knowledge approach. Results show that the fuzzy knowledge base derived using our approach performs better than Genetic Algorithm based approach. Key words- Fuzzy knowledge, Rule set, Membership function, Evolutionary Algorithms (EA), Novel Evolutionary Algorithm (NES), Crossover, Mutation. 1. Introduction The construction of a reliable knowledge-based system, especially for complex application problems, usually requires the integration of multiple knowledge inputs as related domain knowledge is usually distributed among multiple sites. The use of knowledge integrated from multiple knowledge sources is 1 thus important to ensure comprehensive coverage. Most knowledge sources or actual instances in realworld applications, especially in medical and control domains, contain fuzzy or ambiguous information. Expressions of domain knowledge in fuzzy terms are thus seen more and more frequently. Several researchers have recently investigated automatic generation of fuzzy classification rules and fuzzy membership functions using evolutionary algorithms [1]. Many knowledge acquisition and integration systems [4] use genetic algorithms to derive knowledge from training instances. In this study we have proposed an approach that uses a novel evolutionary algorithm for fuzzy knowledge integration process. In this technique, the emphasis is not on the acquisition of a structure with high fitness, as in GA, but on modeling evolution at the granularity of an individual, ESs considers an individual to be composed of a set of behaviors, each of which is a feature. In this approach we used insertion deletion mutation as a variation (genetic) operator in order to introduce variable length string for each individual. Therefore the integration process shows fast convergence with considerable accuracy as comparing with GA approach. The remainder of this paper is organized as follows. A NES based fuzzy knowledge integration framework is proposed in section 2. The fuzzy knowledge encoding approach is explained in section 3. The fuzzy knowledge integration approach is proposed in section 4. Experiments on two application domains- Hepatitis Diagnosis and Sugarcane Breeding Prediction are stated in section 5. Conclusion and future research directions are given in section 6. 2. Fuzzy Knowledge Integration Framework Here, we propose a NES-based fuzzy-knowledge integration framework. The proposed framework can integrate multiple fuzzy rule sets and membership function sets at the same time [4]. Fuzzy rule sets, membership functions, and test objects including instances and historical records may be distributed among various sources. Knowledge from each site might be directly obtained by a group of human experts using knowledge-acquisition tools or derived using machine-learning methods. Here, we assume that all knowledge sources are represented by fuzzy rules since almost all knowledge derived by 2 knowledge-acquisition tools or induced by machine-learning methods may easily be translated into or represented by rules. The proposed framework maintains a population of fuzzy rule sets with their membership functions and uses a Novel Evolutionary Algorithm to automatically derive the resulting fuzzy knowledge base. This integration framework operates in two phases: fuzzy knowledge encoding and fuzzy knowledge integration. The encoding phase first transforms each fuzzy rule set and its associated membership functions into an intermediary representation, which is further encoded as a variable-length string [4]. The integration phase then chooses appropriate strings for “mating,” gradually creating good offspring fuzzy rule sets and membership function sets. The offspring fuzzy rule sets with their associated membership functions then undergo recursive “evolution” until an optimal or nearly optimal set of fuzzy rules and membership functions has been obtained. Fig. 1 shows the two-phase process, where are the fuzzy rule sets with their associated membership function sets, as obtained from different sources for integration. Initial population Generation 0 ~~ R S 1 + MFS 1 Chromosome 1 ~~ R S 2 + MFS 2 Chromosome 2 Generation k Chr 1 Chr 1 Chr 2 Chr 2 The best one Genetic ~~ R S + MFS Operator ~~ R S m + MFSm Chromosome m Knowledge encoding Chr m Chr m Knowledge integrating Fig. 1: The genetic-fuzzy knowledge integration process. 3. Knowledge Encoding The encoding phase first transforms each fuzzy rule set and its associated membership functions into an intermediary representation in a few steps [4], further encoded as a variable-length string. Fig 2 shows µ individuals constructing a generation. Each individual has rule set and membership functions that represent the feature values and corresponding membership criteria for each feature respectively. 3 Ind 1 Rule 1 Rule 2 MF 1 MF 2 Ind 2 Rule 1 Rule 2 MF 1 MF 2 Ind µ Rule 1 Rule 2 MF 1 MF 2 Fig.2: The rule set and membership function encoding of a generation having µ individuals. To effectively encode the associated membership functions, we use two parameters to represent each membership function, as Parodi and Bonelli [4] did. Membership functions applied to a fuzzy rule set are then assumed to be isosceles-triangle functions as shown in Fig. 3, where aij denotes the jth linguistic value of Feature Ai , uaij denotes the membership function of aij , cij indicates the center abscissa of membership function uaij and wij represents half the spread of membership function uaij . u aij 1 uaij w ij cij Ai Fig.3: Membership functions of feature Ai . 4. Knowledge Integration After each fuzzy rule set with its associated membership functions has been encoded as a string, the fuzzy-knowledge integration process starts. It chooses good individuals in the population for “mating,” gradually creating better offspring fuzzy rule sets and membership function sets. In order to make the integration process more effective and to increase accuracy of the resultant knowledge base we used a Novel Evolutionary Strategy (NES) Algorithm in place of Genetic Algorithm (GA). There are µ individuals are created initially by the program using the function urnN_N (a, b) for input variables and 4 output variables. Their corresponding evaluated fitness f value for each individual is also calculated in the evaluated function. Two important factors are used in evaluating process. They are: accuracy (RS) and ~~ complexity (RS) of the resulting knowledge structure. Accuracy of a fuzzy rule set RS is evaluated using test objects as follows [4]: ~~ Accuracy (R~S~ )= total number of objects correctly matched by R S ....................1 total number of objects The more data used the more objective and accurate the evaluation is. The complexity of the resulting rule ( ~ ~ ) is the ratio of rule increase, defined as follows: set R S ~~ Complexity (R~S~ )= Number of rules in the integrated rule set R S .......................2 p ( ) ~~ [ ∑ Number of rules in the initial R S i ] p i =1 ~~ Where R S i is the ith initial fuzzy rule set, and P is the number of initial rule sets. Accuracy and complexity are combined to represent the fitness value of the rule set. Since the goal is to increase the accuracy and decrease the complexity of the resulting rule sets, the evaluation function is then defined as follows: Fitness (RS ) =. [Accuracy(RS )] .............................................................................3 [Complexity(RS )]α Where α is a control parameter, representing a tradeoff between accuracy and complexity. Two genetic operators, crossover and mutation, are applied on individuals of each generation to create new offspring. A. SBMAC In the SBMAC (Sub-population Based Max-mean Arithmetical Crossover) each subpopulation’s elite and the mean –individual (the virtual parent) created from that subpopulation excluding the elite are used. This SBMAC is supposed to explore promising areas in the search space with different directions towards the optimum point [3]. Thus, the algorithm is exposed to less possibility of being trapped in local optima. 5 B. Modified SBMAC For an instance of the evolution, a modified version of the SBMAC operation is shown in fig 4. Firstly, the minimum length individual within each subpopulation is determined. Secondly, the offspring variables are produced up to this length [5].Then the remaining part of the parent subpopulation variables is appended to the offspring population variables to form the full length chromosome. a a1 a2 a3 a4 a5 b b1 b2 b3 b4 b5 Crossover operation a′ 2 a′ 3 a4 a5 b ′ b′ 1 b′ 2 b′ 3 b4 b5 a′ a′ 1 Non crossover part Fig 4: Modified SBMAC operation. C. TVM In the real world, a rapid change is observed at the early stages of life and a slow change is observed at latter stages of life in all kind of animals/plants. These changes are more often occurred dynamically depending on the situation exposed to them. A special dynamic Time-Variant Mutation (TVM), which follows this natural evidence [3], is applied here. D. Insertion Deletion Mutation In order to generate variable length chromosome for each individual insertion deletion mutation [5] is performed on each individual. For insertion mutation, a new randomly generated steering angle value (gene) is inserted at a selected position based on a probability of insertion Pim .This will increase the length of a chromosome. On the other hand the deletion mutation is the reverse action than that of the insertion mutation. Fig 5 shows the insertion deletion mutation procedure. a ′ a′ 1 a′ 2 a′ a′ 3 a′ 1 a′ 2 a′ 3 Deletion mutation at 5th parameter. a4 b5 a5 b′ b′ 1 b′ 2 b′ 3 b′ b′ 1 b′ 2 b′ 3 b4 b5 b′ 4 b5 Insertion mutation at 4th parameter. Fig. 5: Insertion deletion mutation operation. 6 5. Simulation result To demonstrate the effectiveness of the proposed fuzzy knowledge integration approach, we applied it to four application domains. The first one was “The Hepatitis Diagnosis Domain”, which contained 155 cases obtained from Carnegie-Mellon University [4]. The second was “The Sugarcane Breeding Prediction Domain”, which contained 699 cases obtained from Taiwan Sugar Research Institute (TSRI) [4]. The flexibility of this approach is proved by applying successfully in two new application domains, where the GA approach is not applied yet. They are: “The Iris Plants Classification Domain” [7] and “The Tic- Tac-Toe Endgame Domain” [7]. 5.1 The Hepatitis Diagnosis Domain: The hepatitis diagnosis problem was first used here to test the performance of the proposed fuzzyknowledge integration approach. The goal of the experiments was to identify two possible classes, Die or Live, from a set of instances. Table I shows an actual case expressed in term of 19 features and one class. Among these features, Age, Bilirubin, Alk Phosphate, SGOT, Albumin, and Protime are numerical and have membership functions with them. Table I A case for hepatitis diagnosis Features Features value Billirubin 0.90 Alk Phosphate 95 SGOT 28 Albumin 4.0 Protime 75 Class: Live Each rule, consisting of 5 feature tests and a class pattern, was encoded into a record having 5 rule values and 10 membership function value (center, width). The parameter α was set at 0.01. Experimental results show that executing the proposed approach over more generations initially yielded more accurate results and finally gradually converged to a constant. The structure of the best individual after convergence is shown in fig.6. Fig. 6(a), 6(b), 6(c), 6(d), 6(e) are for five numerical parametersBillirubin, Alk Phosphate, SGOT, Albumin, Protime. 7 Fig. 6 (a): Membership function of Billirubin. Fig. 6 (c): Membership function of SGOT. Fig. 6 (b): Membership function of Alk phosphate. Fig. 6 (d): Membership function of Albumin. Fig. 6 (e): Membership function of Protime. Fig 7 shows the relationship between the accuracy of the resulting rule sets with the number of generations by the proposed approach. As the numbers of generations increased, the accuracy as well as resulting fitness value also increased and finally converged to a specific value. 8 Accuracy vs Generation 100 95 Accuracy 90 85 80 75 70 0 50 100 150 200 250 300 350 400 450 500 Generation Fig.7: Relationship between accuracy and number of generations for the hepatitis domain . Table II compares the accuracy of our proposed approach with that of the above learning methods [4]. It can easily be seen that our approach has a higher accuracy than some other learning methods. Moreover, in our approach convergence occurs after 200 generation whereas in GA approach, it takes 4000 generation [4]. Table II A comparison with some other learning methods for hepatitis domain Methods Our approach GA approach Accuracy 96.33% 91.61% 5.2The Sugarcane Breeding Prediction Domain In this experiment, we present a real application to the sugarcane breeding prediction [4] based on our proposed approach. The goal of this application was to breed good sugarcane plants with high sugar contents and strong disease-resistance abilities. Table III A Case for Sugarcane Breeding Prediction Features Values Stalk diameter 3.5cm Stalk length 5.2m Stalk number 96 Cane yield 198 Offspring with high sugar-ingredients and disease-resistant abilities 9 Table III shows an actual case expressed in terms of 36 features and a class report. Here, the parameter α was set at 0.001. Our approach obtained an accuracy rate of 89.52% after 500 execution generations. In fig. 8 the resultant structure of the best individual is represented. Fig 8 (a), 8 (b), 8 (c), 8 (d) shows the resultant membership function for Stalk diameter, Stalk length, Stalk number and Cane yield, respectively. Fig 8 (a): Membership function of Stalk diameter. Fig 8 (c): Membership function of Stalk number. Fig 8 (b): Membership function of Stalk length. Fig 8 (d): Membership function of Cane yield. Fig.9 shows the results for different generations with respect to accuracy in the sugarcane domain. Table IV compares the accuracy of our proposed approach with that of the sugarcane breeding assistant system (SCBAS) learning method [4] in the sugarcane domain. It can easily be seen that our approach has a higher accuracy than the GA approach or SCBAS learning method. Moreover, in our approach convergence occurs after 400 generation whereas in GA approach, it takes 4500 generation[4]. 10 Fig 9: Relationship between accuracy and generation for the sugarcane domain. Table IV A comparison with some other learning methods for sugarcane breeding prediction domain. Methods Our approach GA approach Accuracy 89.01% 76.02% In case of Iris plants classification our approach shows the accuracy of 76% after 100 generation and it is 67% after 300 generation in case of Tic-tac-toe endgame. 6. Concluding Remarks In this paper, we have proposed an appropriate representation to encode the fuzzy knowledge and have shown how fuzzy-knowledge integration can be effectively processed using this representation. Experimental results have also shown that our genetic fuzzy-knowledge integration framework is valuable for simultaneously combining multiple fuzzy rule sets and membership function sets. Our approach needs no human experts’ intervention during the integration process. The time required by our approach is thus dependent on computer execution speed, but not on human experts. Much time can thus be saved since experts may be geographically dispersed. Also, our approach is a scalable integration method that can be applied as well when the number of rule sets to be integrated increases. Integrating a large number of rule sets may increase the validity of the resulting knowledge base. It is also objective since human experts are not involved in the integration process. References [1] Zimmermann, H.J. “Fuzzy Set Theory and Its Application,” 2nd edition, Allied Publishers Ltd, New Delhi, 1996. [2] Ching-Hung Wang, Tzung-Pei Hong and Shian-Shyong Tseng, “Integrating Fuzzy Knowledge by Genetic Algorithms”, IEEE Transaction on Eolutionary Computation Vol-2, No.4, November 1998. [3] K. Watanabe and M.M.A. Hashem, “Evolutionary Computations: New Algorithms and their Applications to Evolutionary Robots,” Series: Studies in Fuzziness and Soft Computing, Vol. 147, Springer-Verlag, Berlin/New York, ISBN: 3-540-20901-8, (2004). [4] M.M.A. Hashem,“Global Optimization Through a new class of Evolutionary Algorithms”, PhD Dissertation, Saga University, Japan, 1999. [5] Marco Russo, “Genetic Fuzzy Learning,” IEEE Transaction on Eolutionary Computation Vol-4, No.3, September 2000. [6] O. Cordon, F. Herrera, F. Hoffmann, F. Gomide and L. Magdalena, “Ten Years of Genetic Fuzzy Systems: Current Framework and New Trends”. [7] http://ics.uci.edu/~mlearn/MLRepository/ 12 

Locally Robust Semiparametric Estimation Juan Carlos Escanciano† Hidehiko Ichimura‡ MIT Indiana University University of Tokyo arXiv:1608.00033v1 [] 29 Jul 2016 Victor Chernozhukov∗ Whitney K. Newey§ MIT July 27, 2016 Abstract This paper shows how to construct locally robust semiparametric GMM estimators, meaning equivalently moment conditions have zero derivative with respect to the first step and the first step does not affect the asymptotic variance. They are constructed by adding to the moment functions the adjustment term for first step estimation. Locally robust estimators have several advantages. They are vital for valid inference with machine learning in the first step, see Belloni et. al. (2012, 2014), and are less sensitive to the specification of the first step. They are doubly robust for affine moment functions, so moment conditions continue to hold when one first step component is incorrect. Locally robust moment conditions also have smaller bias that is flatter as a function of first step smoothing leading to improved small sample properties. Series first step estimators confer local robustness on any moment conditions and are doubly robust for affine moments, in the direction of the series approximation. Many new locally and doubly robust estimators are given here, including for economic structural models. We give simple asymptotic theory for estimators that use cross-fitting in the first step, including machine learning. Keywords: Local robustness, double robustness, semiparametric estimation, bias, GMM. JEL classification: C13; C14; C21; D24. ∗ Department of Economics, MIT, Cambridge, MA 02139, U.S.A E-mail: [email protected]. Department of Economics, Indiana University, Bloomington, IN 47405–7104, U.S.A E-mail: [email protected]. ‡ Faculty of Economics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-0033. E-mail: [email protected]. § Department of Economics, MIT, Cambridge, MA 02139, U.S.A E-mail: [email protected]. † 1 1 Introduction There are many economic parameters that depend on nonparametric or large dimensional first steps. Examples include games, dynamic discrete choice, average consumer surplus, and treatment effects. This paper shows how to construct GMM estimators that are locally robust to the first step, meaning equivalently that moment conditions have a zero derivative with respect to the first step and that estimation of the first step does not affect their influence function. Locally robust moment functions have several advantages. Belloni, Chen, Chernozhukov, and Hansen (2012) and Belloni, Chernozhukov, and Hansen (2014) showed that local robustness, also referred to as orthogonality, is important for correct inference about parameters of interest when machine learning is used in the first step. Locally robust moment conditions are also nearly correct when the nonparametric part is approximately correct. This robustness property is appealing in many settings where it may be difficult to get the first step completely correct. Furthermore, local robustness implies the small bias property analyzed in Newey, Hsieh, and Robins (1998, 2004; NHR henceforth). As a result asymptotic confidence intervals based on locally robust moments have actual coverage probability closer to nominal than for other moments. Also, bias is flatter as a function of first step bias for locally robust estimators than for other estimators. This tends to make their mean-square error (MSE) flatter as a function of smoothing also, so their performance is less sensitive to smoothing. In addition, by virtue of their smaller bias, locally robust estimators have asymptotic MSE that is smaller than other estimators in important cases and undersmoothing is not required for root-n consistency. Finally, asymptotic variance estimation is straightforward with locally robust moment functions, because the first step is already accounted for. Locally robust moment functions are constructed by adding to the moment functions the terms that adjust for (or account for) first step estimation. This construction gives moment functions that are locally robust. It leads to new estimators for games, dynamic discrete choice, average surplus, and other important economic parameters. Also locally robust moments that are affine in a first step component are globally robust in that component, meaning the moments continue to hold when that component varies away from the truth. This result allows construction of doubly robust moments in the sense of Scharfstein, Rotnitzky, and Robins (1999) and Robins, Rotnitzky, and van der Laan (2000) by adding to affine moment conditions an affine adjustment term. Here we construct many new doubly robust estimators, e.g. where the first step solves a conditional moment restriction or is a density. Certain first step estimators confer the small bias property on moment functions that are not locally robust, including series estimators of mean-square projections (Newey, 1994), sieve 2 maximum likelihood estimators (Shen, 1996, Chen and Shen, 1997), bootstrap bias corrected first steps (NHR), and higher-order kernels (Bickel and Ritov, 2003). Consequently the inference advantages of locally robust estimators may be achieved by using one of these first steps. These first steps only make moment conditions locally robust in certain directions. Locally robust moments have the small bias property in a wider sense, that the moments are nearly zero as the first step varies in a general way. This property is important when the first step is chosen by machine learning or in other very flexible, data based ways; see Belloni et. al. (2014). First step series estimators have some special robustness properties. Moments without the adjustment term can be interpreted as locally robust because there is an estimated adjustment term with average that is identically zero. This property corresponds to first step series estimators conferring local robustness in the direction of the series approximation. Also, first step series estimators are doubly robust in those directions when the moment functions and first step estimating equations are affine. The theoretical and Monte Carlo results of NHR show the bias and MSE advantages of locally robust estimators for linear functionals of a density. The favorable properties of a twicing kernel first step versus a standard first step found there correspond to favorable properties of locally robust moments versus original moments, because a twicing kernel estimator is numerically equivalent to adding an estimated adjustment term. The theoretical results show that using locally robust moment conditions increases the rate at which bias goes to zero but only raises the variance constant, and so leads to improved asymptotic MSE. The Monte Carlo results show that the MSE of the locally robust estimator is much flatter as a function of bandwidth and has a smaller minimum than the original moment functions, even with quite small samples. Advantages have also been found in the literature on doubly robust estimation of treatment effects, as in Bang and Robins (2005) and Firpo and Rothe (2016). These results from earlier work suggest that locally robust moments provide a promising approach to improving the properties of semiparametric estimators. This paper builds on other earlier work. Locally robust moment conditions are semiparametric versions of Neyman (1959) C(α) test moments for parametric models, with parametric extensions to nonlikelihood settings given by Wooldridge (1991), Lee (2005), Bera et. al. (2010), and Chernozhukov, Hansen, and Spindler (2015). Hasminskii and Ibragimov (1978) suggested an estimator of a functional of a nonparametric density estimator that can be interpreted as adding the first step adjustment term. Newey (1990) derived the form of the adjustment term in some cases. Newey (1994) showed local robustness of moment functions that are derivatives of an objective function where the first step has been ”concentrated out,” derived the form of the adjustment term for many important cases, and showed that moment functions based on series nonparametric regression have small bias. General semiparametric model results on doubly robust estimators were given in Robins and Rotnitzky (2001). 3 NHR showed that adding the adjustment term gives locally robust moments for functionals of a density integral and showed the important bias and MSE advantages for locally robust estimators mentioned above. Robins et. al. (2008) showed that adding the adjustment term gives local robustness of explicit functionals of nonparametric objects, characterized some doubly robust moment conditions, and considered higher order adjustments that could further reduce bias. The form of the adjustment term for first step estimation has been derived for a variety of first step estimators by Pakes and Olley (1995), Ai and Chen (2003), Bajari, Chernozhukov, Hong, and Nekipelov (2009), Bajari, Hong, Krainer, and Nekipelov (2010), Ackerberg, Chen, and Hahn (2012), Ackerberg, Chen, Hahn, and Liao (2014), and Ichimura and Newey (2016), among others. Locally and doubly robust moments have been constructed for a variety of estimation problems by Robins, Rotnitzky, and Zhao (1994, 1995), Robins and Rotnitzky (1995), Scharfstein, Rotznitzky, and Robins (1999), Robins, Rotnitzky, and van der Laan (2000), Robins and Rotnitzky (2001), Belloni, Chernozhukov, and Wei (2013), Belloni, Chernozhukov, and Hansen (2014), Ackerberg, Chen, Hahn, and Liao (2014), Firpo and Rothe (2016), and Belloni, Chernozhukov, Fernandez-Val, and Hansen (2016). Contributions of this paper are a general construction of locally robust estimators in a GMM setting, a general nonparametric construction of doubly robust moments, and deriving bias and other large sample properties. The special robustness properties of first step series estimators are also shown here. We use these results to obtain many new locally and doubly robust estimators, such as those where the first step allows for endogeneity or is a conditional choice probability in an economic structural model. We expect these estimators to have the advantages mentioned above, that machine learning can be used in the first step, the estimators have appealing robustness properties, smaller bias and MSE, are less sensitive to bandwidth, have closer to nominal coverage for confidence intervals, and standard errors that can be easily computed. Section 2 describes the general construction of locally robust moment functions for semiparametric GMM. Section 3 shows how the first step adjustment term can be derived and shows the local robustness of the adjusted moments. Section 4 introduces local double robustness, shows that affine, locally robust moment functions are doubly robust, and gives new classes of doubly robust estimators. Section 5 describes how locally robust moment functions have the small bias property and a smaller remainder term. Section 6 considers first step series estimation. Section 7 characterizes locally robust moments based on conditional moment restrictions. Section 8 gives locally robust moment conditions for conditional choice probability estimation of discrete game and dynamic discrete choice models. Section 9 gives asymptotic theory based on cross fitting with easily verifiable regularity conditions for the first step, including machine learning. 4 2 Constructing Locally Robust Moment Functions The subject of this paper is GMM estimators of parameters where the sample moment functions depend on a first step nonparametric or large dimensional estimator. We refer to these estimators as semiparametric. We could also refer to them as GMM where first step estimators are “plugged in” the moments. This terminology seems awkward though, so we simply refer to them as semiparametric GMM estimators. We denote such an estimator by β̂, which is a function of the data z1 , ..., zn where n is the number of observations. Throughout the paper we will assume that the data observations zi are i.i.d. We denote the object that β̂ estimates as β0 , the subscript referring to the parameter value under the distribution that generated the data. To describe the type of estimator we consider let m(z, β, γ) denote an r×1 vector of functions of the data observation z, parameters of interest β, and a function γ that may be vector valued. The function γ can depend on β and z through those arguments of m. Here the function γ represents some possible first step, such as an estimator, its limit, or a true function. A GMM estimator can be based on a moment condition where β0 is the unique parameter vector satisfying E[m(zi , β0 , γ0 )] = 0, (2.1) and γ0 is the true γ. Here it is assumed that this moment condition identifies β. Let γ̂ denote some first step estimator of γ0 . Plugging in γ̂ to obtain m(zi , β, γ̂) and averaging over zi gives the P estimated sample moments m̂(β) = ni=1 m(zi , β, γ̂)/n. For Ŵ a positive semi-definite weighting matrix a semiparametric GMM estimator is β̂ = arg min m̂(β)T Ŵ m̂(β), β∈B where AT denotes the transpose of a matrix A and B is the parameter space for β. √ As usual a choice of Ŵ that minimizes the asymptotic variance of n(β̂ − β0 ) will be √ a consistent estimator of the inverse of the asymptotic variance of nm̂(β0 ). Of course that efficient Ŵ may include adjustment terms for the first step estimator γ̂. This optimal Ŵ also gives an efficient estimator in the wider sense shown in Ackerberg, Chen, Hahn, and Liao (2014). The optimal Ŵ makes β̂ efficient in a semiparametric model where the only restrictions imposed are equation (2.1). To explain and analyze local robustness we consider limits when the true distribution of a single observation zi is F , and how those limits vary with F over a general class of distributions. This kind of analysis can be used to derive the asymptotic variance of semiparametric estimators, as in Newey (1994), and is also useful here. Let γ(F ) denote the limit of γ̂ when F is the true distribution of zi . Here γ(F ) is understood to be the limit of γ̂ under general misspecification where F need not satisfy the conditions used to construct γ̂. We also consider parametric models Fτ where τ denotes a vector of parameters, with Fτ equal to the true distribution F0 at τ = 0. 5 We will restrict each parametric model to be regular in the sense used in the semiparametric efficiency bounds literature, so that Fτ has a score S(z) (derivative of the log-likelihood in many cases, e.g. see Van der Vaart, 1998, p. 362) at τ = 0 and possibly other conditions are satisfied. We also require that the set of scores over all regular parametric family has mean square closure that includes all functions with mean zero and finite variance. Here we are assuming that the set of scores for regular parametric models is unrestricted, the precise meaning of the domain of γ(F ) being a general class of distributions. We define local robustness in terms of such families of regular parametric models. Definition 1: The moment functions m(z, β, γ) are locally robust if and only if for all regular parametric models ∂E[m(zi , β0 , γ(Fτ ))] = 0. ∂τ τ =0 This zero pathwise derivative condition means that moment conditions are nearly zero as the first step limit γ(F ) departs from the truth γ0 along any path γ(Fτ ). Below we use a functional derivative condition, but for now this pathwise derivative definition is convenient. Throughout the remainder of the paper we evaluate derivatives with respect to τ at τ = 0 unless otherwise specified. In general, locally robust moment functions can be constructed by adding to moment functions the term that adjusts for (or accounts for) first step estimation. Under conditions discussed below there is a unique vector of functions φ(z, β, γ) such that E[φ(zi , β0 , γ0 )] = 0 and √ n n 1 X 1 X nm̂(β0 ) = √ m(zi , β0 , γ̂) = √ {m(zi , β0 , γ0) + φ(zi , β0 , γ0)} + op (1). n i=1 n i=1 (2.2) Here φ(z, β0 , γ0 ) adjusts for the presence of γ̂ in m(z, β0 , γ̂). Locally robust moment functions can be constructed by adding φ(z, β, γ) to m(z, β, γ) to obtain new moment functions g(z, β, γ) = m(z, β, γ) + φ(z, β, γ). For ĝ(β) = Pn i=1 (2.3) g(zi , β, γ̂)/n, a locally robust semiparametric GMM estimator is obtained as β̂ = arg min ĝ(β)′Ŵ ĝ(β). β In a parametric setting it is easy to see how adding the adjustment term for first step estimation gives locally robust moment conditions. Suppose that the first step estimator γ̂ is a function of a finite dimensional vector of parameters λ where there is a vector of functions h(z, λ) satisfying E[h(zi , λ0 )] = 0 and the first step parameter estimator λ̂ satisfies n 1 X √ h(zi , λ̂) = op (1). n i=1 6 (2.4) For H = ∂E[h(zi , λ)]/∂λ|λ=λ0 the usual expansion gives √ n 1 X n(λ̂ − λ0 ) = −H −1 √ h(zi , λ0 ) + op (1). n i=1 For notational simplicity let the moment functions depend directly on λ (rather than γ (λ)) and so take the form m(z, β, λ). Let Mλ = ∂E[m(zi , β0 , λ0 )]/∂λ. Another expansion gives √ n √ 1 X m(zi , β0 , λ0 ) + Mλ n(λ̂ − λ0 ) + op (1) nm̂(β0 ) = √ n i=1 n 1 X {m(zi , β0 , λ0 ) − Mλ H −1 h(zi , λ0 )} + op (1). =√ n i=1 Here we see that the adjustment term is φ(z, β, λ) = −Mλ H −1 h(z, λ). (2.5) We can add this term to the original moment functions to produce new moment functions of the form g(z, β, λ) = m(z, β, λ) + φ(z, β, λ) = m(z, β, λ) − Mλ H −1 h(z, λ). Local robustness of these moment functions follows by the chain rule and ∂E[g(zi , β0 , λ)] ∂λ = λ=λ0 ∂E[m(zi , β0 , λ) − Mλ H −1 h(zi , λ)] ∂λ λ=λ0 = Mλ − Mλ H −1 H = 0. Neyman (1959) used scores and the information matrix to form such g(z, β, λ) in a parametric likelihood setting, where g(z, β0 , λ0 ) has an orthogonal projection interpretation. There the purpose was to construct tests, based on g(z, β, λ), where estimation of the nuisance parameters λ did not affect the distribution of the tests. The form here was given in Wooldridge (1991) for nonlinear least squares and Lee (2005), Bera et. al (2010), and Chernozhukov et. al. (2015) for GMM. What appears to be new here is construction of locally robust moment functions by adding the adjustment term to original moment functions. In general the adjustment term φ(z, β, γ) may depend on unknown components that are not present in the original moment functions m(z, β, γ). In the above parametric example the matrix Mλ H −1 is unknown so its elements should really be included in γ, along with λ. If we do this local robustness will continue to hold with these additional components of γ because E[h(zi , λ0 )] = 0. For notational simplicity we will take γ to be the first step for the locally robust moment functions g(z, β, γ) = m(z, β, γ) + φ(z, β, γ), with the understanding φ(z, β, γ) will generally depend on first step functions that are not included in m(z, β, γ). In general semiparametric settings the form of the adjustment term φ(z, β, γ) and local robustness of g(z, β, γ) can be explained in terms of influence functions. We will do so in the 7 next Section. In many interesting cases the form of the adjustment term φ(z, β, γ) is already known, allowing construction of locally robust estimators. We conclude this section with an important class of examples. The class of examples we consider is one where the first step γ̂1 is based on a conditional moment restriction E[ρ(zi , γ10 )|xi ] = 0 for a residual ρ(z, γ1 ) and instrumental variables x. The conditional mean or median of yi given xi are included as special cases where ρ(z, γ1 ) = y −γ1 (x) and ρ(z, γ1 ) = 2·1(y < γ1 (x))−1 respectively, as are versions that allow for endogeneity where γ1 depends on variables other than x. We take γ̂1 to have the same limit as the nonparametric twostage least squares (NP2SLS) estimator of Newey and Powell (1989, 2003) and Newey (1991). Thus, γ̂1 has limit γ1 (F ) satisfying γ1 (F ) = arg min EF [{EF [ρ(zi , γ1 )|xi ]}2 ], γ1 ∈Γ and EF denotes the expectation under the distribution F . Suppose that there is γ20 (x) in the mean square closure of the set of derivatives ∂E[ρ(zi , γ1 (Fτ ))|xi ]/∂τ as Fτ varies over regular parametric models such that ∂E[m(zi , β0 , γ1(Fτ ))] ∂E[ρ(zi , γ1 (Fτ ))|xi ] = −E[γ20 (xi ) ]. ∂τ ∂τ (2.6) Then from Ichimura and Newey (2016) the adjustment term is φ(z, β, γ) = γ2 (x, β)ρ(z, γ1 ). (2.7) A function γ20 (x) satisfying equation (2.6) exists when the set of derivatives ∂E[ρ(zi , γ1 (Fτ ))|xi ]/∂τ is linear as Fτ varies over parametric models, ∂E[m(zi , β0 , γ1 (Fτ ))]/∂τ is a linear functional of ∂E[ρ(zi , γ1(Fτ ))|xi ]/∂τ, and that functional is continuous in mean square. Existence of γ20 (x) then follows from the Riesz representation theorem. Special cases of this characterization of γ20 (x) are in Newey (1994), Ai and Chen (2007), and Ackerberg, Chen, Hahn, and Liao (2014). When ∂E[m(zi , β0 , γ1 (Fτ ))]/∂τ is not a mean square continuous functional of ∂E[ρ(zi , γ1 (Fτ ))|xi ]/∂τ then first step estimation should make the moments converge slower √ than 1/ n, as shown by Newey and McFadden (1994) and Severini and Tripathi (2012) for special cases. The adjustment term given here includes Santos (2011) as a special case with R m(z, β, γ1 ) = v(x)γ1 (x)dx − β, though Santos (2011) is more general in allowing for nonidentification of γ10 . There are a variety of ways to construct an estimator φ(zi , β, γ̂) of the adjustment term to be used in forming locally robust moment functions, see NHR and Ichimura and Newey (2016). A relatively simple and general one when the first step is a series or sieve estimator is to treat the first step as if it were parametric and use the parametric formula in equation (2.5). This approach to estimating the adjustment term is known to be asymptotically valid in a variety 8 of settings, see Newey (1994), Ackerberg, Chen, and Hahn (2012), and Ichimura and Newey (2016). For completeness we give a brief description here. We parameterize an approximation to γ1 as γ1 = γ(λ) where λ is a finite dimensional vector of parameters as before. Let m(zi , β, λ) = m(zi , β, γ1(λ)) and λ̂i denote an estimator of λ0 solving E[h(zi , λ0 )] = 0, that is allowed to depend on observation i. Being a series or sieve estimator the dimension of λ and hence of h(z, λ) will increase with sample size. Also, let I(i) be a set of observation indices that can also depend on i. An estimator of the adjustment term is give by φ(zi , β, γ̂) = −M̂iλ (β)Ĥi−1 h(zi , λ̂i ), M̂iλ (β) = X ∂h(zj , λ̂i ) X ∂m(zj , β, λ̂i ) , Ĥi = . ∂λ ∂λ j∈I(i) j∈I(i) This estimator allows for cross fitting where λ̂i , M̂iλ , and Ĥi depend only on observations other than the ith . cross fitting is known to improve performance in some settings, such as in ”leave one out” kernel estimators of averages, e.g. see NHR. This adjustment term will lead to locally robust moments in a variety of settings, as further discussed below. 3 Influence Functions, Adjustment Terms, and Local Robustness Influence function calculations can be used to derive the form of the adjustment term φ(z, β, γ) and show local robustness of the adjusted moment functions g(z, β, γ) = m(z, β, γ) + φ(z, β, γ). To explain influence functions note that many estimators are asymptotically equivalent to a sample average. The object being averaged is the unique influence function. For example, in equation (2.2) we are assuming that the influence function of m̂(β0 ) is g(z, β0 , γ0 ) = m(z, β0 , γ0 )+ φ(z, β0 , γ0). This terminology is widely used in the semiparametric estimation literature. In general an estimator µ̂ of a true value µ0 and its influence function ψ(z) satisfy √ n 1 X ψ(zi ) + op (1), E[ψ(zi )] = 0, E[ψ(zi )ψ(zi )′ ] exists. n(µ̂ − µ0 ) = √ n i=1 The function ψ(z) can be characterized in terms of the functional µ(F ) that is the limit of µ̂ under general misspecification where F need not satisfy the conditions used to construct µ̂. As before, we allow F to vary over a family of regular parametric models where the set of scores for the family has mean square closure that includes all mean zero functions with finite variance. As shown by Newey (1994) the influence function ψ(z) is then the unique solution to a derivative equation of Van der Vaart (1991), ∂µ(Fτ ) = E[ψ(zi )S(zi )], E[ψ(zi )] = 0, ∂τ 9 (3.1) as Fτ (and hence S(z)) varies over the general family of regularly parametric models. Ichimura and Newey (2016) also showed that when ψ(z) has certain continuity properties it can be computed as ∂µ(Fτh ) h , Fτ = (1 − τ )F0 + τ Ghz , (3.2) ψ(z) = lim h−→0 ∂τ where Ghz is constructed so that Fτh is in the domain of µ(F ) and Ghz approaches the point mass at z as h −→ 0. These results can be used to derive the adjustment term φ(z, β, γ) and to explain local robustness. Let γ(F ) denote the limit of the first step estimator γ̂ under general misspecification when a single observation has CDF F, as discussed above. From Newey (1994, pp. 1356-1357) we know that the adjustment term φ(z, β0 , γ0 ) is the influence function of µ(F ) = E[m(zi , β0 , γ(F ))] where E[·] denotes the expectation at the truth. Thus φ(z, β, γ) can be calculated as in equation (3.2) for that µ(F ). Also φ(z, β, γ) satisfies equation (3.1) for µ(F ) = E[m(zi , β0 , γ(F ))], i.e. for the score S(z) at τ = 0 for any regular parametric model {Fτ }, ∂E[m(zi , β0 , γ(Fτ ))] = E[φ(zi , β0 , γ0 )S(zi )], E[φ(zi , β0 , γ0 )] = 0. ∂τ (3.3) Also, φ(z, β0 , γ0) can be computed as ∂E[m(zi , β0 , γ(Fτh ))] h , Fτ = (1 − τ )F0 + τ Ghz , h−→0 ∂τ The characterization of φ(z, β0 , γ0) in equation (3.3) can be used to specify another local robustness property that is equivalent to Definition 1. We have defined local robustness as the derivative on the left of the first equality in equation (3.3) being zero for all regular parametric models. If that derivative is zero for all parametric models then φ(z, β0 , γ0 ) = 0 is the unique solution to this equation, by the set of scores being mean square dense in the set of mean zero random variables with finite variance. Also, if φ(z, β0 , γ0) = 0 then the derivative on the left is always zero. Therefore we have φ(z, β0 , γ0 ) = lim Proposition 1: φ(z, β0 , γ0 ) = 0 if and only if m(z, β, γ) is locally robust. Note that φ(z, β, γ) is the term in the influence function of m̂(β0 ) that accounts for the first step estimator γ̂. Thus Proposition 1 gives an alternative characterization of local robustness, that first step estimation does not affect the influence function of m̂(β0 ). This result is a semiparametric version of Theorem 6.2 of Newey and McFadden (1994). It also formalizes the discussion in Newey (1994, pp. 1356-1357). Local robustness of the adjusted moment function g(z, β, γ) = m(z, β, γ) + φ(z, β, γ) follows from Proposition 1 and φ(z, β, γ) being a nonparametric influence function. Because def φ(z, β, γ) is an influence function it has mean zero at all true distributions, i.e. µ(F ) = 10 R φ(z, β0 , γ(F ))F (dz) ≡ 0 identically in F. Consequently the derivative in equation (3.1) is zero, so that (like Proposition 1) the influence function of µ(F ) is zero. Consequently, under P appropriate regularity conditions φ̄ = ni=1 φ(zi , β0 , γ̂)/n has a zero influence function and so √ nφ̄ = op (1). It then follows that n n √ √ √ 1 X 1 X √ g(zi , β0 , γ̂) = nm̂(β0 ) + nφ̄ = nm̂(β0 ) + op (1) = √ g(zi , β0 , γ0 ) + op (1), (3.4) n i=1 n i=1 where the last equality follows by equation (2.2). Here we see that the adjustment term is zero for the moment functions g(z, β, γ). From Proposition 1 with g(z, β, γ) replacing m(z, β, γ) it then follows that g(z, β, γ) is locally robust. Proposition 2: For the influence function φ(z, β0 , γ0 ) of µ(F ) = E[m(zi , β0 , γ(F ))] the adjusted moment function g(z, β, γ) = m(z, β, γ) + φ(z, β, γ) is locally robust. R Local robustness of g(z, β, γ) also follows directly from the identity φ(z, β0 , γ(F ))F (dz) ≡ 0 as discussed in the Appendix. Also, the adjusted moments ĝ(β0 ) have the same asymptotic variance as the original moments, as in the second equality of equation (3.4). That is, adding φ(z, β, γ) to m(z, β, γ) does not affect the asymptotic variance. Thus the asymptotic benefits of the locally robust moments are in their higher order properties. Other modifications of the moments may also improve higher-order properties of estimators, such as the cross fitting described above (like ”leave on out” in NHR) and the higher order bias corrections in Robins et. al. (2008) and Cattaneo and Jansson (2014). 4 Local and Double Robustness The zero derivative condition in Definition 1 is an appealing robustness property in and of itself. Mathematically a zero derivative is equivalent to the moments remaining closer to zero than τ as τ varies away from zero. This property can be interpreted as local robustness of the moments to the value of γ being plugged in, with the moments remaining close to zero as γ varies away from its true value. Because it is difficult to get nonparametric functions exactly right, especially in high dimensional settings, this property is an appealing one. Such robustness considerations, well explained in Robins and Rotnitzky (2001), have motivated the development of doubly robust estimators. For our purposes doubly robust moments have expectation zero if just one first stage component is incorrect. When there are only two first stage components this means that the moment conditions hold when only one of the first stage components is correct. Doubly robust moment conditions allow two chances for the moment conditions to hold. 11 It turns out that locally robust moment functions are automatically doubly robust in a local sense that the derivative with respect to each individual, distinct first stage component is zero. In that way the moment conditions nearly hold as each distinct component varies in a neighborhood of the truth. Furthermore, when locally robust moment functions are affine functions of a distinct first step component they are automatically globally robust in that component. Thus, locally robust moment functions that are affine in each distinct first step are doubly robust. These observations suggest a way to construct doubly robust moment functions. Starting with any two step semiparametric moment function we can add the adjustment term to get a locally robust moment function. When we can choose a first step of that moment function so that it enters in an affine way the new moment function will be doubly robust in that component. To give these results we need to define distinct components of γ. A distinct component is one where there are parametric models Fτ with that component varying in an unrestricted way but the other components of γ not varying. For a precise definition we will focus on the first component γ1 of γ = (γ1 , ..., γJ ). Definition 2: A component γ1 of γ is distinct if and only if there is Fτ such that γ(Fτ ) = (γ1 (Fτ ), γ20 , ..., γJ0 ), and γ1 (Fτ ) is unrestricted as Fτ varies across parametric models. An example is the moment function g(z, β, γ1, γ2 ) = m(z, β, γ1 ) + γ2 (x, β)ρ(z, γ1 ), where E[ρ(zi , γ10 )|xi ] = 0. In that example the two components γ1 and γ2 are often distinct because γ1 depends only on the conditional distribution of zi given xi and γ20 (x, β) depends on the marginal distribution of xi in an unrestricted way. Local robustness means that the derivative must be zero for any model, so in particular it must be zero for any model where only the distinct component is varying. Thus we have Proposition 3: If γ1 is distinct then for g(z, β, γ) = m(z, β, γ) + φ(z, β, γ) and regular parametric models Fτ as in Definition 2, ∂E[g(zi , β0 , γ1(Fτ ), γ20 , ..., γJ0)] = 0. ∂τ This result is an application of the simple fact that when a multivariable derivative is zero the partial derivative must be zero when the variables are allowed to vary in an unrestricted way. Although this fact is simple, it is helpful in understanding when local robustness holds for individual components. This means that locally robust moment functions automatically have a local double robustness property, that the expectation of the moment function remains nearly zero as each distinct first stage component varies away from the truth. For example, for 12 a first step conditional moment restriction where g(z, β, γ) = m(z, β, γ1 ) + γ2 (x, β)ρ(z, γ1 ), the conclusion of Proposition 3 is ∂E[m(zi , β0 , γ1 (Fτ )) + γ20 (xi )ρ(zi , γ1 (Fτ ))] = 0. ∂τ In fact, this result is implied by equation (2.6), so by construction g(z, β, γ) is already locally robust in γ1 alone. Local robustness in γ2 follows by the conditional moment restriction E[ρ(zi , γ10 )|xi ] = 0. Moments that are locally robust in a distinct component γ1 will be globally robust in γ1 if γ1 enters the moments in an affine way, meaning that for any γ1 and γ = (γ1 , γ20 , ..., γJ0 )′ and any z, g(z, β, (1 − τ )γ0 + τ γ) = (1 − τ ) · g(z, β, γ0) + τ · g(z, β, γ). (4.1) Global robustness holds because an affine function with zero derivative is constant. For simplicity we state a result when Fτ can be chosen so that γ(Fτ ) = (1 − τ )γ0 + τ γ though it will hold more generally. Note that here E[g(zi , β0 , γ(Fτ ))] = (1 − τ )E[g(zi , β0 , γ0 )] + τ · E[g(zi , β0 , γ)] = τ · E[g(zi , β0 , γ)]. Here the derivative of the moment condition with respect to τ is just E[g(zi , β0 , γ)] so Proposition 3 gives the following result: Proposition 4: If equation (4.1) is satisfied and there is Fτ with γ(Fτ ) = ((1 − τ )γ10 + τ γ1 , γ20 , ..., γJ0)′ then E[g(zi , β0 , γ1 , γ20 , ..., γJ0 )] = 0. Thus we see that locally robust moment functions that are affine in a distinct first step component are globally robust in that component. This result includes many existing examples of doubly robust moment functions and can be used to construct new ones. A general class of doubly robust moment functions that appears to be new and includes many new and previous examples has first step satisfying a conditional moment restriction E[ρ(zi , γ10 )|xi ] = 0 where ρ(z, γ1 ) and m(z, β0 , γ1 ) are affine in γ1 . Suppose that E[m(zi , β0 , γ1 )] is a mean-square continuous linear functional of E[ρ(zi , γ1 )|xi ] for γ1 in a linear set Γ. Then by the Riesz representation theorem there is γ ∗ (x) in the mean square closure Π of the image of E[ρ(zi , γ1 )|xi ] such that E[m(zi , β0 , γ1 )] = −E[γ ∗ (xi )E[ρ(zi , γ1 )|xi ]] = −E[γ ∗ (xi )ρ(zi , γ1 )], γ1 ∈ Γ. (4.2) Let γ20 (x) be any function such that γ20 (xi ) − γ ∗ (xi ) is orthogonal to Π and g(z, β, γ) = m(z, β, γ1 ) + γ2 (x, β)ρ(z, γ1 ). Then E[g(zi , β0 , γ1 , γ20 )] = 0 by the previous equation. It also follows that E[g(zi , β0 , γ10 , γ2 )] = 0 by E[ρ(zi , γ10 )|xi ] = 0. Therefore g(z, β, γ1, γ2 ) is doubly robust, showing the following result: 13 Proposition 5: If m(zi , β0 , γ1 ) and ρ(zi , γ1) are affine in γ1 ∈ Γ with Γ linear and E[m(zi , β0 , γ1 )] is a linear, mean square continuous functional of E[ρ(zi , γ1 )|xi ] then there is γ20 (x) such that g(z, β, γ1, γ2 ) = m(z, β, γ1 ) + γ2 (x, β)ρ(z, γ1 ) is doubly robust. Section 3 of Robins et. al. (2008) gives necessary and sufficient conditions for a moment function to be doubly robust when γ1 and γ2 enter the moment functions as functions evaluated at observed x. Proposition 5 is complementary to that work in deriving the form of doubly robust moment functions when the first step satisfies a conditional moment restriction and m(z, β, γ1 ) can depend on the entire function γ1 . It is interesting to note that γ20 such that E[g(z, β0 , γ1, γ20 )] = 0 for all γ1 ∈ Γ is not unique when Π does not include all functions of x, the overidentified case of Chen and Santos (2015). This nonuniqueness can occur when there are multiple ways to estimate the first step γ10 using the conditional moment restrictions E[ρ(zi , γ10 )|xi ] = 0. As discussed in Ichimura and Newey (2016), the different γ20 (xi ) correspond to different first step estimators, with γ20 (xi ) = γ ∗ (xi ) corresponding to the NP2SLS estimator. An important case is a linear conditional moment restrictions setup up like Newey and Powell (1989, 2003) and Newey (1991) where ρ(z, γ1 ) = y − γ1 (w), E[yi − γ10 (wi )|xi ] = E[ρ(zi , γ10 )|xi ] = 0. (4.3) Consider a moment function equal to m(z, β, γ1 ) = v(w)γ1(w) − β for some known function v(w), where the parameter of interest is β0 = E[v(wi )γ10 (wi )]. If there is γ̄(x) such that v(wi ) = E[γ̄(xi )|wi ] then we have E[m(zi , β0 , γ1 )] = E[v(wi ){γ1(wi ) − γ10 (wi )}] = E[E[γ̄(xi )|wi ]{γ1 (wi ) − γ10 (wi )}] = E[γ̄(xi ){γ1(wi ) − γ10 (wi )}] = −E[γ̄(xi )ρ(zi , γ1 )]. It follows that g(z, β, γ) = m(z, β, γ1 ) + γ2 (x)ρ(z, γ1 ) is doubly robust for γ20 (x) = γ̄(xi ). Interestingly, the existence of γ̄ with v(wi ) = E[γ̄(xi )|wi ] is a necessary condition for root-n consistent estimability of β0 as in Severini and Tripathi’s (2012, Lemma 4.1). We see here that a doubly robust moment condition can always be constructed when this necessary condition is satisfied. Also, similarly to the above, the γ20 (x) may not be unique. Corollary 6: If m(z, β, γ1 ) = v(w)γ1(w) − β, equation ( 4.3) is satisfied, and there is γ̄(x) such that v(w) = E[γ̄(x)|w] then g(z, β, γ1, γ2 ) = v(w)γ1(w) − β + γ2 (x)[y − γ1 (w)] is doubly robust for γ20 (x) − γ̄(x) orthogonal to Π. A new example of a doubly robust moment condition corresponds to the weighted average derivative of γ10 (w) of Ai and Chen (2007). Here m(z, β, γ1 ) = v̄(w)∂γ1 (w)/∂w − β for some 14 function v̄(w). Let f0 (w) be the pdf of wi . Assuming that v̄(w)γ1(w)f0 (w) is zero on the boundary of the support of wi , integration by parts gives E[m(zi , β0 , γ1 )] = E[v(wi ){γ1(wi ) − γ10 (wi )}], v(w) = f0 (w)−1 ∂[v̄(w)f0 (w)]/∂w. Assume that there exists γ̄(x) such that v(wi ) = E[γ̄(xi )|wi ]. Then as in Proposition 5 a doubly robust moment function is g(z, β, γ) = v̄(w) ∂γ1 (w) − β + γ2 (x)[y − γ1 (w)]. ∂w A special case of this example is the doubly robust moment condition for the weighted average derivative in the exogenous case where wi = xi given in Firpo and Rothe (2016). Doubly robust moment conditions can be used to identify parameters of interest. In general, if g(z, β, γ1, γ2 ) is doubly robust and γ20 is identified then β0 may be identified from E[g(zi , β0 , γ̄1 , γ20 )] = 0, for any fixed γ̄1 when the solution β0 to this equation is unique. Proposition 7: If g(z, β, γ1, γ2 ) is doubly robust, γ20 is identified, and for some γ̄1 the equation E[g(zi , β, γ̄1, γ20 )] = 0 has a unique solution then β0 is identified as that solution. Applying this result to the NPIV setting gives an explicit formula for certain functionals of γ10 (w) without requiring that the completeness identification condition of Newey and Powell (2003) be satisfied, similarly to Santos (2011). Suppose that v(w) is identified, e.g. as for the weighted average derivative. Since both w and x are observed it follows that a solution γ20 (x) to v(w) = E[γ20 (x)|w] will be identified if such a solution exists. Plugging in γ̄1 = 0 in the equation E[g(zi , β0 , γ̄1 , γ20 )] = 0 gives Corollary 8: If v(wi ) is identified and there exists γ20 (xi ) such that v(wi ) = E[γ20 (xi )|wi ] then β0 = E[v(wi )γ10 (wi )] is identified as β0 = E[γ20 (xi )yi ]. Note that this result holds without the completeness condition. Identification of β0 = E[v(wi )γ10 (wi )] for known v(wi ) with v(wi ) = E[γ20 (xi )|wi ] follows from Severini and Tripathi R (2006). Santos (2011) gives a related formula for a parameter β0 = ṽ(w)γ20(w)dw. The formula here differs from Santos (2011) in being an expectation rather Lebesgue integral. An estimator of β0 could be constructed similarly to Santos (2011), though that is beyond the scope of this paper. Another new example of a doubly robust estimator is a weighted average over income values of an average (across heterogenous individuals) of exact consumer surplus bounds, as in Hausman and Newey (2016). Here y is quantity consumed, w = x = (x1 , x2 )′ , x1 is price, x2 is income, 15 γ10 (xi ) = E[yi |xi ], price is changing between x̆1 and x̄1 , and B is a bound on the income effect. Let v2 (x2 ) be some weight function and v1 (x1 ) = 1(x̆1 ≤ x1 ≤ x̄1 )e−B(x1 −x̆1 ) . For the R moment function m(z, β, γ1 ) = v2 (x2 ) v1 (u)γ1 (u, x2 )du − β the true parameter β0 is a bound on the average of equivalent variation over unobserved individual heterogeneity and income. Let f10 (x1 |x2 ) denote the conditional pdf of x1 given x2 . Note that Z E[m(zi , β0 , γ1)] = E[v2 (x2i ) v1 (u)[γ1(u, x2i ) − γ10 (u, x2i )]du] = E[f10 (x1i |x2i )−1 v1 (x1i )v2 (x2i ){γ1 (xi ) − γ10 (xi )}] = −E[γ20 (xi )E[yi − γ1 (xi )|xi ]], γ20 (x) = f10 (x1 |x2 )−1 v1 (x1 )v2 (x2 ). Then it follows by Proposition 5 that a doubly robust moment function is Z g(z, β, γ) = v2 (x2 ) v1 (u)γ1 (u, x2 )du − β + γ2 (x)[y − γ1 (x)]. When the moment conditions are formulated so that they are affine in the first step Proposition 4 applies to many previously developed doubly robust moment conditions. Data missing at random is a leading example. Let β0 be the mean of a variable of interest w where w is not always observed, y ∈ {0, 1} denote an indicator for w being observed, and x a vector of covariates. Assume w is mean independent of y conditional on covariates x. We consider estimating β0 using the propensity score P0 (xi ) = Pr(yi = 1|xi ). We specify an affine conditional moment restriction by letting γ1 (x) = 1/P (x) and ρ(z, γ1 ) = γ1 (x)y − 1. We have β0 = E[γ10 (xi )yi wi ], as is well known. An affine moment function is then m(z, β, γ1 ) = γ1 (x)yw − β. Note that E[m(zi , β0 , γ1 )] = E[E[yi wi |xi ]{γ1 (xi ) − γ10 (xi )}] = −E[γ20 (xi )ρ(zi , γ1 )], γ20 (xi ) = −γ10 (xi )E[yi wi |xi ]. Then Proposition 5 implies that a doubly robust moment function is given by g(z, β, γ) = γ1 (x)yw − β − γ2 (x)[γ1 (x)y − 1]. This is the well known doubly robust moment function of Robins, Rotnitzky, and Zhao (1994). This example illustrates how applying Propositions 4 and 5 require specifying the first step so that the moment functions are affine. These moment conditions were originally shown to be doubly robust when the first step is taken to be the propensity score P (x). Propositions 4 and 5 only apply when the first step is taken to be 1/P (x). More generally we expect that particular formulations of the first step may be needed to make the moment functions affine in the first step and so use Propositions 4 and 5 to derive doubly robust moment functions. Another general class of doubly robust moment functions depend on the pdf γ1 of a subset of variables xi and are affine in γ1 . An important example of such a moment function 16 R is the average where β0 = f0 (x)2 dx and m(z, β, γ1 ) = γ1 (x) − β. Another is the density weighted average derivative (WAD) of Powell, Stock, and Stoker (1989) where m(z, β, γ1 ) = −2y · ∂γ1 (x)/∂x − β. Assume that E[m(zi , β0 , γ1 )] is a function of γ1 − γ10 that is continuous in R 1/2 the norm [γ1 (u) − γ10 (u)]2du . Then by the Riesz representation theorem there is γ20 (x) with Z E[m(zi , β0 , γ1 )] = γ20 (u)[γ1 (u) − γ10 (u)]du. (4.4) The adjustment term for m(z, β, γ), as in Proposition 3 of Newey (1994), is φ(z, β, γ) = γ2 (x) − R γ2 (u)γ1(u)du. The corresponding locally robust moment function is Z g(z, β, γ1, γ2 ) = m(z, β, γ1 ) + γ2 (x) − γ2 (u)γ1(u)du. (4.5) This function is affine in γ1 and γ2 separately so when they are distinct Proposition 4 implies double robustness. Double robustness also follows directly from Z Z Z E[g(zi , β0 , γ)] = γ20 (u)[γ1(u) − γ10 (u)]du + γ2 (u)γ10(u)du − γ2 (u)γ1(u)du Z = − [γ2 (u) − γ20 (u)][γ1 (u) − γ10 (u)]du. Thus we have the following result: Proposition 9: If m(zi , β, γ1) is affine in γ1 and E[m(zi , β0 , γ1)] is a linear function of R 1/2 , then for γ20 (x) from equation γ1 −γ10 that is continuous in the norm {γ1 (x) − γ10 (x)}2 dx R (4.4), g(z, β, γ) = m(z, β, γ1 ) + γ2 (x) − γ2 (u)γ1(u)du is doubly robust. We can use this result to derive doubly robust moment functions for the WAD. Let δ(xi ) = E[yi |xi ]γ10 (xi ). Assuming that δ(u)γ1(u) is zero on the boundary, integration by parts gives Z E[m(zi , β0 , γ1 )] = −2E[yi ∂γ1 (xi )/∂x] − β0 = 2 [∂δ(u)/∂u]{γ1 (u) − γ10 (u)}du, so that γ20 (x) = 2∂δ(x)/∂x. A doubly robust moment condition is then Z ∂γ1 (x) ∂δ(x) ∂δ(u) g(z, β, γ) = −2y −β+2 − 2 γ1 (u)du. ∂x ∂x ∂u The double robustness of this moment condition appears to be a new result. As shown in Newey, Hsieh, and Robins (1998), a ”delete-one” symmetric kernel estimator based on this moment function gives the twicing kernel estimator of NHR. Consequently the MSE comparisons of NHR for twicing kernel estimators with the original kernel estimator correspond to comparison of a doubly (and locally) robust estimator with one based on unadjusted moment conditions, as discussed in the introduction. 17 It is interesting to note that Proposition 9 does not require that γ1 and γ2 are distinct first step components. For the average density γ1 (x) and γ2 (x) both represent the marginal density of x and so are not distinct. Nevertheless the moment function g(z, β0 , γ) = γ1 (x) − β0 + R γ2 (x) − γ1 (u)γ2 (u)du is doubly robust, having zero expectation if either γ1 or γ2 is correct. This example shows a moment function may be doubly robust even though γ1 and γ2 are not distinct. Thus, there are doubly robust moment functions that cannot be constructed using Proposition 4. All of the results of this Section continue to hold with cross fitting. That is true because the results of this Section concern the moment and their expectation at various values of the first step, and not the particular way in which the first step is formed. 5 Small Bias of Locally Robust Moment Conditions Adding the adjustment term improves the higher order properties of the estimated moments though it does not change their asymptotic variance. An advantage of locally robust moment functions is that the effect of first step smoothing bias is relatively small. To describe this advantage it is helpful to modify the definition of local robustness. In doing so we allow F to represent a more general object, an unsigned measure (charge). Let k·k denote a seminorm on F (a seminorm has all the properties of a norm but may be zero when F is not zero). Also, let F be a set of charges where m(z, β0 , γ(F )) is well defined. Definition 3: m(z, β, γ) is locally robust if and only if E[m(zi , β0 , γ(F ))] = o(kF − F0 k) for F ∈ F . Definition 1 requires that µ(F ) have a zero pathwise derivative. Definition 3 requires a zero Frechet derivative for the semi-norm k·k, generally a stronger condition than a zero pathwise derivative. The zero Frechet derivative condition is helpful in explaining the bias properties of locally robust moment functions. Generally a first estimator will depend on some vector of smoothing parameters b. This b could be the bandwidth in a kernel estimator or the inverse of number of terms in a series estimator. Suppose that the limit of γ̂ for fixed b is γ(Fb ) where Fb is a ”smoothed” version of the true distribution that approaches the truth F0 as b −→ 0. Then under regularity conditions m̂(β0 ) will have limit E[m(zi , β0 , γ(Fb ))]. We can think of kFb − F0 k as a measure of the smoothing bias of Fb . Similarly E[m(zi , β0 , γ(Fb ))] is a measure of the bias in the moment conditions caused by smoothing. The small bias property (SBP) analyzed in NHR is that the expectation of the moment functions vanishes faster than the nonparametric bias as b −→ 0. Definition 4: m(z, β, γ) and Fb have the small bias property if and only if E[m(zi , β0 , γ(Fb ))] = o(kFb − F0 k) as b −→ 0. 18 As long as Fb ∈ F , the set F in Definition 3, locally robust moments will have bias that vanishes faster than the nonparametric bias kFb − F0 k as b −→ 0. Thus locally robust moment functions have the small bias property. Proposition 10: If m(z, β, γ) is locally robust then m(z, β, γ) has the small bias property for any Fb ∈ F . Note that the bias of locally robust moment conditions will be flat as a function of the first step smoothing bias kFb − F0 k as that goes to zero. This flatter moment bias can also make the MSE flatter, meaning the MSE of the estimator does not depend as strongly on kFb − F0 k for locally robust moments as for other estimators. By comparing Definitions 3 and 4 we see that the small bias property is a form of directional local robustness, with the moment being locally robust in the direction Fb . If the moments are not locally robust then there will be directions where the bias of the moments is not smaller than the smoothing bias Fb . Being locally robust in all directions can be important when the first step is allowed to be very flexible, such as when machine learning is used to construct the first step. There the first step can vary randomly across a large class of functions making the use of locally robust moments important for correct inference, e.g. see Belloni, Chernozhukov, and Hansen (2014). This discussion of smoothing bias is based on sequential asymptotics where we consider limits for fixed b. This discussion provides useful intuition but it is also important to consider asymptotics where b could be changing with the sample size. We can analyze the precise effect of using locally robust moments by considering an expansion of the average moments. Let m̄ = Pn Pn i=1 m(zi , β0 , γ0 )/n, ḡ = i=1 g(zi , β0 , γ0 )/n, φ(z) = φ(z, β0 , γ0 ), and F̃ denote the empirical distribution. We suppose that γ̂ = γ(F̂ ) for some estimator F̂ of the true distribution F0 . Let µ(F ) = E[m(zi , β0 , γ(F ))]. By adding and subtracting terms we have Z m̂(β0 ) = ḡ + R̃1 + R̂2 + R̂3 , R̃1 = φ(z)[F̂ − F̃ ](dz), (5.1) Z R̂2 = µ(F̂ ) − φ(z)F̂ (dz), R̂3 = m̂(β0 ) − m̄ − µ(F̂ ). R R The object φ(z)F̂ (dz) = φ(z)[F̂ − F0 ](dz) is a linearization of µ(F̂ ) = µ(F̂ ) − µ(F0 ) so R̂2 is a nonlinearity remainder that is second order. Also R̂3 is a stochastic equicontinuity remainder of a type familiar from Andrews (1994) that is also second order. The locally robust counterpart ĝ(β0 ) to m̂(β0 ) has a corresponding remainder term that is asymptotically smaller than R̃1 . To see this let φ̂(z) = φ(z, β0 , γ(F̂ )) and note that the R mean zero property of an influence function will generally give φ̂(z)F̂ (dz) = 0. Then by R ĝ(β0 ) = m̂(β0 ) + φ̂(z)F̃ (dz) we have 19 Proposition 11: If R φ̂(z)F̂ (dz) = 0 then ĝ(β0 ) = ḡ + R̂1 + R̂2 + R̂3 , Z R̂1 = − [φ̂(z) − φ(z)][F̂ − F̃ ](dz). Comparing this conclusion with equation (5.1) we see that locally robust moments have the same expansion as the original moments except that the remainder R̃1 has been replaced by the remainder R̂1 . The remainder R̂1 will be asymptotically smaller than R̃1 under sufficient regularity conditions. Consequently, depending on cross correlations with other terms, the locally robust moments ĝ(β0 ) can be more accurate than m̂(β0 ). For instance, as shown by NHR the locally robust moments for linear kernel averages have a higher order bias term that converges to zero at a faster rate than the original moments, while only the constant term in the higher order variance is larger. Consequently, the locally robust estimator will have smaller MSE asymptotically for appropriate choice of bandwidth. In nonlinear cases the use of locally robust moments may not lead to an improvement in MSE because nonlinear remainder terms may be important, see Robins et. al. (2008) and Cattaneo and Jansson (2014). Nevertheless, using locally robust moments does make smoothing bias small, which can be an important improvement. In some settings it is possible to obtain a corresponding improvement by changing the first step estimator. For example, as mentioned earlier, for linear kernel averages the locally robust estimator is identical to the original estimator based on a twicing version of the original kernel (see NHR). The improvement from changing the first step can be explained in relation to the remainder R̃1 , that is the difference of the integral of φ(z) over the estimated distribution F̂ and its sample average. Note that F̂ − F̃ will be shrinking to zero so that R̃1 − E[R̃1 ] should be a second order (stochastic equicontinuity) term. E[R̃1 ] is the most interesting term. If E[F̂ ] = Fb and integrals can be interchanged then Z Z E[R̃1 ] = φ(z)Fb (dz) = φ(z)[Fb − F0 ](dz). When a twicing kernel or any other higher order kernel is used this remainder becomes second order, depending on the smoothness of both the true distribution F0 and the influence function φ(z), see NHR and Bickel and Ritov (2003). Thus, by using a twicing or higher order kernel we obtain a second order bias, so all of the remainder terms are second order. Furthermore, series estimator automatically have a second order bias term, as pointed out in Newey (1994). Consequently, for all of these first steps the remainders are all second order even though the moment function is not locally robust. The advantage of locally robust moments are that the improvement applies to any first step estimator. One does not have to depend on the particular structure of the estimator, such as having a kernel of sufficiently high order. This feature is important when the first step is 20 complicated so that it is hard to analyze the properties of terms that correspond to E[R̃1 ]. Important examples are first steps that use machine learning. In that setting locally robust moments are very important for obtaining root-n consistency; see Belloni et. al. (2014). Locally robust moments have the advantages we have discussed even for very complicated first steps. 6 First Step Series Estimators First step series estimators have certain automatic robustness properties. Moment conditions based on series estimators are automatically locally robust in the direction of the series approximation. We also find that affine moment functions are automatically doubly robust in these directions. In this Section we present these results. It turns out that for certain first step series estimators there is a version of the adjustment term that has sample mean zero, so that ĝ(β) = m̂(β). That is, locally robust moments are numerically identical to the original moments. This version of the adjustment term is constructed by treating the first step as if it were parametric with parameters given by those of the series approximation, and calculating a sample version of the adjustment described in Section P 2. Suppose that the coefficients λ̂ of the first step estimator satisfy ni=1 h(zi , λ̂)/n = 0. Let P P M̂λ (β) = n−1 ni=1 ∂m(zi , β, λ̂)/∂λ, Ĥ = n−1 ni=1 ∂h(zi , λ̂)/∂λ, and φ(z, β, γ̂) = −M̂λ (β)Ĥ −1h(z, λ̂) (6.1) be the parametric adjustment term described in Section 2, where γ̂ includes the elements of M̂γ (β) and Ĥ and there is no cross fitting. Note that n n 1X 1X φ(zi , β, γ̂) = −M̂λ (β)Ĥ −1 h(zi , λ̂) = 0. n i=1 n i=1 It follows that ĝ(β) = m̂(β), i.e. the locally robust moments obtained by adding the adjustment P term are identical to the original moments. Thus, if ni=1 h(zi , λ̂)/n = 0, we treat the first step series estimator as parametric, and use the parametric adjustment term the locally robust moments are numerically identical to the original moments. This numerical equivalence results is an exact version of local robustness of the moments in the direction of the series approximation. In some settings it is known that φ(z, β, γ̂) in equation (6.1) is an estimated approximation to φ(zi , β, γ0), justifying its use. Newey (1994, p. 1369) showed that this approximation property holds when the first step is a series regression. Ackerberg, Chen, and Hahn (2012) showed that this property holds when the first step satisfies certain conditional moment restrictions or is part of a sieve maximum likelihood estimator. It is also straightforward to show that this approximation holds when the first step is a series approximates to the solution to the conditional moment restriction E[ρ(zi , γ10 )|xi ] = 0. We expect that in general φ(z, β, γ̂) is an estimator of φ(z, β, γ0 ). 21 We note that the result that ĝ(β) = m̂(β) is dependent on λ̂ not varying with the observations and on being constructed from the whole sample. If we use cross fitting in any form then the numerical equivalence of the original moments with their locally robust counterpart will generally not hold. Also ĝ(β) 6= m̂(β) will generally occur when different models are used for different elements of γ. Such different models will often be present when machine learning is used for constructing the estimators of the different elements of γ. See for example Chernozhukov et. al. (2016). There are interesting cases where the original moment functions m(z, β, γ) with a series estimator for γ are doubly robust in certain directions, with E[m(zi , β0 , γ1)] = 0 when γ1 is a series approximation to γ10 . Here we show this directional double robustness property for series estimators of solutions to conditional moment restrictions and orthogonal series density estimators. Consider first a conditional moment restriction where the residual ρ(z, γ1 ) is affine in γ1 , m(z, β, γ1 ) is also affine in γ1 , and the first step is a linear series estimator. Suppose that the series estimator approximates γ10 by a linear combination pK′λ of a vector of functions pK (w) = (p1K (w), ..., pKK (w))′. Let q K (xi ) be a K × 1 vector of instrumental variables and λ̂ P be the instrumental variables estimator solving a moment condition ni=1 q K (xi )ρ(zi , pK′λ̂) = 0. Under standard regularity conditions the limit λ∗ of λ̂ will solve the corresponding population moment condition E[q K (xi )ρ(zi , pK′λ∗ )] = 0. Let γ20 (x) satisfy equation (4.2). Then if γ20 (xi ) = θ′ q K (xi ) for some θ it follows that E[m(zi , β0 , pK′λ∗ )] = −E[γ20 (xi )ρ(zi , pK′λ∗ )] = −θ′ E[q K (xi )ρ(zi , pK′λ∗ )] = 0. Thus we have the result Proposition 12: If m(zi , β, γ1) and ρ(zi , γ1) are affine in γ1 ∈ Γ with Γ linear, E[m(zi , β0 , γ1 )] is a mean square continuous functional of E[ρ(zi , γ1)|xi ], and γ20 (xi ) satisfying E[m(zi , β0 , γ1 )] = −E[γ20 (xi )ρ(zi , γ1 )] also satisfies γ20 (xi ) = θ′ q K (xi ) for some θ then E[m(zi , β0 , pK′λ∗ )] = 0. The property shown in the conclusion of Proposition 12 is a directional double robustness condition that depends on γ1 being equal to a series approximation to γ10 and on γ20 (x) being restricted. These restrictions are not required for double robustness of g(z, β, γ) = m(z, β, γ1 ) + γ2 (x)ρ(z, γ1 ). We will have E[g(zi , β0 , γ20 , γ1 )] = 0 for all γ1 , and not just for the γ1 that are a series approximation to γ10 , and for any γ20 (x) and not just one that is a linear combination of q K (x). For series first steps the the original moment functions will be doubly robust in certain directions just as they are locally robust in certain directions. Previous examples of Proposition 12 are given in Newey (1990, p. 116), Newey (1999), and Robins et. al. (2007). Proposition 12 allows for endogeneity and m(z, β, γ1 ) to depend on the entire function γ1 . The condition that the instruments q K (xi ) have the same dimension as the approximating functions pK (w) allows for more than K instrumental variables. As is well known, 22 any IV estimator of λ can be viewed as having only K instruments q K (x), each one of which is equal to a linear combination of all the instrumental variables. Here the existence of θ such that γ20 (xi ) = θ′ q K (xi ) is restrictive. It is not sufficient that γ20 (xi ) be any linear combination of all the instrumental variables. We must have γ20 (xi ) equal to a linear combination of the instruments q K (xi ) used in estimating λ∗ . This result also extends to the case where an infinite number of instrumental variables are used in the limit. In that case q K (xi ) can also be interpreted as an infinite dimensional linear combination of instrumental variables. To illustrate, consider again the weighted average derivative example discussed above where m(z, β, γ1 ) = v(w)∂γ1 (w)/∂w − β, ρ(z, γ1 ) = y − γ1 (w), and there is γ20 (x) such that E[γ20 (x)|w] = −f0 (w)−1∂[f0 (w)v(w)]/∂w. (6.2) Suppose that the first step is a linear instrumental variables (IV) estimator with right-hand side variables pK (wi ) and instruments q K (xi ) and let λ∗ be the limit of the IV coefficients. From Proposition 12 it follows that if there is a θ such that θ′ q K (x) = γ20 (x) then  E[v(wi ) ∂pK (wi )′ λ∗ /∂w] − β0 = E[m(zi , β0 , pK′ λ∗ )] = 0. Thus, the weighted average derivative of the linear IV estimator will be consistent when γ20 (x) is a linear combination of q K (x). The case where v(w) is 1, wi is Gaussian, and E[xi |wi ] is linear in wi is interesting. Partial out constants and means so that (wi′ , x′i )′ has mean zero. Let pK (wi ) = wi and let qi = q K (xi ) be any linear combination xi such that ζ = E[qi wi′ ] is nonsingular. Normalize wi and qi so that each have an identity variance matrix. Then f0 (w)−1 ∂f0 (w)/∂w = −w. Note that E[qi |wi ] = ζwi , so that equation (6.2) is satisfied with γ20 (x) = −ζ −1 q K (x).Thus the conditions of Proposition 12 are satisfied, giving the following result Corollary 13: If yi = γ10 (wi ) + εi , E[xi εi ] = 0, E[yi2] < ∞, wi is Gaussian, and E[xi |wi ] is linear in wi then for instruments equal to any linear combination qi of xi with cov(qi , wi ) nonsingular , E[∂γ10 (wi )/∂w] = cov(qi , wi )−1 cov(qi , yi ). We can give a simple, direct proof that only uses qi and εi uncorrelated, as is assumed in Corollary 11, rather than the conditional moment restriction we have been focusing on. With means partialed out we have E[wi ] = E[qi ] = 0 so that cov(qi , wi )−1 cov(qi , yi ) = (E[qi wi′ ])−1 E[qi yi ] = (E[qi wi′ ])−1 E[qi γ10 (wi )] = (E[E[qi |wi ]wi′ ])−1 E[E[qi |wi ]γ10 (wi )] = (E[ζwi wi′ ])−1 E[ζwi γ10 (wi )] = (E[wi wi′ ])−1 E[wi γ10 (wi )] = E[∂γ10 (wi )/∂w], 23 where the fourth equality follows by E[qi |wi ] = ζwi and the last equality holds by Stoker (1986) and wi Gaussian. This result generalizes that of Stoker (1986) to NPIV models where the right hand variables are Gaussian. Further generalizations to non Gaussian cases can be obtained by letting q K (x) and pK (w) be nonlinear in x and w. Orthogonal series density estimators have a property analogous to Proposition 12. Suppose now that pK (x) is orthonormal with respect to Lebesgue measure on (−∞, ∞) so that R K p (u)pK (u)′ du = I. An orthogonal series pdf estimator is γ̂1 (x) = pK (x)′ λ̂, where λ̂ = R K Pn K p (x )/n has limit λ = p (u)γ10 (u)du. Suppose that E[m(zi , β0 , γ1 )] is a continuous i ∗ i=1 linear functional of γ1 − γ10 so that by the Riesz representation theorem there is γ20 (x) with R E[m(zi , β0 , γ1 )] = γ20 (u)[γ1(u) − γ10 (u)]du. If there is θ with θ′ pK (x) = γ20 (x) then by pK (u) orthonormal and equation (4.4) we have Z Z K′ K ′ E[m(zi , β0 , p λ∗ )] = γ20 (u)[p (u) λ∗ − γ10 (u)]du = θ′ pK (u)[pK (u)′ λ∗ − γ10 (u)]du Z Z ′ K K ′ ′ pK (u)γ10 (u)du = θ′ λ∗ − θ′ λ∗ = 0. = θ [ p (u)p (u) du]λ∗ − θ Thus we have the following result: Proposition 14: If i) m(z, β0 , γ1 ) is affine in γ1 , ii) E[m(zi , β0 , γ1 )] − β0 is a functional R
1/2 of γ1 (x) − γ10 (x) that is continuous in the norm [γ1 (u) − γ10 (u)]2 du , and iii) γ20 (xi ) = ′ K K′ θ p (xi ) for some θ then E[m(zi , β0 , p λ∗ )] = 0. The orthogonal series estimators of linear functionals of a pdf discussed in Bickel and Ritov (2003) are examples. Those estimators are special cases of the estimator above where R m(z, β, γ1 ) = γ20 (u)γ1 (u)du − β for prespecified γ20 (u). Proposition 14 implies that the orthogonal series estimator of β0 will be consistent if γ20 (u) is a linear combination of the approximating functions. For example, if pK (x) is vector of polynomials of order K − 1 then the orthogonal series estimator of moments of up to order K − 1 is consistent for fixed K. 7 Conditional Moment Restrictions Conditional moment restrictions are widely used in econometrics to identify parameters of interest. In this Section we expand upon the cases already considered to construct a wide variety of locally robust moment conditions. In particular we extend above results to residuals that may depend on parameters of interest with instrumental variables that can differ across residuals. Here we depart from deriving locally robust moments from the adjustment term for first step estimation. Instead we extend the form of previously derived locally robust moments to the more general setting of this Section. 24 To describe these results let j = 2, ..., J index conditional moment restrictions, ρj (z, β, γ1 ) denote a corresponding residual, and xj be corresponding conditioning variables. We will consider construction of locally robust moment conditions when the true parameters of interest β0 and a first step γ10 satisfy conditional moment restrictions E[ρj (zi , β0 , γ10 )|xji ] = 0, (j = 2, ..., J). (7.1) Here γ1 is specified to include all functions that affect any of the residuals ρj (zi , β, γ1). We continue to assume that the unconditional moment restriction in equation (2.1) holds, though m(z, β, γ1 ) could be zero, with identification of β0 coming from the conditional moment restrictions of equation (7.1). We will discuss this case below. In this setting we consider locally robust moment conditions having the form g(z, β, γ) = m(z, β, γ1 ) + J X γj (xj , β)ρj (z, β, γ1 ), (7.2) j=2 where γj (x, β), (j = 2, ..., J) are unknown functions satisfying properties discussed below. These moment functions depend on J first step components γ = (γ1 , ..., γJ ). By virtue of the conditional moment restrictions these moment functions will be doubly robust in (γ2 , ..., γJ ), meaning that E[g(zi , β0 , γ10 , γ2 , ..., γJ ) = 0. They will be locally robust in γ1 if for the limit γ1 (F ) of γ̂1 and all regular parametric models Fτ as discussed in Section 2, J X ∂ ∂ E[m(zi , β0 , γ1 (Fτ ))] + E[ γj0 (xji ) E[ρj (zi , β0 , γ1 (Fτ ))|xji ]] = 0. ∂τ ∂τ j=2 (7.3) If ∂E[m(zi , β0 , γ1(Fτ ))]/∂τ |τ =0 is a linear mean-square continuous function of (∂E[ρ2 (zi , β0 , γ1 (Fτ ))|xji ]/∂τ, ..., ∂E[ρJ (zi , β0 , γ1 (Fτ ))|xji ]/∂τ )|τ =0 and the mean-square closure of the set of such vectors over all parametric submodels is linear then existence of γj0 (xj ), j ≥ 2 satisfying equation (7.3) will follow by the Riesz representation theorem. In addition, if m(z, β0 , γ1 ) and ρj (z, β0 , γ1), (j ≥ 2), are affine in γ1 then we will have double robustness in γ1 similarly to Proposition 12. Summarizing we have Proposition 15: If equation (7.3) is satisfied then g(z, β, γ) from equation (7.2) is locally robust. Furthermore, if i) m(z, β0 , γ1 ) and ρj (z, β0 , γ1 ), (j ≥ 2) are affine in γ1 ∈ Γ with Γ linear; ii) E[m(zi , β0 , γ1)] is a mean square continuous functional of E[ρj (zi , β0 , γ1 )|xij ], (j ≥ 2) then there is γj0 (x), (j ≥ 2), such that g(z, β, γ) is doubly robust. For local identification of β we also require that rank( ∂E[g(zi , β, γ0)]/∂β|β=β0 ) = dim(β). 25 (7.4) A model where β0 is identified from semiparametric conditional moment restrictions with common instrumental variables x is a special case where m(z, β, γ) is zero and xj = x, (j ≥ 2). In this case let ρ(z, β, γ1 ) = (ρ2 (z, β, γ1 ), ..., ρJ (z, β, γ1 ))′ . The conditional moment restrictions of equation (7.1) can be summarized as E[ρ(zi , β0 , γ10 )|xi ] = 0. This model is considered by Chamberlain (1992) and Ai and Chen (2003, 2007, 2012). We allow the residual vector ρ(z, β, γ1 ) to depend on the entire function γ1 and not just its value at some function of the observed data zi . Also let ϕ(x) = [γ2 (x), ..., γJ (x)] denote an r ×(J −1) matrix of functions of x. A locally robust moment function g(z, β, γ) = ϕ(x)ρ(z, β, γ1 ) will be one which satisfies Definition 1 with g(z, β, γ) replacing m(z, β, γ), i.e. where   ∂E[g(zi , β0 , γ(Fτ ))] ∂E[ρ(zi , β0 , γ1 (Fτ ))|xi ] = 0, = E ϕ(xi ) ∂τ ∂τ for all regular parametric models. We also require that equation (7.4) is satisfied. To characterize local robustness here it is helpful to assume that the set of pathwise derivatives of E[ρ(zi , β0 , γ)|xi ] varies over a linear set as the regular parametric model Fτ varies. To be precise we will assume that γ1 ∈ Γ for Γ linear and for ∆ ∈ Γ we let mγ (x, ∆) = ∂E[ρ(zi , β0 , γ10 + τ ∆)|xi ] ∂τ τ =0 denote the (J − 1) × 1 random vector that is Gateaux derivative of the conditional expectation E[ρ(zi , β0 , γ1 )|xi ] with respect to the first step γ1 in the direction ∆. We assume that mγ (x, ∆) is linear in ∆ and that the mean square closure Mγ of the set {mγ (x, ∆) : ∆ ∈ Γ} equals the mean-square closure of the set {∂E[ρ(zi , β0 , γ1(Fτ ))|xi ]/∂τ } as Fτ varies over all regular parametric models. The local robustness condition can then be interpreted as orthogonality of each row ϕ(xi )′ ej of ϕ(x) with Mγ in the Hilbert space of functions of x with inner product ha, bi = E[a(xi )′ b(xi )], where ek is the k th unit vector. Thus the condition for locally robust g(z, β, γ) = ϕ(x)ρ(z, β, γ1 ) is that E[ϕ(xi )mγ (xi , ∆)] = 0 for all ∆ ∈ Γ. We refer to such ϕ(xi ) as being orthogonal. They can be interpreted as instrumental variables where the effect of estimation of γ1 has been partialed out. There are many ways to construct orthogonal instruments. For instance, given a r × (J − 1) matrix of instrumental variables A(x) one could construct corresponding orthogonal ones ϕ(xi ) as the matrix where each row is the residual of the least squares projection of the corresponding row of A(x) on Mγ . We focus on another way of constructing orthogonal instruments that leads 26 to an efficient estimator of β0 . Let Σ(x) denote some positive definite matrix with smallest eigenvalue bounded away from zero, so that Σ(xi )−1 is bounded. Let ha, biΣ = E[a(xi )′ Σ(xi )−1 b(xi )] denote an inner product and note that Mγ is closed in this inner product by Σ(xi )−1 bounded. Let Ãk (xi , A, Σ) denote the residual from the least squares projection of the k th row A (x)′ ek of A(x) on Mγ with the inner product ha, biΣ . Also let ϕ(xi , A, Σ) be the matrix with k th row Ãk (xi , A, Σ)′ Σ(xi )−1 , (k = 1, ..., r). Then for all ∆ ∈ Γ, A (xi )′ ek − Ãk (xi , A, Σ) ∈ Mγ , E[ϕ(xi , A, Σ)mγ (xi , ∆)] = E[Ã(xi , A, Σ)Σ(xi )−1 mγ (xi , ∆)] = 0, so that ϕ(xi , A, Σ) are orthogonal instruments. Also, Ã(x, A, Σ) can be interpreted as the solution to min {M (x):M (x)′ ek ∈Mγ ,k=1,...,r} E[{A(xi ) − M(xi )}Σ(xi )−1 {A(xi ) − M(xi )}′ ] where the minimization is in the positive semidefinite sense. The orthogonal instruments that minimize the asymptotic variance of GMM in the class of GMM estimators with orthogonal instruments are given by ∂E[ρ(zi , β, γ10 )|xi ] ϕ (xi ) = ϕ(xi , A , Σ ), A (xi ) = ∂β ∗ ∗ ∗ ′ , Σ∗ (xi ) = V ar(ρ(zi , β0 , γ10 )|xi ). ∗ β=β0 To see that ϕ∗ (xi ) minimizes the asymptotic variance note that for any orthogonal instrumental variable matrix ϕ(x) G = E[ϕ(xi )A∗ (xi )′ ] = E[ϕ(xi )Ã(xi , A∗ , Σ∗ )′ ] = E[ϕ(xi )ρ(zi , β0 , γ10 )ρ(zi , β0 , γ10 )′ ϕ∗ (xi )′ ], where the first equality defines G and the second equality holds by ϕ(xi ) orthogonal. Since the p instruments are orthogonal the asymptotic variance matrix of GMM estimator with Ŵ −→ W is the same as if γ̂1 = γ10 . Define mi = G′ W ϕ(xi )ρ(zi , β0 , γ10 ) and m∗i = ϕ∗ (xi )ρ(zi , β0 , γ10 ). The asymptotic variance of the GMM estimator for orthogonal instruments ϕ(x) is (G′ W G)−1 G′ W E[ϕ(xi )ρ(zi , β0 , γ10 )ρ(zi , β0 , γ10 )′ ϕ(xi )′ ]W G(G′ W G)−1 −1 ′ ∗ −1′ = (E[mi m∗′ . i ]) E[mi mi ](E[mi mi ]) The fact that this matrix is minimized in the positive semidefinite sense for ϕ(x) = ϕ∗ (x) follows from Theorem 5.3 of Newey and McFadden (1994) and can also be shown using the argument in Chamberlain (1987). Proposition 16: The instruments ϕ∗ (xi ) give an efficient estimator in the class of IV estimators with orthogonal instruments. The asymptotic variance of the GMM estimator with optimal orthogonal instruments is −1 (E[m∗i m∗′ = E[Ã(xi , A∗ , Σ∗ )Σ∗ (xi )−1 Ã(xi , A∗ , Σ∗ )′ ])−1 . i ]) 27 This matrix coincides with the semiparametric variance bound of Ai and Chen (2003). Estimation of the optimal orthogonal instruments is beyond the scope of this paper. The series estimator of Ai and Chen (2003) could be used for this. 8 Structural Economic Examples Estimating structural models can be difficult when that requires computing equilibrium solutions. Motivated by this difficulty there is increasing interest in two step semiparametric methods based on first step estimation of conditional choice probabilities (CCP). This two step approach was pioneered by Hotz and Miller (1993). In this Section we show how locally robust moment conditions can be formed for two kinds of structural models, the dynamic discrete choice model of Rust (1987) and the static model of strategic interactions of Bajari, Hong, Krainer, and Nekipelov (2010, BHKN). It should be straightforward to extend the construction of locally robust moments to other more complicated structural economic models. The use of such moment conditions will allow for conditional choice probabilities that are estimated by modern, machine learning methods. 8.1 Static Models of Strategic Interactions We begin with a static model of interactions where results are relatively simple. To save space we describe the estimator of BHKN while only describing a small part of the motivational economic structure. Let x denote a vector of state variables for a fixed set of individuals and let y denote a vector of binary variables, each one representing a choice of an alternative by an individual. Let the observations zi = (yi , xi ) represent repeated plays of a static game of interaction and γ10 (x) = E[yi |xi = x] the vector of conditional choice probabilities given a value x of the state. In the semiparametric estimation problem described in Section 4.2 of BHKN there is a known function r(x, β, γ1(x)) of the state variable x, a vector of parameters β, and a possible value γ(x) of the conditional choice probabilities such that the true parameter β0 satisfies E[yi |xi = x] = r (x, β0 , γ10 (x)) , This model can be used to form moment functions m(z, β, γ1 ) = A(x)[y − r (x, β, γ1 (x))], where A(x) is a matrix of instrumental variables; see equation (17) of BHKN. To describe locally robust moment functions in this example, let rγ (x, β, γ1) = ∂r(x, β, γ1 )/∂γ1 where γ1 here denotes a real vector representing a possible value of γ10 (x). Then, it follows from 28 Proposition 4 of Newey (1994), as discussed in BHKN, that the adjustment term for first step estimation of γ10 (x) = E[yi |xi = x] is φ(z, β, γ1 ) = −A(x)rγ (x, β, γ1)[y − γ1 (x)]. This expression differs from BHKN in the appearance of γ1 (x) at the end of the expression rather than rγ (x, β, γ1 (x)), which is essential for local robustness. The locally robust moment conditions are then g(z, β, γ1) = A(x){y − r(x, β, γ1 (x)) − rγ (x, β, γ1(x))[y − γ1 (x)]}. For a first step estimator γ̂(x) of the conditional choice probabilities, the locally robust sample moments will be n 1X A(xi ){yi − r(xi , β, γ̂1(xi )) − rγ (xi , β, γ̂1(xi ))[yi − γ̂1 (xi )]} ĝ(β) = n i=1 Here the locally robust moments are constructed by subtracting from the structural residuals r a linear combination of the first step residuals. Using these moment functions should result in an estimator of the structural parameters with less bias and the other improved properties of locally robust estimators mentioned above. The optimal instruments here are the same as discussed in BHKN. Let I denote the identity matrix, set H(x) = I − ∂r(xi , β0 , γ10 (xi ))/∂γ1 , and let Ω(xi ) = H(xi )V ar(yi |xi )H(xi )T denote the conditional variance of H(xi )(yi − γ10 (xi )). The optimal instruments are given by ∂r(xi , β0 , γ10 (xi )) ′ Ω(x)− A (xi ) = ∂β ∗ where A− denotes a generalized inverse of a positive semi-definite matrix A. This model can also be viewed as a special case of the conditional moment restrictions framework with residual vector ρ(z, β, γ1 ) = (y − γ1 (x), y − r(x, β, γ1 (x)))T . An orthogonal instrument that gives the above locally robust moment function is A(x)[−rγ (xi , β, γ1(xi )), I]. Here the locally robust moment function only depend on one first step function γ1 (x). This feature is shared by all setups where the second step residual r(x, β, γ1) depends only on regressors that are included in the first step γ1 (x). The static model of strategic interactions leads to this structure. The situation is not so simple in other structural economic models, as we see next. 8.2 Dynamic Discrete Choice Dynamic discrete choice estimation is important for modeling economic decisions, Rust (1987). In this setting we find it helpful to describe the underlying economic model in order to explain 29 the form of the moment conditions. Here we give locally robust moment conditions moment conditions that depend on first step estimation of the conditional choice probabilities. We do this for the infinite horizon, stationary, dynamic discrete choice model of Rust (1987). It is straightforward to derive locally robust moment conditions for other structural econometric models. We also focus here on the case of data on many homogenous individuals, but discuss how the approach extends to time series on one individual. Suppose that the per-period utility function for an agent making choice j in period t is given by Ujt = vj (xt , β0 ) + ǫjt , (j = 1, ..., J; t = 1, 2, ...) where we suppress the individual subscript i for notational convenience. The vector xt is the observed state variables of the problem (e.g. work experience, number of children, wealth) and the vector β is unknown parameters. The disturbances ǫt = {ǫ1t , ..., ǫJt } are not observed by the econometrician. As in the majority of the literature we assume that ǫt is i.i.d. over time with known CDF that has support RJ and is independent of the state process xt and that xt is Markov of order 1. Let δ denote a time discount parameter, v̄(x) the expected value function, yjt ∈ {0, 1} the indicator that choice j is made and v̄j (xt ) = vj (xt , β0 ) + δE[v̄(xt+1 )|xt , ytj = 1] the expected value function for choice j. Also let ṽj denote a possible realization of ṽj (xt ) = v̄j (xt ) − v̄1 (xt ), so that ṽ1 ≡ 0. Let ṽ = (ṽ2 , ..., ṽJ ) and Pj (ṽ) = Pr(ṽj + ǫjt ≥ ṽk + ǫkt ; ṽ1 = 0; k = 1, ..., J), (j = 1, ..., J) denote the choice probabilities associated with the distribution of ǫt . Here we normalize to focus on the difference with v̄1 (x) throughout. Let ṽ(x) = (ṽ2 (x), ..., ṽJ (x))′ . Let γa (x) = (γa1 (x), ..., γaJ (x))T be a vector of first step functions with true values γaj0(xt ) = Pr(ytj = 1|xt ). From Rust (1987) we know that for γaj0 (xt ) = Pj (ṽ(xt )), (j = 1, ..., J). v̄(xt ) = E[max{v̄j (xt ) + ǫjt }|xt ] = v̄1 (xt ) + E[max{ṽj (xt ) + ǫjt }|xt ]. j j From Hotz and Miller (1993) we know that P (ṽ) = (P1 (ṽ), P2 (ṽ), ..., PJ (ṽ))′ is a one-to-one function of ṽ, so that inverting this relationship it follows that E[maxj {ṽj (xt ) + ǫjt }|xt ] is a function of γa0 (xt ), say E[maxj {ṽj (xt ) + ǫjt }|xt ] = H(γa0 (xt )), for some function H(·) (e.g. for binary logit H(γa0 (xt )) = .5772 − ln(γa10 (xt ))). Then the expected value function is given by v̄(xt ) = v̄1 (xt ) + H(γa0 (xt )) To use these relationships to construct semiparametric moment conditions we normalize v1 (x, β) = 0 and make an additional assumption for v̄1 (x). The additional assumption is that E[v̄(xt+1 )|xt , 1] does not depend on xt . With this normalization and assumption we have a constant choice specific value function for j = 1, that is v̄1 (x) = v̄1 , with v̄1 (xt ) = 0 + δE[v̄(xt+1 )|xt , yt1 = 1] = δE[v̄(xt+1 )|yt1 = 1] = v̄1 . 30 A sufficient condition for constant v̄1 (xt ) is that j = 1 is a ”renewal” choice where the distribution of the future state does not depend on the current state. In the Rust (1987) example this state is the one where the bus engine is replaced. With this normalization and assumption we now have v̄j (x) = vj (x, β0 ) + δE[v̄1 + H(γa0 (xt+1 ))|xt = x, ytj = 1] = vj (x, β0 ) + δv̄1 + δE[H(γa0 (xt+1 ))|xt = x, ytj = 1], ṽj (x) = vj (x, β0 ) + δE[H(γa0 (xt+1 ))|xt = x, ytj = 1] − δE[H(γa0 (xt+1 ))|yt1 = 1], (j = 2, ..., J). The choice specific expected value differences ṽj (x) have a parametric part vj (x, β0 ) and a nonparametric part that depends on J − 1 additional nonparametric regressions γb (x, γa ) = (γb1 (x, γa ), ..., γb,J−1 (x, γa ))T and an unknown parameter γc where γbj0 (x, γa ) = E[H(γa (xt+1 ))|xt = x, yt,j+1 = 1], j = 1, ..., J − 1; γc0(γa ) = E[H(γa (xt+1 ))|yt1 = 1]. Let γ1 (x) = (γa (x)T , γb(x, γa )T , γc (γa ))T be a vector of first step objects and ṽj (x, β, γ1) = vj (x, β) + δ[γb,j (x, γa ) − γc (γa )], ṽ(x, β, γ1) = (ṽ2 (x, β, γ1), ..., ṽJ (x, β, γ1))T denote the semiparametric choice specific expected value differences. Semiparametric moment conditions can then be formed by plugging in a nonparametric γ̂1 estimator of the first step into the expected differences and plugging those into the choice probability. Let yt = (yt1 , ..., ytJ )T denote the vector of choice indicators for period t and z = (y1T , xT1 , ..., yTT , xTT )T be the vector consisting of the observations on choice and state variables xt for each time period t. Also let At (xt ) be an r × 1 vector of functions of xt where r is the dimension of β. Then for each t = 1, ..., T − 1 we can form semiparametric moment conditions as mt (z, β, γ1 ) = At (xt )[yt − P (ṽ(xt , β, γ1))]. To derive locally robust moment functions we can derive the adjustment term for estimation of γ1 . The first step function γ1 is more complicated than previously considered. It depends on two unknown conditional expectations, E[yt |xt ] and E[·|xt , j], (j = 2, ..., J). From Newey (1994, p. 1357) we know that the adjustment term will be the sum of two terms, each adjusting for one of the two conditional expectations while treating the other as if it was equal to the truth. In the Appendix we give a general form for each of these adjustment terms. Here we apply that general form to derive the corresponding locally robust moment functions for dynamic discrete choice. We begin by deriving the adjustment terms for γb and γc because they are simpler than for γa . The adjustment term for γb and γc are obtained by applying Proposition A1 in the 31 Appendix. Let γ̃b (F ), γ̃c (F ) have the same form as γb and γc except that E[·|xt , yt ] is replaced by EF [·|xt , j]. For Ht+1 = H (γa0 (xt+1 )) and π1 = E[yt1 ] let λbj (z, β, γ1) = [yt,j+1/γa,j+1(xt )]{H(γa (xt+1 )) − γb,j+1(xt , γa )}, λb (z, β, γ1) = (λb1 (z, β, γ1 ), ..., λb,J−1(z, β, γ1 ))T , λc (z, β, γ1 , π1 ) = π1−1 yt1 H(γa (xt+1 )) − γc (γa ). Then for e a (J − 1) × 1 vector of 1′ s we have ∂E[mt (zi , β0 , γa0 , γ̃b (Fτ ), γ̃c (Fτ ))] = E[φtb (zi , β0 , γ1 , γ2)S(zi )], ∂τ ∂P (ṽ(xt , β, γ1))λb (z, β, γ1) φtb (z, β, γ1, π) = −δA(xt ) ∂ṽ ∂P + δE[A(xt ) (ṽ(xt , β, γ1))]e · λc (z, β, γ1 , π1 )} ∂ṽ The adjustment term for estimation of γa (x) is obtained by applying Proposition A2. This term is somewhat complicated because γa (x) is evaluated at x = xt+1 rather than the conditioning argument xt of its true value We assume that xt is stationary over time so that xt+1 and xt have the same pdf, eliminating the ratio of pdf’s in the conclusion of Proposition A2. Let γ̄a (F ) have the same form as γ10 except that γ10 (x) is replaced by EFt [yt |xt = x]. Also let   ∂P (ṽ(xt , β, xt )) ytj A(xt ) |xt+1 = x , (j = 1, ..., J − 1), γ2j0(x, β, γ1 ) = E γaj (xt ) ∂ṽ γ30 (x, π1 ) = π1−1 E[yt1 |xt+1 = x]|x=xt . Then we have ∂E[mt (zi , β0 , γa (Fτ ), γb0 , γc0 )] = E[φta (zi , β0 , γ1 , γ2)S(zi )], ∂τ φta (z, β, γ1 , γ2, γ3 , π1 ) = −δ{γ2 (x, β, γ1) − E[A(xt )Pṽ (ṽ(xt , β, γ1))]γ3 (xt , π1 )}e × ∂H(γa (xt )) [yt − γa (xt )]. ∂γa We can now form locally robust moment conditions as gt (z, β, γ1 , γ2 , γ3, π1 ) = mt (z, β, γ1 ) + φta (z, β, γ1 , γ2 , γ3, π1 ) + φtb (z, β, γ1 , π1 ). With data zi that is i.i.d. over individuals these moment functions can be used for any t to estimate the structural parameters β. Also, for data for a single individual we could use a time P −1 gt (z, β, γ1 , γ2 , γ3, π1 )/(T − 1) to estimate β, although the asymptotic theory we average Tt=1 give does not apply to this estimator. Bajari, Chernozhukov, Hong, and Nekipelov (2009) derived the adjustment term for the more complicated dynamic discrete game of imperfect information. Locally robust moment conditions for such games could be formed using their results. We leave that formulation to future work. 32 9 Asymptotic Theory for Locally Robust Moments In this Section we give asymptotic theory for locally robust estimators. In keeping with the general applicability of locally robust moments to a variety of first steps we consider estimation and conditions that have the most general and simplest conditions we can find for the first step. In particular, the construction here only requires that the first step converge at rate slightly faster than n−1/4 in norms specified below, a simpler condition than in most of the literature. This formulation allows the results to be applied in settings where it is challenging to say much about the first step other than its convergence rate, such as when machine learning is used in the first step. The locally robust form of the moment conditions is essential for this formulation, as previously discussed. We use cross fitting in the first step to obtain an estimator that is root-n consistent and asymptotically normal under such simple conditions. Chernozhukov et. al. (2016) gives results with cross fitting that allow for moment functions that are not smooth in parameters. Here we focus on the smooth in parameters case. Cross fitting has been previously used in the literature on semiparametric estimation. See Bickel, Klaasen, Ritov, and Wellner (1993) for discussion. This approach is different than that some previous work in semiparametric estimation, as in Andrews (1994), Newey (1994), Chen, Linton, and van Keilegom (2003), Ichimura and Lee (2010), where cross fitting was not used and the moment conditions need not be locally robust. The approach adopted here leads to general and simple conditions. The estimator is formed by grouping observations into L distinct groups. Let Iℓ , (ℓ = 1, ..., L) partition the set of observation indices {1, ..., n}. Let γ̂−ℓ be the first step constructed from all observations not in Iℓ . Consider sample moment conditions of the form L 1 XX g(zi , β, γ̂−ℓ). g̃(β) = n ℓ=1 i∈I ℓ We consider GMM estimators based on these moment functions. This is a special case of the cross fitting described earlier. Also, leave one out moment conditions are a further special case where each Iℓ consists of a single observation. We focus here on the case where the number of groups L is fixed to keep the conditions as simple as possible. An important intermediate result is that the adjustment term for the first step is zero by virtue of g(z, β, γ) being locally robust, that is √ n 1 X ng̃(β0 ) = √ g(zi , β0 , γ0) + op (1). n i=1 (9.1) With cross fitting this result holds under relatively weak and simple conditions: R p Assumption 1: For each ℓ = 1, ..., L, i) kg(z, β0 , γ̂−ℓ ) − g(z, β0 , γ0 )k2 F0 (dz) −→ 0, ii) for R √ p ζ > 1, C > 0 we have g(z, β0 , γ̂−ℓ )F0 (dz) ≤ C kγ̂−ℓ − γ0 kζ , and iii) n kγ̂−ℓ − γ0 kζ −→ 0. 33 Lemma 17: If Assumption 1 is satisfied then equation (9.1) is satisfied. This Lemma is proved in the Appendix. There are two important components to this result. One component is a stochastic equicontinuity result √ n L 1 X 1 X ng̃(β0 ) − √ nℓ g(zi , β0 , γ0) − √ n i=1 n ℓ=1 Z p g(z, β0 , γ̂−ℓ )F0 (dz) −→ 0, where nℓ is the number of observations with i ∈ Iℓ . Assumption 1 i) is sufficient for this result. Assumption 1 i) is a much weaker stochastic equicontinuity condition than appears in much of the literature, e.g. Andrews (1994). Those other conditions generally involve boundedness of some derivatives of γ̂. In contrast Assumption 1 i) only requires that γ̂−ℓ have a mean square convergence property. The cross fitting is what makes this condition sufficient. Cattaneo and Jansson (2014) have also previously weakened the stochastic equicontinuity condition and established the validity of the bootstrap for kernel estimators under substantially weaker bandwidth conditions than usually imposed. The second component of the result is that Z √ p nḡ(γ̂−ℓ ) −→ 0, ḡ(γ) = g(z, β0 , γ)F0 (dz). This component follows from Assumptions 1 ii) and iii). By comparing Assumption 1 ii) with Definition 2 we see that this condition implies local robustness in the sense that the Frechet derivative of ḡ(γ) is zero at γ0 . Assumption 1 ii) will generally hold with ζ = 2 if ḡ(γ) is twice continuously Frechet differentiable. In that case Assumption 1 iii) become the n1/4 rate condition familiar from Newey and McFadden (1994) and other works. The more general 1 < ζ < 2 case allows for the first Frechet derivative of ḡ(γ) to satisfy a Lipschitz condition. In this case Assumption 1 iii) will require a convergence rate of γ̂ that is faster than n1/4 . We note that previous results suggest that n1/4 convergence of γ̂ may be stronger than is needed. As shown in Robins et. al. (2008) and Cattaneo and Jansson (2014) the variance terms √ in nḡ(γ̂) are of the same order as the variance term of a nonparametric estimator, rather √ than being the order of n times those variance terms. The arguments for these weaker results are quite complicated so we do not attempt to give an account here. Instead we focus on the relatively simple conditions of Assumption 1. Another component of an asymptotic normality result is convergence of the Jacobian term ∂g̃(β)/∂β. The conditions we impose to account for the Jacobian term are standard. Let G̃(β) = ∂g̃(β)/∂β denote the derivative of the moment function. Assumption 2: There is a neighborhood N of β0 such that i) g(zi , β, γ) is differentiable in β on N with probability approaching 1 ii) there is ζ ′ > 0 and d(zi ) with E[d(zi )] < ∞ such that 34 for β ∈ N and kγ − γ0 k small enough ′ ′ ∂g(zi , β, γ) ∂g(zi , β0 , γ0) ≤ d(zi )(kβ − β0 kζ + kγ − γ0 kζ ), − ∂β ∂β p iii) E[k∂g(zi , β0 , γ0 )/∂βk] < ∞; iv) kγ̂−ℓ − γ0 k −→ 0, (ℓ = 1, ..., L). Define G = E[∂g(zi , β, γ0)/∂β|β=β0 ] p Lemma 18: If Assumption 2 is satisfied then for any β̄ −→ β0 , g̃(β) is differentiable at β̄ p with probability approaching one and ∂g̃(β̄)/∂β −→ G. With Lemmas 17 and 18 in place the asymptotic normality of semiparametric GMM follows in a standard way. p p Theorem 19: If Assumptions 1 and 2 are satisfied, β̂ −→ β0 , Ŵ −→ W , G′ W G is nonsingular, and E[kg(zi , β0 , γ0 )k2 ] < ∞ then for Ω = E[g(zi , β0 , γ0)g(zi , β0 , γ0 )′ ], √ d n(β̂ − β0 ) −→ N(0, V ), V = (G′ W G)−1G′ W ΩW G(G′W G)−1 . It is also useful to have a consistent estimator of the asymptotic variance of β̂. As usual such an estimator can be constructed as V̂ = (Ĝ′ Ŵ Ĝ)−1 Ĝ′ Ŵ Ω̂Ŵ Ĝ(Ĝ′ Ŵ Ĝ)−1 , L ∂g̃(β̂) 1 XX Ĝ = g(zi , β̂, γ̂−ℓ )g(zi , β̂, γ̂−ℓ )′ . , Ω̂ = ∂β n ℓ=1 i∈I ℓ Note that this variance estimator ignores the estimation of γ, which works here because the moment conditions are locally robust. Its consistency will follow under the conditions of Theorem 19 and one additional condition that accounts for the presence of β̂ in Ω̂. ˜ with E[d(z ˜ i )2 ] < Theorem 20: If the conditions of Theorem 19 are satisfied and there is d(z) ∞ such that for kβ − β0 k and kβ − β0 k small enough ˜ i )(kβ − β0 kζ ′ + kγ − γ0 kζ ′ ), kg(zi , β, γ) − g(zi , β0 , γ0 )k ≤ d(z p then V̂ −→ V. In this Section we have used cross fitting to obtain relatively simple conditions for asymptotic normality of locally robust semiparametric estimators. It is also known that in some settings 35 some kinds of cross fitting improves the properties of semiparametric estimators. For linear kernel averages it is known that the leave one out method eliminates a bias term and leads to a reduction in asymptotic mean square error; e.g. see NHR and the references therein. Also Robins et. al. (2008) use cross fitting in higher order bias corrections. These results indicate the some kind of cross fitting can lead to estimators with improved properties. For reducing higher order bias and variance it may be desirable to let the number of groups grow with the sample size. That case is beyond the scope of this paper. 10 APPENDIX We first give an alternative argument for Proposition 2 that is a special case of the proof of Theorem 2.2 of Robins et. al. (2008). As discussed above, φ(z, β0 , γ0 ) is the influence function of the functional µ(F ) = E[m(zi , β0 , γ(F ))]. Because it is an influence function it has mean R zero at all true distributions, i.e. φ(z, β0 , γ(F0 ))F0 (dz) ≡ 0 identically in F0 . Since a regular parametric model {Fτ } is just a subset of all true models, we have Z φ(z, β0 , γ(Fτ ))Fτ (dz) ≡ 0, identically in τ . Differentiating this identity at τ = 0 and applying the chain rule gives ∂E[φ(zi , β0 , γ(Fτ ))] ∂τ τ =0 = −E[φ(zi , β0 , γ0 )S(zi )]. (10.1) Summing equations (3.3) and (10.1) we obtain ∂E[g(zi , β0 , γ(Fτ ))] ∂τ = τ =0 ∂E[m(zi , β0 , γ(Fτ ))] ∂τ + τ =0 ∂E[φ(zi , β0 , γ(Fτ ))] ∂τ τ =0 = E[φ(zi , β0 , γ0 )S(zi )] − E[φ(zi , β0 , γ0 )S(zi )] = 0. Thus we see that the adjusted moment functions g(z, β, γ) are locally robust. Next we derive the form of the adjustment term when a first step is E[·|x, y = 1] for some binary variable y with where yi ∈ {0, 1}. Consider a first step function of the form γ1 (x) = E[wi |xi = x, yi = 1]. Let π(xi ) = E[yi |xi ]. Note that E[wi |xi , yi = 1] = E[yi wi |xi ]/E[yi |xi ] so that ∂EFτ [wi |xi , yi = 1] = E[π(xi )−1 {yi wi − E[yi wi |xi ]} − E[wi |xi , yi = 1](yi − π(xi ))}S(zi )|xi ] ∂τ = E[π(xi )−1 yi {wi − E[wi |xi , yi = 1]}S(zi )|xi ]. Suppose that there is δ(xi ) such that ∂EFτ [m(zi , β0 , γ1(Fτ ))] ∂EFτ [wi |xi , yi = 1] = E[δ(xi ) ] ∂τ ∂τ = E[δ(xi )π(xi )−1 yi {wi − E[wi |xi , yi = 1]}S(zi )]. 36 Then taking the limit gives the following result: Proposition A1: If there is δ(x) such that ∂E[m(zi , β0 , γ1(Fτ ))]/∂τ = E[δ(xi )∂EFτ [wi |xi , yi = 1]/∂τ ] then the adjustment term is φ(z, β, γ) = δ(x)π(x)−1 y{w − E[wi |xi = x, yi = 1]}. Next we derive the adjustment term when a nonparametric regression is evaluated at a variable different than the one being conditioned on in the regression. Note that for δ(v) with E[E[δ(vi )|wi ]EFτ [yi |xi = x]|x=wi ] ∂E[δ(vi )EFτ [yi |xi = x]|x=wi ] = ∂τ ∂τ Z = ∂ E[δ(vi )|wi = w]EFτ [yi |xi = w]fw (w)dw/∂τ Z = ∂ [fw (x)/fx (x)]E[δ(vi )|wi = x]EFτ [yi |xi = x]fx (dx)/∂τ = E[{fx (xi )−1 fw (xi )E[δ(vi )|wi = w]|w=xi (yi − E[yi |xi ])}S(zi )] Taking limits gives Proposition A2: If there is δ(v) such that ∂E[m(zi , β0 , γ1 (Fτ ))]/∂τ = ∂E[δ(vi )EFτ [yi |xi = x]|x=wi ]/∂τ then the adjustment term is φ(z, β, γ) = fx (x)−1 fw (x)E[δ(vi )|wi = x](yi − E[yi |xi = x]). Next we give the proofs for the asymptotic normality results. Proof of Lemma 17: Let Z X ˆ iℓ . ˆ iℓ = g(zi , β0 , γ̂−ℓ ) − ḡ(γ̂−ℓ ) − g(zi , β0 , γ0 ), (i ∈ Iℓ ), ∆ ¯ℓ = 1 ∆ ḡ(γ) = g(z, β0 , γ)F0 (dz), ∆ n i∈I ℓ Also, let Z−ℓ denote a vector of all observations zi for i ∈ / Iℓ . Note that by construction ˆ E[∆iℓ |Z−ℓ ] = 0, so for any i, j ∈ Iℓ , i 6= j, it follows by zi and zj independent conditional ˆ′ ∆ ˆ ˆ′ ˆ on Z−ℓ that E[∆ iℓ jℓ |Z−ℓ ] = E[∆iℓ |Z−ℓ ]E[∆jℓ |Z−ℓ ] = 0 Furthermore Z 2 ˆ iℓ |Z−ℓ ] ≤ kg(z.β0 , γ̂−ℓ ) − g(z, β0 , γ0 )k2 F0 (dz). E[ ∆ Therefore, for nℓ equal to the number of observations in group ℓ, Assumption 1 i) implies Z X 2 1 n ℓ ′ ¯ ∆ ¯ ˆ iℓ |Z−ℓ ] ≤ E[∆ E[ ∆ kg(z.β0 , γ̂−ℓ ) − g(z, β0 , γ0 )k2 F0 (dz) = op (nℓ /n2 ). ℓ ℓ |Z−ℓ ] = 2 2 n i∈I n ℓ 37 ¯ ℓ = op (√nℓ /n). It then follows that Standard arguments then imply that for each ℓ we have ∆ # " n L p √ √ X 1X p ¯ ℓ = op ( nℓ /n) −→ g(zi , β0 , γ0 ) − ḡ(γ̂) = n ∆ n g̃(β0 ) − 0. n i=1 ℓ=1 It also follows by Assumption 1 ii) and iii) that √ n kḡ(γ̂)k ≤ √ p nC kγ̂ − γ0 kζ −→ 0. The conclusion then follows by the triangle inequality. Q.E.D. Proof of Lemma 18: Let G̃(β) = ∂g̃(β)/∂β when the derivative exists and Ĝ = ∂g(zi , β0 , γ0)/∂β. p By the law of large numbers and Assumption 2 iii), Ĝ −→ G. Also, by Assumption 2 i), ii), P iii) G̃(β̄) is well defined with probability approaching one ni=1 d(zi )/n = Op (1) by the Markov inequality, and by the triangle inequality, L 1 XX {d(zi )( β̄ − β0 G̃(β̄) − Ĝ ≤ n ℓ=1 i∈I ℓ ! n 1X ≤ d(zi ) ( β̄ − β0 n i=1 ζ′ ζ′ ′ + kγ̂ℓ − γ0 kζ )} + L X ℓ=1 ′ p kγ̂ℓ − γ0 kζ ) = Op (1)op (1) −→ 0. The conclusion then follows by the triangle inequality. Q.E.D. The proofs of Theorems 19 and 20 are standard and so we omit them. Acknowledgements Whitney Newey gratefully acknowledges support by the NSF. Helpful comments were provided by M. Cattaneo, J. Hahn, M. Jansson, Z. Liao, J. Robins, R. Moon, A. de Paula, J.M. Robin, participants in seminars at Cornell, Harvard-MIT, UCL, and USC. REFERENCES Ackerberg, D., X. Chen, and J. 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Towards a Consumer-Centric Grid: A Behavioral Perspective Walid Saad1, Arnold Glass2 , Narayan Mandayam3, and H. Vincent Poor4 1 Wireless@VT, Bradley Deparmtent of Electrical and Computer Engineering, Virginia Tech, Blacksburg, VA, USA, Email: [email protected] 2 Department of Psychology, Rutgers University, New Brunswick, NJ, USA, Email: [email protected] 3 Electrical and Computer Engineering Department, Rutgers University, New Brunswick, NJ, USA, Email: [email protected] 4 Electrical Engineering Department, Princeton University, Princeton, NJ, USA, E-mail: [email protected] arXiv:1511.01065v1 [] 1 Nov 2015 Abstract—Active consumer participation is seen as an integral part of the emerging smart grid. Examples include demandside management programs, incorporation of consumer-owned energy storage or renewable energy units, and active energy trading. However, despite the foreseen technological benefits of such consumer-centric grid features, to date, their widespread adoption in practice remains modest. To shed light on this challenge, this paper explores the potential of prospect theory, a Nobel-prize winning theory, as a decision-making framework that can help understand how risk and uncertainty can impact the decisions of smart grid consumers. After introducing the basic notions of prospect theory, several examples drawn from a number of smart grid applications are developed. These results show that a better understanding of the role of human decisionmaking within the smart grid is paramount for optimizing its operation and expediting the deployment of its various technologies. I. I NTRODUCTION The electric power grid has undergone unprecedented changes over the past few years. The traditional, hierarchical and centralized electric grid has transformed into a large-scale, decentralized, and “smart” grid [1]–[4]. Such a smart grid is expected to encompass a mix of devices, as shown in Fig. 1, that include distributed renewable energy sources, electric vehicles (EVs), and storage units that can be actively controlled and operated via a reliable, two-way communication infrastructure [1]. The effective operation of such a heterogeneous and decentralized system is expected to change the way in which energy is produced and delivered to consumers. One key byproduct of the smart grid evolution is an ability to deliver innovative energy management services to consumers [1]–[4]. Here, energy management refers to the processes using which energy is generated, managed, and delivered to consumers in the grid. For instance, demand-side management (DSM) and demand response mechanisms will be an integral part of the smart grid. The primary goal of such programs is to dynamically shape and manage the supply and demand on the grid in order to maintain a desirable load over various timescales. Indeed, the design of optimized DSM and demand response protocols and associated pricing schemes has led to significant research in this area in recent years [5]– [37]. This research is supported by the U.S. National Science Foundation under Grants CNS-1446621, ECCS-1549894, ECCS-1549900, and ECCS-1549881. Dr. Saad was a corresponding author. Fig. 1. A future smart grid with a heterogeneous mix of storage units, EVs, renewable sources, and other consumer-owned equipment. Moreover, in the smart grid, consumers will be able to individually own energy production units, such as solar panels, as well as storage devices in the form of EVs or small batteries. This can potentially transform every smart grid consumer into an independent energy production and storage source. Consequently, the possibility of energy trading between such well-equipped consumers will undoubtedly become a reality in the next few years. Indeed, many recent works, such as in [8], [38]–[66], have investigated the various challenges of such large-scale energy exchange, which include the development of optimized market mechanisms, the management of the grid operation, and the optimized exploitation of available consumer-owned storage and energy production units. Realizing this vision of a distributed, sustainable, and consumer-centric smart grid will naturally face many challenges. On the one hand, although DSM programs (and related ideas) have been theoretically shown to yield important technological benefits to the grid, their wide-spread deployment still remains insipid [67]–[73]. On the other hand, the impact of energy trading on the smart grid operation and the realistic assumption that every consumer can become a producer of energy is still not well-understood. In addition, how to maximize the amount of energy that stems from renewable sources is yet another important challenge. Last but not least, the design of efficient dynamic pricing mechanisms that go hand-in-hand with DSM and energy trading schemes is seen as a critical enabler for most of the foreseen smart grid features. To widely deploy such grid features, one important chal- lenge, among many others, is to properly incentivize consumers to actively participate in emerging grid features. For example, without effective adoption of DSM schemes by consumers, power companies will not be able to reap the technological benefits of such load-shaping mechanisms. Similarly, the willingness of consumers to own and actively utilize EVs, storage units, or even renewable sources, is an essential milestone for the deployment of a truly sustainable, consumer-centric smart grid. For example, the statistics in [71] show that installed solar generating capacity has increased from about 1000 MW in 2010 to more than 6000 MW in 2014. However, residential capacity has only increased from about 200 MW in 2010 to about 1000 MW in 2014. The reasons for this small growth are touched upon in [72] – the upfront cost of $21,000 - $25,000 is more than most homeowners have to risk on what is still an uncertain venture. Clearly, coupled with a properly designed and cost-effective ICT infrastructure, active consumer participation plays an instrumental role in facilitating the adoption of some of the smart grid technologies and features. However, somewhat remarkably, most of the existing research in this area is still based on formal mathematical constructs, such as game theory or classical optimization, which presume that consumers are objective, rational entities that are uninfluenced by real-world behavioral considerations. While such an assumption will hold in a highly-centralized, traditional power system, it will remain an important barrier that has prevented the widespread adoption of the smart grid. The primary goal of this article is to shed light on the role of consumer participation in the grid, while exposing the role of the Nobel-prize winning framework of prospect theory, in providing a mathematical basis within which to better understand how consumer behavior and realistic smart grid considerations impact the operation and efficiency of smart energy management mechanisms such as DSM, energy trading, and storage management. To this end, we first provide an overview of important energy management services in which consumer participation plays an important role. For each such service, we expose the state of the art and discuss the key assumptions and limitations. Then, we present the basics of prospect theory and discuss the motivation for applying this framework in a smart grid environment. We illustrate the benefits of prospect theory via two simple examples pertaining to energy storage management and DSM. We conclude by providing a future roadmap on how behavioral studies can play an instrumental role in future smart grid designs. The rest of this paper is organized as follows: Section II presents an overview on existing energy management literature. In Section III, we provide a tutorial on the framework of prospect theory and its motivation. Then, in Section IV, we discuss two smart grid examples. Finally, Section V draws some conclusions and future work. II. E NERGY M ANAGEMENT IN R EVIEW THE S MART G RID : A Owing to the deployment of a smart communication infrastructure and to the presence of new devices, such as storage units, smart meters, and renewable sources, the smart grid presents numerous new opportunities for energy production and distribution that were not possible in a classical grid. For instance, the possibility of deploying smart meters at consumer premises, opens the door for enabling consumers to actively manage their energy. In addition, the ability of a smart meter to communicate in real-time with a power company’s control center, provides the latter with various opportunities to actively control and monitor energy usage. Such new capabilities undoubtedly change the way in which energy is generated and distributed to consumers. Clearly, smart energy management protocols and mechanisms are needed to exploit the opportunities brought forward by this new smart grid infrastructure. Deploying efficient energy management mechanisms in future smart grid systems faces many challenges. The first such challenge is to actively utilize smart metering and consumerbased energy management systems to shape the overall grid load over time. Such load shaping is quintessential for an efficient operation of a large-scale smart grid. Enabling such demand-side management requires both increased automation and active participation by the grid consumers. Another important challenge is to properly integrate and exploit storage devices in the grid. For instance, on the one hand, a power company can make use of EVs to store energy reserves so as to regulate the grid operation. In this example, the power company will be submitting an offer for ancillary services to an independent system operator (ISO). On the other hand, a consumer-based storage unit can be used as a means to store or even sell energy back to the power company (rather than an ISO). Last but not least, an important energy management challenge is to properly decide on how to integrate and utilize consumer-based power sources, such as solar panels, within an operating smart grid system. In summary, energy management in the smart grid involves the planning and operation of energy-related production and consumption units, particulary when such units are consumerowned. Next, we summarize some of the main research topics related to energy management in the smart grid. A. Demand-side Management Demand-side management and demand response programs are arguably the most important form of energy management in the smart grid. DSM can entail a broad range of programs. These programs range from classical direct consumer load control to peak shaving programs and ancillary service provisions. Naturally, each such DSM program has its own challenges. Here, we will summarize some of these programs, while emphasizing the role of consumers. For example, in peak shaving or direct load control programs, DSM schemes typically aim at encouraging consumers to change their energy consumption habits during peak hours. In particular, such load control DSM schemes aim at providing consumers with incentives to shift their unnecessary grid load to various times during the day, so as to shape the peak hour load on the grid. For example, a simple DSM scheme can provide monetary benefits to consumers if they use delay-tolerant equipment, such as dish washers or washing machines, during the night, instead of peak hours [5]–[37]. Even if the individual appliance consumption can be small, the participation of consumers at scale, such as within a neighborhood or city, will significantly impact the smart grid. Moreover, DSM will also extend to other types of consumer-owned devices, such as storage units or renewable energy sources whose consumption and usage might be more significant than standard appliances [5]– [37]. Last but not least, consumers of DSM need not only be home users, but they can extend to industry and even small, local energy providers that work hand-in-hand with the power company. In such cases, significant gains for both consumers and power companies can be achieved if DSM is properly implemented [73]. 1) Challenges: The key challenge in DSM is to design realistic incentive mechanisms that can be used by power companies and consumers alike to manage their power consumption over time. The essence of demand-side management revolves around modeling the interactions and decision making processes of various grid players whose goals and actions are largely interdependent. For example, the change in the load of a certain consumer can lead to change in the pricing scheme used by the power company which, in turn, can lead to a change in the behavior of other consumers. This large coupling in the behavior and goals of the grid consumers has led to an abundant literature that applies the mathematical framework of game theory [74] to analyze and design efficient DSM schemes. Game theory is a mathematical framework that enables one to model the decision making processes of a number of players whose objectives are largely interdependent. The merits of a game-theoretic approach for DSM include: 1) ability to capture the heterogeneity of the devices in the grid, 2) effective integration of consumer-based decisions, 3) synergy between game-theoretic designs and the design of incentive mechanisms, and 4) low-complexity learning mechanisms that can characterize the outcome of a game and that can be practically implemented in a real-world smart grid. 2) State-of-the-art: There has been a surge in research activities related to DSM in recent years [5]–[37]. As already mentioned, the majority of these works adopts a gametheoretic approach to demand-side management. One of the earliest works in this area is [5] which presents a DSM model in which the users are able to decide on how to schedule their appliances over a given time horizon. The basic idea is simple: each consumer selects a certain schedule for its appliances, in such a way so as to minimize its overall cost; given a fixed, yet well-designed pricing scheme from the company. Using a game-theoretic model, the authors in [5] characterize the eventual operating system of the grid and show that, under the assumption that users are rational and will act strategically, significant reductions in the overall grid load can be foreseen using such a DSM scheme. The recent work in [6] extends the model of [5] by including the power company as a player in the system. In this regard, following a grid model similar to [5], the authors enable the power company to strategically decide on its pricing depending on the total power. The objective is to reduce the peak-to-average ratio (PAR) of the load demand. Using numerical simulations, the authors establish the merits of such a DSM scheme and show that noticeable reduction in the PAR can be harnessed via dynamic pricing that adapts to the users’ behavior. Another key contribution on DSM is presented in [7]. In this work, the goal is not to reduce peak hour consumption, but rather to match the supply and demand. Depending on whether there is a deficit or excess of energy, the proposed game-theoretic market model incentivizes the consumers to either increase or shed their load so as to match the supply and maintain normal grid operation. Thus, this work highlights an interesting use of DSM for regulating the overall grid operation, rather than just for reducing or shifting load over time. The work in [8] studies, using a game-theoretic and optimization framework, the ability of consumers to coordinate the way in which they defer their grid load, based on the power company’s pricing scheme. In particular, the authors observe that, when DSM protocols leave the smart meters to react independently, in an uncoordinated manner, to pricing fluctuations, new peak hours may be created thus defeating the main purpose of a DSM scheme. To this end, the authors propose a coordination mechanism between a large population of smart grid consumers. In this mechanism, instead of directly reacting to the pricing change, smart meters, acting on behalf of users, aim to adapt the deferment of loads to the changes in the price. One of the key contributions of this work is to consider such a coordination over a large number of consumers. The results, based on realistic, empirical market models from the UK, show that such a coordinated DSM approach can reduce peak hour demand while also reducing carbon emissions. In [9], an interesting game-theoretic framework was developed to answer an important question: what is the value of DSM and demand response mechanisms in the smart grid. Essentially, the system is viewed as a noncooperative, hierarchical game between a number of generation companies and the consumers. On the one hand, the generation companies are controlled by the utility operator who can determine their production level. On the other hand, the consumers are aggregated into one collective decision maker, who responds to a pricing signal sent out by the operator, to determine the overall consumption level. Using this model, it has been shown that the use of a demand response mechanism can, in some cases, be more beneficial to generation companies than to consumers. This benefit is largely dependent on how consumers respond to the pricing scheme. Therefore, this work has yet again shown that the way in which consumers behave must be properly modeled if one is to reap the benefits of a technology such as DSM. Building on those key contributions, a number of equally interesting DSM schemes have emerged more recently [10]– [37] expanding on the aforementioned works by developing more advanced models such as those that integrate additional players or other energy efficiency metrics. 3) Summary and Remarks: Clearly, existing research has established the technological benefits of DSM. Indeed, most of the existing works such as in [5], [6], [8]–[37], have shown that under fairly realistic scenarios, DSM can yield significant reduction in peak hour load and can provide an interesting means for regulating the overall operation of the grid. The nexus of these existing works remains a game-theoretic model in which various interactive scenarios between the users and one or more utility providers are modeled. The outcomes include a broad range of pricing mechanisms and load-shifting scheduling algorithms that can be implemented to optimize various energy efficiency metrics, such as peak hour load, PAR, and load regulation metrics. Undoubtedly, DSM and related ideas are likely to become an important component of the smart grid. Alas, despite this established gains of DSM, the real-world implementation of such programs (and related ideas) has remained below expectations [67]–[73]. One of the underlying reasons is that, in real life, consumers are not behaving the way they are supposed to, as assumed by many existing mathematical models. In this regard, most of these existing models rely on the assumption that players are rational and will act objectively when faced with a DSM decision. In other words, these models presume that, in the real-world, consumer behavior will follow strictly objective measures of benefits and losses, when deciding on whether or not to subscribe to a DSM scheme or when choosing on how to schedule appliance usage. However, realistically, consumers may deviate from the rational behavior to various factors. For example, on a cold winter day, consumers may be reluctant to shift their heating consumption to a later time of the day, even if such a shift can be beneficial to the grid or can bring some economic benefit to the consumers. Clearly, within the context of DSM, there is an urgent need to capture such “behavioral” factors when designing the demand response mechanisms of the future. Without a careful accounting for the behavioral side, the real-world adoption of DSM mechanisms will not live up to the expectations. B. Integration of Storage Units and Consumer-Owned Renewable Sources Beyond DSM and demand response mechanisms, energy management in the smart grid must account for the presence of a variety of new devices that are expected to be deployed in the near future. Such devices include EVs, storage units, and renewable energy sources. While renewable energy sources may be owned by energy providers or consumers, the majority of EVs and storage units are expected to be consumerowned. The presence of such new components in the power grid presents an interesting opportunity for deploying evolved energy trading mechanisms [41], [57]–[63]. In particular, the ability of consumers to store energy or possibly feed energy back into the grid, via either their storage units or their owned renewable sources, will pave the way towards a large-scale exchange of energy within the grid. For example, on the one hand, consumers with a surplus of energy may decide to send this energy back into the grid, to improve grid regulation and reap some possible monetary benefits. On the other hand, the power company may utilize EVs or other storage units as a means to store energy reserves or to regulate the grid frequency [43]–[45], [52]–[56]. Indeed, if properly managed, the charging and discharging behavior of EVs and storage units can yield significant technological and economic benefits for power companies and consumers alike [8], [38]– [66]. In a nutshell, the effective integration and exploitation of storage units and consumer-owned renewable sources will be an essential property of the future smart grid. How to efficiently exploit such devices to improve the delivery, production, and management of smart grid energy is thus an important problem that must be addressed. 1) Challenges: The challenges of integrating energy storage and renewable sources are numerous. From a power system point of view, the intermittent nature of renewable sources will require fundamentally new ways to operate energy production and generation in the grid. How to develop stochastic optimization algorithms that can adapt to this intermittency is thus a key challenge. In addition, the foreseen large-scale deployment of EVs will present an unprecedented increase in the load on the grid. Here, effective DSM mechanisms, as those discussed in the previous section, which can manage the EVs load will be needed. Nonetheless, storage units and EVs also provide an opportunity for the power grid to store any excess or mismatch in the generation and demand, so as to regulate the overall grid operation. More relevant to this article are the challenges pertaining to the use of storage units and consumer-owned renewables within energy trading markets. In particular, it is foreseen that local markets in which consumers may directly exchange energy with one another or with the grid can be set up in the future smart grid. Such markets are enabled by the presence of storage units, EVs, and consumer-owned generation sources. Important challenges here include: 1) devising economic mechanisms that incentivize consumers, power companies, and energy providers to setup such markets, 2) analyzing the impact of such localized markets on grid operation, and 3) integrating such energy trading within existing DSM mechanisms. 2) State-of-the-art: Integrating storage units and renewable energy sources has been a topic of significant interest to the smart grid community in recent years [8], [38]–[66], [75]– [84]. Beyond the works that focus primarily on the power system operation side [75]–[84], there has been a number of interesting works that investigate the usage of storage, EVs, and renewables, to shape the overall grid load and to establish energy trading markets [8], [38]–[66]. The earliest work in this area is in [38] in which a gametheoretic framework is developed to analyze how consumers equipped with storage unit can smartly decide on when to buy or store energy, in a local smart grid area. The presented scheme is essentially a modified DSM protocol which explicitly factors in the presence of storage units. The market price is assumed to be pre-determined using an auction mechanism and, thus, the work does not account for dynamic pricing. Simulation results presented in [8] show that, based on empirical data from the UK market, the use of storage at consumer premises along with game-theoretic DSM protocol can help in reducing the peak demand which also leads to reduced costs and carbon emissions. The results also analyze the benefit of storage and how it impacts the social welfare of the system thus highlighting the possible practical impact of storage unit integration. One of the most interesting works that follows in this direction is presented in [39]. In this contribution, the authors study a DSM-like scheme in which users can be endowed with storage units and renewable sources. In contrast to traditional DSM schemes in which users only decide whether or not to purchase energy from the grid, the model in [39] enables the users to decide on whether to purchase, produce, or store energy in their batteries. By expanding the decision space of the users, it is shown, using a game-theoretic approach, that a smart exploration of the storage and energy production options can reduce the overall aggregate load on the grid while also providing monetary savings to end-users, under the assumption of rational decision making. Such a study thus motivates the penetration of consumer-owned storage and energy production units. The effective integration of EVs into a smart grid system is studied in [40] using a game-theoretic framework that models the interactions between the grid operator and the EVs. The primary goal is to analyze how EVs can provide ancillary services to the grid, once a proper market model between EVs and the grid is setup. The basic idea is to use a smart pricing policy to exploit the EVs for regulating the grid frequency. The idea is to study how EVs (and their owners) can decide on whether to charge, discharge, or remain idle, in a way to optimize the grid frequency regulation while benefiting both consumers and the grid operator. On the one hand, using such a scheme consumers can obtain additional income while, on the other hand, the grid can achieve the required frequency regulation command signal. The impact of energy trading between owners of EVs is further analyzed in our earlier work in [41]. In this work, a local market in which EV owners can decide on whether or not to sell a portion of their stored energy to the smart grid is studied. Using an auction and a game-theoretic model, we have shown that, when EV owners act strategically, they are able to reap significant benefits from selling their surplus of stored energy to potential buyers in the smart grid. These benefits are reflected in terms of revenues that can be viewed as either direct monetary gains or as coupons or other offers provided from the grid owner to active participants. The impact of distributed renewable energy sources on local energy trading markets is analyzed in [42]. The authors essentially propose the use of an aggregator of distributed energy resources allowing the smart grid to engage in an open energy exchange market. The developed mechanism tightly integrates classical DSM ideas with the use of an active management scheme at the end-users side to allow a better utilization of the renewable energy sources. Overall, the results show that a smart exploration of possibly consumer-owned energy sources can lead to a sustainable source of energy and a reduction in the consumers’ energy consumption cost. Beyond the aforementioned contributions, the exploitation of storage and renewable sources at consumer premises has been studied in a broad range of literature [43]–[66]. These works mainly develop variants of the discussed energy trading mechanisms and establish clearly that the use of mathematical optimization frameworks such as game theory to manage the way in which storage devices, EVs, and consumer-owned energy sources are integrated in the smart grid can bring in substantial technological, economic, and environmental benefits to the smart grid players. 3) Summary and Remarks: The use of energy trading between consumer-owned devices will indisputably be an important feature of the smart grid. As demonstrated in the abundant literature, the associated gains, both from a technical (energy reduction, sustainable generation) and an economic (reduced costs on consumers and providers) point of view, are substantial. Yet, despite these established results, beyond some small deployments of EVs and renewable energy sources in Europe and some areas of the United States [85]–[92], the large-scale introduction of such consumer-owned devices remains modest. Similar to the DSM case, one of the primary limitations of existing models is that they often do not explicitly factor in realistic risk considerations of both consumers and power companies. For instance, even though an open, energy trading market can yield economic and technological benefits, power companies may remain risk averse and continue to rely on traditional, largely controlled markets. Similarly, despite the prospective economic savings and environmental benefits of owning renewable sources or EVs, consumers may still be reluctant to change from their current, effective technologies. Therefore, when analyzing energy trading, one must explicitly factor in such risk considerations and their impact on the overall operation of smart energy management mechanisms. III. P ROSPECT T HEORY FOR THE S MART G RID A. Introduction and Motivation As demonstrated in Section II, the existing literature on smart energy management in the smart grid has established significant technological, economic, and environmental benefits for features such as DSM and energy trading. Yet, the real-world deployment of such mechanisms remains largely below expectations. One of the hurdles facing the real-world implementation of the developed DSM and energy management mechanisms, is the lack of a mathematical and empirical framework that can capture the realistic behavioral patterns of consumers and power companies. Indeed, most existing works [5]–[66], [75]–[84] still assume that consumers and power companies are rational and will abide by the objective decision making rules that are derived via frameworks such as game theory or optimization. However, in practice, empirical studies have shown that, in uncertain and risky situations, human players may not act in accordance with the rational behavior established by decision making frameworks such as game theory [93]–[97]. Given that most foreseen smart grid features are consumer-centric, the “human” factor will undoubtedly play an instrumental role in the success or failure of advanced smart grid features. Thus, uncertainty and risk factors must be properly modeled in any DSM or energy management scheme. Examples of risk in the smart grid include the continuous reliance of operators on traditional markets and the interdependence of the decisions between consumers. In terms of uncertainty, when dealing with an energy management scheme, consumers are faced with uncertain outcomes due to a lack of transparency in explaining the rules of dynamic pricing or due to the presence of stochastic elements such as stochastic generation or uncertain presence or absence of EVs and energy storage units. Thus, expediting the smart grid adoption requires new approaches for analyzing the often irrational and nonconforming nature of the energy management decisions of human players under such risk and uncertainty. Such decisionmaking factors that deviate from the objective, rational behavior assumed in existing works [5], [6], [8]–[66], [75]– [84] can be analyzed by the Nobel-prize winning framework prospect theory (PT) [93]–[95], [98]. Originally conceived for modeling decisions during monetary transactions such as lottery outcomes, PT has made its way into many applications [93]–[100], due to the universal applicability of its concepts. In essence, PT provides one with mathematical tools to understand how decision making, in real life, can deviate from the tenets of expected utility theory (EUT), a conventional game-theoretic notion which is guided strictly by objective notions of gains and losses, player rationality, conformity to pre-determined decision making rules that are unaffected by real-life perceptions of benefits and risk. Illustrative Example: Essentially, PT notions have been developed to understand how consumers, when faced with uncertainty of outcome and risky decisions, will behave in real-life. Suppose that an efficient energy management system is constructed for individual home owners to both buy and sell power on the grid and a dynamic pricing DSM mechanism is available to shift consumption to non-peak periods. Furthermore, suppose that it has been proven that under PT as well as conventional game theory, stable prices can be found, so that the smart grid could ultimately result in more efficient power consumption. Under rational analysis, one might believe when these conditions were satisfied, offering the opportunity to buy and sell power to the public would result in widespread participation and an optimal pricing equilibrium would soon be reached. However, an important implication of PT is that these conditions are insufficient to guarantee such a beneficial result. One important implication of PT is that the preferred choice between a pair of uncertain alternatives is not only determined by the values of the two alternatives but also by how the choice is stated. Consider the following example, which is unnatural only in that the alternatives are designed to have equal value, so that a preference is clearly determined by the statement of the choice. A power company wishes to entice its consumers to abandon buying power at a fixed rate and instead join a system where they buy and sell power at variable rates. Here are two ways the alternatives may be presented in a letter to a consumer: • The Gain Scenario: Your average monthly utility bill is now $450 a month. Under our new smart system, your bill will show a debit of $500 a month. In addition you may choose between: a) A 50% chance of a credit of $100 if you join the smart DSM scheme, or b) A 100% chance of a credit of $50 that will keep your bill the same. • The Loss Scenario: Your average monthly utility bill is now $450 a month. Under our new smart system, your bill will show a credit of $400 a month. In addition, you may choose between: c) A 50% chance of a bill for $100 if you join the smart DSM scheme, or d) A 100% chance of a bill for $50 that will keep your bill the same. In fact, the Gain and Loss scenarios describe the identical alternatives in different words. Alternatives a) and c) are identical and alternatives b) and d) are identical. Nevertheless, based on theoretical and empirical foundations, PT predicts that more people will prefer alternative b) to alternative a) B. Basics of Prospect Theory Prospect theory encompasses a broad range of techniques and tools to account for realistic consumer behavior during decision making processes [94]–[98], [101], [102]. The basic underlying idea is that decision makers, in real-life, will have subjective perceptions of losses, gains, and their competitive environment. For example, instead of viewing each others’ actions (e.g., load shifting schedules) objectively as in classical game theory, players could have different subjective assessments about each others behavior, which, in turn can lead to unexpected, irrational decisions. For example, in DSM, even though rational behavior dictates that consumers follow the load shifting mechanisms of the power company, some consumers may turn on certain appliances at unexpected times, since they are unsure on whether participation level is high enough to obtain economic benefits which will hinder the performance of the DSM scheme. The large spread of such unconventional actions can thus be disruptive to any energy management scheme. In such situations, PT provides Weighted probability (subjective perception) because a certain gain is preferred to a 50% chance at a double-gain but will also prefer alternative c) to alternative d) because a 50% chance of a loss is preferred to a certain, albeit smaller, loss. This prediction has been confirmed in [93] and [101]. The point of this example is not just that the level of participation in smart grid services depends on how it is presented to the public. The point is that important behavioral factors outside of the technical specifications of the smart grid will determine the choices of participants and giving them the opportunity to perform optimally does not guarantee that they will. In other words, people cannot be counted on to always choose optimally among alternatives if merely stating the alternatives differently influences their choices. This holds true even if such alternatives, as discussed in Section II, have immense technological and environmental benefits. Indeed, in [102], Kahneman suggests that people behave non-optimally when buying and selling stocks, selling rising stocks too soon to lock in gains and hanging on to losing stocks too long to resist locking in a loss. If people behave non-optimally in the purchase and sale of securities, the default assumption is that they will perform in the same non-optimal manner in the purchase and sale of power and commodities, especially when people are already familiar with the incumbent pricing and energy management mechanism. The obvious solution to the problem of human behavior is to use prospect-theoretic notions to refine existing gametheoretic mechanisms and guide the way in which optimal strategic decisions are derived as well as to improve the presentation of information to buyers and sellers in the grid to encourage optimal behavior in DSM and energy trading. To provide further insights on the mathematical machinery underlying PT, in the next subsection, we provide an introduction to the basics of the framework. 1 0.8 0.6 Subjective α=0.4 Subjective α=0.25 Subjective α=0.1 Subjective α=0.7 Objective/rational 0.4 0.2 0 0 0.2 0.4 0.6 Objective probability 0.8 1 Fig. 2. Illustration of the prospect-theoretic weighting effect: how objective probabilities are viewed subjectively by human participants. The parameter α determines how far the behavior is from the fully rational case. solid analytical tools that directly address how these choices are framed and evaluated, given the subjective observation of players in the decision-making process. 1) Subjective Actions – The Weighting Effect: The first important PT notion is the so-called weighting effect. In particular, in PT [93]–[98], it is observed that in real-life decision-making, people tend to subjectively weight uncertain outcomes. In particular, in energy management mechanisms, the frequency with which a consumer chooses a certain strategy, say a certain schedule of appliances or a certain storage pattern, depends on how other consumers make their own choices. The dependence stems from many factors. For example, in dynamic pricing schemes, the actual price announced by a power company depends on the entire load of the consumers. Therefore, the decision of a consumer, will subsequently depend on the decision of others. Indeed, when faced with a given smart grid scenario, consumers may act differently over time due to the interdependence of their actions and its unpredictability, and, thus, a probabilistic model for decision making is suitable to capture this uncertainty. Such uncertainty can stem not only from the individual decisions of consumers but also from other smart grid factors (e.g., uncertainty of renewable energy). In classical game-theoretic smart grid schemes [5], [6], [8]– [66], [75]–[84], consumer interdependence is captured via the notions of expected utility theory in which a consumer computes an expected value of its achieved gains or losses, under the observation of an objective probability of choice by other consumers. In contrast, using the weighting effect, PT allows one to capture each consumer’s subjective evaluation on the probabilistic strategies of its opponents. Thus, under PT, instead of objectively observing the information given by the other players and computing a classical expected value for the utility, each consumer perceives a weighted version of its observation on the other actions. The weighting is used to express a “distorted” view that a given consumer or player can have on the actions of others. PT studies have shown that most people overweight low probability outcomes and underweight high probability outcomes. To illustrate the weighting effect, in Fig. 2, we show an Framing effect example 10 8 EUT raw utility PT framing utility 6 4 Utility value example of a weighting function, known as the Prelec function [103], which maps an objective utility into a subjective utility. The mapping is controlled by a parameter α which quantifies the level of subjectivity in the observation. For α = 1, we have the fully rational case, while for α close to 0 we get the fully irrational case. Within a smart grid setting, such a weighting can have a cascading effect on the way in which an energy management scheme works. For example, in a DSM context, a highly irrational consumer will have a largely distorted view on how other consumers behave. In turn, this consumer will become more risk averse or more risk seeking, depending on how the opponents actions impact the dynamic pricing mechanism. As a result, this consumer will not follow the actions recommended by classical, rational mechanisms, but, instead will take an unexpected action which, in turn, will yield unexpected DSM results. Indeed, how to model such a weighting effect and how to integrate it into realistic energy management mechanisms is an important topic for research. In addition, how to design weighting functions that are tailored towards the smart grid and that can work in realistic power system setting is a key challenge. In Section IV, we will discuss with specific examples how weighting modifies the results of energy trading and management protocols. 2) Subjective Perceptions of Utility Functions – The Framing Effect: Another important idea brought forward by PT is the notion of utility framing. In engineering designs, one often defines mathematically rigorous objective (utility) functions that are used to optimize a certain metric of the system. For example, when dealing with an optimal energy generation problem, a smart grid system must find the maximum energy output that can meet or match the demand. In such a case, it is sound to assume that the function that must be optimized is based on an objective metric, the energy in this case. However, when dealing with smart energy management mechanisms having human players, the idea of an objective metric for evaluating utility functions might not be a reasonable assumption. For instance, each individual has a different perception on the economic gains from a certain DSM scheme. For example, a saving of $10 per month may not seem like a significant gain for a relatively wealthy consumer. Instead, a poor consumer might view this amount as a highly significant reduction. Clearly, the objective measure of $10, can be viewed differently by different consumers. In PT, such subjective perceptions of utility functions are captured via the idea of framing or reference points. In essence, each individual frames its gains or losses with respect to a possibly different reference point. Back to the aforementioned example, the wealthy consumer will frame the $10 with respect to its initial wealth which could be close to millions and, thus, this consumer views the $10 as insignificant. In contrast, the poor consumer might have a wealth close to 0 and, thus, when framing the $10 with respect to this reference point, the gains are viewed as significant. One 2 0 −2 −4 PT value is: Concave in gains Convex in losses −6 −8 −10 −10 −8 −6 −4 −2 0 2 Raw utility 4 6 8 10 Fig. 3. Illustration of the prospect-theoretic framing effect: how objective utilities are viewed subjectively by human participants. The utility function value changes depending on a certain reference point that highlights the individual perceptions of gains and losses. popular way to capture such framing effects is by observing that losses loom larger than gains, and, thus, PT provides one transformation that maps objective utility functions into socalled subjective value functions - concave in gains, convex in losses - over the possible outcomes. These gains and losses are measured with respect to a reference point that need not be 0 and that may be different between players. An illustrative example is shown in Fig. 3 for one typical PT value function from [93] assuming a zero reference point for gains/losses. Naturally, as consumers change the way in which they compute their utilities, their overall decision making processes will deviate from the conventional, rational thought. Indeed, when applying PT ideas to game-theoretic settings such as in [98], it is shown that the objective results do not hold. For example, in some cases, it is shown that the choice of a reference point can impact whether or not a certain game has an equilibrium solution or not. Clearly, when one decision maker changes the way in which it evaluates its objective function, the overall operation of any optimization mechanisms will be significantly affected. In the smart grid, we can envision many situations in which to incorporate the framing effects. These situations need not be purely economical. For example, during winter, consumers may perceive less prospective gain from turning off high-capacity loads (such as heaters) at night than during day time. How this “frame of reference” transforms the utility will fundamentally change the outcome of an energy management mechanism that is based on classical objective notions. Moreover, in the smart grid, such reference points and framing effects may change over time, space, and even demographics. Clearly, properly designing and developing framing notions in smart grid DSMs is an important direction that must be investigated to better understand its impact on energy management and trading mechanisms. Having defined the two key effects of PT, in the next sec- EUT Player1 EUT Player2 PT Player1 PT Player2 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.02 0.04 0.06 Unit selling price b ($/kWh) 0.08 0.1 Fig. 4. Impact of the weighting effect on consumer behavior in a two-player charginge/discharging setting. tion, we discuss, in detail, two energy management scenarios to highlight, as an example, the impact of weighting on smart grid protocols. IV. P ROSPECT-T HEORETIC S MART G RID A PPLICATIONS A. Example 1: Charging and Discharging of ConsumerOwned Energy Storage To show the impact of prospect-theoretic considerations in smart grid design, we first study a model in which consumers are equipped with storage units and must decide on how to manage the charging and discharging of their storage units, depending on the network state and the pricing incentives. This model is based on our recent work in [104]. In particular, we consider a grid consisting of multiple consumers who own storage units. For illustration purposes, we assume the case in which only two consumers are “active participants” while all other consumers constitute a passive load on the grid. Each consumer has a storage unit which holds a certain initial amount of energy stored. The power company offers these active consumers the option to either charge their storage unit and, thus, act as a load on the grid, or, instead to actively feed back and sell energy to the grid. Note that, any action taken by either of the two consumers affects both the power system as it impacts the overall needed generation as well as the prices set by the utility company. The choices of both consumers are also coupled, since the choice of acting as load or source, will impact the overall generation and distribution of energy in the grid. In this setting, we assume that the consumers need to make a choice between charging or discharging while optimizing a utility function that captures two properties: a) the economic and technical benefits of storing or selling energy and b) the power system regulatory penalties. Indeed, although the power company allows the two active consumers to individually manage their storage units, it still requires the generation to remain within desired operating conditions which are measured based on an initial point. Power company revenues from two consumers ($) Probability of buying (acting as load) 1 3.5 EUT PT α=0.25 PT α=0.45 PT α=0.65 3 2.5 2 1.5 1 0.5 0 0.02 0.04 0.06 Unit selling price b ($/kWh) 0.08 0.1 Fig. 5. Impact of the weighting effect on the revenues of the power company. We formulate and investigate this setting using a PT-based, classical noncooperative game and we study the equilibrium solution of the game. The equilibrium is essentially a point of the system in which neither of the two consumers can improve its utility by changing the frequency with which it chooses to charge or discharge its storage unit. We analyze the results under both classical EUT and PT considerations. For PT, we first consider the weighting effect: each consumer views a subjective observation on the charging/discharging behavior of its opponent in accordance with a Prelec weighting function such as in Fig. 2. We use a numerical example to show the impact of PT considerations on the operation of the system. We consider a standard 4-bus power system with 2 active consumers. The loads and surpluses of active consumer 1 are, respectively, 20 kWh and 10 kWh, while those of consumer 2 are, respectively, 15 kWh and 5 kWh. Fig. 4 shows the impact of the unit selling price b that is used by the two consumers when discharging energy to the grid. This price is assumed to be equal for both consumers. Clearly, as b increases, both consumers have more incentive to sell than to buy, as the gains start outweighing the regulation penalty. More interestingly, Fig. 4 shows that, for both consumers, PT behavior significantly differs from the rational EUT case. For consumer 2, below 0.07$ per kWh, the probability of buying energy at the PT equilibrium is much higher than under EUT. This implies that for low gains and high risks, the consumer follows a conservative, risk-averse strategy under PT and is less interested in reaping the gains of selling energy compared to EUT. However, as the unit selling price crosses the threshold, the probability that consumer 2 acts as load under PT becomes much smaller than EUT. Thus, once the selling benefits are significant (risks decrease), PT predicts that consumer 2 will start selling more aggressively than in EUT. Analogous behavior is seen for consumer 1 with threshold 0.045$ per kWh. Fig. 5 shows the total power company revenues, collected from the two consumers when they charge their storage unit, as the unit selling price increases. Fig. 5 shows that, as b in- EUT PT (γ=2) PT (γ=1.5) PT (γ=1) 0 30 EUT PT (α = 0.25) The expected utility ($) Total average grid load from the two consumers (kWh) 2 35 25 20 15 −4 −6 −8 −10 10 −12 −2.5 5 0 0.03 −2 0.04 0.05 0.06 0.07 0.08 Unit price c ($/kWh) Fig. 6. Expected grid load when the consumers actively participate with their storage devices under rational EUT and irrational PT behavior. creases, the total revenues decrease, as the consumers begin to sell more and buy less. Note that, here, the power company’s revenues pertain to only those revenues that are collected from the two consumers. This does not include any additional sources of revenues that the power company might collect (e.g., taking a percentage on the profits of the consumers). Clearly, deviations from EUT can have major impacts on energy management in a smart grid setting. Consider the case in which the Prelec rationality factor is set to α = 0.25. When b is below 0.06$ per kWh, under PT, the total revenue is much higher than predicted by EUT. In contrast, if consumers set prices greater than 0.06$ per kWh, PT predicts that the revenues will be much smaller than in EUT. It is thus more beneficial for the company to regulate the consumers’ unit selling price to be below 0.06$ per kWh. Fig. 5 also shows that when the company adopts EUT to regulate the consumers’ selling price, it can lose revenues due to real-life consumer behavior. Fig. 5 also shows that, as α increases, the consumers behave more in line with EUT. However, even for a relatively high value, α = 0.65, the company revenues resulting from PT still yield non-negligible deviations from EUT. We further analyze how consumer behavior impacts grid operation by showing the average expected load on the grid in Fig. 6, as the company varies its minimum price 1 . Fig. 6 shows that the expected load on the system will significantly change between PT and EUT. For PT, when the unit price is small, consumers are less interested in selling their stored energy. However, as the price crosses a threshold, the consumers will start selling more aggressively, rendering the average load smaller than expected. Fig. 6 provides guidelines for realistic DSM with storage. For example, assume the company wants to increase its price to drive consumers to sell more and reduce their load to about 10 kWh. Based on classical EUT-based schemes, the company has to increase the price to 0.078$ per kWh. In real life, because consumers behave subjectively under risk, the power company can increase its unit price to 1 This pertains to the rates that the power company will use to directly charge its consumers. −2 −1.5 −1 −0.5 0 0.5 1 The reference point u0 1.5 2 2.5 Fig. 7. The total utility under both EUT and PT as the reference point u0 varies. only 0.06$ per kWh and obtain the desired load reduction. Also, if the power company wants to reduce its price to sustain a load of 23 kWh from the two consumers, based on EUT, it must offer a price of 0.035$ per kWh. In contrast, PT shows that 0.047$ per kWh will achieve the same impact yet yield more profits. Next, we consider the same example in the presence of framing effects, based on our work in [105]. Here, we assume that both consumers frame their utility with respect to a given reference point that reflects how these consumers evaluate the economic gains or losses from charging or discharging their storage unit. For incorporating framing, we adopted the classical model of [98] in which, under framing, the utility function becomes concave in gains and convex in losses, as losses loom larger than gains. To assess the impact of framing, in simulations, the reference point is chosen to coincide with the case in which consumers discharge/sell energy at the same price that is announced by the power company. In Fig. 7, we show the total expected utility (sum for both consumers) under both EUT and PT, as the reference point varies. In this figure, we choose the same reference point for both consumers and we use typical values for γ which is a parameter that represents the loss aversion, i.e., how a consumer values its losses versus its gains. First, we can see that, the expected PT utility will decrease when the reference point value increases. In essence, the reference point is subtracted from the EUT utility to determine the exact values of gains and losses. For a high reference point (i.e., electricity price), PT consumers will value their stored energy more than in cases in which the reference point is smaller. Thus, using a same selling/discharging price under EUT and PT, the payoff obtained by consumers under PT is smaller than under EUT, due to the fact that, in PT, the reference point reduces the gains from selling energy. Second, this figure also shows that, the framing aversion parameter, i.e., γ, would have different impacts on the PT utility. In particular, when γ increases from 1 to 2, the losses viewed by PT consumers will increase. Thus, with an increasing γ, a PT consumer will start valuing its gains less than in the EUT case, which leads to increasing its conservative, charging strategy. Additional results on framing can be found in [105]. In summary, ignoring consumer behavior in storage-based energy management can lead to unexpected results as shown here for a basic setting. These results can be undesirable from both an economic and technological perspective. Therefore, building on the presented model, one can design more elaborate and realistic storage management mechanisms that account for PT-based notions of subjective perceptions. In addition, the power company can utilize these results to properly shape its pricing schemes. B. Example 2: Demand-side Management under ProspectTheoretic Considerations 2 The reader interested in the mathematical formulation is referred to our work in [106]. Fig. 8. The expected nonparticipating load for the 6 consumer game under both EUT and PT over 24 hours, when consumers have different values of α. The nonparticipating load profile at 19:00 34 The expected nonparticipating load (kWh) Another important application for PT is classical demandside management models. Here, we consider a grid in which consumers are given the opportunity to decide on whether or not to participate in DSM. The DSM scheme considered is one in which the participating consumer would shift its load over time, in order to reduce the overall peak hour load. The actual DSM process is in line with classical game-theoretic settings such as those in [5]. However, in our model, it is assumed that consumers have also a choice of not participating in the DSM at all. In addition, consumers can choose the time starting which they will begin their participation. The decisions of the consumers are driven by the goal of minimizing the overall electricity bill while maintaining their desired load to operate their required appliances2 . One important feature of the considered model is that every load shift by a given consumer will automatically impact the way in which prices are set by the power company. Thus, this interdependence in decision making will naturally warrant a game-theoretic approach to modeling the decision making. In essence, we have a model in which every consumer can decide on the time at which it starts to participate in DSM. Alternatively, the consumer may decide not to participate at all. We can then analyze the frequency with which a consumer will participate or not and we analyze the impact of this participation on the grid by deriving equilibrium conditions [106]. This analysis is done for both the rational and irrational case. For PT, we consider mainly a weighting effect. To gain more insights on the impact of weighting on DSM participation, we consider a numerical validation using a realistic load profile in [107] which represents consumers’ initial demands during Spring 2013, from the Miami International Airport. In these numerical examples, each consumer can choose a starting time to participate in DSM from the time period between 18:00 and 20:00. Alternatively, the consumer can decide not to participate. In Fig. 8, we show the expected nonparticipating load profile using different values for the Prelec rationality parameters α. In this example, each consumer has a different EUT PT Actual demand 32 30 28 26 24 22 20 18 0.1 0.2 0.3 0.4 0.5 α 0.6 0.7 0.8 0.9 Fig. 9. The impact of the rationality parameter α on the expected nonparticipating load of all consumers at 19:00. subjective perception on other consumers and, thus, has a different rationality parameter. In particular, we choose α = [0.5 0.5 0.2 0.1 0.1 0.1] for the 6 considered consumers. This implies that consumer 1 is more rational than consumers 36 while consumers 4-6 are the least rational. In this figure, we can see that, when some consumers have a very irrational observation on their opponents, the PT nonparticipating load between 21:00 and 23:00 will be higher for PT than EUT. This implies that, in realty, if some consumers deviate significantly from their rational strategies (for example, a consumer decides not to assist the power company in load shifting despite the economic benefit), the power company will not be able to shift the total load predicted by the rational, objective model. Thus, this simple, yet insightful example shows that one must better understand how consumers behave (here reflected by the rationality parameter) to better design the dynamic pricing and DSM scheme. We further analyze the impact of the consumer rationality on DSM by showing, in Fig. 9, the expected nonparticipating load at a chosen time of the day which is here selected to be 19:00 for illustrative purposes. Here, it is assumed that all consumers have a similar level of rationality. In Fig. 9, we observe that, under EUT, the expected nonparticipating load is 65.7% of the total load. In contrast, under PT, the nonparticipating load is less than EUT when α > 0.56, i.e., when consumers are fairly rational. Thus, the power company can shift more load in practice, compared to an EUT scheme, if the consumers are all of equal rationality level when α > 0.56. Clearly, there exists a rationality threshold, such that, if α is greater (smaller) than the threshold, PT consumers will have lower (higher) nonparticipating loads than EUT cases. A large value of α, which maps to a small deviation from EUT, yields an increased competition thus raising the costs to the consumers. Consequently, the consumers will become risk seeking and more apt to shift their loads and decrease their payments. Thus, the increasing PT costs will force the majority to shift more loads, compared to EUT. In contrast, a relatively small rationality parameter or a large deviation from EUT, will lead to highly irrational behavior from the consumers which will lead to increasingly high competition and decreasing participation, as consumers become extremely risk averse and unwilling to participate in the DSM process. From Fig. 9, we can infer that one of the reasons for which DSM schemes might have not been adopted widely in practice is due to a severely irrational behavior observed from the consumers. Indeed, as per Fig. 9, one can see that a small deviation from EUT (slight irrationality) may in fact be beneficial for the power company as it increases consumers’ participation. In contrast, a significant deviation from EUT will inevitably lead to highly risk averse behavior which will prevent most consumers from participating; thus yielding detrimental results for the grid and preventing the power company from reaping the benefits of DSM. Through this simple, yet realistic DSM example, we are able to see that by only considering the weighting effect of PT, the results of DSM can significantly change. This is due to the fact that the PT model allows to better capture the way in which consumers behave in practice. Consequently, this motivates a deeper investigation of the role of human decision making in practical DSM mechanisms. C. Choice of PT Parameters In this section, we have brought forward several key results that show how realistic consumer behavior can impact smart grid energy management. However, in these models for assessing the rationality (e.g., the parameter α) or risk aversion of the consumers, we have adopted PT models of risk that were conceived in the economics literature such as in [93]–[98], [103]. Naturally, to understand whether those models map directly into the smart grid, there is a need to run analogous behavioral experiments, with real-world smart grid consumers, to generate new empirical models for PT that can be used to further enhance the results of this section. Such experiments can be based on both qualitative surveys and on real-world simulations in which grid consumers (e.g., homeowners or factories) are solicited to participate in simulated experiments on grid scenarios that pertain to DSM, storage, or other consumer-centric features. Such experiments can mimic the gain and loss scenarios presented in Section III-A. Using such behavioral experiments, we can refine the choice of the various PT parameters and we can generate more advanced models and results. V. C ONCLUSIONS AND F UTURE O UTLOOK Realizing the vision of a smart, consumer-centric grid is without any doubt strongly dependent on gathering a better understanding on the impact and role of consumer behavior in energy management processes such as demand-side management, demand response, or energy trading. In this article, we have shown that the use of prospect theory, a powerful framework from operations research and psychology, can provide the first step towards better understanding the impact of consumer behavior on smart grid operation. Indeed, our preliminary investigations have shown that consumer-related deviations from conventional, rational game-theoretic energy management mechanisms can be one of the primary reasons behind the modest adoption of such mechanisms in practical smart grid systems. Nonetheless, in this article, we have only scratched the surface of this emerging area in smart grid research. Indeed, the study of consumer behavior in the smart grid requires significant advances to frameworks such as PT. Many future directions can be envisioned. For example, our results so far have solely relied on the analysis of the weighting effect. However, we anticipate that the use of both framing and weighting can provide deeper insights into how DSM and energy management can operate in the smart grid. Indeed, the fact that smart grid consumers will have time-dependent reference points while measuring their utility functions provides a very interesting and promising research direction. In addition, our study thus far has focused primarily on economic-oriented models, in which the impact on the power system is restricted to load management. Instead, one can envision the use of PT-based behavioral model to better understand how the overall regulation of the power system operation can be modified due to the uncertainty and risk introduced by consumer-based decision making. Such studies can also be extended to explicitly account for communication and security considerations, both of which can involve enduser decisions. Another important direction for future work is to explicitly account for renewable energy sources. In fact, renewable sources will introduce two types of uncertainty: a) uncertainty due to consumer decisions, as captured in the models of this paper and b) uncertainty due to nature and other environmental factors that affect renewable generation. Here, it is of interest to apply PT models to capture both types of uncertainty. Some early works on PT such as in [98] have shown that when both weighting effects and utility uncertainty are considered, one can expect significant deviation from conventional rational results. How such deviations can be applied in a smart grid context remains an open problem. Moreover, the recent surge in the application of big data analytics in various smart grid scenarios will provide an important avenue to explore the differences between EUT and PT. These data that are being constantly collected can, in the future, provide an important source for corroborating the intuition provided by PT while also providing important information to derive more realistic PT models. Last but not least, as mentioned previously, one important challenge is the lack of any large-scale data on how buyers and sellers will in reality behave in the still speculative smart grid market. Though PT provides broad hints about the factors that may affect choices, the tests of it have been so far restricted to basic models such as those presented here, which are still somewhat distant from the context of a large-scale practical smart grid to be determinative. In other areas, the experiments confirming PT have overwhelmingly been single session experiments in which naive participants make choices in speculative scenarios of no consequence to themselves. This is different from regular participation in a smart grid in which their choices have direct financial consequences on themselves. People have the ability to learn from experience and this is known to affect their choices in some contexts. For example, the endowment effect is that amateur collectors will not sell an item they already own for the price they would be willing to pay for it [108]. However, professional merchants do not show an endowment effect [109]. 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arXiv:1708.06090v1 [] 21 Aug 2017 THE STRONG REES PROPERTY OF POWERS OF THE MAXIMAL IDEAL AND TAKAHASHI-DAO’S QUESTION TONY J.PUTHENPURAKAL, KEI-ICHI WATANABE, KEN-ICHI YOSHIDA Dedicated to Craig Huneke on the occasion of his 65th Birthday. Abstract. In this paper, we introduce the notion of the strong Rees property (SRP) for m-primary ideals of a Noetherian local ring and prove that any power of the maximal ideal m has its property if the associated graded ring G of m satisfies depth G ≥ 2. As its application, we characterize two-dimensional excellent normal local domains so that m is a pg -ideal. Finally we ask what m-primary ideals have SRP and state a conjecture which characterizes the case when mn are the only ideals which have SRP. 1. Introduction Let (A, m) be a Noetherian local ring with d = dim A ≥ 1 and I an m-primary ideal of A. The notion of m-full ideals was introduced by D. Rees and J. Watanabe ([22]) and they proved the “Rees property” for m-full ideals, namely, if I is m-full ideal and J is an ideal containing I, then µ(J) ≤ µ(I), where µ(I) = ℓA (I/mI) is the minimal number of generators of I. Also, they proved that integrally closed ideals are m-full if A is normal. fn , the Ratliff-Rush closure of mn , is m-full ([1, Suppose depth A > 0. Then m Proposition 2.2]). Thus mn is m-full for sufficiently large n. Sometimes we need stronger property for µ(I) and we will call it “Strong Rees property” (SRP for short). Definition (Strong Rees Property). Let I be an m-primary ideal of A. Then we say that I satisfies the strong Rees property if for every ideal J ) I, we have µ(J) < µ(I). So it will be natural to ask the following questions. Question 1.1. Let (A, m) be a normal local ring. Then (1) Does mn has the strong Rees property for every n ≥ 1? (2) If I has the strong Rees property, then is I = mn for some n ≥ 1? Date: August 22, 2017. 2000 Mathematics Subject Classification. Primary 13A30; Secondary 13H15, 13B22, 14B05. 1 2 TONY J.PUTHENPURAKAL, KEI-ICHI WATANABE, KEN-ICHI YOSHIDA Actually, both questions are not true in general. But we can show that (1) in the Question 1.1 holds under suitable mild condition. Also, we will give an example of two-dimensional normal local rings where m2 does not satisfy the strong Rees property. As for (2) of Question 1.1, we will discuss it in Section 6. Assume that I is an m-primary ideal. Then the multiplicity (resp. the minimal number of system of generators) of I is denoted by e(I) (resp. µ(I)). The Loewy length ℓℓ(I) is defined by ℓℓ(I) = min{r ∈ Z≥0 | mr ⊂ I} Notice that the notion of Loewy length of an Artinian ring measures the nilpotency of the maximal ideal. It is natural to ask if e(I) is bounded by the product of µ(I) and ℓℓ(I) after adjusting some error terms. The origin of this work was a discussion of second and third authors with Hailong Dao. He presented an inequality between µ(I), the multiplicity e(I) of I and Loewy length ℓℓ(I). Question 5.1 ([2]). Let (A, m) be a d-dimensional Cohen-Macaulay local ring. Let I ⊂ A be an m-primary (integrally closed) ideal. When does the inequality (d − 1)!(µ(I) − d + 1) · ℓℓ(I) ≥ e(I) hold true? Dao and Smirnov [2] proved that the Question 5.1 holds true if A is a twodimensional analytically unramified and the maximal ideal m is a pg -ideal (or, equivalently, the Rees algebra R(m) is normal and Cohen-Macaulay). We are interested in the converse of the Question 5.1 in the case of d = 2. Here, since e(I) does not change after taking integral closure, it is natural to assume that I is integrally closed. Namely, our Question is Question 1.2. Assume that (A, m) is a two-dimensional excellent normal local domain. If an inequality (µ(I) − 1) · ℓℓ(I) ≥ e(I) holds for any m-primary integrally closed ideal I, then is m a pg -ideal? It turns out that this question is related to the “strong Rees property” of powers of the maximal ideal. The main result in this paper is the following theorem. Theorem 3.2. Let (A, m) be a Noetherian local ring. Assume that depth A ≥ 2 1 and HM (G) has finite length, where G = G(m) = ⊕n≥0 mn /mn+1 . If mℓ is RatliffRush closed, then mℓ has the strong Rees property. By Remark 2.5, any power mℓ is Ratliff-Rush closed whenever depth G ≥ 1. Hence we have the following corollary. THE STRONG REES PROPERTY OF POWERS OF THE MAXIMAL IDEAL 3 Corollary 3.5. If depth G ≥ 2, then mℓ has the strong Rees property for every ℓ ≥ 1. In general, we cannot relax the assumption that depth G ≥ 2 even if A is normal. See Section 4 for more details. In Section 5, as an application of the theorem above, we prove the above question has an affirmative answer. Theorem 5.2. Let (A, m) be a two-dimensional excellent normal local domain containing an algebraically closed field. Then the following conditions are equivalent: (1) m is a pg -ideal. (2) For every m-primary integrally closed ideal I, (µ(I) − 1) · ℓℓA (I) ≥ e(I) (∗) holds true. (3) For any power I = mℓ of m, the inequality (∗) holds true. After proving this theorem, Dao and Smirnov informed us that they proved the same theorem independently. 2. Preliminaries Throughout this paper, let (A, m) be a Noetherian local ring with d = dim A ≥ 1, and let I, J ⊂ A be ideals of positive height. We recall the notion which we will need later. 2.1. m-full ideals. Definition 2.1 (m-full, Rees property[22]). An ideal I is called m-full if there exists an element x ∈ m so that mI : x = I. An ideal I is said to have the Rees property if µ(J) ≤ µ(I) for any ideal J containing I. Proposition 2.2 (See [22, Theorem 3]). Any m-primary m-full ideal has the Rees property. Proof. See the proof of [3, Lemma 2.2].  The following result is due to Rees in the case of normal integral domains. Proposition 2.3 (See [22, Theorem 5]). Any integrally closed ideal of positive height is m-full. Proof. See the proof of Theorem [3, Theorem 2.4].  4 TONY J.PUTHENPURAKAL, KEI-ICHI WATANABE, KEN-ICHI YOSHIDA Definition 2.4 (Ratliff-Rush closure [17]). The Ratliff-Rush closure of I is defined by [ Ie = I n+1 : I n . n≥0 The ideal I is called Ratliff-Rush closed if Ie = I. Remark 2.5. Note that I ⊂ Ie ⊂ I, where I is the integrally closure of I. If we L put G = G(m) = n≥0 mn /mn+1 , then M n+1 ∩ mn )/mn+1 . ] H 0 (G) = (m M n≥0 Lemma 2.6. Assume that for some z ∈ mn−1 \ mn we have zm ⊂ mn−1 . Then, putting I = (x) + mn , µ(I) > µ(mn ), hence mn does not have Rees property. Proof. Let {y1 , . . . , yµ } be a minimal set of generators of mn . Then we can see that {z, y1, . . . , yµ } is a minimal set of generators of I. Hence µ(I) = µ(mn ) + 1.  Any m-full ideal has the Rees property. However, in order to prove our theorem, we need stronger inequalities. Recall the definition of the strong Rees property. Definition 2.7. An m-primary ideal I is said to have the strong Rees property (SRP for short) if µ(J) < µ(I) holds true for every ideal J with I ( J. We can show the existence of m-primary ideals with SRP by the following Lemma. Lemma 2.8. Let (A, m) be a Noetherian local ring. Fix an m-primary ideal I with µ(I) = n. (1) Any maximal element of the set of m-primary ideals I = {J ⊂ A | J ⊃ I and µ(J) ≥ n} has the strong Rees property. (2) If, moreover, I is m-full, then there is an ideal I ′ ⊃ I with the strong Rees property and µ(I ′ ) = µ(I). Proof. (1) is obvious by the definition of I. (2) If I ′ ⊃ I has SRP and µ(I ′ ) ≥ µ(I), we should have µ(I ′ ) = µ(I) since I is m-full.  The next example gives a motivation for us to study the strong Rees property of powers of the maximal ideal. See also [22, Theorem 5] and Section 6. Example 2.9. Assume that A is a two-dimensional regular local ring. Then for any m-primary ideal I, the following conditions are equivalent. (1) I has the strong Rees property. THE STRONG REES PROPERTY OF POWERS OF THE MAXIMAL IDEAL 5 (2) I = mn for some integer n ≥ 1. Indeed, assume that I has the strong Rees property. If we put n = ord(I) = max{n ∈ Z | I ⊂ mn }, then we can take an element f ∈ I so that ord(f ) = n. Then since I/(f ) is an m/(f )-primary ideal of A/(f ), we have µ(I/(f )) ≤ n = e(m/(f )) because A/(f ) is a one-dimensional Cohen-Macaulay local ring. In particular, µ(I) ≤ n + 1 = µ(mn ). If I 6= mn , then n + 1 = µ(mn ) < µ(I) by the assumption (1). But this is a contradiction. Conversely, if I ) mn , then ord(I) < n and µ(I) ≤ ord(I) + 1 < n + 1 = µ(mn ). 2.2. pg -ideals. In what follows, let (A, m) be a two-dimensional excellent normal local domain containing an algebraically closed field k = A/m. Let f : X → Spec A be a resolution of singularities. Then pg (A) = dimk H 1 (OX ) is called the geometric genus of A. Note that it does not depend on the choice of resolution of singularities. Let I ⊂ A be an m-primary integrally closed ideal. Let f : X → Spec A be a resolution of singularities on which I is represented, that is, IOX is invertible and IOX = OX (−Z) for some anti-nef cycle Z on X. Okuma and the last two authors [13] proved that dimk H 0 (OX (−Z)) ≤ pg (A) holds true if H 0 (OX (−Z)) has no fixed component. Definition 2.10 (See [13]). An anti-nef cycle Z is a pg -cycle if OX (−Z) is generated and dimk H 0 (OX (−Z)) = pg (A). An m-primary integrally closed ideal I is called a pg -ideal if I is represented as I = H 0 (OX (−Z)) by a pg -cycle Z. The following theorem gives a characterization of pg -ideals in terms of Rees algebras. Proposition 2.11 (See [14]). Let (A, m) be a two-dimensional excellent normal local domain containing an algebraically closed field. Let I ⊂ A be an m-primary ideal. Then the following conditions are equivalent: (1) I is a pg -ideal. (2) I n = I n for every n ≥ 1 and I 2 = QI for some minimal reduction Q ⊂ I. (3) The Rees algebra R(I) = A[It](⊂ A[t]) is a Cohen-Macaulay normal domain, where t is an indeterminate over A. For instance, any integrally closed m-primary ideal in a two-dimensional rational singularity (i.e. pg (A) = 0) is a pg -ideal. On the other hand, any two-dimensional excellent normal local domain containing algebraically closed field has a pg -ideal; see [13, 14]. 6 TONY J.PUTHENPURAKAL, KEI-ICHI WATANABE, KEN-ICHI YOSHIDA 3. Strong Rees Property of powers of the maximal ideal In what follows, let (A, m) be a Noetherian local ring and set R = R(m) and G = G(m) and M = mR + R+ . The main purpose of this section is to consider the following question. Question 3.1. Assume that I is Ratliff-Rush closed. When does I have the strong Rees property? As an answer, we show that some powers of the maximal ideal m have the strong Rees property if depth G ≥ 2; see Corollary 3.5. More generally, we can show the following theorem. 1 Theorem 3.2 (Strong Rees Property). Assume that depth A ≥ 2 and HM (G) has finite length. If mℓ is Ratliff-Rush closed, then mℓ has the strong Rees property. We first need to prove the following lemma. Lemma 3.3. Suppose that for some x ∈ m, mℓ+1 : x = mℓ . Then for every ideal J with J ⊃ mℓ , µ(J) ≤ µ(mℓ ) is always satisfied, and the following conditions are equivalent: (1) µ(J) = µ(mℓ ). (2) mJ = xJ + mℓ+1 . When this is the case, if we put C = JR/mℓ R, C/(xt)C = (C/(xt)C)0 = J/mℓ . In particular, dim C ≤ 1. Proof. Consider the following two short exact sequences: 0 → mℓ ∩ mJ mℓ mℓ → → → 0. mℓ+1 mℓ+1 mℓ ∩ mJ J J mℓ + mJ → → ℓ → 0. mJ mJ m + mJ Since mℓ /(mℓ ∩ mJ) ∼ = (mℓ + mJ)/mJ, combining two exact sequence implies 0 → (3.1) 0 → mℓ ∩ mJ mℓ J J → ℓ+1 → → ℓ → 0. ℓ+1 m m mJ m + mJ It follows that  ℓA (mℓ ∩ mJ/mℓ+1 ) = µ(mℓ ) − µ(J) + ℓA (J/mℓ + mJ). Furthermore, since mJ/(mℓ ∩ mJ) ∼ = (mℓ + mJ)/mℓ , we get  ℓA (mJ/mℓ+1 ) = ℓA (J/mℓ ) + µ(mℓ ) − µ(J) We now consider an R-module C = JR/mℓ R. The assumption that mℓ+1 : x = m implies that the multiplication map ℓ ·x : C0 = J/mℓ → C1 = mJ/mℓ+1 THE STRONG REES PROPERTY OF POWERS OF THE MAXIMAL IDEAL 7 is injective. Hence µ(mℓ ) − µ(J) = ℓA (C1 ) − ℓA (C0 ) ≥ 0. Moreover, equality holds true if and only if the multiplication map by x is isomorphism, which means that mJ = xJ + mℓ+1 . When this is the case, we have mn+1 J = xmn J + mn+ℓ+1 for every n ≥ 1. The last assertion immediately follows from this.  Lemma 3.4. Let J ⊂ A be an ideal with J ) mℓ , where ℓ ≥ 1. Put C = JR/mℓ R. 1 Assume that depth A ≥ 2, and HM (G) has finite length. If mℓ is Ratliff-Rush closed, then i (1) HM (C) has finite length for i = 0, 1. 0 (2) [HM (C)]0 = 0. Proof. First we define an R-module L(−1) as follows: 0 → R → A[t] → L(−1) → 0 (ex), that is, L(−1) = L n≥0 (A/m n )tn . i (L(−1)) has finite length for i = 0, 1. Claim 1. HM By [15, Proposition 4.7] we have 0 HM (L(−1)) = Mm fn n≥0 mn 0 = HM (G), 1 which has finite length and then it is proved in [16, Theorem 6.2] that HM (L(−1)) 1 has finite length, if and only if HM (G) has finite length. For instance, depth G ≥ 2, i then HM (L(−1)) = 0 for each i = 0, 1. L Secondly, we define D = L(−1)≥ℓ (ℓ) = n≥0 (A/mℓ+n )tn . 0 1 Claim 2. [HM (D)]0 = 0 and HM (D) has finite length. By definition, we have 0 → D(−ℓ) → L(−1) → W → 0, where W is an R-module of finite length. Then since in the exact sequence 0 1 1 W = HM (W ) → HM (D(−ℓ)) → HM (L(−1)) 1 the modules of the both sides have finite length, so does HM (D(−ℓ)). Moreover, 0 0 0 as HM (D(−ℓ)) ⊂ HM (L(−1)), HM (D(−ℓ)) is also of finite length. 0 fℓ /mℓ and our assumption. The first assertion follows from the fact [HM (D)]0 ⊂ m L Thirdly, we define an R-module V = n≥0 (A/mn J)tn as follows: 0 → JR → A[t] → V → 0 (ex). 8 TONY J.PUTHENPURAKAL, KEI-ICHI WATANABE, KEN-ICHI YOSHIDA By definition of V , we have 0 0 1 1 0 = HM (A[t]) → HM (V ) → HM (JR) → HM (A[t]) = 0, where two vanishing follows from the fact that depth A ≥ 2. It follows that 1 0 0 (JR)]n = 0 for large enough n. On the other hand, as HM (V ) ⊂ [HM (V )]n ∼ = [HM 0 0 V , [HM (V )]n = 0 for each n ≤ −1. Thus HM (V ) has finite length. 0 1 Claim 3. [HM (C)]0 = 0 and HM (C) has finite length. One can easily obtain the following exact sequence: 0 → C → D → V → 0 (ex). Hence 0 0 0 → HM (C) → HM (D) 0 → HM (V ) 1 1 → HM (C) → HM (D) → ··· 0 0 As [HM (D)]0 = 0 by Claim 2, we get [HM (C)]0 = 0. Moreover, in the sequence 0 0 1 1 HM (D) → HM (V ) → HM (C) → HM (D), 1 1 the both sides of HM (C) have finite length. Hence so does HM (C).  Proof of Theorem 3.2. Choose an m-superficial element x ∈ m \ m2 . By assumption, we have fℓ = mℓ . mℓ ⊂ mℓ+1 : x ⊂ m Then mℓ+1 : x = mℓ In particular, this means mℓ is m-full. Let J be an ideal with J ) mℓ . By Lemma 3.3, µ(J) ≤ µ(mℓ ). We want to show that this inequality is strict. Now suppose that equality holds true: µ(J) = µ(mℓ ). Put C = JR/mℓ R. In Lemma 3.3, we showed that C/(xt)C has finite length. Hence we have dim C ≤ 1. 1 0 (C) has On the other hand, by Lemma 3.4, we have [HM (C)]0 = 0 and HM ℓ 0 finite length. If dim C = 0, then 0 6= J/m = [C]0 = [HM (C)]0 = 0. This is a 1 contradiction. Hence dim C = 1. Then HM (C) is not finitely generated. This 1 contradicts the fact that HM (C) has finite length. Therefore we conclude that µ(J) < µ(mℓ ), as required.  In the case where depth G ≥ 2, then all powers of the maximal ideal have the strong Rees property. Corollary 3.5. If depth G ≥ 2, then mℓ has the strong Rees property for every ℓ ≥ 1. 1 (G) = 0 and Proof. Assume Theorem 3.2 holds true. If depth G ≥ 2, then HM depth A ≥ 2. Hence the ring A satisfies the assumption on Theorem 3.2.  THE STRONG REES PROPERTY OF POWERS OF THE MAXIMAL IDEAL 9 Corollary 3.6. Let (A, m) be a Cohen-Macaulay local ring with d = dim A ≥ 2. If G = G(m) is Cohen-Macaulay, then mℓ has the strong Rees property for each ℓ ≥ 1. Corollary 3.7. If depth A ≥ 2 and m is a normal ideal (i.e., mn is integrally closed for all n ≥ 1) then mℓ has the strong Rees property for every ℓ ≥ 1. Proof. By [8, Theorem 3.1] we get depth G(mn ) ≥ 2 for all n ≫ 0. Here G(mn ) 1 denotes the associated graded ring of mn . It follows that HM (G(m)) has finite r length. Also as m is integrally closed for all r ≥ 1 we get that it is Ratliff-Rush closed. It follows that mr has SRP for all r ≥ 1 by Theorem 3.2.  Example 3.8. Let (A, m) be a two-dimensional excellent normal local domain. If m is a pg -ideal (e.g. A is a rational singularity), then mℓ has the strong Rees property for every ℓ ≥ 1. On the other hand, “depth G ≥ 1” can be characterized by the Rees property. Proposition 3.9. Put G = G(m), where d = dim G ≥ 1. Then the following conditions are equivalent: (1) depth G ≥ 1. (2) mℓ has the Rees property for every ℓ ≥ 1. fℓ = mℓ and mℓ is m-full for every ℓ ≥ 1. Proof. (1) =⇒ (2) : If depth G ≥ 1, then m ℓ In particular, m has the Rees property. 0 (2) =⇒ (1) : Now suppose depth G = 0. Then HM (G) 6= 0. There exist an ℓ ℓ+1 integer ℓ ≥ 1 and an element z ∈ m \ m such that 0 6= z ∗ = z + mℓ+1 ∈ 0 0 ℓ+2 [HM (G)]ℓ ∩ Soc(HM (G)). Then mz ⊂ m . If we put J = (z) + mℓ+1 , then mJ = m(z) + mℓ+2 = mℓ+2 . Hence J/mJ = ((z) + mℓ+1 )/mℓ+2 ) mℓ+1 /mℓ+2 and so µ(J) = ℓA (J/mJ) > ℓA (mℓ+1 /mℓ+2 ) = µ(mℓ+1 ). This contradicts the assumption that mℓ+1 has the Rees property.  We now consider the case of depth G = 1. We need the following lemma. Lemma 3.10. Suppose that depth G = 1. Assume that J ) mℓ so that µ(J) = µ(mℓ ). Then one can find elements x ∈ m \ m2 and y ∈ J \ (mℓ + (x)) such that x∗ 0 is G-regular and 0 6= z ∗ ∈ Soc(HM (G)), where z = y ∈ A/xA and z ∗ denotes the ∗ initial form of z in G = G/x G. Proof. Take an element x ∈ m \ m2 so that x∗ is G-regular and J 6⊂ mℓ + (x); see the remark below for the exitence of x. Choose y ∈ J \ (mℓ + (x)). Then y ∈ (mk + (x)) \ (mk+1 + (x)) for some k with 0 ≤ k ≤ ℓ − 1. By assumption and Lemma 3.3, we have that my ⊂ mJ = xJ + mℓ+1 . Let denote the image of the surjection A → A/xA and put z = y. Then mz ⊂ mℓ+1 ⊂ mk+2 , which means that 0 0 6= z ∗ ∈ [Soc(HM (G/x∗ G))]k .  10 TONY J.PUTHENPURAKAL, KEI-ICHI WATANABE, KEN-ICHI YOSHIDA Remark 3.11. Suppose depth G = 1. For any ideal J ) mℓ , one can find an element x ∈ m \ m2 so that x∗ is G-regular and J is not contained in mℓ + (x). In order to prove this, it suffices to consider the case ℓ(J/mℓ ) = 1, that is, J = (f )+mℓ, where f 6∈ mℓ and mf ⊂ mℓ . If J ⊂ mℓ + (x) for such an element x as above, then f − ax ∈ mℓ for some a ∈ A. As x∗ is G-regular (and thus a∗ x∗ 6= 0), we obtain the relation a∗ x∗ = f ∗ in G. Let G = k[X]/a and let F be the inverse image of f ∗ in the polynomial ring k[X]. As k is an infinite field and dim k[X]/(a + (F )) ≥ 1, one can find a homogeneous element X of degree one which does not vanish on V (a + (F )). The required assertion follows from here. Hence we have the following. Proposition 3.12. Suppose that depth G = 1. Let x∗ ∈ G1 be a nonzero divisor 0 (G)]ℓ = 0 for all ℓ < n, then mℓ has the of G and put G = G/x∗ G. If [Soc(HM strong Rees property for all ℓ ≤ n. 0 Proof. First we note that Soc(HM (G/x∗ G)) is independent of the choice of x. In fact, the short exact sequence x∗ 0 −→ G(−1) −→ G −→ G := G/x∗ G −→ 0 yields a short exact seuence x∗ 0 0 1 1 HM (G) = 0 −→ HM (G) −→ HM (G)(−1) −→ HM (G). Taking a socle, we get x∗ 1 1 0 (G))(−1) −→ Soc(HM (G)). 0 −→ Soc(HM (G)) −→ Soc(HM 0 1 Since the last map is a zero map, Soc(HM (G)) ∼ (G))(−1) is independent = Soc(HM of the choice of x. Now suppose that mℓ does not have SRP. Then we can find an ideal J ) mℓ so that µ(J) = µ(mℓ ). By Lemma 3.10, there exist an element x ∈ m \ m2 such that 0 x∗ is G-regular and 0 6= [Soc(HM (G/x∗ G)]k for some 0 ≤ k ≤ ℓ − 1 ≤ n − 1. This contradicts the assumption.  Proposition 3.13. When depth G = 1, there exists an n ∈ N ∪ {∞} such that mℓ has the strong Rees property if and only if 1 ≤ ℓ ≤ n. In order to prove the proposition, it suffices to show the following lemma. Lemma 3.14. Suppose that depth G = 1. If mℓ does not have the strong Rees property, then neither does mℓ+1 . Proof. Since depth G = 1, there exists an element x ∈ m \ m2 so that x∗ = x + m2 is G-regular. In particular, mℓ and mℓ+1 are m-full. By assumption and Lemma 3.3, THE STRONG REES PROPERTY OF POWERS OF THE MAXIMAL IDEAL 11 we can take an ideal I ) mℓ so that mI = xI + mℓ+1 . Put J = xI + mℓ+1 . Then mJ = m(xI + mℓ+1 ) = m(xI) + mℓ+2 . Moreover, we suppose that J = mℓ+1 . Then xI ⊂ mℓ+1 and thus I ⊂ mℓ+1 : x = mℓ . This contradicts the choice of I. Hence J ) mℓ+1 and µ(J) = µ(mℓ+1 ). This implies that mℓ+1 does not have SRP.  Example 3.15. For any n ≥ 1, there exists a triple (a, b, c) ∈ N3 such that A = k[[s, ta , tb , tc ]] is a two-dimensional Cohen-Macaulay local domain such that mℓ has the strong Rees property if and only if 1 ≤ ℓ ≤ n. Proof. For a given n ≥ 1, we can choose an integer a = 10N > 2n. Set b = a + 1 and c = (a − 1)(a + 1) − an. Then s, ta , tb , tc is a minimal system of generators because ab − a − b = (a − 1)(a + 1) − a ≥ c(> b > a). Since A = k[[s, x, y, z]]/(yz − xa+1−n , xn z − y a−1 , z 2 − xa+1−2n y a−2 ), we get G = G(m) ∼ = k[S, X, Y, Z]/(Y Z, X n Z, Z 2 , Y a ). 0 (G/SG)) is generated by X n−1 Z ∈ [G/SG]n . Then S is an G-regular and Soc(HM ℓ Hence m has SRP if 1 ≤ ℓ ≤ n. If we put I = (xn−1 z) + mn+1 ) mn+1 , then x · xn−1 z = y a−1 ∈ ma−1 ⊆ mn+2 . Similarly, we have that y · xn−1 z ∈ ma ⊆ mn+2 and z · xn−1 z ∈ m2a−n−2 ⊂ mn+2 . Hence mI = (s · xn−1 z) + mn+2 , and this implies that mn+1 does not have SRP.  Example 3.16. Let A = k[[s, t4 , t5 , t11 ]] and m = (s, t4 , t5 , t11 ). Then m2 does not have strong Rees property. In fact, m2 = (s2 , st4 , st5 , st11 , t8 , t9 , t10 ) and thus µ(m2 ) = 7. If we put I = (s2 , st4 , st5 , t8 , t9 , t10 , t11 ) = m2 + (t11 ), then I ) m2 and µ(I) = 7 = µ(m2 ). On the other hand, since G ∼ = k[S, X, Y, Z]/(XZ, Y Z, Z 2 , Y 4 ) (see e.g. [20, f2 is m-full. Moreover, since Section 2]), we have depth G = 1 and thus m2 = m 2 2 11 2 t ∈ m \ m , m is not integrally closed. We can find an example of two-dimensional excellent normal local domains (A, m) for which m2 does not satisfy the strong Rees property and A/sA ∼ = k[[t4 , t5 , t11 ]] for some nonzero divisor s of A. See the next section. 4. Point divisor on a smooth curve – An example mn does not have strong Rees property In this section we treat a class of normal graded rings of dimension 2 and discuss whether mn has the strong Rees property in such rings. 12 TONY J.PUTHENPURAKAL, KEI-ICHI WATANABE, KEN-ICHI YOSHIDA Definition 4.1. Let k be an algebraically closed field and C be a smooth connected projective curve of genus g over k. We take a point P ∈ C and define H = HC,P = {n ∈ Z | h0 (C, OC (nP )) > h0 (C, OC ((n − 1)P ))}, where hi (C, F ) = dimk H i (C, F ). It is easy to see that HC,P is an additive semigroup and N \ HC,P has just g elements. We define R = RH = RC,P = ⊕n≥0 H 0 (C, OC (nP )T n , as a subring of k(C)[T ], where H 0 (C, OC (nP ) = {f ∈ k(C) | divC (f ) + nP ≥ 0}. Namely, f ∈ H 0 (C, OC (nP ), if and only iff has pole of order at most n at P and no other poles. Then R = RC,P is a normal graded ring of dimension 2 as treated in [5], Chapter 5, §2. In the following, we fix H = HC,P and write A = k[H] and R = RH so that R/T R ∼ = A, where T = 1.T ∈ R1 . Pe We write H = hn1 , . . . , ne i if H = { i=1 ai ni | ai ∈ Z≥0 (i = 1, . . . , e)}. In this case, we say that H is generated by e elements. We denote by H+ the set of positive elements of H and denote n ∈ rH+ if n = h1 + . . . + hr with hi ∈ H+ (i = 1, . . . , r). Remark 4.2. Given a semigroup H, sometimes there does not exist the pair (C, P ) such that HC,P = H. But at least we know the existence of (C, P ) such that H = HC,P in the following cases (cf. [10], [11]) ; (1) (2) (3) (4) k[H] is a complete intersection. H is generated by 3 elements. H is generated by 4 elements and H is symmetric or pseudo-symmetric. g(H) ≤ 9, where g(H) is the number of positive integers not in H. We summarize some property of R = RC,P . We put m = R+ . Proposition 4.3. Let R = RC,P . An element of Rn is denoted by f T n , where f ∈ k(C). We denote by v(f ) the order of the pole of f at P . For non zero elements f, g ∈ k(C), v(f g) = v(f ) + v(g). (1) f T n ∈ Rn if and only if v(f ) ≤ n and f has no other poles on C. (2) Hence if v(f ) < n, then f T n ∈ T n−v(f ) R, because f T v(f ) ∈ Rv(f ) . (3) If HC,P = hn1 , . . . , ne i, which are minimal generating system, then there are elements f1 , . . . , fe ∈ k(C) with v(fi ) = ni (i = 1, . . . , e) such that R = k[T, f1 T n1 , . . . , fe T ne ]. (4) If f T n ∈ Rn and v(f ) = n, then f T n ∈ mr if and only if n ∈ rH+ . (5) T ∈ R1 is a super regular element of R. Namely, if T x ∈ mr for some x ∈ R, then x ∈ mr−1 . THE STRONG REES PROPERTY OF POWERS OF THE MAXIMAL IDEAL 13 Theorem 4.4. Let R = RC,P and H = HC,P = hn1 , . . . , ne i. If for some n ∈ rH+ , n 6∈ (r − 1)H+ , n + ni ∈ (r + 2)H+ for i = 1, . . . , e, then mn+1 does not have the strong Rees property. Proof. By the assumption, there is some n ∈ (r − 1)H+ , n 6∈ rH+ such that n + H+ ⊂ (r + 1)H+ . Then we can take f ∈ k(C) so that v(f ) = n and f T n ∈ Rn . Since f T n 6∈ T R, f T n ∈ mr−1 and f T n 6∈ mr+1 . We put I = (mr , f T n ) and show that µ(I) = µ(mr ). Now, let homogeneous minimal generators of m be {T, g1T n1 , . . . , ge T ne }. Then among the homogeneous minimal generators of m(f T n), (f T n )(gi T ni ) ∈ mr+1 by our assumption. Hence we can obtain minimal generating system of I from that of mr , interchanging f T n and f T n+1, obtaining µ(I) = µ(mr ).  Corollary 4.5. Let R = RC,P , H = HC,P and m = R+ . Then the following conditions are equivalent: (1) For all n ≥ 2, mn has the strong Rees property. (2) The associated graded ring of k[H] with respect to k[H]+ is Cohen-Macaulay. (3) The associated graded ring of R with respect to R+ is Cohen-Macaulay. Example 4.6. Let H = h4, 5, 11i, C a smooth curve of genus 5 such that there is a point P with HC,P = h4, 5, 11i. We put R = RC,P and m = R+ . Since 11+4 ∈ 3H+ and 11 + 5 ∈ 4H+ , we see that m2 does not have the strong Rees property. In this example, we can easily see that mn is integrally closed for all n ≥ 2. Remark 4.7. For 3 generated semigroup H = ha, b, ci, we know when the associated graded ring is Cohen-Macaulay (cf, [7],[12]). 5. Takahashi-Dao’s question Dao and Takahashi [21] gave two upper bounds of the dimension of the singularity category dim Dsg (A): dim Dsg (A) ≤ (µ(I) − dim A + 1)ℓℓ(I) − 1 dim Dsg (A) ≤ e(I) − 1 for any m-primary ideal I contained in the sum N A of the Noether differents of A. They posed the following question. Question 5.1 (Takahashi-Dao). Let (A, m) be a d-dimensional Cohen-Macaulay local ring. Let I ⊂ A be an m-primary (integrally closed) ideal. When does an inequality (d − 1)!(µ(I) − d + 1) · ℓℓA (I) ≥ e(I) hold true (cf. [2])? 14 TONY J.PUTHENPURAKAL, KEI-ICHI WATANABE, KEN-ICHI YOSHIDA The following theorem is motivated by the question as above. In fact, (1) ⇒ (2) is due to Dao and Smirnov, and (2) ⇒ (1) is also proved by them independently. Theorem 5.2. Let (A, m) be a two-dimensional excellent normal local domain containing k = k ∼ = A/m. Then the following conditions are equivalent: (1) m is a pg -ideal. (2) For every m-primary integrally closed ideal I, (µ(I) − 1) · ℓℓA (I) ≥ e(I) (∗) holds true. (3) For any power I = mℓ of m, the inequality (∗) holds true. Proof. (1) =⇒ (2) : We give a sketch of proof here for the sake of the completeness. Assume that m is a pg -ideal. Let I ⊂ A be an m-primary integrally closed ideal. Then there exists a resolution of singularities f : X → Spec A so that IOX = OX (−Z) for some anti-nef cycle Z on X. By [13, Theorem 6.1], we have µ(I) = −M Z + 1, where M is an anti-nef cycle on X so that mOX = OX (−M ). Put r = ℓℓA (I), that is, mr ⊂ I and mr−1 6⊂ I. Then mr ⊂ I = I. Thus rM ≥ Z. Since e(I) = −Z 2 , we have (µ(I) − 1)ℓℓA (I) − e(I) = (−M Z)r + Z 2 = −(rM − Z)Z ≥ 0, as required. (2) =⇒ (3) : Trivial. (3) =⇒ (1) : Now assume that inequalities (µ(mℓ ) − 1) · ℓℓA (mℓ ) ≥ e(mℓ ) hold true for all integers ℓ ≥ 1. This shows that (µ(mℓ ) − 1) · ℓ ≥ e(mℓ ) = e(mℓ ) = ℓ2 · e, where e = e(m). Hence µ(mℓ ) ≥ ℓe + 1. In the case of ℓ = 1, we have that µ(m) − 1 ≥ e(m). On the other hand, Abhyankar’s inequality implies that µ(m) − 1 ≤ e(m), and thus equality holds true. That is, A has maximal embedding dimension in the sense of Sally ([19]). Then µ(mℓ ) = ℓe + 1. Moreover, since G = G(m) is Cohen-Macaulay ([18]), we obtain fℓ is m-full for every ℓ ≥ 1. Then µ(mℓ ) ≤ µ(mℓ ) = ℓe + 1 by Rees that mℓ = m property of m-full ideals. Hence µ(mℓ ) = µ(mℓ ). Then Corollary 3.6 yields that mℓ = mℓ because G(m) is Cohen-Macaulay in our case (Sally [18]). Therefore m is a pg -ideal by [14].  THE STRONG REES PROPERTY OF POWERS OF THE MAXIMAL IDEAL 15 Remark 5.3. We can show that I = mℓ satisfies the above inequality if (A, m) has maximal embedding dimension. Note that mℓ does not necessarily have the strong Rees property; see Sections 3 and 4. It is known that the maximal ideal m of any two-dimensional rational singularity is a pg -ideal. So it is natural to ask the following question. Which ring the maximal ideal of which is a pg -ideal? By a similar argument as in [6, Corollary 11.4], we can show the following. Notice that this gives a slight generalization of the fact that any two-dimensional rational singularity is an almost Gorenstein local ring. Proposition 5.4. Let (A, m) be a two-dimensional excellent normal local domain containing an algebraically closed field. Let KA denote the canonical module of A. If m is a pg -ideal, then A is an almost Gorenstein local ring in the sense of [6]. That is, there exists a short exact sequence of A-modules: 0 → A → KA → C → 0 such that mC = xC for some regular element x over C. Example 5.5. In the notation of Section 4, let P ∈ C be such that HC,P = {0, g + 1, g + 2, . . .}. Then the maximal ideal m of RC,P is a pg -ideal. Proof. Take f ∈ k(C) with f T g+1 ∈ R and v(f ) = g + 1. Then putting Q = (T, f T g+1 ), we see that m2 = Qm. Introduce a valuation w on R such that   v(f ) m w(f T ) = (m − v(f )) + g+1 To show that mn is integrally closed it suffices to show that w(gT m ) ≥ n if and only if gT m ∈ mn . If w(gT m ) ≥ n and m − v(g) = r, since v(g) ≥ (g + 1)(n − r), gT v(g) ∈ mn−r and then gT m ∈ mn . Hence mn is integrally closed for all n ≥ 1 and m is a pg -ideal by Proposition 2.11.  In dimension 2, although Takahashi-Dao’s inequality does not hold for general normal ring A, we have an inequality adding a constant depending on A. Also we have converse inequality for e(I) changing ℓℓ(A) to ord(I). Proposition 5.6. Let (A, m) be an excellent normal local ring as in Theorem 5.2 and I be an integrally closed ideal in A. Then we have the following inequalities; (1) If we put c = pg (A) − ℓA (H 1 (X, OX (−M ))), then we have an inequality (µ(I) − 1 + c) · ℓℓ(I) ≥ e(I). Note that c is an invariant depending only on A. (2) If we change ℓℓ(I) to ord(I), we have the converse inequality; e(I) ≥ (µ(I) − 1) · ord(I). 16 TONY J.PUTHENPURAKAL, KEI-ICHI WATANABE, KEN-ICHI YOSHIDA Proof. We modify our argument in the proof of Theorem 5.2. (1) In the situation of the proof of Theorem 5.2, by [13, Theorem 6.1], we have µ(I) ≥ −M Z + 1 − c and then the argument is the same as Theorem 5.2. (2) Since I ⊂ mord(I) , we have Z ≥ ord(I)M and hence −Z 2 ≥ − ord(I)M Z and −M Z ≥ µ(I) − 1.  Example 5.7. Put A = k[[X, Y, Z]]/(f ), where f is homogeneous of degree n ≥ 3
and we assume A is normal. Then c = pg (A) − ℓA (H 1 (X, OX (−M ))) = n−1 in 2 s this case. In fact, if we take I = m with s ≥ n, then we have ℓℓ(I) = s, µ(I) =
sn − (n − 3)n/2 and we see that c = n−1 is best possible. 2 In dimension ≥ 3, we can take A so that there is no constant c for which the inequality (d − 1)!(µ(I) − d + 1 + c) · ℓℓ(I) ≥ e(I) hold for all integrally closed ideal I. See the next example. Example 5.8. Let A = k[[x, y, z, w]]/(f ), where f is a homogeneous polynomial of degree n and we assume A is normal. Then we can see if n ≥ 5, putting I = ms , to have inequality (d − 1)!(µ(I) − d + 1 + c) · ℓℓ(I) ≥ e(I) holds we have to take c arbitrary large when s tends to infinity. 6. What Ideals have SRP ? We have shown under certain condition, mn has SRP and also in regular local rings of dimension 2, mn (n ≥ 1) are only ideals with SRP in Example 2.9. In this section, we ask if that this property characterize regular local rings and some Veronese subrings in dimension 2. We get a partial result for rational singularities. Proposition 6.1. Let (A, m) be a two-dimensional rational singularity and assume that A/m is algebraically closed. Then we have the following results concerning the strong Rees property for integral closed ideals. (1) If an integrally closed ideal I has the strong Rees property, then I is a good ideal in the sense of [4]. (2) If the minimal resolution of Spec(A) has more than two exceptional curves, then there are integrally closed ideals I 6= mn with the property such that µ(I ′ ) < µ(I) for every integrally closed ideal I ′ strictly containing I. Proof. If A is a rational singularity, and if I = IZ as in the proof of Theorem 5.2, we have shown µ(I) = −M Z + 1. (1) If I is not good and I ′ is the minimal good ideal containing I ′ , we have µ(I) = µ(I ′ ). Hence I does not have SRP. (2) If I = IZ has SRP, then Z is defined on the minimal resolution of Spec(A) and the property “µ(I ′ ) < µ(I) for every integrally closed ideal I ′ strictly containing I” is equivalent to say that “for every cycle Z ′ < Z, −M Z ′ < −M Z or M (Z ′ − Z) > 0. If the minimal resolution of THE STRONG REES PROPERTY OF POWERS OF THE MAXIMAL IDEAL 17 Spec(A) contains more than two curves, we can construct such Z 6= nM with that property.  Conjecture 6.2. We believe that if (A, m) is a local ring of dimension d ≥ 2 and if mn are only ideals of A which has the strong Rees property, then dim A = 2 and either A is a regular local ring or  ∼ = k[[X r , X r−1 Y, . . . , Y r ]] for some r. Acknowledgment 1. The authors would like to thank Hailong Dao for giving us a motivation of this work. Moreover, they would like to thank Shiro Goto, Jürgen Herzog, Ryo Takahashi, Junzo Watanabe and Santiago Zarzuela for giving us several valuable comments. The second author is supported by Grant-in-Aid for Scientific Research (C) 26400053. The third author is supported by Grant-in-Aid for Scientific Research (C) 16K05110. References [1] J. Asadollahi and T. J. Puthenpurakal, An analogue of a theorem due to Levin and Vasconcelos, Commutative algebra and algebraic geometry, 9–15, Contemp. Math., 390, Amer. Math. Soc., Providence, RI, 2005. [2] H. Dao and I. Smirnov, The multiplicity and the number of generators of an integrally closed ideal, arXiv:1703.09427. [3] S. Goto, Integral closedness of complete intersection ideals, J. Algebra 108 (1987), 151–160. [4] S. Goto, S. Iai, and K.-i. Watanabe, Good ideals in Gorenstein local rings, Trans. Amer. Math. Soc. 353 (2001), no. 6, 2309–2346 (electronic). [5] S. Goto and K. -i.Watanabe, On graded rings, I, J. Math. Soc. Japan, 30 (1978), 179–213. [6] S. Goto, R. Takahashi and N. Taniguchi, Almost Gorenstein rings–towards a theory of higher dimension, J. Pure Appl. Algebra 219 (2015), 2666–2712. 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The Dynamic Geometry of Interaction Machine: A Call-by-need Graph Rewriter Koko Muroya and Dan Ghica University of Birmingham, UK arXiv:1703.10027v1 [] 29 Mar 2017 Abstract Girard’s Geometry of Interaction (GoI), a semantics designed for linear logic proofs, has been also successfully applied to programming language semantics. One way is to use abstract machines that pass a token on a fixed graph along a path indicated by the GoI. These token-passing abstract machines are space efficient, because they handle duplicated computation by repeating the same moves of a token on the fixed graph. Although they can be adapted to obtain sound models with regard to the equational theories of various evaluation strategies for the lambda calculus, it can be at the expense of significant time costs. In this paper we show a token-passing abstract machine that can implement evaluation strategies for the lambda calculus, with certified time efficiency. Our abstract machine, called the Dynamic GoI Machine (DGoIM), rewrites the graph to avoid replicating computation, using the token to find the redexes. The flexibility of interleaving token transitions and graph rewriting allows the DGoIM to balance the trade-off of space and time costs. This paper shows that the DGoIM can implement call-by-need evaluation for the lambda calculus by using a strategy of interleaving token passing with as much graph rewriting as possible. Our quantitative analysis confirms that the DGoIM with this strategy of interleaving the two kinds of possible operations on graphs can be classified as “efficient” following Accattoli’s taxonomy of abstract machines. 1 1.1 Introduction Token-passing Abstract Machines for λ-calculus Girard’s Geometry of Interaction (GoI) [16] is a semantic framework for linear logic proofs [15]. One way of applying it to programming language semantics is via “token-passing” abstract machines. A term in the λcalculus is evaluated by representing it as a graph, then passing a token along a path indicated by the GoI. Token-passing GoI decomposes higher-order computation into local token actions, or low-level interactions of simple components. It can give strikingly innovative implementation techniques for functional programs, such as Mackie’s Geometry of Implementation compiler [20], Ghica’s Geometry of Synthesis (GoS) high-level synthesis tool [12], and Schöpp’s resource-aware program transformation to a low-level language [25]. The interactionbased approach is also convenient for the complexity analysis of programs, e.g. Dal Lago and Schöpp’s IntML type system of logarithmic-space evaluation [7], and Dal Lago et al.’s linear dependent type system of polynomialtime evaluation [5, 6]. Fixed-space execution is essential for GoS, since in the case of digital circuits the memory footprint of the program must be known at compile-time, and fixed. Using a restricted version of the call-by-name language Idealised Algol [13] not only the graph, but also the token itself can be given a fixed size. Surprisingly, this technique also allows the compilation of recursive programs [14]. The GoS compiler shows both the usefulness of the GoI as a guideline for unconventional compilation and the natural affinity between its space-efficient abstract machine and call-by-name evaluation. The practical considerations match the prior theoretical understanding of this connection [9]. In contrast, re-evaluating a term by repeating its token actions poses a challenge for call-by-value evaluation (e.g. [11, 24, 18, 3]) because duplicated computation must not lead to repeated evaluation. Moreover, in callby-value repeating token actions raises the additional technical challenge of avoiding repeating any associated computational effects (e.g. [23, 22, 4]). A partial solution to this conundrum is to focus on the soundness of the equational theory, while deliberately ignoring the time costs [22]. However, Fernández and Mackie suggest that in a call-by-value scenario, the time efficiency of a token-passing abstract machine could also be improved, by allowing a token to jump along a path, even though a time cost analysis is not given [11]. For us, solving the the problem of creating a GoI-style abstract machine which computes efficiently with evaluation strategies other than call-by-name is a first step in a longer-range research programme. The compilation techniques derived from the GoI can be extremely useful in the case of unconventional computational platforms. 1 But if GoI-style techniques are to be used in a practical setting they need to extend beyond call-by-name, not just correctly but also efficiently. 1.2 Interleaving Token Passing with Graph Rewriting A token jumping, rather than following a path, can be seen as a simple form of short-circuiting that path, which is a simple form of graph-rewriting. This idea first occurs in Mackie’s work as a compiler optimisation technique [20] and is analysed in more depth theoretically by Danos and Regnier in the so-called Interaction Abstract Machine [9]. More general graph-rewriting-based semantics have been used in a system called virtual reduction [8], where rewriting occurs along paths indicated by GoI, but without any token-actions. The most operational presentation of the combination of token-passing and jumping was given by Fernández and Mackie [11]. The interleaving of token actions and rewriting is also found in Sinot’s interaction nets [26, 27]. We can reasonably think of the DGoIM as their abstract-machine realisation. We build on these prior insights by adding more general, yet still efficient, graph-rewriting facilities to the setting of a GoI token-passing abstract machine. We call an abstract machine that interleaves token passing with graph rewriting the Dynamic GoI Machine (DGoIM), and we define it as a state transition system with transitions for token passing as well as transitions for graph rewriting. What connects these two kinds of transitions is the token trajectory through the graph, its path. By examining it, the DGoIM can detect redexes and trigger rewriting actions. Through graph rewriting, the DGoIM reduces sub-graphs visited by the token, avoiding repeated token actions and improving time efficiency. On the other hand, graph rewriting can expand a graph by e.g. copying sub-graphs, so space costs can grow. To control this trade-off of space and time cost, the DGoIM has the flexibility of interleaving token passing with graph rewriting. Once the DGoIM detects that it has traversed a redex, it may rewrite it, but it may also just propagate the token without rewriting the redex. As a first step in our exploration of the flexibility of this machine, we consider the two extremal cases of interleaving. The first extremal case is “passes-only,” in which the DGoIM never triggers graph rewriting, yielding an ordinary token-passing abstract machine. As a typical example, the λ-term (λx.t) u is evaluated like this: 1. A token enters the graph on the left at the bottom open edge. 2. A token visits and goes through the left sub-graph λx.t. λx.t u 3. Whenever a token detects an occurrence of the variable x in t, it traverses the right sub-graph u, then returns carrying the resulting value. 4. A token finally exits the graph at the bottom open edge. Step 3 is repeated whenever term u needs to be re-evaluated. This strategy of interleaving corresponds to call-by-name reduction. The other extreme is “rewrites-first,” in which the DGoIM interleaves token passing with as much, and as early, graph rewriting as possible, guided by the token. This corresponds to both call-by-value and call-by-need reductions, the difference between the two being the trajectory of the token. In the case of call-by-value, the token will enter the graph from the bottom, traverse the left-hand-side sub-graph, which happens to be already a value, then visit sub-graph u even before x is used in a call. While traversing u, it will cause rewrites such that when the token exits, it leaves behind the graph of a machine corresponding to a value v such that u reduces to v. The difference with call-by-need is that the token will visit u only when x is encountered in λx.t. In both cases, if repeated evaluation is required then the sub-graph corresponding now to v is copied, so that one copy can be further rewritten, if needed, while the original is kept for later reference. 1.3 Contributions This work presents a DGoIM model for call-by-need, which can be seen as a case study of the flexibility achieved through controlled interleaving of rewriting and token-passing. This is achieved through a rewriting strategy which turns out to be as natural as the passes-only strategy is for implementing call-by-name. The DGoIM avoids re-evaluation of a sub-term by rewriting any sub-graph visited by a token so that the updated sub-graph represents the evaluation result, but, unlike call-by-value, it starts by evaluating the sub-graph corresponding to the function λx.t first. We chose call-by-need mainly because of the technical challenges it poses. Adapting the technique to call-by-value is a straightforward exercise, and we discuss other alternative in the Conclusion. We analyse the time cost of the DGoIM with the rewrites-first interleaving, using Accattoli et al.’s general methodology for quantitative analysis [2, 1]. Their method cannot be used “off the shelf,” because the DGoIM does not satisfy one of the assumptions used in [1, Sec. 3]. Our machine uses a more refined transition system, 2 in which several steps correspond to a single one in loc. cit.. We overcome this technical difficulty by building a weak simulation of Danvy and Zerny’s storeless abstract machine [10] to which the recipe does apply. The result of the quantitative analysis confirms that the DGoIM with the rewrites-first interleaving can be classified as “efficient,” following Accattoli’s taxonomy of abstract machines introduced in [1]. As we intend to use the DGoIM as a starting point for semantics-directed compilation, this result is an important confirmation that no hidden inefficiencies lurk within the fabric of the rather complex machinery of the DGoIM. 2 The Dynamic GoI Machine 2.1 Well-boxed Graphs The graphs used to construct the DGoIM are essentially MELL proof structures [15] of the multiplicative and exponential fragment of linear logic. They are directed, and built over the fixed set of nodes called “generators” shown in Fig. 1. H | | Ax Cut ` ! ? D ? Cn ! | Figure 1: Generators of Graphs Figure 2: !-box H A Cn -node is annotated by a natural number n that indicates its in-degree, i.e. the number of incoming edges. It generalises a contraction node, whose in-degree is 2, and a weakening node, whose in-degree is 0, of MELL proof structures. In Fig. 1, a bunch of n edges is depicted by a single arrow with a strike-out. Graphs must satisfy the well-formedness condition below. Note that, unlike the usual approach [15], we need not assign MELL formulas to edges, nor require a graph to be a valid proof net. Definition 2.1 (well-boxed). A directed graph G built over the generators in Fig. 1 is well-boxed if: • it has no incoming edges • each !-node v in G comes with a sub-graph H of G and an arbitrary number of ?-nodes ~u such that: – the sub-graph H (called “!-box”) is well-boxed inductively and has at least one outgoing edges – the !-node v (called “principal door of H”) is the target of one outgoing edge of H – the ?-nodes ~u (called “auxiliary doors of H”) are the targets of all the other outgoing edges of H • each ?-node is an auxiliary door of exactly one !-box • any two distinct !-boxes with distinct principal doors are either disjoint or nested Note that a !-box might have no auxiliary doors. We use a dashed box to indicate a !-box together with its principal door and its auxiliary doors, as in Fig. 2. The auxiliary doors are depicted by a single ?-node with a thick frame and with single incoming and outgoing arrows with strike-outs. Directions of edges are omitted in the rest of the paper, if not ambiguous, to reduce visual clutter. 2.2 Pass Transitions and Rewrite Transitions The DGoIM is formalised as a labelled transition system with two kinds of transitions, namely pass transitions 99K and rewrite transitions . Labels of transitions are b, s, o that stand for “beta,” “substitution,” and “overheads” respectively. Let L be a fixed countable (infinite) set of names. The state of the transition system s = (G, p, h, m) consists of the following elements: • a named well-boxed graph G = (G, ℓG ), that is a well-boxed graph G with a naming lG that assigns a unique name α ∈ L to each node of G • a pair p = (e, d) called position, of an edge e of G and a direction d ∈ {↑, ↓} • a history stack h defined by the grammar h ::=  | Axα : h | Cutα : h | ⊗α : h | `α : h | !α : h | Dα : h | Cnα : h, where α ∈ L and n is some positive natural number. • a multiplicative stack m defined by the BNF grammar m ::=  | l : m | r : m. 3 We refer to a node by its name, i.e. we say “a node α” instead of “a node whose name is α.” A pass transition (G, p, h, m) 99Ko (G, p′ , h′ , m′ ) changes a position using a multiplicative stack, pushes to a history stack, and keeps a named graph unchanged. All pass transitions have the label o. Fig. 3 shows pass transitions graphically, omitting irrelevant parts of graphs. A position p = (e, d) is represented by a bullet • (called “token”) on the edge e together with the direction d. Recall that an edge with a strike-out represents a bunch of edges. The transition in the last line of Fig. 3 (where we assume n > 0) moves a token from one of the incoming edges of a Cn -node to the outgoing edge of the node. Node names α ∈ L are indicated wherever needed. (h, m) Ax α ↑ (h, m) ↓ $α (h, l : m) ↑ $α (h, m) ↓ ! α (h, m) (Axα : h, m) 99Ko Ax α ↓ ($α : h, l : m) 99Ko ↓ $α ($α : h, m) 99Ko ↑ $α (!α : h, m) 99Ko ↓ ! (h, m) ↓ Cut α (h, m) $α ↓ (h, r : m) $α ↑ (h, m) ↓ α Dα (Cutα : h, m) 99Ko Cut α ↑ ($α : h, r : m) 99Ko $α ↓ ($α : h, m) 99Ko $α ↑ (Dα : h, m) 99Ko ↓ Dα (Cnα : h, m) | | ↓ Cn α 99Ko Cn α ↓ Figure 3: Pass Transitions ($ ∈ {⊗, `}, n > 0) A rewrite transition (G, (e, d), h, m) x (G′ , (e′ , d), h′ , m) consumes some elements of a history stack, rewrites a sub-graph of a named graph, and updates a position (or, more precisely, its edge). The label x of a rewrite transition x is either b, s or o. Fig. 4 shows rewrite transition in the same manner as Fig. 3. Multiplicative stacks are not present in the figure since they are irrelevant. The ♯-node represents some arbitrary node (incoming edges omitted). We can see that no rewrite transition breaks the well-boxed-ness of a graph. The rewrite transitions (1),(2),(3), and (4) are exactly taken from MELL cut elimination [15]. The rewrite transition (5) is a variant of (1). It acts on a connected pair of a Cut-node and an Ax-node that arises as a result of the transition (6) or (7) but cannot be rewritten by the transition (1). These transitions (6) and (7) are inspired by the MELL cut elimination process for (binary) contraction nodes; note that we assume n > 0 in Fig. 4. The rewrite transition (6) in Fig. 4 deserves further explanation. The sub-graph H ≈ is a copy of the !-box H where all the names are replaced with fresh ones. The thick Cg+f −1 -node and Cg+2f −1 -node represent families m {Cg(j)+f −1 (j) }m ǫ = ǫ0 , . . . , ǫl and j=0 , {Cg(j)+2f −1 (j) }j=0 , of C-nodes respectively. They are connected to ?-nodes ~ µ ~ = µ0 , . . . , µl in such a way that: • the natural numbers l, m satisfy l ≥ m, and come with a surjection f : {0, . . . , l} ։ {0, . . . , m} and a function g : {0, . . . , m} → N to the set N of natural numbers • each ?-node ǫi and each ?-node µi are both connected to the C-node φf (i) • each C-node φj has g(j) incoming edges whose source is none of the ?-nodes ~ǫ, ~µ. Some rewrite transitions introduce new nodes to a graph. We require that the uniqueness of names throughout a whole graph is not violated by these transitions. Under this requirement, the introduced names ν, ~µ and the renaming H ≈ in Fig. 4 can be arbitrary. Definition 2.2. We call a state ((G, ℓG ), p, h, m) rooted at e0 for an open (outgoing) edge e0 of G, if there exists a finite sequence ((G, ℓG ), (e0 , ↑), , ) 99K∗ ((G, ℓG ), p, h, m) of pass transitions such that the position p appears only last in the sequence. 4 Cutα : Axβ : h ↑ Cut α Ax β h o !α : Cutβ : Axγ : h (1) ↑ !α : h ↑ ! Ax γ α o ↑ ! Cut β `α : Cutβ : ⊗γ : h ↑ ↑ γ Cut β b !α : Cutβ : Cn+1 : ♯δ : h γ !ν : Cutη : ♯δ : h (2) Cut ν ν H | H≈ ↑ ?µ ! ~ | `α Cut β Cutβ : h (5) α Cutβ : h | ↑ b Cut β (3) ? ~ǫ ! Cn+1 γ α s ? ~ǫ ! ~ φ Cutβ : h | | Cg+2f −1 | | H ↑ | δ o ! α Dγ Cut β | ?~ H ↑ (4) Cut β !α : Cutβ : C1γ : ♯δ : h α C1 γ Cut β s Cn γ Cut β α ~ φ H ↑ ?~ | | δ ♯δ ! (6) !α : Cutβ : ♯δ : h | | H ↑ ?~ ♯δ | | !α : Cutβ : Dγ : h Cut η | Cg+f −1 Cut ν ♯δ Cut β | γ | ↑ `α Cut β H ↑ | `α : Cutβ : ⊗γ : h δ ♯δ ! (7) α Cut β Figure 4: Rewrite Transitions (n > 0) Lem. 2.3(1) below implies that, the DGoIM can determine whether a rewrite transition is possible at a rooted state by only examining a history stack. The rooted property is preserved by transitions. Lemma 2.3 (rooted states). Let ((G, ℓG ), (e, d), h, m) be a rooted state at e0 with a (finite) sequence ((G, ℓG ), (e0 , ↑), , ) 99K∗ ((G, ℓG ), (e, d), h, m). 1. History stack represents an (undirected and possibly cyclic) path of the graph G connecting edges e0 and e. 2. If a transition ((G, ℓG ), (e, d), h, m) (99K ∪ ) ((G′ , ℓG′ ), p′ , h′ , m′ ) is possible, the open edges of G′ are bijective to those of G, and the state ((G′ , ℓG′ ), p′ , h′ , m′ ) is rooted at the open edge corresponding to e0 . Proof. (Sketch.) The proof of the first part is by induction on the length of the sequence of move transitions. For the second part, rewrite transitions modify open edges of a graph in a bijective way. The edge that a state is rooted at can be modified only by the rewrite transitions (1) and (5) involving Ax-nodes. 2.3 Cost Analysis of the DGoIM The time cost of updating stacks is constant, as each transition changes only a fixed number of top elements of stacks. Updating a position is local and needs constant time, as it does not require searching beyond the next edge in the graph from the current edge. We can conclude all pass transitions take constant time. We estimate the time cost of rewrite transitions by counting updated nodes. The rewrite transitions (1)–(3) involve a fixed number of nodes, and the transition (7) eliminates one C1 -node. Only the transitions (4) and (6) have non-constant time cost. The number of doors deleted in the transition (4) can be arbitrary, and so is the number of nodes introduced in the transition (6). Pass transitions and rewrite transitions are separately deterministic (up to the choice of new names). However, both a pass transition and a rewrite transition are possible at some states. We here opt for the following “rewrites-first” way to interleave pass transitions with as much rewrite transitions as possible: ( s x s′ (if x possible) ′ def. s _x s ⇐⇒ ′ s 99Kx s (if only 99K [x possible). The DGoIM with this strategy yields a deterministic labelled transition system _ up to the choice of new names in rewrite transitions. We denote it by DGoIM_ , making the strategy explicit. Note that there can be other strategies of interleaving although we do not explore them here. Before we conclude, several considerations about space cost analysis. Space costs are generally bound by time costs, so from our analysis there is an implicit guarantees that space usage will not explode. But if a more refined space cost analysis is desired, the following might prove to be useful. 5 Terms Values Evaluation contexts Substitution contexts t ::= x | λx.t | t t | t[x ← t] v ::= λx.t E ::= h·i | E t | E[x ← t] | Ehxi[x ← E] A ::= h·i | A[x ← t] Pure terms Pure values t ::= x | λx.t | t t v ::= λx.t (t u, E)term →o (t, Ehh·i ui)term (8) (x, E1 hE2 [x ← t]i)term →o (t, E1 hE2 hxi[x ← h·i]i)term (v, E)term →o (v, E)ctxt (9) (10) (λx.t, EhA ui)ctxt →b (t, EhAhh·i[x ← u]ii)term (v, E1 hE2 hxi[x ← A]i)ctxt →s (v ≈ , E1 hAhE2 [x ← v]ii)ctxt (if x ∈ FV∅ (E2 )) (11) (12) (v, E1 hE2 hxi[x ← A]i)ctxt →s (v, E1 hAhE2 ii)ctxt (if x ∈ / FV∅ (E2 )) (13) Figure 5: Call-by-need Storeless Abstract Machine (SAM) The space required in implementing a named well-boxed graph is bounded by the number of its nodes. The number of edges is linear in the number of nodes, because each generator has a fixed out-degree and every edge of a well-boxed graph has its source. Additionally a !-box can be represented by associating its auxiliary doors to its principal door. This adds connections between doors to a graph that are as many as ?-nodes. It enables the DGoIM to identify nodes of a !-box by following edges from its principal and auxiliary doors. Nodes in a !-box that are not connected to doors can be ignored, since these nodes are never visited by a token (i.e. pointed by a position) as long as the DGoIM acts on rooted states. Only the rewrite transition (6) can increase the number of nodes of a graph by copying a !-box with its doors. Rewrite transitions can copy !-boxes and eliminate the !-box structure, but they never create new !-boxes or change existing ones. This means that, in a sequence of transitions that starts with a graph G, any !-boxes copied by the rewrite transition (6) are sub-graphs of the graph G. Therefore the number of nodes of a graph increases linearly in the number of transitions. Elements of history stacks and multiplicative stacks, as well as a position, are essentially pointers to nodes. Because each pass/rewrite transition adds at most one element to each stack, the lengths of stacks also grow linearly in the number of transitions. 3 3.1 Weak Simulation of the Call-by-Need SAM Storeless Abstract Machine (SAM) We show the DGoIM_ implements call-by-need evaluation by building a weak simulation of the call-by-need Storeless Abstract Machine (SAM) defined in Fig. 5. It simplifies Danvy and Zerny’s storeless machine [10, Fig. 8] and accommodates a partial mechanism of garbage collection (namely, transition (13)). We will return to a discussion of garbage collection at the end of this section. The SAM is a labelled transition system between configurations (t, E). They are classified into two groups, namely term configurations and context configurations, that are indicated by annotations term, ctxt respectively. Pure terms (resp. pure values) are terms (resp. values) that contain no explicit substitutions t[x ← u]; we sometimes omit the word “pure” and the overline in denotation as long as that raises no confusion. Each evaluation context E contains exactly one open hole h·i, and replacing it with a term t (or an evaluation context E ′ ) yields a term Ehti (or an evaluation context EhE ′ i) called plugging. In particular an evaluation context E ′ hxi[x ← E] replaces the open hole of E ′ with x and keeps the open hole of E. Labels of transitions are the same as those used for the DGoIM (i.e. b, s and o). The transition (11), with the label b, corresponds to the β-reduction where evaluation and substitution of function arguments are delayed. Substitution happens in the transitions (12) and (13), with the label s, that replaces exactly one occurrence of a variable. The other transitions with the label o, namely (t, E) →o (t′ , E ′ ), search a redex by rearranging a configuration. The two pluggings Ehti and E ′ ht′ i indeed yield exactly the same term. We characterise “free” variables using multisets of variables. Multisets make explicit how many times a variable is duplicated in a term (or an evaluation context). This information of duplication is later used in translating terms to graphs. Notation (multiset). A multiset x := [x, . . . , x] consists of a finite number of x. The multiplicity of x in a multiset M is denoted by M (x). We write x ∈k M if M (x) = k, x ∈ M if M (x) > 0 and x ∈ / M if M (x) = 0. A multiset M comes with its support set supp(M ). For two multisets M and M ′ , their sum and difference are 6 denoted by M + M ′ and M − M ′ respectively. Removing all x from a multiset M yields the multiset M \x, e.g. [x, x, y]\x = [y]. Each term t and each evaluation context E are respectively assigned multisets of variables FV(t), FVM (E), with M a multiset of variables. The multisets FV are defined inductively as follows. FV(x) := [x], FV(λx.t) := FV(t)\x, FV(t u) := FV(t) + FV(u), FV(t[x ← u]) := (FV(t)\x) + FV(u). FVM (h·i) := M, FVM (E t) := FVM (E) + FV(t), FVM (E[x ← t]) := (FVM (E))\x + FV(t), FVM (E ′ hxi[x ← E]) := (FV[x] (E ′ ))\x + FVM (E). The following equations can be proved by a straightforward induction on E. Lemma 3.1 (decomposition). FV(Ehti) = FVFV(t) (E) FVM (EhE ′ i) = FVFVM (E ′ ) (E) A variable x is bound in a term t if it appears in the form of λx.u or u[x → u′ ]. A variable x is captured in an evaluation context E if it appears in the form of E ′ [x ← t] (but not in the form of E ′ hxi[x ← E ′′ ]). The transitions (12) and (13) depend on whether or not the bound variable x appears in the evaluation context E2 . If the variable x appears, the value v is kept for later use and its copy v ≈ is substituted for x. If not, the value v itself is substituted for x. The SAM does not assume the α-equivalence, but explicitly deals with it in copying a value. The copy v ≈ in has all its bound variables replaced by distinct fresh variables (i.e. distinct variables that do not appear in a whole configuration). This implies that the SAM is deterministic up to the choice of new variables introduced in copying. A term t is closed if FV(t) = ∅; and is well-named if each variable gets bound at most once in t, and each bound variable x in t satisfies x ∈ / FV(t). An initial configuration is a term configuration (t0 , h·i)term where t0 is closed and well-named. A finite sequence of transitions from an initial configuration is called an execution. A reachable configuration (t, E), that is a configuration coming with an execution from some initial configuration to itself, satisfies the following invariant properties. Lemma 3.2 (reachable configurations). Let (t, E) be a reachable configuration from an initial configuration (t0 , h·i)term . The term t is a sub-term of the initial term t0 up to α-equivalence, and the plugging Ehti is closed and well-named. Proof. (Sketch.) The proof is by induction on the length of the execution (t0 , h·i)term →∗ (t, E). Not only the term t but also the term u′ in any sub-term t′ u′ or t′ [x ← u′ ] of the plugging Ehti is a sub-term of the initial term t0 . The transition (12) renames a value in the way that preserves closedness and the well-named-ness of pluggings. In the transition (13) where an explicit substitution for a bound variable x is eliminated, the induction hypothesis ensures that the variable x does not occur in the plugging E1 hAhE2 hviii. We now conclude with a brief consideration on garbage collection. Transition (13) eliminates an explicit substitution and therefore implements a partial mechanism of garbage collection. The mechanism is partial because only an explicit substitution that is looked up in an execution can be eliminated, as illustrated below. The explicit substitution [x ← λz .z] is eliminated in the first example, but not in the second example because the bound variable x does not occur. ((λx.x) (λz .z), h·i)term →∗ (λz .z, h·i)ctxt ((λx.λy.y) (λz .z), h·i)term →∗ (λy.y, h·i[x ← λz .z])ctxt We incorporate this partial garbage collection to make clear the behaviour of the DGoIM_ , in particular the use of the rewrite transitions (6) and (7). 7 x† := x Ax | t† x Ck FV(t)\x x (if x ∈k FV(t)) | (λx.t)† := ? ! | FV(t)\x t† † (t u) := FV(u) | | FV(t) u† Ax D Cut u† | (if x ∈k FV(t)) | (t[x ← u])† := t† x C FV(t)\x x k | Cut FV(u) Figure 6: Inductive Translation (·)† of Terms to Well-boxed Graphs x† := t† x Ax † (t u) := FV(u) | FV(t) | t† x Ck FV(t)\x x u† Ax D Cut | | (λx.t)† := | (if x ∈k FV(t)) | (t[x ← u])† := ! u† | ? | FV(t)\x t† x C FV(t)\x x k (if x ∈k FV(t)) Cut FV(u) Figure 7: Inductive Translation (·)†M of Evaluation Contexts to Graphs 3.2 Translation and Weak Simulation A weak simulation is built on top of translations of terms and evaluation contexts. The translations (·)† are inductively defined in Fig. 6 and Fig. 7. What underlies them is the so-called “call-by-value” translation of intuitionistic logic to linear logic. This translates all and only values to !-boxes that can be copied by rewrite transitions. t† of a term t is a well-boxed graph, where some edges are annotated with variables The translation | FV(t) to help understanding. We continue representing a bunch of edges by a single edge and a strike-out, with M | annotations denoted by a multiset, and a bunch of nodes by a single thick node. The translation † EM | FVM (E) of an evaluation context E, given a multiset M of variables, is not a well-boxed graph because it has incoming edges. Lem. 3.3 is analogous to Lem. 3.1; their proof is by straightforward induction on E. Lemma 3.3 (decomposition). † t FV(t) | FVFV(t) (E) | † (Ehti) = † EFV(t) (E ′ )†M FVM (E) † EFV ′ M (E ) FVFVM (E ′ ) (E) | (EhE ′ i)†M = | M | 8 The translations (·)† are lifted to a binary relation between reachable configurations of the SAM and rooted states of the DGoIM_ . | Definition 3.4 (binary relation ). A reachable configuration c and a state ((G, ℓG ), p, h, m) satisfies c  ((G, ℓG ), p, h, m) if and only if:  †  t    FV(t) ↑ (if c = (t, E)term )   †  EFV(t)       v⋄ • (G, p) = v⋄  †  FV(v) ↑  (if v = FV(v)   ? ! ? !   FV(v)  FV(v)   †  EFV(v)   and c = (v, E)ctxt ) | | | | • ℓG is an arbitrary naming • ((G, ℓG ), p, h, m) is rooted at the unique open edge of G. Note that the graph G in the above definition has exactly one open edge, because it is equal to the translation Ehti† (Lem. 3.3) and the plugging Ehti is closed (Lem. 3.2). The binary relation  gives a weak simulation, as stated below. It is weak in Milner’s sense [21], where transitions with the label o are regarded as internal. We can conclude from Thm. 3.5 below that the DGoIM_ soundly implements the call-by-need evaluation. Theorem 3.5 (weak simulation). Let a configuration c and a state s satisfy c  s. 1. If a transition c →b c′ of the SAM is possible, there exists a sequence s _2o _b _o s′ such that c′  s′ . 2. If a transition c →s c′ of the SAM is possible, there exists a sequence s _s _o s′ such that c′  s′ . ′ 3. If a transition c →o c′ of the SAM is possible, there exists a sequence s _N o s such that 0 < N ≤ 4 and ′ ′ c s. 4. No transition _ is possible at the state s′ if c′ = (v, A)ctxt . Proof. We start with two basic observations. First, only a pass transition is possible at a state s that satisfies (t, E)term  s, and second, no transition is possible at a state s that satisfies (v, A)ctxt  s. Fig. 8 shows how the DGoIM_ simulates each transition of the SAM. Stacks, annotations of edges, and some annotations of C-nodes are omitted. The equations in Fig. 8 apply the decomposition properties in Lem. 3.3 as well as the other decomposition properties in the following Lem. 3.6. In the application, we exploit the closedness and well-named-ness of reachable configurations (in the sense of Lem. 3.2). Lemma 3.6 (decomposition). Let M0 , M be multisets of variables. M A‡M FVM (A) | 1. The translation A†M of a substitution context A has a unique decomposition . | † 2. If no variables in M0 are captured in an evaluation context E, the translation EM is equal to the graph 0 +M | M † EM | M0 . | FVM (E) † 3. If each variable in M0 is captured in an evaluation context E exactly once, the translation EM has 0 +M | †† EM | M0 M1 CM0 +M1 . The multiset M1 satisfies supp(M1 ) ⊆ supp(M0 ), | | supp(M0 ) Cut | FVM0 +M (E) | a unique decomposition M and the thick CM0 +M1 -node represents a family {CM0 (x)+M1 (x) }x∈supp(M0 ) of C-nodes. 9 v⋄ Ax ? (E2 )†∅ ! ? | | Ax (E2 )†∅ (E2 )†∅ v⋄ | | | | (v⋄ )≈ ↑ ? ! t† | | | | | Cut C | 99K3o | C | (E2 )†∅ t† ↑ (E2 )†∅ Cut v⋄ | | | C | ? 99Ko ! | EhAi†† ↑ | | ! ? Cut v⋄ Cut ! | | ↑ Ck | E1† v⋄ ? o | | s E1† | | (E2 )†∅ Cut Ax ↑ | | C E1† Ax ↑ ! Cut A‡ E1† →o (10) ? EhAi†† Cut == Ck+1 == | Cut ! | ? v⋄ | | Ck+1 ↑ | | == | t† C →o (9) | | E1† Ax E† ↑ (E2 )†[x] Cut E1† D A‡ Ck+1 | Cut Cut E† Ax == A† Ck+1 Ax ↑ | ↑ 99K4 o D Cut u† | t† Ax | | →o (8) u† ! | | | →s (12) t† ↑ | | (E2 )†∅ v⋄ Ax ↑ † EFV(v) (v⋄ )≈ ↑ ? ! † EFV(v) | t† Ax == EhAi† u† | | | | D A‡ Ax (E2 )†∅ | E1† Cut E1† A† v⋄ v⋄ | | o A‡ | | Figure 8: Illustration of Simulation (k > 0) 10 == ! (E2 )†∅ A‡ | | E1† ! | | (E2 )†∅ ? | ? s ↑ | ↑ | E† E1† | Cut A‡ C1 == | E† ↑ Cut == A† C1 u† | A‡ | o | | b ↑ Cut ? | C | o 99Ko C (E2 )†∅ ! | →s (13) | t† u† | t† ? | E† v⋄ Ax ↑ ↑ | Cut E† v⋄ Ax D | Cut ! | u† | A† ? Cut | ? | nnn Ck ` ↑ ! | | ` ↑ ! v⋄ | | C (E2 )†∅ == | | C ? →b (11) t† ! | | M0 M (E2 )†M0 +M t† x FVM0 +M (E2 )\x xCk Cut | | | † EM = 0 +M (E1 )† | | | M2 M (E2 )†M2 +M t† x x Ck x Cut | | M1 | | (by 2.) | = (E1 )† | | M (E2 )†† M x Ck C x Cut | | | | | = (E1 M1 | | | M2 t† | | x Cut (I.H.) )†† | | | C | | Cut † Figure 9: Decomposing EM 0 +M Proof. The proofs for 1. and 2. are by straightforward inductions on A and E respectively. The proof for 3. is by induction on the dimension of M0 , i.e. the size of the support set supp(M0 ). The base case where M0 = ∅ is obvious. In the inductive case, let x satisfy x ∈ M0 . Since x is captured exactly once in E by assumption, the evaluation context E can be decomposed as E = E1 hE2 [x ← t]i such that x is not captured in E1 or E2 . The evaluation context E2 satisfies x ∈k FVM0 +M (E2 ) for some positive multiplicity k. Moreover the multiset M0 can be decomposed as M0 = M1 + x + M2 such that all the variables in M1 (resp. M2 ) are only captured † †† in E1 (resp. E2 ). The translation EM can be decomposed as in Fig. 9. Finally, we let EM consist of 0 +M † †† (E2 )†† , t , (E ) . 1 M 4 Time Cost Analysis of Rewrites-First Interleaving 4.1 Recipe for Time Cost Analysis Our time cost analysis of the DGoIM_ follows Accattoli’s recipe, described in [2, 1], of analysing complexity of abstract machines. This section recalls the recipe and explains how it applies to the DGoIM_ . The time cost analysis focuses on how efficiently an abstract machine implements an evaluation strategy. In other words, we are not interested in minimising the number of β-reduction steps simulated by an abstract machine. Our interest is in making the number of transitions of an abstract machine “reasonable,” compared to the number of necessary β-reduction steps determined by a given evaluation strategy. Accattoli’s recipe assumes that an abstract machine has three groups of transitions: 1) “β-transitions” that correspond to β-reduction in which substitution is delayed, 2) transitions perform substitution, and 3) other “overhead” transitions. We incorporate this classification using the labels b, s, o of transitions. Another assumption of the recipe is that, each step of β-reduction is simulated by a single transition of an abstract machine, and so is substitution of each occurrence of a variable. This is satisfied by many known abstract machines including the SAM, however not by the DGoIM_ . The DGoIM_ has “finer” transitions and can take several transitions to simulate a single step of reduction (hence a single transition of the SAM, as we can observe in Thm. 3.5). In spite of this mismatch we can still follow the recipe, thanks to the weak simulation . It discloses what transitions of the DGoIM exactly correspond to β-reduction and substitution, and gives a concrete number of overhead transitions that the DGoIM_ needs to simulate β-reduction and substitution. The recipe for the time cost analysis is: 1. Examine the number of transitions, by means of the size of input and the number of β-transitions. 11 2. Estimate time cost of single transitions. 3. Derive a bound of the overall execution time cost. 4. Classify an abstract machine according to its execution time cost. The last step is accompanied by the following taxonomy of abstract machines introduced in [1]. Definition 4.1 (classes of abstract machines [1, Def. 7.1]). 1. An abstract machine is efficient if its execution time cost is linear in both the input size and the number of β-transitions. 2. An abstract machine is reasonable if its execution time cost is polynomial in the input size and the number of β-transitions. 3. An abstract machine is unreasonable if it is not reasonable. The input size in our case is given by the size |t| of a term t, inductively defined by: |x| := 1 |λx.t| := |t| + 1 |t u| := |t| + |u| + 1 |t[x ← u]| := |t| + |u| + 1. Given a sequence r of transitions (of either the SAM or the DGoIM_ ), we denote the number of transitions with a label x in r by |r|x . Since we use the fixed set {b, s, o} of labels, the length |r| of the sequence r is equal to the sum |r|b + |r|s + |r|o . 4.2 Number of Transitions We first estimate the number of transitions of the SAM, and then derive estimation for the DGoIM_ . Lemma 4.2 (quantitative bounds for SAM). Each execution e from an initial configuration (t0 , E)term , comes with the following inequalities: |e|s ≤ |e|b |e|o ≤ |t0 | · (5 · |e|b + 2) + (3 · |e|b + 1). Proof. The proof is analogous to the discussion in [2, Sec. 11]. Let |e|(8) , |e|(9) and |e|(10) be the numbers of transitions (8), (9) and (10) in e, respectively. The number |e|o is equal to the sum of these three numbers. The transition (11) introduces one explicit substitution, and the other transitions never increase the number of explicit substitution. In particular the transition (12) copies a pure value, that means no explicit substitutions are copied. Therefore we can bound the number of transitions that concern explicit substitutions, and obtain |e|(9) ≤ |e|b and |e|s ≤ |e|b . Each occurrence of the transition (10) in an execution is either the last transition of the execution or followed by the transitions (11), (12) and (13). This yields the inequality |e|(10) ≤ |e|b + |e|s + 1 and hence |e|(10) ≤ 2 · |e|b + 1. The transition (8) reduces the size of a pure term that is the first component of a configuration. The pure term is always a sub-term of the initial term t0 (Lem. 3.2). This means each maximal subsequence of an execution that solely consists of the transition (8) has at most the length |t0 |. Such a maximal subsequence either occurs at the end of an execution or is followed by transitions other than the transition (8). Therefore the number of these maximal sub-sequences is no more than |e|b + |e|s + |e|(9) + |e|(10) + 1 that can be bounded by 5 · |e|b + 2. A bound of the number |e|(8) can be given by multiplying these two bounds, namely we obtain |e|(8) ≤ |t0 | · (5 · |e|b + 2). Combining these bounds for the SAM with the weak simulation , we can estimate the number of transitions of the DGoIM_ as below. Proposition 4.3 (quantitative bounds for DGoIM_ ). Let r : s0 _∗ s be a sequence of transitions of the DGoIM_ . If there exists an execution (t0 , h·i)term →∗ (t, E) of the SAM such that s0  (t0 , h·i)term and s  (t, E), the sequence r comes with the following inequalities: |r|s ≤ |r|b |r|o ≤ 4 · |t0 | · (5 · |r|b + 2) + (16 · |r|b + 4). Proof. This is a direct consequence of Lem. 4.2 and Thm. 3.5. 12 4.3 Execution Time Cost We already discussed time cost of single transitions of the DGoIM in Sec. 2.3. It is worth noting that the discussion in Sec. 2.3 is independent of any particular choice of a rewriting and token-passing interleaving strategy. Thm. 4.4 below gives a bound of execution time cost of the DGoIM_ . We can conclude that, according to Accattoli’s taxonomy (see Def. 4.1), the DGoIM_ is “efficient” as an abstract machine for the call-by-need evaluation. Theorem 4.4 (time cost). Let C, D be fixed natural numbers, and r : s0 _∗ s be a sequence of transitions of the DGoIM_ . If there exists an execution (t0 , h·i)term →∗ (t, E) of the SAM such that s0  (t0 , h·i)term and s  (t, E), the total time cost T (r) of the sequence r satisfies: T (r) = O((|t0 | + C) · (|r|b + D)). Proof. We estimated in Sec. 2.3 that time cost of single transitions, except for the rewrite transitions (4) and (6), is constant. Time cost of these rewrite transitions (4) and (6) depends on the number of doors and/or nodes of a !-box. Since the sequence r of transitions simulates an execution of the SAM, every !-box concerned in r arises as the translation of a value v. By definition of the translation (·)† (Fig. 6), the graph v † is a !-box, with as many auxiliary doors as occurrences of free variables in v. The number of auxiliary doors is no more than the number of nodes in the !-box, due to the well-boxed-ness condition, that is linear in the size |v| of the value. Moreover the value v appears as the first component of a context configuration, and therefore it is a sub-term of the initial term t0 (Lem. 3.2). As a result, time cost of each occurrence of the rewrite transitions (4) and (6) in the sequence r is linear in the size t0 . The bound of the total time cost T (r) of the sequence r is given by combining these estimations for single transitions with the results of Prop. 4.3. Corollary 4.5. The DGoIM_ is an efficient abstract machine, in the sense of Def. 4.1. 5 Conclusions We introduced the DGoIM, which can interleave token passing with graph rewriting informed by the trajectory of the token. We focused on the rewrites-first interleaving and proved that it enables the DGoIM to implement the call-by-need evaluation strategy. The quantitative analysis of time cost certified that the DGoIM_ gives an “efficient” implementation in the sense of Accattoli’s classification. The proof of Thm. 4.4 pointed out that eliminating and copying !-boxes are two main sources of time cost. Our results are built on top of a weak simulation of the SAM, that relates several transitions of the DGoIM to each computational task such as β-reduction and substitution. The main feature of the DGoIM is the flexible combination of interaction and rewriting. We here briefly discuss how the flexibility can enable the DGoIM to implement evaluation strategies other than the call-by-need. As mentioned in Sec. 1.2, the passes-only interleaving yields an ordinary token-passing abstract machine that is known to implement the call-by-name evaluation. Because no rewrites are triggered, as oppose to the rewrites-first interleaving, a token not only can pass a principal door but also can go inside a !-box and pass an auxiliary door. These behaviours are in fact not possible with the DGoIM presented in this paper; the transitions and data carried by a token are tailored to the rewrites-first interleaving. To recover an ordinary token-passing machine, we therefore need to add pass transitions that involve auxiliary doors and data structures (so-called “exponential signatures”) that deal with !-boxes, for example. The only difference between the call-by-need and the call-by-value evaluations lies in when function arguments are evaluated. In the DGoIM, this corresponds to changing a trajectory of a token so that it visits function arguments immediately after it detects function application. Therefore, to implement the call-by-value evaluation, the DGoIM can still use the rewrites-first interleaving, but it should use a modified set of pass transitions. Further refinements, not only of the evaluation strategies but also of the graph representation could yield even more efficient implementation, such as full lazy evaluation, as hinted in [26]. Our final remarks concern programming features that have been modelled using token-passing abstract machines. Ground-type constants are handled by attaching memories to either nodes of a graph or a token, in e.g. [20, 18, 3] — this can be seen as a simple form of graph rewriting. 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Revisiting Connected Vertex Cover: FPT Algorithms and Lossy Kernels R. Krithika, Diptapriyo Majumdar, and Venkatesh Raman arXiv:1711.07872v1 [] 21 Nov 2017 The Institute of Mathematical Sciences, HBNI, Chennai, India. {rkrithika|diptapriyom|vraman}@imsc.res.in Abstract The Connected Vertex Cover problem asks for a vertex cover in a graph that induces a connected subgraph. The problem is known to be fixed-parameter tractable (FPT), and is unlikely to have a polynomial sized kernel (under complexity theoretic assumptions) when parameterized by the solution size. In a recent paper, Lokshtanov et al. [STOC 2017], have shown an α-approximate kernel for the problem for every α > 1, in the framework of approximate or lossy kernelization. We exhibit lossy kernels and FPT algorithms for Connected Vertex Cover for parameters that are more natural and functions of the input, and in some cases, smaller than the solution size. Our first result is a lossy kernel for Connected Vertex Cover parameterized by the size k of a split deletion set. A split graph is a graph whose vertex set can be partitioned into a clique and an independent set and a split deletion set is a set of vertices whose deletion results in a split graph. Let n denote the number of vertices in the input graph. We show that • Connected Vertex Cover parameterized by the size k of a split deletion set d 2α−1 α−1 e ) vertices and a admits an α-approximate kernel with O(k + (2k + d 2α−1 α−1 e) k O(1) 3 n time algorithm. • For the special case when the split deletion set is a clique deletion set, the algorithm runs in 2k nO(1) time and the lossy kernel has O(k + d 2α−1 α−1 e) vertices. To the best of our knowledge, this (approximate) kernel is one of the few lossy kernels for problems parameterized by a structural parameter (that is not solution size). We extend this lossy kernelization to Connected Vertex Cover parameterized by an incomparable parameter, and that is the size k of a clique cover. A clique cover of a graph is a partition of its vertex set such that each part induces a clique. We show that • Connected Vertex Cover parameterized by the size k of a clique cover is W[1]-hard but admits an α-approximate kernel with O(kd 2α−1 α−1 e) vertices for every α > 1. This is one of the few problems that are not FPT but admit a lossy kernel. Then, we consider the size of a cluster deletion set as parameter. A cluster graph is a graph in which every component is a complete graph and a cluster deletion set is a set of vertices whose deletion results in a cluster graph. We show that • Connected Vertex Cover parameterized by the size k of a cluster deletion set is FPT via an 4k nO(1) time algorithm and admits an α-approximate kernel with α k α d α−1 e O(k2 + d 2α−1 ) vertices for every α > 1. α−1 e · α−1 + d α−1 e · k • For the special case when the cluster deletion set is a degree-1 modulator (a subset of vertices whose deletion results in a graph with maximum degree 1), the FPT k α algorithm runs in 3k nO(1) time and the lossy kernel has O(k2 + α−1 + d α−1 e· α d α−1 e k ) vertices. 1 Finally, we consider Connected Vertex Cover parameterized by the size k of a chordal deletion set. A chordal graph is a graph in which every cycle of length at least 4 has a chord – an edge joining non-adjacent vertices of the cycle. A chordal deletion set of a graph is a subset of vertices whose deletion results in a chordal graph. We show that Connected Vertex Cover parameterized by k is FPT using the known algorithm for Connected Vertex Cover on bounded treewidth graphs. 1 Introduction, Motivation and Our Results A vertex cover in a graph is a set of vertices that has at least one endpoint from every edge of the graph. In the Vertex Cover problem, given a graph G and an integer `, the task is to determine if G has a vertex cover of size at most `. Undoubtedly, Vertex Cover is one of the most well studied problems in parameterized complexity. In this framework, each problem instance is associated with a non-negative integer called parameter. The pair consisting of a decision problem and a parameterization is called a parameterized problem. A common parameter is a bound on the size of an optimum solution for the problem instance. A problem is said to be fixed-parameter tractable (FPT) with respect to parameter k if it can be solved in f(k)nO(1) time for some computable function f, where n is the input size. Such an algorithm is called a parameterized algorithm or FPT algorithm. For convenience, the running time f(k)nO(1) where f grows super-polynomially with k is denoted as O∗ (f(k)). A kernelization algorithm is a polynomial-time algorithm that transforms an arbitrary instance of the problem to an equivalent instance of the same problem whose size is bounded by some computable function g of the parameter of the original instance. The resulting instance is called a kernel and if g is a polynomial function, then it is called a polynomial kernel and we say that the problem admits a polynomial kernel. In order to classify parameterized problems as being FPT or not, the W-hierarchy: FPT ⊆ W[1] ⊆ W[2] ⊆ · · · ⊆ XP is defined. It is believed that the subset relations in this sequence are all strict and a parameterized problem that is hard for some complexity class above FPT in this hierarchy is unlikely to be FPT. A parameterized problem is said to be in XP if it has an algorithm with runtime f(k)ng(k) time for some computable functions f and g. A problem is said to be para-NP-hard if it is not in XP unless P=NP. The complexity classes FPT and para-NP can be viewed as the parameterized analogues of P and NP. Vertex Cover is FPT via an easy O∗ (2` ) algorithm and after a long race, the current fastest algorithm runs in O∗ (1.2738` ) time [6]. Vertex Cover also has a kernel with 2` − O(log `) vertices and O(`2 ) edges [21]. In the Connected Vertex Cover problem, we seek a connected vertex cover, i.e., a vertex cover that induces a connected subgraph. It is easy to observe that Vertex Cover reduces to Connected Vertex Cover implying that the latter is at least as hard as the former in general graphs. In fact, Connected Vertex Cover is NP-hard even on bipartite graphs [13] where Vertex Cover is solvable in polynomial time (Theorem 2.1.1 [11]). However, Connected Vertex Cover is polynomialtime solvable on chordal graphs and sub-cubic graphs [13,26] and has an O∗ (2O(t) ) algorithm when the input graph has treewidth upper bounded by t [2]. The best known parameterized algorithm for Connected Vertex Cover takes O∗ (2` ) time where ` is the size of the connected vertex cover that we are looking for [7]. Further, the problem does not admit a polynomial kernel unless the polynomial hierarchy collapses (specifically, unless NP ⊆ coNP/poly) [12]. As the goal in parameterized algorithms is to eventually solve the given instance of a problem, the application of a (classical) kernelization algorithm is typically followed by an exact or approximation algorithm that finds a solution for the reduced instance. However, the current definition of kernels provide no insight into how this solution relates to a solution 2 for the original instance and the basic framework is not amenable to be a precursor to approximation algorithms or heuristics. Recently, Lokshtanov et al. [22] proposed a new framework, referred to as lossy kernelization, that is a bit less stringent than the notion of polynomial kernels and combines well with approximation algorithms and heuristics. The key new definition in this framework is that of an α-approximate kernelization. The precise definitions are deferred to Section 2. Informally, an α-approximate polynomial kernelization (lossy kernelization) is a polynomial time preprocessing algorithm that takes as input an instance of a parameterized problem and outputs another instance of the same problem whose size is bounded by a polynomial function of the parameter of the original instance. Additionally, for every c ≥ 1, a c-approximate solution for the reduced instance can be turned into a (αc)-approximate solution for the original instance in polynomial time. The authors of [22] exhibit lossy polynomial kernels for several problems that do not admit classical polynomial kernels including Connected Vertex Cover parameterized by the solution size. In this paper, we extend this lossy kernel for Connected Vertex Cover to parameters that are more natural and functions of the input, and in some cases, smaller than the solution size. In the initial work on parameterized complexity, the parameter was almost always the solution size (with treewidth being a notable exception). A recent trend is to study the complexity of the given problem with respect to structural parameters that are more likely to be small in practice. Also, once a problem is shown to be FPT or to have a polynomial sized kernel by a parameterization, it is natural to ask whether the problem is FPT (and admits a polynomial kernel) when parameterized by a smaller parameter. Similarly, once a problem is shown to be W-hard by a parameterization, it is natural to ask whether the problem is FPT when parameterized by a larger parameter. Structural parameterizations of Vertex Cover [3, 18], Feedback Vertex Set [20] and Graph Coloring [19] have been explored extensively. We refer to [18] and [17] for a detailed introduction to the whole program. A parameter that has gained significant attention recently is the size of a modulator to a family of graphs. Let F denote a hereditary graph class (which is closed under induced subgraphs) on which Vertex Cover is polynomial-time solvable. Suppose S is a set of vertices (called a F-modulator) such that G − S ∈ F. Then, Vertex Cover is FPT when parameterized by the size of |S|. Following is an easy O∗ (2|S| ) algorithm: guess the intersection of the required solution with the modulator and solve the problem on the remaining graph in polynomial time. In contrast, a similar generic result does not seem easy for Connected Vertex Cover and a study of such structural parameterizations for Connected Vertex Cover is another goal of this paper. Our Results. The starting point of our study is to extend the lossy kernelization known for Connected Vertex Cover parameterized by solution size to Connected Vertex Cover parameterized by a smaller parameter. A split graph is a graph whose vertex set can be partitioned into a clique and an independent set. Split graphs can be recognized in polynomial time and such a partition can be obtained in the process [16]. A split deletion set is a set of vertices whose deletion results in a split graph. Connected Vertex Cover has an easy polynomial time algorithm in split graphs, and it is easy to verify that the size of minimum connected vertex cover is at least the size of the minimum vertex cover which is at least the size of the minimum split deletion set. We show that • Connected Vertex Cover parameterized by the size k of a split deletion set is FPT via an O∗ (3k ) algorithm and that the α-approximate kernel for Connected Vertex Cover parameterized by solution size can be extended in a non-trivial way to an d 2α−1 e α−1 ) vertices for every α > 1. This α-approximate kernel with O(k + (2k + d 2α−1 α−1 e) adds to the small list of problems for which lossy kernels with respect to structural 3 parameterizations are known. • For the special case when the split deletion set is a clique deletion set (a set of vertices whose deletion results in a complete graph), the algorithm runs in O∗ (2k ) time and the lossy kernel has O(k + d 2α−1 α−1 e) vertices (linear size). Then, we consider Connected Vertex Cover parameterized by the size of a clique cover. A clique cover of a graph is a partition of its vertex set such that each part induces a clique. We show that • Connected Vertex Cover parameterized by the size k of a clique cover is W[1]-hard but admits an α-approximate kernel with O(kd 2α−1 α−1 e) vertices for every α > 1. This is one of the few problems that are not FPT but admit a lossy kernel. Then, we consider a parameter related (structurally) to clique cover size, namely, the size of a cluster deletion set and show that Connected Vertex Cover is FPT with respect to this parameter. A cluster graph is a graph in which every component is a complete graph and a cluster deletion set is a set of vertices whose deletion results in a cluster graph. We show that • Connected Vertex Cover parameterized by the size k of a cluster deletion set is FPT via an O∗ (4k ) algorithm and admits an α-approximate kernel with O(k2 + α d α−1 e k α d 2α−1 ) vertices for every α > 1. α−1 e · α−1 + d α−1 e · k • For the special case when the cluster deletion set is a degree-1 modulator (a subset of vertices whose deletion results in a graph with maximum degree 1), the algorithm α α k runs in O∗ (3k ) time and the lossy kernel has O(k2 + α−1 + d α−1 e · kd α−1 e ) vertices. Finally, we consider Connected Vertex Cover parameterized by the size k of a chordal deletion set. A chordal graph is a graph in which every cycle of length at least 4 has a chord – an edge joining non-adjacent vertices of the cycle. A chordal deletion set of a graph is a subset of vertices whose deletion results in a chordal graph. We show that Connected Vertex Cover parameterized by k is FPT using the known algorithm for Connected Vertex Cover on bounded treewidth graphs. A summary of these results is presented in Figure 1. Techniques. Our FPT algorithms involve a reduction to the Steiner Tree problem on bipartite graphs. Given a graph G, an integer p and a set T (called terminals) of vertices of G, the Steiner Tree problem is the task of determining whether G contains a tree (called Steiner tree) on at most p vertices that contains T . A bipartite graph is a graph whose vertex set can be partitioned into 2 independent sets. Such a partition is called a bipartition. We use the following result known for Steiner Tree on bipartite graphs. Lemma 1. [7, 25] Given a connected bipartite graph G with bipartition (P, Q), there is an algorithm running in O∗ (2|Q| ) time that computes a minimum set X of vertices of G such that Q ⊆ X and G[X] is connected. Our lower bound results are based on the hardness results known for the Set Cover problem. Given a family F of subsets of a universe U and a positive integer `, the Set Cover problem isSthe task of determining whether there is a subfamily F 0 ⊆ F of size at most ` such that X∈F 0 X = U. Set Cover can be solved in O∗ (2|U| ) time by a dynamic programming routine [14] and the Set Cover Conjecture states that no O∗ ((2 − )|U| ) time algorithm exists for any  > 0 [8]. Our lower bound results are based on the following result that is a consequence of this conjecture. 4 Connected Vertex Cover Clique Deletion Set Vertex Cover Split Deletion Set Modulator Cluster Deletion Set Feedback Vertex Set Degree-1 Chordal Deletion Set Degree-2 Modulator Degree-i(i 3) Modulator FPT FPT and admits P SAKS para-N P -hard Figure 1: Ecology of Parameters for Connected Vertex Cover. The parameter values are the minimum possible for a given graph. An arrow from parameter x to parameter y means y ≤ x. Lemma 2. [8] Connected Vertex Cover parameterized by the solution size ` has no O∗ ((2 − )` ) time algorithm under the Set Cover Conjecture. For each of the parameterized problems considered in this paper, we assume that the modulator (whose size is the parameter) under consideration is given as part of the input. In most cases, this assumption is reasonable. For instance, there is an algorithm by Cygan and Pilipczuk [10] that runs in O∗ (1.2732k+o(k) ) time and returns a split deletion set of size at most k (or decides that no such set exists). In the case of cluster deletion set size as parameter, we use the algorithm by Boral et al. [5] that runs in O∗ (1.9102k ) time and returns a cluster deletion set of size at most k, if one exists. Also, a degree-1 modulator S of size at most k, if one exists, can be obtained in O∗ (1.882k ) time using the algorithm by Wu [27]. Finally, there is an algorithm by Marx [23] that runs in O∗ (2O(k log k) ) time and either outputs a chordal deletion set of size k or decides that no such set exists. 2 Preliminaries We refer to [11] for graph theoretic terms and notation that are not explicitly
defined here. A We use [r] to denote the set {1, 2, . . . , r}. Given a finite set A, we use k to denote the
A collection of subsets of A containing exactly k elements and we use ≤k to denote the collection of subsets of A containing at most k elements. For a graph G, V(G) and E(G) denote the set of vertices and edges respectively. Two vertices u, v are said to be adjacent if there is an edge {u, v} in the graph. The neighbourhood of a vertex v, denoted by NG (v), is the set of vertices adjacent to v and its degree dG (v) is |NG (v)|. The subscript in the notation for neighbourhood and degree is omitted if the 5 graph under consideration is clear. For a set S ⊆ V(G), G − S denotes the graph obtained by deleting S from G and G[S] denotes the subgraph of G induced by set S. The contraction operation of an edge e = uv in G results in the deletion of u and v and the addition of a new vertex w adjacent to vertices that were adjacent to either u or v. Any parallel edges added in the process are deleted so that the graph remains simple. This operation is extended to a subset of edges and the resultant graph is oblivious to the contraction sequence. A path is a sequence of distinct vertices where every consecutive pair of vertices are adjacent. A set of pairwise non-adjacent vertices is called as an independent set and a set of pairwise adjacent vertices is called as a clique. A complete graph is a graph whose vertex set is a clique. The number of components of a graph G is denoted by #comp(G) and by convention if G is connected then #comp(G) is 1. Two non-adjacent vertices u and v are called false twins if N(u) = N(v). A tree-decomposition of a graph G is a pair (T, X = {Xt }t∈V(T) ) such that T is a tree whose every node t is assigned a vertex subset Xt , called a bag, satisfying the following properties. S • t∈V(T) Xt = V(G). • For every edge {x, y} ∈ E(G) there is a t ∈ V(T) such that {x, y} ⊆ Xt . • For every vertex v ∈ V(G) the subgraph of T induced by the set {t | v ∈ Xt } is connected. The width of a tree decomposition is maxt∈V(T) |Xt | − 1 and the treewidth of G, denoted by tw(G), is the minimum width over all tree decompositions of G. Lossy Kernelization: Parameterized complexity terminology and definitions not stated here can be found in [9]. We now state terminology and definitions related to lossy kernelization given in [22]. The key definition in this framework is the notion of a parameterized optimization (maximization / minimization) problem which is the parameterized analogue of an optimization problem in the theory of approximation algorithms. A parameterized minimization problem is a computable function Π : Σ∗ × N × Σ∗ 7→ R ∪ {±∞}. The instances of Π are pairs (I, k) ∈ Σ∗ × N and a solution for (I, k) is a string S ∈ Σ∗ such that |S| ≤ |I| + k. The value of a solution S is Π(I, k, S). The optimum value of (I, k) is OPTΠ (I, k) = min Π(I, k, S), and an optimum solution for (I, k) is a solution S such S∈Σ∗ , |S|≤|I|+k that Π(I, k, S) = OPTΠ (I, k). A parameterized maximization problem is defined in a similar way. We will omit the subscript Π in the notation for optimum value if the problem under consideration is clear from context. Next, we define the notion of a strict α-approximate polynomial-time preprocessing algorithm for Π. It is defined as a pair of polynomial-time algorithms, called the reduction algorithm and the solution lifting algorithm, that satisfy the following properties. • Given an instance (I, k) of Π, the reduction algorithm computes an instance (I 0 , k 0 ) of Π. • Given the instances (I, k) and (I 0 , k 0 ) of Π, and a solution S 0 to (I 0 , k 0 ), the solution Π(I,k,S) Π(I 0 ,k 0 ,S 0 ) lifting algorithm computes a solution S to (I, k) such that OPT(I,k) ≤ max{ OPT(I 0 ,k 0 ) , α}. A reduction rule is the execution of the reduction algorithm on an instance, and we say that it is applicable on an instance if the output instance is different from the input instance. An α-approximate kernelization (or α-approximate kernel) for Π is an α-approximate polynomialtime preprocessing algorithm such that the size of the output instance is upper bounded by a computable function g : N → N of k. A reduction rule is said to be α-safe for Π if there 6 is a solution lifting algorithm, such that the rule together with this algorithm constitute a strict α-approximate polynomial-time preprocessing algorithm for Π. A reduction rule is safe if it is 1-safe. Observe that this definition is more strict that the definition of safeness in classical kernelization. A polynomial-size approximate kernelization scheme (PSAKS) for Π is a family of α-approximate polynomial kernelization algorithms for each α > 1. Note that, the size of an output instance of a PSAKS, when run on (I, k) with approximation parameter α, must be upper bounded by f(α)kg(α) for some functions f and g independent of |I| and k. A PSAKS is said to be time efficient if the reduction algorithm and the solution lifting algorithm run in f(α)|I|c , for some computable function f. We encourage the reader to see [22] for a more comprehensive discussion of these ideas and definitions. 3 Connected Vertex Cover parameterized by Split Deletion Set In this section, we describe an FPT algorithm and a lossy polynomial kernel for Connected Vertex Cover parameterized by the size of a split deletion set. Recall that a split deletion set is a set of vertices whose deletion results in a split graph. 3.1 An FPT algorithm In this subsection we describe an O∗ (3k ) time algorithm for Connected Vertex Cover parameterized by the size k of a split deletion set. Theorem 3. Given a graph G, a split deletion set S and a positive integer `, there is an algorithm that determines whether G has a connected vertex cover of size at most ` in O∗ (3|S| ) time. Moreover, no O∗ ((2 − )|S| ) time algorithm exists for any  > 0 for this problem under the Set Cover Conjecture. Further, if S is a clique deletion set, then there is an algorithm that runs in O∗ (2|S| ) time. Proof. Let H denote the split graph G − S whose vertex set can be partitioned into a clique C and an independent set I. Let |S| = k and let X∗ be a connected vertex cover of G of size at most ` (if one exists). We show that once we guess the intersection of X∗ with S and C, the problem reduces to a version of the Steiner tree problem with at most |S| terminals. As X∗ can afford to exclude at most one vertex from C, there are at most |C| + 1 choices for X∗ ∩ C. Also, there are at most 2k choices for X∗ ∩ S. Consider one such choice (Y, Z) where Y = X∗ ∩ C and Z = X∗ ∩ S. Let T denote the set Y ∪ Z. We will extend T into a connected vertex cover of G. If (C \ Y) ∪ (S \ Z) is not an independent set, no such connected vertex cover exists and we skip to the next choice of (Y, Z). Let us first consider the case when I = ∅. That is, S is a clique deletion set of G. In this case, as we have already made our choice in S and C, and so if G[T ] is not connected, we can skip to the next choice of (Y, Z). If none of the choices leads to the required solution, we declare that G has no connected vertex cover of size at most `. The overall running time of the algorithm is O∗ (2k ). When I 6= ∅, observe that G[T ] has at most |Z| + 1 components. Let R denote the set (N(C \ Y) ∪ N(S \ Z)) \ (Y ∪ Z). Clearly, R ⊆ I and has to be included in the solution. See Figure 2. If there is a vertex r ∈ R that has no neighbours in T , then r cannot be connected to T by vertices from I as R ⊆ I. Therefore, if there is such a vertex r, then we skip to the next choice of (Y, Z). Otherwise, by adding R to T , #comp(G[T ]) cannot increase in this process. The problem now reduces to finding a set J ⊆ I \ R such that G[T ∪ R ∪ J] is connected. b with bipartition (T ∗ , I \ R) where each vertex of Now consider the bipartite graph G ∗ T corresponds to a component of G[T ∪ R]. We draw an edge between a vertex p ∈ I \ R 7 S = split deletion set Z H=G Y R S • Y [ Z = Vertices included in solution. • R = Vertices that are forced into solution by choice of Y [ Z. J • J = Vertices that are required to connect the components of G[Y [ Z [ R]. C I Clique Independent Set Figure 2: Connected vertex cover parameterized by split deletion set. and a vertex c in T ∗ if there is an edge in G between p and a vertex in the component b with terminal set corresponding to c. The task now is to find a minimum Steiner tree of G ∗| ∗ ∗ |T T which can be done Pkin Ok
(2i ) time using Lemma 1. Therefore, the overall running time of the algorithm is i=0 i 2 where the first term in the product denotes the number of choices for X∗ ∩ S where |X∗ ∩ S| = i and the second term is the time taken to find a minimum Steiner tree for an instance with at most i terminals. Thus, the algorithm runs in O∗ (3k ) time. As an edgeless graph is a split graph, we have that the size of a minimum connected vertex cover is at least the size of a minimum connected split deletion set. Therefore, the claimed lower bound follows from Lemma 2. In order to obtain a tighter and refined analysis of the running time of the above algorithm, we generalize the following lemma. Here VC(H) denotes the set of vertex covers of H. P Lemma 4. [7] Let H be a connected graph on h vertices. Then, 2#comp(H[C]) ≤ C∈VC(H) 3· 2h−1 . We next generalize Lemma 4 to graphs that are not necessarily connected. P Lemma 5. Let H be a graph on h vertices with #comp(H) = d. Then, 2#comp(H[C]) ≤ C∈VC(H) 3d 2h−d . Proof. Let H1 , · · · , Hd denote the components of H and let hi denote the number of vertices in Hi . Then, any vertex cover of H is the union of the vertex covers of Hi for 1 ≤ i ≤ d. Thus, we have the following bound from Lemma 4. X C∈VC(H) 2#comp(H[C]) = d Y X 2#comp(Hi [C]) ≤ i=1 C∈VC(Hi ) d Y i=1 3 · 2hi −1 = 3d · 2h−d Now, using Lemma 5 and Theorem 3, we have the following result. Corollary 6. Given a graph G, a split deletion set S with #comp(G[S]) = d and a positive integer `, there is an algorithm that determines whether G has a connected vertex cover of size at most ` in O∗ (3d 2|S|−d ) time. 8 Proof. The running time of the presented in Theorem 3 is upper bounded (ignoring P algorithm polynomial factors) by 2#comp(G[Z]) which in turn is upper bounded by 3d 2|S|−d Z∈VC(G[S]) from Lemma 5. Therefore, the claimed bound follows. Using Lemma 2, Corollary 6 and the fact that a vertex cover of size k (if one exists) can be obtained in O∗ (1.2738` ) time [6], we have the following result. Corollary 7. Given a graph G, a vertex cover S with #comp(G[S]) = d and a positive integer `, there is an algorithm that determines whether G has a connected vertex cover of size at most ` in O∗ (3d 2|S|−d ) time. Further, no O∗ ((2 − )|S| ) time algorithm exists for this problem for any  > 0 under the Set Cover Conjecture. 3.2 A Lossy Kernel It is known that Vertex Cover parameterized by the size of a clique deletion set has no polynomial kernel unless NP ⊆ coNP/poly [4]. As Vertex Cover reduces to Connected Vertex Cover by just adding a universal vertex to the graph, we have the following observation. Observation 1. Connected Vertex Cover parameterized by the size of a clique deletion set has no polynomial kernel unless NP ⊆ coNP/poly. In this section, we show that Connected Vertex Cover parameterized by the size of a split deletion set admits a lossy polynomial kernel. As a consequence, we show that Connected Vertex Cover parameterized by the size of a clique deletion set admits a lossy linear kernel. The parameterized minimization problem definition of interest to us is the following. The cases specified in the definition are ordered, i.e., given (G, S), an integer k and a set T ⊆ V(G), CVC((G, S), k, T ) is assigned the first value applicable.   −∞ if |S| > k or G − S is not a split graph  CVC((G, S), k, T ) = ∞ if T is not a connected vertex cover   |T | otherwise Without loss of generality, we assume that G has no isolated vertices as no minimum connected vertex cover contains them. Given α > 1, let d be the minimum integer greater d−1 than 1 such that α ≥ d−2 . That is, d = d 2α−1 α−1 e. Let H denote the split graph G − S whose vertex set can be partitioned into a clique C and an independent set I. First, we bound the number of vertices in C by applying Reduction Rule 3.1. Reduction Rule 3.1. If |C| ≥ d, then add a new vertex uC adjacent to N(C) and delete C to get the graph G 0 . The resulting instance is (G 0 , S). Observe that S continues to be a split deletion set of G 0 as a result of applying this rule since V(G 0 − S) can be partitioned into the clique {uC } and the independent set I. Lemma 8. Reduction Rule 3.1 is α-safe. Proof. Consider a solution D 0 of the reduced instance. If uC ∈ D 0 , then the solution lifting algorithm returns D = (D 0 \ {uC }) ∪ C which is a connected vertex cover of G such that CVC((G, S), k, D) = CVC((G 0 , S), k, D 0 ) − 1 + |C|. Otherwise, we know that N(C) ⊆ D 0 . Let x be a vertex in C such that C \ {x} has a neighbour in V(G 0 ) \ C. Then, D = D 0 ∪ (C \ {x}) is a connected vertex cover of G such that CVC((G, S), k, D) = CVC((G 0 , S), k, D 0 ) + |C| − 1. In any case, we have shown that CVC((G, S), k, D) = CVC((G 0 , S), k, D 0 ) + |C| − 1. Now, consider an optimum solution D∗ for the original 9 instance. Clearly, |D∗ ∩ C| ≥ |C| − 1. Then, (D∗ \ C) ∪ {uC } is a connected vertex cover of G 0 . Hence, OPT((G 0 , S), k) ≤ OPT((G, S), k) − |C| + 1 + 1. Combining these bounds, we have CVC((G,S),k,D) CVC((G 0 ,S),k,D 0 ) |C|−1 CVC((G 0 ,S),k,D 0 ) OPT((G,S),k) ≤ max OPT((G 0 ,S),k) , |C|−2 ≤ max OPT((G 0 ,S),k) , α} as |C| ≥ d. Observe that the application of Reduction Rule 3.1 does not make any vertex isolated in the reduced graph. Further, when S is a clique deletion set and this rule is not applicable, it follows that G − S has at most d − 1 vertices. This leads to the following result. Theorem 9. Connected Vertex Cover parameterized by the size k of a clique deletion set has a time efficient PSAKS with at most k + d 2α−1 α−1 e vertices. Proof. Let (G, S, `) be an instance on which Reduction Rule 3.1 is not applicable. Then, |S| ≤ k and |V(G − S)| ≤ d where d = d 2α−1 α−1 e. Therefore, |V(G)| ≤ k + d. Let us now consider the case when S is not a clique deletion set. Lemma 10. Let (G, S) be an instance on which Reduction Rule 3.1 is not applicable. Then, G has a connected vertex cover of size at most 2k + d − 1. Proof. If Reduction Rule 3.1 is not applicable, then |C| ≤ d − 1. Now, T = S ∪ C is a vertex cover of G such that G[T ] has at most |S| + 1 components. If G[T ] is not connected, then we can add at most |S| vertices from V(G) \ T to T so that G[T ] becomes connected. Such a choice of vertices exists as G is connected. Hence, it follows that G has a connected vertex cover of size at most |T | + |S| ≤ 2k + d − 1. From Lemma 10, we have OPT((G, S), k) ≤ 2k + d − 1. Therefore, any vertex with at least 2k + d neighbours is present in any optimal solution. We define a partition of vertices of G into the following three parts. • B = {u ∈ V(G) | d(u) ≥ 2k + d} • IB = {v ∈ V(G) \ B | N(v) ⊆ B} • R = V(G) \ (B ∪ IB ) We apply the following two reduction rules which are known from [22] to bound IB (which is an independent set). Reduction Rule 3.2. If there exists u ∈ IB ∩ V(G − S) such that dG (u) ≥ d, then delete NG [u] and add a new vertex w adjacent to every vertex in NG (NG (u)) \ {u}. Further, add 2k + d new vertices W adjacent to w to get the graph G 0 . The resulting instance is (G 0 , (S ∪ {w}) \ N(u)). Observe that S 0 = (S ∪ {w}) \ N(u) is a split deletion set of G 0 as V(G 0 − S 0 ) can be partitioned into the clique C ∩ V(G 0 − S 0 ) and independent set (I ∩ V(G 0 − S 0 )) ∪ W. Further, as |C| ≤ d − 1 and dG (u) ≥ d, it follows that N(u) ∩ S 6= ∅. As we add at most one vertex to S and delete at least one vertex from S to get S 0 , we have |S 0 | ≤ |S|. Lemma 11. Reduction Rule 3.2 is α-safe. Proof. Consider a solution D 0 of the reduced instance. If w ∈ / D 0 , the solution lifting algorithm returns a connected vertex cover D of G of size at most 2k + d − 1 obtained from Lemma 10. Then, we have CVC((G 0 , S 0 ), k, D 0 ) ≥ 2k + d since w has at least 2k + d neighbours and CVC((G, S), k, D) ≤ 2k + d − 1 implying CVC((G, S), k, D) ≤ 10 CVC((G 0 , S 0 ), k, D 0 ). Otherwise, we have w ∈ D 0 and the solution lifting algorithm returns D = (D 0 \ (W ∪ {w})) ∪ N[u] which is a connected vertex cover of G such that CVC((G, S), k, D) ≤ CVC((G 0 , S 0 ), k, D 0 ) − 1 + |N[u]|. Next, consider an optimum solution D∗ for the original instance. Clearly, |D∗ | ≤ 2k + d − 1 from Lemma 10. Thus, B ⊆ D∗ . In particular, NG (u) ⊆ D∗ . Then, (D∗ \ NG [u]) ∪ {w} is a connected vertex cover of G 0 . Hence, OPT((G 0 , S 0 ), k) ≤ OPT((G, S), k) − |NG (u)| + 1. Combining these bounds, 0 ,S 0 ),k,D 0 ) CVC((G 0 ,S 0 ),k,D 0 ) |N(u)| we have CVC((G,S),k,D) as ≤ max CVC((G OPT((G,S),k) ≤ max OPT((G 0 ,S 0 ),k) , |N(u)|−1 OPT((G 0 ,S 0 ),k) , α |N(u)| ≥ d. Recall that two non-adjacent vertices u and v are called false twins if N(u) = N(v). Reduction Rule 3.3. If there exists x ∈ IB ∩ V(G − S) such that x has at least 2k + d false twins in IB ∩ V(G − S), then delete x. The resulting instance is (G − {x}, S \ {x}). Observe that S 0 = S \ {x} continues to be a split deletion set of G 0 = G − {x} as a result of applying this rule. Lemma 12. Reduction Rule 3.3 is 1-safe. Proof. Consider a solution D 0 of the reduced instance. If |D 0 | ≥ 2k + d, the solution lifting algorithm returns a connected vertex cover D of G of size at most 2k + d − 1 obtained from Lemma 10. Then, we have CVC((G 0 , S 0 ), k, D 0 ) ≥ 2k+d and CVC((G, S), k, D) ≤ 2k+d−1 implying CVC((G, S), k, D) ≤ CVC((G 0 , S 0 ), k, D 0 ). Otherwise, it follows that one of the false twins of x, say y, is excluded from D 0 . Thus, N(y) ⊆ D 0 implying that N(x) ⊆ D 0 . Then, the solution lifting algorithm returns D = D 0 which is a connected vertex cover of G such that CVC((G, S), k, D) = CVC((G 0 , S 0 ), k, D 0 ). Next, consider an optimum solution D∗ for the original instance. Clearly, |D∗ | ≤ 2k + d − 1 from Lemma 10. Thus, either x and one of its false twins y is excluded from D∗ or two of the false twins of x, say y and z are excluded from D∗ . In any case, we have another optimal connected vertex cover D∗∗ of G 0 that excludes x. Hence, OPT((G 0 , S 0 ), k) ≤ OPT((G, S), k). Combining these bounds, we CVC((G 0 ,S 0 ),k,D 0 ) have CVC((G,S),k,D) OPT((G,S),k) ≤ OPT((G 0 ,S 0 ),k) . Now, we have the following bound. Lemma 13. Suppose S is a split deletion set of size k of G and none of the Reduction Rules 3.1, 3.2 and 3.3 is applicable on the instance (G, S). Then, |V(G)| is O(k + (2k + d)d ). Proof. We will bound B, IB and R separately in order to bound V(G). We know that G has a connected vertex cover T of size at most 2k + d − 1. As B is the set of vertices of degree at least 2k + d, B ⊆ T and so |B| ≤ 2k + d − 1. Every vertex in R has degree at most 2k + d − 1. Therefore, as T ∩ R is a vertex cover of G[R], |E(G[R])| is O((2k + d − 1)2 ). Also, by the definition of IB , every vertex in R has a neighbour in R and hence there are no isolated vertices in G[R]. Thus, |R| is O((2k + d − 1)2 ). Finally, we bound the size of IB . As Reduction Rule 3.2 is not applicable, every vertex in IB ∩ V(G − S) has degree at most d − 1. For every set B 0 ⊆ B of size at most d − 1, there are at most 2k + d vertices in IB ∩ V(G − S) which have B 0 as their neighbourhood. Otherwise, Reduction Rule 3.3 would
have been applied. Hence, there are at most (2k + d) · 2k+(d−1) vertices in IB ∩ V(G − S). d−1 Finally, as none of the reduction rules increases the size of the split deletion set S, we have |IB ∩ S| ≤ k. Therefore, |IB | is O(k + (2k + d)d ). This leads to a PSAKS for the problem as claimed. Theorem 14. Connected Vertex Cover parameterized by the size k of a split deletion d 2α−1 e α−1 ) vertices. set admits a time efficient PSAKS with O(k + (2k + d 2α−1 α−1 e) 11 Proof. Given α > 1, we choose d = d 2α−1 α−1 e and apply the reduction rules as long as they are applicable. Then, from Lemma 13, |V(G)| is O(k + (2k + d)d ). We note that this kernel is one of the few lossy kernels for a problem with respect to a structural parameter. 4 Connected Vertex Cover parameterized by Clique Cover In this section, we first show that some of the ideas from the previous section can be used to give a lossy kernel for Connected Vertex Cover when parameterized by the size of the clique cover. A clique cover of a graph is a partition of its vertex set such that each part induces a clique. We assume that we are given a partition of the vertex set of the input graph into cliques. For this parameterization, the corresponding parameterized minimization problem is defined in a similar way as defined for split deletion set. Theorem 15. Connected Vertex Cover parameterized by the size k of a clique cover admits a time efficient PSAKS with O(kd 2α−1 α−1 e) vertices. Proof. Without loss of generality, we assume that the graph has no isolated vertices as no minimum connected vertex cover contains them. Given α > 0, let d = d 2α−1 α−1 e. Let C denote the set of k cliques in the clique cover of G. We bound the number of vertices in G by applying Reduction Rule 3.1 on each C ∈ C. From Lemma 8, each application of this rule is α-safe. When this rule is no longer applicable, it follows that G has at most k(d − 1) vertices. This leads to the claimed result. Now, we modify a reduction known from [20] used in the hardness of Feedback Vertex Set with respect to the size of a clique cover to show that Connected Vertex Cover is W[1]-hard under the same parameterization. This makes this one of the few problems that are unlikely to be in FPT but have a lossy kernel. Theorem 16. Connected Vertex Cover parameterized by the size of a clique cover is W[1]-hard. Proof. We reduce the well known W[1]-hard problem, Independent Set parameterized by solution size, to our problem. A non-separating independent set of a graph G is an independent set I such that V(G) \ I is a connected vertex cover of G. Let (G, k) be an instance of Independent Set. We construct a graph G 0 on the vertex set {(v, i) | v ∈ V(G), i ∈ [k]} ∪ {x, y}. We add an edge between (v, i) and (u, j) if and only if i = j or u = v or v ∈ NG (u). Further, for every w ∈ V(G 0 ) \ {x}, we add the edge {w, x} to G 0 . Note that G 0 has a clique cover of size k + 1 as for all j ∈ [k], G 0 [{(v, j) | v ∈ V(G)}] is a complete graph and {x, y} is a clique. We claim that G has an independent set of size k if and only if G 0 has a non-separating independent set of size k + 1. Suppose S = {v1 , . . . , vk } is an independent set of size k in G. Define the set S 0 ⊆ V(G 0 ) as {(v1 , 1), (v2 , 2), . . . , (vk , k), y}. Consider any two vertices (vi , i) and (vj , j) in S 0 such that i 6= j. As for each distinct i, j ∈ [k], we have {vi , vj } ∈ / E(G), it follows that there is no edge between (vi , i) and (vj , j). Also, by the construction of G 0 , y is not adjacent to any vertex in G 0 other than x. Therefore, S 0 is an independent set of size k + 1 in G 0 . As x is adjacent to every vertex in G 0 − S 0 , it follows that the deletion of S 0 does not disconnect G 0 . In other words, S 0 is a non-separating independent set of G 0 . Conversely, suppose S 0 is a non-separating independent set of size k + 1 in G 0 . Clearly, different vertices of S 0 have to come from different cliques of the clique cover of G 0 . We may assume that y ∈ S 0 as if x ∈ S 0 , then (S 0 \ {x}) ∪ {y} is another non-separating independent 12 set of size k + 1. This is due to the facts that y is not adjacent to any vertex other than x and x is a universal vertex in G 0 . Define the set S ⊆ V(G) as {v | (v, j) ∈ S 0 \ {y}}. Consider any two vertices (v, i) and (u, j) in S 0 . Then, u 6= v, i 6= j and v ∈ / NG (u). Therefore, S is an independent set in G of size k. From Theorem 16, it follows that Connected Vertex Cover parameterized by k + q is W[1]-hard where k is the size of a set S whose deletion results in a graph for which there exists a clique cover of size q. However, the problem is in XP as the algorithm in Theorem 3 can be easily generalized to solve Connected Vertex Cover in O∗ (2k nq ) time where n is the number of vertices in the input graph. Further, by an easy adaptation of Theorem 15, the problem admits a PSAKS leading to the following result. Theorem 17. Given a graph G on n vertices, a set S ⊆ V(G) such that |S| ≤ k and G − S has a clique cover Q of size q and a positive integer `, there is an algorithm that determines whether G has a connected vertex cover of size at most ` in O∗ (2k nq ) time. Further, the problem admits a time efficient PSAKS with O(k + qd 2α−1 α−1 e) vertices. Proof. Without loss of generality, we assume that the graph has no isolated vertices as no minimum connected vertex cover contains them. We also assume that the set S and the clique cover Q of G − S are part of the input. First, let us describe the XP algorithm. Let X∗ be a connected vertex cover of G of size at most `. S As at least |Q| − 1 vertices from each ∗ ∗ clique Q ∈ QSare contained in X , we guess Y = X ∩ ( Q∈Q Q). Then, we guess Z = X∗ ∩ S such that (( Q∈Q Q) \ Y) ∪ (S \ Z) is an independent set. The number of choices for (Y, Z) is O∗ (nq 2k ). If G[Y ∪ Z] is not connected or has more than ` vertices, we skip to the next choice of (Y, Z). Otherwise, Y ∪ Z is the required solution. If none of the choices leads to a solution, we declare that G has no connected vertex cover of size at most `. The overall running time of the algorithm is thus O∗ (2k nq ). Now, we describe an α-approximate kernel for each α > 1. Given α > 0, let d = d 2α−1 α−1 e. We bound the number of vertices in G by applying Reduction Rule 3.1 on each Q ∈ Q. From Lemma 8, each application of this rule is α-safe. When this rule is no longer applicable, it follows that G has at most k + q(d − 1) vertices. This leads to the claimed result. 5 Connected Vertex Cover parameterized by Cluster Deletion Set In Section 3.1, we gave an O∗ (2k ) time algorithm for Connected Vertex Cover parameterized by the size k of a clique deletion set. Here, we generalize this algorithm to solve Connected Vertex Cover in O∗ (4k ) time where k is the size of a cluster deletion set. Observe that this is a parameter smaller than the clique deletion set size. Further, we also describe a lossy kernel with respect to this parameterization. A classical polynomial kernel is unlikely as the size of a minimum cluster deletion set is at most the size of a minimum connected vertex cover. This is due to the fact that deleting a connected vertex cover from a graph results in a cluster graph in which every component is an isolated vertex. 5.1 An FPT Algorithm Consider an instance (G, S, `) of Connected Vertex Cover where S is a cluster deletion set. Let H denote the cluster graph G − S. Let X∗ be a connected vertex cover of size at most ` (if one exists) that we are looking for. First, we guess the subset S 0 of S such that S 0 = S ∩ X∗ . If S \ S 0 is not an independent set, we skip to the next choice of S 0 . Define the set F as N(S \ S 0 ) ∩ V(H). Initialize the set X to be S 0 . In the latter steps of our algorithm, 13 we will extend X to a connected vertex cover of G. We also update ` to ` − |S 0 | and delete S \ S 0 from G. We will now describe a sequence of reduction and branching rules to be applied. The rules are applied in the order stated and a rule is applied as long as it is applicable on the instance. That is, a rule is applied only when none of the preceding rules can be applied. Let I denote the set of isolated vertices of H and Q denote the set of vertex sets of components of H − I. That is, I is an independent set and each element of Q is a clique in H. These sets are updated accordingly as and when the rules are applied. First, we apply the following preprocessing rule. Preprocessing Rule 5.1. If there is a clique Q ∈ Q with N(Q) ∩ X = ∅ or a vertex v ∈ F ∩ I such that N(v) ∩ X = ∅, then skip to the next choice of S 0 . The correctness of the first part of the rule follows from the fact that at least |Q| − 1 vertices from Q has to be in any vertex cover and any set of such vertices along with X cannot be extended to a connected subgraph by only adding vertices from V(H). The correctness of the second part follows from the facts that F is forced into the solution and G[X ∪ {v}] cannot be extended to a connected subgraph by only adding vertices from V(H). Let F be partitioned into sets Y and Z defined as follows. • Y = {v ∈ F | N(v) ∩ X 6= ∅}, the set of vertices of F that have a neighbour in X. • Z = F \ Y, the set of vertices of F that have no neighbour in X. These sets are updated over the execution of the algorithm according to the current partial solution X. In particular, at any point of time, Z is the set of vertices in F that are not adjacent to any vertex in the current partial solution X. Our first reduction rule is as follows. Reduction Rule 5.1. Add Y ∪ Z to X, update ` to ` − |Z ∪ Y| and delete Y from H. This rule is justified by the fact that X∗ must contain F. Due to Preprocessing Rule 5.1, every vertex v ∈ Z is in some clique Q in Q. When Reduction Rule 5.1 is no longer applicable, we have V(H) ∩ F = Z. For each v ∈ Z, let Qv denote the clique in Q containing v. The next rule is the following. Reduction Rule 5.2. If there is a vertex v ∈ Z with |Qv ∩ Z| = |Qv | − 1, then add Qv \ Z to X, update ` to ` − 1 and delete Qv from H and Z. The correctness of this rule follows from the fact that there is exactly one vertex u in Qv ∩ N(X \ Z). This vertex is forced into the solution since Qv \ {u} ⊆ X and Qv \ {u} has no neighbours in X \ (Qv \ {u}). Now, for any clique Q ∈ Q, there are at least two vertices that are not in Z. For a vertex v ∈ V(H), let C(v) = {A ∈ G[X] | A is a component of G[X] and NG (v) ∩ V(A) 6= ∅}. The next rule is the following. Reduction Rule 5.3. If there is a clique Q in Q that has two vertices u, v ∈ / Z with C(u) ⊆ C(v), then update X to X ∪ (Q \ {u}) and reduce ` by the number of vertices added to X. Delete Q from H and Z. If there exists an optimal connected vertex cover X that does not contain v, then u ∈ X and X 0 = (X \ {u}) ∪ {v} is also a connected vertex cover of G. This justifies the correctness of the rule. The next rule is a branching rule that is applied when H has a triangle or a larger clique Q with vertices not in Z. Further, for each v ∈ Q \ Z, |C(v)| ≥ 1 and for every u, v ∈ Q \ Z, C(u) 6⊆ C(v). 14 Branching Rule 5.1. If there is a triangle {u, v, w} in H with u, v, w ∈ / Z, then branch into adding u, v or u, w or w, v into X. In each of the branches update ` to ` − 2. The branches in this rule are clearly exhaustive as any vertex cover contains at least two vertices from a triangle. The reason for handling such special triangles will be apparent during the application of subsequent rules. When none of the rules described so far is applicable, every clique Q ∈ Q has exactly 2 vertices not in Z. In particular, if Q ∩ Z = ∅, then |Q| = 2. / Z in H and |C(u)| = 1 and Reduction Rule 5.4. If there is an edge {u, v} with u, v ∈ |C(v)| ≥ 1, then update X to X ∪ (Q \ {u}) and reduce ` by the number of vertices added to X where Q is the clique in Q that contains u and v. Delete Q \ {u} from H and Z. Add u to I. If there exists an optimal connected vertex cover X that does not contain v, then u ∈ X and X 0 = (X \ {u}) ∪ {v} is also a vertex cover of G. Further, G[X 0 ] is connected as the only vertex in Q ∩ X that is adjacent to a vertex in X \ {u} is u and |C(u)| = 1. Note that we crucially use the property that Q has exactly two vertices adjacent to a vertex in X \ {u, v} which is achieved by the application of Branching Rule 5.1. This justifies the correctness of the rule. The next rule is a branching rule that is applied until X is a vertex cover (not necessarily connected) of G. Branching Rule 5.2. If there is an edge {u, v} in H, then in one branch we add u into X and in other branch we add v into X. In both branches, update ` to ` − 1 and delete the vertex added to X from H. When this rule is applied on an edge {u, v}, we have |C(x)|, |C(y)| ≥ 2 and {x, y} ∩ Z = ∅. The branches are exhaustive as any vertex cover contains x or y. At this point, when none of the described rules (reduction and branching) is applicable, we have a vertex cover X of G. That is, V(H) is an independent set. However, X may not necessarily induce a connected subgraph. Let G 0 be the graph obtained from G[V(H) ∪ X] by contracting each component b where X b is the of G[X] into a single vertex. G 0 is bipartite and has a bipartition (V(H), X) set of vertices of G 0 corresponding to the set of components of G[X]. The problem is now to b b which can be solved in O∗ (2|X| find a Steiner tree of G 0 that contains X ) time using Lemma 1. This completes the description of the algorithm leading to the following result. Theorem 18. Given a graph G, a cluster deletion set S and a positive integer `, there is an algorithm that determines whether G has a connected vertex cover of size at most ` in O∗ (4|S| ) time. Proof. Given an instance (G, S, `), we first guess a subset S 0 of S that is contained in the solution. Then, we apply the reduction rules and branching rules in the order stated. We reiterate that each rule is applied as long as it is applicable. Also, a rule is applied only when none of the earlier rules is applicable. Let X be the partial solution that we are trying to extend to a connected vertex cover. Initially X is S 0 . The measure that we use to bound the running time is the number #comp(G[X]) of components of G[X] which is at most |S 0 |. On applying either of the branching rules, it is clear that #comp(G[X]) drops in all the branches by at least one. We will now show that the reductions rules do not increase the measure. When Reduction Rule 5.2 is applied, Qv has a neighbour in X due to Preprocessing Rule 5.1. Similarly, when Reduction Rule 5.3 or 5.4 is applied, Q \ {u} has a neighbour in X. Hence, these three rules do not increase #comp(G[X]). Now consider Reduction Rule 5.1. Adding Y to X do not increase #comp(G[X]) while adding Z to X will definitely increase it. However, if Z is non-empty, then in subsequent rules, some neighbour of Z which is also adjacent to some vertex in (current) X is added to the solution. Consider a vertex z ∈ Z on 15 which Reduction Rule 5.2 is applicable. Then, we add a neighbour z 0 of z that is adjacent to X. Therefore, the measure does not increase. In fact, the measure first increases by 1 (due to addition of z to X) and then decreases by at least 1 (due to addition of z 0 to X). Suppose Reduction Rule 5.2 is not applicable on a vertex z ∈ Z. Then, Qz has at least 2 vertices not in Z. If Reduction Rule 5.3 is applicable on Qz , then adding z into X does not increase the measure. Let us consider the case when Reduction Rule 5.3 is not applicable on Qz . Then, either Branching Rule 5.1 or Reduction Rule 5.4 or Branching Rule 5.2 is applicable to vertices of Qz . In any case, as we add a neighbour of z that is adjacent to X, the measure does not increase. When none of the reduction and branching rules is applicable, we have a set X that is a vertex cover of G. That is, at each leaf of the recursion tree, we have a vertex cover X of G that needs to be connected by adding a minimum number of vertices from V(H). We b as described earlier. Here, X b construct the bipartite graph G 0 with bipartition (V(H), X) is the set of vertices of G 0 corresponding to the set of components of G[X]. Observe that #comp(G[X]) is at most |S 0 |. Then, the time taken at each leaf of the recursion tree is upper b b which is O∗ (2|X| bounded by the time taken to find a Steiner tree of G 0 that contains X ) from 0 Lemma 1. Let |S | = i. If a leaf is at depth j where j ≤ i, it follows that the #comp(G[X]) is at most i − j. Then, the Steiner tree algorithm at this leaf runs in O∗ (2i−j ) time. Let T (j) denote the number of leaves of the search tree at depth j. It is easy to verify that the search tree is a ternary tree as there are at most three branches in any rule. Thus, the number of leaves at depth j is at most 3j . Therefore, the running time (ignoring polynomial factors) of the algorithm is upper bounded by the following value. Let k denote |S|. k  X i X k i=0 i j=0 j i−j 32   k   k X X k i k − 1 i−1 ≤ i 3 =3 k 3 = 3k · 4k−1 i i−1 i=0 i=1 Thus, the running time of the algorithm is O∗ (4k ). A degree-i modulator is a set of vertices whose deletion results in a graph with maximum degree at most i. As a degree-1 modulator is also a cluster deletion set, it follows that Connected Vertex Cover can be solved in O∗ (4|S| ) time when given a degree-1 modulator S as part of input. However, observe that this algorithm runs in O∗ (3k ) time as branching rule 5.2 is never applicable (and so the search tree is a binary tree). Thus, we have the following result. Theorem 19. Given a graph G, a degree-1 modulator S and a positive integer `, there is an algorithm that determines whether G has a connected vertex cover of size at most ` in O∗ (3|S| ) time. It is well known that the treewidth (tw) of a graph is not larger than the size of its minimum vertex cover, and hence of its minimum connected vertex cover. Therefore, a naive dynamic programming routine over a tree decomposition of width tw solves Connected Vertex Cover in O∗ (2tw log tw ) time [24]. The running time bound has been improved to O∗ (2O(tw) ) using algebraic techniques [2]. If the maximum degree of a graph G is upper bounded by 2, then every component of G is an induced path or an induced cycle. Hence, the treewidth of G is at most 2. Therefore, if G has a degree-2 modulator of size at most k, then the treewidth of G is at most k + 2. This shows that Connected Vertex Cover is FPT when parameterized by the size of a degree-2 modulator. As Connected Vertex Cover is NP-hard on graphs with maximum degree at most 4, it is clear that Connected Vertex Cover when parameterized by the size of a degree-4 modulator is para-NP-hard (NP-hard for fixed values of parameter). This observation doesn’t immediately carry over 16 when parameterized by the size of a degree-3 modulator as Connected Vertex Cover is polynomial time solvable in sub-cubic graphs [26]. Nevertheless, Connected Vertex Cover is para-NP-hard even when parameterized by the size of a degree-3 modulator. This is due to the fact that Vertex Cover in sub-cubic graphs is NP-complete [1, 15] and Vertex Cover reduces to Connected Vertex Cover by just adding a universal vertex (which is a degree-3 modulator) to the graph. Theorem 20. Connected Vertex Cover parameterized by the size of a degree-i modulator is FPT when i ≤ 2 and para-NP-hard for i ≥ 3. It is intriguing that though Connected Vertex Cover is polynomial-time solvable on sub-cubic graphs, it is NP-hard on graphs that are just one vertex away from being a sub-cubic graph. Note that the related Feedback Vertex Set problem is also solvable in polynomial-time on sub-cubic graphs, but it is a major open problem whether it is polynomial-time solvable on graphs that have a degree-3 modulator of size 1. 5.2 A Lossy Kernel In this section, we give a PSAKS for Connected Vertex Cover parameterized by the size of a cluster deletion set. We formally define the parameterized minimization problem as follows.   −∞ if |S| > k or some component of G − S is not a clique CVC((G, S), k, T ) = ∞ if T is not a connected vertex cover   |T | otherwise We now prove the following main result of this section. Theorem 21. Connected Vertex Cover parameterized by the size k of a cluster deletion α d α−1 e k α set admits a time efficient PSAKS with O(k2 + d 2α−1 ) vertices. α−1 e · α−1 + d α−1 e · k Proof. Let G be a connected graph and S denote its cluster deletion set of size at most k. Given α > 1, define  = α − 1. Let H denote the cluster graph G − S and F0 be the set of isolated vertices in H. Let F1 = V(H) \ F0 and t denote the number of components in G[F1 ]. α Let C1 , · · · , Ct denote the components of G[F1 ]. Let d1 = d 2α−1 α−1 e and d2 = d α−1 e. Define the set T as follows. We first initialize T to S. Then, for each i ∈ [t], we add a set Xi of |V(Ci )|−1 vertices of Ci such that NG (Xi )∩S 6= ∅ to T . Such a set always exists as G is connected. Now, t P T is a vertex cover of G and |T | = |S|+ (|V(Ci )|−1). Further, #comp(G[T ]) = #comp(G[S]). i=1 If G[T ] is not connected, then we add at most |S| − 1 vertices from V(G) \ T to T so that G[T ] becomes connected. Once again, such a set exists as G is connected. Thus, we have t t P P |T | < 2k + (|V(Ci )| − 1). We know that OPT ((G, S), k) ≥ (|V(Ci )| − 1) as each V(Ci ) i=1 i=1 is a clique and any vertex cover excludes at most one vertex from any clique. Therefore, |T | < 2k + OPT ((G, S), k). If t ≥ 2k  , then the reduction algorithm outputs a constant size 0 0 instance (G , S ). Since OPT ((G, S), k) ≥ t, we have k ≤ 2 OPT ((G, S), k) and it follows that |T | is at most (1 + )OPT ((G, S), k). Equivalently, CVC((G, S), k, T ) ≤ (1 + )OPT ((G, S), k). The solution lifting algorithm simply returns T which is obtained in polynomial time. On the other hand, suppose t < 2k  . For each i ∈ [t], we apply Reduction Rule 3.1 with d = d1 to bound the number of vertices in Ci by d1 . From Lemma 8, we know that Reduction Rule 3.1 is α-safe. When this rule is no longer applicable, we have at most d1 2k  vertices in F1 . Now, it remains to bound the number of vertices in F0 . Observe that any optimal connected vertex cover must contain any vertex x ∈ S with |NG (x) ∩ F0 | ≥ 2k. This is because, if some 17 optimal connected vertex cover T ∗ excludes x, then |T ∗ | ≥ 2k + T is a connected vertex cover with |T | < 2k + t P t P (|V(Ci )| − 1). However, i=1 (|V(Ci )| − 1) leading to a contradiction. i=1 Let S1 = {x ∈ S | |NG (x) ∩ F0 | ≥ 2k}. We first partition F0 into A0 = {v ∈ F0 | NG (v) ⊆ S1 } and B0 = F0 \ A0 . Then, we apply Reduction Rule 3.2 to vertices in A0 with d = d2 and Reduction Rule 3.3 to vertices in A0 with d = 0. Reduction Rule 3.2 is α-safe and Reduction Rule 3.3 is 1-safe from Lemmas 11 and 12. Suppose none of the described rules is applicable on the instance (G, S). We will show that |V(G)| = O(k2 + d2 · kd2 + d1 · k). The set V(G) is partitioned into sets S, F1 , A0 and B0 . As G is connected, G has no isolated vertex. As Reduction Rule 3.1 is not applicable, each component in G[F1 ] has at most d1 vertices. Thus, |V(F1 )| ≤ 2k  · d1 . For any vertex v ∈ F0 , there is a vertex x ∈ S such that {x, v} ∈ E(G). Since the reduction rules described are not applicable,
every vertex x ∈ A0 has at most d2 − 1 neighbors in S1 . Further, for every X ∈ ≤dS21−1 , there are at most 2k vertices from A0 such that each of those 2k vertices dP 2 −1
k d2 as |S | ≤ k. has X as its neighborhood in G. Therefore, |A0 | ≤ 2k · 1 i ≤ 2(d2 − 1)k i=1 As G is connected, each vertex in B0 has a neighbour in S \ S1 . Since, the degree of any vertex in S \ S1 is at most 2k, it follows that |B0 | ≤ 2k2 . Thus, |V(G)| = |S| + |F1 | + |A0 | + |B0 | α d α−1 e 2α−1 k α d2 2 2 is at most k + 2k ).  · d1 + 2(d2 − 1)k + 2k which is O(k + d α−1 e · α−1 + d α−1 e · k As the deletion of a degree-1 modulator results in a graph in which every component has at most 2 vertices, we have the following result as a consequence of Theorem 21. Corollary 22. Connected Vertex Cover parameterized by the size k of a degree-1 α k α modulator admits a time efficient PSAKS with O(k2 + α−1 + d α−1 e · kd α−1 e ) vertices. 6 Connected Vertex Cover parameterized by Chordal Deletion Set In this section, we show that Connected Vertex Cover parameterized by the size of a chordal deletion set is FPT. It is well known that the treewidth (tw) of a graph is at most one more than the size of a minimum feedback vertex set (a set of vertices whose removal results in a forest). This immediately implies that Connected Vertex Cover is FPT when parameterized by the size of a feedback vertex set. Recall that a chordal graph is a graph in which every induced cycle is a triangle. As forests, split graphs and cluster graphs are chordal, the minimum chordal deletion set size is at most a minimum feedback vertex set, a minimum split deletion set size and a minimum cluster deletion set. Theorem 23. Given a graph G, a chordal deletion set S and a positive integer `, there is an algorithm that determines whether G has a connected vertex cover of size at most ` in O∗ (2|S| log |S| ) time. Proof. Let G be a connected graph on n vertices and S denote its chordal deletion set. As G − S is a chordal graph, a tree decomposition of G − S of optimum width in which every bag is a clique can be obtained in linear time [16]. From this tree decomposition of G − S, a tree decomposition T = (T, {Xt }t∈V(T ) ) of G can be obtained by adding S to every bag. We will show that the algorithm for Connected Vertex Cover on bounded treewidth graphs [24] runs in O∗ (2O(|S| log |S|) ) time. The algorithm employs a dynamic programming routine over the tree decomposition T . As this approach is largely standard, we only present an outline of the algorithm here. 18 For a node t of T , let Tt denote the subtree of T rooted at t ∈ V(T ) and Vt denote the union of all the bags of Tt . Let Gt denote the subgraph of G induced by Vt . For t ∈ V(T ), X ⊆ Xt and a partition P = {P1 , · · · , Pq } into at most |X| parts, let Γ [t, X, P] denote a minimum vertex cover Z of Gt with Xt ∩ Z = X and Gt [Z] has exactly q connected components C1 , · · · , Cq where Pi = V(Ci ) ∩ Xt for each i ∈ {1, · · · , q}. Further, if X is empty, Z is required to be connected in Gi . Then, Γ [t, ∅, {∅}] is a minimum connected vertex cover of G. Observe that the total number of (valid) states per node is |S|O(|S|) n. This is due to the fact that for each vertex t ∈ V(T ), G[Xt ] has a clique deletion set of size |S|. Further, each entry can be computed in |S|O(|S|) nO(1) time. This leads to the claimed result. We remark that the lossy kernelization status of this problem remains open even when the chordal deletion set is a feedback vertex set. 7 Concluding Remarks We have given lossy kernels and studied the parameterized complexity statuses of Connected Vertex Cover parameterized by split deletion set size, clique cover number and cluster deletion set size. Our FPT running times (for Connected Vertex Cover parameterized by split deletion set and cluster deletion set) have slight gaps between upper bounds and lower bounds (based on well-known conjectures), and tightening the bounds is an interesting open problem. A more general direction is to explore the parameterized and (lossy) kernelization complexity of Connected Vertex Cover in the parameter ecology program. Figure 1 gives a partial landscape of the (parameterized) complexity of Connected Vertex Cover under various structural parameters. 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Session Types Go Dynamic or How to Verify Your Python Conversations Rumyana Neykova Imperial College London, United Kingdom [email protected] This paper presents the first implementation of session types in a dynamically-typed language Python. Communication safety of the whole system is guaranteed at runtime by monitors that check the execution traces comply with an associated protocol. Protocols are written in Scribble, a choreography description language based on multiparty session types, with addition of logic formulas for more precise behaviour properties. The presented framework overcomes the limitations of previous works on the session types where all endpoints should be statically typed so that they do not permit interoperability with untyped participants. The advantages, expressiveness and performance of dynamic protocol checking are demonstrated through use case and benchmarks. 1 Introduction The study of multiparty session types (MPST) has explored a type theory for distributed programs which can ensure, for any typable programs, a full guarantee of deadlock-freedom and communication safety (all processes conform to a globally agreed communication protocol) through static type checking. However, a static verification is not always feasible and dynamic approaches have several advantages. First, when access to the source code is restricted dynamic verification enables to detect and ensure the correctness of external untyped components. Second, constraints on the message payload are easier to check dynamically. Third, as shown in this paper, dynamic checking is less obstructive to the source code, because it does not require extensions of the host language as in the existing works on session types. In this paper we present a toolchain for session-based programming (hereafter conversation programming) in Python that Global Protocol uses MPST-protocols to dynamically verify the communication Specification PROJECTION safety of the running system. Conversation programming in (Scribble) Python resembles the standard development methodology for Local Specification MPST-based frameworks (Fig. 1). It starts by specifying the intended interactions (choreography) as a global protocol in the proImplementation Source Code (Python) Conversation tocol description language Scribble [13]. Then Scribble local proRuntime tocols are generated mechanically for each participant (role) defined in the protocol. After that processes for each role are impleMonitor mented using MPST operations exposed by Python conversation Verification (Dynamic) library. An external monitor is assigned to each endpoint. During communication initiation the monitor retrieves the local SAFE NETWORK protocol for its process and converts it to a finite state machine (FSM). The FSM continuously checks at runtime that each inter- Figure 1: Development methodology action (execution trace) is correct. If all participants comply to their protocols, the whole communication is guaranteed to be safe [6]. If participants do not comply, violations (such as deadlocks and communication mismatch) are detected and optionally ignored. N. Yoshida, W. Vanderbauwhede (Eds.): Programming Language Approaches to Concurrency- and Communication-cEntric Software 2013 (PLACES’13) EPTCS 137, 2013, pp. 95–102, doi:10.4204/EPTCS.137.8 96 Session Types Go Dynamic The presented framework brings several non-trivial contributions to MPST works. First, Scribble is extended with logic assertions (constraints on the message payload). Second, implementing MPST in a dynamic language requires different code augmentation techniques. For that purpose, we have defined a minimal, but sufficient and extendable format for conversation message headers. Third, we show that using FSMs for MPST checking has reasonable overhead. The algorithm used to convert local session types to FSM is based on [7], however we have optimised it to avoid the state explosion for parallel sub-protocols and have extended it for the new Scribble constructs. Finally, the Python API is more flexible compared to other session types language extensions, because it supports different programming styles (event-driven and thread-based, see Fig. 4). From the existing implementations only SJ [9] features event-driven programming, but it has more strict typing rule. To the best of our knowledge, this is the first implementation of session types for decentralised monitoring. Our practical framework is inspired by the formal model of MPST runtime safety enforcement presented in [6, 5]. In the aforementioned works conformance to stipulated global protocols is guaranteed at runtime through local monitoring. The rest of the paper illustrates the key features of our conversation framework, the Python runtime and its API (§ 2), it also gives overview of the monitoring tool, along with its benchmarks (§ 3). § 5 discusses future work and concludes. The code for the runtime and the monitor tool and example applications are available from [14]. 2 Conversation Programming in Python This section illustrates the stages of our framework and its implementation through a use case. Step 1 and 2 illustrate the use case specification in Scribble, while Step 3 presents one of the main contributions of the paper – a python API for conversation programming. We present a use case obtained from our industrial partners Ocean Observatory Institute (OOI) [11] (use case UC.R2.13 ”Acquire Data From Instrument”). OOI aims to establish cyberinfrastructure for the delivery, management and analysis of scientific data from a large network of ocean sensor. Their architecture relies on distributed run-time monitoring to regulate the behaviour of third-party applications within the system. Part of the monitor tool presented in this paper is already integrated in their system as an internal monitor. Step 1: Global Protocol. The Scribble global protocol for the use case is listed in Fig. 2. Scribble describes interactions between session participants through message passing sequences, branches and recursion. Each message has a label (an operator) and a payload. The first line declares the Data Acquisition protocol and three participant roles – a User (U), an Agent service (A) and an Instrument (I). The overall scenario is as follows: U requests via A to start streaming a list of resources from I (line 2–3). At Line 4 I makes a choice wether to continue the interaction or not. If I supports the requested resource the communication continues and A starts to poll resources from I and streams them to U (line 6–15). Line 10 shows the new assertion construct and restricts I to send data packages that are less than 512MB. The presented assertion extension is inspired by [4]. However, we do not stick to a predefined logic, but allow various policy languages to be incorporated inside an assertion construct. Step 2: Global-to-local Protocol Projection. Local protocols specify the communication behaviour for each conversation participant. An example of a local protocol (the local protocol for role A is given in Fig. 2. A local protocol is essentially a view of the global protocol from the perspective of one participant role and as such it is mechanically projected from the global protocol. Projection basically works by identifying the message exchanges where the participant is involved, and disregarding the rest, while preserving the overall interaction structure of the global protocol. The assertions are similarly R. Neykova 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 97 global protocol DataAquisition(role U, role A, role I) { Request(string:info) from U to A; Request(string:info) from A to I; choice at I { Support from I to A; rec Poll{ Poll from A to I; choice at I { @{size(data) ≤ 512} Raw(data) from I to A ; Formatted(data) from I to U; Poll; } or { Stop from I to A; Stop from A to U;}} } or { NotSupported from I to A; Stop from A to I; Stop from A to U;}} 1 2 3 4 5 6 7 8 9 10 11 local protocol DataAquisition at A(role U, role A, role I) { Request(string:info) from U; Request(string:info) to I; choice at I { Support from I; rec Poll{ Poll to I; choice at I { @{size(data) ≤ 512} Raw(data) from I; 12 13 14 15 16 17 18 19 20 Poll; } or { Stop from I; Stop to U;}} } or { NotSupported from I; Stop to I; Stop to U;}} Figure 2: Global Protocol (left) and Local Protocol for role A (right) preserved by projection where relevant. Step 3: Process Implementation. Fig. 4 illustrates the conversation API by presenting two alternative implemen- # session initiation bla tations in Python for the User process. Our Python conver- create(protocol, inv_config.yml) sation API offers a high level interface for safe conversation # accept an invitation join(self, role, principal_name) programming and maps basic session calculus primitives to # send a msg lower-level communication actions on a concrete transport send(self, to_role, op, payload) (AMQP [1] in this case). The implementation is built on top # receive a msg of Pika [12], a widely used AMQP client library for Python. recv(self, from_role) Fig. 3 lists the basic API methods. In short, the API pro- # receive asynchronously recv_async(self, from_role, callback) vides functionality for (1) session initiation and joining and # close the connection (2) basic send/receive. Each message embeds in its payload stop() a conversation header. The header contains session inforFigure 3: Conversation API mation either for monitor initialisation (in case of invitation messages), or session checking (in case of in-session messages). Conversation initiation The Conversation.create method initiates a new conversation. It creates a fresh conversation id and the required AMQP objects (principal exchange and queue), and sends an invitation message for each role specified in the protocol. Invitation mechanism is needed to map the role names to concrete addressable entities on the network (principals) and to propagate this mapping to all participants. Invitation header carries a conversation id, a role, a principal name (resolvable to a network address) and a name for a Scribble local specification file. In our example, the User starts a session and sends invitation to all other participants. Once the invitations are sent and accepted, a session is established and the intended message exchange can start. An invitation for a role is accepted using the Conversation.join method. It establishes an AMQP connection and, if one does not exist, creates an invitation queue on which the invitee waits to receive an invitation. Conversation message passing The API provides standard send/receive primitives. Send is asynchronous, meaning that a basic send does not block on the corresponding receive; however, the basic re- 98 Session Types Go Dynamic class ClientApp(BaseApp): def start(self): c = Conversation.create(’DataAquisition’, ’config.yml’) c.join(’U’, ’alice’) resource_request = c.receive(’U’) c.send(’I’, resource_request) req_result = c.receive(’I’) if (req_result == SUPPORTED): c.send(’I’, ’Poll’) op, data = c.receive(’I’) while (op != ’Stop’): formatted_data = format(data) c.send(’U’, fomratted_data) c.send(’U’, stop) else: c.send(’U, I’, stop) c.stop() class ClientApp(BaseApp): def start(self): c = Conversation.create(’DataAquisition’, ’config.yml’) c.join(’U’, ’alice’) c.receive_async(’U’, on_request_received) def on_request_received(self, conv, op, msg): if (op == SUPPORTED): conv.send(’I’, ’Poll’) conv.receive_async(’I’, ’on_data_received’) else: conv.send(’I, U’, ’Stop’) def on_data_received(self, conv, op, payload): if (operation != ’Stop’): formatted_data = format(payload) c.send(’U’, formatted_data) else: conv.send(’U’, ’Stop’) conv.stop() Figure 4: Python standard (left) and event-driven (right) implementation of the User process ceive does block until the complete message has been received. An asynchronous receive (receive async) is also provided to support event-driven usage of the conversation API. We have demonstrated two different implementations for the the User process (threaded and event-driven). Both versions require the same monitor for checking. The primitives for sending and receiving specify the name of the sender and receiver role respectively. The runtime resolves the role name to the actual network destination by coordinating with the in-memory conversation routing table created as a result of the conversation invitation. All messages are sent/received as a tuple of an operation and a payload. The API does not mandate how the operation field should be treated, allowing the runtime freedom to interpret the operation name various ways, e.g. as a plain message label, an RMI method name, etc. Syntactic sugar such as automatic dispatch on method calls based on the message operation is possible. More examples of programs using the API can be found in [14]. 3 3.1 Dynamic Verification Monitoring Implementation To guarantee global safety our monitoring framework imposes complete mediation of communications: no communication action should have an effect unless the message is mediated by the monitor. We use the AMQP’s functions to reroute each outgoing/incoming message to its associated monitor. Routing is configured during session initialisation. Figure 5 depicts the main components and internal workflow of our prototype monitor. The lower part relates to session initiation. The invitation message carries (a reference to) the local type for the invitee and the session id (global types can be exchanged if the monitor has the facility for projection.) The monitor generates the FSM from the local type following [7]. Our implementation differs from [7] in the treatment of parallel sub-protocols (i.e. unordered message sequences). For efficiency, the monitor generates nested FSMs for each session thread, avoiding the potential state explosion that comes from constructing their product. FSM generation has therefore polynomial time and space cost in the length R. Neykova 99 Figure 5: Monitor components and workflow. The messages are processed depending on their type: (1) Invitation Messages and (2) Conversation Messages. of the local type. The (nested) FSM is stored in a hash table with session id as the key. Due to MPST well-formedness conditions (message label distinction), any nested FSM is uniquely identifiable from any unordered message (i.e. session FSMs are deterministic). Transition functions are similarly hashed, each entry having the shape: (current state, transition) 7→ (next state, assertion, var) where transition is a triple (label, sender, receiver), and var is the variable binder for the message payload. The upper part of the Figure relates to in-session messages, which carry the session id (matching an entry in the FSM hash table), sender and receiver fields, and the message label and payload. This information allows the monitor to retrieve the corresponding FSM (the message signature is matched to the FSM’s transition function). Any associated assertions are evaluated by invoking an external logic engine; a monitor can be configured to use various logic engines, for example, logic engines that support the validation of assertions, automata-based specifications (such as security automata), or state updates. The current implementation uses a Python predicate evaluator, which is sufficient for the example protocol specifications that we have tested so far. 3.2 Benchmarks These benchmarks measure the communication overhead introduced by our prototype monitor implementation. The results show that the core FSM-related functionality of the monitor adds little overhead in comparison to a dummy monitor that performs plain message forwarding. Benchmark framework. We measure the time to complete a session between client and server endpoints connected to a single-broker AMQP network. Three benchmark cases are compared. The main case (Monitor) is fully monitored, i.e. FSM generation and message checking are enabled for both the client and server. The base case for comparison (Forwarder) has the client and server in the same configuration, but with dummy monitors that perform only message forwarding. For reference, the final case (No Monitor) tests direct AMQP communication between the server and client, i.e. messages are routed directly from an exchange to their destination queues (no intermediate forwarding). Naturally, forwarding-based mediation incurs additional latencies; the actual internal overhead of the monitor is given by the first two benchmark cases. This benchmark framework is applied to three scenarios: 1. Increasing session length (number of messages), for protocol: µ X.S → C{OK().C → S{ACK().X}, KO().end} Session length is the number of times the recursion is repeated. 2. Increasing protocol size (increasing number of parallel states). We repeatedly compose the base pattern to construct bigger protocols for nested FSM generation. S → C{OK().end} | C → S{ACK().end} Session Types Go Dynamic No Monitor 10 100 10 Forwarder Monitor 0.04 Hundreds execution time (seconds) 100 1 0.03 0.02 1 0.1 0.1 0.01 0.01 0.01 1 10 100 1000 (a) recursive iterations 1 10 100 (b) parallel states 1000 0 1 10 (c) message size Figure 6: Microbenchmarks comparing end-to-end monitor performance 3. Increasing payload size (message size), using protocol from (1). Benchmark environment and results. The server and client endpoint processes, both monitors and the RabbitMQ broker (2.7.0/R13B03) are all run on separate machines with the same specification: Intel Core2 Duo 2.80 GHz and 4 GB main memory, running 64-bit Ubuntu 11.04 (kernel 2.6.38) and connected via gigabit Ethernet. Latency between each node is measured to be 0.24 ms on average (ping 64 bytes). The benchmark applications are executed using Python 2.7.1. Figure 6 presents the results for the three benchmark scenarios. Each chart gives the mean time (yaxis) for the client and server to complete one session after repeating the benchmark 100 times for each parameter configuration (session length/parallel states/message size). Scenario (3) message size is measured for session length 1. For all three scenarios, the results show that the overhead of the monitor due to FSM generation and FSM-based message checking, the baseline cost in the current framework, are acceptable (around 20%). Non-communication related computation in more realistic applications and higher latency environments will both contribute to decreasing the relative overhead. For scenario (1) in chart (a), note that the relative overhead decreases (from 12% to 9%) as the session length increases, because the one-time FSM generation cost becomes less prominent. Although our implementation work is ongoing, we believe these results confirm the feasibility of our approach. As expected, the forwarding configuration incurs extra latencies (due to the reciprocal shape of the benchmark protocol) in comparison to the (No Monitor) case. The full source code and raw results of these benchmarks, and additional tests using protocols with assertions, can be obtained from the project homepage [14]. 4 Related Work The work closest to ours is that by Ancona et al. [2]. It explores session types protocols as a test framework for multiagent systems (MAS). A global session type is specified as cyclic Prolog terms in Jason (a MAS development platform) and verified through test monitors. Their global types are less expressive in comparison with the language presented in this paper (due to restricted arity on forks and the lack of assertions). Their monitor is centralised and global safety properties are not discussed. Kruger et al. [10] propose a run-time monitoring framework, projecting MSCs to FSM-based distributed monitors. They use aspect-oriented programming techniques to inject monitors into the implementation of the components. Our outline monitoring verifies conversation protocols and does not require such monitoring-specific augmentation of programs. Gan [8] follows a similar but centralised approach to Kruger et al. R. Neykova 101 Works on monitoring BPEL languages can also be compared. Baresi et al. [3] develop a run-time monitoring tool with assertions. However, a major difference is that BPEL approaches do not treat or prove global safety. BPEL is expressive, but does not support distribution and is designed to work in a centralised manner. 5 Conclusion and Future Work We have shown that session types are amendable for dynamic verification. Our implementation automates distributed monitoring by generating FSMs from local protocol projections. Further benchmarks are needed to compare the conversation API with existing network libraries and to investigate its performance. Future work includes also the incorporation of more elaborate handling of error cases into monitor functionality, extending Scribble and automatic generation of services stubs. Although our implementation work is ongoing, the results confirm the feasibility of our approach. We believe this work is an important step towards a better, safer world of easier to speak and easier to understand distributed conversations. Acknowledgments. I would like to dedicate this paper to the memory of Kohei Honda, who is a constant source of inspiration to me and whose guidance was invaluable. I thank my supervisor Nobuko Yoshida for her constant support and ideas, my colleagues Raymond Hu and Pierre-Malo Denilou for the discussions about the framework; the anonymous reviewers for useful comments and corrections; and Tzu-Chun Chen for her valuable feedback and her inspirational work on the formal system behind the presented work. This work is partially supported by VMWare PhD studentship and EPSRC EP/G015635/1. References [1] Advanced Message Queuing Protocols (AMQP) homepage. display/AMQP/Advanced+Message+Queuing+Protocol. http://jira.amqp.org/confluence/ [2] Davide Ancona, Sophia Drossopoulou & Viviana Mascardi (2012): Automatic Generation of Self-Monitoring MASs from Multiparty Global Session Types in Jason. In: DALT’12, Springer. Available at http://dx. doi.org/10.1007/978-3-642-37890-4_5. [3] Luciano Baresi, Carlo Ghezzi & Sam Guinea (2004): Smart monitors for composed services. In: ICSOC ’04, pp. 193–202. Available at http://doi.acm.org/10.1145/1035167.1035195. [4] Laura Bocchi, Kohei Honda, Emilio Tuosto & Nobuko Yoshida (2010): A theory of design-by-contract for distributed multiparty interactions. In: CONCUR, LNCS 6269, pp. 162–176. Available at http://dx.doi. org/10.1007/978-3-642-15375-4_12. [5] Tzu chun Chen (2013): Theories for Session-based Governance for Large-scale Distributed Systems. Ph.D. thesis, Queen Mary, University of London. [6] Tzu-Chun Chen et al. (2012): Asynchronous Distributed Monitoring for Multiparty Session Enforcement. In: TGC’11, LNCS, Springer. Available at http://dx.doi.org/10.1007/978-3-642-30065-3_2. [7] Pierre-Malo Deniélou & Nobuko Yoshida (2012): Multiparty Session Types Meet Communicating Automata. In: ESOP, LNCS, Springer. Available at http://dx.doi.org/10.1007/978-3-642-28869-2_10. [8] Yuan Gan et al. (2007): Runtime monitoring of web service conversations. In: CASCON ’07, ACM, pp. 42–57. Available at http://doi.ieeecomputersociety.org/10.1109/TSC.2009.16. [9] Raymond Hu, Dimitrios Kouzapas, Olivier Pernet, Nobuko Yoshida & Kohei Honda (2010): Type-Safe Eventful Sessions in Java. In: ECOOP’10, LNCS 6183, Springer-Verlag, pp. 329–353. Available at http://dx.doi.org/10.1007/978-3-642-14107-2_16. 102 Session Types Go Dynamic [10] Ingolf H. Krüger, Michael Meisinger & Massimiliano Menarini (2010): Interaction-based Runtime Verification for Systems of Systems Integration. J. Log. Comput. 20(3), pp. 725–742. Available at http: //dx.doi.org/10.1093/logcom/exn079. [11] OOI. http://www.oceanobservatories.org/. [12] AMQP for Python (PIKA). https://github.com/pika/pika. [13] Scribble Project homepage. http://www.scribble.org. [14] Full version of this paper. http://www.doc.ic.ac.uk/~rn710/spy. 

On The Multiparty Communication Complexity of Testing Triangle-Freeness arXiv:1705.08438v1 [] 23 May 2017 Orr Fischer∗ Shay Gershtein† Rotem Oshman‡ May 24, 2017 In this paper we initiate the study of property testing in simultaneous and non-simultaneous multi-party communication complexity, focusing on testing triangle-freeness in graphs. We consider the coordinator model, where we have k players receiving private inputs, and a coordinator who receives no input; the coordinator can communicate with all the players, but the players cannot communicate with each other. In this model, we ask: if an input graph is divided between the players, with each player receiving some of the edges, how many bits do the players and the coordinator need to exchange to determine if the graph is triangle-free, or far from triangle-free? For general communication protocols, we show that Õ(k(nd)1/4 + k2 ) bits are sufficient to test trianglefreeness in graphs of size n with average degree d (the degree need not be known in advance). For simul√ taneous protocols, where there is only one communication round, we give a protocol that uses Õ(k n) √ √ bits when d = O( n) and Õ(k(nd)1/3 ) when d = Ω( n); here, again, the average degree d does not need to be known in advance. We show that for average degree d = O(1), our simultaneous protocol is asymptotically optimal up to logarithmic factors. For higher degrees, we are not able to give lower bounds on testing triangle-freeness, but we give evidence that the problem is hard by showing that finding an edge that participates in a triangle is hard, even when promised that at least a constant fraction of the edges must be removed in order to make the graph triangle-free. ∗ Computer Science Department, Tel-Aviv University. Email: [email protected] Computer Science Department, Tel-Aviv University. Email: [email protected] ‡ Computer Science Department, Tel-Aviv University. Email: [email protected] † 1 1 Introduction The field of property testing asks the following question: for a given property P , how hard is it to test whether an input satisfies P , or is ǫ-far from P , in the sense that an ǫ-fraction of its representation would need to be changed to obtain an object satisfying P ? Property-testing has received extensive attention, including graph properties such as connectivity and bipartiteness [22], properties of Boolean functions (monotonicity, linearity, etc.), properties of distributions, and many others [35, 17, 21]. The usual model in which propertytesting is studied is the query model, in which the tester cannot “see” the entire input, and accesses it by asking local queries, that is by only viewing a single entry in the object representation at a time. The tester typically does not have at its disposal the possibility of making a ”non-local” query whose answer depends on a substantial subset of the object’s representation, which is a primary source of difficulty in property testing. For example, for graphs represented by their adjacency matrix, the tester might ask whether a given edge is in the graph, or what is the degree of some vertex. The efficiency of a property tester is measured by the number of queries it needs to make. One can also distinguish between oblivious testers, which decide in advance on the set of queries, and adaptive testers, which decide on the next query after observing the answers to the previous queries. It is known that for many graph properties, one-sided oblivious testers are no more than quadratically more expensive than adaptive testers [24]. In this paper we study property testing from a different perspective, that of communication complexity. We focus on property testing for graphs, and we assume that the input graph is divided between several players, who must communicate in order to determine whether it satisfies the property or is far from satisfying it. Each player can operate on its own part of the input “for free”, without needing to make queries; we charge only for the number of bits that the players exchange between them. This is on one hand easier than the query model, because players are not restricted to making local queries, and on the other hand harder, because the query model is centralized while here we are in a distributed setting. This leads us to questions such as: does the fact that players are not restricted to local queries make the problem easier, or even trivial? Which useful “building blocks” from the world of property testing can be implemented efficiently by multi-party protocols? Does interaction between the players help, or can we adopt the “oblivious approach” represented by simultaneous communication protocols? Beyond the intrinsic interest of these questions, our work is motivated by two recent lines of research. First, [10, 19], study property testing in the CONGEST model, and show that many graph property-testing problems can be solved efficiently in the distributed setting. As pointed out in [19], existing techniques for proving lower bounds in the CONGEST model seem ill-suited to proving lower bounds for property testing. It seems that such lower bounds will require some advances on the communication complexity side, and in this paper we make initial steps in this direction. Second, recent work has shown that many exact problems are hard in the setting of multi-party communication complexity: Woodruff et al. [38] proved that for several natural graph properties, such as triangle-freeness, bipartiteness and connectivity, determining whether a graph satisfies the property essentially requires each player to send its entire input. We therefore ask whether weakening our requirements by turning to property testing can help. In this work we focus mostly on the specific graph property of triangle-freeness, an important property which has received a wealth of attention in the property testing literature. It is known that in dense graphs (average degree Θ(n)) there is an oblivious tester for triangle-freeness which is asymptotically optimal in terms of the size of the graph (i.e., adaptivity does not help) [2, 18], and [3] also gives an oblivious √ tester for graphs with average degree Ω( n). The closest parallel to oblivious testers in the world of communication complexity is simultaneous communication protocols, where the players each send a single message to a referee, and the referee then outputs the answer. We devote special attention to the question of 2 the simultaneous communication-complexity of testing triangle-freeness. 1.1 Related Work Property testing is an important notion in many areas of theoretical computer science; see the surveys [35, 17, 21] for more background. Triangle-freeness, the problem we consider in this paper, is one of the most extensively studied properties in the world of property testing; many different graph densities and restrictions have been investigated (e.g., [2, 1, 5, 23]). Of particular relevance to us is triangle-freeness in the general model of property testing, where the average degree of the graph is known but no other restrictions are imposed. For √ in advance, 4 2 3 /d 3 }) on testing triangle-freeness, and a lower nd, n this model, [3] showed an upper bound of Õ(min{ √ √ bound of Ω(max{ n/d, min{d, n/d}, min{ d, n2/3 }n−o(1) }), both for graphs with a average degree d ranging from Ω(1) up to n1−o(1) . For specific ranges of d, [34] and [25] improved these upper and lower bounds, respectively, by showing an upper bound of O(max{(nd)4/9 , n2/3 /d1/3 } and a lower bound of Ω(min{(nd)1/3 , n/d), n/d}). Our simultaneous protocols use ideas, and in one case an entire tester, from [3], but implementing them in our model presents different challenges and opportunities. (Our unrestricted-round protocol does not bear much similarity to existing testers.) As for lower bounds, we cannot use the techniques from [3] or other property-testing lower bounds, because they rely on the fact that the tester only has query access to the graph. For example, [3] uses the fact that a triangle-freeness tester with one-sided error must find a triangle before it can announce that the graph is far from triangle-free ([3] also gives a reduction lifting their results to two-sided error). In the communication complexity setting this is no longer true; there is no obvious reason why the players need to find a triangle in order to learn that the graph is not triangle-free. Property testing in other contexts. Recently, the study of property testing has been explored in distributed computing [7, 10, 19]. Among their other results, Censor-Hillel et al. [10] showed that trianglefreeness can be tested in O( ǫ12 ) rounds in the CONGEST model; expanding this, [19] showed that testing H-freeness for any 4 node graph H can be done in O( ǫ12 ) rounds, and showed that their BFS and DFS approach fails for K5 and C5 -freeness, respectively; [19] does not give a general lower bound. There has also been work on property testing in the streaming model [26]. The related problem of computing the exact or approximate number of triangles has also been studied in many contexts, including distributed computing [14, 11, 30, 15], sublinear-time algorithms (see [16] and the references therein), and streaming (e.g., [27]). Specifically, [27] gives a reduction which shows a lower bound on the space complexity of approximating the number of triangles in the streaming model; we apply their reduction here to show the hardness of testing triangle-freeness, by reducing from a different variant of the problem used in [27]. Communication complexity. The multi-party number-in-hand model of communication complexity has received significant attention recently. In [38] it is shown that several graph problems, including exact triangle-detection, are hard in this model. Many other exact and approximation problems have also been studied, including [6, 31, 39, 9, 37, 8] and others. Unfortunately, it seems that canonical lower bounds and techniques in communication complexity cannot be leveraged to obtain property-testing lower bounds; for discussion, see section 4.6. 1.2 Our Contributions The contributions of this work are as follows: 3 Basic building-blocks. We show that many useful building-blocks from the property testing world can be implemented efficiently in the multi-player setting, allowing us to use existing property testers in our setting as well. For some primitives — e.g., sampling a random set of vertices — this is immediate. However, in some cases it is less obvious, especially when edge duplication is allowed (so that several players can receive the same edge from the input graph). We show that even with edge duplication the players can efficiently simulate a random walk, estimate the degree of a node, and implement other building blocks. Upper √ bounds on testing triangle-freeness. For unrestricted communication protocols, we show that Õ(k 4 nd + k2 ) bits are sufficient to test triangle-freeness, where n is the size of the graph, d is its average degree (which is not known in advance), and k is the number of players. When interaction is not allowed √ √ (simultaneous protocols), we give a protocol that uses Õ(k n) bits when d = O( n), and another protocol √ √ using Õ(k 3 nd) bits for the case d = Ω( n). We also combine these protocols into a single degreeoblivious protocol, which does not need to know the average degree in advance. (This is not as simple as might sound, since we are working with simultaneous protocols, where we cannot first estimate the degree and then use the appropriate protocol for it.) Lower bounds. Our lower bounds are mostly restricted to simultaneous protocols, although we first prove lower bounds for one-way protocols for two or three players, and then then “lift” the results to simultaneous protocols for k ≥ 3 players using the symmetrization technique [33]. √ We show that for average degree d = O(1), Ω(k n) bits are required to simultaneously test trianglefreeness, matching our upper bound. For higher degrees, we are not able to give a lower bound on testing triangle-freeness, but we give evidence that the problem is hard: we show that it is hard to find an edge that participates in a triangle, even in graphs that are ǫ-far from triangle free (for constant ǫ), and where every edge participates in a triangle with (small) constant marginal probability. 2 Preliminaries Unrestricted Communication in the Number-in-Hand Model The default model we consider is this work is the number-in-hand model. In this model k players receive private inputs X1 , . . . , Xk and communicate with each other in order to determine a joint function of their inputs f (X1 . . . Xk ). This is the most general model, as the number of rounds of communication is unrestricted. There are three common variants to this model, according to the mode of communication: the blackboard model, where a message by any player is seen by everyone; the message-passing model, where every two players have a private communication channel and each message has a specific recipient; the coordinator model, which is the variant we consider in this paper, and define promptly. In the coordinator model the players communicate over private channels with the coordinator, but they cannot communicate directly with each other. The protocol is divided into communication rounds. In each such round, the coordinator sends a message of arbitrary size to one of the players, who then responds back with a message. Eventually the coordinator outputs the answer f (X1 . . . Xk ). For convenience, we assume that the players and the coordinator have access to shared randomness instead of private randomness. Note that the players will make explicit use of the fact that the randomness is shared for common procedures like sampling, as the players can agree on which elements to sample simply by agreeing in advance (as part of the protocol) on how to interpret the public bits, and no interaction is required. (For protocols that use more than one round, it is possible to get rid of this assumption and use private randomness instead via Newman’s Theorem [32], which costs at most additional O(k log n) bits. For further details see [29, 32]). 4 The communication complexity of a protocol Π, denoted CC(Π), is the maximum over inputs of the expected number of bits exchanged between the players and the coordinator in the protocol’s run. For a problem P , we let CCk,δ (P ) denote the best communication complexity of any protocol that solves P with worst-case error probability δ on any input. The coordinator model is roughly equivalent to the widely used message-passing model. More concretely, every protocol in the message-passing model can be simulated with a coordinator, incurring an overhead factor of at most log k by appending to each message the id of the recipient, to infrom the coordinator whom to forward this message to. The other direction can also be simulated efficiently, as in the message-passing model we can assign an arbitrary player to be the coordinator and run the protocol as it is. Although in this paper for convenience we consider the coordinator model, our results consequently apply for the message-passing model as well, up to a log k factor. Simultaneous Communication Of particular interest to us in this work are simultaneous protocols, which are, in a sense, the analog of oblivious property testers. This is the second primary model we investigate in addition to unrestricted communication. In a simultaneous protocol, there is only one communication round, where each player, after seeing its input, sends a single message to the coordinator (usually called the referee in this context). The coordinator then outputs the answer. Any oblivious graph property tester which uses only edge queries (which test whether a given edge is in the graph or not) can be implemented by a simultaneous protocol, but the converse is not necessarily true. Communication complexity of property testing in graphs. we are given a graph G = (V, E) on n vertices, which is divided between the k players, with each player j receiving some subset Ej ⊆ E of edges. More concretely, each player, j, receives the characteristic vector of Ej , where each entry corresponds to a single edge, such that if the bit is 1 then that edge exists in E, and if the bit is 0 it is unknown to the player whether it exists or not, as this entry might be 1 in the input of a different player. The logical OR of all inputs results in the characteristic vector of the graph edges, E. Note that there is no guarantee for any vertex for a single player to have all its adjacent edges in its input, as is the case in models like CONGEST. To make our results as broad as possible we follow the general model of property testing in graphs (see, e.g., [3]): we do not assume that the graph is regular or that there is an upper bound on the degree of individual nodes. As in [38], edges may be duplicated, that is, the sets E1 , . . . , Ek are not necessarily disjoint. The goal of a property tester for property P is to distinguish the case where G satisfies P from the case where G is ǫ-far from satisfying P , that is, at least ǫ|E| edges would need to be added or removed from G to obtain a graph satisfying P . An important parameter in our algorithms is the average degree, d, of the graph (also referred to as density); for our upper bounds, we do not assume that d is known, but our lower bounds can assume that it is known to the protocol up to a tight multiplicative factor of (1 ± o(1)). Moreover, as in [3], we focus on d = Ω(1) and d ≤ n1−ν(n) , where ν(n) = o(1), since for graphs of average degree d = Θ(n) there is a known solution whose complexity is independent on n in the property-testing query model and consequently in our model as well. The case of d = o(1), although not principally different, is ignored for simplicity, as its extreme sparsity makes it of less interest than any degree which is Ω(1). Information theory. Our lower bounds use information theory to argue that using a small number of communication bits, the players cannot convey much information about their inputs. For lack of space, we give here only the essential definitions and properties we need. Let (X, Y ) ∼ µ be random variables. (In our lower bounds, for clarity, we adopt the convention that bold-face letters indicate random variables.) To measure the information we learned about X after observ5 ing Y , we examine the difference between the prior distribution of X, denoted µ(X), and the posterior distribution of X after seeing Y = y, which we denote µ(X|Y = y). We use KL divergence to quantify this difference: Definition 1 (KL Divergence). For distributions µ, η : X → [0, 1], the KL divergence between µ and η is X D (µ k η) := µ(x) log (µ(x)/η(x)) . x∈X We require the following property, which follows from the superadditivity of information [13]: if (X 1 , . . . , X n , Y ) ∼ µ are such that X 1 , . . . , X n P are independent, and Y can be represented using m bits (that is, its entropy is at most m), then Ey∼µ(Y ) [ ni=1 D (µ(X i |Y = y) k µ(X i ))] ≤ m. Here and in the sequel, µ(X i ) denotes the marginal distribution of X i according to µ, and µ(X i |Y = y) is the marginal distribution of X i given Y = y, and Ey∼µ(Y ) denotes the expectation according to the distribution µ. Graph definitions and notation. We let deg(v) denote the degree of a vertex v in the input graph, and for a player j ∈ [k], we denote by dj (v) the degree of v in player j’s input (the subgraph (V, Ej )). Definition 2. We say that a pair of edges {{u, v} , {v, w}} ⊆ E is a triangle-vee if {u, w} ∈ E, and in this case we call v the source of the triangle-vee. Definition 3. We say that an edge e ∈ E is a triangle edge if G contains a triangle T , such that e is an edge in T . 3 Upper Bounds All the solutions we present have a one-sided error, that is if a triangle is returned then it exists in G with probability 1. This holds even when the input is not ǫ-far from being triangle free. Therefore, by solving the problem of triangle detection, we also solve triangle-freeness, as we never output a triangle in a triangle-free graph, and do output one with high probability whenever the gap guarantee holds. All algorithms have at most a small constant bound on the error, δ. We also prove for some cases an improved complexity for several relaxations such as having the players communicate in the blackboard model, where each message is seen by all players, or the variant where the players are guaranteed there is no edge-duplication, such that each edge of the graph appears in exactly one input. Additionally, we assume that k = O(poly(n)), to simplify the complexity expressions. 3.1 Building Blocks We start by showing that the essential primitives used in the property testing setting (dense, sparse and general models combined) of graph problems are efficiently translatable into our communication complexity model, where the edges of the graph are scattered across k inputs with possible multiplicity and as default the communication is unrestricted. This illustrates the added power packed in our communication complexity model, that can solve many problems with at most a logarithmic overhead factor, by simulating the PT solution, while for some problems, such as the one we will explore here, there is a significantly more efficient solution. • Querying a specific edge (check for its existence) - This is one of the main primitives in the dense model. This can be done in our model in O(k) by having each player send a bit to the coordinator indicating whether it is in its input, and the coordinator sends the answer bit to all players. 6 • Choosing uniformly a random edge adjacent to a given vertex v - This is the main primitive in the sparse model. It can be simulated by utilizing the random bits to fix an random order, P , over all the n−1 potential edges adjacent to v, and have each player send the first edge in his input according to P to the coordinator, who then sends everyone the first edge according to P of all the edges he received. This costs O(k log(n)). A random walk, which is a pivotal procedure in sparse property-testing, can be simulated by taking a random neighbor each step using this primitive. Note that the permutation was necessary so that edges with higher multiplicity would not be favored, as would happen in a naive implementation. • Querying vertex degree - this is an auxiliary query that is sometimes included in the general PT model. Without duplication this can be done trivially in O(k log(d(v)), by having all players send the number of edges adjacent to v in their input, and have the coordinator sum them up to get the result. With duplication, an exact answer costs Ω(kd(v)) as it is at least as hard as solving disjointness, to ensure no over-counting. However an α-approximation, for any α > 1 can be performed in efficiently as we promptly prove. We can also reduce the complexity in the no-duplication case by using an approximation, as we also show, and in many cases, such as triangle-detection, a constant approximation is good enough. Theorem 3.1. For any given vertex, v, the players can compute an α-approximation, for a constant α > 1 with probability at least (1 − τ ) and communication complexity of O(k log log d(v) + k log k log log k log τ1 ). Proof. First each player, Pi , computes locally, di (v), the number of edges adjacent to v in his input, Ei , and sends Ii , the index of the MSB (the leftmost ’1’ bit in the representation of di (v)), P binary 2Ii +1 . This amounts to at most to the coordinator, who then proceeds to compute the sum d′ = i∈[k] P Ii +1 P Ii +1 2 , which is in itself a K-approximation of d(v), where we can only 2 and at least 2 i∈[k] i∈[k] ′ d over-count. Hence 2k ≤ d(v) ≤ d′ . The coordinator announces d′ to each player and they proceed to the next step. The cost so far has been O(k log log d(v)). In the second phase, the players start a O(log k)-round procedure, where in each round their decrease √ their guess, d′′ , of d(v), by a multiplicative factor of α, having the starting guess be d′′ = d′ . In each round they repeat independently an experiment of sampling possible edges (adjacent to v) and checking whether the sample contains an edge in E. They set a threshold for each round, and if the number of samples that contained an edge in E exceeds the threshold, they stop and declare the value of d′′ for that round as the approximated value. If the last guess is reached, the players output it without running the experiment. Note that the case of d′′ = 1, therefore, is never checked, and we can assume that d′′ > 2. In each round, r, we denote d′′ (r) as the value of the guess, d′′ , for that round, and m(r) is the number of experiments they run. A single experiment is choosing into a set, S(r), every neighbor of v with probability d′′1(r) , using public randomness, and then each player sends the coordinator a bit indicating whether S(r) ∩ Ei = ∅. Each round the players assume their guess is correct, and therefore compute ′′ F (r) = (1 − (1 − d′′ (r))d (r) ) as the probability of success in a single experiment, and thus the expected fraction of successes. The actual probability (as well as the expected fraction of successes) is E(r) = (1 − (1 − d′′ (r))d(v) ). 7 We wish to prove that, with high probability, if d′′ (r) > αd(v), then the number of successes does not exceed the threshold, which is F (r) c , where c is a small constant who value we will determine later. In that case we have E(r) < F (r) and F (r) E(r) ≥ (1−(1−d′′ (r))d ′′ (r) ) d′′ (r) (1−(1−d′′ (r)) α ) , which is lower bounded by, (1 + β1 (α)), where β1 is a constant dependent only on α. The players may assume the lowest possible value for d′′ , in order for the expression to be dependent only on α, as for d′′ = 2 we get a small constant bigger than 1, and as d′′ tends to infinity it increases to 1− 1e 1 1−( 1e ) α . Therefore, by choosing c to be small enough (less than the square root of the difference) and running a number of experiments dependent on β1 and τ , a Chernoff bound yields a 1 − τ bound on the probability of exceeding the τ threshold. We wish to reduce the error by a Θ(log k) factor, to 2 log k , so that we may use the union ′′ bound to ensure this deviation doesn’t happen in any round where d (r) > αd(v), and by a Chernoff argument an increase to m(r) by a factor of O(log log k) suffices. √ , we exceed the threshold with On the other hand, we show that the first guess where d′′ (r) < d(v) α probability at least (1 − τ /2). Combined with what we proved in the previous case, this proves that with probability at least (1 − τ ) we stop at a guess which is within the bounds of an alphaapproximation of the d(v), as required. Now we have d′′ (r) < d(v) √ α that implies E(r) > F (r) and F (r) E(r) ≥ ′′ (1−(1−d′′ (r))d (r) ) √ ′′ , (1−(1−d′′ (r)) αd (r) ) which is upper bounded by constant, β2 (α), dependent only on α. Therefore, m(r) = Θ(log log k) is more than enough for a constant bound on a constant deviation small enough not to reduce the number of successes below the threshold. The total complexity of each round, therefore, is O(k log log k), and summing across all rounds we get O(k log k log log k). The overall complexity of the algorithm is O(k log log d(v) + k log k log log k). Lemma 3.2. In the no-duplication variant, for any given vertex, v, the players can compute a αapproximation of d(v), for alpha = O(1), with complexity O(k log log d(v) k ). Proof. Each player,Pi , computes locally, di (v), the number of edges adjacent to v in his input, Ei , and sends to the coordinator the (log α1 ) most significant bits along with the index of the cutoff, which takes O(log log di (v)) bits to represent. The coordinator assumes all the missing bits are zeros, and which makes this an α-approximation of di (v) for each player, when we can only under-count, and thus the sum of all this approximation, is also an α-approximation. Note that since there is no duplication, the worst case, by convexity, is when all players have a k1 -fraction of the edges, which implies the bound stated in the lemma. Note that this approximation procedure can be applied to any subset of vertex pairs, including estimating the total number of edges in the graph, and not only to the specific set of all possible edges adjacent to a given vertex. More generally, this solves the problem of estimating the number of distinct elements in a set. • Choosing uniformly a random edge - this usually can’t be performed efficiently in the PT model (unless the standard model is augmented), and is commonly replaced by choosing a random vertex, and then a random neighbor from the adjacency list. In our model however, we can once again use 8 randomness to fix an order over the edges and each player sends to the coordinator its highest ranked edge, which is O(log(n)), and across all players this sums up to O(k log n). The coordinator chooses the highest ranked edge, and posts it to all players. • Selecting all edges of a subgraph induced by V ′ ⊂ V - this also cannot be performed efficiently in the PT model, as it requires querying all possible vertex pairs, or going over all relevant adjacency lists. In our model, on the other hand, it is possible by having each player post all edges (in the blackboard model the players post in turns so as to not repeat the same edge) of the relevant subgraph in his input. Let m denote the number of edges in the relevant subgraph, then the complexity is O(km log n), and it is the same with simultaneous communication (except that only the referee will know the answer). The complexity is reduced to O(m log n) if there is no edge-duplication, or if the players communicate using a blackboard. If m is significantly smaller than |V ′ |2 , then our procedure is more efficient. This is particularly relevant when implementing a BFS. It can be done in O(n log n) by having all players post all the neighbors of the currently examined vertex. 3.2 Input analysis Prior to discussing our proposed algorithms, we analyze the properties of the input - a graph ǫ-far from being triangle free. Our pivotal tool for this analysis is bucketing. We partition V into buckets, such that for 1 ≤ i ≤ ⌊log3 n + 1⌋ we have Bi = {v ∈ V | 3i−1 ≤ deg(v) < 3i }, whereas B0 is the bucket of singletons. Note that there are less than log n buckets. Let d− (Bi ) = 3i−1 and d+ (Bi ) = 3i denote respectively the minimal and maximal bounds on degrees of vertices in Bi . We use d(Bi ) to mean any degree in that range, when the 3 factor is negligible, and refer to it as the degree of the bucket. We say an edge is adjacent to a bucket, if it is adjacent to at least one of its vertices. Additionally, we call a set of triangle-vees disjoint if any two of them are either edge disjoint or originate from a different vertex. Note that for simplicity we ignore some rounding issues, and avoid using floor or ceiling values. We are interested in buckets that contain many vertices that participate in a large number of triangles. Towards that end, we introduce the following definition, and analyze its properties. Definition 4 (full bucket). We call Bi a full bucket if the edges adjacent to it contain a set of triangle-vees. Let Bmin denote the bucket with the lowest degree of all the full buckets. ǫnd 2 log n disjoint Observation 3.3. By the pigeonhole principle there is at least one full bucket, as there are at least ǫnd disjoint triangle-vees. Lemma 3.4 (size of a full bucket). If Bi is a full bucket then: ǫnd 2nd ≤ |Bi | ≤ min{n, − } + log n · d (Bi ) d (Bi ) when the upper bound holds regardless of Bi being full. Proof. The number of disjoint triangle-vees in a full bucket is at least 2 ǫnd log n . Therefore, the lower bound pertains to the extreme case when all vertices have the maximal degree d+ (Bi ), that consists entirely of ǫnd d+ (Bi )/2 disjoint triangle-vees, thus reaching the sum log n with as few vertices as possible. The upper bound follows from the opposite extreme when each vertex contributes as little as possible which is d− (Bi ), and there are at most d−2nd (Bi ) such vertices as it would amount to nd edges, the total number of edges in the graph (the 2 factor follows from counting each edge twice). 9 Definition 5 (full vertex). We call a vertex, v, a full vertex if at least an to it are a set of disjoint triangle-vees. Additionally, let ǫ 12 log n -fraction of the edges adjacent F (Bi ) = {v ∈ Bi | v is full} , be the set of full vertices in Bi . And let F (V ) = {v ∈ V | v is full} , be the set of all full vertices in V . Full vertices vertices play a vital role in finding a triangles, as they, by definition, participate in many disjoint triangles. We, therefore, prove several useful lemmas about their incidence, as we are interested in identifying such vertices, preferably using sampling. Lemma 3.5. At least an ǫ 12 log n -fraction of the vertices in a full bucket, Bi , are full. Proof. We prove that otherwise there are less than 2 ǫnd log n triangle-vees adjacent to it, which contradicts it being full. This holds even if we disregard any double-counting, and assume the bucket has the maximal , and all its vertex degrees are d+ (Bi ) (both assumptions can not hold simultaneously, but this size of d−2nd (Bi ) only strengthens our proof). The total contribution to the count of triangle-vees coming from non-full vertices is less than: 2nd ǫ 1 1 · − · d+ (Bi ) · = ǫnd 2 d (Bi ) 12 log n 4 log n . + i) vees to the count, and if we assume the fraction of full Each full vertex can contribute at most d (B 2 ǫ vertices is less than 12 log n , it amounts to less than: ǫ 2nd d+ (Bi ) 1 · − · = ǫnd 12 log n d (Bi ) 2 4 log n . overall all vertices combined contribute less than the required ǫnd 2 log n to the disjoint triangle-vees count. Lemmas 3.4 and 3.5 imply the following corollary: Corollary 3.6. The number of full vertices in a full bucket, Bi , is at least: |F (Bi )| ≥ ǫ2 · d ·n 12 · log2 n · d+ (Bi ) Next, we single out a set of buckets in proximity to a given bucket, that will play a special role in our algorithm. Definition 6 (r-neighbourhood of a bucket). Let r ∈ N such that r ≤ log3 n. We call Nr (Bi ) = {Bj | j ≥ (i − log3 r)} the r-neighborhood of bucket Bi , that is the set of all buckets of higher degrees, itself, and the log3 r buckets right below it in the degree ranking. Additionally, we call N (Bi ) = Bi−1 ∪ Bi ∪ Bi+1 the neighborhood of bucket Bi . 10 Lemma 3.7. Let Bi be a full bucket. We prove the following lower bound on the ratio of the number of full vertices in it to the combined size of the buckets in its neighborhood: ǫ2 |F (Bi )| ≥ |N (Bi )| 312 · log2 n (1) on the size of Bj as we have proven in lemma 3.4. Proof. For any j we have the upper bound d−2nd (Bj ) Therefore, we have the following upper bound on the sum of bucket sizes: |N (Bi )| = |Bi−1 | + |Bi | + |Bi+1 | 2nd 2nd 1 2nd ≤3· − + − + · − = d (Bi ) d (Bi ) 3 d (Bj ) 26 · nd 26 · nd = = − 3 · d (Bi ) 3i 2 ·dn Additionally, we have 12·logǫ2 n·d + (B ) as a lower bound on |F (Bi )| (corollary 3.6). Hence, we get the i following lower bound on the ratio: |F (Bi )| ≥ |N (Bi )| ǫ2 ·dn 12·log2 n·3i 26·nd 3i = ǫ2 312 · log2 n (2) Lemma 3.8. Let Bi be a full bucket. We prove the following lower bound on the ratio of the number of full vertices in it to the combined size of the buckets in its r-neighborhood: ǫ2 |F (Bi )| P ≥ |Bj | 108 · log2 n · r (3) Bj ∈Nr (Bi ) Proof. For any j we have the upper bound d−2nd (Bj ) on the size of Bj as we have proven in lemma 3.4. Therefore, we have the following upper bound on the sum of bucket sizes: X Bj ∈Nr (Bi ) |Bj | = X < j≥(i−log3 r) = 9 · ndr 3i ⌊log3 n+1⌋ X j=(i−log3 r) ⌊log3 n+1⌋ X |Bj | ≤ j=(i−log3 r) 2nd 3j−1 2nd 6nd · r X 1 6nd · r 3 ≤ · ≤ · j−1 i m 3 3 3 3i 2 m≥0 2 ǫ ·dn Additionally, we have 12·log as a lower bound on |F (Bi )| (corollary 3.6). Hence, we get the fol2 n·3i lowing lower bound on the ratio: |F (Bi )| P ≥ |Bj | Bj ∈Nr (Bi ) ǫ2 ·dn 12·log2 n·3i 9·ndr 3i 11 = ǫ2 108 · log2 n · r (4) Now we show that we can efficiently sample edges adjacent to a full vertex in order to detect a trianglevee. Lemma 3.9 (Extended Birthday Paradox). Let v be a vertex of degree d(v) ≥ 2, such that at least αd(v), for α ≥ 2/d, of the edges adjacent to it are a set of disjoint triangle-vees. It is enough to sample each edge q independently with probability p = c · √ 1 , where c = 4 · ln δ1′ , in order for the sampled set to contain α·d(v) a triangle-vee with probability at least (1 − δ′ ). Proof. The probability of any specific triangle-vee to be sampled is p2 . By the linearity of expectation the 2 expected number of triangle-vees sampled is p2 · αd(v) = c2 . By a Chernoff bound the probability that less 2 c2 2 2 c2 than one triangle-vee has been sampled is less than e− 4 ·(1− c2 ) ≤ e− 16 = δ′ . Corollary 3.10. Let v beqa full vertex q of degree d(v). By sampling independently every edge adjacent to it log n 6 , we find a triangle-vee with probability at least (1 − δ/6). with probability p = 4 · ln δ · 12ǫ·d(v) Proof. We get this trivially by plugging α = ǫ 12 log n and δ′ = δ/6 in lemma 3.9. √ Finally, we prove that there are many triangle-vees adjacent to vertices of degree O( nd), such that we can focus only on such vertices, and adjust our analysis accordingly. p Definition 7. Let Vh denote the subset of V that contains all the vertices with degree at least dh = nd/ǫ. Let Eh ⊂ E denote all edges with both endpoints in Vh . Finally, let Vl = V \Vh , and let Gl denote the resulting graph when Eh is removed from G. Lemma 3.11. Gl is 2ǫ -far from being triangle-free, and there are at least ǫnd/2 disjoint triangle-vees adjacent to vertices in Vl . √ ǫnd, it follows that Eh < ǫnd Proof. Because |Vh | ≤ nd dh = 2 , hence if all edges in Eh are removed, at least ǫnd 2 additional edges need to be removed from G for it to be triangle-free, as at least ǫnd are required in total by definition. This also implies that ǫnd 2 of the triangle-vees are adjacent to a vertices in Vl . Definition 8. Let dl = ǫd 2 log n . Lemma 3.12. We have the following bound on degree of vertices in Bmin : dl ≤ d− (Bmin ) ≤ dh Proof. The lower bound follows a simple counting argument, as even if all n vertices are in Bmin , if ǫd d− (Bmin ) < dl , then the total number of triangle-vees adjacent to Bmin is less than 2 log n n which contradicts it being full. The upper bound is implied by lemma 3.11 and the pigeonhole principle. Corollary 3.13. We have: 3 1. |Bmin | ≥= ǫ2 3·log n · √ nd, 5 2. |F (Bmin )| ≥ ǫ2 12·log2 n·3 · √ nd. Proof. If plug the upper bound from lemma 3.12 into lemma 3.4 and corollary 3.6 we get the first and second clause of this corollary respectively. 12 3.3 Unrestricted Communication The first protocol we present requires interaction between the players, and exploits the following advantage we have over the query model: suppose that the players have managed to find a set S ⊆ E of edges that contains a “triangle-vee” — a pair of edges {u, v} , {v, w} ∈ S such that {u, w} ∈ E (but {u, w} is not necessarily in S). Then even if S is very large, the players can easily conclude that the graph contains a triangle: each player examines its own input and checks if it has an edge that closes a triangle together with some vee in S, and in the next round informs the other players. Thus, in our model, finding a triangle boils down to finding a triangle-vee. (In contrast, in the query model we would need to query {u, w} for every 2-path {u, v} , {v, w} ∈ S, and this could be expensive if S is large.) Our goal is to find a full vertex. Once that is obtained, we proved in lemma 3.9 we can efficiently sample a relatively small subset of its edges to find a triangle-vee, thus successfully ending the algorithm. More p concretely, If v is a full vertex, then sampling each of its edges with probability pd(v) = Θ( log n/d(v)) will reveal a triangle-vee with constant probability. Note that deg(v) may be significantly higher than the average degree d in the graph, so we cannot necessarily afford to sample each of v’s edges with probability pd(v) ; we need to find a low-degree vertex which is √ full. Towards that end, we proved in lemma 3.11 we can focus only on vertices of degree at most dh = O( nd). With that in mind, we proceed to present our strategy for finding a full vertex. We first describe the core of the algorithm in a relatively detail-free manner, to emphasize the intuitive narrative leading us throughout the procedure. This will be followed by a rigorous analysis of the full algorithm. How can the players find a full vertex? A uniformly random vertex is not always likely to be full — there might be a small dense subgraph of relatively high-degree nodes which contains all the triangles. In order to target such dense subgraphs, we use bucketing: we partition the vertices into buckets, with each bucket Bi containing the vertices with degrees in the range [3i , 3i+1 ). We want to find a full bucket, and sample its vertices, as we proved that many of them are full. Of course, we cannot know in advance which bucket is full; we must try all the buckets. Hence, we iterate over the buckets in an increasing order of their associated vertex degree, up until dh , and for each bucket, assume it is full, and then sampling enough of its vertices, relying on the lower bound we proved for the number of full vertices in a full bucket, for the sample to include a full vertex. Then, we sample the edges of each vertex, which for the full vertex will result in discovering a triangle-vee with high probability. Although, our assumption can be wrong in many cases, we have proven that there is at least one full bucket, Bi , in that range of degrees, hence the assumption will be correct at least once, which is enough for our algorithm to succeed with high probability. It remains to describe, given Bi is a full bucket, how we can sample a random vertex from it, that is, a random vertex with degree in the range [3i , 3i+1 ). We cannot do that, precisely, but we can come close. Because the edges are divided between the players, no single player initially knows the degree of any given vertex. However, by the pigeonhole principle, for each vertex v there is some player that has at least deg(v)/k of v’s edges, and of course no player has more than deg(v) edges for v. Let B̃ij := v ∈ V | 3i /k ≤ dj (v) ≤ 3i+1 be the set of vertices that player j can “reasonably suspect” S belong to bucket i, where dj denotes the degree of vertex v in the input of player j, and let B̃i := j B̃j . By the argument above, Bi ⊆ B̃i . Also, B̃i ⊆ Nk (Bi ), since the total degree of any vertex selected cannot be smaller than 3i /k. Therefore, sampling uniformly from B̃i is a good proxy for sampling from Bi , although we may also hit adjacent buckets. Nevertheless, we have proven that a full bucket must be large, and hence constitutes at least roughly a k1 -fraction of B̃i and Nk (Bi ). Hence a uniformly random sample will yield a vertex from Bi with probability at least roughly 1/k, and Θ̃(k) samples yield a vertex from Bi with high 13 probability. Lemma 3.14. Let Bi be a full bucket. It suffices to sample uniformly with replacement m = ln ( 6δ ) · 108·log2 n·r ǫ2 vertices from Nr (Bi ), in order for the sampled set to contain a vertex from F (Bi ) with probability at least (1 − 6δ ). 2 ǫ Proof. The probability of sampling a vertex from F (Bi ) is p = 108·log according to Lemma 3.8. There2 n·r fore, the probability of not having a vertex from F (Bi ) after m samples is bounded by (1 − p)m ≤ e−mp = δ/6. To implement the sampling procedure we need two components: first, we need to be able to sample uniformly from B̃i . The difficulty here is that each vertex v ∈ B̃i can be known to a different number of players — possibly only one player j has v ∈ B̃ij , possibly all players do. If we try a naive approach, such as having each player j post a random sample from B̃ij , then our sample will be biased in favor of vertices that belong to B̃ij for many players j. Our solution is to impose a random order on the nodes in B̃i by publicly sampling a permutation π on V (this is done by interpreting the random bits as a permutation on V ), and we then choose the smallest node in B̃i with respect to π. This yields a uniformly random sample, unbiased by the number of players that know of a given node. We call this procedure SampleUniformFromB̃i . Algorithm 1 SampleUniformFromB̃i 1: π ← random permutation on V j 2: Each player j sends the coordinator the first vertex in B̃i with respect to π 3: The coordinator outputs the first vertex with respect to π of all the vertices it received Our sample is too large to treat every sampled vertex as if it is a full vertex from Bi . Sampling edges for each vertex is too costly and wasteful, since it is possible that only a k1 -fraction of the sampled vertices are even in Bi . The second component, therefore, verifies that a sampled √ node indeed belongs to Bi . We cannot do that exactly, but we can come sufficiently close. We compute a 3-approximation of the degree of the sampled node, as explained in Theorem 3.1, and discard vertices whose approximate degree does not match N (Bi ). This substantially reduces the size of the sampled set without discarding any vertex from Bi . We call this procedure ApproxDegree(v). The protocol for player j is sketched in Algorithm 2. Here N = Θ̃(k) is the number of samples from B̃i required to produce a sample from Bi with good probability. Following the procedure described in Algorithm 2, the coordinator sends all the edges he received to all the players, and the players then check their own inputs for an edge that closes a triangle with some triangle-vee sent by the coordinator. With high probability, a triangle-vee is discovered, and the protocol completes in the next round. We move on to a more rigorous analysis of the sampling parameters and the complexity. First we compute the number of vertices we need to uniformly sample from N (Bi ) to find a full vertex, so that we can bound the number of vertices we examine while retaining a high probability of preserving a full vertex. 2 n Lemma 3.15. Let Bi be a full bucket. It suffices to sample uniformly with replacement m = ln ( 6δ )· 312·log ǫ2 vertices from N (Bi ), in order for the sampled set to contain a vertex in F (Bi ) with probability at least (1 − 6δ ). 2 n according to Lemma 3.7. ThereProof. The probability of sampling a vertex from F (Bi ) is p = 312·log δ·ǫ2 fore, the probability of not having a vertex from F (Bi ) after m samples is bounded by (1 − p)m ≤ e−mp = δ/6. 14 Algorithm 2 Code for player j For each i = 0, . . . , log n: ℓ←0 Repeat until ℓ ≥ N : v ← SampleUniformFromB̃i ¯ ← ApproxDegree(v) d(v) √ √ ¯ ≤ 3d+ (Bi ): If d− (Bi )/ 3 ≤ d(v) ℓ←ℓ+1 Jointly generate a public random set S ⊆ V , where each u ∈ S with iid probability pd(v) ¯ Send Ej ∩ ({v} × S) to the coordinator 2 n·k . we use q as We now present the procedure GetFullCandidates(Bi ). Let q = ln ( 6δ ) · 108·log ǫ2 a bound on the total number of samples, and we also bound the number of samples that pass the degree approximation criteria. Both bounds are needed to ensure worst case complexity. Algorithm 3 GetFullCandidates(Bi) 1: count ← 0 2: C ← ∅ 2 n 3: Do until count = q or |C| = ln ( 6δ ) · 312·log ǫ2 4: count ← count + 1 5: v ← SampleUniformFrom B̃i √ 6: compute d′ (v), a 3-approximation of d(v), such that the error probability is at most d− (B ) 7: if √3 i 8: output C √ ≤ d′ (v) ≤ 3d+ (Bi ) then add v to C δ 3q Lemma 3.16. The complexity of GetFullCandidates(Bi ) is O(k2 ·log4 n log log n), and in the no-duplication variant it is O(k2 · log3 n). Proof. Each vertex the players send to the coordinator costs O(log n), therefore the overall complexity of choosing uniformly a vertex from U (Bi ) is O(k log n). According to Theorem 3.1, the complexity of a constant approximation of d(v) for vertex v, when the error bound is Θ( 1q ), is O(k · log k · log log k · (log log n + log k)) = O(k · log2 n · log log n), and in the no-duplication variant it is O(k log log nk ). The number of iterations is at most q = O(k log2 n), hence the total complexity is O(k 2 · log4 n log log n), and in the no-duplication variant it is O(k2 · log3 n). √ Lemma 3.17. If Bi is a full, then C contains a vertex from F (Bi ) along with a correct 3-approximation of its degree with probability at least 1 − 2δ 3 . Proof. First, note that a union bound implies that all O(q) vertex approximations were correct with probability at least (1 − 3δ ). Due to the symmetric process of sampling in algorithm 1, the vertices are chosen uniformly from B̃i . Lemma 3.14, when substituting r = k, implies that q uniform samples is enough to sample a vertex from F (Bi ) with probability at least (1 − 6δ ). And since B̃i ⊆ Nk (Bi ), sampling from B̃i can only improve our probability of finding a full vertex. Additionally, lemma 3.15 assures us that 2 n uniform samples from N (Bi ) is enough to encounter a full vertex with probability at least ln ( 6δ ) · 312·log ǫ2 δ (1− 6 ). Note that if all degree approximations are in the guaranteed range, then all vertices sampled from Bi 15 are added to C, and all other vertices that are added are in N (Bi ). Since our sampling process is symmetric and therefore uniform, vertices sampled into C have at least the same probability of containing a full vertex, 2 n samples suffice. By a as in the case of sampling uniformly from all of N (Bi ), and thus ln ( 6δ ) · 312·log ǫ2 union bound argument, the probability of all approximations being correct, and both sampling processes sufficing for finding a full vertex, is at least (1 − 3δ ) − 2 · 6δ ) = (1 − 23 δ). Finally, if both sampling processes would encounter a full vertex, we find it no matter which stopping condition made us halt, therefore it is also the probability of containing a full vertex in the output. By this point we know how to efficiently obtain, given Bi is a full bucket, a small sample of vertices that is likely to contain one full vertex from Bi . All that is left is to iterate over the sampled set, sampling edges adjacent to each vertex, such that for the full vertex the sample will contain a triangle-vee. Algorithm 4 SampleEdges(v) 1: S ← sample every possible edge adjacent to v with probability p = 4 · r q n ln 6δ · 12dlog ′ (v) 2: each player j sends the coordinator S ∩ Ej if this set is of size at most (1 + 3: the coordinator outputs S ∩ E ǫ· 18 d′ (v)·p 3 ln δ6 ) · √ ′ 3d (v)p Algorithm 5 FindTrinagleVee(Bi) 1: C ← GetFullCandidates(Bi ) 2: for each v ∈ C let S ← SampleEdges(v) and then the coordinator posts all the edges to all the players. 3 Lemma 3.18. The communication complexity of FindTrinagleVee(Bi ) is O(k · log 2 n · log4 n log log n) p d(Bi ) + k2 · Proof. The cost of GetFullCandidates is O(k2 · log4 n log log n). Each edge required O(log n) bits to identify, and since there is a limit on the size of the set the players can send, the overall complexity is p 3 O(k · log 2 n · d(Bi ) + k2 · log4 n log log n). Lemma 3.19. If Bi is a full bucket then the players find a triangle with probability at least 1 − δ using procedure FindTrinagleVee(Bi ). Proof. According to lemma 3.17, C contains a full vertex with probability at least (1 − 23 δ), and the approximation procedure of its degree gave a correct output. And according to corollary 3.10, if v is a full vertex, then sampling each of its edges with probability p suffices to sample a triangle vee, with probability at least (1 − 6δ ). Moreover, the expected size of the sampled set is d(v) · p, and by a Chernoff bound the probability of sampling more the cutoff size specified in step 2 of the sampling algorithm is at most δ/6. Overall, with probability at least (1 − δ), one of the vertices sampled will be full, with the players knowing approximately its degree, and hence sampling enough of its edges so that they find a triangle-vee, and do not need to send a set above the cutoff size. When that happens, one of the players will respond to the coordinator with the third edge completing the triangle-vee into a triangle, and the algorithm will end successfully. All that is left is to find a full bucket. This is achieved by iterating all the relevant buckets. Note the common theme which is that the players do not know how successful each sampling process was up until the algorithm terminates. They do not know which bucket is full, which of the sampled vertices are full, 16 or which of its adjacent edges belong to triangles. Since they examine all bucket, all sampled nodes (after preliminary filtering), and send all sampled edges, that knowledge is redundant, as when they encounter a full bucket and a full vertex they will be treated as such as a working assumption. Our analysis culminates in algorithm FindTrinagle(G), which nests inside it all the procedure we presented so far. Algorithm 6 FindTrinagle(G) 1: Run FindTrinagleVee(Bi) for every bucket starting the first bucket such that d− (Bi ) ≥ dl and until the last bucket where d+ (Bi ) ≤ dh Theorem 3.20. If the input graph is ǫ-far from √ being triangle free, then the players can find√a triangle with 4 nd log5/2 n + k2 log5 n log log n) = Õ(k 4 nd + k2 ). The probability at least (1 − δ) and complexity O(k p complexity is in fact Õ(k d(Bmin ) + k2 ) w.p. at least 1 − δ. Proof. According to lemma 3.12, one of the iterations of FindTriangleVee is performed on Bmin , which is a full bucket. According to lemma 3.19, the players find a triangle in that iteration with probability (1−δ). The maximal complexity of an iteration grows as d(Bi ) grows. The number of iterations is O(log n). Therefore, p with probability at least (1 − δ), the overall complexity is O(k d(Bmin ) log5/2 n + k2 log5 n log log n). However, if an error occurs, players will go over all buckets √ in the given range, thus in the worst √ then the 5/2 5 2 case the complexity is O(k dh log n + k log n log log n) = Õ(k 4 nd + k2 ). Corollary 3.21. There is a one-sided error protocol for testing triangle-freeness with communication com√ √ p 4 4 5/2 4 2 2 plexity O(k nd log n+k log n log log n) = Õ(k nd+k ). The complexity is in fact Õ(k d(Bmin )+ k2 ) w.p. at least 1 − δ. Corollary 3.22. The variant where the players are not given d has the same complexity. Proof. The players can compute 2-approximations of dh and dl , denoted by d′h and d′l , respectively, and then use d′l /2 and 2d′h as the boundaries for the iteration condition. The added complexity is negligible, and the error can be reduced to an arbitrarily small constant. The rest of the algorithm remains the same, and does not rely on any knowledge of d. Theorem 3.23. In the blackboard model, if the input graph is ǫ-far from being triangle free, then the players √ 4 can√find a triangle with probability at least (1p − δ) and complexity O( nd log5/2 n + k2 log5 n log log n) = 4 2 Õ( nd + k ). The complexity is in fact Õ( d(Bmin ) + k2 ) w.p. at least 1 − δ. The same protocol also solves testing of triangle-freeness. Proof. In the blackboard model, posting the edges of the sub-procedure SampleEdges(v) can be implemented more efficiently, having each player post the edges on the blackboard in turns, ensuring no edge is posted twice. This saves a k factor in the communication cost with regards to the coordinator model. 3.4 Simultaneous Communication In the simultaneous model, the players cannot interact with each other — they send only one message to the referee, and the referee then outputs the answer. This rules out our previous approach, as exposing a trianglevee does not help us if the players cannot then check their inputs for an edge that completes the triangle. Indeed, the simultaneous model is closer to the query model in spirit. Accordingly, we include trianglefreeness testers of [3], but show that we can implement them more efficiently in our model. Moreover, we achieve roughly the same complexity without knowing the average degree in advance. 17 √ √ We present separate algorithms for the case of d = Ω( n) and d = O( n), referred to as high and low √ degrees, respectively. when d = Θ( n), both algorithms are essentially identical. We conclude with an algorithm that works for the more general case where d is unknown to the players. 3.4.1 high degrees √ Forpgraphs with average degree Ω( n), the tester from [3] samples a uniformly random set S ⊆ V of Θ( 3 n2 /d) vertices, queries all edges in S 2 , and checks if the subgraph exposed contains a triangle. It is shown in [3] that if the graph is ǫ-far from triangle-free, then the subgraph induced by S will contain Θ(1) triangles in expectation, and the variance is small enough to ensure small error. We can implement this tester easily, and in our model it is less expensive: instead of querying all pairs in S 2 , the players simply send all the edges from S 2 in their input, paying only for edges that exist and not for edges that do not exist in the graph. The set S is large enough
that the number of edges in the subgraph does not deviate significantly from its expected value, Θ (nd)1/3 . We present an algorithm of complexity O(k(nd)1/3 ). We later show, in the lower-bounds section, that √ for average degree Θ( n) this is tight for 3 players. Algorithm 7 FindTringleSimHigh(G) q 2 1: S ← a uniformly random set of vertices of size c 3 nǫd , for a sufficiently large c 2: players send all edges in the subgraph induced by the vertices in S. If the number of edges to be sent by a player 2 4 exceeds l = |S| n2 δ nd, send any l edges. 3: The Referee checks whether the union on edges it received contains a triangle, and outputs accordingly. √ Theorem 3.24. The problem of triangle detection, when d = Ω( n) is known to the players, can be solved with communication cost of O(k(nd)1/3 log n) and a constant error. Proof. Let δ denote the required bound on the error, and let VS denote the subgraph of G induced by S. In [3] it was shown that for a sufficiently large c, the probability of VS not containing a triangle is arbitrarily small, and for our purposes taken as δ/2. Additionally, the probability of each edge appearing in VS is |S|·|S−1| n·(n−1) , thus by the linearity of expectation the expected number of edges in VS is |S|·|S−1| n·(n−1) nd < l 2 δ . Finally, by a Markov argument we get that the probability of the number of edges in VS exceeding l is at most δ/2. When that happens, even if every player has all the edges, none of them exceeds l, and overall, by a union bound argument, we get that the referee receives all edges in VS and they contain a triangle, with probability at 1 least (1 − δ). The complexity is at most O(kl log n) = O(k(nd) 3 log n). Corollary 3.25. The problem of triangle-detection in the no-duplication variant can be solved in the simultaneous model with a constant error, δ, such that the complexity is O((nd)1/3 log n) with probability at least (1 − δ), and the worst case complexity is O(k(nd)1/3 log n). Proof. The players would use algorithm 7 as in the general case, thus the worst case complexity is the same. But as we proved, with probability at least 1 − δ the total number of edges in the subgraph is O((nd)1/3 ), and since there is no duplication, that is also the total number of edges sent. 18 3.4.2 low degrees √ For density d = o( n), the approach above no longer works, as the variance is too large. To illustrate this, consider a graph with d vertices of degree Θ(n), which are the sources of Θ(nd) triangle-vees, such that all triangles have at least one such node. If we were to sample vertices uniformly at random, we need to sample Θ(n/d) vertices in order for the subgraph to contain a triangle. However, whereas in the query model we would need to make Θ(n2 /d2 ) queries to learn the entire subgraph induced by the set we sampled, in our model we proceed as follows (using ideas from [3], which require adaptivity there, and deploying them in a different way): let S be the set of Θ(n/d) uniformly-random vertices. We sample another smaller set, R, √ of Θ( n) vertices, and we send all edges in R × (S ∪ R). If indeed there is a small set of high-degree vertices participating in most of the triangles, then with good probability we will have one of them in S, and by the birthday paradox, one of its triangles will have its other two vertices in R. On the other hand, if the triangles are spread out “evenly”, then the subgraph R × R will probably contain one. The expected size of √ √ R × (S ∪ R) is O( n), and we show that w.h.p. the total communication is Õ(k n). √ Note that both our solutions work for d = Θ( n), and for
this density they are essentially the same: √ both sets, S and R, are of size Θ(n/d) = Θ(d) = Θ (nd)1/3 , so for d = Θ( n) the second protocol is not very different from the first. We can also show that if edge duplication is not allowed, a factor of k is saved in the communication complexity with high probability. Algorithm 8 FindTriangleSimLow(G) S ← sample each vertex with probability p1 = min{ dc , 1} 2: R ← sample each vertex with probability p2 = √cn 3: Players send all to the referee edges with one endpoint in R and the second endpoint R ∪ S. If the √ number of such edges in the input of a player exceeds q = 2c2 ( n + d) · 2δ , that player sends any q edges. 4: The Referee checks whether the union of the edges it received contains a triangle, and outputs accordingly. 1: Where c is a constant to be determined later. √ Theorem 3.26. The problem of triangle detection in the simultaneous model when d = O( n) is known to √ the players, can be solved with communication cost of O(k n log n) and with constant error. Proof. We show that algorithm 8 is such a solution. Let δ denote the required constant bound on the error, let VR denote the graph induced by R, and let VRS denote the graph on R ∪ S that includes all edges with at least one endpoint in R. Since each player sends at most q edges, the complexity of the algorithm is √ O(qk log n) = O(k n log n). 2 The probability of a given edge appearing in VR is p22 = cn , therefore by linearity of expectation, the 2 expected number of edges in VR is at most nd · cn = c2 d. The probability of any edge having one endpoint in S and the other in R is at most 2p1 p2 , therefore by linearity of expectation, the expected number of such √ edges is at most 2ndp1 p2 ≤ 2c2 n. Overall, we get that the expected number of edges in VRS , which are the only edges the players may send, is at most ( q2 ) , and by a Markov argument we get that with probability δ at least (1 − 2δ ) the number of edges in VS does not exceed q, thus all players can send all the edges they have in VRS . We show that with probability at least (1 − 2δ ) the edges in VRS contain a triangle, which via a union bound proves that the referee will receive a triangle with probability at least (1 − δ). Recall Gl , the graph 19 defined in definition 7 to be the subgraph on all edges adjacent to at least one vertex of degree at most dh . For the purpose of analysis, ignore all triangle that are not in Gl . According to lemma 3.11, Gl is 2ǫ -far from being triangle-free, and thus it contains a family of 6ǫ nd edge-disjoint triangles. Fix such a family, T . Note that in any triangle in Gl , at most one vertex in can be of degree higher than dh . Further restricting the number of triangles counted, we only count triangles where at least two vertices of degree at most dh were sampled into R, which implies that if the third vertex is of degree higher than dh it must be sampled into S. Let X be a random variable equal to the number of triangles in VRS with the aforementioned restrictions. The probability of such a triangle to be sampled is at least p1 p22 , therefore the ǫ ǫ E[X] ≥ ndp1 p22 = c2 6 6 (5) . We now bound the variance of X. For each t ∈ T , let Xt be an indicator of t being sampled (with our restrictions). Let dT (v) ≤ d(v) denote the degree of vertex v when only edges of T are left in the graph. Observe that if two triangles have no vertices in common, then they are selected independently. Additionally, note that two triangles in T can have at most one vertex in common, as all triangles are edge disjoint. The probability of two triangles with a joint vertex being both sampled can be split into two cases, one where the joint vertex is in S, and the second case is when it is sampled into R. The probability when the joint vertex is sampled into S, is p1 p42 . The number of triangles that can have
P dT (v) ≤ 2nd, by convexity we get vertex v in common and sampled into S is at most dT (v)/2 . Since 2 v∈V dT (v)/2
2 P v∈V ≤ 2d P i=1 n/2
2 ≤ 2d · n2 8 . As for the case when the common vertex, v, is sampled into R (note that it means that dT (v) < dh ), the probability of both triangles being sampled is p21 p32 . The number of vertices of degree dh is at most P dT (v)/2
2nd ≤ 2 dh , and once again by convexity we get that the number of such triangles is smaller than v∈Vh P v∈Vh dh /2
2 ≤ 2nd dh · d2h 8 . Therefore, the variance of X is, for d > c, bounded by: X dT (v)/2 X dT (v)/2 4 V ar[X] ≤ p1 p2 + p21 p32 2 2 v∈V v∈V ∩R n2 2nd d2h 2 3 ≤ 2d · p1 p42 + · p1 p2 8 dh 8 2 n c c 4 2ndǫ nd c 2 c 3 (√ ) + √ · ( ) (√ ) ≤ 2d · 8 d n n dn 8ǫ d 5 5 5 c c c = + √ ≤ 4 2 4 d and similarly for d ≤ c (which implies that p1 = 1, S = V and d = T heta(1)) we get that the variance 5 is also bounded by c2 . We conclude by employing a Chebyshev bound: Pr(X < 1) ≤ Pr(|X − E[X]| ≥ V ar 2 (X) 8ǫc4 4 δ 1 E[X]) ≤ ≤ = < 2 ((1/2) E[X])2 18c5 9c 2 20 Where the last inequality follows by taking c = probability at least (1 − δ/2). 8 9δ . Therefore, VRS contains at least one triangle with √ Corollary 3.27. The complexity of the no-duplication variant is O( n log n) with probability at least (1 − √ δ), and the worst case complexity is O(k n log n). Proof. The players would use algorithm 8 as in the general case, thus the worst case complexity is the same. But as we proved, with probability at least 1 − δ the total number of edges in the subgraph between an √ endpoint of R and S is O( n), and the total number of edges in the subgraph of R is at most O(d) and since √ there is no duplication, that is also the total number of edges sent, which implies complexity O( n). 3.4.3 Degree Oblivious Algorithm We start with a high-level overview of how we combine the protocols above and modify them so that they can be used without advance knowledge of the degree. The challenge here is that no single player can get a good estimate of the degree from their input, and since the protocol is simultaneous, the players must decide what to do without consulting each other. The natural approach is to use log n exponentially-increasing “guesses” for the density, covering the range [1, n], and try them all; however, if we do this we will incur a high cost for guesses that imply examining a larger sample than needed. We, therefore, take a more fine-grained approach. Our first observation is that some players can make a reasonable estimate of the global density, although they do not know that they can. Let d¯j denote the average degree in player j’s input Ej , and let us say that player j is relevant if d¯j ≥ (ǫ/(4k))d, and irrelevant otherwise. If we eliminate all the irrelevant players and their inputs, the graph still remains (ǫ/2)-far from triangle-free, so we can afford to ignore the irrelevant players in our analysis — except for making sure that their messages are not too large. Since players cannot know if they are relevant, all players assume that they are. Based on the degree d¯j that player j observes, it knows that if it is relevant, then the average degree in the graph is in the range  log n Dj = [d¯j , Θ(k d¯j )]. We fix in advance an exponential scale 2i i=0 of guesses for the density, and execute in parallel log n instances of triangle-freeness protocols, one for each degree 2i . However, each player j only participates in the O(log k) instances corresponding to density guesses that fall in Dj , and sends nothing for the other instances. The protocols are the two algorithms we have presented for a known average degree, with some modification. For relevant players, we know that the true density falls in their range Dj , so they will participate in the “correct” instance. For irrelevant players, we do not care, and their message size is also not an issue: their density estimate is too low, and the communication complexity of each instance increases with the density it corresponds to. If we are not careful, we may still incur a blow-up in communication, as relevant players may use guesses lower than the true density by a factor of k, which increases the size of the sample beyond what is necessary. However, by carefully assigning each player j a communication budget depending on d¯j , we can eliminate the blow-up, and match the degree-aware protocol up to polylogarithmic factors. We now move on to a detailed analysis of the algorithm, which relies on an integration of modified versions of the algorithms we presented in theqnon-oblivious sections. First, we alter the algorithm for highdegrees, such that instead of sampling |S| = 3 n2 ǫd 2/3 = Θ( nd1/3 ) vertices without replacement, we sample each −1/3 ). Additionally, we remove vertex independently (with replacement) with probability Θ( |S| n ) = Θ(nd the cap on the number of edges allowed to be sent from both algorithms (high and low degrees). Let Alghigh denote the modified algorithm for high degrees, and Alglow - for low degrees. We provides figures for both. 21 Algorithm 9 AlgHigh (G) 1: S ← a uniformly random set of vertices of size c q 3 n2 ǫd , for a sufficiently large c 2: players send all edges in the subgraph induced by the vertices in S. 3: The Referee checks whether the union on edges it received contains a triangle, and outputs accordingly. Algorithm 10 AlgLow (G) S ← sample each vertex with probability p1 = min{ dc , 1} 2: R ← sample each vertex with probability p2 = √cn 3: Players send all to the referee edges with one endpoint in R and the second endpoint R ∪ S. 4: The Referee checks whether the union of the edges it received contains a triangle, and outputs accordingly. 1: Lemma 3.28. Alghigh detects a triangle with a small constant error. Proof. By choosing sufficiently large constants, we ensure that the expected number of triangles is not lower than the expected value before the alteration. The bound on the deviation, follows the same analysis as in the proof of the original algorithm, and analogous to the correctness proof of Alglow , as when bounding the variance in the number of edge-disjoint triangles, only triangles with one common vertex are dependent. In terms of complexity, we once again use a Markov argument to claim that the total number of edges in the sampled subgraph does not exceed the expectation by a large constant factor with high probability. Alglow obviously also remains correct, as removing the cap could only increase the chances of detecting a triangle. Moreover, both algorithms have the same complexity as before with probability at least (1 − δ), as implied by their respective complexity analysis in the previous section. We will reinstall modified caps further into our analysis. The main sub-procedure the players utilize in both algorithms is choosing jointly, via public randomness, a subset S ⊆ V , such that each vertex is chosen independently with probability p, and then posting all edges in their inputs with both endpoints in S. We show that the number of edges each player has in S does not significantly exceed the expectation, given his average degree. Lemma 3.29. Let S ⊆ V denote a set where each vertex was sampled with probability p. The number of edges player j has in the subgraph VS induced by S is O(nd¯j · p2 · logn log(k log n)), such that this holds for all players with high probability Θ(1). Proof. Recall the bucketing partition we used in the section of input analysis. For a given player j, we partition V into O(log n) buckets as we did before, only this time according to dj (v) and not d(v). Since each vertex is chosen independently, we may utilize the Chernoff bound to claim that when sampling with probability p, the degree of each sampled vertex is reduced from dj (v) to O(p · dj (v) log(nk)) = O(p · 1 ). Therefore, by the union bound this holds true for all dj (v) log n), with probability of error at most O( nk n vertices in the inputs of all k players, with a small constant error. Next, we also claim that the number of vertices chosen from each bucket, B, is at most O(p·|B| log(k log n)), 1 with probability of error at most O( k log n ), once again due to a Chernoff argument. A union bound implies this holds true for all O(log n) buckets for all k players with a small constant error. Overall, if the number of vertices chosen from each bucket deviates by at most a O(log(k log n)) factor from the expectation, and each vertex degree is reduced to a size that deviates by at most an O(log n) factor from its expectation, we 22 get that for all k players the number of edges they have in the sampled subgraph deviates from its expected value (given d¯j ), by at most a O(log n log(k log n)) factor. For a given guess of the average degree d′ , let p(d′ ) denote the probability with which the players need to sample each vertex. Recall that all relevant players will include p(d) in their range of guesses (the 2 factor difference is asymptotically insignificant), hence the union of all the sampled edges includes a triangles with high probability. We note that for our purposes, an increase of the degree guess, d′ , by a factor of 2 decreases the sampling probability, p(d′ ), also by at most a factor of 2, therefore there is no dangerous super-constant blowup in the sampling probability that would otherwise incur an asymptotic overhead on the complexity. Observe that the guess for the average degree varies inversely as the corresponding sampling probability and thus expected sample size. √ √ We first discuss the case where for player j, d¯j = Ω( n), which implies d = Ω( n). The player performs simultaneously O(log k) algorithms each with a different guess, d′ , of the average degree. For √ 3 ′ ′ high degrees this means p(d ) = Θ( nd ). We now prove that the complexity bound on the message for each player can remain roughly the same, with the error remaining constant. More concretely, each player limits separately each of the O(log k) simul√ 3 ¯ j taneous algorithms by sending at most O( nd log n log (k log √ n)) edges. This implies that the complexity bound of this player over all its simultaneous instances is O( 3 nd log2 n log k log (k log n)). √ 3 Lemma 3.30. A complexity bound of sending at most O( nd¯j log n log (k log n)) edges for each instance of Alghigh suffices for the instance, pertaining to the correct guess, to send all the edges in its corresponding subgraph. Proof. Let r(j) = dd¯j denote the ratio between the correct average degree and the player’s observed average degree. The expected number of edges player j has in the sampled subgraph for a correct guess is 1/3 p √ (nd¯j ) 3 nd¯j ¯j ) ≤ O( 3 nd) n ) = Θ( ) ≤ O( Θ( d (nd)2/3 r(j)2/3 where we used the fact that r(j) = Ω(1). Therefore, player j can limit the edge budget of each algorithm √ 3 with a bound of O( nd¯j log n log (k log n)) following lemma 3.29, with all k players not exceeding the bound with high probability. This lemma along with the fact that we’ve shown that the sample pertaining to the correct guess contains a triangle with high probability implies correctness with constant error. √ Now we deal with the case d̄i ≤ n. As in the previous case the player performs O(log k) algorithms covering the relevant degree range. And as before all relevant players include the correct guess in their range of guesses, implying that the union of messages of all players contain a triangle with high probability following the same analysis as in the non-oblivious case. √ The player splits the relevant range of degrees into two cases. For every degree guess, d′ , where n ≤ √ ǫ n ¯j ¯j empty) the player simulates the algorithm for high d′ ≤ 4k ǫ d (Note that when d ≤ 4k this range is √ 3 degrees as we just described (with an edge limit of O( nd¯j log n log (k log n)) for each algorithm). √ √ If indeed d ≥ n then correctness and complexity analysis for that case is the same as when d¯j ≥ n. √ Whereas for every guess, d′ , where d¯j ≤ d′ ≤ n, the player simulates the AlgLow using d′ instead of d. More concretely, the player samples into S each vertex with probability p = min{ dc′ , 1}, and the of sampling into R each vertex with probability Θ( √1n ) as in the original algorithm remains the same (the players can use the same R across all simultaneous instances). 23 √ We use a cap of O( n log n log (k log n)) edges for each instance of AlgLow and, as the we promptly prove, it suffices for the correct instance. √ Lemma 3.31. A complexity bound of sending at most O( n log n log (k log n)) edges for each instance of √ AlgLow , suffices, for the case where d ≤ n, for the protocol pertaining to the correct guess to send all the edges in its corresponding subgraph. √ ¯j Proof. The expected number of edges player j has in R is Θ( nnd ) = Θ(d¯j ) = O( n). It is not surprising that the expected number did not increase, as the sample size does not depend on the average degree, and we have already assumed, in our previous analysis, the worst case of each edge in R appearing in all inputs. √ For the correct guess, d′ = d ≤ n, the expected number of edges player j has connecting S and √ ¯j R is Θ( dn√d n ) = O( n). Therefore, the expected number of edges player j needs to send overall for the √ √ correct guess is O( n), and indeed, for all guesses where d′ ≤ n, player j can limit the edge budget √ of each algorithms with a bound of O( n log n log (k log n)) following lemma 3.29, with all k players not exceeding the bound with high probability. Since the complexity (and the edge cap) is higher for the simulations of AlgLow than for the simulations of AlgHigh , the simulations of AlgHigh do not affect the overall computation of the complexity asymptotically. √ To conclude, when d ≤ n, the message cost of each player, j, is √ 1 O(max{ n, (nd¯j )1/3 } log2 n log k log (k log n)) = O((nd) 3 log2 n log k log (k log n)) 1 thus the overall complexity of the protocol for all players is O(k(nd) 3 log2 n log k log (k log n)). √ √ √ Whereas when d ≤ n then d¯j ≤ n, and the complexity of player j is O( n log2 n logk ), thus for k √ players we get O(k n log2 n logk ). We summarize our complete procedure for all cases in F indT riangleSimOblivious(G). The correctness follows the fact that all relevant players participate in the instance pertaining to the correct guess, and what we have proved about the edge cap not limiting that instance. Theorem 3.32. The problem of d-oblivious triangle detection in the simultaneous model can be solved with √ √ 1 communication cost of O(k n log2 n log k log (k log n)) for d = O( n), and in O(k(nd) 3 log2 n log k log (k log n)) √ for d = Ω( n), by a single algorithm, with constant error in both cases. Algorithm 11 FindTriangleSimOblivious(G) 1: 2: 3: 4: 5: 6: Each player j ∈ [k] runs simultaneously O(log k) protocols - for each degree guess d′ - covering its ¯j relevant degree range, Dj = [d¯j , 4k ǫ d ]: √ for each guess d′ ≥ n: run AlgHigh and send up to O((nd)( 1/3) log n log (k log n)) edges √ for each guess d′ < n: √ run AlgLow and send up to O( n log n log (k log n)) edges The Referee checks whether the union of edges it received contains a triangle, and outputs accordingly. 24 4 Lower Bounds Our main result in this section is the following: √ ǫ be the task of finding a triangle edge in graphs of size n and Theorem 4.1. For any d = O( n), let Tn,d average degree d which are ǫ-far from triangle-free. Then for sufficiently small constant error probability δ < 1/100 we have:   1/6 ǫ . (1) For k > 3 players: CCsim k,δ (Tn,d ) = Ω k · (nd)   ǫ ) = Ω (nd)1/3 . (2) For 3 players: CCsim (T 3,δ n,d √ To show both results, we first prove them for average degree d = Θ( n), and then easily obtain the √ result for lower degrees by embedding a dense subgraph of degree Θ( n) in a larger graph with lower overall average degree. √ To prove (1), we begin by proving that for graphs of average degree Θ( n), three players require ǫ √ in the one-way communication model, where Alice and Bob Ω(n1/4 ) bits of communication to solve Tn, n send messages to Charlie, and then Charlie outputs the answer. In fact, our lower bound is more general, and allows Alice and Bob to communicate back-and-forth for as many rounds as they like, with Charlie observing the transcript. We then “lift” the result to k > 3 players communicating simultaneously, using symmetrization [33]. To prove (2), we show directly that in the simultaneous communication model, three players require √ ǫ in graphs of average degree Θ(√n). Ω( n) bits to solve Tn,d Our lower bounds actually bound the distributional hardness of the problems: we show an input distribution µ on which any protocol that has a small probability of error on inputs drawn from µ requires high communication. This is stronger than worst-case hardness, which would only assert that any protocol that has small error probability on all inputs requires high communication. 4.1 Information Theory: Definitions and Basic Properties We start with an overview of our information theory toolkit, which is our primary technical apparatus for directly proving lower bound (as opposed to reductions, which we also use to derive subsequent results). Definition 9. The mutual information between two random variables is I(X; Y ) = H(X) − H(X|Y ) = Ey∼Y [D (µ(X|Y = y) k µ(X))]. Lemma 4.2 (Super-additivity of information). If X 1 , . . . , X n are independent, then I(X 1 , . . . , X n ; Y ) ≥ n X I(X i ; Y ). i=1 Lemma 4.3. Let p, q ∈ (0, 1), and let D (q k p) denote the KL divergence between Bernoulli(q) and Bernoulli(p). Then for any p < 1/2 we have D (q k p) ≥ q − 2p. Proof. Since the divergence is non-negative, it suffices to show that for p < 1/2, for any q ≥ 2 · p we have D (q k p) ≥ q − 2p. For convenience, let us write q = p + x, where −p ≤ x ≤ 1 − p. Our goal is to show that when x ≥ p, we have D (p + x k p) ≥ x − p. 25 Consider the difference g(x, p) = D (p + x k p) − (x − p) 1−p−x p+x + (1 − p − x) log − (x − p). = (p + x) log p 1−p At x = p we have g(x, p) ≥ 0 (since divergence is always non-negative); we will show that the derivative w.r.t. x is non-negative for x ≥ p, and hence g(x, p) ≥ 0 for any x ≥ p. Taking the derivative with respect to x, we obtain ∂ p+x p 1 1−p−x 1−p 1 g(x, p) = log + (p + x) · · − log − (1 − p − x) · · −1 ∂x p p + x p ln 2 1−p 1 − p − x (1 − p) ln 2     x x = log 1 + − log 1 − −1 p 1−p The derivative is increasing in x, and since we consider only x ≥ p, it is sufficient to show that it is nonnegative at x = p:   p ∂ = log (1 + 1) − log 1 − g(x, p) −1 ∂x 1−p x=p p − 1 ≥ 0. ≥1+ 1−p In the last step we used the fact that log(1 − z) ≤ −z for any z ∈ (0, 1); in our case, since p < 1/2, we have p/(1 − p) < 1. √ 4.2 Random Graph of Degree Θ( n) In this section, we derive our main results, lower bounds for one-way and simultaneous communication, all √ using a single distribution, µ, for graphs of average degree Θ( n), whose edges are shared among 3 players. In the subsequent sections we move on to showcase methods to generalize these results for k players and other average degrees. 4.2.1 The input distribution and its properties √ Our lower bounds for degree Θ( n) use the following input distribution, µ: we construct a tripartite graph √ G = (U ∪ V1 ∪ V2 , E), where each edge appears iid with probability γ/ n for some constant γ. This distribution has very high probability that the input is ǫ-far from being triangle-free, but it does not guarantee it with probability 1. Still, if we can show some task (finding a triangle, or finding a triangle-edge) is hard on µ, then it is also hard on the distribution µ′ obtained from µ by conditioning on the input being ǫ-far from triangle-free. Observation 4.4. Let Π be a protocol for some task T , with error probability at most δ on some distribution µ supported on a class X of inputs. Then for any Y ⊆ X , the error probability of Π on µ|Y is at most δ/ Prµ [Y]. 26 Proof. We can write: δ ≥ Pr [Π errs on X] = Pr [Π errs on X | X ∈ Y] Pr [X ∈ Y] + Pr [Π errs on X | X 6∈ Y] Pr [X 6∈ Y] ≥ Pr [Π errs on X | X ∈ Y] Pr [X ∈ Y] . The claim follows. In our case we have: Lemma 4.5. When γ is sufficiently small, a graph sampled from µ is O(1)-far from triangle-free with probability at least 1/2. Proof. Let T be the random variable of the set of triangles in the graph, and let I be the set of pairs of triangles that share an edge. let   n γ γ3 E[|T |] = ( √ )3 ≥ n3/2 3 n 12   1 n γ E[|I|] = 3 (n − 3)( √ )5 ≤ E[|T |] 2 3 n Where the last inequality follows from choosing a sufficiently small γ. It follows that E[|T | − |I|] ≥ 21 |T |. Let D denote the maximal size of a set disjoint triangles in the graph. Note that D ≥ |T | − |I|, since given the set T of triangles, we can for each pair in I choose one of the intersecting triangles, and remove the other from T . this process halts after |I| steps and we are left with a set disjoint triangles of size at least |T | − |I|. 3 Therefore E[D] ≥ γ24 · n3/2 . Denote X = n2/3 − D, and let |E| be the size of the set of edges in the graph. Trivially |E| ≥ D, therefore 2 /((1−γ)2 ) P r(X ≤ 0) ≤ Pr(n2/3 − |E| ≤ 0) =≤ Pr(n2/3 ≤ |E|) ≤ e−m where the next to last inequality follows from chernoff bound on the number of edges in the graph. Since X gets negative values with exponentially small probability, and is only polynomial in value, it holds that E[X|X > 0] ≤ (1 + o(1)) E[X]. Therefore E[D] γ3 For convenience denote c1 = 2n 2/3 = 48 . It follows that Pr(D ≤ c1 n3/2 ) = Pr(X ≥ (1 − c)n3/2 ) = Pr(X ≥ (1 − c1 )n3/2 |X > 0) Pr(X > 0) + Pr(X ≥ (1 − c1 ) · n3/2 |X ≤ 0) Pr(X ≤ 0) ≤ 2 2 E[X|X > 0] E[X] Pr(X ≥ (1 − c1 ) · n3/2 |X > 0) + e−m /((1−γ) ) ≤ ≤ (1 + o(1)) + o(1) = (1 − c1 )n2/3 (1 − c1 )n2/3 (1 + o(1)) 1 − 2c1 n2/3 − E[D] + o(1) = (1 + o(1)) + o(1) 2/3 1 − c1 (1 − c1 )n 1 (1 − c1 )n2/3 is a constant smaller than 1, meaning (1 + o(1)) 1−2c 1−c1 + o(1) is smaller than some constant c2 < 1. Therefore with constant probability there are at least c1 disjoint triangles. 27 Therefore, any lower bound we prove for µ translates to asymptotically the same bound on a distribution that is ǫ-free from triangle-free, namely, µ conditioned on being ǫ-free from triangle-freeness. Let X e be an indicator variable for the presence of edge e in the input graph. For a transcript t of a communication protocol Π, let √ ∆t (e) := Pr [X e = 1 | Π = t] − 2γ/ n. Lemma 4.6. We have: E t∼π " X e # ∆t (e) ≤ |Π|. √ Proof. For each edge e, the prior probability that e ∈ E is γ/ n, so by Lemma 4.3, for any transcript t, ∆t (e) ≤ D (π(X e |Π = t) k π(X e )) , By super-additivity of information, |Π| ≥ I(Π; E) ≥ = E " ≥ E " t∼π t∼π X I(Π; X e ) e # X D (π(X e |Π = t) k π(X e )) X ∆t (e). e e # Covered and reported edges. Our lower bounds show that it is hard for the players to find an edge that belongs to a triangle. Intuitively, in order to output such an edge, the players need to identify some edge {v1 , v2 } that (a) is in the input, and (b) closes a triangle together with some third vertex u; that is, for some u, the edges {u, v1 } and {u, v2 } are also in the input. We formalize the notion of “finding” an edge satisfying some property using the posterior probability of the edge satisfying this property given the transcript. Definition 10 (Reported edges). Given a transcript t, let Rep(t) = {e ∈ E | Pr [e ∈ E | Π = t] ≥ 9/10} be the set of edges whose posterior probabilities of being in the input increase to at least 9/10 when transcript t is sent. We call the edges in Rep(t) reported. Definition 11 (Covered edges). Given a transcript t, let C (t) = {e ∈ V1 × V2 | Pr [∃u ∈ U : (u, v1 ) ∈ E 1 ∧ (u, v2 ) ∈ E 2 | Πt] ≥ 9/10} be the set of edges in V1 × V2 whose posterior probability of being covered by a vee rises to at least 9/10 upon observing transcript t. We say that edges in C (t) are covered by Alice and Bob. Let Cov (e) be an indicator for the event that e ∈ C (Π). 28 4.2.2 One-Way Communication Consider a protocol Π between three players — Alice, Bob and Charlie — where Alice and Bob communicate back-and-forth for as many rounds as they want, with Charlie observing their transcript, and finally Charlie outputs an edge from his side of the graph. We claim that the total amount of communication exchanged by Alice and Bob must be Ω(n1/4 ). The underlying intuition for our proof is that by the end of the protocol Charlie needs to be informed √ by Alice and bob of at least Ω( n) vertex pairs in V1 × V2 being covered with high certainty by a vee in their input. This is due to the fact, that only a Θ( √1n )-fraction of these pairs is expected to have an edge connecting them. We prove that the number of pairs Alice and Bob can on average inform Charlie of being covered is at most quadratically larger than their bit-budget, which implies that Ω(n1/4 ) bits are required for such communication, that succeeds with high probability. This result is somewhat surprising as the a priori probability of any edge in Charlie’s input belonging to a triangle is already constant, and elevating only one of these probabilities to 1 − delta suffices for solving the problem. This observation is equally valid for simultaneous communication. Theorem 4.7. For any constant γ ∈ (0, 1), if Π solves the triangle-edge-finding problem under µ with error δ ≤ 1/100, then |Π| = Ω(n1/4 ). Proof. Suppose for the sake of contradiction that there is a protocol Π with communication αn1/4 , where α satisfies: (100α2 + 10α) < (9/20)/γ, and error δ ≤ 1/100. √ Say that transcript t of Π is good if |C (t) | ≥ n/(2γ). Lemma 4.8. Pr [Π is good] ≥ 1 − 20δ. Proof. If t is not a good transcript, then because C (t) is independent of E 3 , √ √ E [|E 3 ∩ C (t) |] ≤ (γ/ n) · ( n/(2γ)) = 1/10. By Markov, Pr [E 3 ∩ C (t) 6= ∅] ≤ 1/2. Whenever E 3 ∩ C (t) = ∅, Charlie must output an edge that is either not in his input (E 3 ), or not covered by t; in the first case this is an error, and in the second case, the probability of an error is at least 1/10, independent of E 3 (it depends only on E 1 , E 2 , which are independent of E 3 , even given Π = t). Therefore, conditioned on E 3 ∩ C (t) = ∅, the error probability is at least 1/10; and overall, for any t that is not good, Pr [error | Π = t] ≥ Pr [E 3 ∩ C (t) = ∅] · (1/10) ≥ 1/20. Since the total probability of error is bounded by δ, we obtain X δ ≥ Pr [error] = Pr [error | Π = t] Pr [Π = t] t ≥ ≥ X bad t X Pr [error | Π = t] Pr [Π = t] (1/20) · Pr [Π = t] = Pr [Π is bad] /20. bad t The claim follows. 29 Next, say that t is informative if: X e∈U ×V1 ∪U ×V2 ∆t (e) ≥ 10αn1/4 . Lemma 4.9. Pr [Π is informative] ≤ 1/10. Proof. By super-additivity, αn1/4 = |Π| ≥ I(Π; E 1 ∪ E 2 ) X I(Π; X e ) ≥ e∈U ×V1 ∪U ×V2  = E  X D (π(X e |Π = t) k π(X e )) ≥ E  X ∆t (e) . t∼Π t∼Π The claim follows by Markov.  e∈U ×V1 ∪U ×V2  e∈U ×V1 ∪U ×V2  (By Lemma 4.3) Corollary 4.10. There exists a transcript which is both good and uninformative. Proof. By union bound, the probability that a transcript is either not good or informative is at most 20δ + 1/10 < 1. We will now show that such a transcript cannot exist, as an uninformative transcript cannot cover enough edges to be good. For any particular transcript t of Π, the inputs of the three players remain independent given Π = t. Therefore, for any edge (v1 , v2 ) ∈ V1 × V2 , X Pr [∃u : (u, v1 ) ∈ E 1 ∧ (u, v2 ) ∈ E 2 | Π = t] ≤ Pr [(u, v1 ) ∈ E 1 ∧ (u, v2 ) ∈ E 2 | Π = t] u∈U = X u∈U = X u∈U = X u∈U (Pr [(u, v1 ) ∈ E 1 ] Pr [(u, v2 ) ∈ E 2 | Π = t]) √
√

∆t (u, v1 ) + 2γ/ n ∆t (u, v2 ) + 2γ/ n √ X (∆t (u, v1 )∆t (u, v2 )) + 2(γ/ n) (∆t (u, v1 ) + ∆t (u, v2 )) . u∈U √ Now let t be a transcript that is good, that is, |C (t) | ≥ n/(2γ), and also uninformative. Let S(t) ⊆ √ C (t) be a set of n/(2γ) covered edges (chosen arbitrarily from C (t)), and let W1 (t) ⊆ V1 and W2 (t) ⊆ V2 be the endpoints of the edges in S. Since each edge (v1 , v2 ) ∈ S(t) is covered in t, X √ X (∆t (u, v1 )∆t (u, v2 )) + 2(γ/ n) (∆t (u, v1 ) + ∆t (u, v2 )) ≥ 9/10, u∈U u∈U 30 and together we have X X (v1 ,v2 )∈S(t) u∈U " √ X (∆t (u, v1 )∆t (u, v2 )) + 2(γ/ n) (∆t (u, v1 ) + ∆t (u, v2 )) u∈U √ ≥ (9/10)|S| = (9/20) n/γ. # On the other hand, " # X X √ X (∆t (u, v1 )∆t (u, v2 )) + 2(γ/ n) (∆t (u, v1 ) + ∆t (u, v2 )) (v1 ,v2 )∈S(t) u∈U ≤  X √ ∆t (u, v2 ) + 2(γ/ n) ∆t (u, v1 )  X X X ∆t (u, v1 )  X X u∈U ≤ u∈U  X X    v1 ∈V1 u∈U v1 ∈V1 v2 ∈V2  u∈U v2 ∈V2  √ + 2(γ/ n) · |S(t)| ·  X X u∈U v1 ∈V1 We therefore have   ∆t (u, v1 ) +  2  √ √ ≤ 10αn1/4 + 2(γ/ n) · n/(2γ) · 10αn1/4 √ ≤ (100α2 + 10α) n. (∆t (u, v1 ) + ∆t (u, v2 )) v1 ∈C1 (t) v2 ∈C2 (t)  ∆t (u, v2 ) X X X u∈U v2 ∈V2  ∆t (u, v2 ) √ √ (100α2 + 10α) n ≥ (9/20) n/γ, contradicting our assumption about α. Streaming Lower Bounds There is a known connection between communication complexity, specifically, one-way communication, and space complexity in the data-stream model. In this model the input arrives as an ordered sequence that must be accessed in order and can be read only once, while the space complexity is defined as the maximal size of the memory used at any given point of the computation. As demonstrated in [4], there is a generic reduction which proves that lower bounds on the one-way communication complexity of a problem, are also lower bounds on the space-complexity of the same problem in the data-stream model. Consequently, we get a corresponding lower bound of Ω(n1/4 ) on the space complexity of detecting a triangle edge (with the input graph distribution identical to the one in our model) in the data-stream model. We present here a sketch of the proof, as the data-stream model is not the focus of this work; for more details on the relationship between lower bounds in the two models refer to [4, 20]. Assume to the contrary that there exists an algorithm, A, that solves the triangle-edge detection with space complexity o(n1/4 ) in the data-stream model. This implies a one-way 3-player protocol, Π, of complexity o(n1/4 ), which implies a contradiction (our ”extended” one-way model is even more powerful than the more standard one-way model used in this reduction, where Alice sends one message to Bob, who then sends one message to Charlie, who has to output the answer), proving our initial assumption to be false. More concretely, Π entails Alice running A on the input, which is viewed as the beginning of the stream, 31 then sending the content of the memory (which is limited by o(n1/4 ) bits) to Bob, who continues the computation of A on his input, which is viewed as the continuation of the stream, and once again sends the content of the memory to Charlie, who concludes the computation of A on his input, the final segment of the stream. We can apply the same reduction to the extended one-way lower bounds we derive later in this chapter √ for a more general average degree d = O( n). 4.2.3 Simultaneous Communication For simultaneous protocols, it is not enough to have some covered edge that also appears in Charlie’s input: the referee needs to know (or believe) that it is in Charlie’s input — that is, with good probability, the edge the referee outputs has a large posterior probability of being in Charlie’s input, given Charlie’s message. Say that edge e is reported by a transcript t if PrE∼µ|t [e ∈ E] ≥ 9/10. The goal of the players is to provide the referee with some edge that is covered by Alice and Bob and also reported by Charlie. We show that the “best” strategy for the players is to choose a set T ⊆ V1 × V2 of Θ(n) edges, and have Alice and Bob try to cover edges from T and Charlie report edges from T . The crux of the lower bound is showing that to target a fixed set of edges T , Alice and Bob must give up their quadratic advantage: whereas in for in our analysis of the one-way lower bound, the sum P of the cover probabilities was bounded by the square of the sum-increase of individual edge probabilities ( e ∆t (e)), here we show that we can bound it √ linearly, yielding a lower bound of Ω( n) instead of Ω(n1/4 ). Fix a deterministic simultaneous protocol Π, where the messages sent by the three players are M 1 , M 2 and M 3 , respectively. Let Π(m1 , m2 , m3 ) denote the edge output by the referee upon receiving messages m1 , m2 and m3 from the three players. We freely interchange the messages with the inputs to the respective players, since the protocol is deterministic; e.g., we write Π(E 1 , E 2 , E 3 ) to indicate the referee’s output upon receiving the messages sent by the players on input (E 1 , E 2 , E 3 ). √ Let C = α n be the number of bits sent by each player, where α will be fixed later. Let δ denote the error of Π on µ. Our goal is to show that when γ and δ are sufficiently small, we require α = Ω(1), so the communication complexity of the protocol is Ω(n). In a simultaneous protocol, the messages sent by the players are independent of each other given the input. In our case, because the inputs are also independent of each other, the messages are independent even without conditioning on a particular input. We therefore abuse notation slightly by omitting parts of the transcript that are not relevant to the event at hand. Specifically, we let Rep(mi ) denote the set of edges covered by a message mi of player i (this is independent of the other players’ messages), and we let C (m1 , m2 ) denote the edges covered by messages m1 , m2 of Alice and Bob, respectively (again, this is independent of Charlie’s message). We also sometimes write the player’s input instead of its message; because the protocol is deterministic, the message is a function of the input. In any simultaneous protocol, the goal of the players is to provide the referee with an edge in Charlie’s input that is both reported by Charlie and covered by Alice and Bob: Lemma 4.11. The probability that there exists an edge that is both reported by Charlie and covered by Alice and Bob is at least 1 − 10δ. That is, Pr [Rep(M 3 ) ∩ C (M 1 , M 2 ) 6= ∅] ≥ 1 − 10δ. Proof. If the referee outputs an edge that is both covered and reported, then of course there must exist such an edge. Let us therefore bound the probability that the referee outputs an edge that is either not reported 32 or not covered. Call a triplet (m1 , m2 , m3 ) of messages “bad” if Π(m1 , m2 , m3 ) = e, where e is either not reported (e 6∈ Rep(m3 )) or not covered (e 6∈ C (m1 , m2 )). The protocol errs whenever it outputs an edge e ∈ E3 that is not in Charlie’s input E 3 , or an edge that does not form a triangle together with some node u ∈ U . If e is not reported (in m3 ), then Pr [e ∈ E 3 | M 3 = m3 ] < 9/10, and if e is not covered (in m1 , m2 ), then Pr [∃u ∈ U : (u, v1 ) ∈ E 1 ∧ (u, v2 ) ∈ E 2 | M 1 = m1 , M 2 = m2 ] < 9/10. Therefore, each bad triplet of messages contributes at least 1/10 to the error probability of the protocol. Together we have X δ ≥ Pr [Π errs] ≥ Pr [(M 1 , M 2 , M 3 ) = (m1 , m2 , m3 )] · (1/10) bad (m1 , m2 , m3 ) = Pr [ (M 1 , M 2 , M 3 ) are bad] /10. The claim follows. By Lemma 4.11, we see that the players’ “best strategy” is to try to “coordinate” the edges reported by Charlie with the edges covered by Alice and Bob, so that the referee can find an edge in the intersection. Indeed, as a corollary we obtain: hP i Corollary 4.12. E Pr[Cov (e)] ≥ 1 − 10δ. e∈Rep(E 3 ) Proof. Fix Rep(E 3 ) = R. By union bound and the independence of the players’ inputs, X X Pr [e ∈ C (M 1 , M 2 )] = Pr [Cov (e)] Pr [R ∩ C (M 1 , M 2 ) 6= ∅ | Rep(E 3 ) = R] ≤ e∈R e∈R Therefore,  X E e∈Rep(E 3 ) = X R ≥ ≥ X R X R    Pr[Cov (e)] E  X e∈Rep(E 3 ) X e∈R   Pr[Cov (e)] | Rep(E 3 ) = R Pr [Rep(E 3 ) = R] ! ! Pr [Cov (e)] Pr [Rep(E 3 ) = R] (Pr [R ∩ C (M 1 , M 2 ) 6= ∅ | Rep(E 3 ) = R] Pr [Rep(E 3 ) = R]) ≥ 1 − 10δ. Analyzing Charlie’s messages. First, observe that Charlie (and the other players) cannot report too many edges, except with small probability. Each reported edge is “a little expensive”: Lemma 4.13. Let mi be a message sent by player i. Assume that γ < 1/2. If e ∈ Rep(mi ), then for sufficiently large n we have D (π(X e | M i = mi ) k π(X e )) ≥ 9 log n/40. 33 √ Proof. Since e ∈ Rep(mi ), the posterior probability that X e = 1 is at least 9/10 > γ/ n. Because D (p k q) increases as |p − q| increases, for sufficiently large n, √
D (π(X e | M i = mi ) k π(X e )) ≥ D 9/10 k γ/ n 9/10 1/10 √ = (9/10) log √ + (1/10) log γ/ n 1 − γ/ n √ 1 n √ + (1/10) log = −H(1/10) + (9/10) log γ 1 − γ/ n 9 9/10 log n ≥ log n. ≥ −1 + 2 40 √ We used the fact that γ < 1/2, so (9/10) log(1/γ) > 0, and also that 1 − γ/ n < 1, and hence log(1/(1 − √ γ/ n)) > 0. It follows that with a budget of C bits, Charlie can only report roughly C edges (in fact, somewhat less) in expectation: Corollary 4.14. E [|Rep(E 3 )|] ≤ 40α √ n 9 log n Proof. By the super-additivity of information, X √ α n = |M3 | ≥ I(M 3 ; E 3 ) ≥ I(M 3 ; X e ) e∈E3 = ≥ ≥ E  E  E  m3 ∼M 3 m3 ∼M 3 m3 ∼M 3   X e∈E3  D (π(X e | M 3 = m3 ) k π(X e )) X e∈Rep(m3 )  D (π(X e | M 3 = m3 ) k π(X e ))  9 log n . |Rep(m3 )| · 40 The claim follows. As we said above, since the referee “wants” to output an edge that is both reported and covered, the goal of the players should be to provide it with such an edge. Let us rank the edges in V1 × V2 according to the probability that they are covered by Alice and Bob: we write V1 × V2 = {e1 , . . . , en2 }, where i ≤ j iff Pr [Cov (ei )] ≥ Pr [Cov (ej )], breaking ties arbitrarily. Let Top(E 3 ) denote the set of |Rep(E 3 )| highest-ranking edges in E 3 . Clearly, X X Pr [Cov (e)] ≤ Pr [Cov (e)] . (6) e∈Rep(E 3 ) e∈Top(E 3 ) That is, “it is in Charlie’s interest” to report edges from Top(E 3 ), as this maximizes the probability that some reported edge is also covered. 34 Let T = {e1 , . . . , em } be the m highest-ranking edges in V1 × V2 , where 9 n. 80α m= For any integer k ≥ 1 we have: X e1 ,...,ek·m Therefore,   X Pr[Cov (e)] E Pr [Cov (e)] ≤ k · X Pr [Cov (e)] . e∈T e∈Rep(E 3 )  ≤ E = X e∈Top(E 3 ) ⌉ log(n2 /m)⌉ X i=1 ≤ ⌉ log(n2 /m)⌉ X i=1 Pr[Cov (e)]  E "  X e∈Top(E 3 ) 2i+1 · X    Pr[Cov (e)] 2i · m ≤ |Top(E 3 )| ≤ 2i+1 · m Pr 2i · m ≤ |Top(E 3 )| ≤ 2i+1 · m ! Pr [Cov (e)] e∈T √ X n 40α ≤ log n · 2 · · Pr [Cov (e)] 9 log n m e∈T P Pr [Cov (e)] . = e∈T √ n Analyzing the cover probabilities E [|Top(E 3 )|] · 2i · m # (7) We show that it is not possible for the two other players to have: X √ Pr [Cov (e)] ≥ β · n, e∈T where β is a constant whose value will be fixed later. √ Notation. Let V H be the set of nodes in V1 ∪ V2 whose degree in T is at least n, and let V L be the remaining nodes in V1 ∪ V2 . Also, let Via = V a ∩ Vi , for a ∈ {L, H} and i ∈ {1, 2}. √ 9 . Since |T | ≈ n, we have |V H | ≤ c · n, where c = 80α Let T1 = V1L × V2 ∪ V1 × V2H and let T2 = V1 × V2L ∪ V1H × V2 . For edges in T1 , their endpoints in V1 √ all have low degree in T1 (edges in V1L × V2 have degree at most n in T , and edges in V1 × V2H also have √ low degree in T1 , since |V H | ≤ c · n). We have T = T1 ∪ T2 , so it suffices to bound the sum of the cover probabilities in T1 and the sum in T2 . (The union is not disjoint; e.g., edges in V1L × V2L appear in both sets, so we may be over-counting). Let NS (v) denote the nodes adjacent to node v in S ⊆ V1 × V2 . We let M 1 , M 2 be random variables denoting the messages sent by the two players, respectively. Let M1 , M2 be the set of all possible messages for each player (resp.). 35 Bounding the cover probabilities in T . For any pair of messages m1 , m2 , if e = (v1 , v2 ) ∈ C (m1 , m2 ), then by union bound, X Pr [(u, v1 ) ∈ E 1 ∧ (u, v2 ) ∈ E 2 | M 1 = m1 , M 2 = m2 ] u∈U ≥ Pr [∃u : (u, v1 ) ∈ E 1 ∧ (u, v2 ) ∈ E 2 | M 1 = m1 , M 2 = m2 ] ≥ 9/10. (8) Because the edges in E 1 and E 2 remain independent given M 1 = m2 , M 2 = m2 , and the messages are also independent of each other and of the other player’s input, for each u ∈ U , Pr [(u, v1 ) ∈ E 1 ∧ (u, v2 ) ∈ E 2 | M 1 = m1 , M 2 = m2 ] = Pr [(u, v1 ) ∈ E 1 | M 1 = m1 , M 2 = m2 ] · Pr [(u, v2 ) ∈ E 2 | M 1 = m1 , M 2 = m2 ] = Pr [(u, v1 ) ∈ E 1 | M 1 = m1 ] · Pr [(u, v2 ) ∈ E 2 | M 2 = m2 ] . Consider first the edges in T1 . Plugging the above into (8), and also writing Pr [(u, v1 ) ∈ E 1 | M1 = m1 ] = √ ∆m1 (u, v1 ) + 2γ/ n (where ∆m1 is the L1 difference between the posterior and the prior), we obtain: X  √
∆m1 (u, v1 ) + 2γ/ n Pr [(u, v2 ) ∈ E 2 | M 2 = m2 ] ≥ 9/10. (9) u∈U Multiplying both sides by Pr [M 2 = m2 ], and summing across all m2 such that (v1 , v2 ) ∈ C (m1 , m2 ), we get that for any m1 , X X  √
∆m1 (u, v1 ) + 2γ/ n · Pr [(u, v2 ) ∈ E 2 | M 2 = m2 ] Pr [M 2 = m2 ] m2 :(v1 ,v2 )∈C(m1 ,m2 ) u∈U = X u∈U ≥ (9/10) !  √
∆m1 (u, v1 ) + 2γ/ n ·  X X m2 :(v1 ,v2 )∈C(m1 ,m2 ) Pr [M 2 = m2 ] Pr [(u, v2 ) ∈ E 2 | M 2 = m2 ] Pr [M 2 = m2 ] m2 :(v1 ,v2 )∈C(m1 ,m2 ) = (9/10) Pr [Cov (v1 , v2 ) | M 1 = m1 ] . Notice that for any u ∈ U , X m2 :(v1 ,v2 )∈C(m1 ,m2 ) ≤ X m2 ∈M2 Pr [(u, v2 ) ∈ E 2 | M 2 = m2 ] Pr [M 2 = m2 ] Pr [(u, v2 ) ∈ E 2 | M 2 = m2 ] Pr [M 2 = m2 ] √ = Pr [(u, v2 ) ∈ E 2 ] = γ/ n. Therefore, X u∈U √
∆m1 (u, v1 ) + 2γ/ n ! √ · (γ/ n) ≥ (9/10) Pr [Cov (v1 , v2 ) | M 1 = m1 ] . 36  Now, taking the expectation over all m1 , " # X √ √ (∆M 1 (u, v1 ) + 2γ/ n) ≥ (9/10) E [Pr [Cov (v1 , v2 ) | M 1 ]] γ/ n E M1 M1 u∈U = (9/10) Pr [Cov (v1 , v2 )] . √ Summing across all v2 such that (v1 , v2 ) ∈ T1 , and using the fact that the degree of v1 in T1 is at most c· n, " # " #! X X X √ √ √ √ √ c n · γ/ n E (∆M 1 (u, v1 ) + 2γ/ n) ≥ (∆M 1 (u, v1 ) + 2γ/ n) γ/ n E M1 ≥ (9/10) u∈U X v2 ∈NT1 (v1 ) M1 u∈U Pr [Cov (v1 , v2 )] . v2 ∈NT1 (v1 ) And now, summing over all v1 ∈ V1 ,   X X X √ ∆M 1 (u, v1 ) + 2γ/ n E  cγ v1 ∈V1  M1 v1 ∈V1 u∈U  = cγ  E  M1 X X v1 ∈V1 u∈U Using Lemma 4.6 we obtain:   √ ∆M 1 (u, v1 ) + 2γ n ≥ (9/10) X e∈T1 Pr [Cov (e)] ≤ X Pr [Cov (v1 , v2 )] . (v1 ,v2 )∈T1 ) cγ(α + 2) √ n. 9/10 For edges in T2 the argument is symmetric. Together we have: X e∈T Pr [Cov (e)] ≤ √ 1 2cγ(α + 2) √ n ≤ (α + 2) n, 9/10 25 assuming that γ is a sufficiently small constant. Combining this with (7), we see that we must have α ≥ 23/25. 4.3 Lifting 3-player Lower Bounds to k Players Using symmetrization [33], we “lift” our lower bounds for a constant number of players to general kplayer lower bounds (Symmetrization was developed in [33] to lift unrestricted 2-player lower bounds to unrestricted k-player lower bounds.) Interestingly, our symmetrization reduction transforms a simultaneous k-player protocol into a one-way 3-player (or 2-player) protocol, so in order to obtain lower bounds on simultaneous protocols for k players we need to first prove lower bounds on one-way protocols for a small number of players. This curious behavior turns out to be inherent, at least for large k: a simultaneous protocol can emulate a one-way protocol, by having each player send their entire input to the referee with probability 1/k, and otherwise send their message under the one-way protocol. The referee can, with constant probability, take the role of one of the players, whose input the referee received, and compute the 37 answer using the messages from the other players. When k is sufficiently large, this may be cheaper than the simultaneous protocol. We say that a k-player distribution µ is symmetric if the marginal distribution of each player’s input is the same. Theorem 4.15. Let P be a graph property, Suppose that µ is a symmetric 3-player input distribution such k,sim ǫ ǫ that CC3,→ µ,δ (P ) = C. Then there is a k-player input distribution η such that CCη,δ (P ) ≥ (k/2)C. Proof. Let η be the following distribution: we sample (X 1 , X 2 , X 3 ) ∼ µ; we give X 1 and X 2 to two random players that are not player k, and the remaining players all receive X 3 . We show by reduction from the 3-player case that η is hard for k players. Let Π be a simultaneous protocol for k players that solves P ǫ on η with error probability δ. We construct a 3-player protocol Π′ as follows: Alice and Bob publicly choose two random IDs i, j ∈ [k] (i 6= j), and take on the roles of players i and j, using their actual inputs under µ. Charlie will play the role of all the remaining players, using his input for each one of them, and also the role of the referee (who has no input). Let embed(i, j, X) denote the input thus constructed, where X = (X1 , X2 , X3 ). The resulting k-player input distribution is exactly η. To simulate the execution of Π, Alice and Bob simply send Charlie the messages players i and j would send under Π to player k. Charlie computes the messages that each player ℓ ∈ [k] \ {i, j} would send, and then, using these messages and the messages received from Alice and Bob, computes the output of the referee. The simulation adds no error — on each input, it exactly computes the referee’s output (or rather, it generates the correct distribution for the referee’s output). Therefore,   X   µ(X) Pr Π′ errs on X Pr Π′ errs = µ X   X X 1 Pr [Π errs on embed(i, j, X)] µ(X)  = k(k − 1) i,j X X = η(Y ) Pr [Π errs on Y ] ≤ δ. Y What is the expected communication of Π′ ? Observe that since Π is simultaneous, each player’s transcript is a (random) function of only its own input: in particular, the distribution of player i’s transcripts is the same under any joint input distribution where player i’s input has the same marginal. In η, all players’ inputs 38 have the same marginal distribution — the marginal distribution of each player’s input in µ. Therefore,   E [|Π(embed(i, j, X))|] E |Π′ (X)| = X∼µ i,j∼U [k],X∼µ = = = E [|Πi (embed(i, j, X))| + |Πj (embed(i, j, X))|] E [|Πi (Y )| + |Πj (Y )|] i,j∼U [k],X∼µ i,j∼U [k],Y ∼η E i∼U [k],Y ∼η [|Πi (Y )|] k 1X E [|Πi (Y )|] Y ∼η k i=1 # " k X 2 2 = E |Πi (Y )| = CC(Π). Y ∼η k k =2 i=1 This result implies a Ω(k · n1/4 ) lower bound for the problem of k players trying to find a triangle-edge √ in a graph of average degree d = Θ( n) via simultaneous communication, For deterministic and symmetric protocols we can do a little better, by modifying the reduction: instead of constructing a one-way protocol, we construct a simultaneous protocol — using the fact that the original k-player protocol is deterministic, Charlie can pick one of the players he simulates and send the message of only that one player to the referee, because we know that all the players simulated by Charlie will send the same message (as they receive the same input). 4.4 Lower Bound for Degree O(1) For graphs with average degree O(1), a lower bound was shown in the streaming model in [27] reducing the Hidden Boolean Matching problem, introduced in [28] to triangle counting approximation in streaming. The same reduction yields a lower bound on triangle testing in two players one-way communication complexity. We present the reduction for the sake of completeness, and to show that it indeed holds in our model as well. We use the bound shown in [36] but need only the bound for matchings (rather than hypermatchings), we give here a simplified version of the problem; Definition 12 (Boolean Matching). In the Boolean Matching problem, denoted BM n , Alice receives a vector x ∈ {0, 1}2n , and Bob receives a perfect matching M on 2n vertices {1, . . . , 2n} and a vector w ∈ {0, 1}n . We represent M as an n × 2n matrix, where each row represents one edge of the matching: if the i-th edge of the matching is {j1 , j2 } ⊆ [2n]2 , then the i-th row of the matrix contains 1 in columns j1 and j2 , and 0 elsewhere. The goal of the players is to distinguish the case where → − Mx ⊕ w = 0 from the case where → − Mx ⊕ w = 1 . Theorem 4.16. The randomized one-way communication complexity of testing triangle-freeness in graphs √ with average degree O(1) is Ω( n). 39 Proof. Given inputs X for Alice and M, w for Bob, the players construct the following graph G = (V, E), where V = {u} ∪ ([n] × [2]): • For each bit i ∈ [n] where xi = 0, Alice adds the edge {u, (i, 0)}; for each bit i ∈ [n] where xi = 1, she adds the edges {u, (i, 1)}. • For each edge ej = {j1 , j2 } in his matching, – If wj = 0, Bob adds edges {(j1 , 0), (j2 , 0)} and {(j1 , 1), (j2 , 1)}; – If wj = 1, Bob adds edges {(j1 , 0), (j2 , 1)} and {(j1 , 1), (j2 , 0)}. For each j ∈ [n], let Mj = {j1 , j2 }. A triangle appears in the subgraph induced by vertices {u, (j1 , 0), (j1 , 1), (j2 , 0), (j2 , 1)} iff either wj = 0 and xj1 = xj2 , or wj = 1 and xj1 6= xj2 . In other words, a triangle appears iff (M x ⊕ w)j = 0. No other → − triangles appear in the graph. Therefore, if M x ⊕ w = 0 then G contains n edge-disjoint triangles, and if → − M x ⊕ w = 1 then G is triangle-free. In the first case, G is 1-far from triangle-freeness. 4.5 Other Degrees We now show how we can extend a lower bound for a given average degree, d, to any lower degree, d′ , by embedding dense inputs of degree d into sparse graphs such that the average degree evens out to be d′ . Lemma 4.17. Let d = Θ(nc ) denote the average degree of the graph, and let CC(T ǫ,d,n ) = Θ(f (n)) denote the communication complexity as a function of n, the number of vertices. Then for any d′ ≤ d, we have 1 ′ CC(T ǫ,Θ(d ),n ) = Θ(f ((d′ n) 1+c )). 1 Proof. For graphs with n′ = (d′ n) 1+c vertices and average degree Θ((n′ )c ), the communication complexity is Θ(f (n′ )). We examine the following subset of graphs with n vertices and average degree d′ : any such graph, G, is a union of (n − n′ ) isolated nodes, and a graph, G′ , which is either triangle-free or ǫ-far from ′ being triangle-free, and has n′ vertices and average degree (n′ )c . The average degree of G is Θ((n′ )c ) · nn = Θ(d′ ), and its distance to being triangle-free is identical to that of G′ , as it has no edges outside of G′ . Since any triangle in G must be contained in G′ , solving the problem on G is equivalent to solving it on G′ . And since we asserted that the complexity of the problem for graphs with the stated properties of G is 1 Θ(f (n′ )) = Θ(f (d′ n) 1+c , it is also the complexity of the problem for graphs of average degree Θ(d′ ). Note that lemma 4.17 holds regardless of the model of communication. Therefore, as a corollary, we √ √ can generalize the lower bounds we derived directly for graphs of average degree n to d = O( n) (for 3 √ players in both cases). Specifically, the Ω(n1/4 ) bound for one-way communication and the Ω( n) bound for simultaneous communication extend to Ω((nd)1/6 ) and Ω((nd)1/3 ), respectively. Furthermore, lemma 4.17, combined with theorem 4.15 and the lower bounds we proved in section 4.2 imply Theorem 4.1, the main result of this section. 4.6 Discussion: Lower Bounds on the Communication Complexity of Property-Testing Lower bounds on the “canonical” problems in communication complexity, such as Set Disjointness and Gap Hamming Distance [12], cannot be leveraged to obtain property-testing lower bounds, at least for trianglefreeness. Some classical problems do feature a gap, where we are only interested in distinguishing two cases that are “far” from each other; however, for property-testing lower bounds, the gap needs to be around 40 zero (either we have no triangles, or we have many edge-disjoint triangles), while existing gap problems typically become easy unless the gap is centered far from zero (for example, in Gap Hamming Distance, the players get vectors x, y ∈ {0, 1}n , and they need to determine whether their Hamming distance is greater √ √ than n/2 + n or smaller than n/2 − n). In addition, because triangles are not “independent” of each other (if they share an edge), the direct sum approach to proving lower bounds, which works well when we can break the problem up into many independent pieces, does not apply here. 5 Summary In this work we showed that in the setting of communication complexity, property testing can be significantly easier than exactly testing if the input satisfies the property: exactly determining whether the input graph contains a triangle was shown to require Ω(knd) bits in [38], but we showed that weakening the requirement to property-testing improves the complexity, and even simultaneous protocols can do better than the best exact algorithm with unrestricted communication. However, the problem does not appear to become trivial, as shown by our lower bounds for simultaneous and restricted one-way protocols. Table 1 summarizes our main results. √ √ d = Θ(1) d = O( n) d = Ω( n) △-freeness √ Õ(k 4 nd + k2 ) Unrestricted Communication Upper bound △-freeness √ √ Õ(k n) Õ(k 3 nd) Simultaneous Communication Upper bound △-edge detection √ ”Extended” One-Way Communication Ω( 6 nd) — 3 players Lower bound △-edge detection √  Simultaneous Communication Ω 3 nd — 3 players Lower bound △-edge detection  √  Ω k · 6 nd — Simultaneous Communication Lower bound △-freeness √ Ω( n) — Simultaneous Communication Lower bound Table 1: Results summary; We have provided non-trivial upper bounds for the entire relevant degree range for both simultaneous and unrestricted communication. Our solutions have several desirable qualities. First, they can overcome the obstacle of not knowing the average degree in advance. Additionally, the algorithms solve not only the problem of triangle-freeness, but more specifically, the problem of triangle detection, which can only be 41 harder. Finally, all solutions have a one-sided error - a graph is found to contain triangles only if a triangle is detected with probability 1. We also address other variants and relaxations, such as the case where all inputs are disjoint, or a case where the players communicate via a blackboard visible to everyone, and describe how these guarantees can improve the complexity. In terms of more general contributions, we describe how to efficiently implement typical building blocks used in standard property-testing solutions, of which the most notable is the proposed procedure for approximating a vertex degree up to a constant, which can be used to solve the more general problem of approximating the number of distinct elements in a set. The task of proving non-trivial lower bounds for triangle-freeness is considerably harder. We have discussed the shortcomings of mainstream techniques in communication complexity for tackling this problem. Nevertheless, we have been able to produce a a tight lower-bound for the closely related problem of triangle√ edge detection for d = Θ( n) in the simultaneous model. We have also been able to prove a lower-bound for an extended variation of one-way communication, which enabled us to derive a bound for k players. Moreover, we showed how to extend these bounds, by a rather generic procedure, to lower average degrees. Finally, we demonstrated how to translate our one-way bounds into streaming-lower bounds, once again via a generic (and well known) reduction. We believe that extending the lower bounds to protocols with unrestricted rounds, and strengthening them to apply to testing triangle-freeness rather than finding a triangle edge, will require techniques from Fourier analysis, like the ones used in [28] to show the lower bound on Boolean Hidden Matching (from which we reduce in Section 4.4). In addition, we believe that devising a hard distribution for dense graphs of √ degree d = ω( n), with desirable properties for proving lower bounds, will require some sophisticated utilization of Behrend graphs [3]. Finally, a worthwhile topic for related future research could be generalizing our techniques for detecting a wider class of subgraphs or testing other properties, relying on the propertytesting relaxation. As demonstrated by this work, this relaxation can significantly reduce the complexity of an otherwise maximally hard problem, but not to a degree that it becomes trivial and uninteresting, as suggested by our lower bounds. More generally, there is much room for a more elaborate investigation of the interrelation between the models of communication complexity and property-testing, as alongside innate distinctions there seem to exist non-trivial similarities the extent of which is yet to be determined. 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Improved Inference for Checking Type Annotations arXiv:cs/0507036v1 [] 14 Jul 2005 Peter J. Stuckey1 , Martin Sulzmann2 and Jeremy Wazny1 1 NICTA Victoria Laboratories Department of Computer Science and Software Engineering The University of Melbourne, Vic. 3010, Australia {pjs,jeremyrw}@cs.mu.oz.au 2 School of Computing, National University of Singapore S16 Level 5, 3 Science Drive 2, Singapore 117543 [email protected] Abstract. We consider type inference in the Hindley/Milner system extended with type annotations and constraints with a particular focus on Haskell-style type classes. We observe that standard inference algorithms are incomplete in the presence of nested type annotations. To improve the situation we introduce a novel inference scheme for checking type annotations. Our inference scheme is also incomplete in general but improves over existing implementations as found e.g. in the Glasgow Haskell Compiler (GHC). For certain cases (e.g. Haskell 98) our inference scheme is complete. Our approach has been fully implemented as part of the Chameleon system (experimental version of Haskell). 1 Introduction Type inference for the Hindley/Milner system [Mil78] and extensions [Rém93,Pot98,OSW99] of it is a heavily studied area. Surprisingly, little attention has been given to the impact of type annotations (a.k.a. user-provided type declarations) and userprovided constraints on the type inference process. For concreteness, we assume that the constraint domain is described in terms of Haskell type classes [Jon92,HHPW94]. Type classes represent a user-programmable constraint domain which can be used to code up almost arbitrary properties. Hence, we believe that the content of this paper is of importance for any Hindley/Milner extension which supports type annotations and constraints. The surprising observation is that even for “simple” type classes type inference in the presence of type annotations becomes a hard problem. Example 1. The following program is a variation of an example from [JN00]. Note that we make use of nested type annotations. 1 class Foo a b where foo :: a->b->Int instance Foo Int b instance Foo Bool b 1 For concreteness, we annotate 1 with Int because in Haskell 1 is in general only a number. p y = (let f :: c -> Int f x = foo y x in f, y + (1::Int)) q y = (y + (1::Int), let f :: c -> Int f x = foo y x in f) We introduce a two-parameter type class Foo which comes with a method foo which has the constrained type ∀a, b.F oo a b ⇒ a → b → Bool. The two instance declarations state that F oo Int b and F oo Bool b hold for any b. Consider functions p and q. In each case the subexpression y+(1::Int) forces y to be of type Int. Note that we could easily provide a more complicated subexpression without type annotations which forces y to be of type Int. The body of function f x = foo y x generates the constraint F oo Int tx where tx is the type of x. Note that this constraint is equivalent to T rue due to the instance declaration. We find that f has the inferred type ∀tx .tx → Int. We need to verify that this type subsumes the annotated type f::c->Int which is interpreted as ∀c.c → Int. More formally, we write Cg ⊢ σi ≤ σa to denote that the inferred type σi subsumes the annotated type σa under some constraint Cg . Suppose σi = (∀ā.Ci ⇒ ti ) and σa = (∀b̄.Ca ⇒ ta ) where there are no name clashes between ā and b̄. Then, the subsumption condition is (logically) equivalent to Cg |= ∀b̄.(Ca ⊃ (∃ā.Ci ∧ ti = ta )). In this statement, we assume that |= refers to the model-theoretic entailment relation and ⊃ refers to Boolean implication. Outermost universal quantifiers are left implicit. Note that in our system, we only consider type equality rather than the more general form of subtyping. For our example, we find that the subsumption condition holds. Hence, expressions p and q are well-typed. Let’s see what some common Haskell implementations such as Hugs [HUG] and GHC [GHC] say. Expression p is accepted by Hugs but rejected by GHC whereas GHC accepts q which is rejected by Hugs! Why? In a traditional type inference scheme [DM82], constraints are generated while traversing the abstract syntax tree. At certain nodes (e.g. let) the constraint solver is invoked. Additionally, we need to check for correctness of type annotations (a.k.a. subsumption check). The above examples show that different traversals of the abstract syntax tree yields different results. E.g. Hugs seems to perform a right-first traversal. We visit f::c -> Int; f x = foo y x first without considering the constraints arising out of the left tuple component. Hence, we find that f has the inferred type ∀tx .F oo ty tx ⇒ tx → Int where y has type ty . This type does not subsume the annotated type. Therefore, type inference fails. Note that GHC seems to favor a left-first traversal of the abstract syntax tree. The question is whether there is an inherent problem with nested type annotations, or whether it’s simply a specific problem of the inference algorithms implemented in Hugs and GHC. Example 2. Here is a variation of an example mentioned in [Fax03]. We make use of the class and instance declarations from the previous example. 2 test y = let f :: c->Int f x = foo y x in f y Note that test may be given types Int → Int and Bool → Int. The constraint F oo ty tx arising out of the program text can be either satisfied by ty = Int or ty = Bool. However, the “principal” type of test is of the form ∀ty .(∀tx .F oo ty tx ) ⇒ ty → Int. Note that the system we are considering does not allow for constraints of the form ∀tx .F oo ty tx . Hence, the above example has no (expressible) principal type. We note that the situation is different for (standard) Hindley/Milner with type annotations. As shown by Odersky and Läufer [OL96], the problem of finding a solution such that σi (the inferred type) is an instance of σa (the annotated type) can be reduced to unification under a mixed prefix [Mil92]. Hence, we either find a principal solution φ such that ⊢ φ(σi ) ≤ φ(σa ) or no solutions. Hence, inference for Hindley/Milner with type annotations is complete. We conclude that type inference for Hindley/Milner with constraints and (nested) type annotations is incomplete. The incompleteness arises because the subsumption check not only involves a test for correctness of annotations, but may also need to find a solution. The above example shows that in our general case there might not necessarily be a principal solution. In order to resurrect completeness we could impose a syntactic restriction on the set of programs. E.g., we could simply rule out type annotations for “nested” let-definitions, or require that the types of all lambda-bound variables occurring in the scope of a nested annotation must be explicitly provided (although it’s unclear whether this is a sufficient condition). In any case, we consider these as too severe restrictions. In fact, the simplest solutions seems to be to enrich the language of constraints. Note that the subsumption condition itself is a solution to the subsumption problem. Effectively, we add constraints of the form ∀b̄.(Ca ⊃ (∃ā.Ci ∧ ti = ta )) to our language of constraints. Then, ∀ty .(∀tx .F oo ty tx ) ⇒ ty → Int will become a valid type of test in the above example. This may be a potential solution for some cases, but is definitely undesirable for Haskell where constraints are attached to dictionaries. It is by no means obvious how to construct dictionaries [HHPW94] for “higher-order” constraints. Furthermore, we believe that type inference easily becomes undecidable depending on the underlying primitive constraint domain. In this paper, we settle for a compromise between full type inference and full type checking. We only check for the correctness of type annotations. But before checking we infer as much as possible. Our contributions are: – We introduce a novel formulation of improved inference for checking annotations in terms of Constraint Handling Rules (CHRs) [Frü95]. While inferring the type of some inner expression we can reach the result of inference for some outer expression. – We can identify a class of programs for which inference is complete. – Our approach is fully implemented as part of the Chameleon system [SW]. We can type a much larger class of programs compared to Hugs and GHC. E.g., Example 1 is typable in our system. We refer to [SW] for more examples. 3 (Var-∀E) (x : ∀ā.D ⇒ t) ∈ Γ Pp |= C ⊃ [t̄/ā]D C, Γ ⊢ x : [t̄/ā]t (Abs) C, Γ ⊢ e1 : t1 → t2 C, Γ ⊢ e2 : t1 C, Γ ⊢ e1 e2 : t2 ā = fv(C1 , t1 ) C2 ∧ C1 , Γ.(g : ∀ā.C1 ⇒ t1 ) ⊢ e1 : t1 C2 , Γ.(g : ∀ā.C1 ⇒ t1 ) ⊢ e2 : t2 g :: C1 ⇒ t1 C2 , Γ ⊢ let in e2 : t2 g = e1 C, Γ.x : t1 ⊢ e : t2 C, Γ ⊢ λx.e : t1 → t2 C1 , Γ ⊢ e 1 : t1 ā = fv(C1 , t1 ) − fv(Γ ) (Let) C2 , Γ.(g : ∀ā.C1 ⇒ t1 ) ⊢ e2 : t2 C2 , Γ ⊢ let g = e1 in e2 : t2 (LetA) (App) Fig. 1. Hindley/Milner with Type Annotations We continue in Section 2 where we formally introduce an extension of the Hindley/Milner system with constraints and type annotations. Section 3 is the heart of the paper. We first motivate our approach by example before mapping the entire type inference problem to CHRs. Type inference then becomes CHR solving. In Section 4 we discuss related work and conclude. 2 Types and Constraints We present an extension of the Hindley/Milner system with constraints and type annotations. f :: C ⇒ t in e Expressions e ::= x | λx.e | e e | let f = e in e | let f =e Types t ::= a | t → t | T t̄ Type Schemes σ ::= t | ∀ā.C ⇒ t Constraints C ::= t = t | U t̄ | C ∧ C CHRs R ::= U t̄ ⇐⇒ C | U1 t1 , ..., Un tn =⇒ C We write ō to denote a sequence of objects o1 , ..., on and o : t to denote o1 : t1 , ..., on : tn . W.l.o.g., we assume that lambda-bound and let-bound variables have been a-renamed to avoid name clashes. We record these variables in some environment Γ . Note that we consider Γ as an (ordered) list of elements, though we commonly use set notation. We denote by {x1 : σ1 , . . . , xn : σn }.x : σ the environment {x1 : σ1 , . . . , xn : σn , x : σ}. Our type language consists of variables a, type constructors T and type application, e.g. T a. We use common notation for writing function and list types. We also make use of pairs, integers, booleans etc. in examples. We find two kinds of constraints. Equations t1 = t2 among types t1 and t2 and user-defined constraints U t̄. We assume that U refers to type classes such as F oo. For our purposes, we restrict ourselves to single-headed simplification CHRs U t̄ ⇐⇒ C and multi-headed propagation CHRs U1 t1 , ..., Un tn =⇒ C. We note that CHRs describe logic formula. E.g. U t̄ ⇐⇒ C can be interpreted as ∀ā.U t̄ ↔ (∃b̄.C) where ā = fv(t̄) and b̄ = fv(C) − ā, and U1 t1 , ..., Un tn =⇒ C can be interpreted as ∀ā.(U1 t1 ∧ ... ∧ Un tn ) ⊃ (∃b̄.C) where ā = fv(t1 , ..., tn ) and b̄ = fv(C) − ā. Via CHRs we can model most known type class extensions. We refer the interested reader to [SS04] for a detailed account of translating 4 classes and instances to CHRs. We claim that using CHRs we can cover a sufficiently large range of Hindley/Milner type systems with constraints such as functional dependencies, records etc. Note that CHRs additionally offer multiheaded simplification CHRs which in our experience so far do not seem to be necessary in the type classes context. Due to space limitations, we only give one simple example showing how to express Haskell 98 type class relations in terms of CHRs. We assume that the meaning of user-defined constraints (introduced by class and instance declarations) has already been encoded in terms of some set Pp of CHRs. User-defined functions are recorded in some initial environment Γinit . Example 3. Consider class Eq a where (==) :: a->a->Bool instance Eq a => Eq [a] class Eq q => Ord a where (<) :: a->a->Bool class Foo a b where foo :: a->b->Int instance Foo Int b instance Foo Bool b Note that the declaration class Eq a => Ord a introduces a new type class Ord and imposes the additional condition that Ord a implies Eq a (which is sensible assuming that an ordering relation assumes the existence of an equality relation). We can model such a condition via a propagation rule. Hence, Pp consists of (Super) Ord a =⇒ Eq a (F1) F oo Int b ⇐⇒ T rue (Eq1) Eq a ⇐⇒ Eq [a] (F2) F oo Bool b ⇐⇒ T rue and Γinit = {(==) : ∀a.Eq a ⇒ a → a → Bool, (<) : ∀a.Ord a ⇒ a → a → Bool, f oo : ∀a, b.F oo a b ⇒ a → b → Int}. We introduce judgments of the form C, Γ ⊢ e : t where C is a constraint, Γ refers to the set of lambda-bound variables, predefined and user-defined functions, e is an expression and t is a type. We leave the type class theory Pp implicit. We say a judgment is valid iff there is a derivation w.r.t. the rules found in Figure 1. Commonly, we require that constraints appearing in judgments are satisfiable. We say that a valid judgment is satisfiable iff all constraints appearing in the derivation are satisfiable. A constraint is satisfiable w.r.t. a type class theory iff we find some model satisfying the theory and constraint. We say a theory Pp is satisfiable iff we find some model for Pp . In rule (Var-∀E), we assume that x either refers to a lambda- or let-bound variable. Note that only let-bound variables and primitives can be polymorphic. For convenience, we combine variable introduction with quantifier elimination. We can build an instance of a type scheme if the instantiated constraint is entailed by the given constraint w.r.t. type class theory Pp . In rule (Let) we couple the quantifier introduction rule with the introduction of user-defined functions. In our formulation, C2 does not necessarily guarantee that C1 is satisfiable. However, our rule (Let) is sound for a lazy semantics which applies to Haskell. 5 Rule (LetA) introduces a type annotation. Note that we assume that type annotations are closed, i.e. all variables appearing in C1 ⇒ t1 are assumed to be universally quantified. This is the assumption made for Haskell 98 [Has]. Note that the environment for typing the function body includes the binding g : ∀ā.C1 ⇒ t1 . Hence, we allow for polymorphic recursive functions. The other rules are standard. Note that we left out the rule for monomorphic recursive functions for simplicity. 3 Type Inference via CHRs We introduce our improved inference scheme first by example. Then, we show how to map the typing problem to a set of CHRs. We give a description of the semantics of CHRs adapted to our setting. Finally, we show how to perform type inference in terms of CHR solving. 3.1 Motivating Examples The following examples give an overview of the process by which we abstract the typing problem in terms of constraints and CHRs. Example 4. Consider the following program. g y = let f x = x in (f True, f y) We introduce new predicates, (special-purpose) user constraints, g(t) and f (t) to constrain t to the types of functions g and f respectively. It is necessary for us to provide a meaning for these constraints, which we will do in terms of CHR rules. The body of each rule will contain all constraints arising from the definition of the corresponding function, which represent that function’s type. For the program above we may generate rules similar to the following. g(t) ⇐⇒ t = ty → (t1 , t2 ), f (tf 1 ), tf 1 = Bool → t1 , f (tf 2 ), tf 2 = ty → f2 f (t) ⇐⇒ t = tx → tx The arrow separating the rule head from the rule body can be read as logical equivalence. Variables mentioned only in a rule’s body are implicitly existentially quantified. In the g rule we see that g’s type is of the form ty → (t1 , t2 ), where t1 and t2 are the results of applying function f to a Bool and a ty . We represent f’s type, at both call sites in the program, by the f user constraint. The f rule is much more straightforward. It simply states that t is f ’s type if t is the function type tx → tx , for some tx , which is clear from the definition of f. We can infer g’s type by performing a CHR derivation, solving the constraint g(t) by applying CHRs (removing the constraint matching the lhs with the rhs). Note that we avoid renaming variables where unnecessary. g(t) ֌g t = ty → (t1 , t2 ), f (tf 1 ), tf 1 = Bool → t1 , f (tf 2 ), tf 2 = ty → f2 ֌f t = ty → (t1 , t2 ), tf 1 = tx → tx , tf 1 = Bool → t1 , f (tf 2 ), tf 2 = ty → f2 ֌f t = ty → (t1 , t2 ), tf 1 = tx → tx , tf 1 = Bool → t1 , tf 2 = t′x → t′x , tf 2 = ty → f2 If we solve the resulting constraints for t, we see that g’s type is ∀ty .ty → (Bool, ty ). 6 Example 5. The program below is a slightly modified version of the program presented in Example 4. g y = let f x = (y,x) in (f True, f y) The key difference is that f now contains a free variable y. Since y is monomorphic within the scope of g we must ensure that all uses of y, in all definitions, are consistent, i.e. each CHR rule which makes mention of ty , y’s type, must be referring to the same variable. This is important since the scope of variables used in a CHR rule is limited to that rule alone. In order to enforce this, we perform a transformation akin to lambda-lifting, but at the type level. Instead of user constraints of form f (t) we now use binary constraints f (t, l) where the l parameter represents f’s environment. We would generate rules like the following from this program. g(t, l) ⇐⇒ t = ty → (t1 , t2 ), f (tf 1 ), tf 1 = Bool → t1 , f (tf 2 , htxi), tf 2 = ty → f2 , l = ls f (t, l) ⇐⇒ t = tx → (ty , tx ), l = (hty |tsi) We write ht1 , ..., tn i to indicate a type-level list containing n types. A list with an n-element prefix but an unbounded tail is denoted by ht1 , ..., tn |ti. When unifying such a type against another list, t will be bound to some sublist containing all elements after the nth. As mentioned above, we now use binary predicates to represent the type of a function. The first argument, which we commonly refer to as the t component, will still be bound to the function’s type. The second component, which we call l, represents a list of unbound variables in scope of that function. We have ensured that whenever the f constraint is invoked from the g rule that ty , the type of y, is made available to it. So, in essence, the ty that we use in the f rule will have the same type as the ty in g, rather than simply being a fresh variable known only in g. Example 6. We now return to the program first introduced in Example 1, and generate the CHR rules corresponding to the function p, which is repeated below. For simplicity we will assume that (+) is defined only on Ints, i.e. (+) : Int → Int → Int. p y = (let f :: c -> Int f x = foo y x in f, y + (1::Int)) We generate the following CHRs from this fragment of the program. We also include the rule which represents foo’s type, and the rule which corresponds to the instance F oo Int b. p(t, l) ⇐⇒ t = ty → (t1 , t2 ), f (t1 , (hty i, hty , tx i), t2 = tr , tplus = Int → Int → Int, tplus = ty → Int → tr , l = (ls, hty , tx i) fa (t, l) ⇐⇒ p(t′ , l), t = c → Int, l = (hty |lsi, hty , tx i) f (t, l) ⇐⇒ t = tx → tr , f oo(tf oo , ls′ ), tf oo = ty → tx → tr , fa (t, l), l = (hty |lsi, hty , tx i) f oo(t, l) ⇐⇒ t = a → b → Int, F oo a b F oo Int b ⇐⇒ T rue 7 Here we have extended the scheme which we used to generate the constraints in the previous example. We stick to binary predicates, but have expanded the l component to include two lists. The first list, which we refer to as the local l component contains, as before, a type-level list of all unbound lambda variables in scope of the function. The second list, which we will often denote LT simply contains all of the lambda-bound variables from the top-level definition down, in a fixed order. We introduce a symbol fa and generate a new rule to represent f’s annotated type. Note also that we add a call to the f rule to unify f’s inferred type with the declared type. As demonstrated earlier, in order to check f it is necessary to consider all of the type information available in f’s context. In particular, for this program, it is critical that we know y has type Int, in order to reduce the F oo Int b constraint which arises from the use of foo, but is absent from the annotation. The way we introduce f’s context into f is by adding a call from the fa rule to the immediate parent definition, which in this case is represented by p. In this instance we are not interested in p’s type, only the effect it has on lambda-bound variables, and any type class constraints which may arise. Note that if p were itself embedded within a function definition, then it too would have such a call (to its own parent), and so f would indirectly inherit p’s context. We perform the following simplified derivation to demonstrate that the CHR formulation above captures the necessary context information within f. f (t, l) ֌f f oo(tf oo , ls′ ), tf oo = ty → tx → tr , fa (t, l), l = (hty i, hty , tx i), ... ֌f oo tf oo = a → b → Int, F oo a b, tf oo = ty → tx → tr , fa (t, l), l = (hty i, hty , tx i), ... we can simplify this to: F oo ty tx , fa (t, l), l = (hty i, hty , tx i), ... ֌fa F oo ty tx , p(t′ , l), l = (hty i, hty , tx i), ... ֌p F oo ty tx , tplus = Int → Int → Int, tplus = ty → Int → tr , l = (ls, hty , tx i), l = (hty i, hty , tx i), ... ֌F oo tplus = Int → Int → Int, tplus = ty → Int → tr , l = (ls, hty , tx i), l = (hty i, hty , tx i), ... Through the call to p from fa we introduce sufficient context information to determine that ty is an Int, and to consequently reduce away the F oo constraint using the instance rule. Note that the LT component is necessary here because p is not aware of its own lambda-bound variables. Without the LT component, p would not be able to “export” the required information about ty to f . 3.2 Constraint and CHR Generation In detail, we show how to map expressions to constraints and CHRs. Lambdaabstractions such as λx.e are preprocessed and turned into λx :: tx .e where tx is a fresh type variable. We assume that LT contains all such type variables tx attached to lambda-abstractions. For each function definition f=e we generate a CHR of the form f (t, l) ⇐⇒ C where l refers to a pair (ll , lg ). The constraint C is generated out of the program text of e. We maintain that ll denotes the set of types of lambda-bound variables in the environment and lg refers to LT the types of all lambda-bound variables. 8 Constraint Generation: t, l, lg fresh (f ∈ E or fa ∈ E f non-recursive) (Var-f) C = {f (t, l), l = (htx i, lg )} {x : tx }, E, f ⊢C (C t) (x : t1 ) ∈ Γ t2 fresh (Var-x) Γ, E, x ⊢C (t2 = t1 t2 ) (VarA-f) fa ∈ E f recursive t, l, lg fresh C = {fa (t, l), l = (htx , lg i)} {x : tx }, E, f ⊢C (C t) Γ.x : t1 , E, e ⊢C (C t2 ) (Abs) C ′ = {C, t3 = t1 → t2 } t3 fresh Γ, E, λx :: t1 .e ⊢C (C ′ t3 ) (Let) Γ, E ∪ {g}, e2 ⊢C (C t) Γ, E, let g = e1 in e2 ⊢C (C t) (App) Γ, E, e1 ⊢C (C1 t1 ) Γ, E, e2 ⊢C (C2 t2 ) t3 fresh Γ, E, e1 e2 ⊢C (C1 , C2 , t1 = t2 → t3 t3 ) (LetA) Γ, E ∪ {ga }, e2 ⊢C (C t) g :: C1 ⇒ t1 Γ, E, let in e2 ⊢C (C t) g = e1 CHR Generation: (Var) h, Γ, E, v ⊢R ∅ (App) h, Γ, E, e1 ⊢R P1 h, Γ, E, e2 ⊢R P2 h, Γ, E, e1 e2 ⊢R P1 ∪ P2 (Abs) h, Γ.x : t, E, e ⊢R P h, Γ, E, λx :: t.e ⊢R P Γ = {x1 : t1 , . . . , xn : tn } t, t′2 , l, lr, lg fresh g, Γ, E, e1 ⊢R P1 h, Γ, E ∪ {g}, e2 ⊢R P2 Γ, E, e1 ⊢C (C1′ t′1 ) (Let) P = P1 ∪ P2 ∪ {g(t, l) ⇐⇒ C1′ , t′1 = t, l = (ht1 , . . . , tn |lri, LT ), h(t′2 , l)⊖ } h, Γ, E, let g = e1 in e2 ⊢R P (LetA) Γ = {x1 : t1 , . . . , xn : tn } t, t′2 , l, lr fresh g, Γ, E ∪ {ga }, e1 ⊢R P1 h, Γ, E ∪ {ga }, e2 ⊢R P2 Γ, E ∪ {ga }, e1 ⊢C (C1′ t′1 ) P = P1 ∪ P2 ∪ ( ) ′′ ga (t, l) ⇐⇒ t = t1 , C1′′ , l = (ht1 , ..., tn |lri, LT ), h(t′2 , l)⊖ g(t, l) ⇐⇒ l = (ht1 , . . . .tn |lri, LT ), ga (t, l), C1′ , t = t′1 h, Γ, E, let g :: C1′′ ⇒ t′′1 in e2 ⊢R P g = e1 Fig. 2. Constraint and CHR Generation We make use of list notation (on the level of types) to refer to the types of λ-bound variables. In order to avoid confusion with lists of values, we write hl1 , . . . , ln i to denote the list of types l1 , . . . , ln . We write hl|ri to denote the list of types with head l and tail r. For constraint generation, we employ judgments Γ, E, e ⊢C (C t) where environment Γ , set of predicate symbols E and expression e are input values and constraint C and type t are output parameters. Note that Γ consists of lambdabound variables only whereas E holds the set of predicate symbols referring to primitive and let-defined functions. Initially, we assume that Einit holds all the 9 symbols defined in Pinit which is the CHR representation of all functions in Γinit . The rules can be found in Figure 2. Consider rule (Var-f). If the function does not carry an annotation or the function is not recursive 2 we make use of the definition CHR. However, strictly making use of the definition CHR might introduce cycles among CHRs, e.g. consider polymorphic recursive functions. In such cases we make use of the annotation CHR, see rule (VarA-f). In both rules we set l to the sequence of types of all lambda-variables in scope. Note that we might pass in more types of lambdabound variables than expected by that function. This is safe because we leave the first component of l “open” at definition sites. That is, we expect at least the types of lambda-bound variables in scope at the definition site and possibly some more. The second component which refers to the sequence of all types of lambda-bound variables appearing in the entire program is left unconstrained. This component will be only constrained at definition sites (see CHR generation rules (Let) and (LetA)). Note that in rule (Abs) the order of lambda-bound variables added to type environment matters. Hence, we silently treat Γ as a list rather than a set. In rules (Let) and (LetA) the constraints arising out of e1 might not appear in C unless we use function g in e2 . Note that we do not generate a constraint for the subsumption condition which will be checked separately. For rule generation, we employ judgments of the form h, Γ, E, e ⊢R P where CHR h, environment Γ , set of predicate symbols E and expression e are input values and the set P of CHRs is the output value. As an invariant we maintain that h refers to the surrounding definition of expression e. Initially, we assume that h refers to some trivial CHR h(t, l) ⇐⇒ T rue and E refers to the set of primitive functions. We refer to Figure 2 for details. There are two interesting rules. Rule (Let) deals with unannotated functions. Note that we do not add g to E when generating constraints and rules from e1 . Hence, we assume for simplicity that unannotated functions are not allowed to be recursive. Of course, our system [SW] handles unannotated, recursive functions. Their treatment is described in a forthcoming report The novel idea of our inference scheme is that we reach surrounding constraints within the definition of g via the constraint h(t2 , l)⊖ . The ⊖ marker (left out in Example 6 for simplicity) serves two purposes: (1) We potentially create cycles among CHRs because we might reintroduce renamed copies of surrounding constraints. Markers will allow us to detect such cycles to avoid non-termination of CHRs. (2) Variables occurring in marked constraints are potentially part of the environment. Hence, we should not quantify over those variables. Examples will follow shortly to highlight these points. Rule (LetA) is similar to rule (Let). Here, the annotation CHR includes the surrounding definition h. The actual inference result is reported in the definition CHR. 3.3 CHR Solving We introduce the marked CHR semantics. We assume that each constraint is attached with either a ⊖ marker or a ǫ (pronounced empty) marker. The empty marker is commonly left implicit. We refer to constraints carrying a ⊖ marker as marked constraints. A constraint carrying the empty marker is unmarked. In 2 We can easily check whether a function is recursive or not by a simple dependency analysis. 10 case of CHR rule application on a marked constraint, we mark all constraints in the body before adding them to the constraint store. Definition 1 (Marked CHR Application). Let d = (U t̄)a (or d = f (t̄)a ) be a primitive constraint where a ∈ {⊖ ,ǫ }. We define mark(d) = a. We write d⊖ to denote (U t̄)⊖ (or f (t̄)⊖ ). Let (R) c1 , ..., cn =⇒ d1 , ..., dm ∈ P and C be a constraint. Let φ be the m.g.u. of all equations in C. Let c′1 , ..., c′n ∈ C such that there exists a substitution θ on variables in rule (R) such that θ(ci ) = φ(c′i ) for i = 1...n. Then, C ֌R C, d′1 , ..., d′m where  di if mark(c′j ) 6= ⊖ for j = 1...n d′i = (di )⊖ otherwise Let (R) c ⇐⇒ d1 , ..., dm ∈ P and C be a constraint. Let φ be the m.g.u. of all equations in C. Let c′ ∈ C such that there exists a substitution θ on variables in rule (R) such that θ(c) = φ(c′ ), that is user-defined constraint c′ matches the left-hand side of rule (R). Then, C ֌R C − c′ , d′1 , ..., d′m where d′i are as above. A derivation step from global set of constraints C to C ′ using an instance of rule r is denoted C ֌r C ′ . A derivation, denoted C ֌∗P C ′ is a sequence of derivation steps using either rules in P such that no further derivation step is applicable to C ′ . The operational semantics of CHRs exhaustively apply rules to the global set of constraints, being careful not to apply propagation rules twice on the same constraints (to avoid infinite propagation). We say a set of CHRs is terminating if for each C there exists C ′ such that C ֌∗P C ′ . Example 7. Consider class Erk a where erk :: a class Foo a where foo :: a f = (erk, let g :: Foo a => a; g = foo in g) Here is a sketch of the translation to CHRs. ga (t) ⇐⇒ f (t′ )⊖ , F oo t g(t) ⇐⇒ ga (t), F oo t f (t) ⇐⇒ t = (a, b), Erk a, g(b) Consider the derivation g(t1 ) ֌g ga (t1 ), F oo t1 ֌ga f (t′ )⊖ , F oo t1 ֌f t′ = (a, b), (Erk a)⊖ , g(b)⊖ , F oo t1 ֌g t′ = (a, b), (Erk a)⊖ , ga (b)⊖ , (F oo b)⊖ , F oo t1 In step ֌f we propagate ⊖ to all new constraints. Note that we encounter a cycle among CHRs (see underlined constraints). Indeed, CHRs may be “nonterminating” because we introduce repeated duplicates of surrounding constraints. To avoid non-termination we introduce an additional CHR Cycle Removal step ... ֌g t′ = (a, b), (Erk a)⊖ , ga (b)⊖ , (F oo b)⊖ , F oo t1 ֌CCR t′ = (a, b), (Erk a)⊖ , (F oo b)⊖ , F oo t1 which is defined as follows. 11 Definition 2 (CHR Cycle Removal). Let f (t, l) ∈ C and f (t′ , l′ )⊖ ∈ C ′ and a derivation ... ֌ C ֌ ... ֌ C ′ . Then, ... ֌ C ֌ ... ֌ C ′ ֌CCR C ′ − f (t′ , l′ )⊖ . We assume that ֌CCR is applied aggressively. We argue that this derivation step is sound because any further rule application on ga (b)⊖ will only add renamed copies of constraints already present in the store. Lemma 1 (CCR Soundness). Let P be a set of CHRs and C and C ′ two ¯fv(C) C ′ . constraints such that C ֌∗P C ′ . Then P |= C ↔ ∃ We can also argue that we break any potential cycle among predicate symbols referring to function symbols. Note that we do not consider breaking cycles among two unmarked constraints. Such cases will only occur in case of unannotated, recursive functions which are left out for simplicity. A detailed description of such cases will appear in a forthcoming report. Also note that we never remove cycles in case of user-defined constraints. In such a case, the type class theory might be non-terminating. Hence, we state the the CCR derivation steps preserves termination assuming the type class theory is terminating. Lemma 2 (CCR Termination). Let Pp be a terminating type class theory, h(t, l) ⇐⇒ T rue a CHR, Einit a set of primitive predicate symbols, Pinit a set of CHRs and (Γ, e) a typing problem such that h, Γ, Einit , e ⊢R Pe and (Γinit , Pinit ) models Einit for some Γinit . Then, Pp ∪ Pe ∪ Pinit is terminating. 3.4 Type Inference via CHR Solving Consider type inference for an expression e w.r.t. an environment Γ of lambdabound variables and an environment Γinit of primitive functions and type class theory Pp . We assume (Pint , Eint ) model Γinit such that for each f : ∀ā.C ⇒ t′ we find f (t, l) ⇐⇒ C, t = t′ ∈ Pinit and f ∈ Einit . Then, we generate Γ, e ⊢C (C t) and h, Γ, Einit , e ⊢R Pe . We generally assume that P denotes Pp ∪ Pe ∪ Pinit . For typability we need to check that (1) constraint C is satisfiable, and (2) all type annotations in e are correct. We are now in the position to describe CHR-based satisfiability and subsumption check procedures. Definition 3 (Satisfiability Check). Let P be a set of CHRs and C a constraint such that C ֌∗P C ′ for some constraint C ′ . We say that C is satisfiable iff the unifier of all equations in C ′ exists. Soundness of the above definition follows from results stated in [SS02] in combination with Lemma 1. Of course, decidability of the satisfiability check depends on whether CHRs are terminating. To check for correctness of type annotations we first need to calculate the set of all subsumption problems. Let Esub(e) be the set of all predicate symbols ga where each ga refers to some subexpression (let g :: C1 ⇒ t1 ; g = e1 in e2 ) in e. Let Fsub(e) be a formula such that ∀t, l.(ga (t, l) ↔ g(t, l)) ∈ Fsub(e) for all ga ∈ Esub(e) . It remains to verify that the type annotation is correct under the abstraction of type inference in terms of P . Formally, we need to verify that 12 P |= Fsub(e) where P refers to first-order logic interpretation of the set of CHRs P . In [SS02], we introduced a CNF (Canonical Normal Form) procedure to test for equivalence among constraints ∀t, l.(ga (t, l) ↔ g(t, l)) w.r.t. some set of CHR by executing ga (t, l) and g(t, l) and verify that the resulting final stores are equivalent modulo variables in the initial store (here {t, l}). Thus, we can phrase ¯t,l .C to denote ∃fv(C) − {t, l}.C. the subsumption check as follows. We write ∃ Definition 4 (Subsumption Check). Let ga ∈ Esub(e) and P be a set of CHRs. We say that g’s annotation is correct iff (1) we execute ga (t, l) ֌∗P C1 ¯t,l .C1 ) ↔ (∃ ¯t,l .C2 ). and g(t, l) ֌∗P C2 , (2) we have that |= (∃ Soundness of the above definition follows from results stated in [SS02] in combination with Lemma 1. Example 8. Recall Example 7 and the derivation g(t1 ) ֌∗ t′ = (a, b), (Erk a)⊖ , F oo t1 . A similar calculation shows that ga (t1 ) ֌∗ s′ = (c, d), (Erk c)⊖ , F oo t1 . Note that the resulting constraints are logically equivalent modulo variable renamings. Hence, g’s annotation is correct. Note that in our formulation of type inference, the type of an expression is described by a set of constraints w.r.t. a set of CHRs. The following procedure describes how to build the associated type scheme. Markers attached to constraints provide important information which variables arise from the surrounding scope. Of course, we need to be careful not to quantify over those variables. Definition 5 (Building of Type Schemes). Let P be a set of CHRs, g a function symbol. We say function g has type ∀ā.C ′ ⇒ t′ w.r.t. P iff (1) We have that g(t, l) ֌∗ C, l = (ll , lg ) for some constraint C, (2) φ, the m.g.u. of C, l = (ll , lg ) exists, (3) let D ⊆ φ(C) such that D is maximal and D consists of unmarked user-defined constraints only, (4) let ā = fv(D, φt) − fv(φll ), (5) let C ′ = φ(C) and t′ = φ(t). Example 9. According to Example 8 we find that g has type ∀t1 .(Erk a, F oo t1 ) ⇒ t1 . Note that Erk a arises from f’s program text. We are able to state soundness of our approach. Theorem 1 (Soundness). Let Pe , Pp and Pinit be three sets of CHRs, h a CHR in Pinit , Γ an environment of simply-typed bindings, Γinit an environment of primitive functions, e an expression, Einit a set of predicate symbols, C a constraint and t a type such that (Pinit , Einit ) models Γinit and Γ, Einit , e ⊢C (C t) and h, Γ, Einit , e ⊢R Pe and type checking of all annotations in e is successful. Let C ֌∗Pp ∪Pinit ∪Pe C ′ for some constraint C ′ . Let φ be the m.g.u. of C ′ where we treat all variables in Γ as Skolem constants. Then, φ(C ′ ), φ(Γ ) ∪ Γinit ⊢ e : φ(t). The challenge is to identify some sufficient criteria under which our type inference method is complete. Because we only check for subsumption we need to guarantee that each subsumption condition will be either true or false. E.g. in Example 2 the subsumption condition boils down to the constraint ∀tx .F oo ty tx . Note that we can satisfy this constraint by setting ty to either Int or Bool. Hence, 13 our task is to prevent such situations from happening. In fact, such situations can never happen for single-parameter type classes. But what about multi-parameter type classes? The important point is to ensure that fixing one parameter will immediately fix all the others. That is, in case of ∀tx .F oo ty tx we know that tx is uniquely determined by ty . We can enforce such conditions in terms of functional dependencies [Jon00]. Definition 6. We say a type class T C is fully functional iff we find a class declaration class TC a1 ... an | fd1 ,..., fdn where fdi = ai -> a1 ... ai−1 ai+1 ... an . We argue that for fully functional dependencies solutions (if they exist) must be unique. In fact, this is not sufficient because there are still cases where we need guess. Example 10. Consider the following simplified representation of the Show class. class Show a where show :: a->String read :: String->a f :: Show a => String->String f x = show (read x) The subsumption check boils down to the formula ∀a.Show a ⊃ ∃a′ .Show a′ which is obviously a true statement (take a′ = a). However, in our translation to CHRs we effectively check for Show a ↔ Show a′ which obviously does not hold. There are further sources where we need to take a guess. Example 11. Consider class Foo instance Show Int instance Foo a => Show a where Pp = {(S1) Show Int ⇐⇒ T rue (S2) Show a ⇐⇒ F oo a}. We have that Pp |= Show Int. However, Show Int ֌S2 F oo Int where |= F oo Int 6↔ T rue which suggests that Pp |= Show Int might not hold. Clearly, by guessing the right path in the derivation we find that Show Int ֌S1 T rue. To ensure that our subsumption check (Definition 4) is complete we need to rule out ambiguous types and require that the type class theory is complete. A type is ambiguous iff we can not determine the variables appearing in constraints by looking at the types alone. The annotation f::Show a=>String->String in Example 10 is ambiguous. A type class theory Pp is complete iff Pp is confluent, terminating and range-restricted (i.e. grounding the lhs of CHRs grounds the rhs) and all simplification rules are single-headed. The type class theory Pp in Example 11 is non-confluent. In [SS02] we have identified these conditions as sufficient to ensure completeness of the Canonical Normal Form procedure to test for equivalence among constraints. Theorem 2. Let Pp a complete and fully functional type class theory. Then our CHR-based inference scheme infers principal types if the types arising are unambiguous. 14 4 Related Work and Conclusion Simonet and Pottier [SP04] introduce HMG(X) a refined version of HM(X) [OSW99,Sul00] which includes among others type annotations. Their type inference approach is based on the “allow for more solutions” philosophy. Hence, they achieve complete type inference immediately. However, they only consider tractable type inference for the specific case of equations as the only primitive constraints. An approach in the same spirit is considered by Hinze and Peyton-Jones [HJ00]. They sketch an extension of Haskell to allow for “higher-order” instances which logically correspond to nested equivalence relations. As pointed out by Faxén [Fax03], in such an extended Haskell version it would be possible to type the program in Example 2. We believe this is an interesting avenue to pursue. We are not aware of any formal results nor a concrete implementation of their proposal. Pierce and Turner [PT00] develop a local type inference scheme where userprovided type information is propagated inwards to nodes which are below the annotation in the abstract syntax tree. Their motivation is to remove redundant annotations. Note that Peyton-Jones and Shields [PJS04] describe a particular instance of local type inference based on the work by Odersky and Läufer’s[OL96]. In our approach we are able to freely distribute type information across the entire abstract syntax tree. Currently, we only distribute information about the types of lambda-bound variables and type class constraints. We believe that our approach can be extended to a system with rank-k types. We plan to pursue this topic in future work. In this paper, we have presented a novel inference scheme where the entire type inference problem is mapped to a set of CHRs. Due to the constraint-based nature of our approach, we are able to make available the results of inference for outer expressions while inferring the type of inner expressions. We have fully implemented the improved CHR-based inference system as part of the Chameleon system [SW] Our system improves over previous implementations such as Hugs and GHC. For some cases, e.g. unambiguous Haskell 98 programs, we can even state completeness. We note that our improved inference scheme can host the type debugging techniques described in [SSW03,SSW04]. In future work, we plan to follow the path of Odersky and Läufer [OL96] and compute (non-principal in general) solutions to subsumption problems. We strongly believe that our improved CHR-based inference will be of high value for such an attempt. Another alternative inference approach not mentioned so far is to only generate all necessary subsumption problems σi ≤ σa and wait for the “proper” moment to solve or check them. 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Sulzmann, and M Wehr. Type inference with constrained types. Theory and Practice of Object Systems, 5(1):35–55, 1999. PJS04. S. Peyton-Jones and M. Shields. Practical type inference for arbitrary-rank types. Submitted for publication, 2004. Pot98. F. Pottier. A framework for type inference with subtyping. In Proc. of ICFP’98, pages 228–238. ACM Press, 1998. PT00. B.C. Pierce and D.N. Turner. Local type inference. ACM Transactions on Programming Languages and Systems, 22(1):1–44, 2000. Rém93. D. Rémy. Type inference for records in a natural extension of ML. In Carl A. Gunter and John C. Mitchell, editors, Theoretical Aspects Of Object-Oriented Programming. Types, Semantics and Language Design. MIT Press, 1993. SP04. V. Simonet and F. Pottier. Constraint-based type inference with guarded algebraic data types. Submitted to ACM Transactions on Programming Languages and Systems, June 2004. SS02. P. J. Stuckey and M. Sulzmann. A theory of overloading. In Proc. of ICFP’02, pages 167–178. ACM Press, 2002. SS04. P.J. Stuckey and M. Sulzmann. A theory of overloading. ACM Transactions on Programming Languages and Systems, 2004. To appear. SSW03. P.J. Stuckey, M. Sulzmann, and J. Wazny. Interactive type debugging in Haskell. In Proc. of Haskell Workshop’03, pages 72–83. ACM Press, 2003. SSW04. P. J. Stuckey, M. Sulzmann, and J. Wazny. Improving type error diagnosis. In Proc. of Haskell’04, pages 80–91. ACM Press, 2004. Sul00. M. Sulzmann. A General Framework for Hindley/Milner Type Systems with Constraints. PhD thesis, Yale University, Department of Computer Science, May 2000. SW. M. Sulzmann and J. Wazny. Chameleon. http://www.comp.nus.edu.sg/˜ sulzmann/chameleon. 16 A Variations Our exisiting translation to CHRs is slightly lazier than most inference algorithms in the sense that we do not infer the types of let-bound variables which are never called. Example 12. Consider: f x = let g = x x in x We generate CHR rules that look like the following: f (t) ⇐⇒ t = tx → tx g(t) ⇐⇒ tx = tx → t Since the f rule never calls g, the unsatisfiable constraint is not introduced, and this program is considered well-typed. This “laziness” can be problematic whenever we need to compare the inferred type of some function with its declared type. Consider an annotated function g, nested within the definition of function f , from which we generate an inference rule, g(t, l) ⇐⇒ ga (t, l), Ci , and an annotation rule, ga (t, l) ⇐⇒ f (t′ , l′ )⊖ , Ca . It’s possible that the types of some global variables are affected in g, but not in ga . In order for g(t, l) and ga (t, l) to be equivalent, we depend on g’s context, as called in the ga rule, to in turn call g and introduce those missing constraints. Example 13. Consider the following program. f y = let g :: Bool g = y in ’a’ We generate the following (simplified) CHR rules: f (t, l) ⇐⇒ t = Char, l = (hi, hty i) g(t, l) ⇐⇒ t = ty , l = (hty i, hty i), ga (t, l) ga (t, l) ⇐⇒ t = Bool, l = (hty i, hty i), f (t′ , l) Even though g’s type annotation is acceptable, out subsumption check would fail, because g(t, l) and ga (t, l) are not equivalent wrt the l component. Clearly, the g rule implies ty = Bool, but the ga rule does not. We can remedy this situation by ensuring that all nested functions are called by their parent function. In this way, when we consider a function annotation, the definition of the function becomes part of its context. Example 14. We modify the program of Example 13, forcing f to call g, but disregard its value. f y = let g :: Bool g = y in const ’a’ g 17 The CHR rule associated with f would now look something like: f (t, l) ⇐⇒ tconst = a → b → a, a = Char, t = ty → a, l = (hi, hty i), g(b, l′ ) This solves our immediate problem, in that the constraints arising from g(t, l) and ga (t, l) are now equialent wrt t and l. Unfortunately this does not work in the case where the function we call is recursive. Consider: f y = let g :: Bool g = const y g in ’a’ Here, if f were to call g, the constraint generated to represent g’s type would be ga (t′ , l′ ). We then face the same problem, that from within ga we have no association between ty and Bool. Clearly, a syntactic transformation of the source program to introduce calls to otherwise uncalled functions is not sufficient. We must modify the CHR generation process to directly insert calls to the inference constraints of functions which are not already called. Example 15. We return to the rules generated in Example 13. The following CHR rule is a modified version of the f rule above which now contains a call to g, further constraining the lambda-bound type variables. f (t, l) ⇐⇒ t = Char, l = (hi, hty i), g(t′ , l)⊖ Using this modified rule, we see now that g(t, l) and ga (t, l), are equivalent, since ty = Bool in both. The ⊖ mark on the g constraint is not significant here, though it does accomodates the simpler form of cycle breaking (by simply removing the repeated constraint) than the equivalent unmarked constraint would. B Monomorphic Recursive Functions In case of monomorphic recursive functions (i.e. recursive functions with no type annotation) we need to update our strategy for breaking cycles among CHRs (Definition 2). We denote by NRF the set of all non-recursive functions, by MRF the set of all recursive functions which carry no type annotations and by ARF the set of all annotated recursive functions. Definition 7 (CHR Cycle Removal). Let f (t, l)⊖ ∈ C and f (t′ , l′ )⊖ ∈ C ′ where f ∈ NRF ∪ ARF and a derivation ... ֌ C ֌ ... ֌ C ′ . Then, ... ֌ C ֌ ... ֌ C ′ ֌CCR C ′ − f (t′ , l′ )⊖ . Let f (t, l) ∈ C and f (t′ , l′ )⊖ ∈ C ′ where f ∈ MRF and a derivation ... ֌ C ֌ ... ֌ C ′ . Then, ... ֌ C ֌ ... ֌ C ′ ֌CCR C ′ − f (t′ , l′ )⊖ , t′ = t. Let f (t, l) ∈ C and f (t′ , l′ ) ∈ C ′ where f ∈ MRF and f (t′ , l′ ) is known to arise from the exact same program source location as f (t, l), and given a derivation ... ֌ C ֌ ... ֌ C ′ . Then, ... ֌ C ֌ ... ֌ C ′ ֌Mono C ′ − f (t′ , l′ ), t′ = t. We assume that ֌CCR and ֌Mono are applied aggressively. 18 Note that according to the definition of a ֌Mono step, we should only ever break cycles amongst constraints representing unannotated, recursive function types when they arise from the same program location. Unifying the types of different calls to the same function is overly restrictive, as the following example illustrates. Example 16. Consider the following program. e = let f = f in (f::Int, f::Bool) The two fs in the body of e are calls to the same recursive, unannotated function. i.e. f ∈ M RF . It would be unnecessarily restrictive, however, to aggressively apply ֌Mono here and unify the types of the two fs, resulting in a type error. Indeed, these two fs are not even part of the same cycle. We note that breaking cycles among constraints arising from the exact same program source location is sufficient. Example 17. Consider f = ... f1 ... f2 where we added numbers 1 and 2 to different uses sites of f. Here is a sketch of type inference where we annotate ֌ to refer to the number of CHR steps applied so far. f ֌n f1 , f2 ֌f f1,1 , f1,2 , f2 . In the last step we reduce a call to f at location 1. Note that we make use of a refined marking scheme. We keep track of the original source location and add the location of the constraint introduced to the store to the existing locations. We refer to [SSW03] for the details of such a “location-history” aware refinement of the CHR semantics. Hence, applying rule (Mono) twice will remove f1,1 and f1,2 (and equate the types of f1,1 and f1,2 with f ). Similarly, we remove the cycles created by f2 . We note that for ֌CCR there is no need to equate the l component in case of f ∈ MRF . The l component is always set at the use site of functions (see constraint generation rules (Var-f)). Equating the l component would yield a strictly weaker system. Example 18. Consider f x = ( f1 ..., let g y = ... f2 ...in g) where we added numbers 1 and 2 to different (monomorphic) uses sites of f. Here is a sketch of type inference f ֌n ...f1 , ..., g ֌n+m ...f1 , ..., f2 where natural numbers n and m refer to the number of CHR steps applied so far. Assume we fully equate f1 and f2 . Note that their local ll components differ (because this component is always exact!) Hence, type inference fails. Note that (as before in case of ARF and NRF ) we only break cycles among constraints which carry the same marker. This yields a more precise method. Example 19. Assume we have situation where ... ֌ f (Int), ... ֌ ... ֌ f (t)⊖ , ... Removing f (t)⊖ and adding t = Int might be too restrictive. 19 Note that by construction ֌Mono never applies to ARF and NRF . Any potential cycle will be eventually broken. We can re-establish Lemmas 2 and and state a slightly stronger Lemma 1 which guarantee soundness of our CHRbased inference approach. Lemma 3 (CCR-Mono Soundness). Let P be a set of CHRs and C and C ′ ¯fv(C) C ′ ⊃ C. two constraints such that C ֌∗P C ′ . Then P |= ∃ ¯fv(C) C ′ does not hold anymore. We may reject typable Note that P |= C ⊃ ∃ programs because we stricly enforce the (Mono) rule. Example 20. Consider h x = (h1 ’a’) && (h2 True) Here is a sketch of type inference. ֌ ֌ ֌Mono ֌Mono ↔ h(t) t = tx → Bool, h1 (Char → Bool), h2 (Bool → Bool) t = tx → Bool, Char → Bool = tx′ → Bool, h1,1 (Char → Bool), h1,2 (Bool → Bool), h2 (Bool → Bool) t = tx → Bool, Char → Bool = tx′ → Bool, Char → Bool = Char → Bool, h1,2 (Bool → Bool), h2 (Bool → Bool) t = tx → Bool, Char → Bool = tx′ → Bool, Char → Bool = Char → Bool, Char → Bool = Bool → Bool, h2 (Bool → Bool) F alse It is interesting to note is that our type inference scheme for recursive function is more relaxed compared to the one found in some other established type checkers. Example 21. Consider the following program. e e f f g :: Bool = g :: Bool -> a = g = f e In the case of GHC, the following error reported is: mono-rec.hs:5: Couldn’t match ‘Bool -> a’ against ‘Bool’ Expected type: Bool -> a Inferred type: Bool In the definition of ‘f’: f = g The problem reported here stems from the fact that within the mutually recursive binding group consisting of e, f and g, f is assigned two ununifiable types, Bool and Bool → a: the first because it must have the same type as g, which according to e must be Bool; and the second because of its type declaration. 20 Our translation scheme is more liberal than this, in that g’s type within e and f may be different. Essentially, we only require that the type of a variable be identical at all locations within the mutually recursive subgroup if a type declaration has been provided for that variable. (Simplified) Translation of the above program to CHRs yields. ea (t) ⇐⇒ t = Bool e(t) ⇐⇒ g(t) fa (t) ⇐⇒ t = Bool → a f (t) ⇐⇒ g(t) g(t) ⇐⇒ ea (te ), fa (tf ), tf = te → Bool It is clear from the above that there are no cycles present amongst these rules. We can use them to successfully infer a type for any of the variables in the program. In fact, our handling of binding groups is similar to [Jon99] where type inference of binding groups proceeds as follows: 1. Extend the type environment with the type signatures. In this case f::forall a. Bool -> a and e::Bool. 2. Do type inference on the bindings without type signatures, in this case g = f e. Do generalisation too, and extend the environment, giving g :: forall a. a. 3. Now, and only now, do type inference on the bindings with signatures. 21 

THE STRUCTURE OF DGA RESOLUTIONS OF MONOMIAL IDEALS arXiv:1610.06526v1 [] 20 Oct 2016 LUKAS KATTHÄN Abstract. Let I ⊂ k[x1 , . . . , xn ] be a squarefree monomial ideal a polynomial ring. In this paper we study multiplications on the minimal free resolution F of a S/I. In particular, we characterize the possible vectors of total Betti numbers for such ideals which admit a differential graded algebra (DGA) structure on F. We also show that under these assumptions the maximal shifts of the graded Betti numbers are subadditive. On the other hand, we present an example of a strongly generic monomial ideal which does not admit a DGA structure on its minimal free resolution. In particular, this demonstrates that the Hull resolution and the Lyubeznik resolution do not admit DGA structures in general. Finally, we show that it is enough to modify the last map of F to ensure that it admits the structure of a DG algebra. Introduction Let S be a polynomial ring over a field, I ⊂ S be a monomial ideal, and let F denote the minimal free resolution of S/I over S. The multiplication map S/I ⊗S S/I → S/I can be lifted to a “multiplication” ∗ : F ⊗S F → F, which in general is associative and graded-commutative only up to homotopy. Moreover, ∗ is unique only up to homotopy. It is known that ∗ can always chosen to be graded-commutative “on the nose” (cf. [BE77, Prop 1.1]), but in general it is not possible to choose ∗ such that it is associative (cf. [Avr74, Appendix], [Avr81]). If ∗ is graded-commutative and associative, then it gives F the structure of a differential graded algebra (DGA). In this situation, we say that S/I admits a minimal DGA resolution. Starting with the work of Buchsbaum and Eisenbud [BE77], a lot of research has been devoted to the study of DGA resolutions, see for example [Avr81] or [Kus94] and the references therein. In the present paper, we study the case where I is generated by squarefree monomials and consider only multiplications which respect the multigrading. These restrictions allow us to prove much stronger statements than in the general situation. Our results can be organized along the following questions: (1) Which ideals admit a minimal DGA resolution? Which constructions of (not necessarily minimal) resolutions yield DGA resolutions? (2) What are the consequences if a given ideal admits a minimal DGA resolution? (3) How unique or non-unique are the DGA structures? (4) What can be said about the structure of DGA resolutions as algebras? 2010 Mathematics Subject Classification. Primary: 05E40; Secondary: 13D02,16W50,13F55. Key words and phrases. Monomial ideal; Free resolution; Differential graded Algebra. 1 2 LUKAS KATTHÄN There are a few classes of ideals in local rings which are known to admit minimal DGA resolutions, see [Kus94] for an overview. In the context of monomial ideals, this is the case for stable ideals [Pee96], matroidal ideals [Skö11] and edge ideals of cointerval graphs [Skö16]. Another class of ideals whose minimal free resolutions are well-understood are generic monomial ideals. However, in Theorem 5.1, we present an example of a (strongly) generic monomial ideal which does not admit a minimal DGA resolution. The example also demonstrates that the hull resolution [BS98] and the Lyubeznik resolution [Lyu88] do not admit a DGA structure in general. On the positive side, we show in Theorem 2.1 that for every monomial ideal I there always exists a monomial m ∈ S such that S/(mI) admits a minimal DGA resolution. Note that the minimal free resolutions of S/I and S/(mI) differ only in the last map. Therefore, one can interpret this result as saying that the other maps in the resolution to not contain obstructions to the existence of a DGA structure. This result actually holds in greater generality for homogeneous ideals in S and even for ideals in regular local domains. Concerning the second question, it is a beautiful result by Buchsbaum and Eisenbud [BE77, Proposition 1.4], that if I admits a minimal DGA resolution, then one can conclude certain lower bounds on the Betti numbers of S/I. In the context of squarefree monomial ideals, we sharpen this result by completely characterizing the possible total Betti numbers of ideals admitting a minimal DGA resolution (Theorem 4.1). Moreover, we show that if I is such an ideal, then its graded Betti numbers satisfy the subadditivity condition of [ACI15] (Proposition 4.4). It is well-known that DGA structures on minimal free resolutions do not need to be unique. Our Example 3.2 demonstrates that this even fails for squarefree monomial ideals, thus answering a question of Buchsbaum and Eisenbud [Pee10, Open Problem 31.4]. On the other hand, we show that at least a certain part of the multiplication is indeed unique, see Proposition 3.1 for the precise statement. Concerning the fourth questions, we show that F is essentially generated in degree 1 with respect to any multiplication. See Proposition 3.4 for the exact statement. As a consequence, whenever I admits a minimal DGA resolution F, then it is a quotient of the Taylor resolution (which has a canonical DGA structure) by a DG-ideal, see Theorem 3.6. Thus one can study the possible DGA structures on F by considering DG-ideals in the Taylor resolution. The key idea behind most of our results is the following simple observation: F is a free S-module which is generated in squarefree degrees, so for any homogeneous element f ∈ F, we can find an element f ′ ∈ F of squarefree degree, such that f = mf ′ for a monomial m ∈ S. We call f ′ the squarefree part of f . This paper is structured as follows. In Section 1 we recall some preliminaries about multiplicative structures on resolution, the Taylor resolution and the Scarf complex, and we introduce the squarefree part of an element f ∈ F. In the next section we prove Theorem 2.1 about the existence of minimal DGA resolutions after multiplication with an element. The following Section 3 is devoted to the study of the structure of multiplication on F. After that, in Section 4 we derive two consequences of the existence of a minimal DGA resolution, namely our characterization of the total Betti numbers and the subadditivity of syzygies. STRUCTURE OF DGA STRUCTURES 3 In the penultimate Section 5 we present an example of a generic monomial ideal which does not admit a minimal DGA resolution, and in the last section we study multiplication on resolutions of ideals which are not necessarily squarefree. 1. Preliminaries and notation Throughout the paper, let k be a field and S = k[x1 , . . . , xn ] be a polynomial ring over k, endowed with the fine Zn -grading. We denote by m := hx1 , . . . , xn i the maximal homogeneous ideal of S. Further, let I ⊂ S be a monomial ideal and let ∂ ∂ ∂ F : · · · → F2 → F1 → F0 → S/I → 0 denote the minimal free resolution of S/I. Note that F is Zn -graded and the differential ∂ respect the multigrading. The multigraded Betti numbers of S/I S are denoted by βi,a (S/I) = TorSi (S/I, k)a . We often identify F with the free L multigraded S-module i Fi . For an element f ∈ F we denote its multidegree by deg(f ) and its homological degree by |f |. We consider the componentwise order on Zn and write a ∧ b and a ∨ b for the componentwise minimum and maximum of a, b ∈ Zn , respectively. 1.1. Multiplications on resolutions. We recall the definition of a DGA structure. Definition 1.1. A differential graded algebra (DGA) structure on F is an S-linear map ∗ : F ⊗S F → F satisfying the following axioms for a, b, c ∈ F: (1) ∗ extends the usual multiplication on F0 = S, (2) ∂(a ∗ b) = ∂(a) ∗ b + (−1)|a| a ∗ ∂b (Leibniz rule), (3) |a ∗ b| = |a| + |b| (homogeneity with respect to the homological grading), (4) a ∗ b = (−1)|a|·|b| b ∗ a (graded commutativity), and (5) (a ∗ b) ∗ c = a ∗ (b ∗ c) (associativity). We will also consider non-associative multiplications. To make this precise, we call a map ∗ : F ⊗S F → F a multiplication if it satisfies all the axioms of a DGA except possibly the associativity. Moreover, we make the convention that in this paper, every multiplication respects the multigrading on F, unless specified otherwise. The only occasion when we consider more general multiplication is in Section 5. While F does not always admit a DGA structure, it always admits a multiplication, cf. [BE77, Prop 1.1], and the multiplication is unique up to homotopy. Explicitly, this means that when ∗1 , ∗2 are two multiplications, then there exist a map σ : F ⊗S F → F raising the homological degree by 1 such that a ∗1 b = a ∗2 b + ∂σ(a, b) + σ(∂a, b) + (−1)|a| σ(a, ∂b) for a, b ∈ F; 1.2. The Taylor resolution and the Scarf complex. We recall the definitions of the Taylor resolution and the Scarf complex of I. We refer the reader to p.67 and Section 6.2 of [MS05], respectively, for further information about these constructions. Let G(I) denote the set of minimal monomial generators of I and choose a total order ≺ on G(I). The choice of the order affects only the signs in the computations. 4 LUKAS KATTHÄN Definition 1.2. The Taylor resolution T of S/I is the complex of free S-modules with basis {gW : W ⊆ G(I)}. The basis elements are graded by |gW | := #W and deg gW := deg mW , where mW := lcm(m : m ∈ W ) for W ⊆ G(I). Further, the differential is given by X mW ∂gW = (−1)σ(m,W ) gW \{m} , mW \{m} m∈W where σ(m, W ) := #{m′ ∈ W, m′ ≺ m}. The Taylor resolution is a free resolution of S/I, but typically not a minimal one. It was shown by Gemeda [Gem76] (see also [Pee10, Proposition 31.3]) that it carries a DGA structure with the multiplication given by  (−1)σ(W,V ) mW mV gW ∗ gV =  0 mW ∪V gW ∪V if W ∩ V = ∅, otherwise, where σ(W, V ) := #{(m, m′ ) ∈ W × V : m′ ≺ m}. To simplify the notation we occasionally write gm1 m2 ... instead of g{m1 ,m2 ,... } for m1 , m2 , . . . ∈ G(I). Definition 1.3. The Scarf complex ∆I of S/I is the simplicial complex with vertex set G(I), where a set W ⊂ G(I) is contained in ∆I if and only if there does not exist another set V ⊂ G(I) with V 6= W and lcm(m : m ∈ W ) = lcm(m : m ∈ V ). Moreover, the algebraic Scarf complex F∆I is the subcomplex of the Taylor resolution generated by the generators gW , W ∈ ∆I . The algebraic Scarf complex is always a subcomplex of the minimal free resolution of S/I (cf. [MS05, Proposition 6.12]), but in general it is not acyclic. We will therefore use the notation gW for generators of F which lie in the Scarf complex. 1.3. The squarefree part. The following very basic observation turns out to be the key to many of our results: Lemma 1.4. Let F be a free S-module and a ∈ Nn . Assume that the degree of every generator of F is less or equal to a. Then every element f ∈ F with deg f  a can be written as f = mf ′ for a monomial m ∈ S and f ′ ∈ F with deg f ′ = (deg f ) ∧ a. Both m and f ′ are uniquely determined. This lemma is almost obvious, but we include a proof for completeness. Proof. Choose a basis of F and let G be the set of those basis elements whose degrees are less or equal than deg f . By assumption, f can be written as a linear combination of elements of G, and we write cg for the coefficient of g in this expansion. Then it holds that f= X g∈G cg xdeg(f )−deg(g) g = xdeg(f )−(a∧deg(f )) X cg x(a∧deg(f ))−deg(g) g. g∈G All exponents are non-negative by our hypothesis, so we have shown the existence of the claimed factorization. The uniqueness of m is trivial, because there is only one monomial of the correct multidegree. Using this, the uniqueness of f ′ follows from the fact that S is a domain and F is torsion-free.  STRUCTURE OF DGA STRUCTURES 5 Definition 1.5. Let F be a free S-module, such that the degree every generator is squarefree, i.e. less or equal than (1, . . . , 1) ∈ Nn . In the situation of Lemma 1.4, we call f ′ the squarefree part of f and denote it by |f |sqf . Note that the map f 7→ |f |sqf is k-linear, but not S-linear. We will apply this definition almost exclusively to the minimal free resolution F of a S/I for a squarefree monomial ideal I ⊂ S. 2. Existence of DGA structures up to multiplication with an element Our first result is the following theorem. It implies that it is enough modify the last map of a minimal free resolution to ensure the existence of a DGA structure. Theorem 2.1. Let S be a regular local ring, or a polynomial ring and I ⊆ S an ideal. There exists a homogeneous element s ∈ S, s 6= 0 such that the minimal free resolution of S/(sI) admits a DGA structure. If I is a monomial ideal (and S a polynomial ring), then s can be chosen to be the least common multiple of the generators of I. We need the following lemma for the proof of the Theorem. ± n Lemma 2.2. Let Q = k[t± 1 , . . . , tn ] be a Z -graded Laurent polynomial for n ≥ 0. Let F: 0 → Fp → · · · → F1 → F0 → 0 be an exact complex of graded Q-modules and assume that F0 = Q. Then the multiplication on F0 can be extended to a graded-commutative DGA structure on F. Proof. Let ∂ denote the differential of F. We claim that there exists a map σ : F → F of homological degree 1 such that ∂ ◦ σ + σ ◦ ∂ = idF and σ ◦ σ = 0. Indeed, every graded module over Q is free (cf. [GW78, Theorem 1.1.4]), so we can choose splittings Fi ∼ = Vi ⊕ ∂(Fi+1 ). We define σi |Vi = 0 and σi (∂(f )) = f for f ∈ Fi+1 . It is not difficult to see that this gives indeed a map σ with the claimed properties. We define the DGA structure on F inductively by the formula a ∗ b :=  ab   σ ∂(a) ∗ b + (−1)|a| a ∗ ∂(b) if |a| = 0 or |b| = 0 otherwise, where the multiplication in the first case is the one from F0 = Q. This multiplication clearly satisfies the Leibniz rule and it extends the multiplication on F0 . It remains to show that ∗ is graded-commutative and associative. Note that σ = σ ◦ ∂ ◦ σ. We proceed by induction on the homological degree, the 6 LUKAS KATTHÄN base case being clear. It holds for a, b, c ∈ F:  a ∗ (b ∗ c) = σ ∂(a ∗ (b ∗ c))   = σ ∂a ∗ (b ∗ c) + (−1)|a| a ∗ (∂b ∗ c) + (−1)|a|+|b| a ∗ (b ∗ ∂c) (#)   = σ (∂a ∗ b) ∗ c + (−1)|a| (a ∗ ∂b) ∗ c + (−1)|a|+|b| (a ∗ b) ∗ ∂c  = σ ∂((a ∗ b) ∗ c)   = (a ∗ b) ∗ c where in (#) we use the induction hypothesis. The commutativity is verified analogously.  Proof of Theorem 2.1. Let F be the minimal free resolution of S/I. Let Q be the subring of the field of fractions of S where we adjoin inverses for all homogeneous elements of S, and set FQ := F ⊗S Q. Note that Q is a multivariate Laurent polynomial ring over some field k. Moreover, Q is flat over S and hence FQ is exact. So by Lemma 2.2, it can be endowed with a DGA structure ∗. Choose a basis for each Fi . Then the multiplication on FQ can be represented as a matrix with entries in Q. Hence we can choose an element s ∈ S such that sq ∈ S for each entry q of this matrix. Let F′ be the subcomplex of F defined by F′0 := F0 and F′i := sFi for i ≥ 1. We claim that F′ is closed under multiplication. Indeed, for sa, ab ∈ F′ the choice of s implies that s(a ∗ b) ∈ F and thus (sa ∗ sb) ∈ F′ . Note that F′ is isomorphic to F except in degree 0, so in particular it is exact in every other degree. In degree 0, it holds that H0 (F′ ) = F0 /∂(sF1 ) = S/(sI). Thus F′ is the minimal free resolution of S/(sI). Finally, consider the case that I is a monomial ideal in a polynomial ring S = k[x1 , . . . , xn ]. Let a, b ∈ F be two homogeneous elements. The product a ∗ b ∈ FQ can be written as a sum X λg xdeg(a)+deg(b)−deg(g) g g∈B where B is an S-basis of F and λg ∈ k. Let now s be the lcm of all generators of I. Then the multidegrees of all elements of B are less or equal to deg(s), and hence s(a ∗ b) ∈ F. Now one can argue as above.  3. Properties of multiplications 3.1. Uniqueness. The multiplication on F is in general not unique. However, the part of the multiplication which lives on the algebraic Scarf complex is uniquely determined, as the next proposition shows. Proposition 3.1. Let I ⊂ S be a squarefree monomial ideal with minimal free resolution F, and let further ∗ be a multiplication on F. Then, for V, W ∈ ∆I with V ∪ W ∈ ∆I , it holds that gW ∗ gV =  (−1)σ(W,V ) mW mV mW ∪V 0 gW ∪V if W ∩ V = ∅, otherwise; STRUCTURE OF DGA STRUCTURES 7 where σ(W, V ) is defined as above. In particular, the product gW ∗ gV does not depend on the choice of ∗. Proof. Let B be an S-basis for F. For every g ∈ B there exists a (not necessarily unique) subset Ug ⊂ G(I) such that deg g = deg mUg and |g| = #Ug . This follows from the fact that F is a direct summand of the Taylor resolution. Expand gW ∗ gV in the basis B: gW ∗ gV = X cg g. g∈B Assume that cg 6= 0 for some g ∈ B. As I is squarefree, it holds that deg mUg = deg g ≤ deg |gW ∗gV |sqf = deg(gW )∨deg(gV ) = deg mW ∪V . Hence deg mUg ∪W ∪V = deg mW ∪V . But W ∪ V is contained in the Scarf complex, and hence it follows that Ug ∪ W ∪ V = W ∪ V and hence Ug ⊆ V ∪ W . On the other hand, it holds that #Ug = |g| = #W + #V ≥ #(W ∪ V ). So we can conclude that Ug = V ∪ W and that #W + #V = #(W ∪ V ). Hence if W and V intersect non-trivially then V gW ∗ gV is zero, and if they are disjoint, then gW ∗ gV = λ mmWWm gW ∪V for a scalar ∪V λ ∈ k. It is clear by induction that the scalars λ are uniquely determined by the Leibniz rule. Moreover, the values given in the claim satisfy the Leibniz rule, hence we have determined the product gW ∗ gV .  The following example show that the multiplication is not unique in general. This answers a question of Buchsbaum and Eisenbud [Pee10, Open Problems 31.4]. Example 3.2. Let I ⊂ S = Q[x1 , x2 , x3 , x4 , y, z] be the ideal with generators a := x1 x2 y, b := x2 x3 , c := x3 x4 z and d := x4 x1 . One can verify with Macaulay2 that the minimal free resolution F of S/I equals its algebraic Scarf complex. The Scarf complex ∆I is the following simplicial complex: c d a b Consider the product ga ∗ gc . Both a and c are vertices of ∆I , but {a, c} ∈ / ∆I , so the preceding Proposition does not apply. And indeed, for any λ ∈ k the choice ga ∗ gc := λ(x4 zgab + x1 ygbc ) (1 − λ)(x3 zgad − x2 ygcd ) satisfies the Leibniz rule. This can be extended to a multiplication on F, which is even a DGA structure, because the length of the resolution is three (cf. [BE77, Prop 1.3]) The next example illustrates that one cannot omit the assumption of I being squarefree from Proposition 3.1. Example 3.3. Consider the ideal I ⊂ S = k[x, y, z] generated by a := x2 , b := xy and c := xz. The algebraic Scarf complex of this ideal coincides with its Taylor resolution T. So if I were squarefree, then the DGA structure on T described above were the only possible multiplication. However, we can modify the multiplication by setting gb ∗ gc := zgab − ygac . The Leibniz rule requires that in addition we set gb ∗ gac := gc ∗ gab := 0. See Table 1 for the full multiplication table of this new multiplication. It is even a DGA structure, again because the projective dimension of S/I is less then four. 8 LUKAS KATTHÄN ∗ ga gb gc gab gac gbc gabc ga gb gc 0 xgab xgac −xgab 0 zgab − ygac −xgac −zgab + ygac 0 0 0 0 0 0 0 xgabc 0 0 0 0 0 Table 1. The multiplication table gab gac gbc gabc 0 0 xgabc 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 of Example 3.3. 3.2. Structure. The next result shows that if I is squarefree, then F is essentially generated in degree 1 for any multiplication on it: Proposition 3.4. Let I ⊂ S be a squarefree monomial ideal with minimal free resolution F. Let ∗ be a multiplication on F. Then every homogeneous element f ∈ F of squarefree degree can be written as f= X |g1,j ∗ (g2,j ∗ (· · · (g|f |−1,j ∗ g|f |,j ) · · · )|sqf , j for gℓ,j ∈ F1 for 1 ≤ ℓ ≤ |f | and all j. Proof. Let g1,1 ∈ F1 be a basis element with deg g1,1 ≤ f . Such an element exists, because f is a linear combination of basis elements, and the degree of each basis element is an lcm of degrees of basis elements of F1 . Note that deg g1,1 ≤ deg f implies that f = |∂(g1,1 ) ∗ f |sqf . Hence it holds that r := f − |g1,1 ∗ ∂f |sqf = |∂(g1,1 ∗ f )|sqf = ∂(|g1,1 ∗ f |sqf ) ∈ mF. In the last step we used that F is minimal, so ∂F ⊂ mF. Now, ∂f has a smaller homological degree than f , so it can be decomposed into a product by induction. Moreover, note that r ∈ mF implies that r is a S-linear combination of elements of Fi of strictly smaller degree than f . So these elements can be decomposed by induction as well. To conclude that f can be decomposed as claimed, we note the following fact: Whenever for f ′ ∈ F and a monomial m ∈ S it holds that m|f ′ |sqf is squarefree, then it follows that m|f ′ |sqf = |m(|f ′ |sqf )|sqf = |mf ′ |sqf .  The preceding result does not hold if I not squarefree. Example 3.5. Consider the multiplication constructed in Example 3.3. Under that multiplication, gbc cannot be written as a product of elements of degree one. We will see in Section 6 below that every monomial ideal admits at least one multiplication which satisfies the conclusion of Proposition 3.4. This is no longer true for more general ideals. Indeed, if I ⊂ S is a homogeneous ideal with three generators satisfying the conclusion of Proposition 3.4, then the third Betti number is bounded by the number of non-associative, graded-commutative products of three elements. On the other hand, it is a celebrated result of Bruns [Bru76] that any free resolution can be modified to yield a resolution of an ideal with three generators, so there does not exist a global bound on the third Betti number. The preceding proposition leads to the following structure theorem for DGA structures on F: STRUCTURE OF DGA STRUCTURES 9 Theorem 3.6. Let I ⊂ S be a squarefree monomial ideal. Assume that the minimal resolution F of S/I is a multigraded DGA. Then there exists a DG-ideal J ⊂ T in the Taylor resolution of S/I, such that ∼ F = T/J as multigraded DGAs over S. −1 Proof. Let Q := S[x−1 1 , . . . , xn ] denote the Laurent polynomial ring. As T ⊗S Q is an exterior algebra over Q, we can define a map ϕ : T ⊗S Q → F ⊗S Q of DGAs by setting ϕ(gm ) := gm , where the latter is interpreted as element of the algebraic Scarf complex of S/I. We need to show that ϕ restricts to a map ϕ : T → F, where we consider T and F as subalgebras of T ⊗ Q and F ⊗ Q in the natural way. For this, consider a subset A = {m1 < m2 < . . . < ms } ⊆ G(I). It holds that m1 ∨ · · · ∨ ms gA = gm1 ∗ gm2 ∗ · · · ∗ gms , m1 · · · ms and, consequently, that m1 ∨ · · · ∨ ms ϕ1 (gm1 ) ∗ ϕ1 (gm2 ) · · · ϕ(gms ) ϕ(gA ) = m1 · · · ms = |ϕ1 (gm1 ) ∗ ϕ1 (gm2 ) · · · ϕ(gms )|sqf . The last expression is clearly contained in F, and as T is generated by the elements (gA )A⊆G(I) it follows that ϕ maps T to F. It remains to show that the restriction of ϕ to T is surjective onto F. But this is clear from Proposition 3.4, because ϕ is surjective in homological degree 1 and it commutes with taking the squarefree part (the latter holds for any homogeneous morphism).  The following example shows that one cannot omit the assumption of squarefreeness in the preceding Theorem. Example 3.7. We continue with Example 3.3 and consider the DGAs structure on F constructed in that example. Let ϕ : T ⊗ Q → F ⊗ Q the map from the proof of Theorem 3.6. Then 1 1 z y ϕ(gbc ) = ϕ( gb ∗ gc ) = ϕ(gb ) ∗ ϕ(gc ) = − gab + gac , x x x x which is not contained in F. So ϕ does not restrict to a map T → F in this case. In fact, F is not the image of T under any map of DGAs, because F ⊗ Q is not generated in homological degree 1. The next example illustrates how one can use Theorem 3.6 to prove that a given ideal does not admit a (multigraded) minimal DGA resolution. Example 3.8. Let I ⊆ S = Q[x1 , x2 , x3 , x4 , x5 , x6 ] be the ideal generated by a := x1 x2 , b := x2 x3 , c := x3 x4 , d := x4 x5 and e := x5 x6 . This is a well-known example by Avramov which does not admit a minimal DGA resolution [Avr81, Example I]. Assume for the contrary that its minimal free resolution F admits a DGA structure. Then by Theorem 3.6 there exists a DG-ideal J ⊂ T in its Taylor S resolution such that F = T/J. We have that β3,(1,1,1,1,0,0) (S/I) = 0 and hence gabc ∈ J. As J is a DG-ideal, it also holds that (∂gabc ) ∗ ge ∈ J. Moreover, gabc ∗ gd = x3 gabcd ∈ J. As T/J is free, J is saturated with respect to the 10 LUKAS KATTHÄN S variables and hence also gabcd ∈ J. Similarly, β3,(0,0,1,1,1,1) (S/I) = 0 implies that ga ∗ (∂gcde ) ∈ J and gbcde ∈ J. It follows that f := (∂gabc ) ∗ ge − ga ∗ (∂gcde ) − x1 ∂gbcde − x4 ∂gabcd = x2 x23 gabe − x22 x3 gade + x1 x2 x3 gbde − x2 x3 x4 gabd is contained in J. On the other hand, the sets {a, b, e}, {a, d, e}, {b, d, e} and {a, b, d} lie in the Scarf complex of I, so the images of the corresponding generators of T are part of a basis of F. But this is a contradiction, because f is a relation between these elements. 4. Consequences of the existence 4.1. Betti numbers. In this section, we prove the following characterization of the possible Betti vectors of squarefree monomial ideals admitting minimal DGA resolutions. Theorem 4.1. Let f = (1, f1 , f2 , . . . ) ∈ Nν be a finite sequence of natural numbers. Then the following conditions are equivalent: (1) There exists a squarefree monomial ideal I in some polynomial ring S whose minimal free resolution is a DGA, such that βiS (S/I) = fi for all i. (2) f is the f -vector of a simplicial complex ∆ which is a cone. Recall that the f -vectors of simplicial complexes are characterized by the Kruskal-Katona theorem [Sta96, Theorem 2.1]. From this one can derive an explicit characterization of the f -vectors of cones, see also [Kal85]. For the implication “1) ⇒ 2)” we use the following general result. L Proposition 4.2. Let A = k≥0 Ak be a DGA over a field k. Assume that A0 = k, that A is generated in degree 1, that dimk A1 ≤ ∞ and finally that the differential of A is not identically zero. Then the Hilbert function of A equals the f -vector of a simplicial complex which is a cone. Aramova, Herzog and Hibi [AHH97, Theorem 4.1] characterize the Hilbert functions of graded-commutative algebras which are generated in degree one. These are indeed the same as f -vectors of simplicial complexes. Our result extends this, as we show showing that the existence of a non-zero differential yields a further restriction to the Hilbert series. Proof. Let C ⊂ A be the subalgebra of cycles. We claim that C is also generated in degree 1 and that A ∼ = C ⊕ C[−1] as k-vector space, where C[−1] denotes the vector space C with degrees shifted by one. These two claims are sufficient to prove our result. Indeed, by the above mentioned [AHH97, Theorem 4.1], the Hilbert function of C is the f -vector of some simplicial complex ∆. An elementary computation shows that in this case the Hilbert function of A ∼ = C ⊕ C[−1] is the f -vector of the cone over ∆. Now we turn to the proof of our claims. Let C1 ⊂ A1 be the space of cycles in A1 and write C ′ for the algebra generated by them. By assumption, the differential ∂ of A does not vanish identically. As A is generated in degree one, the Leibniz rule implies that the first component ∂1 : A1 → A0 = k is nonzero, hence there exists an element f ∈ A1 with ∂f = 1 and (thus) A1 = C1 ⊕ kf . The latter implies that A ∼ = C ′ + f C ′. STRUCTURE OF DGA STRUCTURES 11 Hence any element a ∈ A can be written as a = c1 + f c2 with c1 , c2 ∈ C ′ . In particular, if a ∈ C, then 0 = ∂a = c2 , so a = c1 ∈ C ′ and thus C = C ′ is generated in degree one. For the second claim, we first note that A ∼ = C ⊕ f C. Indeed, for any f c ∈ C ′ ∩ ′ f C it holds that 0 = ∂(f c) = c. Finally, it remains to show that f C ∼ = C(−1). For this it is sufficient to show that the multiplication by f is injective on C. But this is clear, because ∂(f c) = c for any c ∈ C, so the differential provides a left-inverse.  Proof of Theorem 4.1. 1) ⇒ 2) Let Q be the field of fractions of S and let F the minimal free resolution of I. It follows from Proposition 3.4 that F ⊗S Q is generated in degree 1 as Q-algebra. Hence the claim is immediate from Proposition 4.2, applied to F ⊗S Q. 2) ⇒ 1) Set k := f1 . Let ∆ ⊂ 2k be a simplicial complex whose f -vector equals f which is a cone with apex 1. We can find a monomial ideal I in some polynomial ring S, such that the lcm-lattice of I equals the face lattice of ∆, augmented by a maximal element (cf. [IKMF14, Theorem 3.4], [Map13]). Then ∆ is the Scarf complex of I. As ∆ is acyclic, its f -vector equals the Betti vector of I (this can be seen by computing the Betti numbers via the lcm lattice, see [GPW99, Theorem 2.1]). It remains to show that the minimal free resolution of I admits a DGA structure. Let T be Taylor resolution of I. We identify the set of generators of I with the set [k] of vertices of ∆. To construct the minimal free resolution of S/I we use discrete Morse theory [JW09; Skö06]. Let us recall the relevant aspects of this technique. Let G be the directed graph with vertex set 2[k] and edges from V to W whenever the coefficient of gV in the expansion of ∂gW is nonzero. A Morse matching M is a collection of edges (V, W ) in this graph with the following properties: (a) M is a matching, i.e. the edges are disjoint, (b) reversing all edges of M in G results in an acyclic graph, and (c) deg gV = deg gW for all (V, W ) ∈ M. Given such a Morse matching, define JM := spank {gW , ∂gW : ∃V ∈ 2[k] , (V, W ) ∈ M}. Then the complex TM := T/JM is a free resolution of S/I, and it has a k-basis is given by the images of gW for those W ∈ 2[k] which do not appear in the matching. In our situation we define the Morse matching: M := {(gW \{1} , gW ) : W ⊂ [k], W ∈ / ∆, 1 ∈ W } Recall that we assumed 1 to be the apex of ∆, so W ∈ / ∆ and 1 ∈ W imply that W \ {1} ∈ / ∆. This is clearly a matching, and it is easy to see that it satisfies property (b). For property (c), recall that the lcm lattice LI of I is ∆, together with an additional maximal element. We are only matching generators which are not in ∆, so the degrees of all those generators correspond to the maximal element of LI , and thus they are equal. Finally, the unmatched generators correspond exactly to the faces of ∆. As we know that the Betti vector of I equals the f -vector of ∆, we can conclude 12 LUKAS KATTHÄN that TM := T/JM is a minimal free resolution of S/I. On the other hand, by our choice of the matching M, JM is in fact a DG-ideal and hence TM is a DG-algebra.  We give an example to show that the assumptions of I being squarefree and admitting a minimal DGA resolution are both necessary in Theorem 4.1. Example 4.3. Consider the ideal I := hx1 x2 , x2 x3 , x3 x4 , x4 x5 , x5 x6 , x6 x1 i ⊂ Q[x1 , . . . , x6 ]. Using Macaulay2 [GS], one can compute that its total Betti numbers are (1, 6, 9, 6, 2). We claim that this is not the f -vector of any simplicial complex. Indeed, such a simplicial complex would have two 3-simplices. As each 3-simplex has three 2-faces, and the two 3-simplices can only share one 2-face, we would need a total of at least seven 2-faces, but we have only six. On the other hand, by Theorem 2.1 there exists a monomial m ∈ S such that mI admits a minimal DGA resolution. However, mI is not squarefree, and indeed its Betti vector still does not satisfy the conclusion of Theorem 4.1, as it conicides with the Betti vector of I. 4.2. Subadditivity of syzygies. For a monomial ideal I ⊂ S and 0 ≤ i ≤ pdim S/I we define S ti := max{j : βi,j (S/I) 6= 0}. We say that the syzygies of I are subadditive when tb ≤ ta + tb−a for all a, b such that 1 ≤ a < b ≤ pdim S/I. Not every ideal has this property [ACI15], but among monomial ideals no counterexample is known. Here, we show that for squarefree monomial ideals, the existence of a DGA structure on the minimal free resolution implies the subadditivity of syzygies. Proposition 4.4. If I ⊂ S is a squarefree monomial ideal which admits a minimal DGA resolution F, then its syzygies are subadditive. The following lemma is needed for the proof of this result. Lemma 4.5. Let f, g ∈ F be homogeneous. If |f ∗ g|sqf ∈ / mF, then there exist f ′ , g ′ ∈ F \ mF of the same homological degrees as f and g, respectively, such that deg(f ) ∨ deg(g) = deg(f ′ ) ∨ deg(g ′ ). P P Proof. By choosing a basis of F, we may write f = i λi ci fi and g = j µj dj gj , with λi , µi ∈ k, ci , dj ∈ S monomials and fi , gj ∈ F \ mF. By our assumption, P / mF. So at least one summand is not contained |f ∗g|sqf = i,j λi µj |cidj fi ∗gj |sqf ∈ in mF, say |c1 d1 f1 ∗ g1 |sqf . It follows easily from the definition of the squarefree part that there exists a monomial m ∈ S such that |c1 d1 f1 ∗ g1 |sqf = m|f1 ∗ g1 |sqf . But this does not lie in mF, so we can conclude that m = 1 and the claim follows with f ′ := f1 and g ′ := g1 .  Proof of Proposition 4.4. Let f ∈ Fb be an element of total degree tb and we may assume that f ∈ / mF. By Proposition 3.4, we can write f as f= X |g1,j ∗ (g2,j ∗ (· · · (gi−1,j ∗ gi,j ) · · · )|sqf , j for gℓ,j ∈ F1 for 1 ≤ ℓ ≤ i and all j. As f ∈ / mF, the same holds for at least one summand, say for j = 1. STRUCTURE OF DGA STRUCTURES 13 Using that the multiplication on F is associative, we may rewrite that summand as |(g1,1 · · · ga,1 ) ∗ (ga+1,1 · · · gb,1)|sqf . Now we apply Lemma 4.5 to this product to obtain f ′ ∈ Fa and g ′ ∈ Fb−a which are both not contained in mF and satisfy deg f = deg(f ′ ) ∨ deg(g ′). Thus we conclude that tb = deg(f ) = deg(f ′ ) ∨ deg(g ′ ) ≤ deg(f ′ ) + deg(g ′) ≤ ta + tb−a  If we do not assume that F admits a DGA structure, then our methods still suffice to show the case a = 1. This was first shown by Herzog and Srinivasan in [HS15, Corollary 4] using a different method. Corollary 4.6. For a monomial ideal I ⊂ S, it holds that ti ≤ t1 + ti−1 for all 2 ≤ i ≤ pdim S/I. Proof. We may replace I by its polarization and so assume it is squarefree. Further, choose any multiplication on F. Now we just note that the proof of Proposition 4.4 does not require the multiplication to be associative if a = 1.  5. Strongly generic ideals A monomial ideal is called strongly generic if no variable appears with the same nonzero exponent in two distinct generators of I. This class of ideals was introduced and studied by Bayer, Peeva and Sturmfels [BPS98] 1, see also Chapter 6 of [MS05]. Strongly generic ideals have a number of desirable properties, and in particular their minimal free resolution is given by the algebraic Scarf complex [BPS98, Theorem 3.2]. However, this condition is not sufficient to ensure that the ideal admits a minimal DGA resolution. Theorem 5.1. The ideal hx2 , xy, y 2z 2 , zw, w 2 i ⊂ k[x, y, z, w] is strongly generic, but its minimal free resolution does not admit the structure of a DGA, whose multiplication respects the standard Z-grading. This example is a variation of Example 3.8. The proof of Theorem 5.1 is very technical, so we postpone it and discuss the result first. Remark 5.2. Corollary 3.6 in [BPS98] states that strongly generic monomial ideals admit a minimal DGA resolution. I was informed by I. Peeva that the authors of that article are aware that the claim does not hold as stated and that the statement and the proof were supposed to include an additional combinatorial condition. Theorem 5.1 implies that various kinds of resolutions do not admit a DGA structure in general. Corollary 5.3. (1) There exists a monomial ideal whose hull resolution [BS98] does not admit a DGA structure. (2) There exists a monomial ideal whose Lyubeznik resolution [Lyu88] does not admit a DGA structure. 1In [BPS98] these ideals are called generic. In the later article [MSY00] a weaker notion of genericity was introduced and the original one is called strongly generic since then. 14 LUKAS KATTHÄN (3) There exists a monomial ideal whose minimal free resolution is supported on a simplicial complex (and in particular cellular) [BS98], but it does not admit a DGA structure. Proof. The ideal of Theorem 5.1 provides the counterexample in all three cases. This ideal is generic, so its minimal free resolution is supported in its Scarf complex and it coincides with its hull resolution [MS05, Theorem 6.13]. The Lyubeznik resolution depends on the choice of a total order on the generators. One can check that the Lyubeznik resolution with respect to the order xy ≺ zw ≺ x2 ≺ y 2z 2 ≺ w 2 is minimal.  In fact, the minimal free resolution of Avramov’s Example 3.8 can also be given the structure of a cellular resolution, though not with a simplicial complex as support. Proof of Theorem 5.1. In this proof we use both the multigrading and the standard Z-grading, so we refer to the latter as the total degree. Let S = k[x, y, z, w] and let I be the ideal of the statement. We give names to its generators as follows: a := x2 , b := xy, c := y 2z 2 , d := zw, e := w 2 The Scarf complex ∆I is the simplicial complex with the facets {b, c, d} and {a, b, d, e}. As mentioned above, the minimal free resolution F of S/I is given by its algebraic Scarf complex. The generators of F are listed in Table 2, together with their total degrees. Assume that F admits an associative multiplication which respects the Zgrading, which we denote by ∗. The map ∗ : F ⊗ F → F can be decomposed into its homogeneous components with respect to the Z4 -grading. We write × for the component of multidegree (0, 0, 0, 0). Because the differential on F respects the multigrading, × also satisfies the Leibniz rule and lifts the multiplication on S/I. Hence, by the uniqueness of the multiplication, there exists a map σ : F ⊗ F → F[1] such that a ∗ b = a × b + δσ(a, b) for a, b ∈ F, where δσ = ∂ ◦ σ + σ ◦ ∂. Moreover, the component of σ in multidegree (0, 0, 0, 0) vanishes, because ∗ and × coincide in this degree. We proceed by proving a series of claims. Claim 1. For i, j, k ∈ {a, b, d, e}, all products of the form gi × gj and gi × gjk coincide with the corresponding products from the Taylor resolution. If I were squarefree, then this would follow from Proposition 3.1. However, as I is not squarefree, we need to verify this. We start in homological degree 2. For any i, j ∈ {a, b, d, e}, i 6= j, one can verify that the product gi × gj is a multiple of gcd(i, j)gij , because there is no other element of the correct multidegree. Then the Leibniz rule implies that gi × gj = ± gcd(i, j)gij . Moreover gi × gi = 0, because ∗ and thus × was assumed to be graded-commutative. Next, gi ×gij = 0 for all i, j because it is clearly a cycle, and the only boundary of homological degree 3 is ∂gabde , which has a different multidegree. Moreover, for pairwise distinct i, j, k ∈ {a, b, d, e}, it follows again by inspection that gi × gjk is a multiple of gcd(i, lcm(j, k))gijk because this is the only element with the STRUCTURE OF DGA STRUCTURES ga , gb , gd , ge gc gab , gde gad , gae , gbd , gbe gbc , gcd gijk 6= gbcd gbcd gabde Table 2. The degrees of the 15 2 3 3 4 5 5 6 6 generators of F. required multidegree, and again the Leibniz rule determines the coefficient to be ±1. In fact, the other products in the subalgebra generated by ga , gb , gd and gd also coincide with the product from the Taylor resolution, but we do not need this. Claim 2. It holds that (ga × gc ) × ge − ga × (gc × ge ) = yz∂gabde By considering the multigrading and using the Leibnitz rule as above, one can show that ga × gc = yz 2 gab + xgbc and gc × ge = wgcd + y 2 zgde . Using this and Claim 1, one can compute that ∂(gbc × ge ) = ∂(wgbcd + yzgbde ). This implies that gbc × ge = wgbcd + yzgbde + ∂(f ) for some f ∈ F4 . But the x-component of the multidegree of every element in F4 is at least 2, hence f = 0. Now we can compute that (ga × gc ) × ge = yz 2 gab × ge + xgbc × ge = yz 2 gabe + xwgbcd + xyzgbde . Using essentially the same reasoning one can also show that ga × (gc × ge ) = y 2zgade + xwgbcd + yzwgabd . Hence it follows that (ga × gc ) × ge − ga × (gc × ge ) = yz∂gabde . Claim 3. For any f ∈ F it holds that σ(1k , f ) = σ(f, 1k ) = 0. For f ∈ F it holds that f = 1k ∗ f = 1k × f + ∂σ(1k , f ) + σ(∂1k , f ) +σ(1k , ∂f ) | {z =0 } and hence ∂σ(1k , f ) + σ(1k , ∂f ) = 0. We proceed now by induction on |f |. Consider the case |f | = 0, i.e f = 1k . Then σ(1k , 1k ) has multidegree (0, 0, 0, 0) and homological degree 1, which is only possible if σ(1k , 1k ) = 0. Assume now that |f | = i and σ(1k , g) = 0 for all g ∈ F with |g| < i. Then ∂σ(1k , f ) = −σ(1k , ∂f ) = 0 by the induction hypothesis. As F is acyclic, there exist an h ∈ F with |h| = |f | + 2 such that ∂h = σ(1k , f ). We may assume that f is a generator of F. Now, inspection of Table 2 shows that for no generator f of F there exist an element h with deg h = deg f and |h| = |f | + 2. Hence σ(1k , f ) = 0. Claim 4. It holds that σ(gi , gj ) = 0 for i, j 6= c. 16 LUKAS KATTHÄN If σ(gi , gj ) 6= 0 for i, j 6= c, then it is an element of total degree 4 and homological degree 3. But by Table 2, no such element exists. Claim 5. It holds that gi ∗ gj = gi × gj for i, j 6= c. In general we have that gi ∗ gj = gi × gj + ∂σ(gi , gj ) + σ(∂gi , gj ) + σ(gi , ∂gj ). All terms involving σ vanish because of Claim 3 and Claim 4. Claim 6. For i, j, k ∈ {a, b, d, e}, it holds that σ(gi , gjk ) = 0. First, assume that i = j or i = k. We only consider the case i = j, the other case is similar. For this, we compute that Cl. 5 Cl. 1 0 = (gi ∗ gi ) ∗ gj = gi ∗ (gi ∗ gj ) = gi ∗ (gi × gj ) = ±µgi ∗ gij Cl. 3 = ±µ(gi × gij +δσ(gi , gij )) = ±µ∂σ(gi , gij ), | {z =0 } where µ = gcd(i, j). The differential is injective in homological degree 4, hence ∂σ(gi , gij ) = 0 implies that σ(gi , gij ) = 0. Now we consider the case that i, j, k are all different. Then {i, j, k} contains either a and b, or d and e. We only consider the first case, the other one is similar. If {j, k} = {a, b} then σ(gi , gab ) = 0 holds for degree reasons, see Table 2. So assume that i ∈ {a, b}. Again, we assume without loss of generality that i = a. Then either j or k equals b, say j. We compute that Cl. 5 (ga ∗ gb ) ∗ gk = (ga × gb ) ∗ gk = xgab ∗ gk = xgab × gk + xδσ(gab , gk ) (∗) = xgabk + xδσ(gab , gk ) Cl. 4 = xgabk + x∂ σ(gab , gk ), | {z =0 } where in (∗) we used Claim 1 and that gcd(x, k) = 1 for k ∈ {d, e}. On the other hand, a similar computation shows that ga ∗ (gb ∗ gk ) = xgabk + ∂σ(ga , gbk ), where we use that gcd(b, k) = 1. The assumption that ∗ is associative implies that ∂σ(ga , gbk ) = 0 and thus σ(ga , gbk ) because ∂ is injective in homological degree 4. Claim 7. It holds that ga ∗ (gc ∗ ge ) − (ga ∗ gc ) ∗ ge 6= 0. Let us compute the associator of ga , gc and ge : ga ∗ (gc ∗ ge ) − (ga ∗ gc ) ∗ ge = ga × (gc × ge ) − (ga × gc ) × ge + δσ(ga , gc × ge ) − δσ(ga × gc , ge )+ ga × δσ(gc , ge ) − δσ(ga , gc ) × ge + δσ(ga , δσ(gc , ge )) − δσ(δσ(ga , gc ), ge ) We know that ga × (gc × ge ) − (ga × gc ) × ge = yz∂gabde and we want to show that this term does not cancel against any other term. For this we need to consider the component of ga ∗ (gc ∗ ge ) − (ga ∗ gc ) ∗ ge in multidegree (0, 0, 0, 0). As σ has no component in this multidegree, the summands having only one STRUCTURE OF DGA STRUCTURES 17 σ cannot contribute. This it remains to consider the last two terms, namely δσ(ga , δσ(gc , ge )) and δσ(δσ(ga , gc ), ge ). By Claim 3 it holds that δσ(ga , δσ(gc , ge )) = ∂σ(ga , ∂σ(gc , ge )). Further, σ(gc , ge ) has homological degree 3, hence it is of the form σ(gc , ge ) = λgbcd + r where λ ∈ S and r is a sum of terms not involving c. Now it holds that ∂σ(gc , ge ) =λ∂gbcd + ∂r =λxgcd + λwgbc + terms not involving c Using Claim 6 it follows that ∂σ(ga , ∂σ(gc , ge )) = λx∂σ(ga , gcd ) + λw∂σ(ga , gbc ) = λ(λ1 x + λ2 w)∂gabde , for some λ1 , λ2 ∈ S. Here we used that gabde is the only generator in homological degree 4. These terms cannot cancel against yz∂gabde , because they are multiples of x or w, respectively. The same arguments apply to δσ(δσ(ga , gc ), ge ).  Remark 5.4. Unfortunately, it does not seem possible to use Avramov’s obstructions from [Avr81] to prove the preceding theorem. At least, I could not find a suitable regular sequence f1 , . . . , fr in the ideal to consider R = S/(f1 , . . . , fs ). 6. The lcm lattice and non-sqarefree ideals The lcm lattice LI of a monomial ideal is the lattice of all least common multiples of subsets of the minimal generators of I. The lcm lattice was introduced in [GPW99], were it is also proven that its isomorphism type determines the minimal free resolution of I. So one might be led to conjecture that it also determines the possible multiplication on F. However, it is immediately clear from Theorem 2.1 that is not true. Nevertheless, we can identify a class of multiplications which is compatible with the lcm lattice: Definition 6.1. A multiplication ∗ on F is called supportive if for all a, b ∈ F there exists a c ∈ F with deg c ≤ (deg a) ∨ (deg b), such that a ∗ b = mc for a monomial m ∈ S. Example 6.2. The DGA structure on the Taylor resolution is supportive, but the multiplication constructed in Example 3.3 is not supportive. For squarefree monomial ideals, being supportive is not a restriction. Lemma 6.3. If I ⊆ S is a squarefree monomial ideal, then every multiplication on its minimal free resolution is supportive. Proof. Let F be the minimal free resolution of S/I and let B be an S-basis for it. Every element of B has a squarefree degree, we may apply Lemma 1.4. It follows that for any g1 , g2 ∈ B it holds that g1 ∗ g2 = mh for a monomial m and an element h ∈ F with deg h ≤ (deg g1 + deg g2 ) ∧ (1, . . . , 1) = deg g1 ∨ deg g2 .  The next result shows that supportive multiplications are essentially determined by the isomorphism type of the lcm-lattice of I. This is our main motivation for introducing this notion. Proposition 6.4. Let I and I ′ be two monomial ideals in two polynomial rings S and S ′ , respectively, whose lcm-lattices are isomorphic. Then every supportive multiplication on the minimal free resolution of S/I can be relabeled to a supportive multiplication on the minimal free resolution of S ′ /I ′ . 18 LUKAS KATTHÄN Proof. Let ν : LI → LI ′ be the isomorphism. We recall the “relabeling” construction, which was introduced in [GPW99]. Fix a basis B of the minimal free resolution F of S/I. We express the differential ∂ of F in this basis: ∂g = X chg xdeg(g)−deg(h) h h∈B with g ∈ B, chg ∈ k. The relabeled resolution ν(F) is the free S ′ -module with the basis {ḡ : g ∈ B} and deg ḡ = ν(deg g). Here, by ḡ we mean a new basis element. The differential on ν(F) is given by X h deg(ḡ)−deg(h̄) ¯ := ∂ḡ c x h̄. g h∈B This is well-defined because deg h ≤ deg g implies that deg h̄ = ν(deg h) ≤ ν(deg g) = deg ḡ. By [GPW99, Theorem 3.3], ν(F) is indeed a minimal free resolution of S ′ /I ′ . Our definition the relabeling of the multiplication is analogous: If X g1 ∗ g2 = chg1 ,g2 xdeg(g1 )+deg(g2 )−deg(h) h, h∈B then we set ḡ1 ∗ ḡ2 := X chg1 ,g2 xdeg(ḡ1 )+deg(ḡ2 )−deg(h̄) h̄. h∈B To see that this is well-defined, we compute that deg ḡ1 + deg ḡ2 ≥ deg ḡ1 ∨ deg ḡ2 = ν(deg g1 ) ∨ ν(deg g2 ) (1) (∗∗) (∗) = ν(deg g1 ∨ deg g2 ) ≥ ν(deg h) = deg h̄. Here, the equality marked with (∗) holds because ν is an isomorphism of lattices. Moreover, the equality marked with (∗∗) holds because the multiplication is supportive. It is clear that the relabeled multiplication is graded-commutative and satisfies the Leibniz rule, because the structure constants are the same as for the multiplication on F. Finally, the computation (1) also shows that it is supportive.  The last two results have a number of immediate corollaries. Corollary 6.5. Every monomial ideal admits a supportive multiplication on its minimal free resolution. Proof. Choose any multiplication on the minimal free resolution of the polarization of I and relabel it.  Corollary 6.6. Any supportive multiplication satisfies the conclusion of Proposition 3.4. Proof. The conclusion of Proposition 3.4 is clearly invariant under relabeling. Hence relabeling the supportive multiplication to the minimal free resolution of the polarization of I yields the claim.  Corollary 6.7. Let I and I ′ be two monomial ideals with isomorphic lcm lattices and assume that I is squarefree. If I admits a minimal DGA resolution, then so does I ′ . Proof. Associativity is invariant under relabeling.  REFERENCES 19 We close the section by giving an example which illustrates that the last corollary cannot be strengthened by considering the Betti poset [CM14a; TV15; CM14b]. Recall that the Betti poset of a monomial ideal is the subposet of the lcm lattice consisting of those multidegrees in which are nonzero Betti numbers. Example 6.8. 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arXiv:1801.02329v2 [] 29 Jan 2018 Grassmannian Codes with New Distance Measures for Network Coding Tuvi Etzion Hui Zhang Dept. of Computer Science Technion-Israel Institute of Technology Haifa 3200003, Israel Email: [email protected] Dept. of Computer Science Technion-Israel Institute of Technology Haifa 3200003, Israel Email: [email protected] Abstract—Subspace codes are known to be useful in errorcorrection for random network coding. Recently, they were used to prove that vector network codes outperform scalar linear network codes, on multicast networks, with respect to the alphabet size. In both cases, the subspace distance is used as the distance measure. In this work we show that we can replace the subspace distance with two other possible distance measures which generalize the subspace distance. We prove that each code with the largest number of codewords and the generalized distance, given the other parameters, has the minimum requirements needed to solve a given multicast network with a scalar linear code. We discuss lower and upper bounds on the sizes of the related codes. I. I NTRODUCTION Network coding has been attracting increasing attention in the last fifteen years. The seminal work of Ahlswede, Cai, Li, and Yeung [1] and Li, Yeung, and Cai [15] introduce the basic concepts of network coding and how network coding outperforms the well-known routing. The class of networks which were mainly studied are the multicast networks and these are also the target of this work. A multicast network is a directed acyclic graph with one source. The source has h messages, which are scalars over a finite field Fq . The network has N receivers, each one demands all the h messages of the source to be transmitted in one round of a network use. An up-to-date survey on network coding for multicast networks can be found for example in [10]. Kötter and Médard [16] provided an algebraic formulation for the network coding problem: for a given network, find coding coefficients (over a small field) for each edge, which are multiplied with the symbols received at the starting node of the edge, such that each receiver can recover all the demanded information from its received symbols on its in-coming edges. This sequence of coding coefficients at each edge is called the local coding vector. Such an assignment of coding coefficients for all the edges in the network is called a solution for the network. The coding coefficients are scalars and the solution is a scalar linear solution. Ebrahimi and Fragouli [4] have extended this algebraic approach to vector network coding. In this setting the messages are vectors of length t over Fq and the coding coefficients are t×t matrices over Fq . A set of coding matrices such that all the receivers can recover their requested information, is called a vector solution. The alphabet size of the solution is an important parameter that directly influences the complexity of the calculations at the network nodes and as a consequence the performance of the network. Jaggi et al. [14] have shown a deterministic algorithm to find a network code (for multicast networks) of a field size which is the first prime power greater or equal to the number of receivers. In general, finding the minimum required alphabet size of a network code for a given multicast network is NPcomplete [19]. Vector network coding solution with vectors of length t over Fq outperforms the scalar linear network coding solution for the same network if the scalar solution requires an alphabet of size qs , where qs > q t . An important step in the evolution of network coding was the introduction of random network coding [12], [13]. Instead of the deterministic algorithm to design the network code, there is a random selection of coefficients for the local coding vectors at each coding point (taken from Fq ). By choosing a field size large enough, the probability that this random selection won’t be a solution for the network tends to zero. Kötter and Kschischang [17] introduced a framework for error-correction in random network coding. They have shown that for this purpose the codewords are taken as subspaces over a finite field Fq , where the distance measure, called the subspace distance is defined as follows. The set of all subspaces of Fnq is called the projective space and denoted by Pq (n). For two subspaces X, Y ∈ Pq (n) the subspace distance dS (X, Y ) is defined by dS (X, Y ) = dim X + dim Y − 2 dim(X ∩ Y ) . It was shown later in [22] that another metric called the injection metric is better suitable for random network coding. For two subspaces X, Y ∈ Pq (n) the injection distance dI (X, Y ) is defined by dI (X, Y ) = max{dim X, dim Y } − dim(X ∩ Y ) . These two papers [17], [22] have motivated an extensive work on subspace codes and in particular on codes in which all the subspaces have the same dimension. These codes are called constant dimension codes or Grassmannian codes since they belong to the Grassmann space. For a given positive integer n and a given integer k, 0 ≤ k ≤ n, the Grassmannian Gq (n, k) is the set of all subspaces in Pq (n) whose dimension is k. The two distance measures coincide for this family of subspaces or more precisely, if X, Y ∈ Gq (n, k) then dS (X, Y ) = 2dI (X, Y ). Since we concentrate on Grassmannian codes in this paper we will define the Grassmannian distance dG (X, Y ) to be dG (X, Y ) = k − dim(X ∩ Y ) . Most of the research on subspace codes motivated by [17] was in two directions – find the largest codes with prescribed minimum subspace distance and looking for designs based on subspaces. To this end, two quantities were defined. The first one Aq (n, d) is the maximum size of a code in Pq (n) with minimum subspace distance d. The second one Aq (n, d, k) is the maximum size of a code in Gq (n, k) with minimum subspace distance d (Grassmannian distance d/2). A related concept is a subspace design or a block design t-(n, k, λ)q which is a collection S of k-subspaces from Gq (n, k) (called blocks) such that each subspace of Gq (n, t) is contained in exactly λ blocks of S. In particular if λ = 1 this subspace design is called a qSteiner system and is denoted by Sq (t, k, n). Note, that such a q-Steiner system is a Grassmannian code in Gq (n, k) with minimum Grassmannian distance k − t + 1. It was shown recently [8], [9] that subspace codes are useful in vector network coding and by using subspace codes it can be shown that vector network codes outperform scalar linear network codes, in multicast networks, with respect to the alphabet size. The comparison between the alphabet size for a vector network coding solution is done as follows. For a given network N , if there exists a vector network coding solution with vectors of length t and a finite field of size q and a scalar linear network coding solution requires a finite field of size at least qs , then the gap in the between the two solutions is qs − q t . The proof that vector network coding outperforms scalar linear network coding used a family of networks called the generalized combination networks, where the combination networks were defined and used in [21] (but were defined informally before too). The goal of this work is to show that there is a tight connection between optimal Grassmannian codes and network coding solutions for the generalized combination networks. We will define two new dual distance measures on Grassmannian codes which generalize the Grassmannian distance. We discuss on the maximum sizes of the new Grassmannian codes with the new distance measures. We explore the connection between these codes and related generalized combination networks. Our exposition will derive some interesting properties of these codes with respect to the traditional subspace codes and some subspace designs. We will show, using two different approaches, that codes in the Hamming space with minimum Hamming distance are a special case of subspace codes. Some interesting connections to subspace designs will be also explored. The rest of this paper is organized as follows. In Section II we present the combination network and its generalization which was defined in [8], [9]. We discuss the family of codes which provide network coding solutions for these networks. In Section III we further consider this family of codes, define two dual distance measures on these codes, and show how these codes and the distance measures defined on them generalize the conventional Grassmannian codes and Grassmannian distance. We show the connection of these codes with subspace designs. We prove that for each such code of the largest size, over Fq , there exists a generalized combination network which is solved only by this code (or a code with more relaxed parameters). We also discuss how codes in the Hamming space form a subfamily of these subspace codes. Finally, we show which subfamily of these codes is useful for vector network coding. In Section IV we present basic results for the bounds on sizes of these codes and concentrate on the subfamily of these codes which are useful to show how vector network coding outperform scalar linear network coding. We analyse this family of codes in general and concentrate on one specific code with specific parameters in particular. In Section V we provide a conclusion and directions for future research. II. G ENERALIZED COMBINATION NETWORKS In this section we will define the generalized combination network which is a generalization of the combination network [21]. This network defined in [8], [9] was used to prove that vector network coding outperforms scalar linear network coding with respect to the alphabet size. The generalized combination network is called the ǫ-direct links ℓ-parallel links Nh,r,s network, in short the (ǫ, ℓ)-Nh,r,s network. The network has three layers. In the first layer there is a source with h messages. In the second layer there are r nodes. The source has ℓ parallel links to each node in the middle layer. From any α = s−ǫ ℓ nodes in the middle layer, there are ℓ parallel links to one
receiver in the third layer, i.e. there are αr receivers in the third layer. Additionally, from the source there are ǫ direct
parallel links to each one of the αr receivers in the third layer. Therefore, each receiver has s = αℓ + ǫ incoming links. The (0, 1)-Nh,r,s network is the combination network defined in [21]. We will assume some relations between the parameters h, α, ǫ, and ℓ such that the resulting network is interesting and does not have a trivial or no solution [8]. Theorem 1. The (ǫ, ℓ)-Nh,r,αℓ+ǫ network has a trivial solution if ℓ + ǫ ≥ h, and it has no solution if αℓ + ǫ < h. Otherwise, the network has a nontrivial solution. The immediate natural question is which codes will solve the generalized combination network over Fq . The answer is quite simple. Since each receiver has ǫ direct links from the source, it follows that the source can send any required ǫ-subspace of Fhq to the receiver. Hence, the receiver must be able to obtain an (h−ǫ)-subspace of Fhq from the middle layer nodes connected to it. The receiver is connected to α nodes in the middle layer and each one can send to the receiver an ℓ-subspace of Fhq . Hence, a solution for the network exists if and only if there exists a code with r ℓsubspaces of Fhq , such that each α codewords (ℓ-subspaces) span a subspace whose dimension is at least h − ǫ. III. C OVERING /M ULTIPLE G RASSMANNIAN C ODES In this section we provide the formal definition for the codes required to solve the generalized combination networks. We define two distance measures on these codes, prove that both Grassmannian codes, codes in the Hamming space, and subspace designs, are subfamilies of this family of codes. We present some basic properties of these codes and their connection to the solution of the generalized combination networks. An α-(n, k, δ)cq covering (generalized) Grassmannian code (code in short) C is a subset of Gq (n, k) such that each α codewords of C span a subspace of dimension at least δ + k in Fnq . The following theorem is easily verified. Theorem 2. A code C in Gq (n, k) with Grassmannian distance δ is a 2-(n, k, δ)cq Grassmannian code. Theorem 2 implies that the Grassmannian codes form a subfamily of the α-(n, k, δ)cq codes. It is also natural to define the quantity δ as the minimum α-covering Grassmannian distance of the code, where the 2-covering Grassmannian distance is just the Grassmannian distance. In other words, the α-covering Grassmannian distance of α subspaces in Gq (n, k) is the dimension of the subspace which they span in Fnq minus k. A natural problem is to find maximum size of such codes? For this we define the quantity Bq (n, k, δ; α) to be the maximum size of an α-(n, k, δ)cq code. From our discussion it can be readily verified that if C is an α-(n, k, δ)cq code which attains Bq (n, k, δ; α) then C solves the (ǫ, k)-Nn,r,αk+ǫ network, where ǫ = n − δ − k and r = Bq (n, k, δ; α). But, clearly we also have from our previous discussion that such a code has parameters with the minimum conditions which are necessary to solve such a network. Thus, each such code of the maximum size is exactly what needed to solve a certain instant of the generalized combination networks. Finally, for a given Grassmannian code C in Gq (n, k) we can define covering hierarchy, where the 1-covering is just k, the α-covering is the α-covering Grassmannian distance of C plus k. This hierarchy is a q-analog for the generalized Hamming weights [24] for constant weight codes. The way we have described the code which solves the generalized combination network is very natural when we consider the definition of the generalized combination network. We have also defined a natural generalization for the Grassmannian distance, although some might argue that it is less natural from a point of view of a code definition. Hence, we will give now a more natural definition from a point of view of coding theory, or more precisely from a point of view of a packing, which is the combinatorial property of an error-correcting code. For this we will need to use the dual subspace V ⊥ of a given subspace V in Fnq , and the orthogonal complement of a given code C. For a code C in Pq (n) the orthogonal complement C⊥ is defined def by C⊥ = {V ⊥ : V ∈ C}. It is well-known [7] that the minimum subspace distance of C and C⊥ are equal. The following lemma is also well known (see for example Lemma 12 in [7]). Corollary 1. For any given set of α subspaces V1 , V2 , . . . , Vα of Fnq we have !⊥ α α X \ ⊥ . Vi Vi = i=1 i=1 Corollary 1 induces a new definition of a distance measure for the orthogonal complements of the Grassmannian codes which solve the generalized combination networks. For a Grassmannian code C ∈ Gq (n, k), the minimum λ-multiple Grassmannian distance is k − τ + 1, where τ is the largest integer such that each τ -subspace of Fnq is contained in at most λ k-subspaces of C. Theorem 3. If C ∈ Gq (n, k) is an α-(n, k, δ)cq code then C⊥ has minimum (α − 1)-multiple Grassmannian distance δ. Proof. By the definition of an α-(n, k, δ)cq code it follows that for each α subspaces V1 , V2 , . . . , Vα of C we
have
Pα Pα ⊥ ≤ that dim i=1 Vi i=1 Vi ≥ δ + k and hence dim n −Tδ − k. Therefore, by Corollary 1 we have that α dim i=1 Vi⊥ ≤ n − δ − k. This implies that each subspace of dimension n − δ − k + 1 of Fnq can be contained in at most α − 1 codewords of C⊥ . Thus, since C⊥ ∈ Gq (n, n − k), it follows by definition that the minimum (α − 1)-multiple Grassmannian distance of C⊥ is (n − k) − (n − δ − k + 1) + 1 = δ. Theorem 3 leads to a new definition for Grassmannian codes (orthogonal complements of α-(n, k, δ)cq codes). An s-(n, k, λ)m q multiple (generalized) Grassmannia code (code in short) C is a subset of Gq (n, k) such that each s-subspace of Fnq is contained in at most λ codewords of C. Similarly, let Aq (n, k, s; λ) denote the maximum size of an s-(n, k, λ)m q code. One can verify that Theorem 4. An s-(n, k, 1)m q code is a Grassmannian code in Gq (n, k) with minimum Grassmannian distance k − s + 1. Clearly, by Theorem 4 we have that the λ-multiple Grassmannian distance is a generalization of the Grassmannian distance (1-multiple Grassmannian distance). The generalization implied by Theorem 4 is for the packing interpretation of an s-(n, k, 1)m q code. If each s-subspace is contained exactly once in an s-(n, k, 1)m q code C, then C is a q-Steiner system Sq (s, k, n). If each s-subspace is contained exactly λ times in an s-(n, k, λ)m q code C, then C is an s-(n, k, λ)q subspace design. Similarly to Theorem 3 we have Theorem 5. If C ∈ Gq (n, k) is an s-(n, k, λ)m q code then C⊥ has minimum (λ + 1)-covering Grassmannian distance k − λ + 1. Corollary 2. C ∈ Gq (n, k) is an α-(n, k, δ)cq code if and only if C⊥ is an (α − 1)-(n, n − k, n − k − δ + 1)m q code. Lemma 1. For any two subspaces U, V in Pq (n) we have that U ⊥ ∩ V ⊥ = (U + V )⊥ . Corollary 3. For any feasible s, k, n, and λ, we have that Aq (n, k, s; λ) = Bq (n, n − k, k − λ + 1; λ + 1). Clearly, by induction we have the following consequence from Lemma 1. Not all Grassmannian codes can be used as solutions in vector network coding. If there are h messages and each message is a vector of length t over Fq , then each link will carry a t-subspace of Fht q . Therefore, the only Grassmannian codes which are used in vector network coding are α-(ht, ℓt, δt)cq codes, where 1 ≤ ℓ ≤ h − 1, 1 ≤ δ ≤ h − 1, and 2 ≤ α. It should be noted that codes in the Hamming space with the Hamming distance form a subfamily of the subspace codes. This can be shown using two different approaches. The first one was pointed in 1957 by Tits [23] who suggested that combinatorics of sets could be regarded as the limiting case q → 1 of combinatorics of vector spaces over the finite field Fq . The second approach is based on the solution for the generalized combination networks. It was proved in [21] that the (0, 1)-Nh,r,s network has a scalar linear solution over Fq if and only if there exists a linear code, over Fq , of length r, dimension h, and minimum Hamming distance r − s + 1. For such a code we have an h × r generator matrix G for which any set with s columns has a subset of h linearly independent columns. These r columns of G form an s-(h, 1, h − 1)cq code. Such codes were already very well studied in coding theory (and also in projective geometry, see [11]) and we omit their discussion here. Linear codes in the Hamming schemes can be used also in other ways as solutions for the generalized combination network. Let C be a linear code, over Fq , of length n, dimension k, and minimum Hamming distance d. For such a code, in the (n− k)× n parity-check matrix H each d− 1 columns are linearly independent. Hence, the n columns of H form an (d− 1)-(n− k, 1, d− 2)cq code. This code solves the (n − k − d + 1, 1)-Nn−k,n,d−1 network. Each s-(h, 1, δ)cq code C which solves the (ǫ, 1)-Nh,r,s network yields also a related vector solution. This solution forms an s-(ht, t, δt)cq code C′ with the sam size as C, but usually much larger codes than |C| can be constructed. Also nonlinear codes can be used as solutions for the generalized combination networks. Some of them (for the combination networks) were discussed in [21] and some will be discussed in the full version of this paper. IV. B OUNDS ON THE S IZES OF C ODES In this section we will present some bounds on the sizes of codes. Clearly, there is a huge ground for research since the parameters of the codes are in a very large range and our knowledge is very limited. Hence, we will give a brief discussion. More will be given in the full version of the paper. We will give some ideas and some insight about the difficulty of obtaining new bounds and especially the exact size of optimal codes. The bounds are on Aq (n, k, s; λ) and on Bq (n, k, s; α) and clearly by Corollary 3, bounds only on one of them is needed since they are equivalent. There is duality between the two types of codes which were considered with two dual distance measures. We start by considering this duality and related four different simple approaches for bounds on the maximum sizes of codes. In an α-(n, k, δ)cq code C, each subset of α codewords (k-subspaces) span a (δ + k)-subspace of Fnq . In other words, each (δ + k − 1)-subspace of Fnq contains at most α − 1 subspaces of C. The orthogonal complement C⊥ is an (α − 1)-(n, n − k, n − δ − k + 1)m q code. In such a code each (n − δ − k + 1)-subspace of Fnq is contained in at most α − 1 codewords. In other words, any set of α codewords intersect in a subspace of dimension at most n−δ−k. Bounds can be obtained based on any one of these four observations. Each one can give another direction to obtain related bounds. The classic bounds for the cases λ = 1 or α = 2, for an c s-(n, k, λ)m q code and an α-(n, k, δ)q code, respectively, can be easily generalized for higher λ (respectively, α), where the simplest ones are the packing bound and the Johnson bounds [7]. It might be easier to generalize the bounds when we consider s-(n, k, λ)m q codes. Theorem 6. If n, k, s, and λ are  positive integers such that 1 ≤ s < k < n and 1 ≤ λ ≤ ns q , then $ n % s q Aq (n, k, s; λ) ≤ λ k . s q Corollary 4. If n, k, δ, and α are positive integers such   that 1 < k < n, 1 ≤ δ ≤ n − k and 2 ≤ α ≤ nk q , then       n  n−δ−k+1 q  Bq (n, k, δ; α) ≤ (α − 1)    . k n−δ−k+1 q Theorem 7. If n, k, s, and λ are  positive integers such that 1 ≤ s < k < n and 1 ≤ λ ≤ ns q , then   n q −1 Aq (n − 1, k − 1, s − 1; λ) . Aq (n, k, s; λ) ≤ k q −1 Corollary 5. If n, k, δ, and α are positive integers such   that 1 < k < n, 1 ≤ δ ≤ n − k and 2 ≤ α ≤ nk q , then   n q −1 Bq (n − 1, k − 1, δ; α) . Bq (n, k, δ; α) ≤ k q −1 Theorem 8. If n, k, s, and λ are  positive integers such that 1 ≤ s < k < n and 1 ≤ λ ≤ ns q , then  n  q −1 Aq (n, k, s; λ) ≤ n−k Aq (n − 1, k, s; λ) . q −1 Corollary 6. If n, k, δ, and α are positive integers such   that 1 < k < n, 1 ≤ δ ≤ n − k and 2 ≤ α ≤ nk q , then  n  q −1 Bq (n, k, δ; α) ≤ n−k Bq (n − 1, k, δ; α) . q −1 Finally, also the following bound was obtained. Theorem 9. If k + δ ≤ n ≤ 2k, then Bq (n, δ, k; α) ≥ (α − 1)Bq (n, δ, k; 2) , and (α−1)q k(n−k−δ+1) ≤ Bq (n, δ, k; α) ≤ (α−1)Q(q)q k(n−k−δ+1) , Q −j where Q(q) = ∞ ). j=1 (1 − q We would also like to mention that some bounds can be obtained from known results on arcs and caps in projective geometry [11]. Discussion on these will be given in the full version of this paper. How good are these bounds? Can the bound of Theorem 6 be attained? If λ = 1 then good constructions and asymptotic bounds are known [2], [6]. But, Theorem 6 might be misleading. Consider the case where n = 6, k = 4, s = 3, λ = 1, and q = 2. By Theorem 6 we have that A2 (6, 4, 3; 1) ≤ 93, while the actual value is A2 (6, 4, 3; 1) = 21. This is not unique for these parameters, and it occurs since Aq (n, k, s; 1) = Aq (n, n − k, n − 2k + s; 1), i.e. we should consider only k ≤ n/2, when λ = 1. For λ > 1 (respectively, α > 2) there is no similar connection between codes of Gq (n, k) and codes of Gq (n, n − k). The big difference is when k > n/2. A good example can be given by considering the (1, 1)-N3,r,4 network which was used in [8] to show that there is a network with three messages for which vector network coding outperforms scalar linear network coding. For a vector solution of this network we need an 3-(3t, t, t)cq code. For simplicity and for explaining the problems in obtaining lower and upper bounds on Aq (n, k, s; λ) we consider the case of t = 2 and q = 2, i.e. a 3-(6, 2, 2)c2 code or its orthogonal complement a 3-(6, 4, 2)m 2 code. In [9] a code with 51 codewords was presented. By using a modification on rank-metric codes we can find a 2-(6, 4, 3)m 2 code of size 64. But, when a 3-(6, 2, 2)c2 code was considered a code of size 82 was obtained [5] using a general method. This general methods implies that B2 (n + 2, 2, 2; 3) ≥ 8B2 (n, 2, 2; 3) + 2, for n ≥ 3. How these lower bounds are compared with the upper bound? By Theorem 6 we have that A2 (6, 4, 3; 2) ≤  186. By Theorem 7 we have that  A (5, 3, 2; 2) . Hence, we have to A2 (6, 4, 3; 2) ≤ 63 2 15 consider the value of A2 (5, 3, 2; 2). An upper bound on A2 (5, 3, 2; 2) involves some more complicated analysis of the way that the 3-subspaces of the related codes are obtained from extensions of subspaces in F42 using also some sets of linear equations. A bound of A2 (5, 3, 2; 2) ≤ 34 [5] was obtained and hence A2 (6, 4, 3; 2) ≤ 142. A code of size 30 [5] was found and hence 30 ≤ A2 (5, 3, 2; 2) ≤ 34. Recently, these last results were improved and some other interesting bounds were obtained by Sascha Kurz [18]. There is a sequence of subspace designs in which each s-subspace is contained in exactly λ times, which form optimal generalized Grassmannian codes. These subspace designs have small block length and very large λ. Hence, they are limited for solutions of generalized combination networks for which the subspaces have dimension close to the one of the ambient space. For example, Thomas [20] have constructed a 2-(n, 3, 7)2 design, where n ≥ 7 and n ≡ 1 or 5 (mod 6). This is the same as a 7-(n, 3, 2)m 2 code and its orthogonal complement is an 8-(n, n − 3, 2)c2 code. Such a code provides a solution for the (1, n − 3)Nn,7·381,8(n−3)+1 network. Note, that even for the smallest number of messages n, i.e. n = 7, the in-degree of each receiver is 33. It looks more interesting to find solutions for generalized combination networks for which the number of parallel links is small compared to the number or messages. Some interesting codes and related networks exist for small parameters. For example, consider the 2-(6, 3, 3)2 design presented in [3], which is a 3-(6, 3, 2)m 2 code and its orthogonal complement is a 4-(6, 3, 2)c2 code. Such a code provides a solution for the (1, 3)-N6,279,13 network. Other designs which lead to optimal codes for related networks can be found in many recent papers on this emerging topic. V. C ONCLUSIONS AND O PEN P ROBLEMS We have introduced a new family of Grassmannian codes with two new distance measures which generalize the traditional Grassmannian codes and Grassmannian distance. There is a correspondence between the set of these codes of maximum size and the set of generalized combination networks. 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Survey on Additive Manufacturing, Cloud 3D Printing and Services arXiv:1708.04875v1 [cs.CY] 7 Aug 2017 Felix W. Baumann, Dieter Roller University of Stuttgart, Stuttgart, Germany Abstract Cloud Manufacturing (CM) is the concept of using manufacturing resources in a service oriented way over the Internet. Recent developments in Additive Manufacturing (AM) are making it possible to utilise resources ad-hoc as replacement for traditional manufacturing resources in case of spontaneous problems in the established manufacturing processes. In order to be of use in these scenarios the AM resources must adhere to a strict principle of transparency and service composition in adherence to the Cloud Computing (CC) paradigm. With this review we provide an overview over CM, AM and relevant domains as well as present the historical development of scientific research in these fields, starting from 2002. Part of this work is also a metareview on the domain to further detail its development and structure. Keywords: Additive Manufacturing; Cloud Manufacturing; 3D Printing Service 1. Introduction Cloud Manufacturing (here CM, in other works also CMfg) as a concept is not new and has been executed in enterprises for many years [275], under different terms, e.g., Grid Manufacturing [50] or Agile Manufacturing [215]. The decision to have a globally distributed and with many contractors or partners interconnected production process and related supply chains is a luxurious one. Large global corporations and competitions makes “expensive” local production nearly impossible. Email addresses: [email protected] (Felix W. Baumann), [email protected] (Dieter Roller) Preprint submitted to arXiv August 17, 2017 CM is based on a strict service orientation of its constituent production resources and capabilities. Manufacturing resources become compartmentalised and connected and worked with as service entities, that can be rented, swapped, expanded, dismantled or scaled up or down just by the use of software. This almost instantaneous and flexible model of resource usage is what made the Cloud Computing (CC) paradigm very successful for a number of companies. Here computing resources and data storage are all virtual, living in large datacentres around the globe, with the user only interfacing these resources through well-defined APIs (Application programming interface) and paying for only the resources utilised – apart from the costs inflicted by the cloud service providers due to their business model and the surcharged or otherwise calculated requirement for profit. With this work we contribute to the dissemination of knowledge in the domain of Additive Manufacturing (AM) and the concept of CM. Cloud Manufacturing can be seen as having two aspects and applications, where the first application is within an industrial environment for which CM provides a concept to embed, connect and utilise existing manufacturing resources, e.g., 3D printers, drilling-, milling- and other machines, i.e., cloud manufacturing is not limited to AM but AM can be utilised within a CM concept. The second application is for end-users that use AM/3D resources over the Internet in lieu acquiring their own 3D printer. The usage in this second application is highly service oriented and has mainly end-users or consumers as target clients. The consumers can profit from online-based services without the requirement of having to own neither hard- nor software resources for 3D printing. We motivate this work by an overview of the historical development of scientific research in these domains starting from 2002. With this we show that the scientific output within these fields has increased by an average of 41.3 percent annually to about 20000 publications per year (see Sect. 1.2). To develop a better understanding of the topic at hand we discuss various terminological definitions found in literature and standards. We give critique on the common definitions of AM and propose a simpler, yet more accurate definition. For the reader to further grasp these domains we study existing journals catering for these communities and discuss reach and inter-connections. Cloud Manufacturing relies on a service oriented concept of production services or capabilities. We extend an existing study on cloud printing ser2 vices as we see such services as integral components for future CM systems. Cloud manufacturing has two aspects which are detailed in this work. First CM is a methodology that is used within industrial settings for the connection of existing resources to form either a virtual assembly line or to acquire access to manufacturing resources in a service oriented manner. Due to the globalisation of the industry, manufacturers face increased challenges with shorter time-to-markets, requirements for mass customisation (MC) and increased involvement of customers within the product development process. In order to stay or become competitive, companies must utilise their resources more efficiently and where not available they must acquire those resources in an efficient and transparent way. These resources must then be integrated into the existing process environment and released when no longer required. The concepts of cloud computing, where resources are available as services from providers that can be leased, rented, acquired or utilised in other ways are proposed to be applied to the domain of manufacturing. Resources like machines and software, as well as capabilities/abilities become transparently available as services that customers or end-users can use through the respective providers and pay for only the services they require momentarily. Most often, no contractual obligations between the provider and the consumer exists (but it can exist, especially for high-value or highvolume usage) which gives the consumer great flexibility at the expense of possible unavailability of resources by the provider. In the end-user segment, or the consumer aspect of CM the user is interested in using AM resources like 3D printers through a web-based interface in order to be able to have objects produced that are designed to be 3D printed without the necessity to purchase and own a 3D printer themselves. The user commonly uses such services in a similar fashion that they would use a (online) photography lab / printing service. The users’ experience and knowledge of AM and 3D printing can vary significantly. Albeit these two aspects seem to be far apart, the commonality between them is, that the service operator must provide AM resources in both cases in a transparent and usable manner. Resources must be provided with clear definitions of the interface to the service, i.e., the data consumed by the service and data rendered by the service. The description and provisioning of the service must be hardware agnostic as the consumer must be able to select the resources required, e.g., have an object manufactured either on a FDM (Fused Deposition Modeling, also Fused Filament Fabrication FFF) machine or and SLA (Stereolithography) machine without the necessity to 3 alter the underlying data and models but by selection. This work is structured as follows: Section 1.1 provides information of the objective we accomplish with this review. Section 1.1.1 presents the research methodology applied for this work. In section 1.1.2 we disseminate the sources that were used to gather information for this work. Section 1.2 provides a dissemination of the scientific research in these fields with a discussion on its historical development. Chapter 2 contains sections on key terminology and their definition throughout literature and standards. We present these terms as well as synonyms and provide an alternative definition as a proposal. The Chapter 3 is an exhaustive collection of scientific journals relevant to the domains discussed in this work. We provide an insight in their interconnectedness and their structure. Chapter 4 provides a meta-review on the subject for the reader to get a further reaching understanding of the subject and its relevant components. In Chapter 4.1 we discuss the audience or target group for CM and 3D printing related cloud services. Chapter 5 extends the study by Rayna and Striukova [204] due to the importance of 3D printing related cloud services for the topic at hand. Section 6 provides the information on the concepts, terminology, methods relevant to the subject as they are disseminated in literature. We conclude this work with a summary in Chapter 7. 1.1. Research Objective This review is performed to establish an overview on the concept and implementation of CM and the utilisation of Additive Manufacturing (AM) therein. For the understanding it is required to become familiar with the various definitions available and the problems arising from inconsistent usage of terminology. For this we compile differing definitions on key terminology. With this work we aim to present an overview over the topic of CM, and its current research findings. We furthermore present a summary overview over existing online and cloud based 3D printing services that can either be regarded as implementations of CM or be utilised in CM scenarios. This part is to extend the knowledge on relevant online services and their orientation towards numerous services. With the presentation of the identified journals that cater for AM, DM, RP, RM and 3D printing research we want to provide other researchers with insight into possible publication venues and a starting point for the identification of relevant sources for their own work. The review work of this article has the objective to identify relevant literature and summarise the key and essential findings thereof. The review also is intended to 4 Search Results for Terms from Google Trends (2004 - 2016) 3D printing Additive Manufacturing Cloud Manufacturing 3D Print 100 number of queries 80 60 40 20 9 -2 01 5- 08 -2 7 7 -1 23 20 15 -0 8- 11 -2 01 5- 01 -0 6 20 15 -0 1- 01 -2 01 4- 06 -2 6 20 14 -0 6- 20 -2 01 3- 10 -1 4 20 13 -1 0- 10 -2 01 3- 03 -0 4 320 13 -0 7- 29 -2 01 2- 08 -2 4 12 101 -2 18 20 12 -0 220 11 -1 5- 08 -2 01 1- 05 -1 2 0 -0 10 001 -2 -0 11 20 20 10 -0 9- 26 -2 01 0- 02 -2 1 9 -1 14 20 10 -0 2- 05 -2 00 9- 07 -2 9 7-0 09 20 20 08 -1 1- 23 -2 00 8- 11 -1 8 -2 00 8- 04 -0 7 13 4-0 08 20 20 07 -0 9- 02 -2 00 7- 09 -2 7 -2 00 7- 01 -1 5 21 20 07 -0 1- 11 -2 00 6- 06 -0 6 6-0 06 20 20 05 -1 0- 30 -2 00 5- 11 -2 03 500 -2 20 3-0 05 04 20 20 -0 8- 08 -2 00 4- 08 -1 4 0 source of data: trends.google.com; 3D printing, sum=13816, avg=43.72; AM, sum=983, avg=3.11; CM, sum=217, avg=0.69; 3D print, sum=6060, avg=19.18 Figure 1: Queries for 3D Printing, AM, CM and 3D Print on google.com for 2004–2016 from trends.google.com provide a high level overview on identified research needs that are considered essential for the evolution of AM and CM. 1.1.1. Methodology The first part of this review is the analysis of other reviews in order to establish a foundation of the existing works and to have a baseline for the analysis of the journals catering to this domain. The journals are identified and presented in order to help researchers in finding a suitable publication venue and to present the recent development in this area. The journals are identified by literature research, web searching (see Sect. 1.1.2), and as a result of the review analysis. This review identified its sources by web search for each of the identified topics depicted in the concept map (See Sect. 6.1), where the first 30 results from the search engines (see Sect. 1.1.2) each are scanned first by title, then by their abstract. For the creation of the topological map an iterative process is applied. The process starts with the analysis of the following works [273, 265, 275, 279, 102] which we had prior knowledge of due to previous engagements in this research area. After the analysis a backward- and forward search is performed. 5 The searches for the content of the review are sorted by relevance, according to the search engine operator. The articles are then analysed and its core concepts are presented in this work. The reviews for the meta-review are identified by a web search and data gathered during our review. For the compilation of the definitions an extraction process is employed where the identified literature for the review is basis for information extraction and dissemination. The compilation is expanded by literature and Internet research for the appropriate keywords and concepts. The extension to the study by Rayna and Striukova [204] is performed following the research methodology applied in the original work. 1.1.2. Sources This review is based on scientific literature acquired through the respective publishers and searched for using the following search engines: • Google Scholar1 • SemanticScholar2 • dblp3 • Web of Science4 • ProQuest5 Microsoft Academic Search6 is not used for the search as the quality and number of results is unsatisfactory. Scopus7 is not used for the research, as we have no subscription for it. The search engines differ in the handling of grouping and selection operators (e.g., OR, +). For each search engine the appropriate operators where selected when not explicitly stated otherwise. As a search engine for scientific literature, Google Scholar, yields the most 1 https://scholar.google.com https://semanticscholar.org 3 https://dblp.uni-trier.de 4 https://webofknowledge.com 5 https://proquest.com 6 http://academic.research.microsoft.com 7 https://www.scopus.com 2 6 results but with a high degree of unrelated or otherwise unusable sources, like the Google search engine8 itself. Furthermore, the search engine enforces strict usage rules thus hindering automated querying and usage. Results from patents and citations are excluded from the result set for Google Scholar. SemanticScholar offers a responsive interface, that allows for automated querying through JSON9 , to “millions” of articles10 from computer science - a statement that we can not verify as we have seen articles from other domains too. The dblp project indexes over 333333311 from computer science in a very high quality. Its interface allows for automated and scripted usage. Web of Science provides an index of a large number (over 5612 millions) of scientific works. The entries in the index are of high quality but the interface is rather restrictive. ProQuest also has a very restrictive and non-scriptable interface and contains over 54 million 13 entries in its corpus, among which are historical news articles and dissertations. The quality of the results is high. ProQuest and Web of Science are subscription based services. 1.2. Development in Scientific Publications The significance and maturity of a research area is reflected in the number of publications available. We perform a keyword based analysis utilising the sources described in Section 1.1.2. The searches are performed with a number of relevant keywords (including various technologies and methods for AM) and a restriction of the time period starting from 2002 to 2016. The queries are also restricted on the type of results with an exclusion to citations and patents, where applicable. For a study on the patents and the development of patent registrations for this domain we refer to Park et al. [190]. Caveat: Searching on search engines for specific keywords like clip and lens in their abbreviated form will lead to a number of skewed results from works that are not significant for this body of work. For example the search for “Additive Manufacturing” and LENS yield articles in the results that 8 https://google.com JavaScript Object Notation 10 https://www.semanticscholar.org/faq#index-size 11 News from 2016-05-03: “Today, dblp reached the wonderful ”Schnapszahl” of 3,333,333 publications” 12 A search for publications with its publication date between 1700 and 2100 yields 56998216 results 13 A search for publications with its publication date after 1700 yields 54266680 results 9 7 Classification of Research in Cloud Manufacturing OTHER ENVIRONMENTAL SCIENCES ECOLOGY CHEMISTRY MATERIALS SCIENCE OPERATIONS RESEARCH MANAGEMENT SCIENCE AUTOMATION CONTROL SYSTEMS COMPUTER SCIENCE ENGINEERING 0 10 20 30 40 50 60 percent source of data: webofknowledge.com; n=460 Figure 2: Classification of articles for Cloud Manufacturing; source of data: webofknowledge.com Classification of Research in Additive Manufacturing OTHER FOOD SCIENCE TECHNOLOGY SCIENCE TECHNOLOGY OTHER TOPICS PHYSICS METALLURGY METALLURGICAL ENGINEERING CHEMISTRY ENGINEERING MATERIALS SCIENCE 0 10 20 30 40 50 percent source of data: webofknowledge.com; n=4582 Figure 3: Classification of articles for Additive Manufacturing; source of data: webofknowledge.com are either fabricating (optical) lenses using AM or are about lenses in lasers that are used in AM. In case the result sets are as large as in our case it is not feasible to remove those erroneous results and adjust the result set accordingly. We make the reader aware to only take the given numbers as an indication. In Fig. 2 to Fig. 7 the classification of scientific articles according to Web of Science is shown. The classifications do not add up to 100 percent as the respective articles can be classified in more than one field. In the figures the number of results per search term is also listed. Domains with less than five percent aggregated classification are grouped together as “OTHER”. In Fig. 8 the accumulated prevalence of the terms 3D printing versus Additive Manufacturing (AM) is displayed. For these numbers queries are made for a combination of search terms and restrictions on the time period. The scale of the Y-Axis is logarithmic due to the large differences in the number of results per search engine. The dblp database returned the lowest number 8 Classification of Research in 3D Printing OTHER COMPUTER SCIENCE PHYSICS CHEMISTRY SCIENCE TECHNOLOGY OTHER TOPICS ENGINEERING MATERIALS SCIENCE 0 10 20 30 40 50 60 percent source of data: webofknowledge.com; n=3369 Figure 4: Classification of articles for 3D Printing; source of data: webofknowledge.com Classification of Research in Rapid Manufacturing OTHER SCIENCE TECHNOLOGY OTHER TOPICS METALLURGY METALLURGICAL ENGINEERING AUTOMATION CONTROL SYSTEMS PHYSICS MATERIALS SCIENCE ENGINEERING 0 10 20 30 40 50 60 70 percent source of data: webofknowledge.com; n=496 Figure 5: Classification of articles for Rapid Manufacturing; source of data: webofknowledge.com Classification of Research in Rapid Prototyping OTHER CHEMISTRY AUTOMATION CONTROL SYSTEMS PHYSICS SCIENCE TECHNOLOGY OTHER TOPICS COMPUTER SCIENCE MATERIALS SCIENCE ENGINEERING 0 10 20 30 40 50 percent source of data: webofknowledge.com; n=5393 Figure 6: Classification of articles for Rapid Prototyping; source of data: webofknowledge.com 9 Classification of Research in Rapid Tooling OTHER POLYMER SCIENCE METALLURGY METALLURGICAL ENGINEERING AUTOMATION CONTROL SYSTEMS MATERIALS SCIENCE ENGINEERING 0 10 20 30 40 50 60 70 80 percent source of data: webofknowledge.com; n=394 Figure 7: Classification of articles for Rapid Tooling; source of data: webofknowledge.com of results with results consistently less than 10. Google Scholar yielded the largest number of results with the accumulated number of results for the term AM gaining on the term 3D printing since 2009. In Fig. 9 the prevalence of certain AM or 3D Printing technologies is studied by the number of articles from four different search engines for the respective combination of search terms. The largest number of results are from Google Scholar for search term combinations with “3D Printing”. Furthermore, a generalised search is performed for the terminology “Laser, Lithography and Powder”, e.g., summarising technologies like SLM (Selective Laser Melting), SLS (Selective Laser Sintering), SLA, LOM (Laminated Object Manufacturing), LENS (Laser Engineered Net Shaping) for the term “Laser”. The search for technologies like CLIP and LENS are problematic due to the non-specificity of the terminology as described before (See note 1.2). 2. Definition and Terminology In general the usage of the terminology within this field is very inconsistent. Commonly and colloquially the terms 3D printing and AM are used as synonyms. Analysing the prevalence of either of these terms we find that 3D printing is slightly more prevalent for results of scientific literature with 68164 results for the sources described in Sect. 1.1.2 during the period of 2002–2016. In the same period there are over 59506 results for the term Additive Manufacturing. SemanticScholar provided significantly more results (7072 over 1211) for 3D printing and Web of Science yielded almost four times the number of results for Additive Manufacturing over 3D Printing (1956 results to 578). There is also no clear trend in the usage of either terms. With this section we exemplify this situation and present 10 Comparison of AM and 3D printing over 4 search engines Google: 3D Printing SemanticScholar: 3D Printing dblp: 3D Printing WoS: 3D Printing ProQuest: 3D Printing Google: Additive Manufacturing SemanticScholar: Additive Manufacturing" dblp: Additive Manufacturing WoS: Additive Manufacturing ProQuest: Additive Manufacturing summative number of publications 100000 10000 1000 100 10 16 20 15 20 14 20 13 20 12 20 11 20 10 20 09 20 08 20 07 20 06 20 05 20 04 20 03 20 20 02 1 year source of data: scholar.google.com, semanticscholar.org, dblp.uni-trier.de, webofknowledge.com, progquest.com Figure 8: Comparison of AM and 3D Printing on selected search engines (2002–2016) 11 Google: 3D Printing SemanticScholar: 3D Printing WoS: 3D Printing ProQuest: 3D Printing Google: Additive Manufacturing SemanticScholar: Additive Manufacturing WoS: Additive Manufacturing ProQuest: Additive Manufacturing 15000 10000 5000 s m dl sff ls dm de r y ph ra ho g lit po w se r la jm m 3d p le ns ip cl eb m m fd m lo a sl m sl s 0 sl summative number of publications (2002-2016) Results for Manufacturing Technologies source of data: scholar.google.com, semanticscholar.org, webofknowledge.com, proquest.com Figure 9: Comparison of 3D printing technologies on selected search engines (2002–2016) 12 common definitions throughout literature and standards. We furthermore add our point of view in the form of a critique at the end of the section. 2.1. Additive Manufacturing and 3D Printing In this section we present established definitions for AM and related terminology as presented in literature and standards. 2.1.1. Definitions of Additive Manufacturing AM is most often regarded as an umbrella term for technology and methods for the creation of objects from digital models from scratch. It is usually in contrast to subtractive and formative methods of manufacturing as defined in the standard [1]. It is also commonly a synonym for 3D printing. Gibson et al. [89] define AM as: “Additive manufacturing is the formalised term for what used to be called rapid prototyping and what is popularly called 3D Printing. [...] Referred to in short as AM, the basic principle of this technology is that a model, initially generated using a three-dimensional Computer-Aided Design (3D CAD) system, can be fabricated directly without the need for process planning. [...]” Gebhardt [88] defines AM as: “Als Generative Fertigungsverfahren werden alle Fertigungsverfahren bezeichnet, die Bauteile durch Auf- oder Aneinanderfügen von Volumenelementen (Voxel’n), vorzugsweise schichtweise, automatisiert herstellen.”, which we translate as “As generative/additive manufacturing processes all production processes are referred that produce components automatically by depositioning of volume elements (Voxels), preferably layer-wise”. The VDI directives VDI 3404 (Version 2009 [4] and 2014 [6]) define additive fabrication as: “Additive fabrication refers to manufacturing processes which employ an additive technique whereby successive layers or units are built up to form a model.”. The 2009 directive “VDI-Richtlinie: VDI 3404 Generative Fertigungsverfahren - Rapid-Technologien (Rapid Prototyping) - Grundlagen, Begriffe, Qualitätskenngrößen, Liefervereinbarungen” and the 2014 directive “VDIRichtlinie: VDI 3404 Additive Fertigung - Grundlagen, Begriffe, Verfahrensbeschreibungen” are both currently in retracted states. The also retracted ASTM standard F2792-12a “Standard terminology for additive manufacturing technologies” defines AM as “A process of joining materials to make objects from 3D model data, usually layer upon layer, 13 as opposed to subtractive manufacturing methodologies.” with the following synonyms listed “additive fabrication, additive processes, additive techniques, additive layer manufacturing, layer manufacturing, and freeform fabrication.”. Bechthold et al. [26] define AM as: “The terms additive manufacturing (AM) and 3D printing describe production processes in which a solid 3D structure is produced layer by layer by the deposition of suitable materials via an additive manufacturing machine.” Thomas and Gilbert [242] define AM as: “Additive manufacturing is the process of joining materials to make objects from three-dimensional (3D) models layer by layer as opposed to subtractive methods that remove material. The terms additive manufacturing and 3D printing tend to be used interchangeably to describe the same approach to fabricating parts. This technology is used to produce models, prototypes, patterns, components, and parts using a variety of materials including plastic, metal, ceramics, glass, and composites” Klocke [136] defines AM as: “Generative Verfahren: Diese Verfahrensgruppe umfasst alle Technologien, mit denen eine aufbauende, schichtweise Fertigung von Bauteilen realisiert wird. Sie werden auch als Additive Manufacturing Technologies oder als Layer based Manufacturing Technologies bezeichnet. Zum Herstellen der Schichten wird häufig Laserstrahlung verwendet. [...].” translation “Generative Processes: This process group contains all technologies, with which an additive, layer-wise generation of parts is realised. They are also referred to as additive manufacturing technologies or layer based manufacturing technologies. For the creation of the layers oftentimes laser emission is used. [...]” In the ASTM F2792-12a [5] standard AM is defined as: “process of joining materials to make objects from 3D model data, usually layer upon layer, as opposed to subtractive manufacturing methodologies. Synonyms: additive fabrication, additive processes, additive techniques, additive layer manufacturing, layer manufacturing, and freeform fabrication.” Gao et al. [83] use the term AM and 3D printing synonymously: “Additive manufacturing (AM), also referred to as 3D printing, [...]”. Sames et al. [214] also use the term AM and 3D printing synonymously: “Additive manufacturing (AM), also known as three-dimensional (3D) printing, [...]” Lachmayer and Lippert [140] define AM as: “Das Additive Manufacturing 14 (AM), als Überbegriff für das Rapid Prototyping (RP), das Rapid Tooling (RT), das Direct Manufacturing (DM) und das Rapid Repair (RR) basiert auf dem Prinzip des additiven Schichtaufbaus in x-, y- und z-Richtung zur maschinellen Herstellung einer (Near-) Net-Shape Geometrie” which translates to: “Additive manufacturing as an umbrella term for Rapid Prototyping (RP), Rapid Tooling (RT), Direct Manufacturing (DM) and Rapid Repair (RR) is based on the principle of the additive layer fabrication in x-, y- and z-direction for the fabrication of a (near-) net-shape geometry by machines” The ISO/ASTM Standard 52900:2015(E) [9] defines AM as: “process of joining materials to make parts (2.6.1) from 3D model data, usually layer (2.3.10) upon layer, as opposed to subtractive manufacturing and formative manufacturing methodologies”. 2.1.2. Definitions of 3D Printing According to Gebhardt [88] 3D Printing is a generic term that is synonymous to AM and is replacing the term AM in the future due to its simplicity. Bechtholdt et al. [26] use the terms 3D Printing and AM synonymously as umbrella terms for technologies and applications. In the VDI directive [7] the term 3D printing is used for a certain additive process but it is acknowledged that it is generally used as a synonym for AM. The ASTM standard F2792-12a (retracted) defines 3D printing as “The fabrication of objects through the deposition of a material using a print head, nozzle, or another printer technology.” but also acknowledges the common synonymous use of this term for AM, mostly of low-end quality and price machines. Gibson [89] uses the term 3D Printing for the technology invented by researches at MIT [212] but also acknowledges that it is used synonymously for AM and will eventually replace the term AM due to media coverage. The ISO/ASTM Standard 52900:2015(E) [9] defines 3D Printing as: “fabrication of objects through the deposition of a material using a print head, nozzle, or another printer technology”. It is also noted in this standard that the term 3D printing is often used as a synonym for AM, mostly in non-technical context. Furthermore, it is noted that 3D printing is associated with low price and capability machines. 2.1.3. Definitions of Rapid Prototyping In Hopkinson and Dickens [106] Rapid Prototyping (RP) is defined as: “RP refers to a group of commercially available processes which are used to 15 create solid 3D parts from CAD, from this point onwards these processes will be referred to as layer manufacturing techniques (LMTs)” The VDI directive 3405 defines RP as: “Additive fabrication of parts with limited functionality, but with sufficiently well-defined specific characteristics.” Weber et al. [267] define RP as: “Early AM parts were created for the rapid prototyping market and were first employed as visual aids and presentation models. Many lower cost AM systems are still used in this way.” 2.1.4. Definitions of Rapid Manufacturing Hopkinson et al. [108] define Rapid Manufacturing (RM) as: “the use of a computer aided design (CAD)-based automated additive manufacturing process to construct that are used directly as finished products or components.” Previously Hopkinson and Dickens [106] defined RM as: “Rapid manufacturing uses LMTs for the direct manufacture of solid 3D products to be used by the end user either as parts of assemblies or as stand-alone products.” The VDI directive 3404 Version 2009 [4] defines RM as: “Additive fabrication of end products (often also described as production parts). Characteristics: Has all the characteristics of the end product or is accepted by the customer for “series production readiness”. Material is identical to that of the end product. Construction corresponds to that of the end product.” The VDI directive 3405 [7] defines RM as a synonym for direct manufacturing, which is defined as: “Additive fabrication of end products.” 2.1.5. Definitions of Rapid Tooling King and Tansey [135] define Rapid Tooling (RT) as an extension of RP as such: “Rapid tooling is a progression from rapid prototyping. It is the ability to build prototype tools directly as opposed to prototype products directly from the CAD model resulting in compressed time to market solutions.” The VDI directive 3405 [7] defines RT as: “The use of additive technologies and processes to fabricate end products which are used as tools, moulds and mould inserts.” Weber et al. [267] define RT as: “Another class of applications for AM parts is patterns for tooling or tooling directly made by AM. AM processes can be used to significantly shorten tooling time and are especially useful for low-run production of products.” 16 2.1.6. Definitions of Cloud Manufacturing The work by Li et al. [112] appears to be the first to introduce the concept and definition of Cloud Manufacturing (CM), but unfortunately this article is only available in Chinese and could therefore not be considered. The article is cited by more than 450 publications according to Google Scholar. Wu and Yang [280] define CM as such: “Cloud manufacturing is an integrated supporting environment both for the share and integration of resources in enterprise. It provides virtual manufacturing resources pools, which shields the heterogeneousness and the regional distribution of resources by the way of virtualisation. cloud manufacturing provides a cooperative work environment for manufacturing enterprises and individuals and enables the cooperation of enterprise.” Tao et al. [237] define CM indirectly by the following description: “Cloud manufacturing is a computing and service-oriented manufacturing model developed from existing advanced manufacturing models (e.g. ASP, AM, NM, MGrid) and enterprise information technologies under the support of cloud computing, IoT, virtualisation and service-oriented technologies, and advanced computing technologies” Xu [283] defines CM similar to the NIST definition of CC as: “a model for enabling ubiquitous, convenient, on-demand network access to a shared pool of configurable manufacturing resources (e.g., manufacturing software tools, manufacturing equipment, and manufacturing capabilities) that can be rapidly provisioned and released with minimal management effort or service provider interaction”. This definition is also used in the work by Wang and Xu [266]. Zhang et al. [297] describe CM as: “Cloud manufacturing (CMfg) is a new manufacturing paradigm based on networks. It uses the network, cloud computing, service computing and manufacturing enabling technologies to transform manufacturing resources and manufacturing capabilities into manufacturing services, which can be managed and operated in an intelligent and unified way to enable the full sharing and circulating of manufacturing resources and manufacturing capabilities. CMfg can provide safe, reliable, high-quality, cheap and on-demand manufacturing services for the whole life cycle of manufacturing.” 2.1.7. Synonyms for AM As with the previous definitions for AM, RP, RT, RM and 3D printing there is no consensus in the terminology for synonyms of AM in general. The 17 following synonyms can be found in literature and are used in existing works. • direct layer manufacturing or layer manufacturing or additive layer manufacturing • direct digital manufacturing is a synonym for rapid manufacturing [89] • solid freeform fabrication (SFF), three dimensional printing [267] • 3D printing, Additive Techniques, Layer Manufacturing, and Freeform fabrication [183] • additive fabrication, additive processes, additive techniques, additive layer manufacturing, layer manufacturing, and freeform fabrication [5]14 • “The technical name for 3D printing is additive manufacturing [...]” [154] 2.1.8. Critique The existing definitions fall short on their focus on the layer-wise creation of objects as technologies like LENS and multi-axis (n > 3) are not bound and defined by a layer structure but can regarded as a form of AM as they create objects based on 3D (CAD) models from scratch without any of the characteristics of traditional subtractive or formative fabrication methods. Through a systematic decomposition of the existing definitions of AM we conclude that the basic commonality of AM is described as the creation of a physical object from a digital model by a machine. Furthermore, we propose the term AM as an umbrella term that signifies industrial, commercial or professional application and usage whereas 3D printing can be colloquially used for technologies and methods for the creation of physical objects from 3D (CAD) models in other situations. For the actual building machines of additively manufactured parts we recommend the synonymous use of AM fabricator or 3D printer. The first as it describes the functionality in a precise way and the second as it is commonly used and understood by a broad audience. 14 Also https://wohlersassociates.com/additive-manufacturing.html 18 3. Journals related to the Subject We have identified a number of journals specialising in the domain of AM. In this section we explain their foci and their scientific scope. The following journals cater partially or solely for the academic dissemination of works based in or related to the domains of AM, RM, RP and 3D Printing. These journals are identified using the service of the Directory of Open Access Journals15 , Thomson Reuters Web of Science16 and the articles used for this review. Only journals with indication for AM, RM, RP or 3D Printing in either the title or the scope are listed below. In the following overview the abbreviations EiC for Editor in Chief, ImpactF for Impact Factor and SJR for SCImago Journal Rank Indicator17 are used. The Impact Factor is either acquired from the journal’s home page directly when available or looked up from Thomson Reuters InCites Journal Citation Reports18 . For a number of journals neither a SJR nor the IF could be found. The numbers for the available volumes, issues and articles are directly extracted from the respective journal’s website. The listing contains a full list of all members of the board and editors per journal for an assessment of the interconnection between the various journals. Editors and members of the board that are involved in more than one journal are indicated by italicised text and the indication in which other journal they are involved. The journals are ordered by their number of articles published and if two or more journals have an equal number of publications the ordering is chronological. The journals without publications and age available are sorted by their ISSN. The 20 journals have an accumulated 22616 articles published (respectively 17877 articles, when only considering articles from Journal 2 after it was renamed). The median of the first publication date is 2014. Under the assumption that the articles are published equally since the first Journal (Journal 1) started in 1985, 31 years ago, this results in an average number of 576 articles per year, which accounts for approximately 18 % of the average accumulated results of 3197 scientific works indexed by 15 https://doaj.org http://webofknowledge.com 17 “It expresses the average number of weighted citations received in the selected year by the documents published in the selected journal in the three previous years”, see http: //www.scimagojr.com/SCImagoJournalRank.pdf for more details 18 https://jcr.incites.thomsonreuters.com 16 19 http://scholar.google.com for the time frame of 2002 to 2016 (See also Section 1.1). The information on the journals is accurate as of 2016-08-10 according to the respective websites. 1. The International Journal of Advanced Manufacturing Technology Publisher Springer ISSN 1433-3015 URL http://www.springer.com/engineering/production+engineering/ journal/170/PSE ImpactF 1.568 H-Index 7119 SJR 0.9120 Since 1985 Volumes 85 Issues 432 Articles 11727 EiC Andrew Y. C. Nee (See also Journal 3) Board and Editors 1) Kai Cheng 2) David W. Russell 3) M. S. Shunmugam 4) Erhan Budak 5) D. Ben-Arieh 6) C. Brecher 7) H. van Brussel 8) B. Çatay 9) F. T. S. Chan (See also Journal 3) 10) F. F. Chen 11) G. Chryssolouris 12) Chee Kai Chua (See also Journals 4, 6, 11) 13) M. Combacau 14) A. Crosnier 15) S. S. Dimov 16) L. Fratini 17) M. W. Fu 18) H. Huang 19) V. K. Jain 20) M. K. Jeong 21) P. Ji 22) W.-Y. Jywe 23) R. T. Kumara 24) A. Kusiak 25) B. Lauwers 26) W. B. Lee 27) C. R. Nagarajah 28) E. Niemi 29) D. T. Pham 30) S. G. Ponnambalam 31) M. M. Ratnam 32) V. R. Rao 33) C. Saygin 34) W. Steen 35) D. J. Stephenson 36) M. K. Tiwari (See also Journal 18) 37) E. Vezzetti 38) G. Vosniakos 39) X. Xu (See also Journal 3) 40) Y. X. Yao 41) A. R. Yildiz 42) M. Zoe (See also Journals 7, 12) 43) H.-C. Zhang 44) L. Zhang 45) A.G. Mamalis 2. Journal of Manufacturing Science and Engineering Publisher The American Society of Mechanical Engineers 19 20 http://www.scimagojr.com/journalsearch.php?q=20428&tip=sid&clean=0 http://www.scimagojr.com/journalsearch.php?q=20428&tip=sid&clean=0 20 ISSN 1087-1357 URL http://manufacturingscience.asmedigitalcollection.asme. org/journal.aspx ImpactF 1.087 H-Index 6821 SJR 0.822 Since 1996 Volumes 138 Issues 101 (Since “Journal of Engineering for Industry” was renamed to its current title) Articles 7066 (2327 since the renaming in May 1996) EiC Y. Lawrence Yao Board and Editors 1) Sam Anand 2) Wayne Cai (See also Journal 5) 3) Jaime Camelio 4) Hongqiang Chen 5) Dragan Djurdjanovic 6) Guillaume Fromentin 7) Yuebin Guo 8) Yong Huang (See also Journal 9) 9) Yannis Korkolis 10) Laine Mears 11) Gracious Ngaile (See also Journal 5) 12) Radu Pavel 13) Zhijian Pei 14) Xiaoping Qian 15) Tony Schmitz 16) Jianjun (Jan) Shi 17) Daniel Walczyk 18) Donggang Yao 19) Allen Y. Yi 3. Robotics and Computer-Integrated Manufacturing Publisher Elsevier B.V. ISSN 0736-5845 URL http://www.journals.elsevier.com/robotics-and-computer-integrated-manufa ImpactF 2.077 H-Index 6123 SJR 1.6124 Since 1984–1994, 1996 ongoing Volumes 44 Issues 145 Articles 2191 EiC Andre Sharon Board and Editors 1) M. Haegele 2) L. Wang 3) M. M. Ahmad 4) K. 21 http://www.scimagojr.com/journalsearch.php?q=20966&tip=sid&clean=0 http://www.scimagojr.com/journalsearch.php?q=20966&tip=sid&clean=0 23 http://www.scimagojr.com/journalsearch.php?q=18080&tip=sid&clean=0 24 http://www.scimagojr.com/journalsearch.php?q=18080&tip=sid&clean=0 22 21 Akella 5) H. Asada 6) J. Baillieul 7) T. Binford 8) D. Bossi 9) T. Broughton 10) M. Caramanis 11) F. T. S. Chan (See also Journal 1) 12) G. Chryssolouris 13) J. Deasley 14) S. Dubowsky 15) E. Eloranta 16) K. C. Fan 17) J. Y. H. Fuh (See also Journals 4, 15) 18) J. X. Gao 19) M. Gevelber 20) Y. Ito 21) K. Iwata 22) T. Kanade 23) F. Liu 24) L. Luong 25) K. L. Mak 26) K. McKay 27) A. Meng 28) N. Nagel 29) A. Y. C. Nee (See also Journal 1) 30) G. Reinhardt 31) R. D. Schraft 32) W. P. Seering 33) D. Spath 34) H. C. G. Spur 35) N. Suh 36) M. K. Tiwari 37) H. Van Brussel 38) F. B. Vernadat 39) A. Villa 40) M. Weck 41) H. Worn 42) K. Wright 43) C. Wu 44) X. Xu (See also Journal 1) 4. Rapid Prototyping Journal Publisher Emerald Group Publishing, Ltd ISSN 1355-2546 URL http://www.emeraldinsight.com/loi/rpj ImpactF 1.352 H-Index 4925 SJR 0.8126 Since 1995 Volumes 22 Issues 113 Articles 882 EiC Ian Campbell Board and Editors 1) David Bourell (See also Journals 10, 6) 2) Ian Gibson (See also Journals 8, 6) 3) James Martin 4) Sung-Hoon Ahn 5) Paulo Jorge da Silva Bártolo (See also Journals 17, 6, 11) 6) Deon de Beer 7) Alain Bernard (See also Journals 13, 8, 6) 8) Richard Bibb (See also Journal 11) 9) U. Chandrasekhar 10) Khershed Cooper (See also Journal 10) 11) Denis Cormier (See also Journals 9, 8) 12) Henrique de Amorim Almeida (See also Journal 12) 13) Phill Dickens 14) Olaf Diegel (See also Journal 10) 15) Jerry Fuh (See also Journals 15, 3) 16) Jorge Ramos Grez 17) Chua Chee Kai (See also Journals 1, 6, 11) 18) Jean-Pierre 25 26 http://www.scimagojr.com/journalsearch.php?q=21691&tip=sid&clean=0 http://www.scimagojr.com/journalsearch.php?q=21691&tip=sid&clean=0 22 Kruth 19) Gideon N. Levy 20) Toshiki Niino 21) Eujin Pei (See also Journal 12) 22) B. Ravi 23) David Rosen (See also Journals 10, 9) 24) Monica Savalani 25) Tim Sercombe (See also Journals 8, 15) 26) Brent Stucker (See also Journals 10, 9) 27) Wei Sun (See also Journal 10) 28) Jukka Tuomi 29) Terry Wohlers (See also Journal 10) 5. Journal of Manufacturing Processes Publisher Elsevier B.V. ISSN 1526-6125 URL http://www.journals.elsevier.com/journal-of-manufacturing-processes ImpactF 1.771 H-Index 2427 SJR 1.0928 Since 1999 Volumes 24 Issues 47 Articles 620 EiC Shiv G. Kapoor Board and Editors 1) M. Annoni 2) W. Cai (See also Journal 2) 3) G. Cheng 4) J. Dong 5) Z. Feng 6) G. Y. Kim 7) A. S. Kumar 8) X. Li 9) G. Ngaile (See also Journal 2) 10) S. S. Park 11) M. Sundaram 12) B. Wu 13) H. Yamaguchi Greenslet 14) Y. Zhang 6. Virtual and Physical Prototyping Publisher Taylor & Francis ISSN 1745-2767 URL http://www.tandfonline.com/loi/nvpp20 ImpactF N/A H-Index 1529 SJR 0.4230 27 http://www.scimagojr.com/journalsearch.php?q=27677&tip=sid&clean=0 http://www.scimagojr.com/journalsearch.php?q=27677&tip=sid&clean=0 29 http://www.scimagojr.com/journalsearch.php?q=5800173379&tip=sid&clean= 28 0 30 http://www.scimagojr.com/journalsearch.php?q=5800173379&tip=sid&clean= 0 23 Since 2006 Volumes 11 Issues 42 Articles 294 EiC Paulo Jorge da Silva Bártolo (See also Journals 4, 17, 11), Chee Kai Chua (See also Journals 1, 4, 11) Board and Editors 1) Wai Yee Yeong (See also Journal 11) 2) Alain Bernard (See also Journals 4, 13, 8) 3) Anath Fischer (See also Journal 12) 4) Bopaya Bidanda 5) Cijun Shuai (See also Journal 11) 6) David Bourell (See also Journals 10, 4) 7) David Dean (See also Journal 12) 8) Dongjin Yoo (See also Journal 11) 9) Jack Zhou 10) Ian Gibson (See also Journals 4, 8) 11) Jiankang He (See also Journal 11) 12) John Lewandowski 13) Martin Dunn 14) Ming Leu 15) Peifeng Li 16) Shoufeng Yang (See also Journals 17, 11) 17) Shlomo Magdassi 18) Yong Chen (See also Journal 8) 7. RTejournal Publisher University Library of the FH-Aachen University of applied Science ISSN 1614-0923 URL http://www.rtejournal.de ImpactF N/A H-Index N/A SJR N/A Since 2004 Volumes 13 Issues 13 Articles 155 EiC Andreas Gebhardt Board and Editors 1) Ralf Eckhard Beyer 2) Dietmar Drummer 3) KarlHeinrich Grote 4) Sabine Sändig 5) Gerd Witt (See also Journal 12) 6) Michael Zäh (See also Journals 1, 12) 8. International Journal of Rapid Manufacturing Publisher Inderscience Enterprises Ltd. ISSN 1757-8825 URL http://www.inderscience.com/ijrapidm ImpactF N/A 24 H-Index N/A SJR N/A Since 2009 Volumes 5 Issues 20 Articles 93 EiC Bahram Asiabanpour (See also Journal 13) Board and Editors 1) Ali K. Kamrani 2) Denis Cormier (See also Journals 9, 4) 3) Ismail Fidan 4) Ian Gibson (See also Journals 4, 6) 5) Wei Jun 6) Allan Rennie 7) Joseph J. Beaman Jr. (See also Journal 9) 8) Alain Bernard (See also Journals 4, 13, 6) 9) Georges Fadel 10) Mo Jamshidi 11) Behrokh Khoshnevis (See also Journal 10) 12) John M. Usher 13) Richard A. Wysk 14) Abe Zeid 15) Abdulrahman M. Al-Ahmari 16) Manfredi Bruccoleri 17) Satish T. S. Bukkapatnam 18) Yong Chen (See also Journal 6) 19) Fred Choobineh 20) L. Jyothish Kumar (See also Journals 10, 13) 21) Mehdi Mojdeh 22) Benoit Montreuil 23) Kamran Mumtaz 24) Hossein Tehrani Niknejad 25) Pulak Mohan Pandey (See also Journal 13) 26) Prahalad K. Rao 27) Sa’Ed M. Salhieh 28) Tim Sercombe (See also Journals 4, 15) 29) Kathryn E. Stecke 30) Albert Chi To 31) Shigeki Umeda 32) Omid Fatahi Valilai 33) Nina Vojdani 34) Micky R. Wilhelm 35) Stewart Williams 9. Additive Manufacturing Publisher Elsevier B.V. ISSN 2214-8604 URL http://www.journals.elsevier.com/additive-manufacturing ImpactF N/A H-Index 531 SJR 1.0432 Since 2014 Volumes N/A Issues 12 31 http://www.scimagojr.com/journalsearch.php?q=21100349533&tip=sid& clean=0 32 http://www.scimagojr.com/journalsearch.php?q=21100349533&tip=sid& clean=0 25 Articles 93 EiC Ryan Wicker Board and Editors 1) E. MacDonald 2) M. Perez 3) A. Bandyopadhyay 4) J. Beaman (See also Journal 8) 5) J. Beuth 6) S. Bose 7) S. Chen 8) J. W Choi 9) K. Chou 10) D. Cormier (See also Journals 4, 8) 11) K. Creehan 12) C. Elkins 13) S. Fish 14) D. D. Gu 15) O. Harrysson 16) D. Hofmann 17) N. Hopkinson 18) Y. Huang (See also Journal 2) 19) K. Jurrens 20) K. F. Leong 21) J. Lewis (See also Journal 10) 22) L. Love 23) R. Martukanitz 24) D. Mei 25) R. Resnick (See also Journal 10) 26) D. Rosen (See also Journals 10, 4) 27) C. Spadaccini 28) B. Stucker (See also Journals 10, 4) 29) C. Tuck 30) C. Williams 10. 3D Printing and Additive Manufacturing Publisher Mary Ann Liebert, Inc ISSN 2329-7662 URL http://www.liebertpub.com/overview/3d-printing-and-additive-manufacturin 621 ImpactF N/A H-Index N/A SJR N/A Since 2014 Volumes 3 Issues 10 Articles 86 EiC Skylar Tibbits Board and Editors 1) Hod Lipson 2) Craig Ryan 3) Anthony Atala 4) David Benjamin 5) Lawrence J. Bonassar 6) David Bourell (See also Journals 4, 6) 7) Adrian Bowyer 8) Glen Bull 9) Adam Cohen 10) Khershed P. Cooper (See also Journal 4) 11) Scott Crump 12) Olaf Diegel (See also Journal 4) 13) Richard Hague 14) John F. Hornick 15) Weidong Huang 16) Takeo Igarashi 17) Bryan Kelly 18) Behrokh Khoshnevis (See also Journal 8) 19) Matthias Kohler 20) L. Jyothish Kumar (See also Journals 13, 8) 21) Melba Kurman 22) Jennifer A. Lewis (See also Journal 9) 23) Jos Malda 24) Gonzalo Martinez 25) Neri Oxman 26) Bre Pettis 27) Sharon Collins Presnell 28) Phil Reeves 29) Avi N. Reichental 30) Ralph Resnick (See also Journal 9) 31) David W. Rosen (See also Jour26 nals 9, 4) 32) Jenny Sabin 33) Carolyn Conner Seepersad 34) Brent Stucker (See also Journals 9, 4) 35) Wei Sun (See also Journal 4) 36) Hiroya Tanaka 37) Thomas Toeppel 38) Peter Weijmarshausen 39) Terry Wohlers (See also Journal 4) 11. International Journal of Bioprinting Publisher Whioce Publishing Pte Ltd ISSN 2424-8002 URL http://ijb.whioce.com/index.php/int-j-bioprinting ImpactF N/A H-Index N/A SJR N/A Since 2015 Volumes 2 Issues 3 Articles 31 EiC Chee Kai Chua (See also Journals 1, 4, 6) Board and Editors 1) Wai Yee Yeong (See also Journal 6) 2) Aleksandr Ovsianikov (See also Journal 17) 3) Ali Khademhosseini (See also Journal 15) 4) Boris N. Chichkov (See also Journal 17) 5) Charlotte Hauser 6) Cijun Shuai (See also Journal 6) 7) Dong Jin Yoo (See also Journal 6) 8) Frederik Claeyssens 9) Geun Hyung Kim 10) Giovanni Vozzi (See also Journals 17, 15) 11) Ibrahim Tarik Ozbolat 12) Jiankang He (See also Journal 6) 13) Lay Poh Tan 14) Makoto Nakamura 15) Martin Birchall 16) Paulo Jorge Da Silva Bartolo (See also Journals 17, 4, 6) 17) Peter Dubruel 18) Richard Bibb (See also Journal 4) 19) Roger Narayan (See also Journals 20, 15) 20) Savas Tasoglu (See also Journals 17, 15) 21) Shoufeng Yang (See also Journals 6, 17) 22) Vladimir Mironov 23) Xiaohong Wang 24) Jia An 12. Progress in Additive Manufacturing Publisher Springer ISSN 2363-9520 URL http://www.springer.com/engineering/production+engineering/ journal/40964 ImpactF N/A H-Index N/A 27 SJR N/A Since 2016 Volumes 1 Issues 1 Articles 14 EiC Martin Schäfer, Cynthia Wirth Board and Editors 1) Henrique A. Almeida (See also Journal 4) 2) David Dean (See also Journal 6) 3) Fernando A. Lasagni 4) Eujin Pei (See also Journal 4) 5) Jan Sehrt 6) Christian Seidel 7) Adriaan Spierings 8) Xiaoyong Tian 9) Jorge Vilanova 10) Anath Fischer (See also Journal 6) 11) Russell Harris 12) Dachamir Hotza 13) Bernhard Müller 14) Nahum Travitzky 15) Gerd Witt (See also Journal 7) 16) Michael Friedrich Zäh (See also Journals 7, 1) 13. International Journal on Additive Manufacturing Technologies Publisher Additive Manufacturing Society of India ISSN 2395-4221 URL http://amsi.org.in/homejournal.html ImpactF N/A H-Index N/A SJR N/A Since 2015 Volumes 1 Issues 1 Articles 7 EiC Pulak M. Pandey (See also Journal 8), David Ian Wimpenny, Ravi Kumar Dwivedi Board and Editors 1) L. Jyothish Kumar (See also Journals 10, 8) 2) Keshavamurthy D. B. 3) Khalid Abdelghany 4) Suman Das 5) Alain Bernard (See also Journals 4, 8, 6) 6) C. S. Kumar 7) Bahram Asiabanpour (See also Journal 8) 8) K. P. Raju Rajurkar 9) Ehsan Toyserkani 10) Wan Abdul Rahman 11) Sarat Singamneni 12) Vijayavel Bagavath Singh 14. 3D Printing in Medicine Publisher Springer ISSN 2365-6271 URL http://www.springer.com/medicine/radiology/journal/41205 28 ImpactF N/A H-Index N/A SJR N/A Since 2015 Volumes 2 Issues 4 Articles 3 EiC Frank J. Rybicki Board and Editors 1) Leonid L. Chepelev 2) Andy Christensen 3) Koen Engelborghs 4) Andreas Giannopoulos 5) Gerald T. Grant 6) Ciprian N. Ionita 7) Peter Liacouras 8) Jane M. Matsumoto 9) Dimitrios Mitsouras 10) Jonathan M. Morris 11) R. Scott Rader 12) Adnan Sheikh 13) Carlos Torres 14) Shi-Joon Yoo 15) Nicole Wake 16) William Weadock 15. Bioprinting Publisher Elsevier B.V. ISSN 2405-8866 URL http://www.journals.elsevier.com/bioprinting ImpactF N/A H-Index N/A SJR N/A Since 2016 Volumes 1 Issues N/A Articles 1 EiC A. Atala Board and Editors 1) S. V. Murphy 2) T. Boland 3) P. Campbell 4) U. Demirci (See also Journal 17) 5) B. Doyle 6) J. Fisher 7) J. Y. H. Fuh (See also Journals 4, 3) 8) A. K. Gaharwar 9) P. Gatenholm 10) K. Jakab 11) J. Jessop 12) A. Khademhosseini (See also Journal 11) 13) S. J. Lee 14) I. Lelkes 15) J. Lim 16) A. G. Mikos 17) R. Narayan (See also Journals 20, 11) 18) T. Sercombe (See also Journals 4, 8) 19) A. Skardal 20) S. Tasoglu (See also Journals 17, 11) 21) D. J. Thomas 22) G. Vozzi (See also Journals 17, 11) 23) I. Whitaker (See also Journal 17) 24) S. K. Williams II 16. 3D Printing – Science and Technology 29 Publisher DE GRUYTER OPEN ISSN 1896-155X URL http://www.degruyter.com/view/j/3dpst ImpactF N/A H-Index N/A SJR N/A Since 2016 Volumes 0 Issues 0 Articles 0 EiC Haim Abramovich Board and Editors 1) Christopher A. Brown 2) Paolo Fino 3) Amnon Shirizly 4) Frank Walther 5) Kaufui Wong 17. Journal of 3D Printing in Medicine Publisher Future Medicine Ltd ISSN 2059-4755 URL http://www.futuremedicine.com/page/journal/3dp/editors. jsp ImpactF N/A H-Index N/A SJR N/A Since 2016 Volumes 0 Issues 0 Articles 0 EiC Dietmar W Hutmacher Board and Editors 1) Peter Choong 2) Michael Schuetz 3) Iain S. Whitaker (See also Journal 15) 4) Shoufeng Yang (See also Journals 6, 11) 5) Paulo Jorge Bártolo (See also Journals 4, 6, 11) 6) Luiz E. Bertassoni 7) Faiz Y. Bhora 8) Boris N. Chichkov (See also Journal 11) 9) Utkan Demirci (See also Journal 15) 10) Michael Gelinsky 11) Ruth Goodridge 12) Robert E. Guldberg 13) Scott J. Hollister 14) Zita M. Jessop 15) Jordan S. Miller 16) Adrian Neagu 17) Aleksandr Ovsianikov (See also Journal 11) 18) Katja Schenke-Layland 19) Ralf Schumacher 20) Jorge Vicente Lopes da Silva 21) Chris Sutcliffe 22) Savas Tasoglu (See also Journals 15, 11) 23) Daniel Thomas 24) Martijn van Griensven 25) Giovanni 30 Vozzi (See also Journals 15, 11) 26) David J. Williams 27) Chris J. Wright 28) Jing Yang 29) Nizar Zein 18. Smart and Sustainable Manufacturing Systems Publisher ASTM ISSN N/A URL http://www.astm.org/SSMS ImpactF N/A H-Index N/A SJR N/A Since 2017 Volumes 0 Issues 0 Articles 0 EiC Sudarsan Rachuri Board and Editors 1) Darek Ceglarek 2) Karl R. Haapala 3) Yinlun Huang 4) Jacqueline Isaacs 5) Sami Kara 6) Soundar Kumara 7) Sankaran Mahadevan 8) Lihong Qiao 9) Roberto Teti 10) Manoj Kumar Tiwari (See also Journal 1) 11) Shozo Takata 12) Tetsuo Tomiyama 13) Li Zheng 14) Fazleena Badurdeen 15) Abdelaziz Bouras 16) Alexander Brodsky 17) LiYing Cui 18) Bryony DuPont 19) Sebti Foufou 20) Pasquale Franciosa 21) Robert Gao 22) Moneer Helu 23) Sanjay Jain 24) I. S. Jawahir 25) Sagar V. Kamarthi 26) Jay Kim 27) Minna Lanz 28) Kincho H. Law 29) Mahesh Mani 30) Raju Mattikalli 31) Michael W. McKittrick 32) Shreyes N. Melkote 33) P. V. M. Rao 34) Utpal Roy 35) Christopher J. Saldana 36) K. Senthilkumaran 37) Gopalasamudram R. Sivaramakumar 38) Eswaran Subrahmanian 39) Dawn Tilbury 40) Conrad S. Tucker 41) Anahita Williamson 42) Paul William Witherell 43) Lang Yuan 44) Rakesh Agrawal 45) Dean Bartles 46) Gahl Berkooz 47) Jian Cao 48) S. K. Gupta 49) Timothy G. Gutowski 50) Gregory A. Harris 51) Rob Ivester 52) Mark Johnson 53) Thomas Kurfess 54) Bahram Ravani 55) William C. Regli 56) S. Sadagopan 57) Vijay Srinivasan 58) Ram D. Sriram 59) Fred van Houten 60) Albert J. Wavering 19. Powder Metallurgy Progress Publisher DE GRUYTER OPEN 31 ISSN 1339-4533 URL http://www.degruyter.com/view/j/pmp ImpactF N/A H-Index N/A SJR N/A Since N/A Volumes 0 Issues 0 Articles 0 EiC Beáta Ballóková Board and Editors 1) Katarı́na Ondrejová 2) Herbert Danninger 3) Eva Dudrová 4) Marco Actis Grande 5) Abolghasem Arvand 6) Csaba Balázsi 7) Sergei M. Barinov 8) Frank Baumgärtner 9) Paul Beiss 10) Sigurd Berg 11) Michal Besterci 12) Jaroslav Briančin 13) Francisco Castro 14) Andrzej Cias 15) Ján Dusza 16) Juraj Ďurišin 17) Štefan Emmer 18) Sergei A. Firstov 19) Christian Gierl-Mayer 20) Eduard Hryha 21) Pavol Hvizdoš 22) Jan Kazior 23) Jacob Kübarsepp 24) Alberto Molinari 25) John R. Moon 26) Ľudovı́t Parilák 27) Doan Dinh Phuong 28) Raimund Ratzi 29) Wolf D. Schubert 30) František Simančı́k 31) Marin Stoytchev 32) Andrej Šalak 33) José M. Torralba 34) Andrew S. Wronski 35) Timothy Martin 36) Radovan Bureš 20. 3D-Printed Materials and Systems Publisher Springer ISSN 2363-8389 URL http://www.springer.com/materials/journal/40861 ImpactF N/A H-Index N/A SJR N/A Since N/A Volumes 0 Issues 0 Articles 0 EiC Roger J. Narayan (See also Journals 15, 11) Board and Editors 1) Vipul Dave 2) Mohan Edirisinghe 3) Sungho Jin 4) Soshu Kirihara 5) Sanjay Mathur 6) Mrityunjay Singh 7) Pankaj Vadgama 32 Furthermore, the following journals are identified from the literature relevant to this review. Journals catering specifically or explicitly to AM, RM, RP and 3D printing are listed above. The list contains only journals with more than 2 publications. The goal for composing this list is to enable other researches to identify possible publication venues for their work. The list is sorted by the number of publications in our bibliography for each identified journal. The number of each entry indicates the number of publications for the journal. 11 The International Journal of Advanced Manufacturing Technology (See Journal 1) 11 Rapid Prototyping Journal (See Journal 4) 7 Computer-Aided Design33 , ISSN: 0010-4485 6 Robotics and Computer-Integrated Manufacturing (See Journal 3) 6 Journal of Manufacturing Science and Engineering (See Journal 2) 5 Journal of Materials Processing Technology34 , ISSN: 0924-0136 4 International Journal of Computer Integrated Manufacturing35 , ISSN: 1362-3052 4 CIRP Annals - Manufacturing Technology36 , ISSN: 0007-8506 3 Journal of Manufacturing Systems37 , ISSN: 0278-6125 3 Computers in Industry38 , ISSN: 0166-3615 2 Virtual and Physical Prototyping (See Journal 6) 2 Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture39 , ISSN: 2041-2975 33 http://www.journals.elsevier.com/computer-aided-design http://www.journals.elsevier.com/journal-of-materials-processing-technology 35 http://www.tandfonline.com/toc/tcim20/current 36 http://www.journals.elsevier.com/cirp-annals-manufacturing-technology 37 http://www.journals.elsevier.com/journal-of-manufacturing-systems 38 http://www.journals.elsevier.com/computers-in-industry 39 http://pib.sagepub.com 34 33 2 Journal of Intelligent Manufacturing40 , ISSN: 1572-8145 2 International Journal of Production Research41 , ISSN: 1366-588X 2 International Journal of Machine Tools and Manufacture42 , ISSN: 08906955 2 IEEE Transactions on Industrial Informatics43 , ISSN: 1551-3203 2 Enterprise Information Systems44 , ISSN: 1751-7583 2 Applied Mechanics and Materials45 , ISSN: 1662-7482 2 Advanced Materials Research46 , ISSN: 1662-8985 4. Reviews on the Subject The topic of AM in general and its special applications, technologies and directions is extensively researched and results published in literature. The growth of the number of publications as found by Google Scholar and Proquest is illustrated in the following figures (See Fig. 10 and Fig. 11). An analysis of literature within this domain from sources (See Sect. 1.1.2) for scientific literature shows an increase in the number of published works from 2002 to 2016 of 41.3 % on average (See Fig. 10), respectively 26.1 % for the search engine Proquest. This number is from the average of the average growth of results found for keywords related to specific AM topics and AM related literature in general from http://scholar.google.com. In this section we will present the findings of the analysis of available data on the scientific publications. Specific aspects of AM, 3D printing and associated areas are topic of a number of reviews listed below. The list of reviews is compiled by searching on the previously mentioned search engines (See sect. 1.1.2) using a keyword 40 http://link.springer.com/journal/10845 http://www.tandfonline.com/toc/tprs20/current 42 http://www.journals.elsevier.com/international-journal-of-machine-tools-and-manufacture 43 http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=9424 44 http://www.tandfonline.com/toc/teis20/current 45 http://www.scientific.net/AMM 46 http://www.scientific.net/AMR 41 34 Average number of publications per year (Google Scholar) Average annual growth 128 112 10000 96 8000 80 6000 64 48 4000 32 2000 16 16 15 20 14 20 13 20 12 20 11 20 10 20 09 20 08 20 07 20 06 20 05 20 04 20 20 20 03 0 02 0 percent; avg=41.32 144 12000 20 publications; avg=3196.81 Average number of publications 14000 period source of data: scholar.google.com Figure 10: Average number of publications and annual average growth for the combined results from scholar.google.com for 2002–2016 Average number of publications per year (ProQuest) Average annual growth 117 99 6000 81 4000 63 45 2000 27 9 0 percent; avg=26.14 135 8000 6 5 20 1 4 20 1 3 20 1 2 20 1 1 20 1 0 20 1 9 20 1 8 20 0 7 20 0 6 20 0 5 20 0 20 0 4 20 0 20 0 20 0 3 -9 2 publications; avg=441.93 Average number of publications period source of data: proquest.com Figure 11: Average number of publications and annual average growth for the combined results from proquest.com for 2002–2016 35 search. The keywords are “3D Printing” +Review/Survey/“State of the Art”, “Additive Manufacturing” +Review/Survey“State of the Art”, “Rapid Manufacturing” +Review/Survey“State of the Art”. The time range for the search for reviews is restricted from 2005 to 2016. Following this literature search a backward search on the results is performed. From the 70 reviews identified we calculate the average number of authors per review to be 3.3 with an average length of 15.2 pages. The list is sorted chronologically with the general theme or domain of the review provided. 1. Dimitar Dimitrov, Kristiaan Schreve and N. de Beer [61]; General Introduction, Applications, Research Issues 2. Vladimir Mironov, Nuno Reis and Brian Derby [172]; Bioprinting, Technology 3. Ben Utela et al. [252]; New Material Development (Mainly Powders) 4. Abbas Azari and Sakineh Nikzad [21]; Dentistry, Applications in Dentistry 5. Hongbo Lan [143]; Rapid Prototyping, Manufacturing Systems 6. Daniel Eyers and Krassimir Dotchev [72]; Rapid Manufacturing, Mass Customisation 7. Ferry P. W. Melchels, Jan Feijen and Dirk W. Grijpma [170]; Stereolithography, Biomedical Engineering 8. Fabian Rengier et al. [207]; Medicine, Data Acquisition (Reverse-Engineering) using Image Data 9. R. Sreenivasan, A. Goel and D. L. Bourell [227]; Energy Consumption, Sustainability 10. Rupinder Singh [224]; Rapid Prototyping, Casting 11. R. Ian Campbell, Deon J. de Beer and Eujin Pei [44]; Application and Development of AM in South Africa 12. Benjamin Vayre, Frédéric Vignat and François Villeneuve [256]; Metal Components, Technology 36 13. Dongdong Gu et al. [93]; Metal Components, Technology, Terminology 14. Ferry P. W. Melchels et al. [169]; Medicine, Tissue and Organ Engineering 15. Kaufui V. Wong and Aldo Hernandez [271]; General, Technology 16. Lawrence E. Murr et al. [182]; Metal Components, EBM, Laser Melting 17. Shawn Moylan et al. [179]; Quality, Test Artifacts 18. Timothy J. Horn and Ola L. A. Harrysson [109]; General, Applications, Technology 19. Xibing Gong, Ted Anderson and Kevin Chou [90]; EBM, Powder Based AM 20. Flavio S. Fogliatto, Giovani J.C. da Silveira and Denis Borenstein [77]; Mass-Customization 21. K. P. Karunakaran et al. [125]; Rapid Manufacturing, Metal Object Manufacturing 22. Carl Schubert, Mark C. van Langeveld and Larry A. Donoso [217]; General 23. Irene J. Petrick and Timothy W. Simpson [195]; Economics, Business 24. Iulia D. Ursan, Ligia Chiu and Andrea Pierce [251]; Pharmaceutical Drug Printing 25. Jasper Cerneels et al. [47]; Thermoplastics 26. Mohammad Vaezi, Hermann Seitz and Shoufeng Yang [253]; MicroStructure AM 27. Nannan Guo and Ming C. Leu [97]; General, Technology, Materials, Applications 28. Olga Ivanova, Christopher Williams and Thomas Campbell [118]; NanoStructure AM 29. Robert Bogue [30]; General 37 30. Samuel H. Huang et al. [113]; Socio-Ecological and Economy 31. Zicheng Zhu et al. [305]; Hybrid Manufacturing 32. Dazhong Wu et al. [272]; Cloud Manufacturing 33. Bethany C. Gross et al. [92]; Biotech, Chemistry 34. Brett P. Conner et al. [57]; Classification, Object Complexity 35. Brian N. Turner, Robert Strong and Scott A. Gold [248]; Thermoplastics, Physical Properties 36. David W. Rosen [210]; Design for Additive Manufacturing 37. Dimitris Mourtzis, Michael Doukas and Dimitra Bernidaki [178]; Simulation 38. Douglas S. Thomas and Stanley W. Gilbert [242]; Economy, Cost 39. Gustavo Tapia and Alaa Elwany [239]; Process Monitoring, Quality 40. Hae-Sung Yoon et al. [291]; Energy Consumption 41. Jan Deckers, Jef Vleugels and Jean-Pierre Kruth [59]; Ceramics AM 42. Rouhollah Dermanaki Farahani, Kambiz Chizari and Daniel Therriault [74]; Micro-Structure AM 43. Siavash H. Khajavi, Jouni Partanen and Jan Holmström [131]; Supply Chain, Application 44. William E. Frazier [79]; Metal Components 45. Wu He and Lida Xu [102]; Cloud Manufacturing 46. Syed Hasan Massod [163]; Fused Deposition Modeling (FDM) 47. Brian N. Turner and Scott A Gold [247]; Thermoplastic AM, Material Properties 48. Carlos Mota et al. [177]; Medicine, Tissue Engineering 49. C. Y. Yap et al. [290]; SLM 38 50. Donghong Ding et al. [62]; Metal Components, Wire Fed Processes 51. Adamson et al.[11]; Cloud Manufacturing, Terminology 52. Jie Sun et al. [231]; Food Printing, Technology 53. Jin Choi et al. [54]; 4D Printing 54. K. A. Lorenz et al. [158]; Hybrid Manufacturing 55. Merissa Piazza and Serena Alexander [197]; General, Terminology, Academic 56. Omar A. Mohamed, Syed H. Masood and Jahar L. Bhowmik [176]; Process Parameter Optimization (FDM) 57. Seyed Farid Seyed Shirazi et al. [220]; Tissue Engineering, Powder Based AM 58. Sheng Yang and Yaoyao Fiona Zhao [288]; Design for AM, Complexity 59. Sofiane Guessasma et al. [95]; Design for AM, Process Parameter Optimization 60. Wei Gao et al. [83]; General, Technology, Engineering 61. Yong Huang et al. [115]; General, Technology, Research Needs 62. Zhong Xun Khoo et al. [133]; Smart Materials, 4D Printing 63. Hammad H. Malik et al. [162]; Medicine, Surgery 64. Jie Sun et al. [232]; Food Printing 65. Behzad Esmaeilian, Sara Behdad and Ben Wang [71]; Manufacturing 66. H. Bikas, P. Stavropoulos and G. Chryssolouris [28]; General, Technology 67. Julien Gardan [84]; Technology, Engineering, Manufacturing 68. Swee Leong Sing et al. [223]; Metal Components, Medicine, Implants, Materials 39 69. William J. Sames et al. [214]; Metal Components, Materials 70. Andrew J. Pinkerton [198]; Laser-technology 4.1. Stakeholder Distinction Different 3D printing technologies, machines and manufacturers as well as services target different clients for which we propose the following classification. Generally the discerning factors are 1. cost per machine 2. quality of print (e.g., surface quality, physical properties of object) 3. reliability of machine and 4. materials available. From literature the three classes of audience are apparent: • consumer/end user • professional user • industrial application For the consumer a very important factor is the cost of the printer itself with 45 % of consumers are not willing to pay more than $US 299 for a 3D printer [164]. In recent years the price of entry level consumer 3D printers, especially for build-kits, decreased to about $US 30047 . Open-source projects like RepRap have contributed to the decline of costs for these machine [225]. In Fig. 12 we differentiate between the user groups of end-users/consumer, professional users and industrial users. Industrial users rely on high quality available with a large selection of processable materials. Machines for these users are expensive and out of reach of most end-users and professional users. The quality these machines produce is very high and the objects can be used for integration in a product or be a product themselves. Due to these restrictions the availability of such machines is not very wide spread but limited to highly specialised enterprises. On the other end of the spectrum the end-user/consumer has a large choice of 3D printers to select from, they are relatively inexpensive, produce objects of acceptable quality, work on a much lower number materials (typically thermoplastics) and have a reliability that is lower than the reliability 47 XYZPrinting da Vinci Jr. 1.0, XYZprinting-Vinci-Jr-1-0-Printer $US 297.97, 40 https://www.amazon.com/ of professional equipment. In the middle of the spectrum we see professional users, e.g., from design bureaus or architects, that use such machines in a professional manner, draw benefits from the usage of such technology but it is mostly not their main concept of business. In an example, an architect makes use of a 3D printer for the creation of a high-quality model of a building he designed, which is faster and easier than making such a model by hand. Reliability Material Availability Consumer Professional Industrial Cost per Machine Quality Availabilty Figure 12: Audience classification and Expectations 5. 3D Printing Services There are numerous dedicated 3D printing services available to end-users, professionals and industrial users. They differ in the clients they address, the services they offer, the quality they can provide and the cost they charge. In this section we give an overview of a selection of available 3D printing services. The list is not conclusive as a number of enterprises does offer 3D printing services in their portfolio but they are not necessarily to be considered 3D printing services due to either their local mode of operation or the number of 3D printers the user can chose from. This overview is closely based on the work of [204] and extends its findings. We use the following list of properties to distinguish the services: • The target group (End-users, industrial users or professional users) • The local reach (Local or global) 41 • Availability of an API • Services rendered (Design, 3D printing, marketplace, other) Rayna and Striukova [204] base their exploratory study on the following list of services they have identified. For the original list of services we add the following information. • 3D Burrito48 - Pre-Launch Phase • 3D Creation Lab49 • 3DLT50 - Shut down on 2015-12-31 • 3DPrintUK51 • Additer.com52 - Unreachable • Cubify Cloud53 - Acquired by 3D Systems, Service no longer available • i.Materialise54 • iMakr55 • Kraftwürx.com56 • MakerBot/Thingiverse57 • MakeXYZ58 • Ponoko59 48 http://3dburrito.com http://www.3dcreationlab.co.uk 50 http://3dlt.com 51 https://www.3dprint-uk.co.uk 52 http://additer.com 53 http://cubify.com 54 https://i.materialise.com/ 55 http://imakr.co.uk 56 http://www.kraftwurx.com 57 http://thingiverse.com 58 https://www.makexyz.com 59 https://www.ponoko.com/ 49 42 • Sculpteo60 • Shapeways61 For this study we extend the selection with the additional services listed in Tabs. 1 and . 2. Services omitted in these two tables are described in the original study. In contrast to the authors of the original work we think that an exhaustive list of such services is impossible to compile as a large number of local businesses do offer 3D printing services over the Internet and would therefore qualify to be included in such a list. These (local) businesses are hard to identify due to their limited size and reach. Also, an exhaustive list would need to contain 3D printing services and repositories of which many similar and derivative services exist. Further, we extend the classification and study to the provisioning of an API by the respective service. An API should provide methods to use the service programmatically. With an API such printing services can be used as a flexible production means in CM settings. The range of functionality of such APIs can vary significantly and range from the possibility of having a widget displayed on a website with a 3D model viewer, to upload and store digital models in a repository, request quotes for manufacturing or digital fabrication. A commonality for these APIs is the requirement for the thirdparty user to have an account with the service, which is indicated in Tabs. 3 and 4 by Implementer in the column Required for registration. The indication User in this column indicates that the user must be registered with this service too. The implementer registration is intended for scenarios where the API is embedded in a service or website that a third party user then uses. The findings of this study are presented in Tabs. 3 and 4, where we state whether the service provides an API and if it is publicly available or only accessible for business partners, who needs to be registered for the usage of the API and what capabilities the API provides (See Tab. 5). This explorative extension study is performed as described by the original authors. 60 61 https://www.sculpteo.com http://www.shapeways.com/ 43 Table 1: 3D printing platforms and services included in this study – Part 1 Company/Service Name URL 3Faktur http://3faktur.com 3DaGoGo 3DExport 3DHubs 3DPrinterOS 3DShook 3D Warehouse Autodesk 123D Clara.io CreateThis Cults Grabcad La Poste Libre3D Makershop ClassificationEstablished Locati Modeling 2014 Germa Service https://www.3dagogo.com Marketplace 2013 USA https://3dexport.com Marketplace, 2004 USA Repository http://3dhubs.com Crowd 2013 USA Printing Provider https://www.3dprinteros.com Crowd 2014 USA Printing Provider http://www.3dshook.com Marketplace, 2014 Israel Subscription Service https://3dwarehouse.sketchup.com Marketplace, 2006 USA Community, Repository http://www.123dapp.com Software, 2009 USA Marketplace, Repository https://clara.io Repository, 2013 Canad Modeling http://www.createthis.com Marketplace 2013 USA France https://cults3d.com Marketplace, 2013 Repository, Design Service https://grabcad.com Software, 2009 USA Marketplace, Repository http://impression3d.laposte.fr Print 2013 France Provider, 44 Marketplace http://libre3d.com Marketplace, 2014 USA Repository https://www.makershop.co Marketplace, 2013 USA Repository Table 2: 3D printing platforms and services included in this study – Part 2 Company/Service Name NIH 3D Print Exchange URL http://3dprint.nih.gov p3d.in Pinshape REPABLES Rinkak https://p3d.in https://pinshape.com http://repables.com https://www.rinkak.com shapeking http://www.shapeking.com Shapetizer https://www.shapetizer.com Sketchfab https://sketchfab.com stlfinder http://www.stlfinder.com STLHive http://www.stlhive.com Stratasys Direct Express https://express.stratasysdirect.com Threeding https://www.threeding.com Tinkercad https://www.tinkercad.com Treatstock https://www.treatstock.com 45 trinckle https://www.trinckle.com Trinpy https://www.trinpy.com Classification CoCreation, Repository Modeling Marketplace Repository Marketplace, Repository, Crowd Printing Provider Marketplace, Repository Marketplace, Repository, Print Provider Marketplace, Repository Search Engine Marketplace, Repository, Design Service Print Provider Marketplace, Print Provider Design, Repository Marketplace, Community, Crowd Printing Provider Print Provider Marketplace, Table 3: 3D printing platforms and services and their APIs - Part 1 Company / Service Name 3Faktur 3DaGoGo 3DExport 3DHubs 3DPrinterOS 3DPrintUK 3DShook 3D Creation Lab 3D Warehouse Autodesk 123D Clara.io CreateThis Cults Grabcad iMakr i.Materialise Kraftwürx.com La Poste Libre3D MakerBot / Thingiverse Makershop MakeXYZ Materflow MeltWerk Provides Required Capabilities Reach Target Group an API for registration No N/A N/A Regional Consumer No N/A N/A Global Consumer No N/A N/A Global Consumer + Professional Yes Implementer Upload Global Consumer + User No N/A N/A Global Consumer No N/A Global Consumer No N/A N/A Global Consumer No N/A N/A Global Consumer No N/A N/A Global Consumer N/A No N/A Global Consumer Yes Implementer Upload, Global Consumer + Professional Modify, Retrieve No N/A N/A Global Consumer Yes Implementer View, Global Consumer (not Republic) trieve No N/A N/A Global Consumer + Professional No N/A N/A Global Consumer Yes Implementer Upload, Global Consumer + Professional Quoting, Order Yes Implementer Upload, Global Consumer (not Order public) No N/A N/A Regional Consumer No N/A N/A Global Consumer Yes Implementer Upload, Global Consumer Retrieve Yes Implementer Search, Global Consumer + Professional 46 Retrieve Yes Implementer Order Global Consumer + Professional + User No N/A N/A Global Consumer Yes Implementer Upload, Global Consumer Table 4: 3D printing platforms and services and their APIs - Part 2 Company / Service Name NIH 3D Print Exchange p3d.in Pinshape Ponoko REPABLES Rinkak Sculpteo shapeking Shapetizer Shapeways Sketchfab stlfinder STLHive Stratsys Direct Express Threeding Tinkercad Treatstock trinckle Trinpy TurboSquid UPS Yeggi Provides Required Capabilities Reach an API for registration Yes Implementer Upload, Global Retrieve No N/A N/A Global No N/A N/A Global No N/A N/A Global No N/A N/A Global Yes Implementer View, Global Order, Modeling Yes Implementer Upload, Global + User Retrieve, Quoting, Order No N/A N/A Global No N/A Global N/A Yes Implementer Upload, Global + User Quoting, Order Yes Implementer Upload, Global View No N/A N/A Global No N/A N/A Global No N/A N/A Regional No N/A N/A Global No N/A N/A Global Yes Implementer Upload, Global Retrieve 47 No N/A N/A Global No N/A N/A Global No N/A N/A Global No N/A N/A Regional Yes Implementer Search, Global Re- Target Group Consumer Consumer Consumer Consumer Consumer Consumer Consumer + Professional Consumer Consumer Consumer + Professional Consumer Consumer Consumer + Professional Professional Consumer Consumer Consumer Consumer + Professional Consumer Consumer + Professional Consumer Consumer Table 5: Categorising 3D printing online platforms Company / Service Name Design Design marreposiket tory place 3Faktur 3DaGoGo + 3DExport + 3DHubs 3DPrinterOS 3DPrintUK 3DShook + 3D Creation Lab 3D Ware- + house Autodesk + 123D Clara.io + CreateThis + Cults + Grabcad + iMakr i.Materialise + Kraftwürx.com + La Poste + Libre3D + MakerBot + / Thingiverse Makershop + MakeXYZ Materflow + MeltWerk MyMiniFactory + NIH 3D + Print Exchange p3d.in + Pinshape + Ponoko REPABLES + Rinkak + Sculpteo Design Printing Printing Printer Crowd sermarket service sale sourcvice place ing platform + + Editor + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + p + + + + + p + + + + + + + + + + + + + + 48 + + + + + + + As analysed in Tab. 5, the services surveyed offer a different range of services each. No provider could be identified that offers a complete set of service for 3D printing and related tasks. In the table, the indication of p marks companies that do not themselves offer printers through this service but their parental companies do. The o character in the column for printing service for Tinkercad and YouMagine, indicates that the service itself does not render printing services, but has a cooperation with a third party for the provisioning of this service. With the exception of La Poste, UPS and iMakr all the services render their business completely on the Internet without the requirement for physical interaction. La Poste and UPS offer an Internet interface with the physical delivery of the objects in certain shops of theirs. Services that offer a design market place can offer designs and other files costless or for a fee, no distinction is made for this study. Yeggi and stlfinder are search engines for 3D model data that work on the data from other sources. Albeit a search engine, Yeggi provides the integration of printing services and cloud printing services for models available from third party services, thus Yeggi can be classified as a service of services. The service rendered by Trinpy is subscription based with various membership options. Grabcab provides 3D printing planning and control services, and integration with an online editor. 6. Review Cloud Manufacturing is mainly an overlapping manufacturing or engineering concept with application and grounding the development of parts or objects in “traditional” manufacturing. With traditional manufacturing we denote all technologies and methods to create or fabricate objects or parts other than AM. For a distinction between manufacturing methods see Klocke [136], Nee [185] and the DIN Standard 8580 [1]. In this sense all subtractive or formative manufacturing methods are summarises as “traditional manufacturing” methods. As AM offers a large degree of flexibility due to short lead times as well as other beneficial properties, we see that AM is the ideal technology to be considered within CM scenarios. Taking the properties of AM into account we do not predict that AM will replace other manufacturing methods, not even within CM scenarios. Rather AM will fill niches for special applications like mass-customisation, rapid replacement production capabilities or RT, especially within CM scenarios. With this work we aim to contribute to the development of AM methodology and 49 technology in the CM paradigm. 6.1. Topological Map In Fig. 13 the relationship and connection of various concepts relevant to CM is described. This map forms the basis of the following review, where the nodes from the map represent sections from the review where we present the current state of research and elaborate on open research questions. The topics are extracted from literature. This topological map displays the relationship of CM with a variety of connected and enabling technologies and concepts. Additive Manufacturing (See Sect. 6.10) enables CM to be more modular, flexible and offers new capabilities and business opportunities. The Rapid Technology (See Sect. 6.8) and its composition Rapid Prototyping (RP, see Sect. 6.8.3), Rapid Manufacturing (RM, see Sect. 6.8.2) and Rapid Tooling (RT, see Sect. 6.8.1) are areas in which CM can be applied. The topic of Service Orientation (Sect. 6.7) and its composition “as-a-Service” of which Design-as-a-Service (DaaS, see Sect. 6.7.2), Testing-as-a-Service (TaaS, see Sect. 6.7.3) and Manufacturingas-a-Service (MaaS, see Sect. 6.7.1) are explored as examples, are concepts that enable the efficient application of CM. For a broader understanding it is required to research the stakeholders involved in this technology which makes Sect. 6.6. The topics of Scheduling (See Sect. 6.12) and Resource Description (See Sect. 6.13) are to be discussed for the universal and efficient application of CM. The domain of Simulation (See Sect. 6.5) with its composition of Optimisation (See Sect. 6.5.2) and Topological Optimisation (See Sect. 6.5.1) enable a more rapid, more flexible and more robust usage of the technology. For AM technology the application of Topology Optimisation enables the benefits of this technology. Similar to AM is 3D printing (See Sect. 6.4) with its subtopic of Accuracy and Precision (See Sect. 6.4.1) as this technology is a appropriate basis for CM systems. The topic of Hybrid Manufacturing (See Sect. 6.14) gains importance in flexible and agile manufacturing systems which warrants and requires its research. In the topic of Technology (See Sect. 6.2) the general principles and technologies of CM and AM are discussed as these are basic principles for the efficient implementation of these systems. The topic of Cloud Computing (CC, see Sect. 6.11) with its sub-components Internet of Things (IoT, see Sect. 6.11.1) and Cyberphysical Systems (CPS, see Sect. 6.11.2) is the conceptual progenitor of CM and therefore requires careful studying. IoT and CPS are key enabling technologies for CM. The topic of Security 6.3 is of increasing importance with the 50 Figure 13: Topological Map for Cloud Manufacturing spreading application of AM and CM as attack surfaces grow and potential damage increases. 6.2. Technology A large number of technologies and technological advances have made AM possible to evolve from its origin as a RP method to its current state where it is used for end-part manufacturing (RM) and available to consumers [88, 269]. All 3D printed objects are based on a digital model. This model can either be created using CAD software, 3D sculpting software or acquired using reverseengineering methods (e.g., object scanning or photo reconstruction) [89]. Albeit direct slicing from a CAD model has been proposed by Jamieson [119] in 1995 it is still rarely performed. Direct slicing requires implementation in the CAD software for each printer type and printer manufacturer, which is not feasible. Further shortcomings of the de-facto standard file format for AM, i.e., STL, namely the possibility to contain mis-aligned facets, holes or to be non-watertight, as well as being to large in file size are reported by [281]. 51 Besides a Steiner-patch based [193] file format to replace the STL file format the ASTM Committee F42 has published an ISO Standard [10] for the AMF (Additive Manufacturing File Format) with the same intention. Both file formats are created to increase the accuracy for the models described and therefore increase the quality of the resulting printed objects. STL seems to be the prevalent file format for AM with 25700 results on Google Scholar compared to 8230 results for AMF. Further investigation in the file support for different hard- and software vendors is warranted but out of the scope of this work. The review by Dimitrov et al. [61] presents further information on the technology that AM is based on with an overview of applications for it. In the review by Esmaeilian et al. [71] the authors present the relationship of AM and Manufacturing in general as well its benefits. With the emergence of Internet or cloud based CAD Modelling software the creation of models for AM becomes easier as direct integration of 3D printing providers is possible. Furthermore, the collaborative aspect of 3D modelling is enhanced as studied by Jou and Wang [122]. This study used a group of college students as a test group and investigated the adoption of an online CAD modelling ( Autodesk AutoCAD 62 ) software in the curriculum. The authors Andreadis et al. [18] present a case study on the adoption of an unnamed cloud based CAD system in comparison to traditional software, as well as an exhaustive list of benefits of cloud based software. Wu et al. [277] present an economic analysis of cloud based services for design and manufacturing. This work also explores a number of cloud based services along with their pricing. Communities are of great importance to enterprises as shown in West and Kuk [269]. One form of community is a repository for 3D printable digital models that collects and curates models supplied by users for collaboration, exchange, co-creation and sale. In this work the authors conduct a study to research the profit of catering for such a community/repository (Thingiverse) by a former open-source company (Makerbot). Wittbrodt et al. [270] performed experiments to determine the ROI (Return on Investment) of 3D printers for common households and their feasibility in application in end-user scenarios. With their experiment they concluded that an average household can achieve between 40 and 200 percent 62 http://autodesk.com/products/autocad 52 ROI on average usage of such machines. 6.3. Security Security for 3D Printing, AM or CM can be discussed from at least three perspectives. The first perspective would be the legal security of data and models processed within such a scenario. Discussions can range from whether it is legal to manufacture an existing object (replication) which might be protected by intellectual copyright laws to questions regarding product liability in case of company supplied model data. The second perspective is closely related to intellectual property (IP) as it is the technological discussion about the safeguarding of digital model files and data. The third perspective is about the data and process security itself in scenarios with malicious thirdparties (e.g., Hackers, Cyber-criminals). This third perspective is not limited to AM but shares many problems with CC and computing in general. Dolinksy [65] analyses the copyright and its application to 3D Printing for the jurisdiction of the USA. Because legal systems are different to each other such an analysis can not be exhaustive. Grimmelmann [91] further exemplifies the legal status of 3D printing and model creation in the USA with fictitious characters from literature and theatre. He states that the creation of an object irregardless of the source of model for such a creation is infringing on copyright if the object that is replicated is protected by copyright. In [260] the author discusses the current situation of 3D printing in regard to gun laws. This discussion was started by the media in 2013 as models for a functional plastic gun were distributed and the gun manufactured. The author states that current gun control laws are adequate to control 3D printed weapons and that this is currently not a big issue. On a broader scope the authors McNulty et al. [167] research the implications of AM for the national security of the USA where the authors present the benefits of bio or tissue 3D printing for the treatment of battlefield wounds as well as the implications of AM technologies for criminal misconduct. For the analysis of data security the authors Wu et al. [276] present the importance of such technologies within a CM environment. They propose the development of trust models for cyber-physical systems respectively the actors within such systems. The authors of Yampolskiy et al. [284] provide a full risk analysis of a scenario for outsourcing AM under consideration of IP. The risk assessment 53 does not include malicious behaviour other than IP infringement. To secure printed objects against counterfeiting the authors of [75] study and recommend the use of chemical components for authentication. Possible attacks on the 3D printing process by third parties is researched in [294] where one scenario is about the introduction of wilfully integrated material differences into an object in order to weaken the object under load. If the printing process itself is secured the question remains if a printed object is the original, a genuine replicate or a faked replicate. For the identification of genuine objects the authors of [14] research the applicability of physical signatures to 3D printed objects. In [110] the authors present a watermarking technique for 3D printed objects that is resilient against repeated digital scanning of the manufactured object. For a generalised discussion on security of cloud services and cloud computing we refer to [230] where the authors present issues ranging from data integrity to confidentiality. The concepts and terminology of CC security are also discussed in [306] of which the concept of confidentiality, trust and privacy are most relevant to scenarios of cloud based AM where users have physical objects created from digital models by third parties. Sturm et al. [229] present attack scenarios and mitigation strategies for attacks on AM system. The authors see rising CPS implementations in AM as potential intrusion vectors for attacks. The authors discuss various attacks for each of the manufacturing process phases. Furthermore, the authors identify the STL file format as a potential risk for tampering and attacking. Among the recommendations for mitigation is file hashing and improved process monitoring. Bridges et al. [39] briefly explore possible attacks on the cyber-physical systems that are used for AM. Among the attack scenarios the authors identify theft and tampering. 6.4. 3D Printing Following the distinction between AM and 3D printing given in the definition of 3D printing (See Sect. 2.1.2) by some authors into high-quality professional or industrial usage and lower-quality end-user or semi-professional usage 3D printing could not be part of CM. As we relax the definition of AM and 3D printing and use the terms as synonyms, we survey technological developments within this chapter. Technological progress and development 54 are essential to the widespread use and application of 3D Printing or AM in the CM paradigm. In the short article by Hansen et al. [100] the authors propose a measurement method for the correction or calibration of FDM printers. For this purpose the authors develop a measurement plate that is printed with specified parameters. In their experiment the authors recorded roundness errors of up to 100 µm. The calibration could not be applied due to the printer control software being closed-source. Anitha et al. [19] analyse the process variables layer thickness, bead width and deposition speed for their influence on the quality of objects manufactured using FDM. The authors find that the layer thickness is contributing with approximately 50 % to the surface roughness of the manufactured objects. Balogun et al. [22] describe and an experiment on the energy consumption and carbon footprint of models printed using FDM technology. They define three specimens of 9000 mm3 and 18000 mm3 volume which are printed on a Stratasys dimension SST FDM. Their experiment also captures the energy consumption of the post processing with and ultrawave precision cleaning machine. The energy consumed for the print is approximately 1 kWh. Over 60 % of the energy is consumed in non-productive states, e.g., pre-heating. This energy consumption profile warrants high utilisation of 3D printers when aiming for a low ecological impact and penalises frequent and long idle times of the 3D printer. Brajlih et al. [37] propose a comparison method for the speed and accuracy of 3D printers. As a basis the authors introduce properties and capabilities of 3D printers. A test-object designed by the authors is used to evaluate the average manufacturing speed of an Objet EDEN330 Polyjet and 3D Systems SLA3500 SLA manufacturing machine in an experiment. Furthermore, the experiment includes an EOS EOSINT P385 SLS and Stratasys Prodigy Plus FDM machine. The experiment concludes that the SLS machine is capable of the highest manufacturing speed (approx. 140 cm3 /h). In the experiment the angular and dimensional deviations are significant (up to 2.5° for a 90° nominal, and 0.8 mm for a 10 mm nominal). Roberson et al. [208] develop a ranking model for the selection of 3D printers based on the accuracy, surface roughness and printing time. This decision making model is intended to enable consumers and buyers of such hardware to select the most appropriate device. Utela et al. [252] provide a review on the literature related to the devel55 opment of new materials for powder bed based 3D printing systems. They decompose the development into five steps, for which they provide information on the relevant literature. Brooks et al. [40] perform a review on the history and business implications of 3D printing. They argue that the most promising approach for companies to benefit from 3D printing technology is to invest in and adapt current business models to support supplementary printing for the users. They also present the importance of the DMCA (Digital Millennium Copyright Act) in the USA under the aspect of 3D printing for current and upcoming businesses and services in the USA. Bogue [30] aims to provide an introduction into 3D printing with this review. The historical development of the various printing technologies is presented and furthermore, applications with examples are explored. Petrick and Simpson [195] compare traditional manufacturing, which they classify as “economy of scale”, with AM. AM is classified by the authors as “economy of one”. They base their future hypotheses on the traditional design-build-deliver model and current patterns in supply chains from which they draw logical conclusion for future developments. These hypotheses are sparsely supported by literature. They predict that in the future the boundaries between the design-build-deliver paradigm will be less clear and that design and production will be closely coupled with experiments. One obvious prediction is that the supply chains will get shorter and the production will be more localised both geographically and in regard to time planning. Matias and Rao [164] conduct an exploratory study on the business and consumer markets of 3D printing. This study consists of a survey based part for consumers within the area of 3D printing with a sample size of 66 participants conducted in 2014. One of their findings for the consumers is the willingness of 45 % of the participants to spend only $US 299 on this technology. They also found out that a large number of consumers is not proficient with the technology and the required software. This finding was backed by five interviews conducted with business persons from five different companies. Their interviewees also expressed concerns that there will not be mass market for 3D printing within the next five to ten years. 6.4.1. Accuracy/Precision The accuracy, precision and geometrical fidelity of 3D printed objects has been researched in many works for over 20 years [116, 73] due to the necessity to produce objects that match their digital models closely. This topic is of 56 general relevance for AM as only precise objects are usable for the various applications. Increased precision and accuracy enables AM and CM to be a valid manufacturing technology. Dimitrov et al. [61] conducted a study on the accuracy of the 3DP (3DPrinting) process with a benchmark model. Among the three influencing factors for the accuracy is the selected axis and the material involved. Turner and Gold [247] provide a review on FDM with a discussion on the available process parameters and the resulting accuracy and resolution. Boschetto and Bottini [32] develop a geometrical model for the prediction of the accuracy in the FDM process. They predict the accuracy based on process parameters for a case study for 92 % of their specimens within 0.1 mm. Armillotta [20] discusses the surface quality of FDM printed objects. The author utilises a non-contacting scanner with a resolution of 0.03 mm for the assessment of the surface quality. Furthermore, the work delivers a set of guidelines for the FDM process in respect to the achievable surface quality. Equbal et al. [69] present a Fuzzy classifier and neural-net implementations for the prediction of the accuracy within the FDM process under varying process parameters. They achieve a mean absolute relative error of 5.5 % for the predictor based on Fuzzy logic. Sahu et al. [213] also predict the precision of FDM manufactured parts using a Fuzzy prediction, but with different input parameters (Signal to noise ratio of the width, length and height). Katatny et al. [126] present a study on the dimensional accuracy of FDM manufactured objects for the use as medical models. The authors captured the geometrical data with a 3D Laser scanner at a resolution of 0.2 mm in the vertical direction. In this work a standard deviation of 0.177 mm is calculated for a model of a mandible acquired from Computer Tomography (CT) data. To counter expected deviations of the object to the model, Tong et al. [243] propose the adaption of slice files. For this adaption the authors present a mathematical error model for the FDM process and compare the adaption of slice files to the adaption of STL (STereoLitography) files. Due to machine restrictions the corrections in either the slice file and the STL file are comparable, i.e., control accuracy of the AM fabricator is not sufficient to distinguish between the two correction methods. Boschetto and Bottini [33] discuss the implications of AM methods on the process of design. For this discussion they utilise digitally acquired images 57 to compare to model files. Garg et al. [87] present a study on the comparison of surface roughness of chemically treated and untreated specimens manufactured using FDM. They conclude that for minimal dimensional deviation from the model the objects should be manufactured either parallel or perpendicular to the main axis of the part and the AM fabricator axis. 6.5. Simulation Simulation in the area of AM is of great importance even though the process of object manufacturing itself is relatively cheap and fast when compared to other means of production. But even 3D printed objects can take many hours to be manufactured in which the AM resource is occupied. Furthermore, with specialised and high value printing materials the costs can be prohibitively expensive for misprinted parts. In [104] the authors describe a voxel based simulation for 3D printing for the estimation of the precision of AM objects. Pal et al. [189] propose a finite-element based simulation for the heattransfer of powder based AM methods. With this simulation the authors claim that the general quality of the printed objects can be enhanced and post-processing/quality-control can be reduced. The work of Zhou et al. [301] proposed a numerical simulation method for the packing of powder in AM. This research is conducted to provide a better understanding of the powder behaviour in methods like SLS or SLM. Alimardani [15] propose another numerical simulation method for the prediction of heat distribution and stress in printed objects for Laser Solid Freeform Fabrication (LSFF), a powder based process similar to LENS. They compare their numerically computed predictions with experimental specimens, in one finding the maximum time-dependent stress could be reduced by eight percent by improvements made by the simulation. Ding et al. [64] discuss a FEM based simulation model for wire and arc based AM. Their simulation of the thermo-mechanical properties during this process is performed in the ABAQUS63 software. Chan [49] presents graphical simulation models for the use in manufacturing scenarios not limited to AM. Such models must be adopted to contain virtual production entities like the ones provided by CM. 63 http://www.3ds.com/products-services/simulia/products/abaqus 58 Mourtzis et al. [178] present a review on the aspects of simulation within the domain of product development (PD). For this work they give introduction to concepts and technologies supporting and enabling PD. The concepts are explained in sections of two paragraphs each and supported by existing literature. They link the concepts to simulation research within these areas, where applicable. The concepts and technology introduced includes Computer Aided Design (CAD), Computer Aided Manufacturing (CAM), Computer Aided Process Planning (CAPP), Augmented and Virtual Reality (AR and VR), Life Cycle Assessment (LCA), Product Data Management (PDM) and Knowledge Management (KM), Enterprise Resource Planning (ERP), Layout Planning, Process-, Supply Chain- and Material Flow Simulation, Supervisory Control and Data Acquisition (SCADA) and Manufacturing Systems and Networks Planning and Control. Their review is based on over 100 years of research in the area of simulation and 15954 scientific articles from 1960 to 2014. The articles are aligned with the product and production lifecycle. Furthermore, this review includes a comparison of commercially available simulation software. The authors conclude their work with a detailed analysis of research opportunities aligned to the concepts introduced within this work. 6.5.1. Topological Optimisation One of the key benefits of AM is the ability to create almost any arbitrarily complex object which makes topological optimisation ideal for AM. In CM scenarios such optimisations can be embedded in the digital process chain and be offered and applied as services. In this section we present a number of research works on topological optimisation for AM. In [48] the authors discuss the application of topology optimisation for AM in a general manner giving an overview of the current state. Galantucci et al. [80] present experimental results of topology optimised and FDM printed objects from compression tests. In their experiment the reduction of filling material reduced the material consumption but also the maximum stress of the object. Almeida and Bártolo [58] propose a topology optimisation algorithm for the use in scaffold construction for bio-printing. This optimisation strategy is aimed to create scaffolds that are more bio-compatible due to their porosity but yield high structural strength. The authors conducted an experiment for the comparison of the topological optimised structures and un-optimised structures with reduced infill. Their approach yields structurally stable scaf59 folds up to 60 % porosity. The work by Leary et al. [146] focuses on the topological optimisation to create objects without the requirement for additional support structures. For this approach the authors perform a general topological optimisation first, then orient the part optimally to reduce the required support material and apply a strategy to add part structures to remove the required support material. In an experiment conducted the authors create an object that requires significantly less material (89.7 cm3 compared to 54.9 cm3 ) and is manufactured in 2.6 hours compared to 5.7 hours for the optimised part with support structures. Tang et al. [233] propose a design method for AM with the integration of topological optimisation. For this work the authors analyse existing design methods for AM. Bracket et al. [36] provide an overview and introduction of topology optimisation for AM. The authors identify constraints and restrictions for the usage of topology optimisation, e.g., insufficient mesh-resolution or insufficient manufacturing capabilities. Among the identified opportunities of topology optimisation in AM is the ability create lattice structures and design for multi-material objects. Gardan [85] proposes a topology optimisation method for the use in RP and AM. The work is focused on the inner-part of the object. In non-optimised objects this is filled with a pre-defined infill pattern of a user-selectable density. The authors implement the method in a plugin for Rhinoceros 3D64 and provide an experiment with SLA and SLS. The article does not provide detailed information on the implementation of the software and the algorithm. Gardan and Schneider [86] expand on the prior work by Gardan by slightly expanding the previous article. In this work the authors apply the optimisation method to prosthetic hip implants and additionally experiment on a FDM 3D printer. Hiller and Lipson [105] propose a genetic algorithm (GA) for the multimaterial topological optimisation. With this approach the authors demonstrate the optimisation of varying degrees of stiffness within a part. They utilise a high-level description of the parts properties to design the desired object automatically in its optimised composition. 64 https://rhino3d.com 60 6.5.2. Optimisation For AM and CM as processes a number of optimisations is possible and necessary. The optimisations can relate to the optimisation of process parameters for quicker manufacturing, higher quality manufacturing or increased utilisation of hard- and software resources. The optimisation can furthermore, regard the embeddability and integration of AM and CM within existing production processes. Optimisation for AM is a topic that is researched for a long time, as illustrated by the following two articles. Cheng et al. [52] propose an optimisation method for the criteria of manufacturing time, accuracy and stability. This optimisation is based on the calculation of the optimal part orientation. As a basis for the optimisation, the authors analyse the sources for errors in AM processes, e.g., tessellation errors, distortion and shrinkage, overcuring or stair-stepping effects. For their model they weight input parameters according to the inflicting errors and perform a multi-objective optimisation. Lin et al. [153] propose a mathematical model to reduce the process error in AM. In the first part the authors analyse the different process errors for various types of AM, e.g., under- and overfill, stair-stepping effects. Their model optimises the part orientation for minimal errors. Albeit this work is more than 15 years old, optimal orientation and placement of objects is not widely available in 3D printing control software. More recently, Rayegani and Onwubolu [203] present two methods for the process parameter prediction and optimisation for the FDM process. The authors provide an experiment to evaluate the tensile strength of specimens for the optimised process parameters. For the optimised parameters the authors provide the solution of 0° part orientation, 50° raster angle, 0.2034 mm raster width and -0.0025 mm air-gap. These parameters yield a mean tensile strength of 36.8603808 MPa. Paul and Anand [192] develop a system for the calculation of the energy consumption in SLS processes and process parameter optimisation to minimise the laser energy. For the model the authors neglect the energy consumption of all elements (e.g., heating bed) but the laser system. Paul and Anand [194] propose an optimisation for AM for the reduction of support material. Their approach is to optimise the part orientation for the minimum of support material. Furthermore, they provide optimisation for minimum cylindricity and flatness errors. Jin et al. [17] propose a method to optimise the path generation of ex- 61 trusion based AM, e.g., FDM. The optimisation goals for this approach are machine utilisation and object precision. The authors perform a study with their approach but comparison data on the quality and time consumption of other algorithms is missing. Khajavi et al. [131] present an optimisation for the spare-parts industry of fighter jets by the utilisation of AM. This work is on a systemic optimisation with AM being one strategy to achieve the optimum solution. The authors analyse the current situation in this specific application and propose an optimised solution based on distributed manufacturing or AM. Ponche et al. [199] present an optimisation for the design for AM based on a decomposition of functional shapes and volumes. The authors argue that objects designed for traditional manufacturing are not necessarily suitable for AM but require partial or complete re-design to adjust for the specifics of a certain AM process, e.g., in the inability to produce sharp corners and edges. In their work one optimisation goal is the reduction of material and therefore cost. Hsu and Lai [111] present the results of an experiment and the resulting process parameter optimisation for the 3DP manufacturing method. The authors improved the dimensional accuracy of each axis to under 0.1 mm. Furthermore, the authors improved on the building time by approximately 10 % and on the flexural stress by approximately 30 %. The authors experimented on the four process parameters that are layer height, object location, binder saturation and shrinkage. 6.6. Stakeholder In CM systems or cloud based printing systems naturally a number of stakeholders is involved. With this section we are presenting the current state in research on the identification of stakeholders in this domain as well as research regarding their agendas. In Rayna and Striukova [204] the authors identify the requirements of end-users for online 3D printing services. They base their study on concepts relevant to these stakeholders like user-participation and co-creation. Park et al. [190] provide a statistical analysis of patents and patent filings in the domain of 3D printing and bioprinting that can serve as a basis for decision making in the investment and R&D in these fields. The stakeholders in this case are the investors and managers. Hämäläinen and Ojala [99] applied the Stakeholder Theory by Freeman on the domain of AM and performed a study on eight companies with semi62 structured interviews. They identified five companies that use AM for prototyping (RP). They further analysed the benefits of AM for the interviewed companies. Buehler et al. [42] created a software called GripFab for the use in special needs education. For this software and the use of 3D printing in special needs education they performed a stakeholder analysis. The analysis is based on observations and found beneficial uses for this technology, e.g., in the form of assistive devices. Munguı́a et al. [181] analyse what influence missing standards have on the stakeholders of RM and develop a set of best practises for RM scenarios. They identified the four main contributors to RM cost as Operation times, Machine costs, Labour costs and Material costs. Lehmhus et al. [149] analyse the usage of data acquisition technologies and sensors within AM from the perspectives of the identified stakeholders designer, producer, user and regulatory or public bodies. They argue that the producers in such scenarios might become obsolete if AM is utilised in a complete CM sense. Fox [78] introduces the concept of virtual-social-physical (VSP) convergence for the application to product development. Within this concept he argues that AM can play an integral part to enhance product development. He identifies requirements from stakeholders in the product development process and addresses them in this work. Flatscher and Riel [76] propose a method to integrate stakeholder in an integrated design-process for the scenarios of next-generation manufacturing (Industry 4.0). In their study, a key challenge was the integration of all stakeholders in a team structure which they solved by integrating influential persons from different department in the joint operation. Maynard [166] discusses the risks and challenges, that come with the paradigm of Industry 4.0. Industry 4.0 as a concept is incorporating other concepts like AM and CM. The author briefly identifies the possible stakeholders of this technology as consumers, CEOs and educators. In the report by Thomas [241] the author performs a very detailed and thorough analysis of stakeholders for AM technology in the USA. The list of identified stakeholders is 40 entries long and contains very specific entries (e.g., Air Transport Providers or Natural Gas Suppliers) as well as generalised stakeholder groups (e.g., Professional Societies or Consumers). 63 6.7. Service Orientation Service orientation denotes a paradigm from the domain of programming (Service Oriented Architecture, SOA). Within this paradigm the functionality or capability of a software is regarded and handled as a consumable service. The services offer encapsulated capabilities that can be consumed by users or other services in an easy to use, well-defined and transparent manner. The services are inter- and exchangeable within business processes as their inner-working is abstracted and the services act like black boxes with well-defined interfaces. With CM or AM in general, this service orientation can be expanded to the physical resources of manufacturing. Similar to service orientation from the programming domain, it must be bound by the same stringency of well-defined interfaces and transparent or abstract execution of functionality or capability. In the review on service-oriented system engineering (SOSE) by Gu and Lago [94], the authors propose the hypotheses that the challenges in this domain can be classified by topic and by type. SOSE is the engineering discipline to develop service oriented systems. The authors identified 413 SOSE challenges from the reviewed set of 51 sources. The authors furthermore identified quality, service and data to be the three top challenges in this domain. Wang et al. [263] provide a review on the CC paradigm. For this work the authors classify CC into the three layers ( Hardware-as-a-Service - HaaS, Software-as-a-Service - SaaS and Data-as-a-Service - Daas). In this early work on CC the authors establish the importance of SoA for the CC paradigm. Tsai et al. [246] present an initial survey on CC architectures and propose a service architecture for CC (Service-Oriented Cloud Computing Architecture - SOCCA) for the interoperability between various cloud systems. Among the identified problems with CC architectures are tight coupling, lack of SLA (Service Level Agreement) support, lack of multi-tenancy support and lack of flexibility in user interfaces. The authors utilise SOA for the implementation of their prototype that is deployed on Google App Engine65 . Alam et al. [13] present a review on impact analysis in the domains of Business Process Management (BPM) and SOA. In their work the authors discuss the relationship and convergence of the two methods. From a set of 60 reviewed studies the authors conclude that BPM and SOA are becoming 65 https://appengine.google.com 64 dominant technologies. Zhang et al. [296] propose a management architecture for resource service composition (RSC) in the domain of CM. For this work the authors analyse and define the flexibility of resources and their composition. The implementation by the authors supports resource selection based on QoS and flexibility. Shang et al. [219] propose a social manufacturing system for the use in the apparel industry. Their implementation connects existing logistics and manufacturing systems with a strong focus on the consumer. For this architecture the authors rely heavily on SOA technology and describe the implementation of various layers required. Tao et al. [235] analyse the development of Advanced Manufacturing Systems (AMS) and its trend towards socialisation. In this work the authors establish the relationship between service orientation and manufacturing. The authors identify three phases for the implementation of service-oriented manufacturing (SOM), namely “Perception and Internet connection of Manufacturing resource and capability and gathering, Aggregation, management, and optimal allocation of Manufacturing resource and capability in the form of Manufacturing service and Use of Manufacturing service”. Thiesse et al. [240] analyse economic implications of AM on MIS (Management Information Systems) and the service orientation of these systems. In this work the authors analyse the economic, ecological and technological potential of AM and its services. The authors conclude that the services for the product development will be relocated upstream. For the service composition of cloud applications standards and definitions are essential. In the work by Binz et al. [29] the authors introduce the TOSCA (Topology and Orchestration Specification for Cloud Applications) standard. Albeit this standard is focused on the deployment and management of computing and other non-physical resources its architectural decisions and structures are of relevance to CM systems. Support for encapsulation and modelling as described in this work is sparse for other CM systems. As an extension to the previous work, the authors Soldani et al. [226] propose and implement a marketplace (TOSCAMART) for TOSCA for the distribution of cloud applications. Such a marketplace would be highly beneficial to CM systems as it can foster innovation, collaboration, re-use and competition. 65 6.7.1. Manufacturing-as-a-Service Described in Sect. 6.7 the service orientation regards capabilities as services that can be consumed. Such a class of services is the manufacturing of products. As a consumer of such a service, one is not necessarily interested in the process of manufacturing (e.g., what type of machine is used) or the location of manufacturing as long as a pre-agreed upon list of qualities of the end-product is complied with. As an example it can be said that a user wants two parts made from a certain metal, within a certain tolerance, certain properties regarding stress-resistance and within a defined time frame. The input of this service would then be the CAD model and the properties that must be fulfilled. The parts could then either be milled or 3D printed in any part of the world and then shipped to the user. The user must pay for the service rendered, i.e., the manufacturing of objects, but is not involved with the manufacturing itself as this is performed by a service provider. In the seventh EU Framework Programme, the project ManuCloud66 was funded that consolidated research on this topic. In this section we present current research articles on the subject in order to illustrate the concept of MaaS, its role for CM and applications. Tao et al.[236] propose an algorithm for a more efficient service composition optimal-selection (SCOS) in cloud manufacturing systems. Their proposed method is named FC-PACO-RM (full connection based parallel adaptive chaos optimisation with reflex migration) and it optimises the selection of manufacturing resources for the quality properties time, cost, energy, reliability, maintainability, trust and function similarity. In an experiment they proof that their implementation performs faster than a genetic algorithm (GA), an adaptive chaos optimisation (ACO) algorithm for the objectives of time, energy and cost, but not for the objective of reliability. Veiga et al. [258] propose a design and implementation for the flexible reconfiguration of industrial robot cells with SMEs in mind. These robot cells are mostly reconfigurable by design but with high barriers for SMEs due to the requirement of experts. The system proposed enables an intuitive interface for reconfiguration of the cells in order to enhance the flexibility of manufacturing. The implementation draws heavily on SOA concepts. The implementation supports the flexible orchestration of robotic cells as services. Zhang et al. [297] provide an introduction into the paradigm of CM. 66 http://www.manucloud-project.eu 66 Within this work the authors discuss issues arising from the implementation and the architecture itself. The authors present the decomposition of this paradigm into its service components, that are “design as a service (DaaS), manufacturing as a service (MFGaaS), experimentation as a service (EaaS), simulation as a service (SIMaaS), management as a service (MANaaS), maintain as a service (MAaaS), integration as a service (INTaaS)”. The authors implement such a CM system as a prototype for evaluation and discussion. Moghaddam et al. [175] present the development of MaaS and its relationship to the concepts of CM, Cloud Based Design and Manufacture (CBDM) and others. The authors propose SoftDiss [221] as an implementation platform for CM systems. Van Moergestel et al. [255] analyse the requirements for and propose an architecture for a manufacturing system that enables low-volume and lowcost manufacturing. The authors identify customer requirements for lowvolume and flexible production of products as a driver for the development of the CM concept or other MaaS implementations. The architecture relies on cheap reconfigurable production machines (equiplet). For the implementation of the system the authors utilise open source software like Tomcat and have a strong focus on the end-user integration via Web technology. Sanderson et al. [216] present a case study on distributed manufacturing systems which the authors call collective adaptive systems (CAS). The example in their case study is a manufacturing plant by Siemens in the UK which is part of the “Digital Factory” division. The authors present the division, structure and features of the company which is compared to CAS features. Among the identified challenges the authors list, physical layouting, resource flow through supply chains and hierarchical distributed decision making. For the integration of MaaS (which is called Fabrication-as-a-Service, FaaS, in this work) into CM, Ren et al. [206] analyse the service provider cooperative relationship (CSPR). Such a cooperation of MaaS/FaaS providers within a CM system is essential for the task completion rate and the service utilisation as demonstrated by the authors in an experiment. Guo [96] proposes a system design method for the implementation of CM systems. Within this work the MaaS layer of the CM system is further divided into “product design, process design, purchasing, material preparing, part processing and assembly and marketing process”. In the generalised five-layer architecture for the implementation of CM systems, the MaaS is located in the fifth layer. Yu and Xu [293] propose a cloud-based product configuration system 67 (PCS) for the implementation within CM systems. Such systems interface with the customer enabling the customer to configure or create products for ordering. Within a CM such a system can be employed to directly prepare objects that can be manufactured directly utilising MaaS capabilities. In the implementation within an enterprise the authors utilise the STEP file format for information exchange. 6.7.2. Design-as-a-Service Besides physical and computational resources that are exposed and utilized as services the concept of CM allows for and requires traditional services to be integrated. Such a service is for example the design of an object, which is traditionally either acquired as a service from a third-party company or rendered in-house. As with the physical Manufacturing-as-a-Service the service rendered here must be well-defined and abstract. The service in this section is that of the design for AM or traditional manufacturing. This paradigm can lead to increased involvement of the user as described by Wu et al. [279]. The authors provide an introduction to social product development - a new product development paradigm rooted in Web 2.0 and social media technology. They conducted a study on their students in a graduate level course on product development. They structure the process in four phases beginning with acquisition of user requirement through social media. With the social product development process (PDP) the product development involves the users or customers more directly and more frequently than with traditional PDP. This increased degree of integration requires support through technology which is provided by social media and Web 2.0 technology for communication and management. Unfortunately the scientific literature on DaaS is sparse and mostly only mentioned as part of architectural or systematic descriptions or implementations of CM systems. In Tao et al. [237] the authors place DaaS among other capabilities services that are part of the CM layer. Other capabilities services are Manufacturing, Experimentation, Simulation, Management, Maintenance and Integration-as-a-Service. In Adamson et al. [12] the same classification is used (but without the Integration and Maintenance, and a combination of the Simulation and Experimentation service). The authors also briefly review literature in the domain of collaborative design for CM systems. In Yu et al. [292] DaaS is also identified as a capability of CM systems and part of its layered structure. 68 Johanson et al. [121] discuss the requirements and implications of distributed collaborative engineering or design services. According to the authors the service orientation of design and its collaborative aspects will render enterprise more competitive due to reduced costs for software, decreased design times and innovative design. Furthermore, such services promote tighter integration and cooperation with customers. Laili et al. [141] propose an algorithm for a more efficient scheduling of collaborative design tasks within CM systems. As collaborative design task scheduling is NP-hard, the authors propose a heuristic energy adaptive immune genetic algorithm (EAIGA). In an experiment the authors prove that their implementation is more stable with higher quality results than compared to an genetic algorithm (GA) and a immune GA (IGA). Duan et al. [66] explore the servitization of capabilities and technologies in CC scenarios. The authors explore and discuss a variety of service offerings as described in literature. Among the identified as-a-Service offerings is Designas-a-Service which is referenced to Tsai et al. [246]. Duan et al. provide a large collection of “aaS” literature. Contrary to the indication by Duan et al. the work of Tsai et al. does not cover DaaS. It however covers the architectural design of CC systems and service provisioning as well as an analysis of potential drawbacks and limitations of CC systems. 6.7.3. Testing-as-a-Service Similar to the Design as a-Service (See Sect. 6.7.2) this exposition of a capability as a service can play an important role within CM systems. In general the QA for AM is not sufficiently researched and conducted as the traceability of information from the original CAD model to the manufactured part is insufficient due to the number of conversion steps, file formats and systems involved. Albeit mentioned in a number of publications on the design and implementation of CM architectures, designs or systems, e.g., Ren et al. [205] or Gao et al. [82], the research on Testing-as-a-Service (TaaS) in CM systems is sparse and the authors are not aware of any dedicated works on this topic. In contrast, TaaS as a concept for software testing in the cloud is researched by a number of authors, see e.g., Gao et al. [81] or Mishra and Tripathi [173] for an introductory overview, Yan et al. [285] for the special application of load testing, Tsai et al. [245] for service design or Llamas et al. [157] for a software implementation. Extrapolating from the benefits that TaaS brings to software quality, 69 e.g., transparency, scalability, concurrency, cost-reduction and certification by third parties, research on this area in CM scenarios is warranted. In contrast to software QA, physical testing has an extended set of requirements and limitations, e.g., object under test must be physically available, higher likelihood that standardised test protocols exist, inability to scale without hardware investment or inability to scale beyond minimum time required for testing. With this section we want to motivate further discussion and research into this area. 6.8. Rapid Technology As an umbrella term in accordance to the definition “General term to describe all process chains that manufacture parts using additive fabrication processes.” by [4] we examine the relevance of this technology for CM with this chapter. This technology is integral to the product development especially with its sub-technology that is RP (See Sect. 6.8.3). This and the following sections extend on the definitions provided in Sect. 2 by examples and research findings. For a brief introduction we refer to the following articles. Li et al. [151] propose a method for rapid new product ramp-up within large multi-national companies relying on disperse supply chain networks and out-sourcing partners. In this work the authors consider large-volume product development. For the conceptual framework the authors identified critical members and defined a ramp-up process as a flowchart. Mavri [165] describes 3D printing itself as a rapid technology and analyses the impact of this technology on the production chain. The author performs an analysis on the influences on the phases of product design, production planning, product manufacturing as well as the topics material utilisation, inventory and retail market. The findings of the author include that AM enables companies to act more agile, cater for smaller markets, limit potential inventory issues and can sustain smaller and slimmer supply chain networks. Muita et al. [180] discuss the evolution of rapid production technologies and its implications for businesses. The authors investigate business models and processes, transitions as well as materials and logistics. A decomposition of rapid technology into the phases or layers (Rapid Prototyping, 3D Printing, Rapid Tooling, Rapid Product Development and Rapid Manufacturing) is provided and discussed. The authors recommend the adaption of AM by all companies. 70 In the book by Bertsche and Bullinger [27] the authors present the work of a research project on RP and the various problems addressed within the topic of Rapid Product Development (RPD). One aspect of this research is the development and integration of systems to efficiently store and retrieve information required throughout the process. Information required in the process is knowledge on construction, quality, manufacturing, cost and time. In Lachmayer et al. [140] the authors present current topics of AM and its application in the industry. In the chapter by Zghair [295] the concept of rapid repair is discussed. This concept is intended to prolong the life-time of high-investment parts as well as modification of parts in academic settings. The authors perform an experiment for this approach with three objects and conclude that there is no visible difference between additional object geometry in the case of previously SLM manufactured objects. Differences are visually detectable for cast objects that are repaired. 6.8.1. Rapid Tooling The use case of RT for AM is that the required tools or moulds for the (mass-) production of other parts or objects is supported by provisioning of said tools or moulds. See the definitions of RT in Sect. 2.1.5. RT as a concept is researched and applied for at least 26 years [212]. Conceptually little has changed since the early publications but the number of available AM technologies, materials and support by other concepts like CC has increased. Since its start the idea of RT is to create tools or tooling directly from CAD models thus saving time and material. In this section we present articles from this research to give an overview to the reader and present its relevance and relationship to the concept of CM. In the review by Boparai et al. [31] the authors thoroughly analyse the development of RT using FDM technology. FDM manufactured objects commonly require post-processing for higher-quality surfaces which is discussed by the authors in a separate section of their work. The authors present a variety of applications of RT with FDM which include casting and injection moulds and scaffolds for tissue engineering. Furthermore, the authors discuss material selection and manufacturing, as well as testing and inspection. The review by Levy et al. [150] on RT and RM for layer oriented AM from 2003 already states that AM is not just for RP anymore. According to the definition of RT by the authors tools are supposed to last a few thousand to millions applications. The authors focus mainly on plastic injection moulds for tooling and survey a large number of different technologies and materials. 71 Similarly, the definition of RT by King and Tansey [135], is focused on injection moulds, a definition that has since been expanded to other tooling areas. In this work the authors present research on material selection for SLS manufactured moulds. In this work the authors analyse RapidSteel and copper polyamide for the use in time-critical RT scenarios. Lušić et al. [161] present a study on the applicability of FDM manufactured moulds for carbon fibre reinforced plastic (CFRP) objects. The authors achieved up to 84 % material saving or 47 % time saving for optimised structures compared to a solid mould at a comparable stiffness. The authors experimented with varying shell thicknesses and infill patterns. Nagel et al. [184] present the industrial application of RT in a company. The authors present at a high level the benefits and thoughts leading to the creation of flexible grippers for industrial robots utilising 3D printing. The authors also present a browser based design tool for the creation of the individual grippers with which the company is able to reduce the time required for product design by 97 %. Chua et al. [56] present a thorough introduction to RT as well as a classification into soft- and hard tooling, with a further divide into direct soft tooling, indirect soft tooling, direct hard tooling and indirect hard tooling. Among the benefits of RT the authors see time and cost savings as well as profit enhancements. The authors discuss each of the classifications with examples and the relevant literature. Examples from industry given support the benefits proposed by the authors. Rajaguru et al. [201] propose a method for the creation of RT moulds for the production of low-volume plastic parts. With this indirect tooling method, the authors are able to produce low-cost moulds in less than 48 hours. The authors present an experiment where the mould is used for 600 repetitions. The method uses electroless plating of nickel and phosphorous alloy for the micro-pattern moulds. In the introduction to RT, Equbal et al. [70] start with the basics of various AM technologies. The authors provide a classification schema for RT and discuss each class with the appropriate examples. According to the authors, RT is a key technology for globally active companies in respect to flexibility and competitiveness. In the review by Malik et al. [162] the authors investigate the use of 3D printing in the field of surgery. The authors discuss the fabrication of medical models for education and operation planning as well as drill-guides and templates as RT technology. In contrast, the direct fabrication of implants 72 or prosthetics as described by the authors is regarded RM. 6.8.2. Rapid Manufacturing In contrast to RP the goal of RM is the creation of parts and objects directly usable as end-products or part of end-products (See Sect. 2.1.4). To achieve this usability the requirements on the quality of the parts is higher, therefore the quality control and quality assurance are stricter. Hopkinson and Dickens [107] provides findings on cost analysis for the manufacturing of parts for traditional manufacturing and AM. The authors identify the current and potential future benefits for RM as the ability to manufacture with less lead time, increased geometric freedom, manufacture in distributed environments and potentially the use of graded material for production. The authors compared the costs incurred for the creation of two objects with injection moulding (IM), SLA, FDM and SLS. For IM the tool costs are high (27360 and 32100 Euro) whereas the unit costs are low (0.23 and 0.21 Euro). In their calculation the equilibrium for the cost of IM and SLS for one of the objects is at about 14000 units and for the other part at around 600 units. This finding validates RM for certain low-volume production scenarios. Ruffo et al. [211] also present a cost estimation analysis and is an extension and update to the previous work. The authors calculated with a much lower utilisation of the machines (57 % compared to 90 %), higher labour cost as well as production and administrative overhead costs. Furthermore, the authors took other indirect costs like floor/building costs and software costs into consideration. The authors calculated a higher unit cost for the object (3.25 Euro compared to 2.20 Euro), and a non-linear costing function due to partial low-utilisation of the printing resources which is due to incomplete rows for unit counts not equal or multiple of maximum unit packing. The comparison of these two works illustrates the necessity to use the most up-to date and complete models for costing estimation. Ituarte et al. [117] propose a methodology to characterise and asses AM technologies, here SLS, SLA and Polyjet. The methodology proposed is an experimental design for process parameter selection for object fabrication. The authors find that surface quality is the hardest quality to achieve with AM and might not suffice for RM usage with strict requirements. Such an analysis is of value in order to asses the feasibility of certain manufacturing methods in RM scenarios. In the review by Karunakaran et al. [125], the authors survey and classify 73 technologies capable of manufacturing metallic objects for RM. The technologies surveyed are CNC-machining, laminated manufacturing, powder bed processes, deposition processes, hybrid processes and rapid casting. The authors develop different classification schemes for RM processes based on various criteria, e.g., material or application. Furthermore, the authors compile a list of RM process capabilities to be used for the selection of appropriate RM processes. Simhambhatla and Karunakaran [222] survey build strategies for metallic objects for RM. The authors focus on the issues of overhangs and undercut structures in metallic AM. The work concludes with a comparative study on the fabrication of a part using CNC-machining and a hybrid layered manufacturing (HLM) method. With the hybrid approach the authors build the part in 177 minutes compared to 331 minutes at a cost of 13.83 Euro compared to 24.32 Euro. Hasan et al. [101] present an analysis of the implications of RM on the supply chain from a business perspective. For this study the authors interviewed 10 business representatives and 6 RP or RM service providers. The authors propose both reverse-auctioning as well as e-cataloguing as modes for business transactions. With rapid changing production the need arises for rapid fixture design and fabrication for the RM provider itself. This issue is discussed by Nelaturi et al. [186], as they propose a mechanism to synthesise fixture designs. The method analyses the models to be manufactured and supported by fixtures as STL files for possible fixture application areas. The algorithm furthermore calculates possible fixture positions and inflicting forces. The authors select existing fixtures from in-house or online catalogues of fixtures for application. Gupta et al. [98] propose an adaptive method to slice model files of heterogeneous objects for the use with RM. For this the authors decompose the slicing process into three phases (Slicing set up, Slices generation and Retrieving data). The work also surveys other existing slicing techniques for various optimisation goals, e.g., quality, computing resources or part manufacturing time. For the extraction of geometric and material information the authors utilise a relational database for efficient storage. The authors find that utilising the appropriate slicing technique the fabrication time can be reduced by up to 37 %. Hernández et al. [103] present the KTRM (Knowledge Transfer of Rapid Manufacturing) initiative which is created to improve training and knowledge transfer regarding RM in the European Union. For the requirement analysis 74 of such a project, the authors conducted a study with 136 participants of which the majority (70 %) are SMEs. Such training initiatives are beneficial to the growth in application and increased process majority as the authors find that the knowledge of RM is low but the perceived benefits of this technology include higher quality parts, lower time to markets and increased competitiveness. With the chapter by Paul et al. [191], the authors provide a thorough overview over laser-based RM. The authors discuss classifications of such systems as well as composition of these systems in general. Process parameters are presented and located in literature. Furthermore, the authors discuss materials available for this class of RM and applications. This work is a comprehensive overview, covering all relevant aspects of the technology, including monitoring and process control. 6.8.3. Rapid Prototyping Following the definitions in Sect. 2.1.3 Rapid Prototyping (RP) is the concept to speed-up the creation of prototypes in product development. These prototypes can be functional, visual, geared towards user-experience or of any other sort. RP was one of the first uses for AM and oftentimes the terms AM and RP are used synonymously. The quick or rapid creation of prototypes does not necessarily mean fast in absolute terms but rather a more rapid way to create prototypes than traditionally created using skilled or expert labour (e.g., wooden models created by carpenters) or subtractive or formative manufacturing methods oftentimes requiring specialised tooling or moulds. Pham and Gault [196] provide an overview of commonly used methods to rapidly create prototypes with information on the achievable accuracy, speed and incurred costs of each technology from a very early perspective. A number of technologies, e.g., Beam Interference Solidification (BIS), has since been disused. The accuracy for Fused Deposition Modeling (FDM) stated with 127 µm has not been improved significantly since then. Masood [163] reviews the technology of FDM and examines the usability of it for RP. Among the advantages of this technology is “Simplicity, Safety, and Ease of Use” as well as “Variety of Engineering Polymers” which makes it suitable for the creation of functional prototypes. A number of limitations, like “Surface Finish and Accuracy”, can diminish the suitability of this technology for certain aspects of prototyping. In their keynote paper Kruth et al. [137] survey the technologies used for 75 RP and produce examples for the technologies. Furthermore, they briefly explain the developmental bridge from RP to RT and RM. The authors Yan et al. [286] present the historical development of RP from its roots in the analogue and manual creation of prototypes to digital fabrication methods. The also present a list of current limitations for digital RP. Among the five limitations they place high-manufacturing cost, for the manufacturing resources, and the insufficient forming precision. The first argument of cost is often put forward in its reversed statement as RP is proposed as a low-cost production method, when compared to traditional prototyping. Azari and Nikzad [21] present a review on RP in dentistry with a distinction of models in dentistry and its general meaning. They discuss the problems in data-capture for RP due to the nature of living patients. They further discuss the use of AM for drill-guides which is an application for RT. Liu et al. [155] present a study on profit mechanisms associated with cloud 3D printing platforms predominantly in China. They argue that such services can enable small and medium sized enterprises (SME) to produce prototypes more rapidly and cheaper thus increasing their competitiveness. Roller et al. [209] introduce the concept of Active Semantic Networks (ASN) as shared database systems for the storage of information for the product development process. 6.9. Design In traditional (subtractive or formative) manufacturing the design is driven by the capabilities provided by the manufacturing equipment. This is described as Design for Manufacturing or Design for Manufacturability (DFM) which means that the parts designed must be easy and cheap to manufacture. Especially in large volume production the parts must be machinable in a simple way as tooling, tool changes and complex operations are expensive. Furthermore, with traditional manufacturing certain operations like hollowed or meshed structures are not possible to produce or incur large costs. With AM the design of objects or parts is not strictly limited by these considerations as flexibility comes for free and a number of operations (e.g., intersecting parts, hollowed structures) become possible. The designer can chose more freely from available designs and is less restricted. The design itself can concentrate on the functionality of the part, rather than its manufacturability. 76 In the review by Rosen [210] the author proposes principles that are relevant for design for AM (DFAM) as they exist in literature. The suitability of AM is declared for parts of high complexity and low production volume, high production volume and high degree of customisation, parts with complex or custom geometries or parts with specialised properties or characteristics. Within this review the author proposes a prototypical design process for AM that is derived from a European Union standardisation project by the name of SASAM67 . Kerbrat et al. [129] propose a multi-material hybrid design method for the combination of traditional manufacturing and AM. For this method the object is decomposed based on the machining difficulty. The authors implemented their method in a CAD System (Dassault Systems SolidWorks68 ) for evaluation. This hybrid design method is not limited to a specific AM or manufacturing technology. The authors omit information on how the decomposed part or parts are fused together and how to compensate for inaccuracies within the manufacturing process. Throughout the design process and later for manufacturing it is necessary to convey and transport information on design decisions and other specifications. Brecher et al. [38] provide an analysis of the STEP [8] and STEP-NC [2] file formats. This analysis is used to propose extensions necessary for the use in an interconnected CAD-CAM-NC (CAx) platform. Buckner and Love [41] provide a brief presentation of their work on the automatic object creation using Multi-Objective Constrained Evolutionary Optimisation (MOCEO) on a high-performance computing (HPC) system. With their software, utilising Matlab69 and driving SolidWorks, the objects are created automatically following a given set of restrictions and rules. Cai et al. [43] propose a design method for the personalisation of products in AM. Their work defines basic concepts ranging from Design Intent to Consistency Models. The design method is intended to convey design intentions from users or designers in a collaborative design or CAD environment. Vayre et al. [257] propose a design method for AM with a focus on the constraints and capabilities. This design method consists of four steps (Initial Shape Design, Parameter Definition, Parametric Optimisation, Shape 67 http://www.sasam.eu http://www.solidworks.com 69 http://mathworks.com/products/matlab 68 77 Validation). For the initial shape design, the authors propose the use of topological optimisation. The authors illustrate this process with an example of the re-design of an aluminium alloy square bracket. Diegel et al. [60] discuss the value of AM for a sustainable product design. The authors explore the benefits (e.g., Mass customisation, freedom of design) and design considerations or restrictions for AM (e.g., surface finish, strength and flexibility). The authors argue that AM offers create potential for the creation of long-lasting, high-quality objects and parts that can save resources throughout their lifetime by optimised design. Ding et al. [63] analyse existing slicing strategies for the creation of objects with AM. Besides the analysis, the authors propose a strategy to create multi-directional slicing paths to be used with AM machines that support multi-axis deposition or fabrication. By the authors’ analysis the existing software for slice creation is insufficient and leaves uncovered areas (hole or gap). This work is not on the design for AM but rather on the design of the resulting machine-paths for the manufacture with AM fabricators. In Wu et al. [274] the authors discuss the concept of Cloud-Based Design and Manufacturing (CBDM, see also [278]) in which the whole design and manufacturing process chain is executed in a cloud environment. CBDM is an extension to the CM concept as it expands the process chain horizontally into the collaborative and cooperative domain of the cloud. CBDM utilises Web 2.0 technology, service-oriented architecture (SOA) concepts, semantic web technologies and has an inherent connection to social networking applications. In this article concepts like collaboration, cooperation and crowdsourcing for design for AM are discussed and exemplified. 6.10. Additive Manufacturing We see AM an integral component in CM and Industry 4.0 settings due to the benefits it provides. Among those benefits are flexibility, resource efficiency and the freedom in and of design. In this section we survey scientific literature regarding AM, especially works that provide an overview (e.g., reviews, surveys), present important aspects or exhibit common characteristics of this domain. Le Bourhis et al. [35] develop the concept of design for sustainable AM (DFSAM) to minimise the yet unknown environmental impact of AM. According to the authors about 41 % of the total energy consumption globally is attributed to industry. The authors further provide a division for the French industry in 2010 where about 12 % percent are attributed to manufacturing. 78 The authors claim that AM can reduce the energy required as it limits waste material. The authors experiment on the energy and resource consumption of the Additive Laser Manufacturing (ALM) process and present a method to calculate electricity, powder and gas consumption for an object based on the respective GCode. In their work Kim et al. [134] present a federated information systems architecture for AM. This architecture is intended to facilitate an end-to-end digital implementation of AM, i.e., “digital thread”, design-to-product process. The authors analyse, for each phase (Part geometry/design, Raw/tessellated data, Tessellated 3D model, Build file, Machine data, Fabricated Part, Finished Part, Validated Part) the available and used data formats and supporting software. The focus of their conceptual architecture is interoperability by an open architecture. Balogun et al. [23] perform an experiment on the electricity consumption of the FDM process. The authors divide the manufacturing process into its components (Start-up, warm-up, ready-state, build-state). In an experiment they analyse three different FDM machines (Stratasys Dimension SST FDM, Dentford Inspire D290 and PP3DP) for their power consumption profile during manufacturing. The machines differ significantly in the energy demand with the Dentford machine requiring 1418 Wh and the PP3DP only requiring 66 Wh. Furthermore, the authors compare the energy consumption and manufacturing duration of a FDM machine to a milling machine. In the experiment the AM process consumed 685 Wh and the Mikron HSM 400 milling machine only 114 Wh. The AM cycle time was 3012 s (without 3600 s for support structure removal in an ultrasonic cleaning tank) and the milling machine cycle time was 137 s. Weller et al. [268] discuss the implications of AM on the company and industry level. Economic characteristics, i.e., opportunities like acceleration and possible price premiums, lower market entry barriers and limitations like missing economy of scale, missing quality standards are discussed in this analysis. The authors perform modelling of various scenarios and propositions for the market under the influence of AM. Their prediction for first adoption is within markets with an overall lower economy of scale. Efthymiou et al. [67] present a systematic survey on the complexity in manufacturing systems. Albeit not directly referencing AM this study is relevant to understand the implications of AM on manufacturing systems. Turner et al. [248] survey melt extrusion AM processes. This work is part of a two piece series (See also [247]) with this part focusing on the design 79 and process modelling. The authors provide a short market analysis in their introduction. The authors discuss literature relating to various processing steps and problems, e.g., die swelling, with melt extrusion processes. The authors provide a thorough overview on the literature for this topic. Mitev [174] approaches the topic of AM in a very uncommon manner, namely with a philosophical approach. This is the sole publication with this approach found by the authors. The author discusses AM for the question on what matter is and how 3D printing affects our concept of matter and material. In contrast to the previous author, Bayley et al. [25] present a model for the understanding of error generation in FDM. This work consists of two parts with experiments. The first part analyses actual errors in FDM manufactured parts (e.g., Roundness error, geometrical deviation). In the second part the authors construct a framework for error characterisation and quantification. In the review by Kai et al. [123] the authors evaluate the relationship of manufacturing systems and AM briefly. The authors also provide an overview over one possible decomposition of AM and its academic relevance through numbers of published works from 1975 to 2015. 6.11. Cloud Computing Cloud Computing (CC) is the concept of virtualized computing resources available to consumers and professionals as consumable services without physical restraints. Computing, storage and other related tasks are performed in a ubiquitous cloud which delivers all these capabilities through easy to use and interface front-ends or APIs. These concepts enable enterprises to acquire computing capacities as required while, often paying only for the resources consumed (Pay-as-you-go) in contrast to payment for equipment and resources in stock (e.g., leasing, renting or acquisition). Concepts developed for this computing paradigm are of importance for the CM domain, as many problems stated or solved are interchangeable within domains. What CC is to computing resources (e.g., storage, computing, analysis, databases) CM is to physical manufacturing resources (e.g., Tools, 3D printer, drills). In the definition of Cloud Computing, Mell and Grance [171] from NIST develop and present the characteristics and services models for CC. Truong and Dustdar [244] present a service for estimating and monitoring costs for cloud resources from the domain of scientific computing. This model is also suitable for the monitoring of costs in other cloud based computing 80 scenarios as CM with adaptions. The authors present an experiment where they analyse the cost of scientific workflows on with on-premise execution and deployment to the Amazon Web Service (AWS) cloud system. Stanik et al. [228] propose the cloud layer that is Hardware-as-a-Service for the remote integration of distinct hardware resource into the cloud. The authors argue from the point of embedded systems development and testing but the concepts described are universally applicable for any hardware that is intended to be exposed as a service. Mehrsai et al. [168] propose a cloud based framework for the integration of supply networks (SN) for manufacturing. The authors discuss the basics of supply networks and CC in order to develop a concept to integrate CC for the improvement of SNs. This modular approach is demonstrated in an experimental simulation. Oliveira et al. [187] research the factors influencing the adoption of CC in general and for the manufacturing sector. The authors test their hypothesis on a survey of 369 companies from Portugal with 37.94 % of the companies from the domain of manufacturing. The authors find that security concerns do not inhibit the adoption of CC in the manufacturing domain sub-sample of their survey group. Ramisetty et al. [202] propose an ontology based architecture for the integration of CC or cloud platforms in advanced manufacturing. The authors claim that adoption of CC in manufacturing is less than in comparable industries due to the lack of social or collaborative engagement. The authors implement three services (Ontology, Resource Brokering and Accounting) for an evaluation in the WheelSim App. The authors propose an “App Marketplace” for manufacturing services to further the adoption of CC in the manufacturing industry. Um et al. [250] analyse the benefit of CC on the supply chain interactions in the aerospace industry. The authors propose a manufacturing network for contracting and subcontracting based on CC. In this architecture the basis for information exchange is the STEP-NC file format. Valilai and Houshmand [254] propose a service oriented distributed manufacturing system on the basis of CC. For their work the authors analyse the requirements and basics of globally distributed manufacturing systems. The proposed system (XMLAYMOD) utilises the STEP file format for the information exchange and enables a collaborative and distributed product development process as well as process planning and execution. 81 6.11.1. Internet of Things Internet of Things (IoT) is a term used to describe a network consisting of physical objects connected to the Internet. These physical objects can be tools, parts, machines, actuators or sensors. The concept of IoT is integral to the CM paradigm as it is necessary to control the AM resources transparently and monitor the resources for efficient utilisation planning and scheduling. In IoT scenarios the use of open-standards helps to avoid vendor lock-in. Tao et al. [234] present a very high-level description of the possible integration of IoT in CM scenarios. In four proposed layers (IoT, Service, Application, Bottom Support) of CM systems, they declare IoT and the corresponding layer as core enabling technology. Tao et al. [238] propose a five layer (Resource layer, perception layer, network layer, service layer and application layer) architecture for a CM system. The authors propose the utilisation of IoT technology as a method to interface the manufacturing resources into the architecture. This work is very similar to [234]. Qu et al. [200] present a case study on the integration of CM and IoT technology into an enterprise to synchronise the production logistics (PL) processes. For the implementation they propose a five tier (Physical resource layer, Smart object layer, Cloud manufacturing resource management layer, Cloud manufacturing core service layer, Cloud manufacturing application layer) decomposition. The system uses AUTOM [299] as a backbone for the IoT integration. Baumann et al. [24] propose the development of flexible sensor boards for the use in the monitoring of AM processes. The authors analyse existing sensors available and provide an architectural overview over a system for the incorporation of these sensor boards into a manufacturing control and monitoring system. With these sensors AM resources can be bridged to control systems or services thus enabling IoT functionality for the resources. Caputo et al. [46] perform a review on IoT from a managerial perspective with the application of AM. The authors develop a four staged (Radical, Modular, Architectural and Incremental) conceptual framework to classify innovation and research on the topic. Within this framework’s description AM resource will become digitally represented by sensors and IoT technology. In the review by Kang et al. [124], the authors focus on global research on smart manufacturing and its enabling technologies and underlying concepts. In the section on IoT the authors link this concept to other technologies like 82 SoA, CM and smart sensors. Vukovic [259] discusses the importance of APIs for the IoT deployment and usage. The author discusses the common architectural patterns in IoT scenarios and the arising requirement for APIs to further this technology. In the work by Kubler et al. [138], the authors discuss the evolution of manufacturing paradigms and the origins of CM along its relationship with IoT technology. The authors conclude that CM is not widely adopted because of security concerns but research in AM and IoT will drive CM forward. 6.11.2. Cyber-physical Systems Cyber-physical Systems (CPS) are one of the key enabling technologies for the Internet of Things. CPS is a term coined by the NSF (National Science Foundation) to describe systems, e.g., machines, parts or tools that have capabilities to sense and interact with their physical environment while being connected to the Internet in order to relay state and environment information to an Internet based control system. The first occurrence in scientific literature can be found in Lee [147]. In the domain of 3D Printing, AM and CM such systems are required to enable seamless integration of systems. With CPS, it is possible for an AM hardware resource to signal its current status or utilisation to a centralised or cloud-based control infrastructure in order to participate in scheduling endeavours and become part of a controllable system. In the work by Chen and Tsai [51] the authors propose the concept of ubiquitous manufacturing. This concept is similar to CM but with a stronger focus on mobility of users and manufacturing resources. For this concept ubiquitous sensor and IoT technology are key enabling technologies. Lee et al. [148] propose a five layer architecture for CM based on IoT technology. The layers in this architecture are from bottom to top: Smart Connection Layer, Data-to-Information Conversion Layer, Cyber Layer, Cognition Layer and Configuration Layer. The goal of this work is to provide a guideline for implementation of such a CPS backed manufacturing systems and to improve the product quality as well as the system’s reliability. Sha et al. [218] provide a general introduction into CPS and the related research challenges. The authors identify QoS composition, knowledge engineering, robustness and real-time system abstraction as the four main research questions for this technology. In the survey by Khaitan and McCalley [130] the authors study the design, development and application of CPS. In the list of identified application 83 scenarios (Surveillance, Networking Systems, Electric Power Grid and Energy Systems, Data Centres, Social Networks and Gaming, Power and Thermal Management, Smart Homes, Medical and Health Care, Vehicular and Transportation Systems) manufacturing systems are missing. Despite this lack of mention, application in e.g., transportation and power management is relevant for CM systems. Kuehnle [139] proposes a theory of decomposition for manufacturing resources in distributed manufacturing (DM) systems. DM is similar in concept to CM in relation to the decomposition of manufacturing resources in vitalised services. According to the author IoT technology and CPS are among the enabling technologies for this smart manufacturing concept. Yao and Lin [289] expand the concept of CPS into socio-cyber-physicalsystems (SCPS) with this study on smart manufacturing. This extension to social aspects of manufacturing (e.g., collaboration and cooperation) is expected to be an integral part of the next industrial revolution (Industry 4.0). Turner et al. [249] discuss the risk of attacks and their implications on CPS in the domain of manufacturing. The authors present a number of attack vectors, e.g., attack on the QA process and counter or mitigation strategies. According to the authors CPS provide an additional attack surface for malicious third parties. 6.12. Scheduling In CM as in CC a number of resources must be provisioned on demand. In contrast to CC the requirements for the execution resource can be more complex than just a computing resource. With CM manufacturing resources must first be described in an abstract way (See Sect. 6.13) to be schedulable. In this section we present current research on the challenges that come from scheduling. Cheng et al. [53] introduce the concept of CM in their work and perform a brief review over possible criteria for scheduling in such scenarios. The authors provide four scheduling modes based on the three identified stakeholders (Operator, Consumer and Provider) and the system as a whole. The proposed modes consider energy consumption, cost and risk. The proposed system-centred cooperative scheduling mode yields the highest utilisation in their experiment. Liu et al. [156] propose a scheduling method for CM systems for multiple enterprise and services scenarios. The authors use the criteria time, cost and 84 pass-rate for the task selection. Based on these criteria constraints are constructed for the decomposition of tasks into subtasks and their distribution onto resources. The authors take geographical distance, respectively delivery times between CM locations into consideration. Their simulation concludes that for a 50 task scenario, with 10 enterprises offering 10 services in total, the utilisation is 49.88 % compared to 10 tasks (17.07 %). The authors provide no specific scheduling solution with their work. Laili et al. [142] define the problem of optimal allocation of resources based on a 3-tier model (Manufacturing Task Level, Virtual Resource Layer and Computing Resource Layer). The authors prove that the optimum resource allocation is NP-complete. For the reason of NP-completeness of the scheduling problem this and other authors propose heuristics based algorithms to provide near-optimal scheduling. Heuristics based scheduling algorithms provide near-optimum solutions for most of the scheduling instances without the guarantee to achieve an optimum solution but at greater speed than exact computation. In this work the authors propose a heuristic algorithm inspired by the immune system (Immune Algorithm, IA). In an experiment they compare their algorithm against three other heuristic algorithms and it performed comparable. Wang [261] proposes a web-based distributed process planning (WebDPP) system that performs process planning, machining job dispatching and job execution monitoring. The system is implemented as a prototype and connects to legacy machine controllers. The proposed system acts directly on the manufacturing resource and interfaces with the Wise-ShopFloor framework [262]. The author does not provide information on scheduling algorithms or methods used. Huang et al. [114] propose a scheduling algorithm based on Ant Colony Optimisation (ACO). In an experiment they compare the algorithm with and without a serial schedule generation scheme (SSGS) against another heuristic Genetic Algorithm (GA). Their algorithm for conflict resolution performs faster and with better quality results than the GA when used with the SSGS. Lartigau et al. [144] present an 11-step framework for scheduling and order decomposition within a CM system. This scheduling is deadline oriented and implemented in a company environment for evaluation. The paper lacks validation and conclusive results for the proposed algorithm. In the work by Zhang et al. [300] the authors propose a queue optimisation algorithm for the criteria lowest cost, fastest finished time, cleanest environ85 ment and highest quality. The proposed CM system relies on active and realtime environment and machine-status sensing through heterogeneous sensors. Furthermore, they utilise semantic web (Ontology) technology for the system. Cao et al. [45] refine an ACO algorithm for efficient scheduling within a CM. This algorithm optimises for time, quality, service or cost (TQSC). With the addition of a selection mechanism to ACO their ACOS algorithm performs with better quality results and faster convergence in comparison to Particle Swarm Optimisation (PSO), GA and Chaos Optimisation (CO). Jian and Wang [120] propose an adapted PSO algorithm (Improved Cooperative Particle Swarm Optimisation, ICPSO) for the use in batch processing within a CM system. Batch tasks are indivisible units of work to be executed with manufacturing resources. The authors present an experiment for the comparison of the proposed algorithm with an PSO and a cooperative PSO scheduling algorithm in respect to the cost and time criteria. The algorithm performs better than the other two algorithms. 6.13. Resource Description For the usage of manufacturing resources within CM there must be an abstract definition or description of the resources. Open-standards are preferable where available in order to avoid vendor lock in. Luo et al. [160] propose a six step framework for the description of Manufacturing Capabilities (MC). The representation of this information utilises ontology and Fuzzy technology. Within the framework the authors represent information on the manufacturing equipment, computing resources, intellectual resources, software and other resources. Wang et al. [264] also propose an ontology based representation for manufacturing resources. The information and ontology is derived from manufacturing task descriptions. The authors implement their algorithm in an enterprise setting in a medium-sized Chinese company for evaluation. As a more general approach to CC scheduling Li et al. [152] propose an ontology based scheduling algorithm with PSO. The authors motivate their work by an example in a logistics centre which is relevant to the domain of CM. For this algorithm the selection is restricted based on the Quality of Service (QoS) with time, cost, availability and reliability as criteria. Zhu et al. [302] develop an XML based description for manufacturing resources oriented at the Web Service Description Language (WSDL) for webservices. The authors separate the resource description into two parts (Cloud End, CE and Cloud Manufacturing Platform, CMP). In their approach, they 86 reflect static data, e.g., physical structure or input data types, in the CE layer whereas the CMP layer reflects the dynamic data, e.g., function parameters. Wu et al. [282] propose an ontology based capability description for industrial robot (IR) systems. IR are regarded as manufacturing resources and described as such. Besides manufacturing machines such IR systems enable CM to perform as a flexible and agile manufacturing system. 6.14. Hybrid Manufacturing Hybrid Manufacturing is a term used for the combination of AM and traditional manufacturing methods. The combination of these methods promises to provide benefits from both, e.g., speed and accuracy of a milling machine with the low material input from AM. Lu et al. [159] propose an architecture for a hybrid manufacturing cloud. Their definition of hybrid refers to cloud operation modes (private, community and public cloud). Besides the architecture they present a cloud management engine (CME) which is implemented for evaluation purposes on Amazon Web Service (AWS). In the work by Kenne et al. [128] a model for a hybrid manufacturingremanufacturing system is proposed. The authors refer the term hybrid to manufacturing and remanufacturing in combination. Remanufacturing denotes an alternative use of products at the end of their product lifecycle for value generation. In an experiment the authors calculate the cost for a mixture of parameters, e.g., return rates and come to the conclusion that the system is applicable with customisation to various industries. In the review by Chu et al. [55] the authors discuss 57 hybrid manufacturing processes. These micro- and nanoscale processes are categorised in three different schemes (concurrent, main/assistive separate and main/main separate). The authors survey a combination of 118 processes in this work. The review by Zhu et al. [305] provides a classification of hybrid manufacturing processes. The authors present an extensive list of mainly two-process combination manufacturing processes. For this work the authors explore the existing definitions of manufacturing and hybrid manufacturing processes in literature. In another work by Zhu et al. [304] the authors propose a build time estimation for the iAtractive [303] process that combines additive, subtractive and inspection processes. This process is based on FDM and the build time prediction is based on the same parameters as normal FDM build time prediction. The authors provide a discussion on an experiment for which 87 their estimation ranged from approximately -12 % – 12 % to the real build time. The authors only provide a build estimation method for the additively manufactured part of the process. Lauwers et al. [145] propose a definition and classification of hybrid manufacturing processes with their work. They define these processes as acting simultaneously on the same work area or processing zone. This definition excludes processes that combine processing steps sequentially. Elmoselhy [68] proposes a hybrid lean-agile manufacturing system (HLAMS). The author develops the system for the requirements in the automotive industry. The definition of hybridity in this work refers to the school of thinking for manufacturing. The work by Kendrick et al. [127] proposes a solution to the problems associated with distributed manufacturing through the utilisation of hybrid manufacturing processes. The authors propose four options for the usage of distributed hybrid manufacturing systems (Local factories, manufacturing shops, community areas, personal fabrication). The described usage of hybrid MS can be further utilised in CM systems. Yang et al. [287] propose a hybrid system for the integration of multiple manufacturing clouds. The definition of hybridity used in this work refers to the mixture of diverse manufacturing clouds and not on the manufacturing process itself. The architecture proposed links the various clouds together for a single point of interaction integration. The authors define adaptors and a broker system and implement these for evaluation purposes. In the overview by Zhang et al. [298] the authors use the term hybrid to describe the cloud management. Their definition for the three cloud types used in CM is private/enterprise cloud, public/industry cloud and a mixture of both as hybrid cloud. 6.15. Research Implications From the provided literature we have identified the following number of open research questions. The listing compiled is non-exhaustive due to the nature of scientific research. Bourell et al. [3] provide a report on the “Roadmap for Additive Manufacturing” workshop that took place in 2009 and resulted in proposal for research of the coming 10 to 12 years. The recommendations are grouped into 1. Design 2. Process Modelling and Control 3. Materials, Processes and Machines 4. Biomedical Applications 5. Energy and Sustainability Applications 6. Education 7. Development and Community and 8. National Testbed 88 Center. The recommendations include the proposal to create design methods for aiding designers with AM, creation of closed-loop printing systems and the design and implementation of open-architecture controllers for AM fabricators. The authors reflect on their proposed roadmap in an article [34] five years later. In this analysis the authors state that the direct influence of the Roadmap is hard to quantify. The authors remark that the report is referenced about 50 times in scientific literature but only one project can be clearly attributed to the Roadmap. Lan [143] identifies the following four tasks for future research in his review. 1. Combination of Web services and software agents 2. Collaborative network environment with the focus on integration and interoperability 3. Focus on Web technology integration in RM systems and 4. Collaborative product commerce and collaborative planning and control. In the review by Fogliatto et al. [77] on Mass Customisation (MC), the authors identify the following research needs: 1. Research on Rapid Manufacturing (RM) to support MC 2. Research on the value of MC for consumers as well as environmental, economic and ethic value 3. Research on Quality Control 4. Research on Warranty for MC objects and 5. Case Studies and empirical validation. Khan and Turowski [132] perform a survey on challenges in manufacturing for the evolution to Industry 4.0. The authors identify six current and future challenges which are the following topics 1. Data integration (IoT, Big-Data, real-time data, data management) 2. Process flexibility (Adaption, Change management) 3. Security (Connectivity, monitoring, compliance) 4. Process integration within and across enterprise boundaries (Integrated processes, logistics, optimisation) 5. Real-time information access on hand-held devices (Web technology, ERP integration) and 6. Predictive Maintenance (Machine data, sensors). Among the research needs identified by Adamson et al. [11] in their review are the following 1. Capabilities, information and knowledge integration and sharing as well as cloud architectures 2. Definitions and standards for CM 3. Intelligent, flexible and agile, distributed monitoring and control systems 4. Business models 5. Intellectual properties and 6. Cost, security and adoption of CM systems. Furthermore, the authors identify and predict 1. The emergence of cloud service providers 2. Real world connectivity (IoT) 3. New collaboration and cooperation scenarios (Customer-manufacturer and manufacturing collaboration) 4. Increased competitiveness 5. Cloud closed89 loop manufacturing 6. Manufacturing of feature function blocks 7. Increased awareness and research on sustainable operations. In the work by Oropallo and Piegl [188] the authors specifically researched and compiled ten challenges in current AM systems that require research. The challenges are 1. Shape optimisation (Cellular structures and topology optimisation) 2. Design for 3D printing (Software support, design methodology) 3. Pre- and Postprocessing (File formats, model preprocessing, part postprocessing) 4. Printing methodologies (Layered manufacturing, voxel and digital material, non-layer oriented methods) 5. Error control (Before and during printing) 6. Multi material printing (Modelling and manufacturing support) 7. Hardware and Maintenance issues (Process and material based issues) 8. Part orientation 9. Slicing (Adaptive and direct slicing) 10. Speed Wu et al. [274] explicitly identify the following research needs for the evolution of CM 1. Cloud-Based Manufacturing (Modeling and simulation of material flow, concurrency and synchronisation for scalability) 2. CloudBased Design (Social media integration and leveraging, CAx convergence and cloud enablement) 3. Information, Communication, and Cyber Security (IoT, Security-as-a-Service) and 4. Business Model. The work by Huang et al. [115] examines the state of the art of AM and names the following research areas for future investigation: 1. Materials 2. Design (Methods and tools, complex geometries, lifecycle cost analysis) 3. Modeling, Sensing, Control, and Process Innovation (Multi-scale modelling simulation, error and failure detection, optical geometry reconstruction, faster hardware, bioprinting) 4. Characterization and Certification and 5. System Integration and Cyber Implementation (Knowledge management integration, cloud based systems). 7. Summary This article provides an overview over the topic that is CM and 3D Printing services. With the overview of the existing definitions (See Sect. 2) and the extension of the definition proposed we create the foundation for the following work. The review is based on the topological map presented in Sect. 6.1. Concepts, techniques, methods and terminology is presented by exploring different authors work. We perform an explorative extension study to [204] due to relevance for this domain (See Sect. 5). In this study we cover and analyse 90 48 publicly available services. extension considers APIs of such services a further distinction to be made. This work also gives an overview on available journals in the domain of AM or 3D printing in general as to support other researchers’ in finding suitable audiences for their work. One journal was established 31 years ago and provides a catalogue of over 21000 articles with no exclusivity to AM or 3D Printing. In recent years a number of new journals were established or are currently in the process of being established. Their focus is solely on AM or related domains like bioprinting. The domain of AM, CM, 3D Printing, RM and related domains is thoroughly presented in this work by means of literature analysis in scientometrical sense (See Sect. 1.1.2 and Sect. 1.2). The results presented in this work illustrate the scientific development of various techniques and methods from these domains in a time period ranging from 2002 to 2016 (See Sect. 6). Disclosure This work is not funded by any third party. The authors do not have any conflicting interests in the topic and provide an unbiased and neutral article on the topic contained within this article. References [1] DIN 8580 Fertigungsverfahren - Begriffe, Einteilung. Din, 2003. URL: http://www.din.de/de/mitwirken/normenausschuesse/ natg/normen/wdc-beuth:din21:65031153. [2] Iso 10303-238:2007 - industrial automation systems and integration – product data representation and exchange – part 238: Application protocol: Application interpreted model for computerized numerical controllers. Iso, 2007. URL: http://www.iso.org/iso/catalogue_ detail.htm?csnumber=38036. [3] Roadmap for Additive Manufacturing Identifying the Future of Freeform Processing. 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arXiv:1708.08118v1 [] 27 Aug 2017 MERGE DECOMPOSITIONS, TWO-SIDED KROHN-RHODES, AND APERIODIC POINTLIKES SAMUEL J. V. GOOL AND BENJAMIN STEINBERG Abstract. This paper provides short proofs of two fundamental theorems of finite semigroup theory whose previous proofs were significantly longer, namely the two-sided KrohnRhodes decomposition theorem and Henckell’s aperiodic pointlike theorem, using a new algebraic technique that we call the merge decomposition. A prototypical application of this technique decomposes a semigroup T into a two-sided semidirect product whose components are built from two subsemigroups T1 , T2 , which together generate T , and the subsemigroup generated by their setwise product T1 T2 . In this sense we decompose T by merging the subsemigroups T1 and T2 . More generally, our technique merges semigroup homomorphisms from free semigroups. Introduction Eilenberg’s variety theorem [3] provides a dictionary between formal language theory and finite semigroup theory. In particular, membership problems in certain Boolean algebras of regular languages (languages accepted by finite automata) are equivalent to membership problems in varieties of finite semigroups. Other natural problems in language theory transform into questions about pointlikes with respect to a variety of finite semigroups, a notion introduced by Henckell and Rhodes [4]. An important problem in language theory is the separation problem: given disjoint regular languages, determine whether they can be separated by a language from a given variety of regular languages. The separation problem is equivalent to decidability of pointlike pairs [1], which is strictly stronger than the membership problem [10, 2]. Decidability of pointlikes can be used to obtain decidability of membership problems of related varieties. For instance, the second author showed, using the decidability of aperiodic pointlikes and Zelmanov’s solution to the restricted Burnside problem, that the join of the variety of aperiodic semigroups with any variety of finite groups of bounded exponent has decidable membership problem, answering a question of Rhodes and Volkov [13]. The first decidability result on pointlikes was Henckell’s theorem on the decidability of aperiodic pointlikes [4], which for a long time was considered one of the most difficult results in the subject. Henckell not only provided a decidability algorithm: he also gave an elegant structural description of the aperiodic pointlike sets that we call Henckell’s formula. Henckell’s original proof idea is a variation on the holonomy proof [5] of the Krohn-Rhodes theorem [7] for directly decomposing semigroups into wreath products. The difficult part of Henckell’s proof is to prove that a certain semigroup is aperiodic, which he does by wreath product embeddings. In [6], Henckell, Rhodes and the second author provided a direct proof that Henckell’s semigroup is aperiodic, leading to a simpler and shorter proof of his main theorem. They also extended the theorem beyond aperiodic pointlikes to the variety of semigroups Date: August 29, 2017. The first-named author was supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant #655941; the second-named author was supported by United States - Israel Binational Science Foundation #2012080 and NSA MSP #H98230-16-1-0047. 1 2 SAMUEL J. V. GOOL AND BENJAMIN STEINBERG whose subgroups have prime divisors belonging to a fixed set π of primes (the restriction of this proof to the aperiodic case can be found in [11, Ch. 4]). Although simpler than the original proof of Henckell [4], the proof in [6] is still non-trivial. Recently, Place and Zeitoun [9] gave a new proof of the decidability of aperiodic pointlikes, which, unlike the previous proofs, is inductive. They use a language theoretic reformulation of the problem of computing pointlike sets and the McNaughton-Schützenberger theorem that the aperiodic languages are precisely the first order definable languages [14]. The PlaceZeitoun approach follows the inductive proof scheme of the Krohn-Rhodes theorem (the socalled ‘V ∪ T ’ argument [8, 11]) later used by Wilke in the logic context [15], but done in the power set of the semigroup. This paper introduces a new algebraic tool, that we call the merge decomposition, in Section 2. In Section 3, we use this tool to give a short proof of the inductive step of the two-sided Krohn-Rhodes decomposition theorem (cf. [11, Ch. 5]). Then, in Section 4, we use the merge decomposition in the inductive step of the Place-Zeitoun inductive scheme to give a short algebraic proof of Henckell’s formula for the aperiodic pointlikes. We feel that our approach has several advantages over previous approaches [4, 6, 9]. First of all, it leads to a significantly shorter proof than the previous ones. Secondly, we obtain the best known bound on the length of a two-sided Krohn-Rhodes decomposition of the aperiodic semigroup witnessing pointlikes (or, equivalently, quantifier-depth of the first order formula giving separation). An advantage of our approach is that it is potentially extendable beyond the realm of first order logic on words. For instance, decidability of pointlikes for some larger varieties than aperiodics is obtained in [6]. We leave this to future work. 1. Preliminaries We assume familiarity with notions from the theory of semigroups, in particular, relational morphisms and divisions, the wreath product (denoted ≀), and the two-sided semidirect product of semigroups (denoted ⊲⊳) and of varieties of finite semigroups (denoted ∗∗); see, e.g., [11, Ch. 1]. Throughout the paper, we call ‘variety’ what is called ‘pseudovariety’ in [11]. Augmented semigroups. Let T be a finite semigroup. Let T I be the monoid obtained by adjoining a new identity, I, to T and T 0 the semigroup obtained by adjoining a new zero to T . We denote by SL the variety of finite semilattices and by U1 the two-element semilattice. Fact 1.1. If a variety V contains T and SL, then T 0 ∈ V. If a variety V contains T , SL, and is generated by monoids, then T I ∈ V. Proof. The first statement is true because T 0 is a homomorphic image of T × U1 [3, Ex. I.9.2]. For the second statement, we distinguish two cases. If T is a monoid, then T I embeds in T × U1 , where U1 denotes the two-element semilattice [3, Ex. I.9.1]. If T is not a monoid, then T divides some monoid M ∈ V, and since T is not a monoid, it follows that T I divides the same monoid M .  The semigroup T acts faithfully on T I by multiplication on the right, and thus T embeds into the semigroup of total functions on T I ; we identify every element t ∈ T with the corresponding right multiplication map. Further, for every t ∈ T I , we denote by t♯ the function with constant value t. We define T ♯ := T ∪{t♯ : t ∈ T I }, the semigroup consisting of the right multiplication maps and the constant maps. Thus, T ♯ naturally acts on T on the right.1 Dually, T ♭ denotes 1Note that our definition of T ♯ for a semigroup T deviates slightly from the definition of M ♯ for a monoid M in [11, Subsec. 4.1.2]. MERGE DECOMPOSITIONS, TWO-SIDED KROHN-RHODES, AND APERIODIC POINTLIKES 3 the semigroup consisting of left multiplication maps for every t ∈ T and constant maps t♭ for every t ∈ T I . Note that T ♭ = ((T op )♯ )op , and T ♭ acts on T on the left. Fact 1.2. For any finite semigroup T , let Te denote the monoid obtained from T by adjoining an identity and a zero and let M be any monoid with |M | > |T |. Then T ♭ embeds in M ≀ Te. Proof. Fix a bijection t 7→ mt between T and a subset of M \ {1M }. We define a function e i : T ♭ → M ≀ Te. For every t ∈ T , define i(t) := (c1 , t), where c1 ∈ M T denotes the function e with constant value 1M , the identity of M , and i(t♭ ) := (ft , 0), where ft ∈ M T denotes the function defined by ft (0) := 1M and ft (t′ ) := mt′ t for all t′ ∈ T I . It is straightforward to verify that i is an injective homomorphism.  Triple product. Let (S, +) be a (not necessarily commutative) semigroup equipped with two actions on it, a left action of a semigroup (SL , ·) and a right action of a semigroup (SR , ·), which commute. The triple product 2 T = (SR , S, SL ) is the semigroup of triples (sR , s, sL ), with multiplication defined by (sR , s, sL ) · (s′R , s′ , s′L ) := (sR s′R , ss′R + sL s′ , sL s′L ). Fact 1.3. If S ∈ V and SL , SR ∈ W, then (SR , S, SL ) ∈ V ∗∗ W. Proof. Define an action of SL × SR on S by hsL , sR is := sL s and shsL , sR i := ssR . Then T is isomorphic to the two-sided semidirect product S ⊲⊳ (SL × SR ). (Cf., e.g., [3, Sec. V.9].)  2. The merge decomposition Throughout this section, we fix: • a finite alphabet A and two disjoint subalphabets A1 , A2 such that A = A1 ∪ A2 ; + • two homomorphisms ψ1 : A+ 1 → T1 and ψ2 : A2 → T2 , with T1 and T2 finite; + • a homomorphism χ : (T1 × T2 ) → T0 . + For any w1 ∈ A+ 1 , w2 ∈ A2 , define µ(w1 w2 ) := (ψ1 (w1 ), ψ2 (w2 )) ∈ T1 × T2 . Since the + + + + subsemigroup (A1 A2 ) of A+ is freely generated by the infinite set of generators A+ 1 A2 , + + + the function µ extends uniquely to a homomorphism µ : (A1 A2 ) → (T1 × T2 )+ . We define + + → T0 to be the composition χ ◦ µ. For i = 0, 1, 2, we denote the external ψ0 : (A+ 1 A2 ) identity of TiI by Ii , and we also denote by ψi the homomorphism from the corresponding free monoid to the finite monoid TiI ; i.e., ψi (ε) := Ii . + ∗ ∗ For any word w in A+ , uniquely write w = v2 uv1 , with v2 ∈ A∗2 , u ∈ (A+ 1 A2 ) , and v1 ∈ A1 , I I + I and define τ (w) := (ψ2 (v2 ), ψ0 (u), ψ1 (v1 )). The function τ : A → T2 × T0 × T1 is not a homomorphism in general. The aim in this section is to show that the kernel of τ can be refined to a semigroup congruence of finite index in a well-controlled variety. To this end, we will define a semigroup TM and a homomorphism ψM : A+ → TM . Let I I S := (T0I )T1 ×T2 , with the pointwise product of T0I , written additively. We define a left action of T1♯ and a right action of T2♭ on S. For s ∈ S, sL ∈ T1♯ and sR ∈ T2♭ , let sL ssR ∈ S be defined by [sL ssR ](t1 , t2 ) := s(t1 sL , sR t2 ) for every (t1 , t2 ) ∈ T1I × T2I . Let TM := (T2♭ , S, T1♯ ) be the triple product; we call TM the merge semigroup associated to ψ1 , ψ2 and χ. Fact 2.1. Let V be a variety, and W a variety generated by monoids and containing SL. I I If T0 ∈ V , T1 , T2 ∈ W, and TM is any triple product of T2♭ , (T0I )T1 ×T2 , and T1♯ , then TM ∈ V ∗∗ (SL ∗∗ W). 2We follow the notation of [3, Sec V.9]; note the positions of the semigroups acting on the left and on the right. Also note that the multiplication  can be  viewed as matrix multiplication, if we represent an element (sR , s, sL ) by the lower triangular matrix ssR s0L . 4 SAMUEL J. V. GOOL AND BENJAMIN STEINBERG Proof. Applying Fact 1.2 with M a semilattice (e.g., a chain) with |T2 | + 1 elements and using f T2 ∈ W by Fact 1.1 yields T2♭ ∈ SL ∗∗ W. Similarly, T1♯ ∈ SL ∗∗ W. Fact 1.3 gives the result.  For any w1 ∈ A+ 1 , we define an element sw1 ∈ S by sw1 (t1 , I2 ) := I0 and sw1 (t1 , t2 ) := χ(t1 ψ1 (w1 ), t2 ), for all t1 ∈ T1I and t2 ∈ T2 . Now let ψM : A+ → TM be the unique homomorphism defined by ψM (a1 ) := (I2♭ , sa1 , ψ1 (a1 )) for a1 ∈ A1 , ψM (a2 ) := (ψ2 (a2 ), i0 , I1♯ ) for a2 ∈ A2 , where i0 denotes the identity of S, i.e., the function with constant value I0 . We call the homomorphism ψM : A+ → TM the merge decomposition of A+ along χ, ψ1 and ψ2 . The crucial property of the merge decomposition is the following. Proposition 2.2. There exists a function f : TM → T2I × T0I × T1I such that f ◦ ψM = τ . Proof. For any (t2 , s, t1 ) ∈ TM , define f (t2 , s, t1 ) := (t2 I2 , s(I1 , I2 ), I1 t1 ). We show f ◦ψM = τ . ♭ We first prove, for all w1 ∈ A+ 1 , ψM (w1 ) = (I2 , sw1 , ψ1 (w1 )). By induction, assume that this + holds for all shorter words in A1 . Then, writing w1 = a1 w1′ , the left and right coordinates are clearly as stated, and the middle coordinate of ψM (w1 ) = ψM (a1 )ψM (w1′ ) is sa1 I2♭ +ψ1 (a1 )sw1′ . From the definition of the right action and of sa1 we get that sa1 I2♭ = i0 . From the definition of the left action and of sw1′ and sw1 , we get that ψ1 (a1 )sw1′ = sw1 . For w2 ∈ A+ 2 , we easily obtain ψM (w2 ) = (ψ2 (w2 ), i0 , I1♯ ), since sL i0 sR = i0 for all sL , sR , because i0 is a constant map. ♯ + ♭ Multiplying these two results, for any w1 ∈ A+ 1 and w2 ∈ A2 , ψM (w1 w2 ) = (I2 , sw1 w2 , I1 ), where sw1w2 (I1 , I2 ) = sw1 (I1 , ψ2 (w2 )) = χ(ψ1 (w1 ), ψ2 (w2 )) = ψ0 (w1 w2 ). + + We next prove, by induction on the length of u ∈ (A+ 1 A2 ) as a word in the free semigroup ♯ + ♭ generated by A+ 1 A2 , that ψM (u) = (I2 , su , I1 ), where su (I1 , I2 ) = ψ0 (u). We have already + + + + ′ ′ established the base case. If u = (w1 w2 )u for some w1 ∈ A+ 1 and w2 ∈ A2 with u ∈ (A1 A2 ) , then, for the middle coordinate su of ψM (u) = ψM (w1 w2 )ψM (u′ ), we have su (I1 , I2 ) = [sw1 w2 I2♭ + I1♯ su′ ](I1 , I2 ) = ψ0 (w1 w2 ) · ψ0 (u′ ) = ψ0 (u). + + Finally, to prove that f ◦ ψM = τ , let w ∈ A+ . Suppose that w = v2 uv1 with u ∈ (A+ 1 A2 ) , + + v1 ∈ A1 and v2 ∈ A2 . Then, using our previous calculations, we get ψM (v2 uv1 ) = (ψ2 (v2 ), i0 , I1♯ ) · (I2♭ , su , I1♯ ) · (I2♭ , sv1 , ψ1 (v1 )) = (ψ2 (v2 )♭ , s, ψ1 (v1 )♯ ), where s(I1 , I2 ) =    i0 I2♭ + I1♯ su I2♭ + I1♯ sv1 (I1 , I2 ) = I0 · su (I1 , I2 ) · I0 = ψ0 (u). Thus, in this case, f (ψM (w)) = τ (w). If one or more of the factors in the factorization w = v2 uv1 are empty, then the proof is similar but simpler.  We end with a prototypical application of the technique, to be used in the next section. Corollary 2.3. Let S be a finite semigroup and let T1 , T2 be subsemigroups of S such that T1 ∪ T2 generates S. Denote by T0 := hT1 T2 i, the subsemigroup generated by T1 T2 . Then the I I semigroup S divides a triple product of T2♭ , (T0I )T1 ×T2 , and T1♯ . Proof. Let Ai := Ti × {i} for i = 1, 2 and A := A1 ∪ A2 . Denote by ψ : A+ ։ S the surjective homomorphism defined on generators (ti , i) ∈ A by ψ(ti , i) := ti . For i = 1, 2, let ψi be + the restriction of ψ to A+ i , and let χ : (T1 × T2 ) → T0 be the homomorphism defined by χ(t1 , t2 ) := t1 t2 for (t1 , t2 ) ∈ T1 × T2 . Note that ψ0 , as defined above, in this case turns MERGE DECOMPOSITIONS, TWO-SIDED KROHN-RHODES, AND APERIODIC POINTLIKES 5 + + I I I I out to be the restriction of ψ to (A+ 1 A2 ) . Hence, writing m : T2 × T0 × T1 → T for the multiplication map m(t2 , t0 , t1 ) := t2 t0 t1 , we have ψ = m ◦ τ . Let ψM : A+ → TM be the merge decomposition along χ, ψ1 , and ψ2 . By Proposition 2.2, pick f : TM → T2I × T0I × T1I such that τ = f ◦ ψM . Then ψ = m ◦ f ◦ ψM , so S divides TM since ψ is surjective.  3. Two-sided Krohn-Rhodes theorem In this section, we apply the merge decomposition technique of Section 2 to give a short proof of the crucial step in the two-sided Krohn-Rhodes theorem. For any finite semigroup S, define VS to be the smallest variety which is closed under twosided semidirect products, and which contains SL and all simple groups that divide S. Theorem 3.1 (Two-sided Krohn-Rhodes). Let S be a finite semigroup. Then S ∈ VS . Proof. By induction on |S|. Case 1. S is a group. Any finite group embeds in an iterated wreath product of its simple group divisors, cf., e.g., [11, Cor. 4.1.6]. Case 2. S is cyclic. Any finite cyclic semigroup divides an iterated wreath product of a subgroup and copies of U1 , cf., e.g., [11, Cor. 4.1.28]. Case 3. S is not a group and S is not cyclic. Let A be a minimal generating set for S and note that |A| ≥ 2. Since S is not a group, without loss of generality, S is not right simple (cf., e.g., [11, Lem. A.3.3]). Therefore, there exists a ∈ A such that aS ( S. Let A1 := {a}, A2 := A \ A1 , Ti := hAi i for i = 1, 2, and T0 := hT1 T2 i. By minimality of A, T1 and T2 are strictly contained in S. By the induction hypothesis, Ti ∈ VTi , which is contained in VS , since any simple group dividing Ti also divides S. Moreover, T0 ⊆ aS, so T0 is also strictly contained in S. By the induction hypothesis again, T0 ∈ VT0 ⊆ VS . Since T1 ∪ T2 generates I I S, by Corollary 2.3, S divides a triple product of T2♭ , (T0I )T1 ×T2 , and T1♯ . Hence, by Fact 2.1, S ∈ VS ∗∗ (VS ∗∗ VS ) = VS .  4. Henckell’s theorem on aperiodic pointlikes Recall that any element s in a finite semigroup S has a unique idempotent power, sω . A semigroup S is called aperiodic if every subgroup of S is trivial, or, equivalently, sω s = sω for every s ∈ S. For k ≥ 1, define SLk+1 := SL ∗∗ SLk . A semigroup S is aperiodic if, and only if, S ∈ SLk for some k; indeed, the necessity follows from Theorem 3.1.3 Fact 4.1. For any m, n ≥ 1, SLm ∗∗ SLn ⊆ SLm+n . Proof. By induction on m. The case m = 1 is true by definition. By the lax associativity of double semidirect product [11, Cor. 2.6.26], (SL ∗∗ SLm−1 ) ∗∗ SLn ⊆ SL ∗∗ (SLm−1 ∗∗ SLn ). By the induction hypothesis, SL ∗∗ (SLm−1 ∗∗ SLn ) ⊆ SL ∗∗ SLm+n−1 = SLm+n .  Let V be a variety. A subset X of a finite semigroup S is called V-pointlike if, for any relational morphism ρ : S → 7 T with T ∈ V, X ⊆ ρ−1 (t) for some t ∈ T . Any singleton set is V-pointlike, and the collection of V-pointlike subsets of a semigroup S forms a downward closed subsemigroup, PLV (S), of the power semigroup 2S , partially ordered by inclusion, and with multiplication of subsets of S. The following observation is specific to the variety A of aperiodic semigroups: if X is an AS pointlike set in S, then so is the set X ω+∗ := n≥0 X ω X n . Indeed, for any ρ : S → 7 T with T 3A finite semigroup S lies in SLk if, and only if, every language recognized by S can be defined by a first-order sentence of quantifier depth ≤ k; this result is contained in [14, Ch. VI], and relates our work in Section 4 to the logical approach of [9]. 6 SAMUEL J. V. GOOL AND BENJAMIN STEINBERG aperiodic, X ⊆ ρ−1 (t) for some t ∈ T , which gives X m ⊆ ρ−1 (tm ) for all m ≥ 1. Aperiodicity of T then yields X ω X n ⊆ ρ−1 (tω ) for all n ≥ 0. We will call a subset U of 2S saturated, if it is a subsemigroup that is closed downward in the inclusion order and closed under the operation X 7→ X ω+∗ . Clearly, any subset U of 2S is contained in a smallest saturated set, which we call its saturation, and denote by Sat(U ). We will need the following lemma, which was essentially already in [4]; see also [6]. S Lemma 4.2. Let G be a subgroup of 2S . Then G ∈ Sat(G). Proof. Let C1 , . . . , Ck be anSexhaustive S list of the cyclic subgroups of G. Note that, for any generator X of Ci , X ω+∗ = Ci , so Ci ∈ Sat(G) forS every i.SAlso noteSthat G = C1 · · · Ck . Therefore, since multiplication distributes over union, G = ( C1 ) · · · ( Ck ) ∈ Sat(G).  We will use the merge decomposition (Section 2) to give a short proof of the following theorem. Theorem 4.3 (cf. [4, 6, 9]). Let S be a semigroup. The set PLA (S) is the saturation of the set of singletons in 2S . Moreover, if A is a generating set for S, then PLA (S) = PLSLk (S), |S| where k = (|A| − 1)2( 2 ) + 2|A| − 1. Proof. Throughout the proof, for any finite alphabet A, semigroup S, and homomorphism S (|S2ϕ |) + 2|Sϕ | − 1. ϕ : A+ → 2S , define Uϕ := im(ϕ), Sϕ := Uϕ , and k(ϕ) := (|ϕ(A)| − 1)2 Claim. For any homomorphism ϕ : A+ → 2S \{∅}, there exists a homomorphism ψ : A+ → T S with T ∈ SLk(ϕ) and ϕ(ψ −1 t) ∈ Sat(Uϕ ) for every t ∈ T . Proof of Claim. The construction of ψ : A+ → T with T ∈ SLk(ϕ) is by induction on the parameter (|Sϕ |, |ϕ(A)|) in N2 , ordered lexicographically. Case 1. For every a ∈ A, ϕ(a)Sϕ = Sϕ = Sϕ ϕ(a). Let e = ϕ(w) be an idempotent in the minimal S ideal of Uϕ . Then G := eUϕ e is a subgroup of Uϕ , see, e.g., [11, App. A]. By Lemma 4.2, G lies in Sat(eUϕ e), and hence also in Sat(Uϕ ), since eUϕ e ⊆ Uϕ . Using the assumption in this case and the fact that multiplication distributes over union, we have [  [ G. Sϕ = ϕ(w)Sϕ ϕ(w) = e Uϕ e = Thus, Sϕ lies in Sat(Uϕ ), and we choose ψ to be the trivial homomorphism A+ → {1} ∈ SL. Case 2. |ϕ(A)| = 1. Denote the unique element of ϕ(A) by X. Since Uϕ is a finite cyclic semigroup, pick m ≤ |Uϕ | such that X m is idempotent, i.e., X m = X ω . Let T := hx | xm = xm+1 i, the finite aperiodic cyclic semigroup of order m, and let ψ : A+ → T be the homomorphism defined by a 7→ x for every letter a ∈ A. Note that T ∈ SLm [11, Lem. 4.1.27], and, since Uϕ ⊆ 2Sϕ \ {∅}, we have S m ≤ |Uϕ | ≤ 2|Sϕ | −1 = k(ϕ). From the definitions, note that, for 1 ≤ i < m, ϕ(ψ −1 xi ) = X i , S −1 i ω+∗ which lies in Uϕ , and for i ≥ m, ϕ(ψ x ) = X , which lies in Sat(Uϕ ). Case 3. |ϕ(A)| ≥ 2, and there is a0 ∈ A such that ϕ(a0 )Sϕ ( Sϕ or Sϕ ϕ(a0 ) ( Sϕ . Without loss of generality, we may assume ϕ(a0 )Sϕ ( Sϕ . Let A1 := {a ∈ A | ϕ(a) = ϕ(a0 )}, and A2 := A \ A1 . Note that, since |ϕ(A)| ≥ 2, ϕ(A1 ) and ϕ(A2 ) are non-empty proper + subsets of ϕ(A). For i = 1, 2, denote by ϕi the restriction of ϕ to A+ i , and pick ψi : Ai → Ti S S with Ti ∈ SLk(ϕi ) and ϕ(ψi−1 t) = ϕi (ψi−1 t) ∈ Sat(Uϕi ) ⊆ Sat(Uϕ ), for all t ∈ Ti . Without loss of generality, we may assume the ψi are surjective. MERGE DECOMPOSITIONS, TWO-SIDED KROHN-RHODES, AND APERIODIC POINTLIKES 7 Let ϕ0 : (T1 ×ST2 )+ → 2S \ {∅} be the unique homomorphism defined, for (t1 , t2 ) ∈ T1 × T2 , by ϕ0 (t1 , t2 ) := ϕ(ψ1−1 t1 · ψ2−1 t2 ). Note that Sϕ0 ⊆ ϕ(a0 )Sϕ , since any w ∈ ψ1−1 t1 · ψ2−1 t2 starts with a letter from the subalphabet A1 . Since ϕ(a0 )Sϕ ( Sϕ by assumption, |Sϕ0 | < |Sϕ |, so the induction hypothesis applies to ϕ0 : pick a homomorphism χ : (T1 × T2 )+ → T0 with S T0 ∈ SLk(ϕ0 ) such that ϕ0 (χ−1 (t)) ∈ Sat(Uϕ0 ) ⊆ Sat(Uϕ ), for every t ∈ T0 . + + Define µ : (A+ → (T1 × T2 )+ and ψ0 := χ ◦ µ, as in Section 2. Note that, for any 1 A2 ) + + w1 ∈ A1 , w2 ∈ A2 , we have ϕ(w1 w2 ) ⊆ ϕ0 (µ(w1 w2 )), and, hence, ϕ(w) ⊆ ϕ0 (µ(w)) for all S S w ∈ (A+ A+ )+ . Therefore, by the definition of ψ0 , ϕ(ψ0−1 t) ⊆ ϕ0 (χ−1 t) for all t ∈ T0 , so S 1 2−1 also ϕ(ψ0 t) ∈ Sat(Uϕ ). Applying the construction of Section 2, let ψM : A+ → TM be the merge homomorphism, and pick f : TM → T2I × T0I × T1I such that f ◦ ψM = τ . Let t ∈ TM , and write f (t) = (t2 , t0 , t1 ) ∈ T2I × T0I × T1I . If (t2 , t0 , t1 ) ∈ T2 × T0 × T1 , then [ [ [ [ [ −1 t) ⊆ ϕ(τ −1 (t2 , t0 , t1 )) = ϕ(ψ2−1 t2 ) · ϕ(ψ0−1 t0 ) · ϕ(ψ1−1 t1 ) ∈ Sat(Uϕ ), ϕ(ψM and, if tS i = Ii for one or more i ∈ {0, 1, 2}, a similar inclusion holds, omitting the corresponding factors ϕ(ψi−1 ti ) from the final product. Let us write m := |Sϕ |. Note that, since Sϕ0 is strictly contained in Sϕ and ϕ0 (T1 × T2 ) is contained in 2Sϕ0 \ {∅}, we have m m−1 m m−1 k(ϕ0 ) ≤ (2m−1 − 2)2( 2 ) + 2m−1 − 1 = 2( 2 ) − 2( 2 )+1 + 2m−1 − 1 ≤ 2( 2 ) − 1,
m−1 m−1
and 2m−1 ≤ 2( 2 )+1 . using that m 2 2 =m−1+ By Facts 2.1 and 4.1, TM ∈ SLk , where k = k(ϕ0 ) + max{k(ϕ1 ), k(ϕ2 )} + 1. Using that |ϕ(Ai )| < |ϕ(A)|, we have  m    m k(ϕ0 ) + max{k(ϕ1 ), k(ϕ2 )} + 1 ≤ 2( 2 ) − 1 + (|ϕ(A)| − 2)2( 2 ) + 2m − 1 + 1 = k(ϕ).  Now, to prove the theorem, let A be a generating set for S, define ϕ : A+ → 2S by ϕ(a) := {a} for a ∈ A, and pick ψ : A+ → T as in the claim. Then Uϕ is the set of singletons, |ϕ(A)| = |A|, |S| and Sϕ = S, so that k(ϕ) = (|A| − 1)2( 2 ) + 2|S| − 1 =: k. Define the relational morphism S ρ: S → 7 T by ρ−1 (t) := ϕ(ψ −1 t). Then, for any SLk -pointlike X ⊆ S, we have X ⊆ ρ−1 (t) for some t ∈ T , and therefore, since ρ−1 (t) lies in Sat(Uϕ ) by the claim, so does X. We have proved that PLSLk (S) ⊆ Sat(Uϕ ), while the remarks at the beginning of this section imply Sat(Uϕ ) ⊆ PLA (S), which is clearly contained in PLSLk (S), since SLk ⊆ A. Thus, PLSLk (S) = Sat(Uϕ ) = PLA (S).  Acknowledgements (|S2ϕ |) In an earlier version of the proof of Theorem 4.3, we proved the claim for k(ϕ) := |ϕ(A)|2 . We acknowledge the help of MathOverflow [12] for guiding us to the slightly better bound given in the paper. References 1. J. Almeida, Some algorithmic problems for pseudovarieties, Publ. Math. Debrecen 54 (1999), no. suppl., 531–552, Automata and formal languages, VIII (Salgótarján, 1996). 2. K. Auinger and B. Steinberg, On the extension problem for partial permutations, Proc. Amer. Math. Soc. 131 (2003), no. 9, 2693–2703. 3. S. Eilenberg, Automata, languages, and machines. Vol. B, Academic Press, New York, 1976, With two chapters by Bret Tilson, Pure and Applied Mathematics, Vol. 59. 8 SAMUEL J. V. GOOL AND BENJAMIN STEINBERG 4. K. Henckell, Pointlike sets: the finest aperiodic cover of a finite semigroup, J. Pure Appl. Algebra 55 (1988), 85–126. 5. K. Henckell, S. Lazarus, and J. Rhodes, Prime decomposition theorem for arbitrary semigroups: general holonomy decomposition and synthesis theorem, J. Pure Appl. Algebra 55 (1988), no. 1-2, 127–172. 6. K. Henckell, J. Rhodes, and B. Steinberg, Aperiodic pointlikes and beyond, Internat. J. Algebra Comput. 20 (2010), no. 2, 287–305. 7. K. Krohn and J. Rhodes, Algebraic theory of machines. I. Prime decomposition theorem for finite semigroups and machines, Trans. Amer. Math. Soc. 116 (1965), 450–464. 8. K. Krohn, J. Rhodes, and B. Tilson, Algebraic theory of machines, languages, and semigroups, Edited by Michael A. Arbib. With a major contribution by Kenneth Krohn and John L. Rhodes, Academic Press, New York, 1968, Chapters 1, 5–9. 9. T. Place and M. Zeitoun, Separating regular languages with first-order logic, Logical Methods in Computer Science 12 (2016), no. 1:5, 1–30. 10. J. Rhodes and B. Steinberg, Pointlike sets, hyperdecidability and the identity problem for finite semigroups, Internat. J. Algebra Comput. 9 (1999), no. 3-4, 475–481. 11. , The q-theory of Finite Semigroups, Springer, 2009. 12. B. Steinberg, A strange two-variable recursion, MathOverflow question, answered by M. Fischler, https://mathoverflow.net/q/278517. , On pointlike sets and joins of pseudovarieties, Internat. J. Algebra Comput. 8 (1998), no. 2, 13. 203–234. 14. H. Straubing, Finite automata, formal logic, and circuit complexity, Progress in Theoretical Computer Science, Birkhäuser Boston Inc., Boston, 1994. 15. T. Wilke, Classifying discrete temporal properties, STACS’99, Lec. Notes. Comp. Sci., vol. 1563, Springer, 1999, pp. 32–46. Department of Mathematics, City College of New York, Convent Avenue at 138th Street, New York, New York 10031, USA E-mail address: [email protected] and [email protected] 

Inverse of a Special Matrix and Application Thuan Nguyen arXiv:1708.07795v1 [cs.DM] 25 Aug 2017 School of Electrical and Computer Engineering, Oregon State University, Corvallis, OR, 97331 Email: [email protected] Abstract—The matrix inversion is an interesting topic in algebra mathematics. However, to determine an inverse matrix from a given matrix is required many computation tools and time resource if the size of matrix is huge. In this paper, we have shown an inverse closed form for an interesting matrix which has much applications in communication system. Base on this inverse closed form, the channel capacity closed form of a communication system can be determined via the error rate parameter α. Keywords: Inverse matrix, convex optimization, channel capacity. I. M ATRIX C ONSTRUCTION In Wireless communication system or Free Space Optical communication system, due to the shadow effect or the turbulent of environment, the channel conditions can be flipped from “good” to “bad” or “bad” to “good” state such as Markov model after the transmission time σ [1] [2]. For simple intuition, in “bad” channel, a signal will be transmitted incorrectly and in “good” channel, the signal is received perfectly. Suppose a system has total n channels, the “good” channel is noted as “1” and “bad” channel is “0”, respectively, the transmission time between transmitter and receiver is σ, the probability the channel is flipped after the transmission time σ is α. We note that if the system using a binary code such as On-Off Keying in Free Space Optical communication, then the flipped probability α is equivalent to the error rate. Consider a simple case for n = 2, suppose that at the beginning, both channel is “good” channel, the probability of system has both of channels are “good” after transmission time σ, for example, is (1−α)2 . Let call Aij is the probability of system from the state has i − 1 “good” channels and n − i + 1 “bad” channels transfers to state has j − 1 “good” and n − j + 1 “bad” channels. Obviously that 1 ≤ i ≤ n + 1 and 1 ≤ j ≤ n + 1. For example, the transition matrix A2 and A3 for n = 2 and n = 3 are constructed respectively as follows: (1 − α)2  A2 = α(1 − α) α2   (1 − α)3  (1 − α)2 α A3 =  (1 − α)α2 α3 2α(1 − α) (1 − α)2 + α2 2α(1 − α) 3(1 − α)2 α 2(1 − α)α2 + (1 − α)3 2(1 − α)2 α + α3 3α2 (1 − α)  α2 α(1 − α) . (1 − α)2 3α2 (1 − α) 2(1 − α)2 α + α3 2(1 − α)α2 + (1 − α)3 3(1 − α)2 α given by Proposition 2. Moreover, this matrices are obviously central symmetric matrix. Proposition 1. For n channels system, the transition matrix An has size (n + 1) × (n + 1) and all entries An ij in row i column j will be established by An ij = s=max(i−j,0) n+1−i s i−1  j−i+2s n−(j−i+2s) α (1 − α) Proof. From the definition, Anij is the probability from state has i − 1 “good” channels or i − 1 bit “1” transfer to state has j − 1 “good” channels or j − 1 bit “1”. Therefore, suppose s is the number channels in i − 1 “good” channels that is flipped to “bad” channels after the transmission time σ and 0 ≤ s ≤ i − 1. Thus, to maintain j − 1 “good” channels after the time σ, the number of “bad” channels in n + 1 − i “bad” channels must be flipped to “good” channels is: (j − 1) − ((i − 1) − s) = j − i + s Therefore, the total number of channels are flipped their state after transmission time σ is: s + (j − i + s) = j − i + 2s and the total number of channels that preserves their state after transmission time σ is n−(j −i+2s). However, 0 ≤ s ≤ i−1. Similarly, the number of “bad” channels in n + 1 − i “bad” channels must be flipped to “good” channels should be in 0 ≤ j − i + s ≤ n + 1 − i. Hence: ( max s = min(n + 1 − j; i − 1) min s = max(0; i − j) Therefore, An ij can be determined by below form: An ij =  j − i + s  s=min(n+1−j,i−1) X s=max(i−j,0) n+1−i s i−1  j−i+2s n−(j−i+2s) α (1 − α) Proposition 2. All the entries of inverse matrix A−1 n given in Proposition 1 can be determined via original transition matrix  A n for ∀ α 6= 1/2. α3  (1 − α)α2  . (1 − α)2 α (1 − α)3 These transition matrices are obviously size (n+1)×(n+1) since the number of “good” channels can achieve n+1 discrete values from 0, 1, . . . , n. Moreover, these class matrices have several interesting properties: (1) all entries in matrix An can be determined by Proposition 1; (2) the inverse of matrix An is s=min(n+1−j,i−1)  j − i + s  X An −1 ij = (−1)i+j An (1 − 2α)n ij Due to the pages limitation, we will show the detailed proof at the end of this paper. To illustrate our result, an example of the inverse matrix A2 are shown as follows: A−1 2  (1 − α)2 1  = −α(1 − α) (1 − 2α)2 α2 −2α(1 − α) (1 − α)2 + α2 −2α(1 − α)  α2 −α(1 − α) . (1 − α)2 Next, base on the existence of inverse matrix closed form, we will show that a capacity closed form for a discrete memory-less channel can be established. We note that in [3], the authors said that haven’t closed form for channel capacity problem. However, with our approach, the closed form can be established for a wide range of channel with error rate α is small. II. O PTIMIZE SYSTEM CAPACITY A discrete memoryless channel is characterized by a channel matrix A ∈ Rm×n with m and n representing the numbers of distinct input (transmitted) symbols xi , i = 1, 2, . . . , m, and output (received) symbols yj , j = 1, 2, . . . , n, respectively. The matrix entry Aij represents the conditional probability that given a symbol xi is transmitted, the symbol xj is received. Let p = (p1 , p2 , . . . , pm )T be the input probability mass vector, where pi denotes the probability of transmitting symbol xi , then the probability mass vector of output symbols q = (q1 , q2 , . . . , qn )T = AT p, where qi denotes the probability of receiving symbol yi . For simplicity, we only consider the case n = m such that the number of transmitted input patterns is equal the number of received input patterns. The mutual information between input and output symbolsis: I(X; Y ) = H(Y ) − H(Y |X), where H(Y ) = H(Y |X) = j=n X qj j=1 m X n X − log qj pi Aij log Aij . i=1 j=1 Thus, the mutual information function can be written as: I(X; Y ) = − j=n X (AT p)j log (AT p)j + m X n X pi Aij log Aij , i=1 j=1 j=1 where (AT p)j denotes the jth component of the vector q = (AT p). The capacity C of a discrete memoryless channel associated with a channel matrix A puts a theoretical maximum rate that information can be transmitted over the channel [3]. It is defined as: C = max I(X; Y ). p (1) Therefore, finding the channel capacity is to find an optimal input probability mass vector p such that the mutual information between the input and output symbols is maximized. For a given channel matrix A, I(X; Y ) is a concave function in p [3]. Therefore, maximizing I(X; Y ) is equivalent to minimizing −I(X; Y ), and the capacity problem can be cast as the following convex problem: Minimize: m X n n X X pi Aij log Aij (AT p)j log (AT p)j − i=1 j=1 j=1 Subject to: ( pi  0 1T p = 1 Optimal numerical values of p∗ can be found efficiently using various algorithms such as gradient methods [4] [5]. However, in this paper, we try to figure out the closed form for optimal distribution p via KKT condition. The KKT conditions state that for the following canonical optimization problem: Problem Miminize: f (x) Subject to: gi (x) ≤ 0, i = 1, 2, . . . n, hj (x) = 0, j = 1, 2, . . . , m, construct the Lagrangian function: L(x, λ, ν) = f (x) + n X i=1 λi gi (x) + m X νj hj (x), (2) j=1 then for i = 1, 2, . . . , n, j = 1, 2, . . . , m, the optimal point x∗ must satisfy:   gi (x∗ ) ≤ 0,    ∗   hj (x ) = 0, dL(x,λ,ν) (3) |x=x∗ ,λ=λ∗ ,ν=ν ∗ = 0, dx   ∗ ∗  λi xi = 0,    λ∗ ≥ 0. i Our transition matrix that is already established in previous part can represent as a channel matrix. In the optical transmission, for example, the transmission bits are denoted by the different levels of energy, for example, in On-Off Keying code bit “1” and “0” is represented by high and low power level. This energy is received by a photo diode and converse directly to the voltage for example. However, these photo diode work base on the aggregate property when collecting all the incident energy, that said, if two channels transmit a bit “1” then the photo diode will receive the same energy “2” even though this energy comes from a different pair of channels. Therefore, the received signal is completely dependent to the number of bits “1” in transmission side. Hence, in receiver side, the photo diode will recognize n + 1 states 0, 1, 2, . . . , n. From this property, the transition matrix A is the previous section is exactly the system channel matrix. The channel capacity of system, therefore, is determined as an optimization problem in (1). Next, we will show that the above optimization problem can be solved efficiently by KKT condition. We note that our method can establish the closed form for general channel matrix and then the results are applied to special matrix An . First, we try to optimize directly with input distribution p, however, the KKT condition for input distribution is too complicated to construct the first derivation. On the other hand, base on the existence of inverse channel matrix, the output variable is more suitable to work with KKT condition since. Due to 0 ≤ qj ≤ 1, the Lagrange function from (2) with output variable q is: L(qj , λj , νj ) = I(X, Y ) + j=n X j=n X qj λj + ν( qj − 1) j=1 j=1 Using KKT conditions, at optimal point qj∗ , λ∗j , ν ∗ :  ∗ qj ≥ 0   Pj=n ∗     j=1 qj = 1   dI(X, Y ) ν ∗ − λ∗j − =0 dqj∗    λ∗ ≥ 0     j∗ ∗ λj qj = 0 P Because 0 ≤ pi ≤ 1, i = 1, . . . , (n) and ni=1 pi = 1, so Pi=n always exist pi > 0. From qj = i=1 pi Aij with ∀ Aij > 0, we can see clearly that qj > 0 with ∀qj or qj∗ > 0 with ∀qj∗ . Therefore with fifth condition, λ∗j = 0 with ∀λ∗j . Then, we have simplified KKT conditions: Pj=n ∗  j=1 qj = 1 dI(X, Y ) =0 ν ∗ − dqj∗ The derivations are determined by: j=n i=n dI(X, Y ) X −1 X Aij log Aij − (1 + log qj ) Aji = dqj j=1 i=1 Let call: i=n X A−1 ji j=n X Aij log Aij = Kj From the second KKT simplified condition, we can compute ∀ qj∗ : ∗ qj∗ = 2Kj −ν −1 And finally: ∗ Due to the channel matrix is a closed form of α, the optimal input vector p and output vector q also is a function of α. However, we note that since the KKT condition works directly to the output variable q, the optimal input p can be invalid pi > 1 or pi < 0. In next step, our simulations shown that for n ≤ 10 and α ≤ 0.2, both output and input vector are valid. That said, our approach will be worked with a good system where the error probability α is small. In case of the invalid optimal input vector, the upper bound of channel capacity, of course, will be established. III. C ONCLUSION In this paper, our contributions are twofold: (1) establish an inverse closed form for a class of channel matrix based on the error probability α; (2) figure out the closed form for channel matrix with small error rate α and determine the upper bound system capacity for a high error rate channel. R EFERENCES [1] Jeff McDougall and Scott Miller. Sensitivity of wireless network simulations to a two-state markov model channel approximation. In Global Telecommunications Conference, 2003. GLOBECOM’03. IEEE, volume 2, pages 697–701. IEEE, 2003. [2] Hong Shen Wang and Nader Moayeri. Finite-state markov channel-a useful model for radio communication channels. IEEE transactions on vehicular technology, 44(1):163–171, 1995. [3] Thomas M Cover and Joy A Thomas. Elements of information theory. John Wiley & Sons, 2012. [4] Michael Grant, Stephen Boyd, and Yinyu Ye. Cvx: Matlab software for disciplined convex programming, 2008. [5] Stephen Boyd and Lieven Vandenberghe. Convex optimization. Cambridge university press, 2004. j=1 i=1 A PPENDIX Next, using derivation of I(X,Y) at qj = qj∗ and last KKT condition: ν ∗ = Kj − (1 + log qj ∗ ) Hence: qj∗ = 2Kj −ν ∗ −1 Next, using first KKT simplified condition, we have the sum of all output states is 1. j=n X 2Kj −ν ∗ −1 =1 j=1 2 ν∗ = j=n X 2 Kj −1 j=1 Therefore, ν ∗ can be figured out by: ∗ ∗ pT = q T A−1 ij ν = log j=n X j=1 2Kj −1 Proof for Proposition 2. Proof. To simplify our notation, the “good” and “bad” channel are represented by bit “1” and “0”, respectively. Next, we will use the definition to show that: An An −1 = I If matrix A∗n is constructed by A∗nij = (−1)i+j An ij , then we need to show that: An A∗n = B = (1 − 2α)n I Firstly, we note that the An ij and An ∗ij is only different by sign of the first index (−1)i+j . Therefore, Bij which is computed by product of row i in matrix An ij and column j in matrix An ∗ij , can be computed by: Bij = k=n+1 X k=1 An ik A∗nkj Note that the An ik is the probability from state i “good” channels (with i − 1 bit “1” and n − i + 1 bit “0”) to medium state has k “good” channels (with k − 1 bit “1” and n − k + 1 bit “0”). Moreover, if the sign is ignored, then An ∗kj also is the probability going from state k to state j, too. However,
n the state k includes C k−1 sub-states which have a same number of “good” and “bad” channels. For example with n = 2, state k = 2 includes two sub-states that contains one “good” and one “bad” channels are “10” an “01”. Therefore, the total number
of sub-states while k runs from 1 to n is P k=n+1 n C k−1 = 2n sub-states. Let compute Bij by divided k=1 into two subsets: Compute Bij for i = j: This means that Bii is the sum of the probability from state i − 1 bit “1” go to states has k − 1 bit “1” then come back to state has i − 1 bit “1”. In 2n sub-states, we can divide back to n + 1 categories by the number of different position between i and k. • If all the bit in i and k are the same, then the probability is: ! ! C n n (1 − α)2n (1 − α)n (1 − α)n = C 0 0 • If all the bit in i and k different at only one position, then the probability is: ! n (1 − α)2(n−1) (1 − α)2 C 1 • If all the bit in i and k different at only two positions, then the probability is: ! n C (1 − α)2(n−2) (1 − α)2×2 2 • If all the bit in i and k different at all positions, then the probability is: ! C n (1 − α)2n n Therefore, Bii can be determined by the probability of all n + 1 categories such as: ! n 2t C α (1−α)2n−2t = ((1 − α)2 − α2 )n = (1−2α)n Bii = t t=0 t=n X A∗nkj into two subsets: Compute Bij for i6= j: Let divide k + j is odd and A∗nkj < 0 or k + j is even and A∗nkj > 0, Pk=n respectively. Therefore, Bij = k=1 An ik A∗nkj also is distributed into positive or negative subsets. Next, we will show that the positive subset in Bij is equal the negative subset then Bij = 0 for i 6= j. Indeed, suppose that state i with i − 1 bit “1” go to state k1 and then to back to state j with j −1 bit “1” and Bik1 is positive value. Next, we will show that existence a state k2 such that Bik2 is negative value and Bik1 = −Bik2 . Let call s is the number of positions where state i and j have a same bit. Obviously that s ≤ n − 1 due to i 6= j. For example if n = 4 and i = 1111 and j = 0001, we have s = 1 because i and j share a same bit “1” in the positions fourth. Suppose that an arbitrary state k1 are picked, we will show how to chose the state k2 with Bik1 = −Bik2 . Consider two follows cases: • If (n − s) is odd. k2 is constructed by maintain s position of k1 where i and j have same bit and flip bit in the n − s rest positions. • If (n − s) is even. k2 is constructed by maintain s + 1 position of k1 where s position are i and j have a same bit and one position where i and j have a different bit, next n − s − 1 rest positions will be flipped. Note that since s ≤ n − 1 then we are able to flip n − s − 1 rest positions. We obviously can see that k1 and k2 satisfied the probability condition |Bik1 | = |Bik2 | due to the number of flipped bit between i and k1 equals the number of flipped bit between k2 and j and the number of flipped bit between j and k1 equals the number of flipped bit between k2 and i. Next, we will prove that k1 and k2 make Bik1 and Bik2 in different subsets. Indeed, call number of bit “1” in k1 is b1 , number of bit “1” in k2 is b2 , number of bit “1” in s bit same of i and j is bs , respectively. Therefore, the number of bit “1” of k1 in (n − s) rest positions is (k1 − ks ), the number of bit “1” of k2 in (n − s) rest positions is (k2 − ks ). • If (n − s) is odd. Since all bit in (n − s) rest positions of k1 is flipped to create k2 , then total number of bit “1” in n − s bit of k1 and k2 is (k1 − ks + k2 − ks = n − s) is odd. So, (k1 + k2 ) should be an odd number. That said (k1 − k2 ) is odd or (k1 + j) − (k2 + j) is odd. Therefore, Bik1 and Bik2 bring the contradict sign. • If (n − s) is even. Because, we fix one more position to create k2 , then number of flipped bit (n−s−1) is odd number. If one more bit is fixed in k1 is “0”, we have a same result with case (n − s) is odd. If fixed bit is “1”, similarly in first case (k1 − ks − 1) + (k2 − ks − 1) = n − s − 1 is odd number, therefore (k1 + k2) is odd number. That said (k1 − k2 ) is odd or (k1 + j) − (k2 + j) is odd. Therefore, Bik1 and Bik2 bring the contradict sign. Therefore, the state k2 always can be created from a random state k1 and Bik1 and Bik2 bring a contradict sign. That said for i 6= j, Bij = 0. Therefore: B = (1 − 2α)n I The Proposition 2, therefore, are proven. 

2016 ICSEE International Conference on the Science of Electrical Engineering Randomized Independent Component Analysis Matan Sela Ron Kimmel arXiv:1609.06942v1 [stat.ML] 22 Sep 2016 Department of Computer Science Technion - Israel Institute of Technology Abstract—Independent component analysis (ICA) is a method for recovering statistically independent signals from observations of unknown linear combinations of the sources. Some of the most accurate ICA decomposition methods require searching for the inverse transformation which minimizes different approximations of the Mutual Information, a measure of statistical independence of random vectors. Two such approximations are the Kernel Generalized Variance or the Kernel Canonical Correlation which has been shown to reach the highest performance of ICA methods. However, the computational effort necessary just for computing these measures is cubic in the sample size. Hence, optimizing them becomes even more computationally demanding, in terms of both space and time. Here, we propose a couple of alternative novel measures based on randomized features of the samples - the Randomized Generalized Variance and the Randomized Canonical Correlation. The computational complexity of calculating the proposed alternatives is linear in the sample size and provide a controllable approximation of their Kernelbased non-random versions. We also show that optimization of the proposed statistical properties yields a comparable separation error at an order of magnitude faster compared to Kernel-based measures. I. I NTRODUCTION Independent component analysis (ICA) is a well-established problem in unsupervised learning and signal processing, with numerous applications including blind source separation, face recognition, and stock price prediction. onsider the following scenario. A couple of speakers are located in a room. Each of them plays a different sound . Two microphones which are arbitrarily placed in the same room record unknown linear combinations of the sounds. The goal of ICA is to process the signals recorded by the mics for recovering the soundtracks played by the speakers. More precisely, the basic idea of ICA is to recover ns statistically independent components of a non-Gaussian random T vector s = (s1 , ..., sns ) from ns observed linear mixtures of its elements. That is, we assume that some unknown matrix A ∈ Rns ×ns mixes the entries of s such that x = As. From samples of x, the goal is to estimate an un-mixing matrix W such that y = W x and the components of y are statistically independent. The matrix W is found by a minimization process over a contrast function which measures the dependency between the unmixed elements. Ideally, finding the matrix W which minimizes the mutual information (MI) provides the theoretically most accurate reconstruction. The MI is defined as the Kullback-Liebler divergence between the joint distribution of y, p(y1 , ..., yns ), and the product of its marginal distributions, ns Y p(yi ). It is a non-negative function which vanishes if and i=1 only if the components of y are mutually independent. Unfortunately, in practical applications, the joint and the marginal distributions are unknown. Estimating and optimizing the MI directly from the samples is difficult. Fitting a parametric or a nonparametric probabilistic model to the data based on which MI is calculated is problem dependent and is often inaccurate. Many researches proposed alternative contrast functions. One of the most robust alternative is the F-correlation proposed in [1]. This function is evaluated by mapping the data samples into a reproducing kernel Hilbert space, F where a canonical correlation analysis is performed. The largest kernel canonical correlation (KCC) and the product of the kernel canonical correlations, known as the kernel generalized variance (KGV), are two possible contrast functions. Despite their superior performance, algorithms based on minimizing the KCC or the KGV (denoted as kernelized ICA algorithms) are less attractive for practical uses since the complexity of exact evaluation of these functions is cubic in the sample size. A recent strand of research suggested randomized nonlinear feature maps for approximating the reproducing kernel Hilbert space corresponding to kernel functions. This technique enables revealing nonlinear relations in a data by performing linear data analysis algorithms, such as Support Vector Machine [2], Principal Component Analysis and Canonical Correlation Analysis [3]. These methods approximate the solution of kernel methods while reducing their complexity from cubic to linear in the sample size. Here, we propose two alternative contrast functions, Randomized Canonical Correlation (RCC) and Randomized Generalized Variance (RGV) which approximate KCC and KGV, respectively, yet require just a fraction of the computational effort to evaluate. Furthermore, the proposed random approximations are smooth, easy to optimize and converge to their kernelized version as the number of random features grows. Finally, we propose optimization algorithms similar to those proposed in [1], for solving the ICA problem. We demonstrate that our method has a comparable accuracy as KICA but runs 12 times faster while separating components of real data. II. BACKGROUND AND R ELATED W ORKS A. Canonical Correlation Analysis Canonical correlation analysis is a classical linear analysis method introduced in [4], which generalizes the Principal Component Analysis (PCA) for two or more random vectors. In PCA, given samples of a random vector x ∈ Rd , the idea is to search for a vector w ∈ Rd , which maximizes the variance of the projection of x onto w. In practice, the principal components are the eigenvectors corresponding to the largest eigenvalues of the empirical covariance matrix 1 XX T , where X is a matrix containing samples of C = N x as its columns, and N is the sample size. In canonical correlation analysis (CCA), given a couple of random vectors x ∈ Rdx and y ∈ Rdy , one looks for a pair of vectors wx ∈ Rdx and wy ∈ Rdy , that maximize the correlation between the projection of x onto wx and the projection of y onto wy . More formally, CCA can be formulated as max wx ∈Rdx ,wy ∈Rdy = corr(wxT x, wyT y) = cov(wxT x, wyT y) (var(wxT x))1/2 (var(wyT y))1/2 Let X ∈ Rdx ×N and Y ∈ Rdy ×N be matrices of samples of x and y, respectively. The empirical Canonical Correlation Analysis problem is given by minimize Wx ∈Rdx ×r ,Wy ∈Rdy ×r subject to kd corr(WxT X, WyT Y ) − Ik cd orr(Wx , Wx ) = I, cd orr(Wy , Wy ) = I, where cd orr(·, ·) is the empirical correlation. The canonical correlations λ1 , ..., λr are found by solving the following generalized eigenvalue problem    0 CXY wx = CY X 0 wy    CXX + γI 0 wx λ , 0 CY Y + γI wy 1 XY T is the empirical cross covariance where CXY = N matrix, and γI is added to the diagonal for stabilizing the solution. As discussed next, the kernelized ICA method use a kernel formulation of CCA for evaluating its contrast functions which measures the independence of random variables. B. Kernelized Independent Component Analysis In ICA, we minimize a contrast function which is defined as any non-negative function of two or more random variables that is zero if and only if they are statistically independent. By definition, a pair of random variables x1 and x2 are said to be statistically independent if p(x1 , x2 ) = p(x1 )p(x2 ). As a result, for any two functions f1 , f2 ∈ F The F-correlation ρF is defined as the maximal correlation among all the functions f1 (x1 ), f2 (x2 ) in F. That is = f1 ,f2 ∈F |cov (f1 (x1 ), f2 (x2 ))| max f1 ,f2 ∈F 1/2 (varf1 (x1 )) (varf2 (x2 )) 1/2 Obviously, if x1 and x2 are independent, then ρF = 0. As proven in [1], if F is a functional space that contains the Fourier basis (f (x) = eiωx , ω ∈ R), then, the opposite is also true. This property implies that ρF can replace the mutual information while searching for a matrix W that transforms the vector (x1 , x2 )T into a vector with independent components. Computing the F-correlation directly for an arbitrary space F requires estimating the correlation between every possible pair of functions in the space, making the calculation impractical. However, if F is a reproducing kernel Hilbert space (RKHS), the F-correlation can be evaluated. Let k(x, y) be a kernel function associated with the inner product between functions in a reproducing kernel Hilbert space F. Denote Φ(x) as the feature map corresponding to the kernel k(x, y), such that k(x, y) = hΦ(x), Φ(y)i. The feature map is given by Φ(x) = k(·, x). Then, from the reproducing property of the kernel, for any function f (x) ∈ F, f (x) = hΦ(x), f i, ∀f ∈ F, ∀x ∈ R. It follows that for every functional space F associated with a kernel function k(x, y) and a feature map Φ(x), the Fcorrelation between a pair of random variables x1 and x2 can be formulated as ρF = max |corr (hΦ(x1 ), f1 i, hΦ(x2 ), f2 i)| . f1 ,f2 ∈F i N Let {xi1 }N i=1 and {x2 }i=1 be the samples of the random variables x1 and x2 , respectively. Any function f ∈ F N X can be represented by f = αi Φ(xi ) + f ⊥ , where f ⊥ i=1 is a function in the subspace of functions orthogonal to span{Φ(x1 ), ..., Φ(xN )}. Thus, c (hΦ(x1 ), f1 i, hΦ(x2 ), f2 i) = cov N 1 X hΦ(xk1 ), f1 ihΦ(xk2 ), f2 i = N k=1 N N N X X 1 X = hΦ(xk1 ), α1i Φ(xi1 ) + f1⊥ ihΦ(xk2 ), α2i Φ(xi2 ) + f2⊥ i N i=1 i=1 = E{f1 (x1 )f2 (x2 )} = E{f1 (x1 )}E{f2 (x2 )} 1 N k=1 N X hΦ(xk1 ), N X α1i Φ(xi1 )ihΦ(xk2 ), N X α2i Φ(xi2 )i i=1 i=1 k=1 N X N X N X α1i K1 (xi1 , xk1 )K2 (xj2 , xk2 )α2i k=1 i=1 j=1 = 1 N = 1 T α K1 K2 α2 , N 1 Equivalently, cov (f1 (x1 ), f2 (x2 )) = 0. max |corr (f1 (x1 ), f2 (x2 ))| ρF = where K1 and K2 are the empirical kernel Gram matrices of x1 and x2 , respectively. With a similar development for the empirical variance, one conclude c (hΦ(x1 ), f1 i) = var 1 T α K1 K1 α1 N 1 c (hΦ(x2 ), f2 i) = var 1 T α K2 K2 α2 N 2 Therefore, the empirical F-correlation between x1 and x2 is given by ρ̂F (x1 , x2 ) = max α1 ,α2 ∈RN where P is the rank of Kκ . Figure II-B demonstrates the Kernel Generalized Variance versus the Mutual Information for a Gaussian kernel and for different σ’s. The complexity of a naive implementation of both KCC and KGV is O(N 3 ). However, an approximate solution can be computed using Cholesky decomposition which reduces the complexity to be roughly quadratic in the sample size. α1T K1 K2 α2
1/2 T 2
1/2 α2 K2 α2 α1T K12 α1 which can be evaluated by solving the following generalized eigenvalue problem    0 K1 K2 α1 = K2 K1 0 α2  2   K1 0 α1 λ . 0 K22 α2 The calculation of the F-correlation is thus equivalent to solving a kernel CCA problem. For ensuring computational stability, it is common to use a regularized version of KCCA given by    0 K1 K2 α1 = K2 K1 0 α2     0 α1 (K1 + N2κ I)2 λ . α2 0 (K2 + N2κ I)2 For m random variables, the regularized KCCA amounts to finding the largest generalized eigenvalue of the following system    0  K2 K1  ..  . Km K1   λ  K1 K2 0 .. . Km K2 (K1 + N2κ I)2 0 .. . 0 ··· ··· ··· α1 K1 Km K2 Km   α2   .  = ..   ..  . 0 αm 0 (K2 + N2κ I)2 .. . 0 ··· ··· ··· P for different σ values as a function of the angle of rotation of the orthogonal matrix W . C. Randomized Features In kernel methods, each data sample x is mapped to a function in the reproducing kernel Hilbert space Φ(x), where the analysis is performed. The inner product between two representational functions Φ(x) and Φ(y) can be evaluated directly on the samples x and y using the kernel function k(x, y). For real valued, normalized (k(x, y) ≤ 1), shift invariant kernels Rd × Rd Z T k(x, y) = p(w)e−jw (x−y) dw ≈ Rd   0 α1 0   α2   .  ..   ..  . 2 Nκ αm (Km + 2 I) or in short, Kκ α = λDκ α. This is the first contrast function proposed in [1]. The second one, denoted as the kernel generalized variance, depends upon not only the largest generalized eigenvalue of the problem above but also on the product of the entire spectrum. This is derived from the Gaussian case where the mutual information is equal to minus half the logarithm of the product of generalized eigenvalues of the regular CCA problem. In summary, the kernel generalized variance is given by I(x1 , ..., xm ) = − Fig. 1. The mutual information and the Kernel Generalized Variance 1X log λi , 2 i=1 m X 1 −jwiT x jwiT y ≈ e e m i=1 = m X 1 cos(wiT x + bi ) cos(wiT y + bi ), m i=1 where p(w) is the inverse Fourier transform of k and bi ∼ U (0, 2π), and wi ’s are independently drawn from p(w). Thus, the kernel function can be approximated by transforming the data samples into an m-dimensional random space z(x) =  1  T √ cos(w1T x + b1 ), ..., cos(wm x + bm ) and taking the inm ner product between the maps. k(x, y) ≈ hz(x), z(y)i. The following theorem shows that the reproducing kernel Hilbert space corresponding to k can be approximated by a random map, in the inner product sense. Let k(x, y) = k(x − y) be a shift-invariant kernel function such that k(x, y) ≤ 1. Denote p(w) as the inverse Fourier Z transform of k, that is, k(x) = p(w)e−i2πw T x dx. Inde- Rd {wi }m i=1 pendently sample m variables from the distribution p(w) and m random numbers bi uniformly from the section [−π, r π] and construct the r andom Fourier feature map z(x) =  2  T cos(w1T x + b1 ), ..., cos(wm x + bm ) . We define the Zm correlation of a pair of random variables x1 and x2 as ρz (x1 , x2 ) = max w1 ∈Rm ,w2 ∈Rm corr(w1T z(x1 ), w2T z(x2 )) The empirical covariance between w1T z(x1 ) and w2T z(x2 ) is given by c 1T z(x1 ), w2T z(x2 )) = cov(w Fig. 2. The analytic and empirical error of the kernel approximation using random Fourier features as a function of the number of features m. N 1 X hw1 , z(xk1 )ihw2 , z(xk2 )i = N k=1 N 1 X T w1 z(xk1 )z(xk2 )T w2 = N k=1 ! N X 1 = w1T z(xk1 )z(xk2 )T w2 N d×N Theorem II.1. Let X ∈ R be a matrix containing samples of x ordered in its columns. Denote z(X) ∈ Rm×N as a matrix containing the Fourier random features of each column of X as its columns. Denote K̂ = z(X)T z(x) and K as the empirical kernel matrix corresponding to the same kernel as K̂. Then r 3n2 log n 2n log n EkK̂ − Kk ≤ + , m m k=1 z = w1T Ĉ12 w2 Similarly, the empircal variances can be computed as z vd ar(w1T z(x1 )) = w1T Ĉ11 w1 where the norm is an operator norm. z vd ar(w2T z(x2 )) = w1T Ĉ22 w2 , Proof. See [3]. Figure II-C shows the analytic bound versus the empirical error of an approximate kernel K̂ evaluated on 104 data samples. Introduced in [2] for approximating the solution of kernel Support Vector Machine, these random feature maps are also useful for solving kernel Principal Component Analysis (kPCA), and kernel Canonical Correlation Analysis (kCCA), [3]. The main purpose of using these features is the fact that they enable reducing the complexity of kernel methods to linear in the sample size, at the expense of a mild gap in accuracy. Here, we extend this idea for solving the problem of ICA. III. R ANDOMIZED I NDEPENDENT C OMPONENT A NALYSIS As demonstrated in [1], the kernel canonical correlation and the kernel generalized variance measure the statistical dependence between a set of sampled random variables. In addition, optimization over these functions for decomposing mixtures of these variables into independent ones is more accurate and robust. However, evaluating these functions requires O(N 3 ) operations. Fortunately, for certain types of kernels, these contrast functions can be approximated using random features in linear time. where z Ĉ11 = N 1 X z(xk1 )z(xk1 )T N and z Ĉ22 = k=1 N 1 X z(xk2 )z(xk2 )T . N k=1 Thus the empirical Z-correlation is the largest generalized eigenvalue of the system    z 0 Ĉ12 w1 = z w Ĉ21 0  z  2  Ĉ11 + γI 0 w1 λ . (1) z w2 + γI 0 Ĉ22 As demonstrated in Figure 3, as the number of random features grows, the Z-correlation converges to the F-correlation. Notice that the size of the generalized eigenvalue system in the random feature space is ns m × ns m which is much smaller than the kernel case where it is of size ns N × ns N . The kernel canonical correlation provide less accurate results than the kernel generalized variance since it takes into account only the largest generalized eigenvalue. This can also be justified by the Gaussian case where the kernel generalized variance is shown to be a second order approximation of the mutual information around the point of independence as σ and comparing the accuracy of our algorithm in estimating the unmixing matrix from the mixed data, we used the Amari distance given by ! Pns ns 1 X j=1 |aij | d(V, W ) = −1 + 2ns i=1 maxj |aij  ns  Pns 1 X i=1 |aij | + −1 , 2ns j=1 maxi |aij | where aij = (V W −1 )ij . This metric, introduced in [5], is invariant to scaling and permutation of the rows or columns of the matrices. Fig. 3. Approximation of the Kernel Canonical Correlation using Ran- domized Canonical Correlation with different numbers of features. goes to zero. Thus, we propose to approximate the kernel generalized variance by P δz (x1 , ..., xns ) = − 1X log λk 2 k=1 where λk are the generalized eigenvalues of 1. We define this function as the random generalized variance (RGV). As demonstrated in Figure 4, the more features we take for calculating the empirical covariances in the random space, the better RGV approximates KGV. Fig. 5. Probability density functions used in our tests. These pdf-s are identical to those used in [1]. Approximation of the Kernel Generalized Variance using Randomized Generalized Variance with different numbers of features. Fig. 4. IV. E XPERIMENTAL R ESULTS To evaluate the performance of optimization over our novel contrast functions for solving ICA problems, we independently draw samples from two or more probability functions given in Figure 5. Then, we applied some random transformation with condition number between one to two. For measuring First, we evaluated the performance of our randomized ICA approach (RICA) on two mixtures of independent sources drawn in independently identically distributed fashion from the pdfs in 5. We repeated the experiments for 250 and 1000 samples, and the Amari distance for various ICA algorithms are summerized in Tables I and II, respectively. As expected, the our results are comparable to those of KICA [1], while our algorithm is strictly linear in the sample size. We compared the robustness of our contrast function to outlier samples. The evaluation was done by choosing at random a certain number of points in the dataset and adding 5 or −5 at random with probability 0.5. The Amari distance between the estimated matrix and the true one for several algorithms is demonstrated in Figure 6. We repeated the experiments 1000 times and averaged the results. The most robust algorithms are KICA [1], while RICA algorithms are still more robust then the others. Since our framework imitates the kernel ICA methods in the random feature space, we evaluated the accuracy and the runtime of the methods in unmixing real audio signal. The results are given in Table III. With comparable accuracy, our methods run more than ten times faster. TABLE II T HE A MARI ERRORS ( MULTIPLIED BY 100) AND THE RUNTIME Fig. 6. Robustness to outliers. TABLE I T HE A MARI ERRORS ( MULTIPLIED BY 100) OF STATE - OF - THE - ART ICA ALGORITHMS EVALUATED ON A COUPLE OF MIXTURES OF INDEPENDENT SOURCES DRAWN FROM THE DENOTED PDFS . I N THIS EXPERIMENT WE USED 250 SAMPLES AND REPEATED THE EXPERIMENT 1000 TIMES . T HE KGV, KCC, RCC, AND RGV ALGORITHMS WERE INITIALIZED WITH THE SIMILAR MATRICES . E ACH ROW REPRESENT THE PDFS FROM WHICH THE SAMPLES WERE DRAWN . T HE MEAN ROW REPRESENT THE AVERAGE PERFORMANCE OF EACH METHOD AND RAND ROW DENOTES THE PERFORMANCE OF THE ALGORITHMS FOR A MIXTURE OF A PAIRS OF SOURCES RANDOMLY SELECTED FROM THE PDFS . W E REPEATED THIS EXPERIMENT 1000 TIMES . pdfs a b c d e f g h i j k l m n o p q r mean rand F-ica 8.1 12.5 4.9 12.9 10.6 7.4 3.6 12.4 20.9 16.3 13.2 21.9 8.4 12.9 9.8 9.1 33.4 13.0 12.8 10.5 Jade 7.2 10.1 3.6 11.2 8.6 5.2 2.8 8.6 16.5 14.2 9.7 18.0 5.9 9.7 6.8 6.4 31.4 9.3 10.3 9.0 KCC 9.6 12.2 4.9 15.4 3.8 3.9 3.0 17.6 33.7 3.4 9.3 18.1 4.3 8.4 15.5 4.9 13.0 13.5 10.8 8.0 RCC 8.8 10.7 4.5 13.6 3.7 4.2 2.9 14.4 28.9 3.2 8.1 14.4 11.1 11.4 14.5 6.0 16.5 11.7 10.5 8.7 KGV 7.3 9.6 3.3 12.8 2.9 3.2 2.7 14.4 31.1 2.9 7.0 15.5 3.2 4.7 10.9 3.4 8.5 9.7 8.5 5.9 RGV 6.9 8.7 3.6 12.0 3.1 3.5 2.7 12.2 29.0 2.9 6.5 13.5 7.2 7.5 10.8 4.7 11.9 9.2 8.7 6.8 V. C ONCLUSIONS We proposed two novel pseudo-contrast functions for solving the ICA problem. The functions are evaluated using randomized Fourier features and thus can be computed in linear time. As the number of feature grows, the proposed functions converge to the renowned kernel generalized variance the kernel canonical correlation which require a computational effort that is cubic in the sample size. The accuracy of the proposed ICA methods is comparable to the state-of-the-art but runs over ten times faster. The proposed functions could pdfs a b c d e f g h i j k l m n o p q r mean rand F-ica 4.2 5.9 2.2 6.8 5.2 3.7 1.7 5.4 9.0 5.8 6.3 9.4 3.7 5.3 4.2 4.0 16.4 5.7 5.8 5.8 Jade 3.6 4.7 1.6 5.3 4.0 2.6 1.3 3.9 6.6 4.2 4.4 6.7 2.6 3.7 3.1 2.7 12.4 4.0 4.3 4.6 KCCA 5.3 5.4 2.0 8.4 1.6 1.7 1.4 6.5 12.9 1.5 4.3 7.0 1.7 2.8 5.2 2.0 3.8 5.3 4.4 3.6 RCC 4.9 5.3 1.7 7.4 1.6 1.9 1.2 6.0 12.1 1.3 3.6 6.6 3.1 3.2 4.8 2.4 4.5 4.5 4.2 3.7 KGV 3.0 3.0 1.4 5.3 1.2 1.3 1.2 4.4 10.6 1.3 2.6 4.9 1.3 1.8 3.3 1.4 2.1 3.2 3.0 2.5 RGV 3.5 3.9 1.4 5.9 1.3 1.4 1.1 4.4 10.0 1.2 2.7 4.9 2.0 2.3 3.4 1.7 3.0 3.3 3.2 2.8 TABLE III T HE A MARI ERRORS ( MULTIPLIED BY 100) AND THE RUNTIME ANALYSIS IN SEPARATING A PAIR OF INDEPENDENT AUDIO SIGNALS FROM TWO RECORDED MIXTURES . Method KCC RCC KGV RGV # repl 100 100 100 100 Amari Distance 3.2 3.3 1.2 1.3 Runtime (seconds) 337.7 28.4 258.0 23.6 also be evaluated using Nystrom extension for kernel matrices. R EFERENCES [1] F. R. Bach and M. I. Jordan, “Kernel independent component analysis,” J. Mach. Learn. Res., vol. 3, pp. 1–48, Mar. 2003. [Online]. Available: http://dx.doi.org/10.1162/153244303768966085 [2] A. Rahimi and B. Recht, “Weighted sums of random kitchen sinks: Replacing minimization with randomization in learning,” in Advances in Neural Information Processing Systems 21, D. Koller, D. Schuurmans, Y. Bengio, and L. Bottou, Eds. Curran Associates, Inc., 2009, pp. 1313–1320. [Online]. Available: http://papers.nips.cc/paper/ 3495-weighted-sums-of-random-kitchen-sinks-replacing-minimization-with-randomizat pdf [3] D. Lopez-Paz, S. Sra, A. Smola, Z. Ghahramani, and B. Schölkopf, “Randomized nonlinear component analysis,” arXiv preprint arXiv:1402.0119, 2014. [4] H. Hotelling, “Relations between two sets of variates,” Biometrika, vol. 28, no. 3/4, pp. 321–377, 1936. [5] S.-i. Amari, A. Cichocki, H. H. Yang et al., “A new learning algorithm for blind signal separation,” Advances in neural information processing systems, pp. 757–763, 1996. 

arXiv:1508.05813v3 [] 7 Apr 2016 DUAL OF BASS NUMBERS AND DUALIZING MODULES MOHAMMAD RAHMANI AND ABDOLJAVAD TAHERIZADEH Abstract. Let R be a Noetherian ring and let C be a semidualizing R-module. In this paper, we impose various conditions on C to be dualizing. For example, as a generalization of Xu [22, Theorem 3.2], we show that C is dualizing if and only if for an R-module M , the necessary and sufficient condition for M to be C-injective is that πi (p, M ) = 0 for all p ∈ Spec (R) and all i 6= ht (p), where πi is the invariant dual to the Bass numbers defined by E.Enochs and J.Xu [8]. 1. introduction Throughout this paper, R is a commutative Noetherian ring with non-zero identity. A finitely generated R-module C is semidualizing if the natural homothety map R −→ Hom R (C, C) is an isomorphism and Ext iR (C, C) = 0 for all i > 0. Semidualizing modules have been studied by Foxby [9], Vasconcelos [20] and Golod [10] who used the name suitable for these modules. Dualizing complexes, introduced by A.Grothendieck, is a powerful tool for investigating cohomology theories in algebraic geometry. A bounded complex of Rmodules D with finitely generated homologies is said to be a dualizing complex for R, if the natural homothety morphism R → RHom R (D, D) is quasiisomorphism, and id R (D) < ∞. These notion has been extended to semidualizing complexes by L.W. Christensen [5]. A bounded complex of R-modules C with finitely generated homologies is semidualizing for R if the natural homothety morphism R → RHom R (C, C) is quasiisomorphism. He used these notion to define a new homological dimension for complexes, namely GC -dimension, which is a generalization of Yassemi’s G-dimension [23]. The following, is the translation of a part of [5, Proposition 8.4] to the language of modules: Theorem 1. Let (R, m, k) be a Noetherian local ring and let C be a semidualizing Rmodule. The following are equivalent: (i) C is dualizing. (ii) GC -dim R (M ) < ∞ for all finite R-modules M . (iii) GC -dim R (k) < ∞. In particular, the above theorem recovers [4, 1.4.9]. Note that k is a Cohen-macaulay Rmodule of type 1. R.Takahashi, in [17, Theorem 2.3], replaced the condition G-dim R (k) < ∞ in [4, 1.4.9] by weaker conditions and obtained a nice characterization for Gorenstein rings. 2000 Mathematics Subject Classification. 13C05, 13D05, 13D07, 13H10. Key words and phrases. Semidualizing modules, dualizing modules, GC -dimension, Bass numbers, dual of Bass numbers, minimal flat resolution, local cohomology. 1 2 M. RAHMANI AND A.- J. TAHERIZADEH Indeed, he showed that R is Gorenstein, provided that either R admits an ideal I of finite Gdimension such that R/I is Gorenstein, or there exists a Cohen-Macaulay R-module of type 1 and of finite G-dimension. The following is the main result of section 3, which generalizes Theorem 1 as well as [17, Theorem 2.3]. See Theorem 3.4 below. Theorem 2. Let (R, m) be a Noetherian local ring and let C be a semidualizing R-module. The following are equivalent: (i) C is dualizing. (ii) There exists an ideal a with GC -dim R (aC) < ∞ such that C/aC is dualizing for R/a. (iii) There exists a Cohen-Macaulay R-module M with rR (M ) = 1 and GC -dim R (M ) < ∞. (iv) rR (C) = 1 and there exists a Cohen-Macaulay R-module M with GC -dim R (M ) < ∞. E.Enochs et al. [1], solved a long standing conjecture about the existence of flat covers. Indeed, they showed that if R is any ring, then all R-modules have flat covers. E.Enochs [6], determined the structure of flat cotorsion modules. Also, E.Enochs and J.Xu [8, Definition 1.2], defined a new invariant πi , dual to the Bass numbers, for modules related to flat resolutions. J.Xu [22], studied the minimal injective resolution of flat R-modules and minimal flat resolution of injective R-modules. He characterized Gorenstein rings in terms of vanishing of Buss numbers of flat modules, and vanishing of dual of Bass numbers of injective modules. More precisely, the following theorem is [22, Theorems 2.1 and 3.2]. Theorem 3. Let R be a Noetherian ring. The following are equivalent: (i) R is Gorenstein. (ii) An R-module F is flat if and only if µi (p, F ) = 0 for all p ∈ Spec (R) whenever i 6= ht (p). (iii) An R-module E is injective if and only if πi (p, E) = 0 for all p ∈ Spec (R) whenever i 6= ht (p). In section 4, we give a generalization of Theorem 3. Indeed, in Theorem 4.3, we prove the following result. Theorem 4. Let R be a Noetherian ring and let C be a semidualizing R-module. The following are equivalent: (i) C is pointwise dualizing. (ii) An R-module M is C-injective if and only if πi (p, M ) = 0 for all p ∈ Spec (R) whenever i 6= ht (p). (iii) An R-module M is injective if and only if πi (p, Hom R (C, M )) = 0 for all p ∈ Spec (R) whenever i 6= ht (p). Theorem 4 has several applications. Let (R, m) be a d-dimensional Cohen-Macaulay local ring possessing a canonical module. In this section, we give the structure of the minimal flat resolution of Hdm (R), the top local cohomology of R. More precisely, the following theorem is Corollary 4.7. DUAL OF BASS NUMBERS AND DUALIZING MODULES 3 Theorem 5. Let (R, m) be a d-dimensional Cohen-Macaulay local ring possessing a canonical module. The minimal flat resolution of Hdm (R) is of the form cm −→ · · · −→ Q Tp −→ Q Tp −→ Hdm (R) −→ 0, 0 −→ R ht (p)=1 ht (p)=0 in which Tp is the completion of a free Rp -module with respect to pRp -adic topology. In this section, by using the above resolution, we obtain the following isomorphism for a d-dimensional Cohen-Macaulay local ring (See Corollary 4.8). ( Hdm (R) i = d, d d ∼ Tor R i (Hm (R), Hm (R)) = 0 i= 6 d. 2. preliminaries In this section, we recall some definitions and facts which are needed throughout this paper. By an injective cogenerator, we always mean an injective R-module E for which Hom R (M, E) 6= 0 whenever M is a nonzero R-module. For an R-module M , the injective hull of M , is always denoted by E(M ). Definition 2.1. Let X be a class of R-modules and M an R-module. An X -resolution of M is a complex of R-modules in X of the form ∂X ∂X 1 n X0 −→ 0 Xn−1 −→ . . . −→ X1 −→ X = . . . −→ Xn −→ ∼ such that H0 (X) = M and Hn (X) = 0 for all n ≥ 1. Also the X -projective dimension of M is the quantity X -pd R (M ) := inf{sup{n ≥ 0|Xn 6= 0} | X is an X -resolution of M } . So that in particular X -pd R (0) = −∞. The modules of X -projective dimension zero are precisely the non-zero modules in X . The terms of X -coresolution and X -id are defined dually. Definition 2.2. A finitely generated R-module C is semidualizing if it satisfies the following conditions: (i) The natural homothety map R −→ Hom R (C, C) is an isomorphism. (ii) Ext iR (C, C) = 0 for all i > 0. For example a finitely generated projective R-module of rank 1 is semidualizing. If R is Cohen-Macaulay, then an R-module D is dualizing if it is semidualizing and that id R (D) < ∞ . For example the canonical module of a Cohen-Macaulay local ring, if exists, is dualizing. Definition 2.3. Following [12], let C be a semidualizing R-module. We set FC (R) = the subcategory of R–modules C ⊗R F where F is a flat R–module. IC (R) = the subcategory of R–modules Hom R (C, I) where I is an injective R– module. 4 M. RAHMANI AND A.- J. TAHERIZADEH The R-modules in FC (R) and IC (R) are called C-flat and C-injective, respectively. If C = R, then it recovers the classes of flat and injective modules, respectively. We use the notations C-fd and C-id instead of FC -pd and IC -id , respectively. Proposition 2.4. Let C be a semidualizing R-module. Then we have the following: (i) Supp (C) = Spec (R), dim (C) = dim (R) and Ass (C) = Ass (R). (ii) If R → S is a flat ring homomorphism, then C ⊗R S is a semidualizing S-module. (iii) If x ∈ R is R–regular, then C/xC is a semidualizing R/xR-module. (iv) If, in addition, R is local, then depth R (C) = depth (R). Proof. The parts (i), (ii) and (iii) follow from the definition of semidualizing modules. For (iv), note that an element of R is R-regular if and only if it is C-regular since Ass (C) = Ass (R). Now an easy induction yields the equality.  Definition 2.5. Let C be a semidualizing R-module. A finitely generated R-module M is said to be totally C-reflexive if the following conditions are satisfied: (i) The natural evaluation map M −→ Hom R (Hom R (M, C), C) is an isomorphism. (ii) Ext iR (M, C) = 0 = Ext iR (Hom R (M, C), C) for all i > 0. For an R-module M , if there exists an exact sequence 0 → Gn → · · · → G1 → G0 → M → 0, of R-modules such that each Gi is totally C-reflexive, then we say that M has GC -dimension at most n, and write GC -dim R (M ) ≤ n. If there is no shorter such sequence, we set GC dim R (M ) = n. Also, if such an integer n does not exist, then we say that M has infinite GC -dimension, and write GC -dim R (M ) = ∞. The next proposition collects basic properties of GC -dimension. For the proof, see [10]. Proposition 2.6. Let (R, m) be local, M a finitely generated R-module and let C be a semidualizing R-module. The following statements hold: (i) If GC -dim R (M ) < ∞, and x ∈ m is M -regular, then GC -dim R (M ) = GC -dim R (M/xM ) − 1. If, also, x is R-regular, then GC -dim R (M ) = GC/xC -dim R/xR (M/xM ). (ii) If GC -dim R (M ) < ∞ and x is an R-regular element in Ann R (M ), then GC -dim R (M ) = GC/xC -dim R/xR (M ) + 1. (iii) Let 0 → K → L → N → 0 be a short exact sequence of R-modules. If two of L, K, N are of finite GC -dimension, then so is the third. (iv) If GC -dim R (M ) < ∞, then GC -dim R (M ) = sup{i ≥ 0 | Ext iR (M, C) 6= 0} = depth (R) − depth R (M ). Definition 2.7. Let C be a semidualizing R-module. The Auslander class with respect to C is the class AC (R) of R-modules M such that: i (i) Tor R i (C, M ) = 0 = Ext R (C, C ⊗R M ) for all i ≥ 1, and DUAL OF BASS NUMBERS AND DUALIZING MODULES 5 (ii) The natural map M → Hom R (C, C ⊗R M ) is an isomorphism. The Bass class with respect to C is the class BC (R) of R-modules M such that: (i) Ext iR (C, M ) = 0 = Tor R i (C, Hom R (C, M )) for all i ≥ 1, and (ii) The natural map C ⊗R Hom R (C, M )) → M is an isomorphism. The class AC (R) contains all R-modules of finite projective dimension and those of finite Cinjective dimension. Also the class BC (R) contains all R-modules of finite injective dimension and those of finite C-projective dimension (see [18, Corollary 2.9]). Also, if any two Rmodules in a short exact sequence are in AC (R) (resp. BC (R)), then so is the third (see [13]). Proposition 2.8. Let (R, m) be a local ring and let C be a semidualizing R-module. b is a dualizing R-module b . (i) C is a dualizing R-module if and only if C ⊗R R (ii) Let x ∈ m be R-regular. Then C is a dualizing R-module if and only if C/xC is a dualizing R/xR-module. Proof. Just use the definition of dualizing modules.  Theorem 2.9. Let C be a semidualizing R-module and let M be an R-module. (i) C-id R (M ) = id R (C ⊗R M ) and id R (M ) = C-id R (Hom R (C, M )). (ii) C-fd R (M ) = fd R (Hom R (C, M )) and fd R (M ) = C-fd R (C ⊗R M ). Proof. For (i), see [18, Theorem 2.11] and for (ii), see [19, Proposition 5.2].  Lemma 2.10. Let C be a semidualizing R-module, E be an injective cogenerator and M be an R-module. (i) One has C-id R (M ) = C-fd R (Hom R (M, E)). (ii) One has C-fd R (M ) = C-id R (Hom R (M, E)). Proof. (i). We have the following equalities C-id R (M ) = id R (C ⊗R M ) = fd R (Hom R (C ⊗R M, E)) = fd R (Hom R (C, Hom R (M, E)) = C-fd R (Hom R (M, E)), in which the first equality is from Theorem 2.9(i), and the last one is from Theorem 2.9(ii). (ii). Is similar to (i).  Remark 2.11. Let (R, m) be a local ring and let M be a finitely generated R-module. We use νR (M ) to denote the minimal number of generators of M . More precisely, νR (M ) = vdim R/m (M ⊗R R/m). It is easy to see that if x ∈ m, then νR (M ) = νR/xR (M/xM ). In particular, if x ∈ Ann R (M ), then νR (M ) = νR/xR (M ). Assume that depth R (M ) = n. The type of M , denoted by rR (M ), is defined to be vdim R/m (Ext nR (R/m, M )). If x ∈ m, then rR (M/xM ) = rR/xR (M/xM ) by [2, Exercise 1.2.26]. Also, if x ∈ m is M - and R-regular, then rR (M ) = rR/xR (M/xM ) by [2, Lemma 3.1.16]. Assuma that C is a semidualizing 6 M. RAHMANI AND A.- J. TAHERIZADEH R-module. Then rR (C) | rR (R). Indeed, by reduction modulo a maximal R-sequence, we can assume that depth R (C) = 0 = depth (R). Then we have rR (R) = vdim R/m Hom R (R/m, R) = vdim R/m Hom R (R/m, Hom R (C, C)) = vdim R/m Hom R (R/m ⊗R C, C) = vdim R/m Hom R (R/m ⊗R C ⊗R/m R/m, C) = vdim R/m Hom R/m (R/m ⊗R C, Hom R (R/m, C)) = νR (C)rR (C). In particular, if rR (R) = 1 (e.g. R is Gorenstein local), then νR (C) = 1 and then C ∼ = R. Definition 2.12. Let M be an R-module and let X be a class of R-modules . Following [7], a X -precover of M is a homomorphism ϕ : X → M , with X ∈ X , such that every homomorphism Y → M with Y ∈ X , factors through φ; i.e., the homomorphism Hom R (Y, ϕ) : Hom R (Y, X) → Hom R (Y, M ) is surjective for each module Y in X . A X -precover ϕ : X → M is a X -cover if every ψ ∈ Hom R (X, X) with ϕψ = ϕ is an automorphism. Definition 2.13. Following [6], an R-module M is called cotorsion if Ext 1R (F, M ) = 0 for any flat R-module F. Remark 2.14. In [1], E. Enochs et al. showed that if R is any ring, then every R-module has a flat cover. It is easy to see that flat cover must be surjective. By [6, Lemma 2.2], the kernel of a flat cover is always cotorsion. So that if F → M is flat cover and M is cotorsion, then so is F . Therefore for an R-module M , one can iteratively take flat covers to construct a flat resolution of M . Since flat cover is unique up to isomorphism, this resolution is unique up to isomorphism of complexes. Such a resolution is called the minimal flat resolution of M . Note that the minimal flat resolution of M is a direct summand of any other flat resolution of M . Assume that · · · → Fi → · · · → F1 → F0 → M → 0, is the minimal flat resolution of M . Then Fi is cotorsion for all i ≥ 1. If, in addition, M is cotorsion, then all the flat modules in the minimal flat resolution of M are cotorsion. E. Enochs [6], determined the structure of flat cotorsion modules. He showed that if F is flat ∼ QTp where Tp is the completion of a free Rp -module with respect and cotorsion, then F = p to pRp -adic topology. So that we can determine the structure of the minimal flat resolution of cotorsion modules. Definition 2.15. Let M be a cotorsion R-module and let · · · → Fi → · · · → F1 → F0 → M → 0, be the minimal flat resolution of M . Following [8], for a prime ideal p of R and an integer i ≥ 0, the invariant πi (p, M ) is defined to be the cardinality of the basis of a free Rp -module Q whose completion is Tp in the product Fi ∼ = Tp . By [8, theorem 2.2], for each i ≥ 0, p Rp πi (p, M ) = vdim Rp /pRp Tor i
Rp /pRp , Hom R (Rp , M ) . DUAL OF BASS NUMBERS AND DUALIZING MODULES 7 Remark 2.16. Let M be a finitely generated R-module. There are isomorphisms Hom R (M, E(R/p)) ∼ = Hom R (M, E(R/p) ⊗R Rp ) ∼ = Hom R (M, E(R/p)) ⊗R Rp ∼ = Hom Rp (Mp , ERp (Rp /pRp )), where the the first isomorphism holds because E(R/p) ∼ = ERp (Rp /pRp ), and the second isomorphism is tensor-evaluation [7, Theorem 3.2.14]. 3. Finiteness of GC -dimension Throughout this section, C is a semidualizing R-module. We begin with three lemmas that are needed for the main result of this section. It is well-known that a local ring over which there exists a non-zero finitely generated injective module, must be Artinian. Our first lemma generalizes this fact by replacing the injectivity condition with weaker assumption. Lemma 3.1. Let (R, m) be local and let M be a finitely generated R-module with depth (M ) = 0. If Ext 1R (R/m, M ) = 0, then R is Artinian. In particular, M is injective. Proof. We show that dim (R) = 0. Assume, on the contrary, that dim (R) > 0. Note that if N is an R-module of finite length, then by using a composition series for N in conjunction with the assumption, we have Ext 1R (N, M ) = 0. Now an easy induction on ℓR (N ) yields the equality ℓR (Hom R (N, M )) = ℓR (N )ℓR (Hom R (R/m, M )). Next, note that ℓR (R/mi ) < ∞ i i+1 for any i ≥ 1, and that the sequence {ℓR (R/mi )}∞ for i=1 is not bounded since m 6= m 2 any i ≥ 1. Hence {ℓR (Hom R (R/mi , M ))}∞ i=1 is not bounded. But 0 :M m ⊆ 0 :M m ⊆ · · · is a chain of submodules of M , and hence is eventually stationary. This is a contradiction. Therefore R is Artinian. Finally, the assumption Ext 1R (R/m, M ) = 0 implies that M is injective.  Lemma 3.2. Let (R, m) be a local ring and let M be a Cohen-Macaulay R-module with GC dim R (M ) < ∞. Then rR (C) | rR (M ). Proof. We use induction on n = depth (R). If n = 0, then by Proposition 2.6(iv), we have ∼ Hom R (Hom R (M, C), C), and GC -dim R (M ) = 0, and hence there is an isomorphism M = the equalities depth R (C) = 0 = depth R (M ). Hence we have rR (M ) = vdim R/m Hom R (R/m, M ) = vdim R/m Hom R (R/m, Hom R (Hom R (M, C), C)) = vdim R/m Hom R (R/m ⊗R Hom R (M, C), C) = vdim R/m Hom R (R/m ⊗R Hom R (M, C) ⊗R/m R/m, C) = vdim R/m Hom R/m (R/m ⊗R Hom R (M, C), Hom R (R/m, C)) = νR (Hom R (M, C))rR (C). Therefore rR (C) | rR (M ). Now, assume inductively that n > 0. We consider two cases: Case 1. If depth R (M ) = 0, then M is of finite length since it is Cohen-Macaulay. Hence we can take an R-regular element x such that xM = 0. Set (−) = (−) ⊗R R/xR. Then by 8 M. RAHMANI AND A.- J. TAHERIZADEH Proposition 2.6(ii), we have GC -dim R (M ) < ∞. Also, note that M is a Cohen-Macaulay R-module. Hence by induction hypothesis we have rR (C) | rR (M ). Thus rR (C) | rR (M ). Case 2. If depth R (M ) > 0, then we can take an element y ∈ m to be M - and R-regular. Set (−) = (−) ⊗R R/yR. Now M is a Cohen-Macaulay R-module, and that GC -dim R (M ) = GC -dim R (M ) < ∞, by Proposition 2.6(i). Therefore, by induction hypothesis, we have rR (C) | rR (M ), whence rR (C) | rR (M ). This complete the inductive step.  Lemma 3.3. Let (R, m) be local and that rR (C) = 1. If there exists a totally C-reflexive R-module of finite length, then C is dualizing. Proof. Assume that M is a finite length C-reflexive R-module. Then depth R (M ) = 0, and hence depth (R) = GC -dim (M ) = 0 by Proposition 2.6(iv). Therefore, we have depth R (C) = 0 by Proposition 2.4(iv). Now assume, on the contrary, that C is not dualizing. Hence, by Lemma 3.1, we have Ext 1R (R/m, C) 6= 0. Let 0 = M0 ⊂ M1 ⊂ · · · ⊂ Mr = M , be a composition series for M . Thus the factors are all isomorphic to R/m, and we have exact sequences 0 → Mi−1 → Mi → R/m → 0, for all 1 ≤ i ≤ r. Applying the functor Hom R (−, C), we get the exact sequence 0 → Hom R (R/m, C) → Hom R (Mi , C) → Hom R (Mi−1 , C), for each 1 ≤ i ≤ r−1. Now since depth R (C) = 0 and rR (C) = 1, we have Hom R (R/m, C) ∼ = R/m. Hence we have the inequality ℓR (Hom R (Mi , C)) ≤ ℓR (Hom R (Mi−1 , C)) + 1 for each 1 ≤ i ≤ r − 1. On the other hand, application of the functor Hom R (−, C) on the exact sequence 0 → Mr−1 → M → R/m → 0, yields an exact sequence 0 → Hom R (R/m, C) → Hom R (M, C) → Hom R (Mr−1 , C) → Ext 1R (R/m, C) → Ext 1R (M, C) = 0. Therefore ℓR (Hom R (M, C)) = ℓR (Hom R (Mr−1 , C)) + 1 − ℓR (Ext 1R (R/m, C)). But since ℓR (Ext 1R (R/m, C)) > 0, we have ℓR (Hom R (M, C)) < ℓR (Hom R (Mr−1 , C)) + 1 ≤ ℓR (Hom R (Mr−2 , C)) + 2 ≤ ··· ≤ ℓR (Hom R (M0 , C)) + r =r = ℓR (M ). Now since Hom R (M, C) is again a totally C-reflexive R-module of finite length, the same
argument shows that ℓR Hom R (Hom R (M, C), C) ≤ ℓR (Hom R (M, C)). But since M is ∼ Hom R (Hom R (M, C), C), which implies that ℓR (M ) < totally C-reflexive, we have M = ℓR (M ), a contradiction. Hence C is dualizing. The following theorem is a generalization of [17, Theorem 2.3]. Theorem 3.4. Let (R, m) be local. The following are equivalent:  DUAL OF BASS NUMBERS AND DUALIZING MODULES 9 (i) C is dualizing. (ii) There exists an ideal a with GC -dimR (aC) < ∞ such that C/aC is dualizing for R/a. (iii) There exists a Cohen-Macaulay R-module M with rR (M ) = 1 and GC -dimR (M ) < ∞. (iv) rR (C) = 1 and there exists a Cohen-Macaulay R-module M of finite GC -dimension. Proof. (i)=⇒(ii). Choose a = 0. (ii)=⇒(iii). We show that C/aC has the desired properties. First, the exact sequence 0 → aC → C → C/aC → 0, in conjunction with Proposition 2.6(iii), show that GC -dim R (C/aC) < ∞. On the other hand, C/aC is a Cohen-Macaulay R/aR-module and hence is a Cohen-Macaulay R-module. Finally, by [2, Exercise 1.2.26], we have rR (C/aC) = rR/a (C/aC) = 1. (iii)=⇒(iv). By Lemma 3.2, we have rR (C) = 1. (iv)=⇒(i). Assume that M is a Cohen-Macaulay R-module with GC -dim R (M ) < ∞. We use induction on m = depth R (M ). If m = 0, then M is of finite length since it is Cohenp Macaulay. Since Ann R (M ) = m, we can choose a maximal R-sequence from elements of Ann R (M ), say x. In view of Proposition 2.8(ii) and Proposition 2.6(ii), we can replace C by C/xC and R by R/xR, and assume that M is totally C-reflexive. In this case, C is dualizing by Lemma 3.3. Now assume inductively that m > 0. Hence depth (R) > 0 by Proposition 2.6(iv), and we can take an element x ∈ m to be M - and R-regular. Set (−) = (−)⊗R R/xR. Now M is a Cohen-Macaulay R-module and rR (C) = rR (C) = 1. Also, by Proposition 2.6(i), we have GC -dim R (M ) =GC -dim R (M ) < ∞. Hence, by induction hypothesis, C is dualizing for R, whence C is dualizing for R by Proposition 2.8(ii).  It is well-known that the existence of a finitely generated (resp. Cohen-Macaulay) module of finite injective (resp. projective) dimension implies Cohen-Macaulyness of the ring. But, in the special case that C is dualizing, the proof is easy, as the following relations show dim (R) = dim R (C) ≤ id R (C) = depth (R), where the first equality is from Proposition 2.4(i), and the remaining parts are from [2, Theorem 3.1.17]. Therefore, in view of Theorem 3.4, we can state the following corollary. Corollary 3.5. Let (R, m) be local. If there exists a Cohen-Macaulay R-module of type 1 and of finite GC -dimension, then R is Cohen-Macaulay. 4. C-injective modules In this section, our aim is to extend two nice results of J.Xu [22]. It is well-known that a Noetherian ring R is Gorenstein if and only if µi (p, R) = δi,ht(p) (the Kronecker δ). As a generalization, J.Xu [22, Theorem 2.1], showed that R is Gorenstein if and only if for any R-module F , the necessary and sufficient condition for F to be flat is that µi (p, F ) = 0 for all p ∈ Spec (R) and all i 6= ht (p). Next, in [22, Theorem 3.2], he proved a dual for this theorem. Indeed, he proved that R is Gorenstein if and only if for any R-module E, the necessary and sufficient condition for E to be injective is that πi (p, E) = 0 for all p ∈ Spec (R) and 10 M. RAHMANI AND A.- J. TAHERIZADEH all i 6= ht (p). In the present section, first we generalize the mentioned results. Next, we use our new results to determine the minimal flat resolution of some top local cohomology of a Cohen-Macaulay local rings and their torsion products. Lemma 4.1. The followings are equivalent: (i) C is pointwise dualizing. (ii) C-fd R (E(R/m)) = ht (m) for any m ∈ Max (R). (iii) C-fd R (E(R/m)) < ∞ for any m ∈ Max (R). (iv) C-fd R (E(R/p)) = ht (p) for any p ∈ Spec (R). (v) C-fd R (E(R/p)) < ∞ for any p ∈ Spec (R). (vi) C-id R (Tm ) = ht (m) for any m ∈ Max (R). (vii) C-id R (Tm ) < ∞ for any m ∈ Max (R). (viii) C-id R (Tp ) = ht (p) for any p ∈ Spec (R). (ix) C-id R (Tp ) < ∞ for any p ∈ Spec (R). Proof. (i)=⇒(ii). Assume that m ∈ Max (R). There are equalities C-fd R (E(R/m)) = fd R (Hom R (C, E(R/m))) = fd Rm (Hom Rm (Cm , ERm (Rm /mRm )) = id Rm (Cm ) = dim (Rm ) = ht (m), in which the first equality is from Theorem 2.9(ii), and the second one is from Remark 2.16. (ii)=⇒(iii). Is clear. (iii)=⇒(i). We can assume that (R, m) is local. Now one can use Theorem 2.9(ii), to see that id R (C) = fd R (Hom R (C, E(R/m))) = C-fd R (E(R/m)) < ∞, whence C is dualizing. (i)=⇒(iv). Let p be a prime ideal of R. Note that E(R/p)q 6= 0 if and only if q ⊆ p. Now as in (i)=⇒(ii), we have C-fd R (E(R/p)) = dim (Rp ) = ht (p). (iv)=⇒(v). Is clear. (v)=⇒(i). Again, we can assume that R is local. Now the proof is similar to that of (iii)=⇒(i).
(ii)⇐⇒(vi) and (iii)⇐⇒(vii). Note that Tm = Hom R E(R/m), E(R/m)(X) for some set X. Now we have the equalities C-id R (Tm ) = id R (C ⊗R Tm ) = id R C ⊗R Hom R E(R/m), E(R/m)(X)

= id R Hom R Hom R (C, E(R/m)), E(R/m)(X) = fd R (Hom R (C, E(R/m))) = C-fd R (E(R/m)),

DUAL OF BASS NUMBERS AND DUALIZING MODULES 11 in which the first equality is from Theorem 2.9(i), the fourth equality is from Remark 2.16 and the fact that E(R/m)(X) is an injective cogenerator in the category of Rm -modules, and the last one is from Theorem 2.9(ii). (iv)⇐⇒(viii) and (v)⇐⇒(ix). Are similar to (ii)⇐⇒(vi).  The following theorem is a generalization of [21, theorem 2.1]. Theorem 4.2. The following are equivalent: (i) C is pointwise dualizing. (ii) An R-module M is C-flat if and only if µi (p, M ) = 0 for all p ∈ Spec (R) whenever i 6= ht (p). (iii) An R-module M is flat if and only if µi (p, C ⊗R M ) = 0 for all p ∈ Spec (R) whenever i 6= ht (p). Proof. (i)=⇒(ii). First assume that M is C-flat. Set M = C ⊗R F , where F is a flat R-module. Since C is pointwise dualizing, we have µi (p, C) = 0 for all p ∈ Spec (R) with i 6= ht (p). Assume that 0 → C → E 0 (C) → E 1 (C) → ... → E i (C) → ... is the minimal injective resolution of C. By applying the exact functor − ⊗R F to this resolution, we find an exact complex 0 → M = C ⊗R F → E 0 (C) ⊗R F → E 1 (C) ⊗R F → ... → E i (C) ⊗R F → ..., (∗) which is an injective resolution for M . By [7, Theorem 3.3.12] the injective R-module E(R/p) ⊗R F is a direct sum of copies of E(R/p) for each p ∈ Spec (R). Now, since the minimal injective resolution of M is a direct summand of the complex (∗), we get the result. Conversely, suppose that M is an R-module such that µi (p, M ) = 0 for all p ∈ Spec (R) whenever i 6= ht (p). In order to show that M is C-flat, it is enough to prove that Mm is Cm flat Rm -module for all m ∈ Max (R). For if Mm is Cm -flat Rm -module for all m ∈ Max (R), then Hom R (C, M )m ∼ = Hom Rm (Cm , Mm ) is flat as an Rm -module for all m ∈ Max (R) by Theorem 2.9(ii). Hence Hom R (C, M ) is a flat R-module and thus M is C-flat by Theorem 2.9(ii). Hence, replacing R by Rm , we can assume that (R, m) is local. Clearly we may assume that M 6= 0. In this case we have id R (M ) < ∞ since by assumption µi (p, M ) = 0 for all p ∈ Spec (R) and all i > dim (R) . Hence the assumption in conjunction with Lemma 4.1, imply that M has a bounded injective resolution all of whose terms have finite C-flat dimensions. More precisely, by Lemma 4.1, if E i is the i-th term in the minimal injective resolution of M , then C-fdR (E i ) = i for all 0 ≤ i ≤ id (M ). Breaking up this resolution to short exact sequences and using [19, Corollary 5.7], we can conclude that C-fdR (M ) = 0. Hence M is C-flat, as wanted. (ii)=⇒(iii). Assume that M is a flat R-module. Then C ⊗R M ∈ FC and µi (p, C ⊗R M ) = 0 for all p ∈ Spec (R) whenever i 6= ht (p) by assumption. Conversely, suppose that µi (p, C ⊗R M ) = 0 for all p ∈ Spec (R) whenever i 6= ht (p). Then, by assumption, C ⊗R M is C-flat. Set C ⊗R M = C ⊗R F , where F is flat. Therefore C ⊗R M ∈ BC (R), whence 12 M. RAHMANI AND A.- J. TAHERIZADEH M ∈ AC (R) by [18, Theorem 2.8(b)]. Thus we have the isomorphisms M ∼ = Hom R (C, C ⊗R M ) ∼ = Hom R (C, C ⊗R F ) ∼ = F, where the first and the last isomorphism hold since both M and F are in AC (R). (iii)=⇒(i). Note that R is a flat R-module. Hence by assumption, if m ∈ Max (R), then i µ (m, C ⊗R R) = 0 for all i > ht (m). Thus id Rm (Cm ) < ∞, as wanted.  Theorem 4.3. The following are equivalent: (i) C is pointwise dualizing. (ii) An R-module M is C-injective if and only if πi (p, M ) = 0 for all p ∈ Spec (R) whenever i 6= ht (p). (iii) An R-module M is injective if and only if πi (p, Hom R (C, M )) = 0 for all p ∈ Spec (R) whenever i 6= ht (p). Proof. (i)=⇒(ii) . Assume that M is a nonzero C-injective R-module. Set M = Hom R (C, E) with E is injective. First, we show that M is cotorsion. Assume that F is a flat R-module. Then, by [7, Theorem 3.2.1], we have Ext 1 (F, Hom R (C, E)) ∼ = Hom R (Tor R (F, C), E) = 0, R 1 and hence M is cotorsion. Fix a prime ideal p of R and set k(p) = Rp /pRp . Note ∼ ⊕ E(R/q), where that Hom R (Rp , E) is an injective R-module and that Hom R (Rp , E) = q∈X X ⊆ Ass R (E) and each element of X is a subset of p. There are isomorphisms

R R Tor i p k(p), Hom R (Rp , Hom R (C, E)) ∼ = Tor i p k(p), Hom R (Cp , E)
R ∼ = Tor i p k(p), Hom Rp (Cp , Hom R (Rp , E)
R ∼ = Tor i p k(p), Hom Rp (Cp , ⊕ E(R/q) q∈X
∼ = Hom Rp Ext iRp (k(p), Cp ), ⊕ E(R/q) , q∈X where the last isomorphism is from [7, Theorem 3.2.13]. Now since Cp is dualizing for Rp , we have Ext iRp (k(p), Cp ) = 0 for all i 6= ht (p). Therefore πi (p, M ) = 0 for all i 6= ht (p). Conversely, assume that M is a non-zero R-module with πi (p, M ) = 0 for all i 6= ht (p). By assumption, the minimal flat resolution of M is of the form Q · · · −→ Fi −→ · · · −→ F1 −→ F0 −→ M −→ 0, in which Fi = Tp for all i ≥ 1. Also, in view of [22, Lemma 3.1], we have ht (p)=i Q Tp . Hence the minimal flat resolution of M is of the form F0 = ht (p)=0 Q Q Q · · · −→ Tp −→ · · · −→ Tp −→ Tp −→ M −→ 0. (∗) ht (p)=i ht (p)=1 ht (p)=0 Let E be an injective cogenerator. According to Lemma 2.10(i), it is enough to show that Hom R (M, E) is C-flat. In fact, by Theorem 4.2, we need only to show that i µ (p, Hom R (M, E)) = 0 for all i 6= ht (p) and all i ≥ 0. Applying the exact functor Hom R (−, E) on (∗), we get an injective resolution   Q 0 −→ Hom R (M, E) −→ Hom R Tp , E −→ ht (p)=0 DUAL OF BASS NUMBERS AND DUALIZING MODULES Hom R  Q   Tp , E −→ · · · −→ Hom R ht (p)=1 for Hom R (M, E). Note that Hom R  Q   Q Tp , E ht (p)=i  Q ht (p)=i 13  Tp , E −→ · · · , is an injective R-module for all i ≥ 0. ∼ ⊕E(R/q). We show that ht (q) = i. Since C is pointwise Tp , E = ht (p)=i dualizing, by Lemma 4.1, we have C-fd R (E(R/q)) = ht (q). On the other hand, we have Set Hom R the equalities C-fd R (E(R/q)) = C-fd R  (⊕E(R/q))   Q = C-fd R Hom R Tp , E ht(p)=i  Q = C-id R Tp ht (p)=i = i, in which the third equality is from Lemma 2.10(i), and the last one is from Lemma 4.1. Hence µi (p, Hom R (M, E)) = 0 for all i ≥ 0 with i 6= ht (p), as wanted. (ii)=⇒(iii). Assume that M is an injective R-module. Then Hom R (C, M ) ∈ IC and i µ (p, Hom R (C, M )) = 0 for all p ∈ Spec (R) whenever i 6= ht (p) by assumption. Conversely, suppose that µi (p, Hom R (C, M )) = 0 for all p ∈ Spec (R) whenever i 6= ht (p). Then, by assumption, Hom R (C, M ) is C-injective. Set Hom R (C, M ) = Hom R (C, I), where I is injective. Therefore Hom R (C, M ) ∈ AC (R), whence M ∈ BC (R) by [18, Theorem 2.8(a)]. Thus we have the isomorphisms M ∼ = C ⊗R Hom R (C, M ) ∼ = C ⊗R Hom R (C, I) ∼ = I, where the first and the last isomorphism hold since both M and I are in BC (R). (iii)=⇒(i). Assume that m is a maximal ideal of R. Set k(m) = Rm /mRm . Since E(R/m)
is injective, by assumption, we have πi m, Hom R (C, E(R/m)) = 0 fo all i 6= ht (m). On the other hand, there are isomorphisms
Hom Rm Ext iRm (k(m), Cm ), E(k(m))
m ∼ k(m), Hom Rm (Cm , E(k(m)) = Tor R i
∼ Tor Rm k(m), Hom R (Cm ⊗R Rm , E(k(m)) = m m i
m ∼ k(m), Hom R (Rm , Hom Rm (Cm , E(k(m)) = Tor R i
∼ = Tor Rm k(m), Hom R (Rm , Hom R (C, E(R/m)) , i where the first isomorphism is from [7, Theorem 3.2.13], and the last one is from Remark
2.16. From this isomorphisms, it follows that Hom Rm Ext iRm (k(m), Cm ), E(k(m)) = 0 for all i 6= ht (m), from which we conclude that Ext iRm (k(m), Cm ) = 0 for all i 6= ht (m), since E(k(m)) is an injective cogenerator in the category of Rm -modules. Thus Cm is dualizing for Rm , as required.  Corollary 4.4. Let C be pointwise dualizing. Then flat cover of any C-injective R-module is C-injective. 14 M. RAHMANI AND A.- J. TAHERIZADEH Proof. By Lemma 4.1, C-id R (Tp ) = 0 for any prime ideal p with ht (p) = 0. Hence Tp is C-injective for any prime ideal p with ht (p) = 0. Assume that M is a C-injective R-module. Q By Theorem 4.3, we have F (M ) = Tp . Now the result follows since the class IC ht (p)=0 closed under arbitrary direct product.  Corollary 4.5. The R-module C is pointwise dualizing if and only if for any prime ideal p of R,
πi p, Hom R (C, E(R/p)) = ( 1 i = ht (p), 0 i 6= ht (p). Proof. Assume that p ∈ Spec (R). Set k(p) = Rp /pRp . We have the following equalities

R πi p, Hom R (C, E(R/p)) = vdim k(p) Tor i p k(p), Hom R (Rp , Hom R (C, E(R/p))
R = vdim k(p) Tor i p k(p), Hom R (Cp , Hom Rp (Rp , E(R/p))
R = vdim k(p) Tor i p k(p), Hom R (Cp , E(R/p))
= vdim k(p) Hom Rp Ext iRp (k(p), Cp ), E(R/p) , where the second equality is from Remark 2.16, and the last equality is from [7, Theorem 3.2.13]. Now, C is pointwise dualizing if and only if Cp is the dualizing module of Rp for all p ∈ Spec (R), and this is the case if and only if ( k(p) Ext iRp (k(p), Cp ) ∼ = 0 i = ht (p), i 6= ht (p). for all p ∈ Spec (R). Thus we are done by the above equalities and the fact that ∼  Hom Rp (k(p), E(R/p)) = k(p). In the following corollaries, we are concerned with the local cohomology. For an R- module M , the i-th local cohomology module of M with respect to an ideal a of R, denoted by Hia (M ), is defined to be Hia (M ) = limExt iR (R/an , M ). −→ n≥1 For the basic properties of local cohomology modules, please see the textbook [3]. Corollary 4.6. Let (R, m) be a Cohen-Macaulay local ring with dim (R) = d possesing

a canonical module ωR . Then πi m, Hdm (R) = δi,d , and πi q, Hdm (R) = 0 for any nonmaximal prime ideal q whenever i 6= ht (q). Proof. By [3, Theorem 11.2.8], we have Hdm (R) ∼ = Hom R (ωR , E(R/m)), and hence Hdm (R) is ωR -injective. Assume that q is a non-maximal prime ideal of R. Then by the Theorem 4.3, we

have πi q, Hdm (R) = 0 for all i 6= ht (q). Finally, by corollary 4.5, we have πi m, Hdm (R) = 0
 for all i 6= d and that πd m, Hdm (R) = 1, as wanted. If (R, m) is a Cohen-Macaulay local ring with dim (R) = d, then by [3, Corollary 6.2.9] the only non-vanishing local cohomology of R with respect to m is Hdm (R). Also, if R admits a DUAL OF BASS NUMBERS AND DUALIZING MODULES 15 canonical module, then by [7, Proposition 9.5.22], we have fd R (Hdm (R)) = d. The following corollary describes the structure of the minimal flat resolution of Hdm (R). Corollary 4.7. Let (R, m) be a d-dimensional Cohen-Macaulay local ring possessing a canonical module. The minimal flat resolution of Hdm (R) is of the form cm −→ · · · −→ Q Tp −→ Q Tp −→ Hdm (R) −→ 0. 0 −→ R ht (p)=1 ht (p)=0 In the following corollary, we give another proof of [16, Corollary 3.7]. Our approach is direct, and uses the well-known fact that the homology functor Tor can be computed by a flat resolution. Corollary 4.8. Let (R, m) be a d-dimensional Cohen-Macaulay local ring. Then ( Hdm (R) i = d, d R d ∼ Tor i (Hm (R), Hm (R)) = 0 i 6= d. b is a d-dimensional complete Cohen-Macaulay local ring, and hence Proof. Note that R admits a canonical module ωRb . The R-module Hdm (R) is Artinian by [3, Theorem 7.1.6], and b thus naturally has a R-module structure by [3, Remark 10.2.9]. Hence Tor R (Hd (R), Hd (R)) i m m is Artinian for all i ≥ 0 by [14, Corollary 3.2]. Thus there are isomorphisms d d Tor R i (Hm (R), Hm (R)) d d ∼ b = Tor R i (Hm (R), Hm (R)) ⊗R R b d ∼ b b d = Tor R i (Hm (R) ⊗R R, Hm (R) ⊗R R)
b d d ∼ b , b H b (R) = Tor R H b (R), i mR mR in which the second isomorphism is from [7, Theorem 2.1.11], and the last one is flat base change [3, Theorem 4.3.2]. Also, we have the isomorphisms b Hdm (R) ∼ = Hdm (R) ⊗R R d ∼ b = H b (R) mR
∼ b R) b , = Hom Rb ωRb , ERb (R/m in which the first isomorphism holds because Hdm (R) is Artinian, the second isomorphism is the flat base change, and the last one is local duality [3, Theorem 11.2.8]. Thus Hdm (R) is b a ωRb -injective R-module. Hence, by Corollary 4.7, the minimal flat resolution of Hdm (R), as b an R-module, is of the form [ b b −→ · · · −→ Q TQ −→ Q TQ −→ Hdm (R) −→ 0, 0 −→ R mR ht (Q)=1 ht (Q)=0 b bQ -adic topology, in which TQ is the completion of a free RQ -module with respect to QR b Observe that the above resolution is a flat resolution of Hdm (R) as an for Q ∈ Spec (R). R-module since the modules in the above resolution are all flat R-modules. Therefore, we b and assume that R is complete. So that, the minimal flat resolution of can replace R by R, Hdm (R) is of the form cm −→ · · · −→ 0 −→ R Q Tp −→ ht (p)=1 Q Tp −→ Hdm (R) −→ 0, ht (p)=0 16 M. RAHMANI AND A.- J. TAHERIZADEH in which Tp is the completion of a free Rp -module with respect to pRp -adic topology, for p  ∈ Spec(R). Next, note that for each prime ideal p with p 6= m, we have Q d Tp = 0. Indeed, we can write Hdm (R) = limMα , where Mα is a finitely Hm (R) ⊗R −→ generated submodule of Hdm (R). α∈I
Also Tp = Hom R E(R/p), E(R/p)(X) for some set X. Now since Mα is of finite length by [3, Theorem 7.1.3], we can take an element x ∈ m r q such that xMα = 0. But multiplication of x induces an automorphism on E(R/p) and hence Q  Q Tp is both an isomorphism and on Tp . Consequently, multiplication of x on Mα ⊗R Q  Q  Tp = 0 since Tp = 0, from which we conclude that Hdm (R) ⊗R zero. Hence Mα ⊗R d d tensor commutes with direct limit. Thus Tor R i (Hm (R), Hm (R)) = 0 for i 6= d. Finally, we have d d Tor R d (Hm (R), Hm (R)) ∼ cm ⊗R Hd (R) =R m ∼ cm ) (R = HdmR d m ∼ c c = Hom d (ωd Rm , E d (Rm /mRm )) Rm Rm ∼ cm = Hom Rm (ωRm , ERm (Rm /mRm )) ⊗Rm R ∼ Hom R (ωR , ER (Rm /mRm )) = m m m ∼ = Hom R (ωR , E(R/m)) ⊗R Rm ∼ = Hom R (ωR , E(R/m) ⊗R Rm ) ∼ = Hom R (ωR , E(R/m)) ∼ = Hd (R), m in which the second isomorphism is the flat base change [3, Theorem 4.3.2], the third isomorphism is local duality [3, Theorem 11.2.8], and the fifth one is from [3, Remark 10.2.9], since Hom Rm (ωRm , ERm (Rm /mRm )) is an Artinian Rm -module and hence has a natural structure cm -module. as an R  The following theorem is a slight generalization of [22, Theorem 3.3]. Theorem 4.9. The following are equivalent: (i) C is pointwise dualizing. (ii) If M is a cotorsion R-module such that C-id R (M ) = n < ∞, then M admits a minimal flat resolution such that πi (p, M ) = 0 for all p ∈ Spec (R) whenever ht (p) ∈ / {i, ..., i + n}. Proof. (i) =⇒ (ii). We use induction on n. If n = 0, then we are done by Theorem 4.3. Now assume inductively that n > 0 and the case n ie settled. Fix a prime ideal p of R. Assume that M is a cotorsion R-module with C-id R (M ) = n + 1. Hence M ∈ AC (R), and so the IC -preenvelope of M is injective by [18, Corollary 2.4(b)]. Thus there exists an exact sequence 0 → M → Hom R (C, I) → L → 0, (∗) in which I is injective, and L = Coker (M → Hom R (C, I)). Note that L is cotorsion since both M and Hom R (C, I) are cotorsion. Also, since both M and Hom R (C, I) are in AC (R), we have L ∈ AC (R), and therefore Tor R 1 (C, L) = 0. On the other hand DUAL OF BASS NUMBERS AND DUALIZING MODULES 17 C ⊗R Hom R (C, I) ∼ = I, by [7, Theorem 3.2.11]. Hence application of C ⊗R − on (∗) yields an exact sequence 0 → C ⊗R M → I → C ⊗R L → 0. By Theorem 2.9(i), we have id R (C ⊗R M ) = n + 1. Therefore id R (C ⊗R L) = n, whence C-id R (L) = n. Now induction hypothesis applied to Hom R (C, I) and L yields that
/ {i, ..., i + n}. πi p, Hom R (C, I) = 0 for all i 6= ht (p), and that πi (p, L) = 0 fo all ht (p) ∈ Note that Ext 1R (Rp , M ) = 0 since M is cotorsion. Hence the exact sequence (∗) yields an exact sequence 0 → Hom R (Rp , M ) → Hom R (Rp , Hom R (C, I)) → Hom R (Rp , L) → 0, and the later exact sequence, by applying k(p) ⊗Rp −, yields the long exact sequence

Rp Rp k(p), Hom R (Rp , L) → k(p), Hom R (Rp , Hom R (C, E)) → Tor i+1 · · · → Tor i+1

R R Tor i p k(p), Hom R (Rp , M ) → Tor i p k(p), Hom R (Rp , Hom R (C, E)) → · · · .
R From the above long exact sequence, it follows that Tor i p k(p), Hom R (Rp , M ) = 0 for all ht (p) ∈ / {i, ..., i + n + 1}, as wanted. This completes the inductive step. (ii) =⇒ (i). Let m be a maximal ideal of R. Now Hom R (C, E(R/m)) is C-injective and
hence by assumption πi m, Hom R (C, E(R/m)) = 0 for all i 6= ht (m). Now by the same argument as in the proof of Theorem 4.3, we have Ext iRm (k(m), Cm ) = 0 for all i 6= ht (m), whence Cm is dualizing for Rm .  Corollary 4.10. The following statements hold true: (i) If C is pointwise dualizing, then C-id R (F (M )) ≤ C-id R (M ) for any cotorsion Rmodule M . (ii) If C-id R (F (M )) ≤ C-id R (M ) for any R-module M , then C is pointwise dualizing. Proof. (i). Assume that M is a cotorsion R-module. If C-id R (M ) = ∞, then we are done. Q So assume that C-id R (M ) = n < ∞. Then by Theorem 4.9, we have F (M ) = Tp where 0 ≤ ht (p) ≤ n. Now the result follows by Lemma 4.1. (ii). Assume that m is a maximal ideal of R. We have to show that Cm is dualizing for Rm . Assume that x is a maximal R-sequence in m. Then fd R (R/xR) < ∞, and Ass R (C/xC) = {m} since x is also a maximal C-sequence. Hence we have the equalities C-fd (C/xC) = fd R (Hom R (C, C/xC)) = fd R (Hom R (C, C ⊗R R/xR)) = fd R (R/xR) < ∞, in which the first equality is from Theorem 2.9(ii), and the third one holds because R/xR ∈ AC (R). Assume that E is an injective cogenerator. Set (−)∨ = Hom R (−, E). Then Cid R ((C/xC)∨ ) < ∞ by Lemma 2.10(ii). Now if F is the flat cover of (C/xC)∨ , then by assumption, we have C-id R (F ) < ∞. Therefore, we have C-fd R (F ∨ ) < ∞ by Lemma 2.10(i). Next, note that we have C/xC ֒→ (C/xC)∨∨ ֒→ F ∨ . Hence, the injective envelope of C/xC is a direct summand of F ∨ . Thus, in fact, E(R/m) 18 M. RAHMANI AND A.- J. TAHERIZADEH is a direct summand of F ∨ , since R/m ֒→ C/xC. It follows that C-fd R (E(R/m)) < ∞, and hence we are done by Lemma 4.1, since m was arbitrary.  Acknowledgments. We thank the referee for very careful reading of the manuscript and also for his/her useful suggestions. References 1. L. Bican, R. El Bashir and E. Enochs, All modules have flat covers, Bull. London Math. Soc. 33 (2001), 385–390. 2. W. Bruns and J. Herzog , Cohen-Macaulay rings, Cambridge University Press, Cambridge, 1993. 3. M.P. Brodmann and R.Y. Sharp , Local cohomology: an algebraic introduction with geometric applications, Cambridge University Press, Cambridge, 1998. 4. L.W. Christensen, Gorenstien Dimensions, Lecture Notes in Math., vol. 1747, Springer, Berlin, 2000. 5. L.W. Christensen, Semi-dualizing complexes and their Auslander categories, Trans. Amer. Math. Soc. 5 (2001), 1839–1883. 6. E. Enochs, Flat covers and flat cotorsion modules, Proc. Amer. Math. Soc. 92 (1984), 179–184. 7. E. Enochs and O. Jenda, Relative Homological Algebra, de Gruyter Expositions in Mathematics 30, 2000. 8. E. Enochs and J. Xu, On invariants dual to the Bass numbers, Proc. Amer. Math. Soc. 125 (1997), 951–960. 9. H.-B. Foxby, Gorenstein modules and related modules, Math. Scand. 31 (1973), 267–284 . 10. E. S. Golod, G-dimension and generalized perfect ideals, Trudy Mat. Inst. Steklov. Algebraic geometry and its applications 165 (1984), 62–66. 11. R. Hartshorne, Local cohomology, A seminar given by A. Grothendieck, Harvard University, Fall, vol. 1961, Springer-Verlag, Berlin, 1967. 12. H. Holm, P. Jørgensen, Semi-dualizing modules and related Gorenstein homological dimensions, J. Pure Appl. Algebra, 205(2006) 423–445. 13. H. Holm, D. White, Foxby equivalence over associative rings, J. Math. Kyoto Univ. 47 no.4, (2007), 781–808. 14. B. Kubik, M. J. Leamer, and S. Sather-Wagstaff, Homology of artinian and Matlis reflexive modules I, J. Pure Appl. Algebra, 215 (2011), no. 10, 2486–2503. 15. B. Kubik, M. J. Leamer, and S. Sather-Wagstaff, Homology of artinian and mini-max modules, J. Algebra. 403 (2014), no.1, 229–272. 16. M. Rahmani and A.-J. Taherizadeh, Tensor product of C-injective modules, preprint, 2015, available from http://arxiv.org/abs/1503.05492. 17. R. Takahashi, Some characterizations of Gorenstein local rings in terms of G-dimension, Acta Math. Hungar. 104 no. 4, (2004), 315–322. 18. R. Takahashi and D.White, Homological aspects of semidualizing modules, Math. Scand. 106 (2010) 5–10. 19. M. Salimi, E. Tavasoli, S. Sather-Wagstaff and S. Yassemi, Relative tor functor with respect to a semidualizing module, Algebras and Representation Theory, 17, no.1, (2014), 103–120. 20. W. V. Vasconcelos, Divisor theory in module categories, North-Holland Math. Stud. 14, North-Holland Publishing Co., Amsterdam (1974). 21. J. Xu, Flat covers of modules, Lecture Notes in Mathematics, Vol. 1634, Springer, New York, (1996). 22. J. Xu, Minimal injective and flat resolutions of modules over Gorenstein rings, J. Algebra 175 (1995), 451–477. 23. S. Yassemi, G-dimension, Math. Scand. 77 (1995), no. 2, 161–174. DUAL OF BASS NUMBERS AND DUALIZING MODULES Faculty of Mathematical Sciences and Computer, Kharazmi University, Tehran, Iran. E-mail address: [email protected] E-mail address: [email protected] 19 

Practical estimation of rotation distance and induced partial order for binary trees Jarek Duda arXiv:1610.06023v1 [] 19 Oct 2016 Jagiellonian University, Golebia 24, 31-007 Krakow, Poland. Email: [email protected] Abstract—Tree rotations (left and right) are basic local deformations allowing to transform between two unlabeled binary trees of the same size. Hence, there is a natural problem of practically finding such transformation path with low number of rotations, the optimal minimal number is called the rotation distance. Such distance could be used for instance to quantify similarity between two trees for various machine learning problems, for example to compare hierarchical clusterings or arbitrarily chosen spanning trees of two graphs, like in SMILES notation popular for describing chemical molecules. There will be presented inexpensive practical greedy algorithm for finding a short rotation path, optimality of which has still to be determined. It uses introduced partial order for binary trees of the same size: t1 ≤ t2 iff t2 can be obtained from t1 by a sequence of only right rotations. Intuitively, the shortest rotation path should go through the least upper bound or the greatest lower bound for this partial order. The algorithm finds a path through candidates for both points in representation of binary tree as stack graph: describing evolution of content of stack while processing a formula described by a given binary tree. The article is accompanied with Mathematica implementation of all used procedures (Appendix). Keywords: tree rotation, algorithmics, machine learning I. I NTRODUCTION Tree is a basic tool of computer science, used for example to represent hierarchical structure in data, e.g. in hierarchical clustering [1]. It is a natural question to evaluate similarity between such trees, maybe propose a path of local deformations to get a transformation between such two structures. A natural candidate for the required elementary local deformation of a binary tree, maintaining order of leaves, are tree rotations (left and right), which switch levels of two nodes and reconnect their subtrees. It is used for example to balance binary search tree in popular AVL method [2]. The minimal number of rotations to transform between two binary trees is a metric called rotation distance. It was shown that for any two N -node trees, for n ≥ 11, this distance is at most 2N − 6 and there exist pairs of trees fulfilling this 2N − 6 distance ([3], [4]). Binary trees have multiple equivalent representations, for example as bracketing or triangularization of polygon used in these two cited articles. There will be discussed greedy algorithm for finding a rotation path which base on a different representation: of evolution of stack content while processing such bracketed formula, which can be found for example in Chapter 7 of ”Concrete mathematics” book [5]. We will refer to this function as stack graph, it is visualized if Fig. 1 and left/right rotations become Figure 1. An example of binary tree (top), corresponding bracketing (center) and stack graph (bottom): describing evolution of content of stack while processing a formula described by a given binary tree (bracketing). It can be obtained by post-order tree traversal: visiting leaf corresponds to +1 for the graph (’x’ in bracketing, push value to stack), visiting internal node to -1 (closing bracket: pop two values then push their function). simple lower/lift procedures, visualized in Fig. 2, which turn out convenient for searching for a short path. This representation allows to conclude that a natural relation is a partial order: t1 ≤ t2 iff there exists a series of right rotations transforming tree t1 to t2 . This partial order for 4 and 5 node trees is presented in Fig. 3 and 4. These figures evaluating distance between different hierarchical clusterings. Another example of situation where we need to quickly evaluate distance between trees is for comparing graphs for which we can define a spanning tree in an unique way. We cannot do it for general graphs, as it would solve the graph isomorphism problem, but such spanning trees are a popular tool to describe e.g. chemical molecules in so called SMILES notation [6]. Such trees are more complex than binary unlabeled: degree of nodes can be larger than two and both vertices and edges may have types worth including in the definition of distance - will require a generalization of the discussed method. II. PARTIAL ORDER AND GREEDY SEARCH FOR COMMON LIFT Denote the
set of all n-leaf unlabeled binary trees as Tn , |Tn | = 2n−2 n−1 /n is Catalan number. Each of such trees has n − 1 internal nodes of degree 2, denote the number of all its nodes as N = 2n − 1. Let us define its stack graph as in Fig. 1: Figure 2. Tree rotations (left) and their analogue for corresponding bracketing (gray) and stack graph (right). The α, β and γ are nonegative graphs of even width (can be zero), changing by ±1 per position. Presented lift will be denoted by lij , where i, j are the marked positions. Figure 3. Partial order for all 4 leaf binary trees (top) and their stack graphs (bottom). Each arrow denotes possibility of single right rotation (or lift). suggest that the shortest path between two trees has to go through their least upper bound (∧) or their greatest lower bound (∨) for this order, or in other words: the path can be chosen (sorted) as first a sequence of right rotations, then of left rotations (∧) or oppositely (∨). However, this intuition has still to be verified. There will be presented a natural inexpensive (O(n · ’path length’)) greedy algorithm to find the common lift (CL) for stack graphs of two trees, which corresponds to a candidate for the least upper bound (∧) of these trees. By taking mirror image of the trees, which switches left and right rotations and so reverses the order, this algorithm can also provide a candidate for the greatest lower bound (∨). Its optimality has still to be verified, but currently it can be used to quickly find the upper bound for the rotation distance or to approximate this distance and find a short path, especially for situations where the optimality is not crucial. For example for various machine learning situations, like Definition 1. Stack graph st : {0, . . . , N } → N for tree t ∈ Tn is function defined by the following conditions: • st (0) = 0, st (N ) = 1, • for i = 1, . . . , N , st (i) − st (i − 1) = 1 if i-th node in post-order traversal of t is leaf, −1 otherwise Stack graph is fixed for 0 and 1, and is at least 1 for i ∈ {1, . . . , N }. For better visualization, it will be depictured with points joined by lines, sometimes without the fixed [0, 1] range (Fig. 3, 4). Denote Sn = s(Tn ) as the set of all stack graphs for n-leaf binary trees. The tree rotation operations are presented in Fig. 2. In bracketing notation, right rotation changes ((αβ)γ) into (α(βγ)) and left rotation the opposite, where α, β, γ represent some formulas (subtrees), which can be degenerated into single variables (leaves). As we can see in this figure, right rotation corresponds to lift by (−1, 1) vector of some {i, . . . , j} range. It can be degenerated: i = j for leaves. Definition 2. For two stack graphs: s1 , s2 ∈ Sn we will say that s2 is (i, j) lift of s1 , denoted as s2 = lij (s1 ), if the following conditions are fulfilled: 1) i ≤ j, s1 (i) = s1 (j) 2) s1 (i − 1) = s1 (i + 1) = s1 (j − 1) = s(i) + 1 3) for i < k < j, s1 (k) > s1 (i) 4) for k < i and j < k, s1 (k) = s2 (k) 5) for i ≤ k ≤ j, s2 (k − 1) = s1 (k) + 1 The value of stack graph will be referred as level. Condition 1) says that i ≤ j are on the same level of s1 , 2) says that there is first ’∨’ shape in i, then ’\’ shape in j. Condition 3) enforces s1 being above the i, j level. Finally 4) and 5) say that s1 and s2 differ only on the [i, j] range, in which s2 is lift by (−1, 1) vector of s1 . Obviously, lifting cannot decrease value in any position: Observation 3. If t2 is a right rotation of t1 , then for all i ∈ {0, . . . , N }, st1 (i) ≤ st2 (i). Figure 4. Partial order for all 5 leaf binary trees (top) and their stack graphs (bottom). Each arrow denotes possibility of single right rotation (or lift). Observe symmetry of the top diagram - taking mirror image of all trees switches left and right rotation, reversing the order It allows to conclude antisymmetry of the following relation, its reflexivity and transitivity are obvious, making it a partial order in Tn : Definition 4. For t1 , t2 ∈ Tn we define partial order: t1 ≤ t2 iff t1 = t2 or there exists a sequence of right rotations from t1 to t2 . Examples of this partial order for 4 and 5 leaf trees are presented in Fig. 3 and 4 correspondingly. These examples suggest that the shortest rotation path has to go through the least upper bound (∧) or the greatest lower bound (∨), however, it does not have to be generally true. Having algorithm searching for a short path going through the least upper bound (∧), we could use it to find path going through the greatest lower bound (∨) by taking mirror images of both trees, which switch left and right rotations, reversing this order. There is a nontrivial relation between stack graph of a tree and of its mirror image, which examples are presented in various figures of this paper, but it can found in O(n) time and memory (see mirror[] in Appendix). To find a candidate of the the greatest lower bound, we can take both stack graphs and use a sequence of lifts applied to a locally lower one, to finally equalize them, getting a candidate for the lowest common lift and of path to them. Example of such process is presented in Fig. 5. It is presented (without tracking the changes) as Algorithm 1 and 2. Its naive implementation (see findrotationpath[] in Appendix) is linear in n and path length. Figure 7 shows Figure 5. Example of finding a common lift using the greedy algorithm. Its step finds the lowest level where the two stack graphs disagree, find the most-right position of such points in this level (i), find the closest position in this level right to this point (j), then lift (i, j) segment. Such steps continue until equalizing both stack graphs, getting a common lift. examples of common lifts found by this algorithm, Figure 8 shows example of found rotation path for tree with 30 leaves. III. C ONCLUSIONS AND FURTHER PERSPECTIVES There was presented a practical way to search for a short rotation path between two unlabeled binary trees of the same size. Its optimality is not guaranteed in this moment, nor counterexamples were found. It can be used to find a rotation path and bound from above or approximate the rotation path. Algorithm 1 Greedy common lift for s1, s2 stack graphs while s1 6= s2 do lvl = min{s1[k] : s1[k] 6= s2[k]} i = max{k : s1[k] 6= s2[k], lvl = min(s1[k], s2[k])} if s1[i] < s2[i] then j = min{k > i: s1[k] = lvl}; lift(s1,i,j) else j = min{k > i: s2[k] = lvl}; lift(s2,i,j) end if end while Algorithm 2 Greedy shortest path for s1, s2 find common lift for s1 and s2 find common lift for mirror(s1) and mirror(s2) take the shorter one The basic question to investigate is optimality of this algorithm. If it is not optimal, maybe find its improvements, find bounds for its inaccuracy, try to characterise counterexamples. If it is already optimal, the proof might go through the following steps, each of which might turn out false: 1) The shortest rotation path has to go through the least upper bound or the greatest lower bound of the two trees. In other words, the shortest rotation path can be chosen (sorted) as only right rotations first then only left rotations, or oppositely, 2) The common lift found by the greedy algorithm corresponds to the least upper bound, 3) The greedy algorithm finds the shortest path to the obtained common lift - the number of lifts cannot be reduced. Another large topic is applying this or extended methods for various problems, especially in machine learning, where not being optimal seems not crucial. There will be needed some expansions, like generalization to non-binary trees, which can be realized for example by splitting a higher degree node into a few binary nodes. Also, in some applications we should distinguish types of nodes and edges (e.g. atoms and molecular bonds), which might require modifying the metric and so the optimization problem. A related line of work is trying to apply this method to evaluate similarity of graphs by comparing some their arbitrarily chosen spanning trees. Choosing an isomorphism independent spanning tree is generally a difficult problem, as it would allow to solve the graph isomorphism problem. However, it can be useful for some families of graphs, used for example in SMILES representation of chemical molecules. A PPENDIX This appendix contains Wolfram Mathematica sources used to generate figures in this article. It generates random binary tree assuming uniform probability distribution among trees of given number of leaves. For this purpose, it uses enumerative decoding [7] to get a random sequence of +1 and -1 which sums to 1, then finds its only cyclic rotation giving a stack graph. (* enumerative decoding *) dec[n_, w_, X_] := (l = w; x = X; Table[b = Binomial[n - i, l]; If[x < b, 0, x -= b; l--; 1], {i, n}]); (* generate stack graph of n leaf random tree *) randtree[n_] := (nn = 2 n - 1; s = 2 dec[2 n - 1, n, RandomInteger[Binomial[2 n - 1, n - 1] - 1]] - 1; Do[s[[i]] += s[[i - 1]], {i, 2, nn}]; min = Min[s]; div = Max[Table[If[s[[i]] == min, i, 0], {i, nn}]]; Join[s[[div ;; nn]], 1 + s[[1 ;; div]]] - min); (* draw tree from stack graph *) drawtree[s_] := (st = Table[0, {i, Length[s]}]; stp = 1; vn = 1; edg = {}; Do[If[s[[i + 1]] - s[[i]] == 1, st[[stp++]] = vn++, AppendTo[edg, {st[[stp - 2]] -> vn, st[[stp - 1]] -> vn}]; st[[stp - 2]] = vn++; stp--], {i, 1, Length[s] - 1}]; TreePlot[Sort[Flatten[edg]], Automatic, Length[s] - 1, VertexLabeling -> False]); (* mirror image of tree from stack graph *) mirror[s_] := (ls = Length[s]; p = 0; pbr = Table[i, {i, ls/2}]; nbr = Table[0, {i, ls/2}]; Do[If[s[[i + 1]] - s[[i]] == 1, p++, brac = pbr[[pbr[[p]] - 1]]; nbr[[pbr[[p]] = brac]]++], {i, ls - 1}]; S = Table[0, {i, ls}]; p = 1; Do[p++; S[[p]] = S[[p - 1]] + 1; Do[p++; S[[p]] = S[[p - 1]] - 1, {j, 1, nbr[[i]]}] ,{i, ls/2, 1, -1}]; S); (* the greedy algorithm to find path *) lift[s_, f_, t_] := Join[s[[1 ;; f - 1]], s[[f + 1 ;; t]] + 1, {s[[t]]}, s[[t + 1 ;; Length[s]]]]; findrotationpath[S1_, S2_] := ( s1 = S1; s2 = S2; ll = {s1}; rl = {s2}; While[s1 != s2, min = Infinity; Do[If[s1[[i]] != s2[[i]], m = Min[s1[[i]], s2[[i]]]; If[m <= min, min = m; pm = i]], {i, Length[s1]}]; px = pm + 2; If[s1[[pm]] < s2[[pm]], While[s1[[px]] > min, px++]; AppendTo[ll, s1 = lift[s1, pm, px]], While[s2[[px]] > min, px++]; PrependTo[rl, s2 = lift[s2, pm, px]]]]; Join[ll, rl[[2 ;; Length[rl]]]]) (* example of application *) n = 10; S1 = randtree[n]; S2 = randtree[n]; path1 = findrotationpath[S1, S2]; path2 = findrotationpath[mirror[S1], mirror[S2]]; Print[Row[Table[drawtree[path1[[i]]], {i, Length[path1]}]]] Row[Table[drawtree[path2[[i]]], {i, Length[path2]}]] R EFERENCES [1] J. H. Ward Jr, “Hierarchical grouping to optimize an objective function,” Journal of the American statistical association, vol. 58, no. 301, pp. 236– 244, 1963. [2] G. Adelson-Velsky and E. Landis, “An algorithm for the organization of information,” Proceedings of the USSR Academy of Sciences, vol. 146, pp. 263–266, 1962. [3] D. D. Sleator, R. E. Tarjan, and W. P. Thurston, “Rotation distance, triangulations, and hyperbolic geometry,” Journal of the American Mathematical Society, vol. 1, no. 3, pp. 647–681, 1988. [4] L. Pournin, “The diameter of associahedra,” Advances in Mathematics, vol. 259, pp. 13–42, 2014. [5] D. E. Knuth, R. L. Graham, O. Patashnik et al., “Concrete mathematics,” Adison Wesley,, 1989. [6] D. Weininger, “Smiles, a chemical language and information system. 1. introduction to methodology and encoding rules,” Journal of chemical information and computer sciences, vol. 28, no. 1, pp. 31–36, 1988. [7] L. Oktem, Hierarchical enumerative coding and its applications in image compression. Tampere University of Technology, 1999. Figure 6. Example of rotation path found by the greedy algorithm for common lift, for the original trees (top) and their mirror images (bottom). The marked trees correspond to the common lift. The upper path is shorter so it will be the final answer. Figure 7. Five examples (columns) of the found common lift by greedy algorithm (highest orange line) for stack graphs of two random trees (lower: blue and green lines). The upper row corresponds to stack graphs of the original trees: common lift is a candidate for the least upper bound. The lower row corresponds to stack graphs of mirror images of the trees. Figure 8. Example of found rotation path for two random 30 vertex trees. The marked tree corresponds to the common lift. 

1 A Statistical Characterization of Localization Performance in Wireless Networks Christopher E. O’Lone, Graduate Student Member, IEEE, arXiv:1710.07716v1 [cs.NI] 20 Oct 2017 Harpreet S. Dhillon, Member, IEEE, and R. M. Buehrer, Fellow, IEEE Abstract Localization performance in wireless networks has been traditionally benchmarked using the CramérRao lower bound (CRLB), given a fixed geometry of anchor nodes and a target. However, by endowing the target and anchor locations with distributions, this paper recasts this traditional, scalar benchmark as a random variable. The goal of this work is to derive an analytical expression for the distribution of this now random CRLB, in the context of Time-of-Arrival-based positioning. To derive this distribution, this work first analyzes how the CRLB is affected by the order statistics of the angles between consecutive participating anchors (i.e., internodal angles). This analysis reveals an intimate connection between the second largest internodal angle and the CRLB, which leads to an accurate approximation of the CRLB. Using this approximation, a closed-form expression for the distribution of the CRLB, conditioned on the number of participating anchors, is obtained. Next, this conditioning is eliminated to derive an analytical expression for the marginal CRLB distribution. Since this marginal distribution accounts for all target and anchor positions, across all numbers of participating anchors, it therefore statistically characterizes localization error throughout an entire wireless network. This paper concludes with a comprehensive analysis of this new network-wideCRLB paradigm. Index Terms Cramér-Rao lower bound, localization, order statistics, Poisson point process, stochastic geometry, Time of Arrival (TOA), mutual information, wireless networks. The authors are with Wireless@VT, Bradley Department of Electrical and Computer Engineering, Virginia Tech, Blacksburg, VA, USA. Email: {olone, hdhillon, buehrer}@vt.edu. This paper was presented in part at the IEEE ICC 2017 Workshop on Advances in Network Localization and Navigation (ANLN), Paris, France [1]. 2 I. I NTRODUCTION The Global Positioning System (GPS) has for decades been the standard mechanism for position location anywhere in the world. However, the deployment locations of recent and emerging wireless networks have begun to put a strain on the effectiveness of GPS as a localization solution. For example, as populations increase, precipitating the expansion of urban environments, cell phone use in urban canyons, as well as indoors, is continually increasing. The rise of these GPSconstrained environments highlight the need to fall back on the existing network infrastructure for localization purposes. Additionally, with the emergence of Wireless Sensor Networks (WSNs) and their increased emphasis on energy efficiency and cost-effectiveness, [2], [3], the possibility of equipping each potential target node with a GPS chip quickly becomes impractical. Furthermore, the deployment of these networks in GPS-constrained environments further necessitates a reliance on the terrestrial network for a localization solution. Thus, localization within a network, performed by the network itself in the absence of GPS, has begun to garner attention, [4], [5]. Benchmarking localization performance in wireless networks has traditionally been done using the Cramér-Rao lower bound, which provides a lower bound on the position error of any unbiased estimator [6]. Common practice has been to analyze the CRLB in fixed scenarios of anchor nodes and a target, e.g., [7], [8], [9], [10]. This strategy produces a scalar/fixed value for the CRLB and is specific to the scenario being analyzed. While this idea does provide insight into fundamental limits of localization performance, it is rather limited in that it does not take into account all possible setups of anchor nodes and target positions within a network. In order to account for all possible setups, it is useful to appeal to the field of stochastic geometry. Whereas in the past stochastic geometry has been applied towards the study of “connectivity, capacity, outage probability, and other fundamental limits of wireless networks,” [11], [12], we now however apply it towards the study localization performance in wireless networks. Modeling anchor node and target placements with point processes opens the possibility of characterizing the CRLB over all setups of anchor nodes and target positions. Thus, the CRLB is no longer a fixed value, but rather a random variable (RV) conditioned on the number of participating anchors, where the randomness is induced by the inherent randomness of the anchor positions. Upon marginalizing out the number of participating anchors, the resulting marginal distribution of the CRLB will characterize localization performance throughout an entire wireless 3 network. A. Related Work The quest for a network-wide distribution of localization performance comprises two main steps. The first step involves finding the distribution of the CRLB conditioned on the number of participating anchor nodes, and the second step involves finding the probability that a given number of anchors can participate in a localization procedure. With regards to the first step, there have been several attempts in the literature to obtain this conditional distribution. An excellent first attempt can be found in the series of papers, [13], [14], [15], in which approximations of this conditional distribution were presented for ReceivedSignal-Strength (RSS), Time-of-Arrival (TOA), and Angle-of-Arrival (AOA) based localization, respectively. These approximations were obtained through asymptotic arguments by driving the number of participating anchor nodes to infinity. While these approximate distributions are very accurate for larger numbers of participating anchors, they are less than ideal for lower numbers. However, it is desirable that this conditional distribution be accurate for lower numbers of participating anchors, since this is the dominate case in terrestrial networks, e.g., cellular. This conditional distribution of the CRLB was also explored in [16]. Here, the authors were able to derive the true expression for this conditional distribution through a clever re-writing of the CRLB using complex exponentials. This distribution was then used to derive and analyze the so-called “localization outage probability” in scenarios with a fixed number of randomly placed anchor nodes. While this expression represents the true conditional distribution, its complexity puts it at a disadvantage over simpler approximations (discussed further in Section III-D). The second step, which involves finding the participation probability of a given number of anchor nodes, was explored in [17]. In this work, the authors modeled a cellular network with a homogeneous Poisson point process (PPP), which consequently allowed them to derive bounds on L-localizability, i.e., the probability that a mobile device can hear at least L base stations for participation in a localization procedure.1 Further, by employing a “dominant interferer analysis,” they were able to derive an accurate expression for the probability of L-localizability, which can easily be extended to give the probability of hearing a given number of anchor nodes for participation in a localization procedure. 1 The term hearability (or hearable) is used to describe anchor nodes that are able to participate in a localization procedure, i.e., their received SINR at the target is above some threshold. 4 B. Contributions This paper proposes a novel statistical characterization of a wireless network’s ability to perform TOA-based localization. Using stochastic geometry to model target and anchor node placements throughout a network, and using the CRLB as the localization performance benchmark, this paper presents an analytical derivation of the CRLB’s distribution.2 This distribution offers many insights into localization performance within wireless networks that previously were only attainable through lengthy, parameter-specific, network simulations. Thus, this distribution 1) offers a means for comparing networks in terms of their localization performance, by enabling the calculation of network-wide localization statistics, e.g., avg. localization error, 2) unlocks insight into how changing network parameters, such as SINR thresholds, processing gain, frequency reuse, etc., affect localization performance throughout a network, and 3) provides network designers with an analytical tool for determining whether a network meets, for example, the FCC E911 standards [18]. In pursuit of this distribution, this paper makes four key contributions. First, this work presents an analysis of how the CRLB is affected by the order statistics of the internodal angles. This analysis reveals an intimate connection between the second largest internodal angle and the CRLB, which leads to an accurate approximation of the CRLB. Second, this approximation is then used to obtain the distribution of the CRLB conditioned on the number of hearable anchors. Although this is a distribution of the CRLB approximation, its simplicity and accuracy offer clear advantages over the true distribution presented in [16] and the approximate distribution presented in [14]. These advantages are discussed further in Section III-D. Third, this work then takes a major step by combining this conditional distribution of the CRLB with the distribution of the number of participating anchors. This eliminates the conditioning on a given number of anchors, allowing for the analytical expression of the marginal CRLB distribution to be obtained. Since this marginal distribution now simultaneously accounts for all possible target and anchor node positions, across all numbers of participating anchor nodes, it therefore statistically characterizes localization error throughout an entire wireless network, and thus signals a departure from the existing literature. Additionally, since the two component distributions are parameterized by various network parameters, our resulting marginal distribution 2 We will be using the square root of the CRLB as our performance benchmark, however, we just state the CRLB here as to not unnecessarily clutter the discussion. This will be described further in Section III-A. 5 of the CRLB will also be parameterized by these network parameters. Consequently, our final contribution involves a comprehensive analysis of this new network-wide CRLB paradigm, where we examine how varying network parameters affects the distribution of the CRLB — thereby revealing how these network parameters affect localization performance throughout the network. II. P ROBLEM S ETUP This section details the assumptions we propose in determining the network layout as well as the localization procedure. Additionally, we describe important notation and definitions used throughout the paper and conclude with how the assumptions impact the network setup. A. Network Setup and Localization Assumptions Assumption 1. We assume a ubiquitous, two-dimensional wireless network with anchor nodes distributed according to a homogeneous PPP over R2 . Further, we assume potential targets to be distributed likewise, where the anchor and target point processes are assumed to be independent. Remark. This assumption for modeling wireless networks is common in the literature, e.g., [19], [20], [21], [22], [23]. Assumption 2. We assume a 2-Way TOA-based positioning technique is used within the twodimensional network. Remark. Although Time-Difference-of-Arrival (TDOA) is also commonly implemented, 2-Way TOA represents a viable approach, and additionally offers a lower bound on TDOA performance. Furthermore, the 2-Way assumption eliminates the need for clock synchronization between the target and anchor nodes. Assumption 3. Range measurements are independent and exhibit zero-mean, normally distributed range error. Remark. A reader familiar with wireless positioning will recognize this as a classic Line-of-Sight (LOS) assumption. However, those familiar with localization in terrestrial networks will realize that Non-Line-of-Sight (NLOS) measurements are more common. Thus, while we move forward under this LOS assumption in order to make progress under this new paradigm, we will see in Section IV that we can adapt our model to accommodate NLOS measurements by selecting ranging errors consistent with NLOS propagation. 6 TABLE I S UMMARY OF N OTATION Symbol fX (·) h(X) u(x) k·k 1[·] tr(X) L S Θk ∠k X(k) α β K Description Probability distribution function (pdf) of X Differential entropy of X Unit step function, 0 when x < 0, 1 when x ≥ 0 `2 -norm Indicator function; 1[A] = 1 if A true, 0 otherwise Trace of the matrix X Number of participating anchors, L ∈ {0, 1, 2, . . . } The localization performance benchmark Anchor node angle k, a random variable Internodal angle k, a random variable k th order statistic of a sequence {Xi } of RVs Path loss exponent, α > 2 Post-proc. SINR threshold Frequency reuse factor Symbol FX (·) I(X; Y ) N (µ, σ 2 ) P[A] b·c XT N M θk ϕk λ γ q σr2 Description Cumulative distribution function (cdf) of X Mutual Information between X and Y Normal distribution, mean µ, variance σ 2 Probability of event A Floor function Transpose of the matrix X Max anchors tasked to perform localization Fixed localization error, in meters, M ∈ R+ A realization of Θk A realization of ∠k Density of PPP of anchor locations Processing gain Average network load (i.e., network traffic) Common range error variance Assumption 4. The range error variance, σr2 , is common among measurements from participating anchor nodes and is considered a known quantity. Remark. This assumption is often made in the literature for range-based localization, e.g., [10], [25], and although not realistic in every scenario, it allows us to gain initial insight into the problem and will be relaxed in future work. B. Notation The notation used throughout this paper can be found in Table I. C. Localizability We now define the terms localizable and unlocalizable, which were introduced in [17]. Definition 1. We say that a target is localizable if it detects localization signals from a sufficient number of anchor nodes such that its position can be determined without ambiguity. Remark. Under Assumption 2, this implies that L ≥ 3. We also define unlocalizable to be the negation of Definition 1. For the purposes of this setup and subsequent derivations, we will initially only consider scenarios in which the target is localizable, to avoid unnecessary complication. Later in Section 7 III-E and those which follow, we will account for scenarios in which the target is unlocalizable, and will modify our results accordingly. D. Impact of Assumptions With these assumptions in place, we now describe how they impact the network setup. From Assumption 1, since the anchors and potential targets are distributed by independent, homogeneous PPPs over R2 (i.e., stationary), then without loss of generality, we may perform our analysis for a typical target placed at the origin of the xy-plane [21]. This is due to the fact that the independence and stationarity assumptions imply that no matter where the target is placed in the network, the distribution of anchors relative to the target appears the same. Next, we assume that the number of hearable anchors is some fixed value, L, and begin by numbering these anchors in terms of their increasing distance from the origin (target position). This is depicted in Fig. 1 for a particular realization of a homogeneous PPP in which there are four hearable anchor nodes. Fig. 1 also depicts how their corresponding angles, measured counterclockwise from the +x-axis, are labeled accordingly. Assumption 1 further implies that these angles of the hearable anchors are i.i.d. random variables that come from a uniform distribution on [0, 2π). Definition 2. If the target is placed at the origin of an x y-plane, then the term anchor node angle, Θ k , is defined to be the angle corresponding to hearable anchor node k, measured i.i.d. counterclockwise from the +x-axis. Note that Θ k ∼ unif[0, 2π), ∀k ∈ {1, . . . , L}. In later sections we will see that the distances between the anchor nodes and the target are important for determining how many anchors are able to participate in a given localization procedure. However, under Assumption 4, once particular anchor nodes are identified as participating in a localization procedure, then they are endowed with the common range error variance. As we will see in the following section, this assumption, along with Assumptions 2 and 3 will lead to the CRLB (with L fixed) as being dependent on only the angles between participating anchor nodes (i.e., internodal angles). Thus, since the CRLB expression is only dependent on the internodal angles, the distances between participating anchor nodes and the target need not be considered. Hence, we may view the participating anchors as being placed on a circle about the origin. This is depicted in Fig. 2, under the same PPP realization as in Fig 1. 8 y 1 y θ1 θ3 2 θ2 θ (3) 4 θ4 x ϕ(2) θ (2) ϕ(1) θ (1) ϕ(3) ϕ(4) x θ (4) 3 Fig. 1. I NITIAL L ABELING S CHEME . The dots represent Fig. 2. E QUIVALENT S ETUP. This is the realization from a particular realization of anchors placed according to a Fig. 1, where only participating anchor nodes and the in- homogeneous PPP. The origin represents the location of the ternodal angles they trace out are considered, whereas their target. The anchors participating in the localization procedure distances from the target are not. The realizations of the RVs are labeled in increasing order w.r.t. their distance from the are given by Θ(k) = θ (k) and ∠(k) = ϕ(k) . Note, the anchor origin. Their corresponding anchor node angles are labeled node angle order stats “renumber” the participating anchors accordingly. Note the realization of the RV Θk is Θk = θ k . in terms of increasing angle c.c.w. starting from the +x-axis. Next, we formally define the term internodal angle. Since the anchor node angles are such i.i.d. that Θ k ∼ unif[0, 2π), for k ∈ {1, . . . , L}, we may examine their corresponding order statistics, Θ(1), Θ(2), . . . , Θ(L) , where 0 ≤ Θ(1) ≤ Θ(2) ≤ · · · ≤ Θ(L) ≤ 2π by definition. Thus, the order statistics of the participating anchor node angles effectively “renumber” the nodes in terms of increasing angle, starting counterclockwise from the +x-axis. This is also depicted in Fig. 2. Definition 3. If participating anchor nodes are considered according to their anchor node angle order statistics, then an internodal angle, ∠ k , is defined to be the angle between two consecutive participating anchor nodes. That is, ∠ k = Θ(k+1) − Θ(k) for 1 ≤ k < L and ∠ L = 2π − (Θ(L) − Θ(1) ). Remark. Since the internodal angles are functions of RVs, they themselves are RVs. Thus, we may also consider their order statistics, ∠(1), ∠(2), . . . , ∠(L) . These order statistics of the internodal angles are depicted for the particular PPP realization in Fig. 2. In summary, Fig 2 depicts an example of a typical setup realization, given L = 4, once all of the assumptions are taken into consideration. 9 III. D ERIVATION OF THE N ETWORK -W IDE CRLB D ISTRIBUTION In this section, we first formally define our localization performance benchmark: the square root of the CRLB. Using this definition and assuming a random placement of anchors, as well as a random L, we then describe how this work generalizes localization performance results currently in the literature. In what follows, we present the steps necessary to derive the marginal distribution of our localization performance benchmark. A. The Localization Performance Benchmark Consider the traditional localization scenario, where the number of participating anchor nodes (L) and their positions, as well as the target position, are all fixed. We represent the set of coordinates of these anchors by n o  T ΨL = ψi ∈ R2 ψi = xi, yi , i ∈ {1, 2, 3, ..., L} .  T The coordinates of the target are denoted by ψt = xt, yt . Next, under Assumptions 2, 3, and 4, the 2-Way range measurements between the target and the L participating anchors are given by ri = di + ni , where ri is the measured distance divided by 2 (i.e., the measured 1-Way distance), q
2
2 xi − xt + yi − yt di = kψi − ψt k = i.i.d. is the true 1-Way distance, and ni ∼ N (0, σr2 ) for all i ∈ {1, 2, . . . , L}. Remark. Note that under Assumption 4, σr2 is common among the range measurements. For 2 2 2-Way TOA we set this to be σr2 = σ2-Way /4. If 1-Way TOA is considered, then σr2 = σ1-Way . Thus, moving forward we may consider the range measurements to always be 1-Way, regardless of whether 1-Way or 2-Way TOA is used. Continuing, Assumption 3 enables the likelihood function to be easily written as a product. Denoting the vector of range measurements as r = [r1, . . . , rL ]T , the likelihood function is ! L   Ö 2 (r − d ) 1 i i exp − . L ψt r, ΨL, σr2 = √ 2 2σ 2πσ r r i=1 From this likelihood function, we obtain the following Fisher Information Matrix (FIM) ÍL  ÍL  2  i=1 cos θ i i=1 cos θ i sin θ i  1  J L (φt ) = 2  , ÍL  σr Í L cos θ sin θ 2 sin θi  i i i=1 i=1   10 where cos θi = xi −xt di and sin θi = yi −yt di . Remark. Note that if the target is placed at the origin, then the angles here, i.e., the θi ’s, are a particular realization of the anchor node angles from Definition 2. Taking the inverse of the FIM above, then the CRLB for any unbiased estimator, ψ̂t = [ x̂t, ŷt ]T , of the target position, ψt = [xt, yt ]T , is given by   CRLB = tr J−1 (ψ ) t , L (1) as defined in Ch. 2.4.1 of [24]. Definition 4. We define our localization performance benchmark, S, to be the square root of the CRLB in (1): r   S , tr J−1 (ψ ) t . L (2) Remark. This benchmark is often referred to as the position error bound (PEB) in the literature, [25], [26]. To conclude, notice that from (2), a closed-form expression for S can be obtained: √ L S = σr v u !2 , t L L L Õ Õ Õ cos2 θi sin2 θi − cos θi sin θi i=1 i=1 (3) i=1 which is a function of the anchor node angles. B. Departure From the Traditional Localization Setup In the previous section, we assumed a traditional setup, in which the number of participating anchor nodes (L) and their positions (ΨL ), as well as the target position (ψt ), were all fixed. In this section however, we now assume that both ΨL and L are random and briefly describe how the random placement of anchors impacts our localization performance benchmark, S, and how the randomness of L signals a departure from the existing literature. We begin by describing how the random placement of anchors affects our localization performance benchmark. This is accomplished by now invoking Assumption 1 and examining the impact that this has on the expression for S given by (3). Recall from Section II-D that Assumption 1 implies that ψt = [0, 0]T and that the anchor node angles are i.i.d. on [0, 2π). Thus, the θi realizations in (3) are now replaced with the random variables, Θi , from Definition 11 2. Since S is now a function of random variables, it itself becomes a random variable, and we may now seek its distribution. While work in the past has sought this distribution for S, e.g., [14], [16], there always remained one implicit assumption: a fixed L. Therefore, the results presented thus far in the literature have only applied to localization setups with a fixed number of anchor nodes, and hence were not applicable network-wide. To address this issue, we now consider L to be a random variable, whose distribution statistically quantifies the number of anchor nodes participating in a localization procedure. This new interpretation of L will consequently allow us to decouple S from L, thereby enabling the marginal distribution of S to account for all possible positioning scenarios within a network. In addition to the contributions outlined in Section I-B, taking advantage of this new interpretation of L, to subsequently obtain the marginal distribution of S, is the main contribution setting this work apart from the existing literature. C. Approximation of the CRLB Before deriving the conditional distribution of S given L, we use this section to acquire an approximation of our expression for S in (3). This will consequently allow for an accurate, tractable, closed-form expression for the conditional distribution of S to be obtained. 1) Approximation Preliminaries and Goals: To facilitate the search for an accurate approximation, we recall that (3) is now in terms of the random variables Θi , and thus can be rewritten using the internodal angles from Definition 3. This given by √ L S = σr v u ! , tÕ j−1 L−1 Õ L Õ ∠k sin2 i=1 j=i+1 (4) k=i Definition 5. We define the terms underneath the square-root in the denominator in (4) by the random variable D. That is, D= L−1 Õ L Õ i=1 j=i+1 sin2 j−1 Õ ! ∠k . (5) k=i Thus, we would like to find an approximation for D which comprises two key traits: (1) it allows for a straightforward transformation of random variables, i.e., the number of sin2 (·) terms does not change with L (and would ideally only involve a single term), and (2) it simultaneously does not sacrifice accuracy, i.e., the approximation should preserve as much “information” 12 as possible about the setup of anchors, implying that the approximation should dominate (or contribute the most to) the total value of D. 2) Initial Approach and Intuition: In trying to find an approximation that satisfies both traits, we consider the following possibilities. First, consider approximating D with the sine squared of an arbitrary internodal angle, sin2 (∠ k ), or with a sum of consecutive internodal angles, sin2 (∠ k +∠ k+1 +...), where the starting angle, ∠ k , is arbitrary. While these possible approximations may seem like reasonable candidates for satisfying the first trait, they unfortunately fall short of satisfying the second trait. To see why, it is illustrative to examine Fig. 2 under different realizations/placements of anchor nodes. By only looking at the same unordered internodal angle on every realization, little knowledge is gained about the total setup of anchors. For example, on one realization, the arbitrary internodal angle being examined might be large and would therefore give a strong indication of how the rest of the anchors are placed, however, on another realization, this same internodal angle might be small, thus giving little information about the placement of the remaining anchors. Hence, in general, arbitrary internodal angles do not provide accurate approximations due to their inconsistency in describing the anchor node setup, which consequently leads to their (sine squared terms’) inability to capture the total value of D across all realizations of anchors for a given L. 3) A Quantitative Approach Using Mutual Information: Taking advantage of the intuition gained above, it should now be clear that we would like an approximation that utilizes angles which tend to consistently dominate any given setup. Therefore, it makes sense to examine larger internodal angles, and thus, the use of internodal angle order statistics follows naturally. Since we ideally desire a single-term approximation, we examine the possibility of using the sine-squared of the largest, second largest, and third largest internodal angles, i.e.,3 W(L) = sin2 (∠(L) ), W(L−1) = sin2 (∠(L−1) ), and W(L−2) = sin2 (∠(L−2) ). (6) Note that these larger internodal angles should intuitively contain more “information” about the setup of anchors, and consequently D, since they greatly restrict the placement of the remaining anchors. Since each of the order statistic approximations above might seem viable under this 3 We will not examine sums of the larger internodal angle order statistics, e.g., sin2 (∠(L) + ∠(L−1) ) or sin2 (∠(L) ) + sin2 (∠(L−1) ), for the following two reasons: (1) they would lead to a more complex expression for the conditional distribution of S than desired, and (2) they offer no gain in accuracy over using the sine squared of just a single internodal angle order statistic, as evidenced through simulation. 13 1.6 I(D; W(L) |L) I(D; W(L−1) |L) I(D; W(L−2) |L) Mutual Information (Bits) 1.4 1.2 1 0.8 0.6 0.4 0.2 0 3 4 5 6 7 Number of Participating Anchor Nodes (L) Fig. 3. J USTIFYING A PPROXIMATIONS T HROUGH M UTUAL I NFORMATION. The mutual informations were calculated numerically by computing (8) and (9), where the necessary distributions were generated using a Monte Carlo simulation of 10 million anchor node realizations. The bin width of these distributions was chosen to be 0.01, Matlab’s ‘spline’ option was used to interpolate the integrands in (8) and (9), and the supports of D and W(i) are given by D = [0, L 2 /4] and W(i) = [0, 1], where i ∈ {L, L − 1, L − 2}. Furthermore, we adopt the convention: 0 log2 0 = 0 “based on continuity arguments” [29]. qualitative notion of information, we thus turn towards a quantitative notion, in order to justify the use of one of these approximations. Towards this end, we utilize the concept of mutual information. The reason behind this choice, over that of correlation for example, is because mutual information captures both linear and nonlinear dependencies between two random variables, since it is zero if and only if the two random variables are independent [28]. Hence, we examine the mutual information between D and the random variables W(L) , W(L−1) , and W(L−2) , so that we may quantify which approximation carries the most information about D. Thus, we condition on L equaling some integer `(≥ 3) and calculate: I(D; W(i) |L = `) = h(D|L = `) − h(D|W(i), L = `), where i ∈ {L, L − 1, L − 2}, and the differential entropies are given by ∫ 1 h(D|L = `) = fD (d|`) log2 dd, and fD (d|`) D ∫ ∫ 1 h(D|W(i), L = `) = fDW(i) (d, w|`) log2 dd dw, fD (d|w, `) (7) (8) (9) W(i) D where W(i) and D are the supports of W(i) and D, respectively [29]. The mutual information between the approximations in (6) and D are given in Fig. 3, versus L. From Fig. 3, it is evident 14 that the mutual information between D and the approximation W(L−1) is the highest across all numbers of participating anchors shown. 4) Investigating the High Mutual Information of D and W(L−1) : To explore the reasoning behind this result, we examine the effect that W(L−1) has on the total value of D. We begin by rewriting D as follows: Proposition 1. The random variable D from Definition 5 can be equivalently expressed as ! j+i−1 L−2 Õ L−i L Õ Õ Õ
sin2 ∠(k) + sin2 ∠k . D= i=2 j=1 k=1 k= j  Proof. See Appendix A. By separating the sine-squared terms of the internodal angle order statistics from the total sum, Proposition 1 makes it clearer as to how our approximations from (6) may affect the total value of D. To reveal the effects of W(L−1) in particular, we present the following lemma along with its corollaries: Lemma 2. The cdf of the second largest order statistic of the internodal angles, ∠(L−1) , conditioned on L, is given by F∠(L−1)   X Õ nϕ n−1 L ϕ|L = (−1) (n − 1) 1 − 2π n n=0
! L−1 , (10)  where X = min L, b2π/ϕc and the support is 0 ≤ ∠(L−1) ≤ π. (Note L ≥ 2, since if L < 2, ∠(L−1) would not exist.) Proof. We refer the reader to the conference version of this paper, i.e., Appendix B of [1].  Corollary 2.1. Given a finite L, the expected value of the second largest order statistic of the internodal angles, ∠(L−1) , conditioned on L, is given by !   L Õ L 2π(n − 1) E[∠(L−1) |L] = (−1)n n nL n=2 Proof. See Appendix B. Corollary 2.2. Given a finite L, the variance of ∠(L−1) , conditioned on L, is given by !" #   L 2 Õ 4π L n−1 2 c VAR[∠(L−1) |L] = (−1)n + , L n=2 n n n(L − 1) L (11)  (12) 15 where c = 
m+1 L m−1 m=2 (−1) m m ÍL  .  2 2 2 Proof. Note: VAR[∠(L−1) |L] = E[∠(L−1) |L] − E[∠(L−1) |L] , where the derivation of E[∠(L−1) |L] is analogous to that of E[∠(L−1) |L] in the proof of Corollary 2.1.  Next, we plot Corollary 2.1, plus/minus one and two standard deviations of ∠(L−1) . This is given in Fig. 4, versus L. Here, we can see that the second largest internodal angle is centered and concentrated around π/2, suggesting that W(L−1) = sin2 (∠(L−1) ) will be concentrated about its maximum of one. This implies that, for the majority of anchor node placements, sin2 (∠(L−1) ) will be a dominant term in the expression for D in Prop. 1. Thus, the sin2 (∠(L−1) ) term will tend to contribute the most, that a given sin2 (·) term could contribute, to the total value of D. This is especially true for small values of L, which is our focus. Also for low L, ∠(L−1) is intuitively the dominant angle in that it places the greatest constraints on the remaining angles. That is, once ∠(L−1) is determined, it restricts both ∠(L) and ∠(L−2) , and thus gives the greatest sense of the total setup of anchors. Note, when considering order statistics other than ∠(L−1) , the constraints placed on the remaining angles are not as pronounced. Furthermore, by examining different realizations of anchors (Fig. 2 as an example), along with Prop. 1, one can see that when W(L−1) is small (∠(L−1) ≈ 0 or π), than so is D, and when W(L−1) is large (∠(L−1) ≈ π/2), a large value of D follows. Thus, W(L−1) ’s consistency as a dominant term in Prop. 1, along with its intuitive correlation with D, offer supporting evidence as to why I(D; W(L−1) |L = `) is higher than both I(D; W(L) |L = `) and I(D; W(L−2) |L = `) for low L. In summary, mutual information has proved its utility by revealing that W(L−1) is perhaps the best approximation of D, for the desirable lower values of L. Since W(L−1) possesses the two desirable traits for an approximation, discussed at the beginning of this section, we henceforth use W(L−1) in our approximation of D. 5) Completing the Approximation: To complete the approximation of D, and consequently S, all that now remains is to ensure that D and W(L−1) have the same range of possible values, i.e., the same support. This will ensure that our ultimate approximation for S will produce the same range of values as the true S. In order to accomplish this, we approximate D with a scaled version of W(L−1) , i.e., D ≈ k W(L−1) , and thus search for the value of the constant k so that kW(L−1) yields the desired support. Since D = [0, dmax ] and W(L−1) = [0, 1] (which follows from the support of ∠(L−1) , Lemma 2), then in order to have the support of kW(L−1) equal that of D, we simply need to set k = dmax . The value of dmax is presented in the following lemma: 16 1 π E[6 E[6 E[6 7 π /8 (L−1) (L−1) (L−1) | L] | L] ± σ | L] ± 2σ 0.9 0.8 P[S ≤ abscissa | L] 0.7 5 π /8 π /2 3 π /8 E[6 (L−1) |L] (Radians) 6 π /8 Increasing L (L = 3, 5, 7, 9) 0.6 0.5 0.4 0.3 2 π /8 0.2 π /8 Solid: True Dashed: Theorem 4 0.1 0 3 4 5 6 7 8 9 0 10 20 30 Number of Participating Anchor Nodes (L) 40 50 60 70 Meters Fig. 4. T HE S ECOND L ARGEST I NTERNODAL A NGLE O R - Fig. 5. ACCURACY OF T HEOREM 4. The true conditional cdf S TATISTIC . This figure gives a sense of the concentration of S given L was generated using a Monte Carlo simulation of of the distribution of ∠(L−1) around π/2. E[∠(L−1) |L] is given q by Cor. 2.1 and σ = VAR[∠(L−1) |L] is from Cor. 2.2. (4) over 1 million random setup realizations of the internodal DER angles. Note, σr = 20 m. Lemma 3. Let L be finite. If D is given as in (5), then its maximum value is dmax = L 2 /4. Proof. See Appendix C.  Thus, we now have D ≈ (L 2 /4) · W(L−1) , which completes our approximation of D. Lastly, substituting this approximation for D into the expression for S in (4) finally yields: Approximation 1. The localization performance benchmark, S, can be approximated by r 4 1 S ≈ σr · · , L sin(∠(L−1) ) (13) where 0 ≤ ∠(L−1) ≤ π, as stated in Lemma 2. D. The Conditional CRLB Distribution Theorem 4. If the localization performance benchmark, S, is given by Approximation 1, then the cdf of S conditioned on L is ! L−1 X ! L−1     X2 1 Õ Õ L nϕ L nϕ 2 1 FS (s | L, σr ) = (−1)n−1 (n − 1) 1 − − (−1)n−1 (n − 1) 1 − , (14) n 2π n 2π n=0 n=0 p  where ϕ1 = sin−1 (a/s), ϕ2 = π − sin−1 (a/s), a = σr · 4/L, X1 = min L, b2π/ϕ1 c , X2 =  min L, b2π/ϕ2 c , and the support is S ∈ [a, ∞). 17 Proof. See Appendix D.  Remark. Although Theorem 4 is the conditional distribution of our approximation of S, it provides two clear advantages over the true conditional distribution presented in [16] and over the approximate conditional distribution presented in [14]. First, Theorem 4 offers a simple, closed-form, algebraic expression involving only finite sums, as opposed to the rather complex expression in [16] involving an improper integral of products of scaled Bessel functions. Second, Theorem 4 is remarkably accurate for lower numbers of participating anchor nodes, see Fig. 5. This comes in contrast to the approximate distribution presented in [14], which was derived asymptotically and therefore only accurate for higher numbers of participating anchors. This selective accuracy of Theorem 4 is desirable since a device is more likely to hear lower numbers of participating anchors, especially in infrastructure-based, terrestrial wireless networks.4 E. The Distribution of the Number of Participating Anchors The next step needed to achieve our goal is to find the distribution of the number of participating anchors, fL (`). In this section, we build upon the localizability results from [17] in order to obtain this distribution. Towards this end, we present the relevant theorems from this work and modify them for our use here. Finally, we conclude with a discussion on the applicability of these results. 1) Overview of Localizability Work: Recall from Section I-A that the goal of [17] was to derive an expression for the probability that a mobile can hear at least ` base stations for participation in a localization procedure in a cellular network, i.e., P[L ≥ `]. To derive this expression, the authors assumed that base stations were placed according to a homogeneous PPP, and then examined the SIRs of the base station signals received at a “typical user” placed at the origin.5 Specifically, they examined the SIR of the ` th base station (denoted SIR` ), since this was used directly to determine P[L ≥ `]. Since SIR` depends on the locations of interfering base stations, then the base stations’ placement according to a PPP implies that SIR` becomes a random variable. Consequently, its distribution also becomes a function of the PPP density, λ. Additionally, the authors incorporate 4 By infrastructure-based wireless networks, we mean any wireless network setup with mobile devices, fixed access points, and separate uplink/downlink channels. 5 SIR = Signal to Interference Ratio. Noise is ignored since [17] assumes an interference-limited network. 18 a network loading parameter, q, into SIR` , where 0 ≤ q ≤ 1. This means that any given base station can be considered active (i.e., interfering with base station `’s signal) with probability q. Furthermore, SIR` is also a function of pathloss, α, where α > 2, and the distances of base stations to the target. With SIR` statistically characterized, the authors were able to determine P[L ≥ `] by noting that P[L ≥ `] = P[SIR` ≥ β/γ], where β/γ is the pre-processing SIR threshold for detection of a signal.6 Here, γ is the processing gain at the mobile (assumed to also average out the effect of small scale fading), and β is the post-processing threshold. Thus, since P[L ≥ `] = P[SIR` ≥ β/γ], then P[L ≥ `] must also depend on all of the network parameters described above. We denote this dependency by P[L ≥ ` | α, λ, q, γ, β]. Before continuing to the localizability results, we mention one last caveat regarding the PPP density. That is, when shadowing is present, it can easily be incorporated into the PPP network model through small displacements of the base station locations. This results in a new PPP density, which accounts for this effect of shadowing. This new shadowing-transformed density is given by λ̃ = λ E[Sz2/α ], where Sz is assumed to be a log-normal random variable representing the effect of shadowing on the signal from base station z to the origin [31].7 Thus, by using λ̃, we incorporate shadowing under the log-normal model presented in Section II of [31]. 2) The Localizability Results: In this section, we present the main theorem which will enable us to obtain fL (`). Lemma 5. (Theorem 2, [17]) The probability that a mobile device can hear at least ` base stations for participation in a localization procedure is given by P[L ≥ ` | α, λ̃, q, γ, β] = `−1 © Õ α−2 ­1− e− 2qβ/γ ­ n=0 «  α−2 2qβ/γ n! n  ∫ ∞∫ r`  `−1 −α  ` Õ ª r β 4( λ̃π)   ` ® fΩ (0) + 1 ≥ f (ω)   Ω 2−α −r 2−α ® r   (` − 1)! ω=1 γ 2πq λ̃ 2(ω−1) 2−α ` 1 0 0  r −α + · r 2 −r 2 + α−2 r`  2−α 1 ¬   ` 1 × r1 (r`2 − r12 )ω−1r`2(`−ω)−1 ωe−λ̃πr` dr1 dr`, 2 where ` ∈ {1, 2, . . . }, and Ω is a random variable denoting the number of active participating 6 Note that for q < 1, this equality holds for all but a few rare corner cases. However, the probability of these cases occurring is vanishingly small and thus has little to no impact on the accuracy of the subsequent localizability results [17]. 7 The Sz are assumed to be i.i.d. ∀z. Sz ’s log-normal behavior implies it is distributed normally when expressed in dB [31]. 19 base stations interfering with the ` th . Note, Ω ∼ Binomial(` − 1, q) = fΩ (ω). Additionally, for the trivial case of ` = 0, we define P[L ≥ 0] = 1. (Note: γ and β are in linear terms, not dB.) Remark. This theorem was derived under the following assumptions: (1) a dominant interferer and (2) interference-limited networks. We refer the reader to Section III-D of [17] for further details regarding these assumptions and the consequent derivation of this theorem. 3) The Distribution of L: With these localizability results, we now finally present the distribution of the number of participating anchors. Theorem 6. The pdf of L is given by fL (` | α, λ̃, q, γ, β) = P[L ≥ ` | α, λ̃, q, γ, β] − P[L ≥ ` + 1 | α, λ̃, q, γ, β], where the support is L ∈ {0, 1, 2, . . . } and the probabilities are given by Lemma 5. 4) The Distribution of L with Frequency Reuse: Using Theorem 6, we may obtain another expression for fL (`) which incorporates a frequency reuse parameter, K. This parameter models the ability of base stations to transmit on K separate frequency bands, thereby limiting interference to a per-band basis. This can easily be incorporated into the model by considering K independent PPPs whose densities are that of the original PPP divided by K. Thus, if nk is the number of participating base stations in band k, then the total number of participating base Í stations is given by L = K k=1 n k . Thus, to find P[L = `] under frequency reuse, we simply need to account for all of the per-band combinations of participating base stations such that their sum equals `. This is given in the following corollary, which is a modification of Theorem 3 of [17]. Corollary 6.1. The pdf of L, given a frequency reuse factor of K, is ! K Õ Ö λ̃ fL (` | α, λ̃, q, γ, β, K) = fL nk α, , q, γ, β , K k=1 {nÍ1,...,n K } i ni =` where the multiplicands are given by Thm. 6, K ∈ {1, 2, . . . }, and the support is L ∈ {0, 1, 2, . . . }. Remark. When K = 1, this corollary reduces to Theorem 6. Further, this corollary may be evaluated numerically through the use of a recursive function. 5) Applicability of the Results: Now that we have obtained the pdf of L, we conclude with a brief discussion regarding its applicability. We begin by taking note of the support of fL (`). Whereas up until this section we have proceeded under the assumption that the target is 20 localizable, the support of fL (`) now allows us to consider cases where the target is unlocalizable, i.e., L = 0, 1, 2. Thus, as will be addressed in the following section, we may use these cases to determine the percentage of the network where a target is unlocalizable. Lastly, we note that while the localizability results in [17] (Lemma 5) were presented in the context of cellular networks, these results are actually applicable to any infrastructure-based wireless network using downlink measurements, so long as the distribution parameters are altered accordingly. This implies that the distribution for L (Corollary 6.1) also has this applicability, since it was derived using Lemma 5. Thus, since we use Corollary 6.1, along with a modified Theorem 4, to derive the marginal distribution of S, then this final distribution will also be applicable to any infrastructure-based wireless network employing a TOA localization strategy. F. The Marginal CRLB Distribution In this section, we modify Theorem 4 and combine this with Corollary 6.1 to obtain the marginal distribution of S. First, we state one last network assumption often used in practice: Assumption 5. For a given localization procedure, only a finite number of anchor nodes, N (≥ 3), are ever tasked to transmit localization signals. Remark. Just because N anchors are tasked, does not mean that all N signals are necessarily heard. Under this assumption, and further considering scenarios where the target is unlocalizable, we modify Theorem 4 as follows:       FS (s | L = N, σr ) 0 FS (s | L, σr ) = FS (s | L, σr )     u(s − M)  L≥N L = 3, . . . , N − 1 (15) L = 0, 1, 2 where FS0 is the new modified conditional distribution of S, FS is the previous conditional distribution given in Theorem 4, and M ∈ R+ is a predetermined localization error value used to account for unlocalizable scenarios (described in more detail below). Remark. While Theorem 4 only accounted for scenarios in which the target was localizable, this modified form, however, now accounts for unlocalizable scenarios. For these scenarios, this modified conditional distribution yields a step function, which is a valid cdf and corresponds to a deterministic value for the localization error, i.e., P[S = M | L] = 1. Thus, we account for 21 cases where the target is unlocalizable by assigning an arbitrary localization error value for M, which is chosen to represent cases where there is ambiguity in a target’s position estimate.8 Remark. It is possible that a mobile may hear more anchors than are tasked to perform the localization procedure, i.e., L > N. In this case, the N participating anchors are likely those with the highest received SIR at the target, if connectivity information is known a priori. Therefore, in this scenario, localization performance will only be based on the N anchors tasked. This is clearly reflected in the modified conditional distribution of S in (15). Using this modified Theorem 4 in (15), along with Corollary 6.1, we may now obtain the distribution of localization error for an entire wireless network: Theorem 7. The marginal cdf of the localization performance benchmark, S, is h i FS (s | σr , α, λ̃, q, γ, β, K, M, N) = FS (s | ` = N, σr ) 1 − P[L ≤ N − 1|α, λ̃, q, γ, β, K] + N−1 Õ FS (s | `, σr ) fL (` | α, λ̃, q, γ, β, K) `=3 + u(s − M) P[L ≤ 2|α, λ̃, q, γ, β, K], where FS (s | L = `, σr ) is given by Theorem 4, fL (` | α, λ̃, q, γ, β, K) is given by Corollary 6.1, Íx and P[L ≤ x|α, λ̃, q, γ, β, K] = `=0 fL (` | α, λ̃, q, γ, β, K). Proof. Multiplying the modified conditional distribution of S, given in (15), by the marginal distribution of L from Corollary 6.1 gives the joint distribution of S and L. Then, setting L equal to a particular realization, `, and summing over all realizations, gives us the marginal cdf of S, as desired.  Remark. First, recall that the conditional distribution of our approximation of S, FS (s | L = `, σr ) given by Theorem 4, is accurate for lower ` values. Next, note that: (1) fL (`) declines rapidly as ` increases (an intuitive result, since the probability of hearing many anchor nodes should be small in infrastructure-based wireless networks), and (2) only a maximum of N nodes are tasked to perform a localization procedure (Assumption 5). Thus, these two facts validate the use of our approximation, since the cases where our approximation is less than ideal (i.e., for 8 In this paper, we choose a value for M that applies for all L < 3. Thus, a large enough M allows for a clear distinction between the localizable and unlocalizable portions of the network through a quick examination of the cdf of S. Note however, that one could account for the L = 0, 1, 2 scenarios separately. For example, if L = 1 then in a cellular network one may want to choose M to be the cell radius, since the user equipment typically knows in which cell it is located [32]. 22 large L) will now either be multiplied by fL (`) ≈ 0, or considered invalid in a realistic network under Assumption 5. As a consequence, Theorem 7 will also retain this accuracy. Remark. We conclude by noting that this distribution accounts for localization error over all setups of anchor nodes, numbers of participating anchors, and placements of a target anywhere in the network. Hence, this distribution completely characterizes localization performance throughout an entire wireless network and represents the main contribution of this work. IV. N UMERICAL A NALYSIS In this section, we examine the accuracy of Theorem 7 and investigate how changing network parameters affects localization performance throughout the network. A. Description of Simulation Setup Here, we discuss the parameters that are fixed across all simulations and include a description of how the simulations were conducted. 1) Fixed Parameter Choices and their Effect on Model Assumptions: For these simulations, we consider the case of a cellular network, and thus, we place anchor nodes such that the PPP density matches that of a ubiquitous hexagonal grid with 500m intersite distances (i.e., √
λ = 2/ 3 · 5002 m2 ) [17]. Furthermore, we choose a shadowing standard deviation of 8 dB, which defines our shadowing-transformed density parameter, λ̃. Next, we set our pathloss exponent, α ≈ 4, which was chosen to represent a pathloss similar to that seen in a typical cellular network. Note that this pathloss value is indicative of NLOS range measurements, which are inherently a part of localization in cellular networks. Recall however, that Assumption 3 implied the use of Line-of-Sight (LOS) measurements. Thus, we attempt to mimic NLOS in our simulations by selecting a range error to account for a reasonable delay spread under NLOS conditions. This is described further in the following section. We note that a subject of future work will be to seek a refinement of this model by incorporating NLOS directly into the range measurements. The last parameter that remains fixed across simulations is M. This parameter was chosen to be large enough such that an examination of the cdf of S will reveal which percentage of the network the target is unlocalizable. Towards this end, it was sufficient to choose M ≈ 200 m for our simulations here. Again, the choice of this value is arbitrary and left up to network designers and how they would like to treat the unlocalizable cases. 23 1 1 Increasing Frequency Reuse (K = 1, 2, 3, 6) 0.9 0.8 0.9 0.8 0.7 P[S ≤ abscissa] P[S ≤ abscissa] 0.7 0.6 0.5 0.4 0.3 0.6 Decreasing q (q = 1, 0.75, 0.50, 0.25) 0.5 0.4 0.3 0.2 0.2 Solid: True Dashed: Theorem 7 0.1 Solid: True Dashed: Theorem 7 0.1 0 0 0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 Meters Fig. 6. T HE E FFECT OF F REQUENCY R EUSE. This result 25 30 35 40 45 50 Meters Fig. 7. T HE I MPACT OF D ECREASING N ETWORK L OAD. exposes the large impact that frequency reuse has on local- This plot demonstrates the improvement in localization perfor- ization performance throughout the network. The parameters mance due to a decrease in network loading. The parameters are: N = 10, β = 10 dB, γ = 20 dB, q = 1, and σr = 20 m. were chosen as follows: N = 10, β = 10 dB, γ = 20 dB, The plots’ “piece-wise” appearance is due to L being discrete. K = 2, and σr = 20 m. 2) Conducting the Simulations: The true marginal cdf of S was generated through a simulation over 100,000 positioning scenarios. Each scenario consisted of an average placement of 1,000 anchor nodes, placed according to a homogeneous PPP, with the target located at the origin. Next, the anchor nodes whose SIRs surpassed the detection threshold were deemed to participate in the localization procedure, and their corresponding coordinates were used to calculate the true value of S given by Definition 4. If more than N anchor nodes had signals above the threshold, then the N anchors with highest SIRs were used to calculate S. B. The Effect of Frequency Reuse on the Network-Wide Distribution of Localization Error In this section, we explore how frequency reuse impacts localization performance throughout the network. This simulation, as well as all subsequent simulations, compares the true (simulated) distribution of S with our analytical model from Theorem 7. All parameters were fixed at the levels stated in Fig. 6, while the only parameter varied was the frequency reuse factor, K. The range error standard deviation, σr , was chosen according to the detection threshold, β, and the CRLB of a range estimate (see [6], equ. 3.40), assuming a 10MHz channel bandwidth. Approximately 10m was added to account for a reasonable delay spread in NLOS conditions. From Fig. 6, the most notable impact that frequency reuse has on localization performance is 24 that of localizability. That is, with just a small increase in frequency reuse from K = 1 to K = 2, the portion of the network with which a target is localizable increases from only ≈ 25% to an astonishing ≈ 85%. Furthermore, localization error is also reduced, although the improvement is not as drastic as the increase in localizability. Additionally, as frequency reuse increases, the gains in localizability stop after K = 3, with the gains in localization error also declining after K = 3 as well. Thus, we can conclude that an increase in frequency reuse is strongly advisable if once desires an increase in localization performance within a network, a result which coincides with what has been seen in practice, viz. 3GPP. Lastly, we note the excellent match between the true simulated distribution and our analytically derived distribution given by Theorem 7. We will see that this accuracy of Theorem 7 is retained across all of our results in this section. C. Examining the Effects of Network Loading Here we examine the effect that network loading has on localization performance throughout the network. This is accomplished by varying the percentage of the network, q, actively transmitting (interfering) during a localization procedure. All parameter values other than q were fixed and σr was chosen in the same manner as in the frequency reuse case. From the distributions plotted in Fig. 7, we can see that a decrease in network load leads to an improvement in localizability, as well as an improvement in localization error. However, the improvement is not as pronounced as in the frequency reuse case. Further, examining the 80th percentile for example, it is evident that the rate of improvement in localization error declines as the network load declines as well. Thus, since low network traffic is usually never desirable, a network designer looking to optimize localization performance may find solace in the fact that gains in performance begin to decline as network loading decreases also. D. The Impact of Processing Gain In this section, we examine the effects of changing the processing gain, since it is perhaps the easiest parameter for a network designer to change in practice. (Note that we choose σr here as in the previous simulations.) From Fig. 8, it is evident that as the processing gain increases, there is a corresponding improvement in localizability across the network, as well as an improvement in localization error. As a consequence, there exists a clear trade-off between sacrificing processing time for gains in localization performance. However, it appears that these improvements begin to level off at a processing gain of ≈25 dB. This is promising, as processing gains higher than 1 1 0.9 0.9 0.8 0.8 0.7 0.7 P[S ≤ abscissa] P[S ≤ abscissa] 25 0.6 Increasing Processing Gain (γ = 15 dB, 20 dB, 25 dB) 0.5 0.4 0.3 Increasing σr (σr = 20m, 40m, 60m, 80m) 0.6 0.5 0.4 0.3 0.2 0.2 Solid: True Dashed: Theorem 7 0.1 0 Solid: True Dashed: Theorem 7 0.1 0 0 10 20 30 40 50 60 70 0 Meters Fig. 8. T HE E FFECT OF I NCREASING 20 40 60 80 100 120 140 160 Meters P ROCESSING G AIN. Fig. 9. T HE I MPACT OF I NCREASING R ANGE E RROR. Inject- This figure highlights the trade-off that exists between pro- ing range error results in a predictable effect on localization cessing time and localization performance. The distribution performance throughout the network, yet has no effect on parameter values are: N = 10, β = 5 dB, q = 1, K = 2, and localizability. The parameter values are: N = 10, β = 10 dB, σr = 30 m. γ = 20 dB, K = 3, and q = 1. this can quickly become impractical. Examining the 80th percentile, we can see that a 10 dB increase in processing gain can lead to about a 30m improvement in localization error throughout the network. Thus, increasing the processing gain can be an easily implementable solution for achieving moderate gains in localization performance. E. Range Error and its Impact on Localization Performance within the Network With this last result, we attempt to mimic the effect of an increasing NLOS bias by injecting additional range error into the measurements. In examining Fig. 9, we note that the first σr value, i.e., σr = 20 m, was chosen as in the frequency reuse simulation. Therefore, the subsequent choices represent that of mimicking the impact of increasing NLOS bias. From Fig. 9, we can see that increasing range error has no effect on localizability within the network. This should be clear from the analytical model, since σr does not appear as a parameter in Corollary 6.1. Additionally, injecting more range error into the measurements results in a predictable effect on the distribution of localization performance. This is also evidenced by examining Theorem 4, where σr appears as a scale parameter. Thus, mimicking the effects of NLOS measurements results in a scaling of the distribution of S, implying a predictable reduction in localization performance. 26 V. C ONCLUSION This paper presents a novel parameterized distribution of localization error, applicable throughout an entire wireless network. Invoking a PPP network model, as well as common assumptions for TOA localization, has enabled this distribution to simultaneously account for all possible positioning scenarios within a network. Deriving this result involved two main steps: (1) a derivation of an approximate distribution of our localization performance benchmark, S, conditioned on the number of hearable anchor nodes, L, which yielded a conditional distribution with desirable accuracy and tractability properties, and (2) a modification of results from [17] to attain the distribution of L. Thus, using these two distributions, we arrived at the final, marginal distribution of S, which also retained the same parameterizations as its two component distributions. What followed was a numerical analysis of this network-wide distribution of localization performance. This analysis revealed that our distribution can offer an accurate, baseline tool for network designers, in that it can be used to get a sense of localization performance within a wireless network, while also providing insight into which network parameters to change in order to meet localization requirements. Since this marginal distribution of S is a distribution of the TOA-CRLB throughout the network, it consequently provides a benchmark for describing localization performance in any network employing an unbiased, efficient, TOA-based localization algorithm. The amount of insights that this network-wide distribution of localization error reveals are numerous, and the results presented in this paper have only begun to explore this new paradigm. It is our hope that this work spawns additional research into this new concept, and that future work, such as incorporating NLOS measurements, accounting for collaboration, adding other localization strategies (TDOA/AOA), etc., can further refine the model presented here. In closing, this work presents an initial attempt to provide network designers with a tool for analyzing localization performance throughout a network, freeing them from lengthly simulations by offering an accurate, analytical solution. 27 A PPENDIX A P ROOF OF P ROPOSITION 1 It first helps to visualize the sin2 (·) terms of the sum in (5) on a 2-D grid, where the i’s represent the rows and the j’s represent the columns. As an example, the case of L = 4 gives j=1 j=2 i=1 j=3 j=4 ∠1 + ∠2 ∠1 + ∠2 + ∠3 ∠1 i=2 ∠2 + ∠3 ∠2 i=3 ∠3 where just the arguments of the sin2 (·) terms are represented in the grid for clarity. From this arrangement, it is evident that the sum in (5) represents the process of summing each row sequentially, starting at i = 1. Now, however, we choose to sum the terms diagonally, starting with the lowest diagonal and working our way upward. This yields D= L−1 Õ L−i Õ i=1 j=1 sin2 j+i−1 Õ ! ∠k . (16) k= j Considering the cases i = 1 and i = L − 1 separately, we may rewrite (16) as ! ! j+i−1 L−1 L−1 L−2 Õ L−i Õ Õ Õ
Õ 2 2 2 ∠ k + sin ∠k . D= sin ∠ k + sin i=2 j=1 k=1 k= j k=1 Next, note that 2 sin L−1 Õ ! ∠ k = sin 2 2π − L−1 Õ !

∠ k = sin2 2π − (Θ(L) − Θ(1) ) = sin2 ∠ L , (17) k=1 k=1 where the last two equalities follow from Definition 3. Hence, D= L Õ k=1 sin2 ∠ k +
L−2 Õ L−i Õ sin2 i=2 j=1 j+i−1 Õ ! ∠k . k= j Lastly, we complete the proof by noting that the first sum may be equivalently expressed by replacing the internodal angles with their order statistics. A PPENDIX B P ROOF OF C OROLLARY 2.1 Using Lemma 2 and our assumption of a finite L, the pdf of ∠(L−1) conditioned on L is ! ! L−2   X Õ d L n(n − 1)(L − 1) nϕ f∠(L−1) (ϕ|L) = F∠ (ϕ|L) = (−1)n 1− , dϕ (L−1) n 2π 2π n=0 28  where X = min L, b2π/ϕc and the support is 0 ≤ ∠(L−1) ≤ π, as in Lemma 2. Next, note that the lower summation limit may be rewritten as n = 2 and that the upper summation limit, X, can be simplified to just L by appending an indicator function to the summand. This gives the logically equivalent expression: ! ! L−2 " #   L Õ n(n − 1)(L − 1) nϕ 2π n L f∠(L−1) (ϕ|L) = (−1) 1− 1 ϕ≤ . n 2π 2π n n=2 Using the above expression for the pdf, the expectation is derived as follows: ∫ π E[∠(L−1) |L] = ϕ f∠(L−1) (ϕ|L) dϕ 0 ! ! L−2   L n(n − 1)(L − 1) nϕ ϕ (−1) 1− dϕ = n 2π 2π 0 n=2 ! ∫ 2π ! L−2   L Õ n nϕ n(n − 1)(L − 1) n L = (−1) ϕ 1− dϕ n 2π 2π 0 n=2 !   L Õ L 2π(n − 1) = (−1)n , nL n n=2 ∫ 2π n L Õ n (18) (19) (20) where (18) follows from absorbing the indicator function into the integration limits since n ≥ 2, (19) follows from our assumption of a finite L, and (20) is derived through integration by parts. A PPENDIX C P ROOF OF L EMMA 3 This is a straightforward application of the lowest Geometric Dilution Of Precision (GDOP) presented in [30]. First, since L is finite, then D is a continuous real function defined on a compact subset of R L . Thus, its maximum must exist. Call it dmax . Next, under our Assumptions 2, 3, and 4, GDOP, presented in [27], can be written as GDOP = p L/D, where D follows from the lemma assumption. Further, under our assumptions, [30] asserts √ that the lowest GDOP is given by GDOPmin = 2/ L. Since GDOPmin must occur when D is at p √ its maximum, we have that 2/ L = L/dmax , and the lemma follows. A PPENDIX D P ROOF OF T HEOREM 4 Under Approximation 1, we have S = a/sin(∠(L−1) ), where a is defined in Theorem 4. To determine the support of S conditioned on L, we know from Lemma 2 that if 0 ≤ ∠(L−1) ≤ π, 29 then the support for the RV sin(∠(L−1) ) must be 0 ≤ sin(∠(L−1) ) ≤ 1. From here, we see that 0 ≤ sin(∠(L−1) ) ≤ 1 ⇒ 1≤ 1 sin(∠(L−1) ) ⇒ a≤ a sin(∠(L−1) ) ⇒ a ≤ S, and hence, S ∈ [a, ∞). Next, to find the cdf of S conditioned on L, consider the following FS (s | L, σr ) = P[S ≤ s | L, σr ] " a ≤s =P sin(∠(L−1) ) " = P ∠(L−1) ≤ π−sin−1 # (21) L a s ! # " L − P ∠(L−1) ≤ sin−1 a s ! # L = P[∠(L−1) ≤ ϕ2 | L] − P[∠(L−1) ≤ ϕ1 | L]

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Solving Markov decision processes for network-level post-hazard recovery via simulation optimization and rollout (Invited Paper) arXiv:1803.04144v1 [math.OC] 12 Mar 2018 Yugandhar Sarkale1 , Saeed Nozhati2 , Edwin K. P. Chong1 , Bruce R. Ellingwood2 , Hussam Mahmoud2 Abstract— Computation of optimal recovery decisions for community resilience assurance post-hazard is a combinatorial decision-making problem under uncertainty. It involves solving a large-scale optimization problem, which is significantly aggravated by the introduction of uncertainty. In this paper, we draw upon established tools from multiple research communities to provide an effective solution to this challenging problem. We provide a stochastic model of damage to the water network (WN) within a testbed community following a severe earthquake and compute near-optimal recovery actions for restoration of the water network. We formulate this stochastic decisionmaking problem as a Markov Decision Process (MDP), and solve it using a popular class of heuristic algorithms known as rollout. A simulation-based representation of MDPs is utilized in conjunction with rollout and the Optimal Computing Budget Allocation (OCBA) algorithm to address the resulting stochastic simulation optimization problem. Our method employs nonmyopic planning with efficient use of simulation budget. We show, through simulation results, that rollout fused with OCBA performs competitively with respect to rollout with total equal allocation (TEA) at a meagre simulation budget of 5-10% of rollout with TEA, which is a crucial step towards addressing large-scale community recovery problems following natural disasters. I. INTRODUCTION Natural disasters have a significant impact on the economic, social, and cultural fabric of affected communities. Moreover, because of the interconnected nature of communities in the modern world, the adverse impact is no longer restricted to the locally affected region, but it has ramifications on national or international scale. Among other factors, the occurrence of such natural disasters is on the rise owing to population growth and economic development in hazard-prone areas [1]. Keeping in view the increased frequency of natural disasters, there is an urgent need to address the problem of community recovery post-hazard. Typically, the resources available to post-disaster planners are limited and relatively small compared to the impact of 1 Yugandhar Sarkale and Edwin K. P. Chong are with Department of Electrical and Computer Engineering, Colorado State University, Fort Collins, CO 80523-1373, USA Yugandhar.Sarkale, [email protected] 2 Saeed Nozhati, Bruce R. Ellingwood and Hussam Mahmoud are with the Department of Civil and Environmental Engineering, Colorado State University, Fort Collins, CO 805231372, USA Saeed.Nozhati, Bruce.Ellingwood, [email protected] This work was supported by the National Science Foundation under Grant CMMI-1638284. This support is gratefully acknowledged. Any opinions, findings, conclusions, or recommendations presented in this material are solely those of the authors and do not necessarily reflect the views of the National Science Foundation. the damage. Under these scenarios, it becomes imperative to assign limited resources to various damaged components in the network optimally to support community recovery. Such an assignment must also consider multiple objectives and cascading effects due to the interconnectedness of various networks within the community and must also successfully adopt previous proven methods and practices developed by expert disaster management planners. Holistic approaches addressing various uncertainties for networklevel management of limited resources must be developed for maximum effect. Civil infrastructure systems, including power, transportation, and water networks, play a critical part in post-disaster recovery management. In this study, we focus on one such critical infrastructure system, namely the water networks (WN), and compute near-optimal recovery actions, in the aftermath of an earthquake, for the WN of a test-bed community. Markov decision processes (MDPs) offer a convenient framework for representation and solution of stochastic decision-making problems. Exact solutions are intractable for problems of even modest size; therefore, approximate solution methods have to be employed. We can leverage the rich theory of MDPs to model recovery action optimization for large state-space decision-making problems such as our. In this study, we employ a simulation-based representation and solution of MDP. The near-optimal solutions are computed using an approximate solution technique known as rollout. Even though state-of-the-art hardware and software practices are used to implement the solution to our problem, we are faced with the additional dilemma of computing recovery actions on a fixed simulation budget without affecting the solution performance. Therefore, any prospective methodology must incorporate such a limitation in its solution process. We incorporate the Optimal Computing Budget Allocation (OCBA) algorithm into our MDP solution process [2], [3] to address the limited simulation budget problem. II. TESTBED CASE STUDY A. Network Characterization This study considers the potable water network (WN) of Gilroy, CA, USA as an example to illustrate the proposed methodology. Gilroy, located 50 kilometers (km) south of the city of San Jose, CA is approximately 41.91 km2 in area, with a population of 48,821 [4]. We divide our study area into 36 grid regions to define the properties of infrastructure systems, household units, and the population. Our model of the community maintains adequate detail to study the Fig. 1. The modeled Water Network of Gilroy performance of the WN at a community level under severe earthquakes. The potable water of Gilroy is provided only by the Llgas sub-basin [5]. The potable water wells, located in wood-frame buildings, pump water into the distribution system. The Gilroy municipal water pipelines range from 102 mm to 610 mm in diameter [5]. In this study, a simplified aggregated model of WN of Gilroy adopted from [5] is modeled. This model shown in Fig. 1, includes six water wells, two booster pump stations (BPS), three water tanks (WT), and the main pipelines. the water main, and G(·) is the moment-generating function of εPGV (the residual of the PGV). The term C for water pipe segment i is C = K × 0.00187 × PGVi , where K is a coefficient determined by the pipe material, diameter, joint type, and soil condition based on the guidelines prepared by the American Lifeline Alliance [9]. Adachi and Ellingwood [8] demonstrated that the Upper Bound (UB) and exact solutions (1) are close enough so that in practical applications the UB assessment (conservative evaluation) can be used. Repair crews, replacement components, and tools are considered as available units of resources to restore the damaged components of WN following the hazard. One unit of resources is required to repair each damaged component [10], [11]. However, the available units of resources are limited and depend on the capacities and policy of the entities within the community. To restore the WN, the restoration curves based on exponential distributions synthesized from HAZUS-MH [7] are used, as summarized in Table I. The pipe-restoration time in the WN is based on repair rate or number of repairs per kilometer. TABLE I T HE EXPECTED REPAIR TIMES (U NIT: DAYS ) Component Water tanks Wells Pumping plants Damage States Minor 1.2 0.8 0.9 Moderate 3.1 1.5 3.1 Extensive 93 10.5 13.5 Complete 155 26 35 B. Seismic Hazard Simulation The San Andreas Fault (SAF), which is near Gilroy, is a source of severe earthquakes. In this study, we assume that a seismic event of moment magnitude Mw = 6.9 occurs at one of the closest points on the SAF projection to downtown Gilroy with an epicentral distance of approximately 12 km. Ground motion prediction equations (GMPE) determine the conditional probability of exceeding ground motion intensity at specific geographic locations within Gilroy for this earthquake. The Abrahamson et al. [6] GMPE is used to estimate the Intensity Measures (IM) and associated uncertainties. Peak Ground Acceleration (PGA) is considered for the above-ground WN facilities and wells, whereas Peak Ground Velocity (PGV) is considered as IM of pipelines. C. Fragility and Restoration Assessment of Water Network The physical damage to WN components can be assessed by seismic fragility curves. We use the fragility curves presented in HAZUS-MH [7] for wells, water tanks, and pump stations based on the IM of PGA. This study adopts the assumptions in [8] for water pipelines. The failure probability of a pipe is bounded as follows: 1 − GεPGV (−CLµPGV ) ≤ E[Pf ] ≤ 1 − E[exp(−CLµPGV )] (1) where Pf is the failure probability of a pipe, L is the length of pipe, µPGV is the average PGV for the entire length of III. PROBLEM DESCRIPTION AND SOLUTION A. MDP Framework We provide a brief description of MDP [12] for the sake of completeness. An MDP is a controlled dynamical process useful in modelling of wide range of decision-making problems. It can be represented by the 4-tuple hS, A, T, Ri. Here, S represents the set of states, and A represents the set of actions. Let s, s0 ∈ S and a ∈ A; then T is the state transition function, where T (s, a, s0 ) = P(s0 | s, a) is the probability of going into state s0 after taking action a in state s. R is the reward function, where R(s, a, s0 ) is the reward received after transitioning from s0 to s as a result of action a. In this study, we assume that |S| and |A| are finite. R is bounded and realvalued and a deterministic function of s, a and s0 . Implicit in our presentation are also the following assumptions: First order Markovian dynamics (history independence), stationary dynamics (reward function is not a function of absolute time), and full observability of the state space (outcome of an action in a state might be random, but we know the state reached after action is completed). In our study, we assume that we are allowed to take recovery actions (decisions) indefinitely until all the damaged components of our modelled problem are repaired (infinite-horizon planning). In this setting, we have a stationary policy π, which is defined as π : S → A. Let’s say that decisions are made at discrete-time t; then π(s) is the action to be taken in state s (regardless of time t). Our objective is to find an optimal policy π ∗ . For the infinite-horizon case, π ∗ is defined as C. Simulation-Based Representation of MDP π ∗ = arg max V π (s0 ), (2) π where " π V (s0 ) = E ∞ ∑γ # t R(st , π(st ), st+1 ) (3) t=0 is called the value function for a fixed policy π, and 0 < γ < 1 is the discount factor. Note that the optimal policy is independent of the initial state s0 . Also, note that we maximize over policies π, where at each time t the action taken is at = π(st ). Stationary optimal policies are guaranteed to exist for discounted infinite-horizon optimization criteria [13]. To summarize, our presentation is for infinite-horizon discretetime MDPs with the discounted value as our optimization criterion. We now briefly explain the simulation-based representation of an MDP [17]. Such a representation serves well for large state and action spaces, which is a characteristic feature of many real-world problems. When |S| or |A| is large, it is not feasible to represent T and R in a matrix form. A simulation-based representation of an MDP is a 5tuple hS, A, R, T, Ii, where S and A are as before, except |S| and |A| are large. Here, R is a stochastic real-valued bounded function that stochastically returns a reward r when input s and a are provided, where a is the action applied in state s. T is a simulator that stochastically returns a state s0 when state s and action a are provided as inputs. I is the stochastic initial state function that stochastically returns a state according to some initial state distribution. R, T , and I can be thought of as any callable library functions that can be implemented in any programming language. B. MDP Solution A solution to an MDP is the optimal policy π ∗ . We can obtain π ∗ with linear programming or dynamic programming. In the dynamic programming regime, there are several solution strategies, namely value iteration, policy iteration, modified policy iteration, etc. Unfortunately, such exact solution algorithms are intractable for large state and actions spaces. We briefly mention here the method of value iteration because it illustrates the Bellman’s equation [14]. Studying Bellman’s equation is useful for defining Q value function. Q value function will play a critical role in describing the ∗ rollout algorithm. Let V π denote the optimal value function ∗ for some π ∗ ; Bellman showed that V π satisfies: ( ) h ∗ i ∗ V π (s) = max γ · ∑ P(s0 | s, a) · V π (s0 ) + R(s, a, s0 ) . a∈A(s) s0 (4) Equation (4) is known as the Bellman’s optimality equation, where A(s) is the set of possible actions in any state s. The value iteration algorithm solves (4) by using Bellman backup repeatedly, where Bellman backup is given by: ( )   Vi+1 (s) = max γ ∑ P(s0 | s, a) · Vi (s0 ) + R(s, a, s0 ) . (5) a∈A(s) D. Problem Formulation After an earthquake event occurs, the components of the water network remain undamaged or exhibit one of the damage states as shown in Table I. Let L0 be the total number of damaged component at t. Let tc represent the decision time when all components are repaired. There is a fixed number of resource units (M) available to the decision maker. At each discrete time t, the decision maker has to decide the assignment of unit of resource to the damaged locations; each component cannot be assigned more than one resource unit. When the number of undamaged locations is greater than the number of units of resources (because of sequential application of repair actions, or otherwise), we retire the extra unit of resources so that M is less than or equal to the total damaged locations. • • s0 ∗ Bellman showed that limi→∞ Vi = V π , where V0 is initialised arbitrarily.1 Next, we define the Q value function of policy π:   Qπ (s, a) = γ · ∑ P(s0 | s, a) · V π (s0 ) + R(s, a, s0 ) . (6) s0 The Q value function of any policy π gives the expected discount reward in the future after starting in some state s, taking action a and following policy π thereafter. Note that this is the inner term in (4). 1 On a historical note, Lloyd Shapely’s paper [15] included the value iteration algorithm for MDPs as a special case, but this was recognised only later on [16]. • States S: Let st be the state of the damaged components of the system at time t; then st is a vector of length L0 , 0 st = (st1 , . . . , stL ), and stl is one of the damaged state in Table I where l ∈ {1, . . . , L0 }. Actions A: Let at denote the repair action to be carried out at time t. Then, at is a vector of length L0 , at = 0 (at1 , . . . , atL ), and atl ∈ {0, 1} ∀l,t. When atl = 0, no repair work is to be carried out at l. Similarly, when atl = 1, repair work is carried out at l. Simulator T: The repair time associated with each damaged location depends on the state of the damage to the component at that location (see Table I). This repair time is random and is exponentially distributed with expected repair times shown in Table I. Given st and at , T gives us the new state st+1 . We say that a repair action is complete as soon as at least one of the locations where repair work is carried out is fully repaired. Let’s denote this completion time at every t by tˆ. Note that it is possible for the repair work at two or more damaged locations to be completed simultaneously. Once the repair action is complete, the units of resources at remaining locations, where repair work was not complete, are also available for reassignment • • • along with unit of resources where repair was complete. The new repair time at such unrepaired locations is calculated by subtracting tˆ from the time required to repair these locations. It is also possible to reassign the unit of resource at the same unrepaired location if it is deemed important for the repair work to be continued at that location by the planner. Because of this reason, preemption of repair work during reassignment is not a restrictive assumption, on the contrary, it allows greater flexibility to the decision maker for planning. Because the repair times are random, the outcomes of repair actions are random as not the same damaged component will be repaired first every time (random repair time) even if the same repair action at is applied in st . Hence, our simulator T is stochastic. Alternative formulation where outcome of repair action is deterministic is also an active area of research [18], [19]. Rewards R: We wish to optimally plan decisions so that maximum people will get water in minimum amount of time. We combine these two competing objectives to define our reward as: r , (7) R(st , at , st+1 ) = trep where r is the number of people who have water after action at is completed, and trep is the total repair time (days) required to reach st+1 from any initial state s0 . Note that our reward function is stochastic because the outcome of our action at is random. Initial State I: We have already described the stochastic damage model of the components for the modeled network in Section II-B and Section II-C. The initial damage states associated with the components will be provided by these models. Discount factor γ: In our simulation studies, γ is fixed at 0.99. E. Rollout The rollout algorithm was first proposed for stochastic scheduling problems by Bertsekas and Castanon [20]. Instead of the dynamic programming formalism [20], we motivate the rollout algorithm in relation to the simulation-based representation of our MDP. Suppose that we have access to a non-optimal policy π, and our aim is to compute an improved policy π 0 . Then, we have: π 0 (st ) = arg max Qπ (st , at ), at (8) where the Q function is as defined in (6). If the policy defined in (8) π 0 is non-optimal, it is a strict improvement over π [13]. This result is termed as policy improvement theorem. Note that the improved policy π 0 is generated as a greedy policy w.r.t. Qπ . Unlike the exact solution methods described in Section III-B, we are interested here in computing π 0 only for the current state. Methods that use (8) as the basis for updating the policy suffer from the curse of dimensionality. Before performing the policy improvement step in (8), we have to first calculate the value of Qπ . Calculating the value of Qπ in (8) is known as policy evaluation. Policy evaluation is intractable for large or continuous state and action spaces. Approximation techniques alleviate this problem by calculating an approximate Q value function. Rollout is one such approximation technique that utilises montecarlo simulations. Particularly, rollout can be formulated as an approximate policy iteration algorithm [17], [21]. An implementable (programming sense) stochastic function (simulator) SimQ(st , at , π, h) is defined in such a way that its expected value is Qπ (st , at , h), where h is a finite number representing horizon length. In the rollout algorithm, SimQ is implemented by simulating action at in state st and following π thereafter for h − 1 steps. This is done for all the actions at ∈ A(st ). A finite horizon approximation of Qπ (st , at ) (termed as Qπ (st , at , h)) is required; our simulation would never finish in the infinite horizon case because we would have to follow policy π indefinitely. However, V π (st ), and consequently Qπ (st , at ), is defined over the infinite horizon. It is easy to show the following: |Qπ (st , at ) − Qπ (st , at , h)| = γ h Rmax . 1−γ (9) The approximation error in (9) reduces exponentially fast as h grows. Therefore, the h-horizon results apply to the infinite horizon setting, for we can always choose h such that the error in (9) is negligible. To summarize, the rollout algorithm can be presented in the following fashion for our problem: Algorithm 1 Uniform Rollout (π,h,α,st ) for i = 1 to n do for j = 1 to α do ãi, j ← SimQ(st , ati, j , π, h) . See algorithm 2 end for end for ãti ← Mean(ãi, j ) k ← arg max ãti return atk Algorithm 2 Simulator SimQ(st , ati, j , π, h) st+1 ← T (st , ati, j ) r ← R(st , ati, j , st+1 ) for p = 1 to h − 1 do st+1+p ← T (st+p , π(st+p )) r ← r + γ p R(st+p , π(st+p ), st+1+p ) end for return r In Algorithm 1, n denotes |A(st )|. Note that Algorithm 2 returns the discounted sum of rewards. When h = tc , we term the rollout as complete rollout, and when h < tc , the rollout is called truncated rollout [20]. It is possible to analyse the performance of uniform rollout in terms of uniform allocation α and horizon depth h [17], [22]. F. Optimal Computing Budget Allocation In the previous section, we presented the rollout method for solving our MDP problem. In the case of uniform rollout, we allocate a fixed rollout sampling budget α to each action, i.e., we obtain α number of rollout samples per candidate action to estimate the Q value associated with the action. In the simulation optimization community, this is analogous to total equal allocation (TEA) [23] with a fixed budget α for each simulation experiment (a single simulation experiment is equivalent to one rollout sample). In practice, we are only interested in the best possible action, and we would like to direct our search towards the most promising candidates. Also, for large real-world problems, the simulation budget available is insufficient to allocate α number of rollout samples per action. We would like to get a rough estimate of the performance of each action and spend the remaining simulation budget in refining the accuracy of the best estimates. This is the classic exploration vs. exploitation problem faced in optimal learning and simulation optimization problems. Instead of a uniform allocation α for each action, nonuniform allocation methods have been explored in the literature pertaining to Algorithm 1 called as adaptive rollout [24]. An analysis of performance guarantees for adaptive rollout remains an active area of research [24]–[26]. These nonuniform allocation methods guarantee performance without a constraint on the budget of rollouts. Hence, we explore an alternative non-uniform allocation method that would not only fuse well into our solutions (adaptively guiding the stochastic search) but would also incorporate the constraint of simulation budget in its allocation procedure. Numerous techniques have been proposed in the simulation optimization community to solve this problem. We draw upon one of the best performers [27] which naturally fits into our solution framework—OCBA. Moreover, the probability of correct selection of an alternative in OCBA mimics finding the best candidate action at each stage in Algorithm 1. Formally, the OCBA problem [28] for Section III-D can be stated as : n max P{CS} such that ∑ Ni = B, N1 ,...,Nn (10) i=1 where B represents the simulation budget for determining optimal at for st at any t. At each OCBA allocation step, barring the best alternative, the OCBA solution assigns an allocation that is directly proportional to the variance of each alternative and inversely proportional to the squared difference between the mean of that alternative and the best alternative. Here, we only provide information required to initialize the OCBA algorithm. For a detailed description of OCBA, including the solution to the problem in (10), see [28]. The key initialization variables, for the OCBA algorithm [28], are k, T (not to be confused with T in this paper), ∆, and n0 . The variable k is equal to variable n in our problem. The value of n changes at each t and depends on the number of damaged components and units of resources. The variable T is equal to per-stage budget B in our problem. More information about Fig. 2. Performance comparison of rollout vs base policy for 3 units of resources. the exact value assigned to B is described in Section IV. We follow the guidelines specified in [29] to select n0 and ∆; n0 in the OCBA algorithm is selected equal to 5, and ∆ is kept at 15% of n (within rounding). IV. SIMULATION RESULTS We simulate 100 different initial damage scenarios for each of the plots presented in this section. There will be a distinct recovery path for each of the initial damage scenarios. All the plots presented here represent the average of 100 such recovery paths. Two different simulation plots of rollout fused with OCBA are provided in Fig. 2 and Fig. 3. They are termed as rollout with OCBA1 and rollout with OCBA2. The method applied is the same for both cases; only the per-stage simulation budget is different. A per-stage budget (budget at each decision time t) of B = 5 · n + 5000 is assigned for rollout with OCBA1 and B = 5 · n + 10000 for rollout with OCBA2. Fig. 2 compares the performance of rollout fused with OCBA and base policy. The rollout algorithm is known to have the “lookahead property” [20]. This behavior of the rollout algorithm is evident in the results in Fig. 2, where the base policy initially outperforms the rollout policy, but after about six days the former steadily outperforms the later. Recall, that our objective is to perform repair actions so that maximum people will have water in minimum amount of time. Evaluating the performance of our method in meeting this objective is equivalent to checking the area under the curve of our plots. This area represents the product of the number of people who have water and the number of days for which they have water. A larger area represents that greater number of people were benefitted as a result of the recovery actions. The area under the curve for recovery with rollout (blue and red plots) is more than its base counterpart (black). A per-stage budget increase of 5000 simulations in rollout with OCBA2 with respect to rollout with OCBA1 shows improvements in the recovery process. In the plots shown in Fig. 3, we use M = 5. In the initial phase of planning, it might appear that the base policy outperforms the rollout for a substantial amount of time. However, this is not the case. Note that the number of days for which the base policy outperforms rollout, in Fig. 3. Performance comparison of rollout vs base policy for 5 unit of resources. rollout. Therefore, these two algorithms form a powerful combination together, where each algorithm consistently and sequentially reinforces the performance of the other. Such synergistic behavior of the combined approach is appealing. Lastly, our simulation studies show that increments in the simulation budget of rollout results in marginal performance improvement for each increment. Beyond a certain increment in the simulation budget, the gain in performance might not scale with the simulation budget expended. A possible explanation is that small simulation budget increase might not dramatically change the approximation of Q value function associated with a state-action pair. Thus, π 0 in (8) might not show a drastic improvement compared to the one computed by a lower simulation budget (policy improvement based on Q approximation that utilises lower simulation budget). V. F UTURE W ORK Fig. 4. Performance comparison of uniform rollout (TEA), rollout with OCBA and base policy for 3 units of resources. both Fig. 2 and Fig. 3, is about six days, but because the number of resource units has increased from three to five, the recovery is faster, giving an illusion that the base policy outperforms rollout for a longer duration. It was verified that the area under the curve for recovery with rollout (blue and red curves) is more than its base counterpart (black curve). Because OCBA is fused with rollout here, we would like to ascertain the exact contribution of the OCBA approach in enhancing the rollout performance. For the rollout with OCBA in Fig. 4, B = 5 · n + 20000, whereas α = 200 for uniform rollout. The recovery as a result of these algorithms outperforms the base policy recovery in all cases. Also, rollout with OCBA performs competitively with respect to uniform rollout despite a meagre simulation budget of 10% of uniform rollout. The area under the recovery process in Fig. 4, as a result of uniform rollout, is only marginally greater than that due to rollout with OCBA. Note that after six days, OCBA slightly outperforms uniform rollout because it prioritizes the simulation budget on the most promising actions per-stage. Rollout exploits this behavior in each stage and gives a set of sequential recovery decisions that further enhances the outcome of the recovery decisions. We would like to once again stress that such an improvement is being achieved at a significantly low simulation budget with respect to uniform For future work, we would like to leverage the availability of multiple base polices in the aftermath of hazards in our framework and incorporate parallel rollout in the solution method [30]. We anticipate further improvements to the performance demonstrated here when OCBA is fused with parallel rollout. 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Ultrafast photonic reinforcement learning based on laser chaos Makoto Naruse1, Yuta Terashima2, Atsushi Uchida2 & Song-Ju Kim3 1 Strategic Planning Department, National Institute of Information and Communications Technology, 4-2-1 Nukui-kita, Koganei, Tokyo 184-8795, Japan 2 Department of Information and Computer Sciences, Saitama University, 255 Shimo-Okubo, Sakura-ku, Saitama city, Saitama 338-8570, Japan 3 WPI Center for Materials Nanoarchitectonics, National Institute for Materials Science, 1-1 Namiki, Tsukuba, Ibaraki 305-0044, Japan * Corresponding author. Email: [email protected] 1 ABSTRACT Reinforcement learning involves decision making in dynamic and uncertain environments, and constitutes one important element of artificial intelligence (AI). In this paper, we experimentally demonstrate that the ultrafast chaotic oscillatory dynamics of lasers efficiently solve the multi-armed bandit problem (MAB), which requires decision making concerning a class of difficult trade-offs called the explorationexploitation dilemma. To solve the MAB, a certain degree of randomness is required for exploration purposes. However, pseudo-random numbers generated using conventional electronic circuitry encounter severe limitations in terms of their data rate and the quality of randomness due to their algorithmic foundations. We generate laser chaos signals using a semiconductor laser sampled at a maximum rate of 100 GSample/s, and combine it with a simple decision-making principle called tug-of-war with a variable threshold, to ensure ultrafast, adaptive and accurate decision making at a maximum adaptation speed of 1 GHz. We found that decision-making performance was maximized with an optimal sampling interval, and we highlight the exact coincidence between the negative autocorrelation inherent in laser chaos and decision-making performance. This study paves the way for a new realm of ultrafast photonics in the age of AI, where the ultrahigh bandwidth of photons can provide new value. 2 INTRODUCTION Physical unique attributes of photons have been utilized in information processing in the literature of optical computing1. New photonic processing principles have recently emerged to solve complex time-series prediction problems2-4, and issues in spatiotemporal dynamics5 and combinatorial optimization6, which coincide with the rapid shift to the age of artificial intelligence (AI). These novel approaches exploit the ultrahigh bandwidth attributes of photons and their enabling device technologies2,3,6. This paper experimentally demonstrates the usefulness of ultrafast chaotic oscillatory dynamics in semiconductor lasers for reinforcement learning, which is among the most important elements in machine learning. Reinforcement learning involves adequate decision making in dynamic and uncertain environments7. It forms the foundation of a variety of applications, such as information infrastructures8, online advertisements9, robotics10, transportation11, and Monte Carlo tree search12, which is used in computer gaming13. A fundamental of reinforcement learning is known as the multiarmed bandit problem (MAB), where the goal is to maximize total reward from multiple slot machines, the reward probabilities of which are unknown7,14,15. To solve the MAB, one needs to explore better slot machines. However, too much exploration may result in excessive loss, whereas too quick a decision, or insufficient exploration, may lead to neglect of the best machine. There is a trade-off, referred to as the explorationexploitation dilemma7. A variety of algorithms for solving 3 the MAB have been proposed in the literature, such as -greedy14, softmax16, and upper confidence bound17. These approaches typically involve probabilistic attributes, especially for exploration purposes. While the implementation and improvements of such algorithms on conventional digital computing are important for various practical applications, understanding their limitations and investigating novel approaches are also important based on perspectives from postsilicon computing. For example, the pseudo-random number generation (RNG) used in conventional algorithmic approaches has severe limitations, such as its data rate, due to the operating frequencies of digital processors (~ GHz range). Moreover, the quality of randomness in RNG has serious limitations18. The usefulness of photonic random processes for machine learning is also discussed by utilizing multiple optical scattering19. We consider that directly utilizing physical irregular processes in nature is an exciting approach with the goal of realizing artificially constructed, physical decision-making machines20. Indeed, the intelligence of slime moulds or amoebae, a single-cell natural organism, has been used in solution searches, whereby complex inter-cellular spatiotemporal dynamics play a key role21. This stimulated the subsequent discovery of a new principle of decision making strategy called tug of war (TOW), invented by Kim et al.22,23. The principle of the TOW method originated from the observation of slime moulds: the dynamic expanding and shrinking of their bodies while maintaining a constant intracellular resource volume allows them to collect environmental information, and the conservation 4 of the volume of their bodies entails a nonlocal correlation within the body. The fluctuation, or probabilistic behaviour, in the body of amoebae is important for the exploration of better solutions. The name “TOW” is a metaphor to represent such a nonlocal correlation while accommodating fluctuation, which enhances decision-making performance23. This principle can be adapted to photonic processes. In past research, we experimentally exhibited physical decision making based on near-field-mediated optical excitation transfer at the nanoscale20,24 and with single photons25. These former studies pursued the ultimate physical attributes of photons in terms of diffraction limit-free spatial resolutions and energy efficiency by near-field photons26,27, and the quantum attributes of single-light quanta28. The nonlocal aspect of TOW is directly physically represented by the wave nature of a single photon or an exciton polariton, whereas fluctuation is also directly represented by their intrinsic probabilistic attributes. However, the fluctuations are limited by practical limitations on the measurements and control systems (second range in the worst case) as well as the single-photon generation rate (kHz range). The ultrafast, high-bandwidth aspect of photons is another promising physical platform for TOW-based decision making to complement diffraction-limit-free and low-energy near-field photon approaches as well as quantum-level single-photon strategies. As demonstrated below, chaotic oscillatory dynamics of lasers that contains negative autocorrelation experimentally demonstrates 1 GHz decision making. In addition to the resultant speed merit, it should be emphasized that the technological maturity of ultrafast photonic devices allows for relatively easy and scalable system 5 implementation through commercially available photonic devices. Furthermore, the applications of the proposed ultrafast photonics-based, and the former near-field/single-photon-based, decision making are complementary: the former targets high-end, data centre scenarios by highlighting ultrafast performance, whereas the latter appeals to low-energy, Internet-of-Things-related (IoT)29, and security30 applications. In this study, we demonstrate ultrafast reinforcement learning based on chaotic oscillatory dynamics in semiconductor lasers31-34 that yields adaptation from zero-prior knowledge at a gigahertz (GHz) range. The randomness is based on complex dynamics in lasers32-34, and its resulting speed is unachievable in other mechanisms, at least through technologically reasonable means. We experimentally show that ultrafast photonics has significant potential for reinforcement learning. The proposed principles using ultrafast temporal dynamics can be matched to applications including an arbitration of resources at data centres35 and high-frequency trading36, where decision making is required at least within milliseconds, and other such high-end utilities. Scientifically, this study paves a way toward the understandings of the physical origin of the enhancement of intelligent abilities (which is reinforcement learning herein) when natural processes (laser chaos herein) are coupled with external systems; this is what we call natural intelligence. Chaotic dynamics in lasers have been examined in the literature32-34, and their applications have exploited the ultrafast attributes of photonics for secure communication37-39, random number generation31,40,41, remote sensing42, and reservoir computing2-4. Reservoir computing is a type of 6 neural network similar to deep learning13 that has been intensively studied to provide recognitionand prediction-related functionalities. The reinforcement learning described in this study differs completely from reservoir computing from the perspective that neither a virtual network nor machine learning for output weights is required. However, it should be noted that reinforcement learning is important in complementing the capabilities of neural networks, indicating the potential for the fusion of photonic reservoir computing with photonic reinforcement learning in future work. Principle of reinforcement learning For the simplest case that preserves the essence of solving the MAB, we consider a player who selects one of two slot machines, called slot machines 1 and 2 hereafter, with the goal of maximizing reward (known as the two-armed bandit problem). Denoting the reward probabilities of the slot machines by Pi (i  1, 2) , the problem is to select the machine with the highest reward probability. The amount of reward dispensed by each slot machine for a play is assumed to be the same in this study. The measured chaotic signal s (t ) is subjected to the threshold adjuster (TA), according to the TOW principle. The output of the TA is immediately the decision concerning the slot machine to choose. If s (t ) is equal to or greater than the threshold value T (t ) , the decision is made to select slot machine 1. Otherwise, the decision is made to select slot machine 2. The reward—the win/lose information of a slot machine play—is fed back to the TA. The chaotic signal level s (t ) is compared with the threshold value T (t ) denoted by 7 T (t )  k  TA(t )  (1) where TA(t ) is the threshold adjuster value at cycle t, TA(t )  is the nearest integer to TA(t ) rounded to zero, and k is a constant determining the range of the resultant T (t ) . In this study, we assumed that TA(t )  takes the values  N ,   1, 0,1,  , N , where N is a natural number. Hence the number of the thresholds is 2 N  1 , referred to as TA’s resolution. The range of T (t ) is limited between kN and kN by setting T (t )  kN when TA(t )  is greater than N , as well as T (t )   kN when TA(t )  is smaller than  N . If the selected slot machine yields a reward at cycle t (in other words, wins the slot machine play), the TA value is updated at cycle t  1 based on TA(t  1)    TA(t ) if slot machine 1 wins TA(t  1)    TA(t ) if slot machine 2 wins (2) where  is referred to as the forgetting (memory) parameter20, and  is the constant increment (in this experiment,   1 and   0.999 ). In this study, the initial TA value was zero. If the selected machine does not yield a reward (or loses in the slot machine play), the TA value is updated by TA(t  1)    TA(t ) if slot machine 1 fails TA(t  1)    TA(t ) if slot machine 2 fails (3) where  is the increment parameter defined below. Intuitively speaking, the TA takes a smaller value if slot machine 1 is considered more likely to win, and a greater value if slot machine 2 is considered more likely to earn the reward. This is as if the TA value is being pulled by the two slot machines at both ends, which coincides with the notion of a tug of war. 8 The fluctuation, necessary for exploration, is realized by associating the TA value with the threshold of digitization of the chaotic signal train. If the chaotic signal level s (t ) is equal to or greater than the assumed threshold T (t ) , the decision is immediately made to choose slot machine 1; otherwise, the decision is made to select slot machine 2. Initially, the threshold is zero; hence, the probability of choosing either slot machine 1 or 2 is 0.5. As time elapses, the TA value shifts (becomes positive or negative) towards the slot machine with the higher reward probability based on the dynamics shown in Eqs. (2) and (3). We should note that due to the irregular nature of the incoming chaotic signal, the possibility of choosing the opposite machine is not zero, which is a critical feature of exploration in reinforcement learning. For example, even when the TA value is sufficiently small (meaning that slot machine 1 seems highly likely to be the better machine), the probability of the decision to choose slot machine 2 is not zero. In TOW-based decision making, the increment parameter  in Eq. (3) is determined based on the history of betting results. Let the number of times when slot machine i is selected in cycle t be Si and the number of wins in selecting slot machine i be Li . The estimated reward probabilities of slot machines 1 and 2 are given by L L Pˆ1  1 , Pˆ2  2 . S1 S2 (4)  is then given by   Pˆ1  Pˆ2 . 2  ( Pˆ1  Pˆ2 ) (5) 9 The initial  is assumed to be unity, and a constant value is assumed when the denominator of Eq. (5) is zero. The detailed derivation of Eq. (5) is shown in Ref. 23. RESULTS The architecture of laser chaos-based reinforcement learning is schematically shown in Fig. 1a. A semiconductor laser is coupled with a polarization-maintaining (PM) coupler. The emitted light is incident on a variable fibre reflector, by which a delayed optical feedback is supplied to the laser, leading to laser chaos40. The output light at the other end of the PM coupler is detected by a highspeed, AC-coupled photodetector through an optical isolator (ISO) and an attenuator, and is sampled by a high-speed digital oscilloscope at a rate of 100 GSample/s (a 10-ps sampling interval). The detailed specifications of the experimental apparatus are described in the Methods section. Figure 1b shows an example of the chaotic signal train. Figures 1c and 1d show the optical and radio frequency (RF) spectra of laser chaos measured by optical and RF spectrum analysers, respectively. The semiconductor laser was operated at a centre wavelength of 1547.782 nm. The standard bandwidth31 of the RF spectrum was estimated as 10.8 GHz. Figure 1e summarizes the histogram of the signal levels of the chaotic trains, which spanned from 0.2 to 0.2, with the level zero at its maximum incidence, and slightly skewed between the positive and negative sides. A remark is that the small incidence peak at the lowest measured amplitude is due to our experimental apparatus (not by the laser), and it does not critically effect in the present study. The rounded TA 10 values, TA(t )  , are also schematically illustrated at the right-hand side of Fig. 1b and the upper side of Fig. 1e, assuming N  10 . This means that TA(t )  ranges from  1 0 to 10 . Signal value s (t ) spans between approximately 0.2 and 0.2. The particular example of TA values shown in Figs. 1b and 1e shows the case when the constant k in Eq. (1) is given by 0.02, so that the actual threshold T (t ) spans from 0.2 to 0.2. In the experimental demonstration of this study, the TA and the slot machines were emulated in offline processing, whereas online processing was technologically feasible due to the simple procedure of the TA (Remarks are given in the Discussion section). [EXPERIMENT-1] Adaptation to sudden environmental changes We first solved the two-armed bandit problems given by the following two cases, where the reward probabilities {P1 , P2 } were given by {0.8, 0.2} and {0.6, 0.4} . We assumed that the sum of the reward probabilities P1  P2 was known prior to slot machine plays. With this knowledge,  is unity by Eq. (5). The slot machine was consecutively played 4,000 times, and this play was repeated 100 times. The red and blue curves in Fig. 2a show the evolution of the correct decision rate (CDR), defined by the ratio of the number of selections of the machines that yielded the higher reward probability at cycle t in 100 trials, with respect to the probability combination of {0.8, 0.2} and {0.6, 0.4} . The chaotic signal was sampled every 10 ps; hence, the total duration of the 4,000 plays of the slot machine was 40 ns. In order to represent sudden environmental changes (or uncertainty), the reward 11 probability was forcibly interchanged every 10 ns, or every 1,000 plays (For example, {P1 , P2 }  {0.8, 0.2} was reconfigured to {P1 , P2 }  {0.2, 0.8} ). The resolution of TA was set to 5 (or N  2 ). We observed that the CDR curves quickly approached unity, even after sudden changes in the reward probabilities, showing successful decision making. The adaptation was steeper in the case of a reward probability combination of {0.8, 0.2} than that of {0.6, 0.4} , since the difference in the reward probability was greater in the former case ( 0.8  0.2  0.6 ) than the latter ( 0.6  0.4  0.2 ). This meant that decision making was easier. The red and blue curves in Fig. 2b represent the evolution of TA values ( TA(t ) ) in the case of {0.8, 0.2} and {0.6, 0.4} , respectively, where the TA value became greater than 2 or 2; hence, it took long for the TA values to be inverted to the other polarity following environmental change. This is the origin of some delay in the CDR in responding to the environmental changes observed in Fig. 2a. One simple method of improving adaptation is to limit the maximum and minimum values of TA(t ) . The forgetting parameter  is also crucial to improving adaptation speed. Figures 2c, 2d and 2e characterize decision-making performance dependencies with respect to the configuration of TA. Figure 2c concerns TA resolutions demonstrated by the seven curves therein, with corresponding TA resolutions of 5 to 255, where the adaptation was quicker in fewer resolutions than not. Figure 2d considers TA range dependencies: while keeping the centre of the TA value at zero and the TA resolution at 5, the three curves in Fig. 2d compare CDRs by a TA range of 12 0.1, 0.2 and 0.4. We observed that full coverage (0.4) of the chaotic signal yields the best performance. Figure 2e shows the TA’s centre value dependencies while maintaining TA ranges of 0.4 and a resolution of 5, where we can clearly observe the deterioration of CDRs by the shift in the centre of the TA from zero. [EXPERIMENT-2] Adaptation from zero prior knowledge Here we considered decision-making problems without any prior knowledge of the slot machines. Hence, parameter  needed to be updated. The reward probabilities of the slot machines were set to {P1 , P2 }  {0.5, 0.1} , and 50 consecutive plays were executed. The colour curves in Fig. 3a depict CDRs at six sampling intervals of chaotic signal trains from 10 to 400 ps. Hence, the time needed to complete 50 consecutive slot machine plays differed among these; it ranges from 10 ps  50  500 ps to 400 ps  50  20 ns . Moreover, the black curve shows the CDR obtained by uniformly distributed pseudo-random numbers generated by the Mersenne Twister for the random signal source, instead of experimentally observed chaotic signals. We observed that CDRs based on chaotic signals exhibited more rapid adaptation than the uniformly distributed pseudo-random numbers. The most prompt adaptation, or the optimal performance, was obtained at a particular sampling interval. Figure 3b compares CDRs at the cycle t  3 ; the sampling interval of 50 ps yielded the best performance, indicating that the original chaotic dynamics of the laser could physically be optimized 13 such that the most prompt decision-making is realized. Indeed, the autocorrelation of the laser chaos signal trains was evaluated as shown in Fig. 3c, its negative maximum value is taken when the time lag is given by 5 or 5, corresponding exactly to the sampling interval of 50 ps ( 5  10 ps ). In other words, the negative correlation of chaotic dynamics enhanced the exploration ability for decision making. Furthermore, this finding suggests that optimal performance is obtained at the maximum sampling rate (or data rate) by physically tuning the dynamics of the original laser chaos, which will be an important and exciting topic for future investigation. The adaptation speed of decision making was estimated as 1 GHz in this optimal case, where CDR was larger than 0.95 at 20 cycles with a 50ps sampling interval (20 GSamples/s) in Fig. 3a (1 GHz = (50 ps  20 cycles)1). Furthermore, we characterized CDRs with normally distributed random numbers (referred to as RANDN) in order to ensure that the statistical incidence patterns of the laser chaos, which were similar to a normal distribution shown in Fig. 1e, were not the origins of the fast adaptation of the decision making. By keeping the mean value of RANDN at zero, the standard deviation (  ) was configured as 0.2, 0.1 and 0.01. As shown in Fig. 4, the CDRs at cycle t  3 by RANDN were inferior to chaotic signals and the uniformly distributed random numbers (denoted by RAND). Moreover, the CDR was evaluated by surrogating the chaotic signal time series sampled at 50 ps intervals, which resulted in poorer performance than the original, as shown in Fig. 4. These evaluations support the claim that laser chaos is beneficial to the performance of reinforcement learning, in addition to its ultrafast data rate for random signals. 14 DISCUSSION The performance enhancement of decision making by chaotic laser dynamics is demonstrated and the impacts of negative autocorrelation is clearly suggested. Further understanding between chaotic oscillatory dynamics and decision making is a part of important future research. The first regards to physical insights. Toomey et al. recently showed that the complexity of laser chaos varies within the coherence collapse region in the given system43. The level of the optical feedback, injection current of the laser becomes an important parameter in determining the complexity of chaos, which is the entrance to thorough insights. We also consider the use of the bandwidth enhancement technique31 with optically injected lasers to improve the adaptation speed of decision making over tens of GHz. Meanwhile, besides negative autocorrelation inherent in laser chaos, other perspectives could address the underlying mechanism such as diffusivity,44 Hurst exponents,45 etc. In the experimental demonstration, the nonlocal aspect of the TOW principle was not directly, physically relevant to the chaotic oscillatory dynamics of the lasers. By combining the chaotic dynamics with the threshold adjustor, the nonlocal and fluctuation properties of the TOW principle emerged, which have not been completely realized in the literature nor in our past experimental studies24,25. Such a hybrid realization of nonlocality in TOW leads to higher likelihood of technological implementability, and better scalability and extension to higher-grade problems. Online post-processing can become feasible through electric circuitry, as already demonstrated for a 15 random-bit generator41,46 since the post-processing is very simple as described in Eqs. (1) to (5). For scalability, a variety of approaches can be considered, such as time-domain multiplication, which exploits the ultrafast attributes of chaotic lasers, and is a frequently used strategy in ultrafast photonic systems. Introducing multi-threshold values31,41 is another simple extension of our proposed scheme. Whole-photonic realization is an interesting issue to explore, and has already been implied by the analysis where the time-domain correlation of laser chaos strongly influences decision-making performances. The mode dynamics of multi-mode lasers47 are very promising for the implementation of nonlocal properties of fully photonic systems required for decision making. Synchronization and its clustering properties in coupled laser networks48,49 are also interesting approaches to physically realizing the nonlocality of the TOW. These systems can automatically tune the optimal settings for decision making (e.g., negative autocorrelation properties), which can lead to autonomous photonic intelligence. We also make note of the extension of the principle demonstrated in this paper to highergrade machine learning problems. The competitive MAB50 with multiple players is an exciting topic for photonic intelligence research as it involves the so-called Nash equilibrium, and is the foundation of such important applications as resource allocation and social optimization. Investigating the possibilities of extending the present method and utilizing ultrafast laser dynamics for competitive MAB is highly interesting. 16 Conclusion We experimentally established that laser chaos provides ultrafast reinforcement learning and decision making. The adaptation speed of decision making reached 1 GHz in the optimal case with the sampling rate of 20 GSample/s (50-ps decision-making intervals) using the ultrafast dynamics inherent in laser chaos. The maximum adaptation performance coincided with the negative maximum of the autocorrelation of the original time-domain laser chaos sequences, demonstrating the strong impact of chaotic lasers on decision making. The origin of the performance was also validated by comparing with uniformly and normally distributed pseudo-random numbers as well as surrogated arrangements of original chaotic signal trains. This study is the first demonstration of ultrafast photonic reinforcement learning or decision making, to the best of our knowledge, and paves the way for research on photonic intelligence and new applications of chaotic lasers in the realm of artificial intelligence. METHODS Optical system The laser used in the experiment was a distributed-feedback (DFB) semiconductor laser mounted on a butterfly package with optical fibre pigtails (NTT Electronics, KELD1C5GAAA). The injection current of the semiconductor laser was set to 58.5 mA (5.85 Ith), where the lasing threshold Ith was 10.0 mA. The relaxation oscillation frequency of the laser was 6.5 GHz. The temperature of the 17 semiconductor laser was set to 294.86 K. The laser output power was 13.2 mW. The laser was connected to a variable fibre reflector, which reflected a fraction of light back into the laser, inducing high-frequency chaotic oscillations of optical intensity32-34. 1.9 % of the laser output power was fed back to the laser cavity from the reflector. The fibre length between the laser and the reflector was 4.55 m, corresponding to the feedback delay time (round trip) of 43.8 ns. Polarization-maintaining fibres were used for all optical fibre components. The optical output was converted to an electronic signal by a photodetector (New Focus, 1474-A, 35 GHz bandwidth) and sampled by a digital oscilloscope (Tektronics, DPO73304D, 33 GHz bandwidth, 100 GSample/s, eight-bit vertical resolution). The RF spectrum of the laser was measured by an RF spectrum analyser (Agilent, N9010A-544, 44 GHz bandwidth). The optical wavelength of the laser was measured by an optical spectrum analyser (Yokogawa, AQ6370C-20). Data analysis [EXPERIMENT-1] A chaotically oscillated signal train was sampled at a rate of 100 GSample/s by 9,999,994 points, which lasted approximately 10 s. As described in the main text, 4,000 consecutive plays were repeated 100 times; hence, the total number of slot machine plays was 400,000. With a 10-ps interval sampling, the initial 400,000 points of the chaotic signal were used for the decision-making experiments. The post-processing required for 400,000 iterations of slot machine plays was approximately 1.2 s (or 3.0 s/decision), on a normal-grade personal computer (Panasonic, CF-SX3, 16 GB RAM, Windows 7, MATLAB R2011b). 18 [EXPERIMENT-2] (1) Sampling methods: A chaotic signal train was sampled at 10-ps intervals with 9,999,994 sampling points. Such a train was measured 120 times. Each chaotic signal train was referred to as chaosi , and there were 120 kinds of such trains: i  1,  ,120 . In demonstrating 10  M ps sampling intervals, where M was a natural number ranging from 1 to 40 (namely, the sampling intervals were 10 ps, 20 ps, , and 400 ps), we chose one of every M samples from the original sequence. (2) Evaluation of CDR regarding a specific chaos sequence: For every chaotic signal train chaosi , 50 consecutive plays were repeated 200 times. Consequently, 10,000 points were used from chaosi . Such evaluations were repeated 100 times. Hence, 1,000,000 slot machine plays were conducted in total. These CDRs were calculated for all signal trains chaosi ( i  1,  ,120 ). (3) Evaluation of CDR of all chaotic sequences: We evaluated the average CDR of all chaotic signal trains ( i  1,  ,120 ) derived in (2) above, which were the results discussed in the main text. (4) Autocorrelation of chaotic signals: The autocorrelation was computed based on all 9,999,994 sampling points of chaosi , and was evaluated for all chaosi ( i  1,  ,120 ). The autocorrelation demonstrated in Fig. 3c was evaluated as the average of these 120 kinds of autocorrelations. (5) Surrogate methods: The surrogate time series of original chaotic sequences were generated by the randperm function in MATLAB which is based on the sorting of pseudorandom numbers generated by Mersenne Twister. 19 Data availability. The data sets generated during the current study are available from the corresponding author on reasonable request. References 1. 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A., Davidson, N. & Kanter, I. Controlling synchronization in large laser networks. Phys. Rev. Lett. 108, 214101 (2012). 49. Williams, C. R. S., Murphy, T. E., Roy, R., Sorrentino, F., Dahms, T. & Schöll, E. Experimental observations of group synchrony in a system of chaotic optoelectronic oscillators. Phys. Rev. Lett. 110, 064104 (2013). 50. Kim, S. -J., Naruse, M. & Aono, M. Harnessing the Computational Power of Fluids for Optimization of Collective Decision Making. Philosophies Special Issue “Natural Computation: Attempts in Reconciliation of Dialectic Oppositions” 1, 245–260 (2016). Acknowledgements This work was supported in part by the Core-to-Core Program, A. Advanced Research Networks from the Japan Society for the Promotion of Science and Grants-in-Aid for Scientific Research from Japan Society for the Promotion of Science. 25 Figure 1 | Architecture of photonic reinforcement learning based on laser chaos. (a) Architecture and experimental configuration of laser chaos-based reinforcement learning. Ultrafast chaotic optical signal is subjected to the tug-of-war (TOW) principle that determines the selection of slot machines. ISO: optical isolator. (b) An example of chaotic signal trains sampled at 100 GSample/s. The signal level is subjected to a threshold adjustment (TA) for decision making. (c) Optical spectrum and (d) RF spectrum of the laser chaos used in the experiment. (e) Incidence statistics (histogram) of the signal level of the laser chaos signal. 26 Figure 2. Reinforcement learning in dynamically changing environments. (a) Evolution of the correct decision rate (CDR) when the reward probabilities of the two slot machines are {0.8, 0.2} and {0.6, 0.4}. The knowledge that the sum of the reward probability, which is unity in these cases, is supposed to be given. The reward probability is intentionally swapped every 10 ns in order to represent sudden environmental changes or uncertainty. Rapid and adequate adaptation is observed in both cases. (b) Evolution of the threshold adjuster (TA) value underlying correct decision making. (c-e) CDR performance dependency on the setting of TA. (c) TA resolution dependency. (d) TA range dependency. (e) TA centre value dependency. 27 Figure 3 | Reinforcement learning from zero prior knowledge. (a) Evolution of CDR with different sampling intervals of the laser chaos signal (10, 20, 30, 40, 50, and 400 ps) and uniformly distributed pseudo-random numbers. CDR exhibits prompt adaptation when the sampling interval is 50 ps. (b) CDR is evaluated as a function of the sampling interval from 10 ps to 400 ps, where the maximum performance is obtained at 50 ps. (c) Autocorrelation of the laser chaos signals exhibits its negative maximum when the time lag is 5 or 5, which exactly coincides with the fact that the optimal adaptation is realized at 50 ps (10 ps  5) sampling intervals. 28 Figure 4 | Comparison of learning performance in laser chaos, uniformly and normally distributed pseudo-random numbers, and surrogate laser chaos signals. The laser chaos sampled at 50 ps interval exhibits the best performance compared with other cases, indicating that the dynamics of laser chaos affects reinforcement learning ability. 29 

Parametric Strategy Iteration Thomas M. Gawlitza1 , Martin D. Schwarz2 , and Helmut Seidl2 1 Carl von Ossietzky Universität Oldenburg, Ammerländer Heerstraße 114-118, D-26129 Oldenburg, Germany [email protected] 2 Technische Universität München, Boltzmannstraße 3, D-85748 Garching, Germany {schwmart,seidl}@in.tum.de arXiv:1406.5457v1 [] 19 Jun 2014 Abstract Program behavior may depend on parameters, which are either configured before compilation time, or provided at runtime, e.g., by sensors or other input devices. Parametric program analysis explores how different parameter settings may affect the program behavior. In order to infer invariants depending on parameters, we introduce parametric strategy iteration. This algorithm determines the precise least solution of systems of integer equations depending on surplus parameters. Conceptually, our algorithm performs ordinary strategy iteration on the given integer system for all possible parameter settings in parallel. This is made possible by means of region trees to represent the occurring piecewise affine functions. We indicate that each required operation on these trees is polynomial-time if only constantly many parameters are involved. Parametric strategy iteration for systems of integer equations allows to construct parametric integer interval analysis as well as parametric analysis of differences of integer variables. It thus provides a general technique to realize precise parametric program analysis if numerical properties of integer variables are of concern. 1 Introduction Since the very beginnings of linear programming, parametric versions of linear programming (LP for short) have already been of interest (see, e.g., [6, 8] for an overview and [16] for recent algorithms). Parametric LP can be applied to answer questions such as: how much does the result of the analysis (optimal value/solution) depend on specific parameters? What is the precise dependency between a parameter and the result? In which regions of parameter values do these dependencies significantly change? Such types of sensitivity and mode analyses are important in order to obtain a better understanding of the problem under consideration and its potential analysis. Sensitivity and mode questions equally apply to programs whose behavior depends on parameters. Such parameters either could be provided at configuration time by engineers, or at runtime, e.g., through sensor data or other kinds of input. The goal then is to determine how the output values produced by the program may be influenced by the parameters. The same software may, e.g., control the break of a truck or a car, but must behave quite differently in the two use cases. Here, we consider the static analysis of parameterized systems and propose methods for inferring, how numerical program invariants may depend on the parameters of the program. These questions cannot be answered by linear or integer linear programming related techniques alone, since the constraints to be solved are necessarily non-convex. Still, such questions can be answered for interval analysis or, more generally, for template-based numerical analyses such as difference bound matrices or octagons, since their analysis results can be expressed in first-order linear real arithmetic. This observation has been exploited by Monniaux [22] who applied quantifier elimination algorithms to obtain parametric analysis results. The resulting system can provide amazing results. However, if the programs under consideration have more complicated control-flow, i.e., do not consist of a single program point only, fixpoint computation realized by means of quantifier elimination does no longer seem appropriate. In this paper, we are concerned with invariants over integers (opposed to rationals in Monniaux’ system). 1 Parametric Strategy Iteration T.M. Gawlitza, M.D. Schwarz, H. Seidl Example 1. Consider the following parametric program: x = p1 ; while (x < p2 ) x = x + 1; where p1 , p2 are the parameters. The parametric invariant for program exit states that x = p1 holds, if p2 ≤ p1 , and x = p2 otherwise. Thus, the analysis should distinguish two modes where the invariant inferred for program exit is significantly different, namely the set of parameter settings where p2 ≤ p1 holds and its complement. In the first mode, the value of x at program exit is only sensitive to changes of the parameter p1 , while otherwise it is sensitive to changes of the parameter p2 . Note that for integer linear arithmetic, quantifier elimination is even more intricate than over the rationals. As shown in [9, 10, 11], non-parametric interval analysis as well as the analysis of differences of integer variables can be compiled into suitable integer equations. In absence of parameters, integer equation systems where no integer multiplication or division is involved, can be solved without resorting to heavy machinery such as integer linear programming. Instead, an iteration over max-strategies suffices [9, 10, 11]. Here, a max-strategy maps each application of a maximum operator to one of its arguments. Once a choice is made at each occurrence of a maximum operator, a conceptually simpler system is obtained. For systems without maximum operators, the greatest solution can be determined by means of a generalization of the Bellman-Ford algorithm. The greatest solutions of the maximum-free systems encountered during the iteration on max-strategies, provide us with an increasing sequence of lower approximations to the overall least solution of the integer system. Given such a lower approximation, we can check whether a solution and thus the least solution has already been reached. Otherwise, the given max-strategy is improved, and the iteration proceeds. In contrast to ordinary program analysis, parametric program analysis infers a distinct program invariant for each possible parameter setting. To solve these parametric systems, we propose to apply strategy iteration simultaneously for all parameter settings (Section 3). We show that this algorithm terminates—given that we can effectively deal with the parametric intermediate results considered by the algorithm. For that, we show that the intermediate parametric values can be represented by region trees (Section 4). Here, a region tree over a finite set C of linear inequalities is a data-structure for representing finite partitions of the parameter space into non-empty regions of parameter settings which are indistinguishable by means of the constraints in C. A value (of a set V of values) is then assigned to each non-empty region of the tree. We also indicate that each basic operation which is required for implementing parametric strategy iteration can be realized with region trees in polynomial time — assuming that the number of parameters is fixed (Section 5). Finally, we apply parametric strategy iteration for parametric integer equations to solve parametric interval equations and thus to perform parametric program analysis for integer variables of programs, and report preliminary experimental results (Section 6). 2 Basic Concepts In this section we provide basic notions and introduce systems of parametric integer equations. By Z we denote the complete linearly ordered set Z∪{−∞, ∞}. Let P and X denote finite sets of parameters and variables (unknowns), which are disjoint. A system E of parametric integer equations is given by x = ex , x∈X where each right-hand side ex is of the form e1 ∨ . . . ∨ er for parametric integer expressions e1 , . . . , er . Here, “∨” denotes the maximum operator. In this paper, an integer expression is built up from constants in Z, variables, parameters, and negated parameters by means of application of operators. As operators, 2 Parametric Strategy Iteration T.M. Gawlitza, M.D. Schwarz, H. Seidl we consider “∧” (minimum), “+” (addition), “;” (test of non-negativity) and multiplication with nonnegative scalars. Addition as well as scalar multiplication is extended to −∞ and ∞ by: x + −∞ = −∞ + x = −∞ x+∞=∞+x=∞ c · (−∞) = −∞ ∀x ∈ Z ∀x ∈ Z\{−∞} ∀c ≥ 0 0·∞=0 c·∞=∞ ∀c > 0 A parametric integer expression is defined by means of the following grammar: e ::= a | p | −p | x | e1 ∧ e2 | e1 + e2 | e1 ; e2 | c · e1 where a ∈ Z, c ∈ N, p ∈ P, x ∈ X. Given a parameter setting π : P → Z and a variable assignment ξ : X → Z, the value of an expression e is determined by: JaKπ ξ = a JxKπ ξ = ξ(x) JpKπ ξ = π(p) J−pKπ ξ = −π(p) Je1 2 e2 Kπ ξ = Je1 Kπ ξ 2 Je2 Kπ ξ Jc · eKπ ξ = c · JeKπ ξ Here, 2 ∈ {∧, +}. For a given parameter setting π, Eπ denotes the (non-parametric) integer equation system obtained from E by replacing every parameter p of E with its value π(p). For a given parameter setting π, a solution to Eπ is a variable assignment ξ ∗ that satisfies all equations of Eπ . That is, for each equation x = e1 ∨ . . . ∨ er in E, ξ ∗ (x) = Je1 Kπ ξ ∗ ∨ . . . ∨ Jer Kπ ξ ∗ Since all operators occurring in right hand sides are monotonic, for every parameter setting π, Eπ has a uniquely determined least solution. Finally, a parametric solution of E is a mapping Ξ which assigns to each possible parameter setting π, a solution of Eπ . Ξ is the parametric least solution of E iff Ξ(π) is the least solution of Eπ for every parameter setting π. Example 2. Consider the parametric system E which consists of the single equation x = p1 ∨ (x + 1 ∧ p2 ). Then the parametric least solution Ξ of E is given by ( π(p1 ) if π(p1 ) ≥ π(p2 ) Ξπx= π(p2 ) if π(p1 ) < π(p2 ) for all parameter settings π. 3 Parametric Strategy Iteration In the following we w.l.o.g. assume that the set of parameters P is given by P = {p1 , . . . , pk }. Accordingly, parameter settings from P → Z can be represented as vectors from Zk . Our goal is to enhance the strategy iteration algorithm from [9, 10, 11] to an algorithm that computes parametric least solutions of systems of parametric integer equations. Conceptually, we do this by performing each operation for all parameter settings in parallel. For that, we lift the complete lattice Z of integer values to the set Zk → Z of parametric values which is again a complete lattice w.r.t. the point-wise extension of the ordering on 3 Parametric Strategy Iteration T.M. Gawlitza, M.D. Schwarz, H. Seidl Z where the least and greatest elements are given by the functions const−∞ and const∞ mapping each vector of parameters to the constant values −∞ and ∞, respectively. Accordingly, sub-expressions of right-hand sides are no longer evaluated one by one for each parameter setting. Instead, each binary operator 2 on Z is lifted to a binary operator 2∗ on parametric values from Zk → Z by defining: (φ1 2∗ φ2 )(π) = φ1 (π) 2 φ2 (π) for all parameter settings π ∈ Zk . In particular, the lifted maximum operator ∨∗ equals the least upper bound of the complete lattice Zk → Z. Likewise, scalar multiplication with a non-negative constant c is lifted point-wise from a unary operator on Z to a unary operator on Zk → Z. For convenience, we henceforth denote the lifted operators with the same symbols by which we denote the original operators. The original system E of parametric equations over Z thus can be interpreted as a system of equations over the domain Zk → Z of parametric values. For all parametric variable assignments ρ : X → Zk → Z, expressions e are interpreted as follows: Jpi K ρ = proji JaK ρ = consta Je1 2 e2 K ρ = Je1 K ρ 2 Je2 K ρ J−pi K ρ = −proji JxK ρ = ρ(x) Jc · eK ρ = c · JeK ρ Here, 2 is a binary operator, consta is a parametric value which maps all arguments to the constant a, and proji denotes the projection onto the ith component of its argument vector. With respect to this interpretation the least solution ρ∗ of E is a mapping of type X → Zk → Z. Let us call ρ∗ the least parametric solution. Let Ξ denote the parametric least solution as defined in the last section. Then Ξ and ρ∗ are not identical — but in one-to-one correspondence. By fixpoint induction, it can be verified that: Ξ π x = ρ∗ x π for all variables x ∈ X, and all parameter settings π ∈ Zk . In the same way as the abstract domain Z and the operators on Z, we also lift the notion of a strategy from [10] to the notion of a parametric strategy. For technical reasons, let us assume that the right-hand side for each variable in E is of the form a ∨ e1 ∨ . . . er where a ∈ Z. This can always be achieved, e.g., by replacing right-hand sides e which are not of the right format with −∞ ∨ e. A parametric strategy σ then assigns to each variable x, a parametric choice. If the right-hand side for x is given by e0 ∨ . . . ∨ er , σ x maps each parameter setting π ∈ Zk to a natural number in the range [0, r] (signifying one of the argument expressions ei ). Moreover, we need an operator “next” which takes a given parametric strategy σ together with a parametric variable assignment ρ : X → Zk → Z and then, for every parameter setting π, switches the choice provided by σ whenever required by the evaluation of subexpressions according to ρ. That is, σ 0 = next(σ, ρ) implies that the following properties hold for all equations x = e0 ∨ . . . ∨ er and all parameter settings π: 1. Jeσ0 x π K ρ π ≥ Jei K ρ π for all i ∈ {0, . . . , r}. 2. If Jeσ x π K ρ π ≥ Jei K ρ π for all i, then σ 0 x π = σ x π. Note that this operator changes the choice given by the argument strategy σ only if a real improvement is guaranteed. An operator “next” with properties 1) and 2) is a locally optimal parametric strategy improvement operator because it chooses for each x a best alternative everywhere (relative to ρ). For the correctness of the algorithm it would be sufficient to choose some strategy that is an improvement compared to the current strategy at ρ. Finally, we need an operator “select” which, based on a parametric choice φ : Zk → N, selects one of the arguments, i.e., select φ (v0 , . . . , vr ) is the parametric value given by: select φ (v0 , . . . , vr ) π = vφ(π) π 4 Parametric Strategy Iteration T.M. Gawlitza, M.D. Schwarz, H. Seidl for all parameter settings π. With these parametric versions of the corresponding operators used by strategy iteration, we propose the algorithm in Fig. 1 for systems of parametric integer equations with n unknowns. The resulting algorithm is called parametric strategy iteration or PSI for short. σ = {x 7→ const0 | x ∈ X}; do { forall (x ∈ X) ρ(x) = const∞ ; for (int i = 0; i < n; i++) forall ((x = e0 ∨ . . . ∨ er ) ∈ E) ρ(x) = select (σ x) (Je0 K ρ, . . . , Jer K ρ); old = σ; σ = next(σ, ρ); } while (old 6= σ); output(ρ); // initial strategy // begin BF // end BF // // strategy improvement termination detection Figure 1: Parametric strategy iteration for a parametric integer equation system with n unknowns. σ = {x 7→ 0 | x ∈ X}; do { forall (x ∈ X) ρ(x) = ∞; for (int i = 0; i < n; i++) forall ((x = e0 ∨ . . . ∨ er ) ∈ E) ρ(x) = Jeσ x K ρ; old = σ; σ = next(σ, ρ); } while (old 6= σ); output(ρ); // initial strategy // begin BF // end BF // // strategy improvement termination detection Figure 2: Ordinary strategy iteration for a non-parametric integer equation system with n unknowns. PSI starts with the initial parametric strategy σ mapping each variable and parameter setting to the constant 0, i.e., it selects for each variable and parameter setting the constant term on the right-hand side. For a given parametric strategy, the Bellman-Ford algorithm is used to determine the greatest solution (the for-loop labeled as BF). This Bellman-Ford iteration amounts to n rounds of round robin iteration (n the number of variables) starting from the top element of the lattice. During round robin iteration, the appropriate integer expression ei as right-hand side for each variable x and each parameter setting π, is selected by means of the auxiliary function select according to the current parametric strategy σ. As a result of Bellman-Ford iteration for all parameter settings in parallel, the next approximation ρ to the least parametric fixpoint is obtained. This parametric variable assignment then is used to improve the current parametric strategy σ by means of the operator “next”. This is repeated until the parametric strategy does not change any more. For a comparison, Fig. 2 shows a version of the non-parametric strategy iteration as presented in [10]. The mappings ρ, σ there have functionalities: ρ:X→Z σ:X→N where the evaluation Jei K of expressions ei results in integer values only. Since the strategy σ specifies a single integer expression ei for any given variable x, the call to “select” in PSI can be simplified 5 Parametric Strategy Iteration T.M. Gawlitza, M.D. Schwarz, H. Seidl to Jeσ x K ρ. This optimization is not possible in the parametric case, since different parameter values may result in different ei to be selected. For systems of integer equalities without parameters strategy iteration computes the least solution as has been shown in [10]. Assume for a moment that we can compute with parametric values and parametric strategies effectively, i.e., can represent them in some data structure, test them for equality, compute the results of parametric operator applications, as well as realize the operations “select” and “next”. Then the algorithmic scheme from Fig. 1 can be implemented, and we obtain: Theorem 1. Let E be a parametric integer equation system with n variables where each right-hand side is a maximum of at most r non-constant integer expressions. The following holds: 1. Parameterized strategy iteration as given by Fig. 1 terminates after at most (r + 1)n strategy improvement steps where each round of improvement requires at most O(n · |E|) evaluations of parametric operators. 2. On termination, the algorithm returns the least parametric solution of E. Proof. First we observe that, when probing the intermediate values of σ and ρ for any given parameter setting π = (p1 , . . . , pk ) ∈ Zk , the algorithm from Fig. 1 for the parametric system E returns the same values and strategic choices as the algorithm from Fig. 2 when run on the integer system Eπ . Moreover upon termination, the strategic choices σ x π as well as the values ρ x π do no longer change, and therefore for each parameter setting π, a least solution of Eπ has been attained. Since the maximal number of strategies considered by strategy iteration for ordinary integer systems is bounded by (r +1)n (independently of the values of the constants in the system), we conclude that the parametric algorithm also performs at most (r + 1)n strategy improvement steps. Therefore, the algorithm terminates. Since then for each parameter setting π, a least fixpoint of Eπ has been obtained, the resulting assignment is equal to the least parametric solution of E. Being able to effectively compute with parametric variable assignments and parametric strategies is crucial for the implementation of parametric strategy iteration as a practical algorithm. In the following we will explore the structure of parametric variable assignments and parametric strategies occurring during the algorithm. A set S ⊆ Zk of integer points is called convex iff S equals the set of integer points inside its convex hull over the rationals. A mapping f : Zk → V (V some set) is called piecewise constant iff there is a finite partition Ψ of Zk into nonempty convex sets together with a mapping Ψf : Ψ → V such that f (π) = Ψf (P ) for all P ∈ Ψ and all π ∈ P . The cardinality of the partition Ψ is called the fragmentation of f . In fact, the fragmentation of f depends on the representation of f rather than the function f itself. Still, we intentionally do not differentiate between the function and its representation here. Let Aff k ⊆ Zk → Z denote the set of functions which are either const−∞ , const∞ or an affine function from Zk → Z. We call f ∈ Zk → Z piecewise affine iff there is a piecewise constant mapping f˜ : Zk → Aff k such that f (π) = f˜(π)(π). If f is piecewise affine, then there is a finite partition Ψ of Zk into non-empty convex sets together with a mapping Ψf : Ψ → Aff k such that f (π) = Ψf (P )(π) for all P ∈ Ψ and π ∈ P . Assume f1 , f2 are piecewise affine where both mappings share the same partition Ψ of the parameter space Zk into convex sets. Then the functions c · f1 as well as f1 + f2 are piecewise affine using the same partition Ψ. The functions f1 ∨ f2 , f1 ∧ f2 , and f1 ; f2 are piecewise affine as well with, however, a possibly different finite partition into convex sets. The fragmentation is increased at most by a factor 2. From that, we conclude that the inner for-loop which implements the BF iteration, may increase the fragmentation of a common partition of values and the parametric strategy σ only by a factor 2nm∧ , where m∧ is the number of occurrences of minimum operators ∧ in E. The applications of the operator “;” do not contribute, since, in this phase, they never evaluate to a parametric value which returns 6 Parametric Strategy Iteration T.M. Gawlitza, M.D. Schwarz, H. Seidl −4 0 1 3 Figure 3: The partition [−4, 0, 1, 3] where the elements of the second region are displayed. −∞. Assume that ρ is the parametric variable assignment computed for the parametric strategy σ after executing the inner for-loop. The parametric strategy σ 0 returned by the call to the next-function for σ and ρ will again be a piecewise constant function whose fragmentation compared with the fragmentation of ρ is increased at most by a factor of 2m∨ +m∧ +m; , where m∨ and m; denote the number of occurrences of ∨-operators and ;-operators, respectively. This holds because we basically have to evaluate right-hand sides in order to apply the function next to σ and ρ. Therefore, compared with the fragmentation of σ, the fragmentation of σ 0 increased by a factor of at most 2m∨ +m∧ +m; ·2nm∧ = 2m∨ +(n+1)m∧ +m; . Since the initial strategy has fragmentation 1 and the total number of strategy improvement steps is bounded, we obtain our second theorem: Theorem 2. Consider the parametric strategy improvement algorithm from Fig. 1. All encountered parametric strategies are piecewise constant. Likewise, all encountered variable assignments are piecewise affine. Additionally: 1. The fragmentation is bounded by 2d·(m∨ +(n+1)m∧ +m; ) , where n is the number of unknowns in the integer system of equations, m2 is the number of 2-operators, where 2 ∈ {∨, ∧, ; }, and d is the maximal number of strategies for any parameter setting. 2. The absolute value of any occurring number is bounded by (c ∨ 2)sn · a where a is the maximum of the absolute values of all constants, c is the maximal occurring constant in a scalar multiplication and s is the maximal size of a right-hand side. In the second part of Theorem 2, we provided bounds for the coefficients occurring in affine functions of parametric values. These bounds follow since each parametric value is determined by means of n round of round robin iteration. Consequently the sizes of the numbers occurring in parametric values are always polynomial in the input size of PSI. Since each inequality used for refining the current partition of the parameter space is obtained from the comparison of two affine functions, we conclude that the sizes of coefficients of all occurring inequalities also remain polynomial. 4 Region Trees The key issue for a practical implementation of PSI is to provide an efficient data-structure for partitions of the parameter space Zk into convex components. In case k = 1, i.e., when the system depends on a single parameter only, the partition Ψ consists of a set of non-empty intervals [−∞, z0 ], [z0 + 1, z1 ] . . . , [zr−1 + 1, zr ], [zr + 1, ∞] whose union equals Z. Thus, it can be represented by a finite ordered list [z0 ; . . . ; zr ] (see Fig. 3). By means of the list representation, all required operations on parametric values as well as on parametric strategies can be realized in polynomial time. The case k > 1 is less obvious. We use a representation based on satisfiable conjunctions of linear inequalities on parameters a1 p1 + . . . + ak pk ≤ b with a1 , . . . , ak , b ∈ Z. Note that the negation of this inequality is given by the inequality −a1 p1 − . . . − ak pk ≤ −b − 1. Disjunctions of satisfiable conjunctions of inequalities are organized into a binary tree t as shown, e.g., in Fig. 4 on top. Each node in the tree is labeled with an inequality c. The left child of a node labeled with c corresponds to the case where c holds while the right child corresponds to the case where ¬c holds. A leaf v of the tree t thus represents the conjunction of the inequalities as provided by the path reaching v. The path 7 Parametric Strategy Iteration T.M. Gawlitza, M.D. Schwarz, H. Seidl −2p1 + p2 ≤ −2 −p1 − p2 ≤ −6 −p1 − p2 ≤ −6 −p1 − p2 ≤ −2 −p1 − p2 ≤ −2 Figure 4: The region tree for the inequalities of Example 3 together with the region represented by the second leaf. (c1 , j1 ) . . . (cr , jr ) in t which successively visits the nodes labeled by c1 , . . . , cr and continues with the j1 , . . . , jr th successors, respectively, (where ji ∈ {1, 2}), represents the conjunction cj11 ∧ . . . ∧ cjrr where c1 = c and c2 = ¬c. As an invariant of t, we maintain that all conjunctions corresponding to paths in t are satisfiable. The leaves of t are annotated with the values attained in the corresponding parameter region. In order to obtain a more canonical representation, we additionally impose a strict linear ordering ≺ on the inequalities (analogous to the linear ordering of variables for OBDDs) where the inequality c at a node in t should be less than all successor inequalities. Moreover, we demand that c ≺ ¬c should hold. We call the corresponding data-structure region tree. Example 3. Consider the following set of linear inequalities: −2p1 + p2 ≤ −2 −p1 − p2 ≤ −6 −p1 − p2 ≤ −2 Assume further that the inequalities are ordered from left to right and that each of them is smaller than its negation. Then we obtain a region tree as depicted in Fig. 4 at the top, where the integer points corresponding to the second leaf are shown at the bottom. The tree is not a full binary tree, since the second inequality −p1 − p2 ≤ −6 implies the third inequality −p1 − p2 ≤ −2. A trivial upper bound for the number of leaves of a region tree over a set of n linear inequalities is given by 2n . However, in our application we assume that the parameter space is of fixed dimensionality k. Therefore many occurring inequalities are at least partially redundant. For the important case where k ≤ n, we can establish the following more precise upper bound on the number of leaves: Lemma 1. The number of a region tree over n linear inequalities over k parameters with
Pk of leaves k ≤ n is bounded by i=0 ni . A similar bound has been inferred for cells of maximal dimension k in arrangements of n hyperplanes (see, e.g., [15]). The intersection of halfspaces as required for our estimation, results in an identical recurrence relation and therefore in an identical solution. As a consequence of Lemma 1, the number of leaves and thus also the number of nodes of a region tree for a fixed set of parameters is polynomial in the number of involved linear inequalities. When maintaining region trees, we repeatedly must verify whether a (growing) conjunction of linear inequalities is satisfiable (over Z). Several algorithms have been proposed to solve this problem (see, e.g., [24, 5]). If the number of parameters is fixed and small, polynomial run-time can be guaranteed, 8 Parametric Strategy Iteration T.M. Gawlitza, M.D. Schwarz, H. Seidl e.g., by relying on the LLL algorithm for lattices in combination with the ellipsoid method [18, 17] or by means of generating functions [19]. Note that for small numbers of variables, even Fourier-Motzkin elimination (though only complete for rational satisfiability) is polynomial. 5 Implementing Parametric Strategy Iteration The efficiency of the resulting algorithm crucially depends on the fragmentation of parametric variable assignments and strategies occurring during iteration. Instead of globally maintaining a single common partition, we allow individual partitions for each intermediately computed value as well as for each variable from X. Before applying a binary operator to parametric argument values t1 , t2 , first a common refinement of the partitions of t1 , t2 is computed by means of a function “align”. Given that 0 a tree is the type of region trees whose leaves are labeled with values of type 0 a, the function “align” has the following type align : 0 a tree → 0 b tree → (0 a ∗ 0 b) tree In case of addition, the operator “+” is applied for each component separately. In case of minimum, the components of the common refinement may be further split into halves in order to represent the result as piecewise affine function. Here, an extra function “normalize” is required which re-establishes the ordering on the inequalities in the tree. Since the number of nodes of a region tree is polynomial in the number n of inequalities, and each required subsumption test is polynomial in n and the maximal size of an occurring number, we obtain: Lemma 2. Assume that C is a set of n linear inequalities over a fixed finite set of parameters where the sizes of all occurring numbers are bounded by m. Then the operations “align” as well as addition, scalar multiplication and minimum lifted to region trees with inequalities from C are polynomial-time in the numbers n and m. We build up the nodes of the resulting trees in pre-order. In order to achieve the given complexity bound, we take care not to introduce nodes which correspond to unsatisfiable conjunctions of inequalities, i.e., empty regions. Similar to parametric variable assignments, also parametric strategies are not determined as a whole. Instead, we maintain for each right-hand side a ∨ e1 ∨ . . . ∨ er a separate piecewise constant mapping from Zk into the range [0, r] of natural numbers which identifies for each parameter setting, the subexpression which is currently selected. The idea is that one variable x should not suffer from the fragmentation required for another unrelated variable of the system. Also for the operations “next” and “select”, we obtain: Lemma 3. Assume that C is a finite set of linear inequalities over a fixed finite set of parameters. Then the operations next and select are polynomial-time in the number n of variables and the size of C. Putting lemmas 2, 3 together we conclude that the running time of PSI is fast, whenever the number of occurring inequalities is small and only few strategies are encountered. Summarizing, we obtain: Theorem 3. Consider a system E of integer equations with k parameters. The least parametric solution of E can be computed in time polynomial in the bit size of E, the maximal number of strategies encountered for any parameter setting, and the maximal number of encountered inequalities. Although in our experiments with interval analysis for the benchmark programs used in Section 6, the sets of involved inequalities stayed reasonably small, this however need not always be the case. Example 4. For each m ≥ 0, consider the following system with a single parameter p: xi = xi+1 ∨ −2i + x0i+1 xm = x ∨ −2m + x0 x = p ∧ −p 9 Parametric Strategy Iteration T.M. Gawlitza, M.D. Schwarz, H. Seidl x0i = x0i+1 ∧ 2i + xi+1 x0m = x0 ∧ 2m + x x0 = −p ∨ p where 1 ≤ i < m. Let ρ∗ be the least parametric solution. Then we have:  −p − 2m if p ≤ −2m − 1    0 if −2m ≤ p ≤ 2m , p even ρ∗ x 1 p = −1 if −2m ≤ p ≤ 2m , p odd    m p−2 if 2m + 1 ≤ p Thus, the fragmentation of the mapping ρ∗ necessarily grows exponentially with m. We conclude that for large values m, any parametric analyzer will exhibit an exponential behavior on the system of equations from Example 4. 6 Parametic Program Analysis and Experimental Evaluation As indicated in the introduction, interval analysis for integer variables can be compiled into a finite + system of integer equations. The set of unknowns of this system are of the forms x− u and xu where u is a program point and x is a program variable of the program to be analyzed, and the superscripts −, + indicate the negated lower bounds and the upper bounds of the respective intervals (see [9, 10] for the details of the transformation). The least solution ρ∗ of the integer system then translates into the program invariant which, for program point u and variable x, asserts that all runtime values of x are in ∗ + the interval [−ρ∗ (x− u ), ρ (xu )]. Here, [∞, −∞] signifies the empty set of values (unreachability of u). The transformation of programs into integer equations is readily extended to programs with parameters. For the sake of the transformation, parameters are treated as constants occurring in the program, thus resulting in a parametric system of equations as introduced in Section 2. For the program of Example 1, e.g., we obtain: − x− 1 = p1 ∨ x3 + − − x− 2 = (x1 + ∞); (x1 + p2 − 1); (x1 ∧ ∞) − x− 3 = x2 + (−1) + − − x− 4 = (x1 + (−p2 )); (x1 + ∞); (x1 ∧ −p2 ) + x+ 1 = p1 ∨ x3 + − + x+ 2 = (x1 + ∞); (x1 + p2 − 1); (x1 ∧ p2 − 1) + x+ 3 = x2 + 1 + − + x+ 4 = (x1 + (−p2 )); (x1 + ∞); (x1 ∧ ∞) + The least parametric solution for the unknowns x− 4 and x4 (signifying the bounds of the values of x at program exit) is given by: ξ(x− 4 ) = if − p1 + p2 ≤ 0 then − p1 else − p2 ξ(x+ 4 ) = if − p1 + p2 ≤ 0 then p1 else p2 The resulting parametric invariant for the program exit states that x equals p1 , if −p1 + p2 ≤ 0, and x equals p2 otherwise. We have provided prototypical implementations of parametric strategy iteration for parametric integer equations, and based on these implementations, also for parametric interval equations. For convenience, the user may additionally specify a boolean combination of linear constraints as a global assumption on the parameter values of interest. Thus, we may, e.g., specify that generally, 0 ≤ p1 ∧ p1 ≤ p2 should hold. Then the analyzer takes only considers values less or equal to the tree in Fig. 5. 10 Parametric Strategy Iteration T.M. Gawlitza, M.D. Schwarz, H. Seidl −p1 ≤ 0 const−∞ p1 − p2 ≤ 0 const∞ const−∞ Figure 5: The topmost value under the assumption 0 ≤ p1 ≤ p2 . One implementation based on lists, deals with the one-parameter case only, while the other implementation, which is based on region trees, can deal with multiple parameters. The total ordering ≺ used by our analyzer orders according to the number of variables, where the lexicographical ordering on the vector of coefficients is used for constraints with the same number of variables. For deciding satisfiability of conjunctions of inequalities, we generally rely on Fourier-Motzkin elimination (with integer tightening). Only in the very end, when it comes to produce the final result, we purge regions containing no integer points by means of an integer solver. We have tried our implementations on the rate limiter example from [22] as well as on several small (about 20 interval unknowns) but intricate systems of equations in order to evaluate the impact of the number of parameters as well as the impact of the chosen method for checking emptiness of integer polyhedrons on the practical performance. The tests have been executed on an Intel(R) Core(TM) i5-3427U CPU running Ubuntu. On that machine, parametric interval analysis of the rate limiter example terminated after less than 5s. The remaining benchmarks are based on programs where interval analysis according to the standard widening/narrowing approach fails to compute the least solution. For each example equation system, we successively introduce parameters for the constants used, e.g., in conditions and initializers. The system of equations nested is derived from a program with two independent nested loops. The systems amatoj correspond to three example programs presented in [1]. The system rupak corresponds to an example program by Rupak Majumdar presented at MOD’11. Both amato2 and rupak do not realize a plain interval analysis but additionally track differences of variables. Interestingly, the number of required strategy improvements does not depend on the number of parameters — with the notable exception amato0 where for three and four parameters, the number increases from 8 to 9. Generally, the number of iterations is always significantly lower than the number of unknowns in the system of equations. 0 params 1 params 2 params 3 params 4 params 20 10 0 nested amato0 amato1 amato2 rupak Figure 6: Fragmentation. Figure 6 shows the number of regions in the results with different behavior. For the 0 parameter case this is always 1. As expected, the fragmentation increases with the number of parameters — but not as excessively as we expected. In case of rupak, the number of regions even decreased for three and four parameters. The reason is that by introducing fresh parameters, also the ordering on inequalities 11 Parametric Strategy Iteration T.M. Gawlitza, M.D. Schwarz, H. Seidl changes. The ordering on the other hand may have a significant impact onto fragmentation. 10 0 params 1 params 2 params 3 params 4 params 1 0.1 0.01 nested amato0 amato1 amato2 rupak Figure 7: Execution time in (s). Figure 7 shows the running times of the benchmarks on a logarithmic scale. We visualize the runtimes for 0 through 4 parameters each. The filled and outlined bars correspond to Fourier-Motzkin elimination and integer satisfiability for testing emptiness of regions, respectively. The inscribed red bars in the single parameter case represent the run-time obtained by using linear lists instead of region trees. The bottom-line case without parameters is fast and the dedicated implementation for single parameters increases the run-time only by a factor of approximately 1.4. Using region trees clearly incurs an extra penalty, which increases significantly with the number of parameters. While replacing FourierMotzkin elimination for testing emptiness of regions with an enhanced algorithm for integer satisfiability increases the run-time only by an extra factor of about 1.5. When considering absolute run-times, however, it turns out that the solver even for four parameters together with full integer satisfiability is not prohibitively slow (a few seconds only for all benchmark equation systems). Details on experimental results can be found at www2.in.tum.de/˜seidl/psi. 7 Related Work Parametric analysis of linear numerical program properties has been advocated by Monniaux [22, 23] by compiling the abstract program semantics to real linear arithmetic and then use quantifier elimination to determine the parametric invariants. We have conducted experiments with Monniaux’ tool M JOLL NIR , by which we tried to solve real relaxations of parametric integer equations. Since M JOLLNIR has no native support for positive or negative infinities these values had to be encoded through formulas with extra propositional variables. This approach, however, did not scale to the sizes we needed. Our conjecture is that the high Boolean complexity of the formulas causes severe problems. Beyond that, fewer calls to quantifier elimination for formulas with many variables (as required by M JOLLNIR) may be more expensive than many calls to an integer solver for formulas with few variables (namely, the parameters as in our approach). All in all, since our integer tool and M JOLLNIR tackle slightly different problems, a precise comparison is difficult. Still, our experiments indicates that our approach behaves better than a quantifier elimination-based approach for applications where the control-flow is complex with multiple control-flow points, but where few parameters are of interest. Since long, relational program analyses, e.g., by means of polyhedra have been around [4, 2] which also allow to infer linear relationships between parameters and program variables. The resulting invariants, however, are convex and thus do not allow to differentiate between different linear dependencies in different regions. In order to obtain invariants as precise as ours, one would have to combine polyhedral domains with some form of trace partitioning [20]. These kinds of analysis, though, must rely on widening and narrowing to enforce termination, whereas our algorithms avoid widening and narrowing 12 Parametric Strategy Iteration T.M. Gawlitza, M.D. Schwarz, H. Seidl completely and directly compute least solutions, i.e., the best possible parametric invariants. Parametric analysis of a different kind has also been proposed by Reineke [25] in the context of worst-case execution time (WCET). They rely on parametric linear programming as implemented by the PIP tool [7] and infer the dependence of the WCET on architecture parameters such as the chache size by means of black box sampling of the WCETs obtained for different parameter settings. Our data structure of region trees is a refinement of the tree-like data-structure Q UAST provided by the PIP tool [7]. Similar data-structures are also used by Monniaux [22] to represent the resulting invariants, and by Mihaila et al. [21] to differentiate between different phases of a loop iteration. In our implementation, we additionally enforce a total ordering on the constraints in the tree nodes and allow arbitrary values at the leaves. Total orderings on constraints have also been proposed for linear decision diagrams [3]. Variants of LDDs later have been used to implement non-convex linear program invariants [13, 12] and in [14] for representing linear arithmetic formulas when solving predicate abstraction queries. In our application, sharing of subtrees is not helpful, since each node v represents a satisfiable conjunction of the inequalities which is constituted by the path reaching v from the root of the datastructure. Moreover, our application requires that the leaves of the data-structure are not just annotated with a Boolean value (as for LDDs), but with values from various sets, namely strategic choices, affine functions or even pairs thereof. 8 Conclusion Solving systems of parametric integer equations allows to solve also systems of parametric interval equations, and thus to realize parametric program analysis for programs using integer variables. To solve parametric integer equations, we have presented parametric strategy iteration. Instead of solving integer optimization and satisfiability problems involving all unknowns of the problem formulation (as an approach based on quantifier elimination), our algorithm is a smooth generalization of ordinary strategy iteration, which applies integer satisfiability to inequalities involving parameters only. Our prototypical implementation indicates that this approach indeed has the potential to deal with nontrivial problems — at least when only few parameters are involved. Introducing further parameters significantly increases the analysis costs. Surprisingly, the number of strategies required as well as the fragmentation observed in our examples increased only moderately. Accordingly, the required running times were quite decent. More experiments are necessary, though, to obtain a deeper understanding of parametric strategy iteration. Also, we are interested in exploring the practical potential of sensitivity and mode analysis enabled by this new algorithm, e.g., for automotive and avionic applications. Acknowledgement. We thank Stefan Barth (LMU) for Example 4, and Jan Reineke (Universität des Saarlandes) for useful discussions. References [1] Gianluca Amato and Francesca Scozzari. Localizing widening and narrowing. In Static Analysis Symposium, volume 7935 of LNCS, pages 25–42. Springer, 2013. [2] Roberto Bagnara, Patricia M. Hill, and Enea Zaffanella. The Parma polyhedra library: Toward a complete set of numerical abstractions for the analysis and verification of hardware and software systems. Sci. Comput. Program., 72(1–2):3–21, 2008. [3] Sagar Chaki, Arie Gurfinkel, and Ofer Strichman. Decision diagrams for linear arithmetic. In Formal Methods in Computer-Aided Design, pages 53–60. IEEE Press, 2009. [4] Patrick Cousot and Nicolas Halbwachs. Automatic discovery of linear restraints among variables of a program. In Principles of Programming Languages, pages 84–96. ACM Press, 1978. 13 Parametric Strategy Iteration T.M. Gawlitza, M.D. Schwarz, H. Seidl [5] Isil Dillig, Thomas Dillig, and Alex Aiken. Cuts from proofs: A complete and practical technique for solving linear inequalities over integers. In Computer Aided Verification, volume 5643 of LNCS, pages 233–247. Springer, 2009. [6] Paul Feautrier. Parametric integer programming. RAIRO Recherche Opérationnelle, 22:243–268, 1988. [7] Paul Feautrier, Jean-François Collard, and Cédric Bastoul. A Solver for Parametric Integer Programming Problems, 2007. [8] Thomas Gal and Harvey J. Greenberg, editors. Advances in Sensitivity Analysis and Parametric Programming. Kluwer Academic Publishers, 1997. [9] Thomas Gawlitza and Helmut Seidl. Precise fixpoint computation through strategy iteration. In Programming Languages and Systems, volume 4421 of LNCS, pages 300–315. Springer, 2007. [10] Thomas Gawlitza and Helmut Seidl. Precise interval analysis vs. parity games. In Formal Methods, volume 5014 of LNCS, pages 342–357. Springer, 2008. [11] Thomas Gawlitza and Helmut Seidl. Abstract interpretation over zones without widening. In Workshop on Invariant Generation, volume 1 of EPiC, pages 12–43. EasyChair, 2012. [12] Khalil Ghorbal, Franjo Ivancic, Gogul Balakrishnan, Naoto Maeda, and Aarti Gupta. Donut domains: Efficient non-convex domains for abstract interpretation. In Verification, Model Checking, and Abstract Interpretation, volume 7148 of LNCS, pages 235–250. Springer, 2012. [13] Arie Gurfinkel and Sagar Chaki. Boxes: A symbolic abstract domain of boxes. In Static Analysis Symposium, volume 6337 of LNCS, pages 287–303. Springer, 2010. [14] Arie Gurfinkel, Sagar Chaki, and Samir Sapra. Efficient predicate abstraction of program summaries. In NASA Formal Methods, volume 6617 of LNCS, pages 131–145. Springer, 2011. [15] D. Halperin. Arrangements. In Joseph O’Rourke Jacob E. Goodman, editor, Handbook of Discrete and Computational Geometry (2nd Edition), chapter 24, pages 529–562. CRC Press, Inc., 2004. [16] A. Holder. Parametric LP analysis. Encyclopedia of Operations Research and Management Science, 2011. [17] Leonid G. Khachiyan. Polynomial algorithms in linear programming. USSR Computational Mathematics and Mathematical Physics, 20(1):53–72, 1980. [18] H.W. Lenstra, A.K. Lenstra, and L. Lovász. Factoring polynomials with rational coefficients. Mathematische Annalen, 261:515–534, 1982. [19] Jesús A. De Loera, David Haws, Raymond Hemmecke, Peter Huggins, and Ruriko Yoshida. Three kinds of integer programming algorithms based on Barvinok’s rational functions. In Integer Programming and Combinatorial Optimization, volume 3064 of LNCS, pages 244–255. Springer, 2004. [20] L. Mauborgne and X. Rival. Trace partitioning in abstract interpretation based static analyzers. In European Symposium on Programming, volume 3444 of LNCS, pages 5–20. Springer, 2005. [21] B. Mihaila, A. Sepp, and A. Simon. Widening as abstract domain. In NASA Formal Methods, volume 7871 of LNCS, pages 170–186. Springer, 2013. [22] David Monniaux. Automatic modular abstractions for linear constraints. In Principles of Programming Languages, pages 140–151. ACM Press, 2009. [23] David Monniaux. Quantifier elimination by lazy model enumeration. In Computer Aided Verification, volume 6174 of LNCS, pages 585–599. Springer, 2010. [24] William Pugh. The omega test: a fast and practical integer programming algorithm for dependence analysis. In Supercomputing, pages 4–13. IEEE Press, 1991. [25] Jan Reineke and Johannes Doerfert. Architecture-parametric timing analysis. In Real-Time and Embedded Technology and Applications Symposium, April 2014. To appear. 14 

arXiv:1604.03115v2 [math.CO] 4 Apr 2017 A BROAD CLASS OF SHELLABLE LATTICES JAY SCHWEIG AND RUSS WOODROOFE Abstract. We introduce a new class of lattices, the modernistic lattices, and their duals, the comodernistic lattices. We show that every modernistic or comodernistic lattice has shellable order complex. We go on to exhibit a large number of examples of (co)modernistic lattices. We show comodernism for two main families of lattices that were not previously known to be shellable: the order congruence lattices of finite posets, and a weighted generalization of the k-equal partition lattices. We also exhibit many examples of (co)modernistic lattices that were already known to be shellable. To begin with, the definition of modernistic is a common weakening of the definitions of semimodular and supersolvable. We thus obtain a unified proof that lattices in these classes are shellable. Subgroup lattices of solvable groups form another family of comodernistic lattices that were already proven to be shellable. We show not only that subgroup lattices of solvable groups are comodernistic, but that solvability of a group is equivalent to the comodernistic property on its subgroup lattice. Indeed, the definition of comodernistic exactly requires on every interval a lattice-theoretic analogue of the composition series in a solvable group. Thus, the relation between comodernistic lattices and solvable groups resembles, in several respects, that between supersolvable lattices and supersolvable groups. 1. Introduction Shellings are a main tool in topological combinatorics. An explicit shelling of a simplicial complex ∆ simultaneously shows the sequentially Cohen-Macaulay property, computes homotopy type, and gives a cohomology basis. Frequently, a shelling also gives significant insight into the homeomorphy of ∆. The downside is that shellings are often difficult to find, and generally require a deep understanding of the complex. In this paper, we describe a large class of lattices whose order complexes admit shellings. The shellings are often straightforward to explicitly write down, and so give a large amount of information about the topology of the order complex. Included in our class of lattices are many examples which were not previously understood to be closely related. The question that first motivated this research project involved shelling a particular family of lattices. The order congruence lattice O(P ) of a finite poset P is the subposet of the partition lattice consisting of all equivalence classes arising as the level sets of an orderpreserving function. Order congruence lattices interpolate between Boolean lattices and partition lattices, as we will make precise later. Such lattices were already considered by Sturm in [39]. More recently, Körtesi, Radeleczki and Szilágyi showed the order congruence 1 A BROAD CLASS OF SHELLABLE LATTICES 2 lattice of any finite poset to be graded and relatively complemented [22], while Jenča and Sarkoci showed such lattices to be Cohen-Macaulay [21]. The Cohen-Macaulay result naturally suggested to us the question of whether every order congruence lattice is shellable. After proving the answer to this question to be “yes”, we noticed that our techniques apply to a much broader class of lattices. Indeed, a large number of the lattices previously shown to be shellable lie in our class. Thus, our main result (Theorem 1.2 below) unifies numerous results on shellability of order complexes of lattices, in addition to proving shellability for new examples. It is our belief that our results will be useful to other researchers. Finding a shelling of a lattice can be a difficult problem. In many cases, showing that a lattice is in our class may be simpler than constructing a shelling directly. All lattices, posets, simplicial complexes, and groups considered in this paper will be finite. 1.1. Modernistic and comodernistic lattices. We now define the broad class of lattices described in the title and introduction. Our work relies heavily on the theory of modular elements in a lattice. Recall that an element m of a lattice L is left-modular if whenever x < y are elements of L, then the expression x ∨ m ∧ y can be written without parentheses, that is, that (x ∨ m) ∧ y = x ∨ (m ∧ y). An equivalent definition is that m is left-modular if m is not the nontrivial element in the short chain of any pentagonal sublattice of L; see Lemma 2.10 for a more precise statement. Our key object of study is the following class of lattices. Definition 1.1. We say that a lattice L is modernistic if every interval in L has an atom that is left-modular (in the given interval). We say that L is comodernistic if the dual of L is modernistic, that is, if every interval has a left-modular coatom. Our main theorem is as follows. (We will recall the definition of a CL-labeling in Section 2.3 below.) Theorem 1.2. If L is a comodernistic lattice, then L has a CL-labeling. Corollary 1.3. If L is either comodernistic or modernistic, then the order complex of L is shellable. The CL-labeling is explicit from the left-modular coatoms, so Theorem 1.2 also gives a method for computing the Möbius function of L. See Lemma 3.7 for details. We find it somewhat surprising that Theorem 1.2 was not proved before now. We speculate that the reason may be the focus of previous authors on atoms and CL-labelings, whereas Theorem 1.2 requires dualizing exactly one of the two. Remark 1.4. The name “modernistic” comes from contracting “atomically modular” to “atomically mod”. Since atomic was a common superlative from the the late 1940’s, and since the mod (or modernistic) subculture was also active at about the same time, we find the name to be somewhat appropriate, as well as short and perhaps memorable. A BROAD CLASS OF SHELLABLE LATTICES 3 1.2. Examples and applications. Theorem 1.2 has a large number of applications, which we briefly survey now. First, we can now solve the problem that motivated the project. Theorem 1.5. If P is any poset, then the order congruence lattice O(P ) is comodernistic, hence CL-shellable. We also recover as examples many lattices already known to be shellable. The following theorem lists some of these, together with references to papers where they are shown to be shellable. Proposition 1.6. The following lattices are comodernistic, hence CL-shellable: (1) Supersolvable and left-modular lattices, and their order duals [2, 23, 26]. (2) Order duals of semimodular lattices [2]. (I.e., semimodular lattices are modernistic.) (3) k-equal partition lattices [7], and their type B analogues [5]. (4) Subgroup lattices of solvable groups [33, 43]. We comment that many of these lattices are shown in the provided references to have EL-labelings. Theorem 1.2 provides only a CL-labeling. Since the CL-labeling constructed is explicit from the left-modular elements, Theorem 1.2 provides many of the benefits given by an EL-labeling. We do not know if every comodernistic lattice has an EL-labeling, and leave this interesting question open. Experts in lattice theory will immediately recognize items (1) and (2) from Proposition 1.6 as being comodernistic. Theorem 1.2 thus unifies the theory of these well-understood lattices with the more difficult lattices on the list. The CL-labeling that we construct in the proof of Theorem 1.2 can moreover be seen as a generalization of the standard EL-labeling for a supersolvable lattice, further connecting these classes of lattices. We will prove that k-equal partition lattices and their type B analogues are comodernistic in Section 6. In Section 6.3 we will show the same for a new generalization of k-equal partition lattices. The proofs show, broadly speaking, that coatoms of (intervals in) these subposets of the partition lattice inherit left-modularity from that in the partition lattice. Although modernism and comodernism give a simple and unified framework for showing shellability of many lattices, not every shellable lattice is (co)modernistic. For an easy example, the face lattice of an n-gon has no left-modular elements when n > 3, so is neither modernistic nor comodernistic. 1.3. Further remarks on subgroup lattices. We can expand on the connection with group theory suggested by item (4) of Proposition 1.6. For a group G, the subgroup lattice referred to in this item consists of all the subgroups of G, ordered by inclusion; and is denoted by L(G). Theorem 1.7. If G is a group, then G is solvable if and only if L(G) is comodernistic. Stanley defined supersolvable lattices in [34] to abstract the interesting combinatorics of the subgroup lattices of supersolvable groups to general lattices. Theorem 1.7 says that comodernism is one possibility for a similar abstraction for solvable groups. A result of a A BROAD CLASS OF SHELLABLE LATTICES 4 similar flavor was earlier proved by Schmidt [30]; our innovation with comodernism is to require a lattice-theoretic analogue of a composition series in every interval of the lattice. We further discuss possible notions of solvability for lattices in Section 5. Shareshian in [33] showed that a group G is solvable if and only if L(G) is shellable. Theorems 1.2 and 1.7 give a new proof of the “only if” direction of this result. For the “if” direction, Shareshian needed a hard classification theorem from finite group theory. Our proof of Theorem 1.7 does not rely on hard classification theorems. On the other hand, it follows directly from Shareshian’s proof that G is solvable if L(G) is sequentially Cohen-Macaulay. Thus, Shareshian’s Theorem gives a topological characterization of solvable groups. The characterization given in Theorem 1.7 is lattice-theoretic, rather than topological. It is an interesting open problem to give a classification-free proof that if L(G) is sequentially Cohen-Macaulay, then G is solvable. 1.4. Organization. This paper is organized as follows. In Section 2 we recall some of the necessary background material. We prove Theorem 1.2 (our main theorem) in Section 3. In the remainder of the paper we show how to apply comodernism and Theorem 1.2 to various classes of lattices. The techniques may be illustrative for those who wish to prove additional classes of lattices to be comodernistic. In Section 4, we examine order congruence lattices, and prove Theorem 1.5. In Section 5 we prove Theorem 1.7, and argue for comodernism as a notion of solvable for lattices. We close in Section 6 by showing that k-equal partition lattices (and variations thereof) are comodernistic. Acknowledgements We would like to thank Vincent Pilaud and Michelle Wachs for their helpful remarks. We also thank the anonymous referee for his or her thoughtful comments. The example in Figure 5.1 arose from a question of Hugh Thomas. The second author is grateful to the University of Miami for their hospitality in the spring of 2016, during which time a part of the paper was written. 2. Preliminaries We begin by recalling some necessary background and terminology. Many readers will be able to skip or skim this section, and refer back to it as necessary. 2.1. Posets, lattices, and order complexes. A poset P is bounded if P has a unique least element 0̂ and greatest element 1̂. Associated to a bounded poset P is the order complex, denoted ∆P , a simplicial complex whose faces consist of all chains (totally ordered subsets) in P \ {0̂, 1̂}. In particular, the vertices of ∆P are the chains of length 0 in P \ {0̂, 1̂}, that is, the elements of P \ {0̂, 1̂}. The importance of the order complex in poset theory arises since the Möbius function µ(P ) (important for inclusion-exclusion) is given by the reduced Euler characteristic χ̃(∆P ). We often say that a bounded poset P possesses a property from simplicial topology (such as “shellability”), by which we mean that ∆P has the same property. A BROAD CLASS OF SHELLABLE LATTICES 5 We say that a poset P is Hasse-connected if the Hasse diagram of P is connected as a graph. That is, P is Hasse-connected if and only for any x, y ∈ P , there is a sequence x = x0 , x1 , . . . , xk = y such that xi is comparable to xi+1 for each i. A poset L is a lattice if every two elements x, y ∈ L have a unique greatest lower bound (the meet x ∧ y) and unique least upper bound (the join x ∨ y). It is obvious that every lattice is bounded, hence has an order complex ∆L. A poset is graded if all its maximal chains have the same length, where the length of a chain is one less than its cardinality. The height of a bounded poset P is the length of the longest chain in P , and the height of an element x is the height of the interval [0̂, x]. An atom of a bounded poset P is an element of height 1. The order dual of a poset P is the poset P ∗ with reversed order relation, so that x <∗ y in P ∗ exactly when x > y in P . Poset definitions may be applied to the dual by prepending a “co”: for example, an element x is a coatom if x is an atom in P ∗ . For more background on poset and lattice theory from a general perspective, we refer to [37]. For more on order complexes and poset topology, we refer to [42]. 2.2. Simplicial complexes and shellings. We assume basic familiarity with homology and cohomology, as exposited in e.g. [18, 27]. A shelling of a simplicial complex ∆ is an ordering of the facets of ∆ that obeys certain conditions, the precise details of which will not be important to us. Not every simplicial complex has a shelling; those that do are called shellable. We remark that in the early history of the subject, shellings were defined only for balls and spheres [29]. Later, shellings were considered only for pure complexes, that is, complexes all of whose facets have the same dimension. Nowadays, shellings are studied on arbitrary simplicial complexes [7]. Shellable complexes are useful for showing a complex to satisfy the Cohen-Macaulay (in the pure case) or sequentially Cohen-Macaulay property (more generally). These properties are important in commutative algebra as well as combinatorics. We refer to [36] for more on shellable and Cohen-Macaulay complexes. 2.3. CL-labelings and EL-labelings. The definition of a shelling is often somewhat unwieldy to work with directly, and it is desirable to find tools through which to work. One such tool is given by a CL-labeling, which we will now define. If x and y are elements in a poset P , we say that y covers x when x < y but there is no z ∈ P so that x < z < y. In this situation, we write x ⋖ y, and may also say that x ⋖ y is a cover relation. Thus, a cover relation is an edge in the Hasse diagram of P . A rooted cover relation is a cover relation x ⋖ y together with a maximal chain from 0̂ to x (called the root). A rooted interval is an interval [x, y] together with a maximal chain r from 0̂ to x. In this situation, we use the notation [x, y]r . Notice that every atomic cover relation of [x, y]r can be rooted by r. A chain-edge labeling of a bounded poset P is a function λ that assigns an element of an ordered set (which will always for us be Z) to each rooted cover relation of P . Then λ assigns A BROAD CLASS OF SHELLABLE LATTICES 6 a word over Z to each maximal chain on any rooted interval by reading the cover relation labels in order, so e.g. the word associated with 0̂ ⋖ x1 ⋖ x2 ⋖ x3 ⋖ . . . is λ(0̂ ⋖ x1 , 0̂)λ(x1 ⋖ x2 , 0̂ ⋖ x1 )λ(x2 ⋖ x3 , 0̂ ⋖ x1 ⋖ x2 ) · · · . Remark 2.1. Since many researchers may be less familiar with CL-labelings and the machinery behind them, it may be helpful to think of a chain-edge labeling via the following dynamical process. Begin at 0̂, and walk up the maximal chain 0̂ = x0 ⋖ x1 ⋖ x2 ⋖ · · · ⋖ 1̂. At each step i, assign a label to the label xi−1 ⋖ xi . In assigning the label, you are allowed to look backwards at where you have been, but are not allowed to look forwards at where you may go. At each step, you add the assigned label to the end of a word associated with the maximal chain. We say that a maximal chain c is increasing if the word associated with c is strictly increasing, and decreasing if the word is weakly decreasing. We order maximal chains by the lexicographic order on the associated words. Definition 2.2. A CL-labeling is a chain-edge labeling that satisfies the following two conditions on each rooted interval [x, y]r : (1) There is a unique increasing maximal chain m on [x, y]r , and (2) the increasing chain m is strictly earlier in the lexicographic order than any other maximal chain on [x, y]r . If a CL-labeling λ assigns the same value to every x ⋖ y irrespective of the choice of root, then we say λ is an EL-labeling. Björner [2] and Björner and Wachs [6, 7] introduced CL-labelings, and proved the following theorem. Theorem 2.3. [8, Theorem 5.8] If λ is a CL-labeling of the bounded poset P , then the lexicographic order on the maximal chains of P is a shelling order of ∆P . In this case, a cohomology basis for ∆P is given by the decreasing maximal chains of P , and ∆P is homotopy equivalent to a bouquet of spheres in bijective correspondence with the decreasing maximal chains. For this reason, bounded posets with a CL- or EL-labeling are often called CL- or ELshellable. Since the order complex of P and that of the order dual of P coincide, either a CL-labeling or a dual CL-labeling implies shellability of ∆P . From the cohomology basis, it is straightforward to compute Euler characteristic, hence also Möbius number. Corollary 2.4. [8, Proposition 5.7] If λ is a CL-labeling of the bounded poset P , then the Möbius number of P is given by µ(P ) = χ̃(∆P ) = #even length decreasing maximal chains in P − #odd length decreasing maximal chains in P. A BROAD CLASS OF SHELLABLE LATTICES 7 2.4. Order congruence lattices. If P and Q are posets, then a map ϕ : P → Q is orderpreserving if whenever x ≤ y, it also holds that ϕ(x) ≤ ϕ(y). The level set partition of a map ϕ : P → Q is the partition with blocks of the form ϕ−1 (q). If π is the level set partition of an order preserving map ϕ : P → Q, then π is an order partition of P . Since every poset has a linear extension, it is easy to see that it would be equivalent to restrict the definition of order partition to the case where Q = Z. As previously defined, the order congruence lattice O(P ) is the subposet of the partition lattice ΠP consisting of all order partitions of P . The cover relations in O(P ) correspond to merging blocks in an order partition, subject to a certain compatibility condition. Example 2.5. Consider P = [3] with the usual order. Then the function mapping 1, 2 to 1 and 3 to 2 is order-preserving, so 12 | 3 ∈ O([3]). Similarly, the partition 1 | 23 ∈ O([3]). It is not difficult to see, however, that there is no order-preserving map with level set partition 13 | 2. Thus, the lattice O([3]) is isomorphic to the Boolean lattice on 2 elements. More generally, an elementary argument shows that the order congruence lattice of a chain on n elements is isomorphic to a Boolean lattice on n − 1 elements. It is obvious that the order congruence lattice of an antichain is the usual partition lattice. Thus, order congruence lattices interpolate between Boolean lattices and partition lattices. It is easy to confuse O(P ) with another closely related lattice defined on a poset P . We say that a subset S ⊆ P is order convex if whenever a ≤ b ≤ c with a, c ∈ S, then also b ∈ S. The order convexity partition lattice of P , denoted Oconv (P ), consists of all partitions where every block is order convex. There is some related literature on the related lattice of all order convex subsets of a poset, going back to [1]. We do not know if order convexity lattices must always be comodernistic or shellable, as intervals of the form [π, 1̂] in Oconv seem difficult to describe. Example 2.6. Consider the bowtie poset B, with elements a1 , a2 , b1 , b2 and relations ai < bj (for i, j ∈ {1, 2}). As B has height 1, all subsets are order convex, so that Oconv (B) ∼ = Π4 . However, the partitions a1 b1 | a2 b2 and a1 b2 | a2 b1 are not order congruence partitions, so are not in O(B). We additionally caution the reader that the notion of order congruence considered here is less restrictive than that considered in [28], where congruences are required to respect lower/upper bounds. In another point of view, it is straightforward to show that order-preserving partitions are in bijective correspondence with certain quotient objects of P . Thus, the order congruence lattice assigns a lattice structure to quotients of P . See [21, Section 3] for more on the quotient view of O(P ). For our purposes, it will be enough to understand intervals above atoms in O(P ). Say that elements x, y of poset P are compatible if either x ⋖ y, y ⋖ x, or x, y are incomparable. If x, y are compatible in P , then Px∼y is the poset obtained by identifying x and y. That is, Px∼y is obtained from P by replacing x, y with w, subject to the relations z < w whenever A BROAD CLASS OF SHELLABLE LATTICES 8 z < x or z < y, and z > w whenever z > x or z > y. We remark in passing that this identification is an easy special case of the quotienting viewpoint discussed above. Lemma 2.7. Let P be a poset. A partition π of P is an atom of O(P ) if and only if π has exactly one non-singleton block consisting of compatible elements {x, y}. In this situation, we have the lattice isomorphism [π, 1̂]O(P ) ∼ = O(Px∼y ). Repeated application of Lemma 2.7 allows us to understand any interval of the form [π, 1̂] in O(P ). Although we will not use need this, intervals of the form [0̂, π] are also not difficult to understand. Let π have blocks B1 , B2 , . . . , Bk . It is well-known (see e.g. [37, Example 3.10.4]) that an interval of this form in the full partition lattice is isomorphic to the product of smaller partition lattices ΠB1 × · · · × ΠBk . It is straightforward to see via the orderpreserving mapping P → Z definition that a similar result holds in the order congruence lattice. That is, [0̂, π] in O(P ) is lattice-isomorphic to O(B1 ) × · · · × O(Bk ), where Bi refers to the induced subposet on Bi ⊆ P . Combining this observation with Lemma 2.7 allows us to write any interval in O(P ) as a product of order congruence lattices of quotients of subposets. We find it simpler to give more direct arguments, but readers familiar with poset products may appreciate this connection. 2.5. Supersolvable and semimodular lattices. We previously defined an element m of a lattice L to be left-modular if (x ∨ m) ∧ y = x ∨ (m ∧ y) for all pairs x < y. A lattice is modular if every element is left-modular. A lattice L is usually defined to be semimodular if whenever a ∧ b ⋖ a in L, then b ⋖ a ∨ b. We prefer the following equivalent definition, which highlights the close connection between semimodularity and comodernism: Lemma 2.8 (see e.g. [38, essentially Theorem 1.7.2]). A lattice L is semimodular if and only for every interval [x, y] of L, every atom of [x, y] is left-modular (as an element of [x, y]). Thus, the definition of a modernistic lattice is obtained from that of a semimodular lattice by weakening a single universal quantifier to an existential quantifier. An M-chain in a lattice is a maximal chain consisting of left-modular elements. A lattice is left-modular if it has an M-chain, and supersolvable if it is graded and left-modular. Supersolvable lattices were originally defined by Stanley [34], in a somewhat different form. The theory of left-modular lattices was developed in a series of papers [10, 23, 24, 26], and it was only in [26] that it was noticed that Stanley’s original definition of supersolvable is equivalent to graded and left-modular. There is an explicit cohomology basis for a supersolvable lattice, which does not seem to be as well-known as is deserved. A chain of complements to an M-chain m = {0̂ = m0 ⋖ m1 ⋖ · · · ⋖ mn = 1̂} is a chain of elements c = {0̂ = cn ⋖ cn−1 ⋖ · · · ⋖ c0 = 1̂} so that each ci is a complement to mi , that is, so that ci ∨ mi = 1̂ and ci ∧ mi = 0̂. A less explicit form of the following appears in [2, 34], and a special case in [41]. A BROAD CLASS OF SHELLABLE LATTICES 9 Theorem 2.9. If L is a supersolvable lattice with a fixed M-chain m, then a cohomology basis for ∆L is given by the chains of complements to m. In particular, the Möbius number of L is (up to sign) the number of such chains. A (strong form of a) homology basis for supersolvable lattices appears in [32]. 2.6. Left-modularity. We now recall some additional basic properties of left-modular elements. First, we state more carefully the equivalent “no pentagon” condition mentioned in the Introduction. Lemma 2.10. [24, Proposition 1.5] An element m of the lattice L is left-modular if and only if for every a < c in L, we have a ∧ m 6= c ∧ m or a ∨ m 6= c ∨ m. The pentagon lattice (usually notated as N5 ) consists of elements 0̂, 1̂, a, b, c with the only nontrivial relation being a < c. Lemma 2.10 says exactly that m is left-modular if and only if m never plays the role of b in a sublattice of L isomorphic to N5 . Thus, Lemma 2.10 is a pleasant generalization of the characterization of modular lattices as those with no pentagon sublattices. Another useful fact is: Lemma 2.11. [23, Proposition 2.1.5] If m is a left-modular element of the lattice L, and x < y in L, then x ∨ m ∧ y is a left-modular element of the interval [x, y]. Finally, we give an alternate characterization of left-modularity of coatoms, in the flavor of Lemma 2.8. This characterization will be useful for us in the proof of Theorems 1.2 and 3.5, and is also often easy to check. Lemma 2.12 (Left-modular Coatom Criterion). Let m be a coatom of the lattice L. Then m is left-modular in L if and only if for every y such that y 6≤ m we have m ∧ y ⋖ y. Proof. If m ∧ y < z < y, then z < y violate the condition of Lemma 2.10 with m, hence m is not left-modular. Conversely, if z < y violate the condition of Lemma 2.10, then y 6≤ m, and m ∧ y = m ∧ z < z < y.  Corollary 2.13. If L is a lattice and m a left-modular coatom of L, then for any x < y in L either x ∨ m ∧ y = y or else x ∨ m ∧ y ⋖ y. Proof. If x 6≤ m or y ≤ m, then x ∨ m ∧ y = y. Otherwise, apply Lemmas 2.11 and 2.12.  2.7. Group theory. We recall that a group G is said to be solvable if either of the following equivalent conditions is met: (1) There is a chain 1 = N0 ⊂ N1 ⊂ N2 ⊂ · · · ⊂ Nk = G of subgroups in G, so that each Ni is normal in G, and so that each factor Ni /Ni−1 is abelian. (2) There is a chain 1 = H0 ⊂· H1 ⊂· H2 ⊂· · · · ⊂· Hn = G of subgroups in G, so that each Hi is normal in Hi+1 (but is not necessarily normal in G). Note that it follows in this case that each factor Hi /Hi−1 is cyclic of prime order. Since every subgroup of a solvable group is solvable, an alternative form of the latter is: A BROAD CLASS OF SHELLABLE LATTICES 10 (2’) For every subgroup H ⊆ G, there is a subgroup K ⊂· H such that K ⊳ H. A subgroup H of G is said to be subnormal if there is a chain H ⊳ L1 ⊳ L2 ⊳ · · · ⊳ G. Thus, Condition (2) says a group is solvable if and only if G has a maximal chain consisting of subnormal subgroups. A group is supersolvable if there is a maximal chain in L(G) consisting of subgroups normal in G. Thus, a group is supersolvable if there is a chain which simultaneously meets the conditions in (1) and (2). One important fact about the subgroup lattice of supersolvable groups is: Theorem 2.14. [20] For a group G, the subgroup lattice L(G) is graded if and only if G is supersolvable. Subgroup lattices were one of the motivations for early lattice theorists in making the definition of (left-)modularity. It follows easily from the Dedekind Identity (see Lemma 5.3) that if N ⊳ G, then N is left-modular in L(G). In particular, if G is a supersolvable group, then L(G) is a supersolvable lattice. Moreover, a normal subgroup N satisfies a stronger condition. An element m of a lattice is said to be modular (or two-sided-modular ) if it neither plays the role of b nor of a in any pentagon sublattice, where a, b are as in the discussion following Lemma 2.10. A second application of the Dedekind Identity shows any normal subgroup to be modular in L(G). The following lemma, whose proof is immediate from the definitions, says that leftmodularity and two-sided-modularity are essentially the same for the purpose of comodernism arguments. Lemma 2.15. If L is a lattice, and m is a maximal left-modular element, then m is modular. We refer to e.g. [12, Chapter A] for further general background on group theory, and to [31] for the reader interested in further background on lattices of subgroups. 3. Proof of Theorem 1.2 3.1. sub-M-chains. As discussed in Section 2.5, a lattice is left-modular if it has an Mchain, that is, a maximal chain consisting of left-modular elements. The reader may be reminded of the maximal chain consisting of normal elements in the definition of a supersolvable group. We extend the notion of M-chain to comodernistic lattices. A maximal chain 0̂ = m0 ⋖ m1 ⋖ · · · ⋖ mn = 1̂ in L is a sub-M-chain if for every i, the element mi is left-modular in the interval [0̂, mi+1 ]. The reader may be reminded of the maximal subnormal chain in a solvable group. It is straightforward to show that a lattice is comodernistic if and only if every interval has a sub-M-chain. Stanley [35] and Björner [2] showed that any supersolvablen lattice has an EL-labeling, and o (ss) (ss) (ss) (ss) Liu [23] extended this to any left-modular lattice. If m = 0̂ = m0 ⋖ m1 ⋖ · · · ⋖ mn = 1̂ A BROAD CLASS OF SHELLABLE LATTICES 11 is an M-chain, then the EL-labeling is defined as follows: (3.1) (ss) λss (x ⋖ y) = max{i : x ∨ mi−1 ∧ y = x} (ss) = min{i : x ∨ mi ∧ y = y}. The essential observation involved in proving Theorem 1.2 is that, if we replace the Mchain used for λss with a sub-M-chain, then we can still label the atomic cover relations of L in the same manner as in λss . More precisely, if m is a sub-M-chain in a lattice L, then let (3.2) λ(0̂ ⋖ a) = 1 + max{i : mi ∧ a = 0̂}. Adding 1 is not essential, and we do so only so that the labels will be in the range 1 through n, rather than 0 through n − 1. 3.2. The CL-labeling. We construct the full CL-labeling recursively from (3.2). We say that a chain c is indexed by a subset S = {i1 < · · · < ik } of the integers if c = {ci1 < · · · < cik }. That is, we associate an index or label with each element of the chain. Notice that we require the indices to (strictly) respect order. We will need the following somewhat-technical lemma to handle non-graded lattices. Lemma 3.1. Let L be a lattice with a sub-M-chain m of length n. Then no chain of L has length greater than n. Proof. We proceed by induction on n. The base case is trivial. Suppose that 0̂ = c0 ⋖ c1 ⋖ · · · ⋖ cℓ = 1̂ is some chain in L. Let mn−1 ⋖ 1 be the unique coatom in m, and i be the greatest index such that ci ≤ mn−1 . Then cj ∨ mn−1 = 1̂ for any j > i, so by the left-modular property ci+1 ∧ mn−1 < ci+2 ∧ mn−1 < · · · < cℓ ∧ mn−1 = mn−1 . Thus, 0̂ = c0 ⋖ c1 ⋖ · · · ⋖ ci = ci+1 ∧ mn−1 < ci+2 ∧ mn−1 < · · · < cℓ ∧ mn−1 = mn−1 is a chain of length ℓ − 1 on [0̂, mn−1 ], and by induction ℓ − 1 ≤ n − 1.  Definition 3.2. Let L be a comodernistic lattice of height n. Take a fixed sub-M-chain m given as 0̂ = m0 ⋖ m1 ⋖ · · · ⋖ mn = 1̂ as the starting point for a recursive construction. Let x ⋖ a, r be a rooted cover relation. Assume by recursion that we are given a subM-chain m(r) on [x, 1̂]. Further assume that the elements of m(r) are indexed by a subset (r) S ⊆ [n] ∪ {0}, and that 1̂ = mn . Label x ⋖ a as in (3.2) that is, as (3.3) (r) λ(x ⋖ a) = 1 + max{i : mi ∧ a = x}. To continue the recursion, it remains to construct an indexed sub-M-chain m(r∪a) on [a, 1̂]. (r) Suppose that λ(x ⋖ a) = 1 + i. It is clear that mi is the greatest element of m(r) such (r) (r) (r) that a 6≤ mi . By abuse of notation, let m>i be the portion of m that is greater than mi , A BROAD CLASS OF SHELLABLE LATTICES 12 (r) and let S>i be the indices greater than i on m(r) . Thus, the labels of m>i are exactly S>i . Let S<i = S \ (S>i ∪ i) similarly be the indices less than i on m(r) . (r) Now by construction, all elements of m>i are greater than a. By the comodernistic (r) property, the submodular chain m>i may be completed to a sub-M-chain m(r∪a) for [a, 1̂]. (r) Preserve the indices on m>i , and index the elements of m(r∪a) \ m(r) by elements of S<i . It follows by applying Lemma 3.1 on [0̂, mi+1 ] that there are enough indices available in S<i to perform such indexing. The recursion can now continue, which completes the definition of the CL-labeling. Notation 3.3. Throughout the remainder of Section 3, we fix L to be a comodernistic lattice of height n, with a sub-M-chain m = {0̂ = m0 ⋖ m1 ⋖ · · · ⋖ mn = 1̂}. Indeed, we select a sub-M-chain on every interval, which uniquely determines a chain-edge labeling λ as in Definition 3.2. Remark 3.4. By repeated application of Lemma 2.11, it follows that if L has an M-chain m = {0̂ = m0 ⋖ m1 ⋖ · · · ⋖ mn = 1̂}, then the set {u ∨ mi ∧ v} is an M-chain for the interval [u, v]. With this choice of (sub)-M-chain on each interval, the labeling λ of Notation 3.3 coincides with λss . We are now ready to prove the following refinement of Theorem 1.2. Theorem 3.5. The labeling λ of Notation 3.3 is a CL-labeling. Proof. It is clear from construction that λ is a chain-edge labeling. By the recursive construction, it suffices to show that an interval of the form [0̂, y] has a unique increasing maximal chain, and that every lexicographically first maximal chain on [0̂, y] is increasing. Let m = {0̂ = m0 ⋖ m1 ⋖ · · · ⋖ mn = 1̂} be the sub-M-chain used to define the labeling. Let ℓ = 1 + max{i : mi ∧ y < y}. It is clear from the construction that every atomic cover relation on [0̂, y] receives a label that is at most ℓ. Since the elements greater than mℓ−1 are preserved until the corresponding labels are used, no chain on [0̂, y] receives any label greater than ℓ. Similarly, the mℓ−1 element in the sub-M-chain is by construction preserved until the ℓ label is used, and a chain receives an ℓ label when it leaves the interval [0̂, mℓ−1 ]. Since y ∈ / [0̂, mℓ−1 ], we see that every maximal chain on [0̂, y] receives an ℓ label. Thus, an increasing chain must have the ℓ label on its last cover relation. But Corollary 2.13 gives mℓ−1 ∧ y to be a coatom of [0̂, y]. It follows by the definition of the labeling that every increasing chain on [0̂, y] must end with mℓ−1 ∧ y ⋖ y. An easy induction now yields the only increasing chain to be 0̂ = m0 ∧ y ≤ m1 ∧ y ≤ · · · ≤ mℓ−1 ∧ y, the “projection” of the sub-M-chain to [0̂, y]. As mℓ−1 ∧ y ⋖ y, the projection chain is in particular maximal. We now show that this chain is the unique lexicographically first chain. In the construction, the least label of an atomic cover relation on [0̂, y] corresponds with the least mi+1 such that A BROAD CLASS OF SHELLABLE LATTICES 13 1̂ 3 2 1 or 2 m2 2 c 1 m1 3 3 a 1 2 1 b 3 0̂ Figure 3.1. A comodernistic labeling of a lattice mi+1 ∧ y > 0̂. But this is the (unique) first non-0̂ element of the increasing chain. The desired follows.  3.3. More details about the CL-labeling. Any chain-edge labeling assigns a word to each maximal chain of L. Since when we label a cover relation with i according to λ, we remove i from the index set (used for available labels), we obtain a result extending one direction of [25, Theorem 1]. Lemma 3.6. The chain-edge labeling λ assigns a word with no repeated labels to each maximal chain in L. Thus, if L is graded of height n, then λ assigns a permutation in Sn to each maximal chain. The decreasing chains are also easy to (recursively) understand. Recall that if x and y are lattice elements, then x is a complement to y if x ∨ y = 1̂ and x ∧ y = 0̂. The following is an extension of Theorem 2.9 for comodernistic lattices. Lemma 3.7. If 0̂ ⋖ c1 ⋖ · · · ⋖ 1̂ is a decreasing chain of L with respect to λ, then c1 is a complement to mn−1 . Proof. As in the proof of Theorem 3.5, every maximal chain on [0̂, 1̂] contains an n label. Thus λ(0̂ ⋖ c1 ) = n, so mn−1 ∧ c1 = 0̂. The result follows.  It is natural to ask whether theorems about supersolvable geometric lattices (see e.g. [34]) extend to comodernistic geometric lattices. The answer to this question is positive, but for rather uninteresting reasons: Proposition 3.8. If L is a geometric lattice, then any sub-M-chain of L is also an Mchain. Thus, a lattice L is geometric and comodernistic if and only if L is geometric and supersolvable. A BROAD CLASS OF SHELLABLE LATTICES 14 Proof. A result of Brylawski [11, Proposition 3.5] says that if m is modular in a geometric lattice L, and x is modular in [0̂, m], then x is also modular in L. The result now follows from Lemma 2.15 and an inductive argument.  In contrast to Proposition 3.8, lattices that are not geometric may have sub-M-chains which are not M-chains. We close this section by working out a small example in detail. Example 3.9. We consider the lattice C in Figure 3.1, which we obtained by removing a single cover relation from the Boolean lattice on 3 elements. It is easy to check that m2 is modular in C, but that m1 is not modular in C. (Indeed, m1 together with b < c generate a pentagon sublattice.) Since any lattice of height at most 2 is modular, the chain 0̂ ⋖ m1 ⋖ m2 ⋖ 1̂ is a sub-M-chain, though not an M-chain. With the exception of c ⋖ 1̂, the label of every cover relation in C is independent of the choice of root. We have indicated these labels in the diagram. But we notice that the interval [a, 1̂] inherits the sub-M-chain a ⋖ m2 ⋖ 1̂, while the interval [b, 1̂] has unique maximal (subM-)chain b ⋖ c ⋖ 1. Thus, the edge c ⋖ 1̂ receives a label of 1 with respect to root 0̂ ⋖ a ⋖ c, but a label of 2 with respect to root 0̂ ⋖ b ⋖ c. The reader may have noticed that the atom a is indeed left-modular. Thus, although we have shown the comodernistic labeling determined by the given sub-M-chain, there is also a supersolvable EL-labeling of C. We will see in Example 4.7 and Figure 4.1 a lattice that is neither geometric nor supersolvable, but that is comodernistic. 4. Order congruence lattices In this section, we examine the order congruence lattices of posets, as considered in the introduction and in Section 2.4. We prove Theorem 1.5, and apply Lemma 3.7 to calculate the Möbius number of O(P ). 4.1. Order congruence lattices are comodernistic. A useful tool for showing certain lattices to be comodernistic is given by the following lemma. Lemma 4.1. Let L be a meet subsemilattice of a lattice L+ . If m ∈ L+ is a left-modular coatom in L+ , and m ∈ L, then m is also left-modular in L. Proof. Since m is a coatom in L+ and therefore in L, the join of x and m is either m (if x ≤ m) or 1̂ (otherwise) in both lattices. In particular, the join operations in L and L+ agree on m, and we already know the meet operations agree by the subsemilattice condition. The result now follows by Lemma 2.10.  The following theorem follows immediately. Theorem 4.2. If L is a meet subsemilattice of the partition lattice ΠS , and m ∈ L is a partition x | (S \ x) for some element x ∈ S, then m is a left-modular coatom of L. We now show: A BROAD CLASS OF SHELLABLE LATTICES 15 Lemma 4.3. If P is any poset, then the order congruence lattice O(P ) is a meet subsemilattice of ΠP . Proof. It is clear from definition that O(P ) is a subposet of ΠP . It suffices to show that if π1 , π2 are in O(P ), then their meet π1 ∧ π2 also is in O(P ). Let f1 , f2 : P → Z be such that π1 and π2 are the level sets of f1 and f2 . But then the product map f1 × f2 : P → Z × Z (where Z × Z is taken with the product order) has the desired level set partition.  That order congruence lattices are comodernistic now follows easily. Proof (of Theorem 1.5). Let P be a poset, and let x be a maximal element of P . Assume by induction that the result holds for all smaller posets. It is straightforward to see m = x | (P \ x) is the level set partition of an order preserving map. By Theorem 4.2 and Lemma 4.3, the element m is a left-modular coatom on the interval [0̂, 1̂]. Since [0̂, m] is lattice-isomorphic to O(P \ x), we get by induction that [0̂, π] is comodernistic when π < m. If π is incomparable to m, then π ∧ m is a left-modular coatom of [0̂, π] by Corollary 2.13. Finally, repeated application of Lemma 2.7 and induction gives that intervals of the form [π ′ , 1̂] are comodernistic. The result follows for general intervals [π ′ , π].  4.2. The Möbius number of an order congruence lattice. We now use the comodernism of the order congruence lattice O(P ) to recover the Möbius number calculation due to Jenča and Sarkoci. Denote by Compat(x) the set of all y ∈ P that are compatible with x. That is, Compat(x) consists of all y such that either y ⋖ x, x ⋖ y, or y is incomparable to x. Our proof is short and simple. Theorem 4.4. [21, Theorem 3.8] For any poset P with maximal element x, the Möbius function of the order congruence lattice satisfies the recurrence X µ(O(P )) = − µ(O(Px∼y ). y∈Compat(x) Proof. By Lemma 3.7 together with the proof of Theorem 1.5, every decreasing chain of O(P ) begins with a complement to the (left-modular) order partition x | P \ x. Such complements are easily seen to be atoms a whose non-singleton block is {x, y}, where y ∈ P is compatible with x. The result now follows by Lemma 2.7.  Jenča and Sarkoci also show in [21] that if P is a Hasse-connected poset, then the number of linear extensions of P satisfies the same recurrence as µ(O(P )). We give a short bijective proof of the same, which has the same flavor as the proof of the main result in [13]. Let LE(P ) denote the set of linear extensions of P . Lemma 4.5. Let P be a poset and x a maximal element of P . If x is not also minimal, then there is a bijection [ LE(P ) → LE(Px∼y ). y∈Compat(x) A BROAD CLASS OF SHELLABLE LATTICES 16 Proof. Since x is not minimal, it cannot be the first element in any linear extension L of P . If y is the element immediately preceding x in L, then it is clear that x and y are compatible. Then Lx∼y is a linear extension of Px∼y . To show this map is a bijection, we notice that the process is reversible. If L is a linear extension of Px∼y , replace the element corresponding to the identification of x and y with x followed by y to get a linear extension of P .  Corollary 4.6. If P is a Hasse-connected poset, then the decreasing chains of O(P ) are in bijective correspondence with the linear extensions of P . In particular, |µ(O(P )| is the number of linear extensions of P , and ∆O(P ) is homotopy equivalent to a bouquet of this number of (|P | − 3)-dimensional spheres. Proof. Since P is Hasse-connected, a maximal element x cannot also be minimal. Moreover, if P is Hasse-connected then Px∼y is also Hasse-connected. The result now follows immediately by Theorem 4.4 and Lemma 4.5.  In the case where P is not Hasse-connected, a similar approach can be followed. Indeed, the same argument as in Lemma 4.5 applies, except that we must discard the linear extensions that begin with x in each recursive step where they arise. This argument identifies the decreasing chains of O(P ) with a recursively-defined subset of the linear extensions of P . We do not have a non-recursive description of this subset. Jenča and Sarkoci give a somewhat different description of |µ(O(P ))| for a non-Hasse-connected poset P in [21, Theorem 4.5]. 4.3. An order congruence lattice that is neither geometric nor supersolvable. Example 4.7. Consider the pentagon lattice N5 , obtained by attaching a bottom and top element 0̂ and 1̂ to the poset with elements a, b, c and relation a < c. In this case, O(N5 ) and Oconv (N5 ) coincide, and are pictured in Figure 4.1. The reader can verify by inspection that no atom of O(N5 ) is left-modular, thus, the lattice O(N5 ) is comodernistic but neither geometric nor supersolvable. As some of the coatoms of O(N5 ) are not left-modular, the dual of O(N5 ) also fails to be geometric. We remark that order congruence lattices that fail to be geometric were examined earlier in [22]. 5. Solvable subgroup lattices In this section, we discuss applications to and connections with the subgroup lattice of a group. 5.1. Known lattice-theoretic analogues of classes of groups. Since the early days of the subject, a main motivating object for lattice theory has been the subgroup lattice of a finite group. Indeed, a (left-)modular element may be viewed as a purely lattice-theoretic analogue or extension of a normal subgroup. Focusing on the normal subgroups characterizing a class of groups then typically gives in a straightforward way an analogous class of lattices with interesting properties. For example, every subgroup of an abelian (or more A BROAD CLASS OF SHELLABLE LATTICES 17 01abc 0ab|1c 0b|1ac 0b|1c|a 0b|1|ac 0ab|1|c 0b|1|a|c 0|1ac|b 0|1c|a|b 0abc|1 0|1c|ab 0a|1c|b 0|1|ab|c 0a|1bc 0|1abc 0|1|abc 0|1|ac|b 0a|1|bc 0|1|a|bc 0ac|1|b 0ac|1b 0|1bc|a 0a|1|b|c 0|1b|ac 0a|1b|c 0|1b|a|c 0|1|a|b|c Figure 4.1. The order congruence lattice O(N5 ) of the pentagon lattice N5 . Nontrivial left-modular elements are shown with rectangles. Group class Lattice class for L(G) Characterizes Self-dual? group class? cyclic distributive Yes Yes abelian, Hamiltonian modular No Yes nilpotent lower semimodular No No supersolvable supersolvable Yes Yes solvable ??? Table 1. Classes of groups and related classes of lattices. generally Hamiltonian) group is normal, so a corresponding class of lattices is that of the modular lattices. We summarize some of these analogies in Table 1. We remark that, although every normal subgroup is modular in the subgroup lattice, not every modular subgroup is normal. Similarly, although every nilpotent group has lower A BROAD CLASS OF SHELLABLE LATTICES 18 semimodular subgroup lattice, group that are not nilpotent may also have lower semimodular subgroup lattice. For example, the subgroup lattice of the symmetric group on 3 elements L(S3 ) has height 2, hence is modular (despite being neither abelian nor nilpotent). As L(S3 ) is lattice isomorphic to L(Z23 ), a subgroup lattice characterization of these classes of groups is not possible. It may then be surprising that a group G is supersolvable if and only if L(G) is a supersolvable lattice. The reason for this is more superficial than one might hope: Iwasawa proved in [20] that a group is supersolvable if and only if its subgroup lattice is graded. However, the definition of supersolvable lattice seems to capture the pleasant combinatorial properties of supersolvable groups much better than the definition of graded lattice does. Semimodular and supersolvable lattices have been of great importance in algebraic and topological combinatorics. In particular, both classes of lattices are EL-shellable, and the EL-labeling gives an efficient method of computing homotopy type, Möbius invariants, etc. 5.2. Towards a definition of solvable lattice. After the previous subsection, it may come as some surprise that there is no widely-accepted definition of solvable lattice. It is the purpose of this subsection to make the case for the definition of comodernistic lattices as one good candidate. It was independently proved by Suzuki in [40] and Zappa in [45] that solvable groups are characterized by their subgroup lattices. Later Schmidt gave an explicit characterization: Proposition 5.1 (Schmidt [30]; see also [31, Chapter 5.3]). For a group G, the following are equivalent: (1) G is solvable. (2) L(G) has a chain of subgroups 1 = G0 ( G1 ( · · · ( Gk = G such that each Gi is modular in L(G), and such that each interval [Gi , Gi+1 ] is a modular lattice. (3) L(G) has a chain of subgroups 1 = G0 ⊂· G1 ⊂· · · · ⊂· Gn = G such that each Gi is modular in the interval [1, Gi+1 ]. The reader will recognize the conditions in Proposition 5.1 as direct analogues of the conditions from Section 2.7. Despite this close correspondence, proving that Conditions (2) and (3) of the proposition imply solvability is not at all trivial. Although Proposition 5.1 combinatorially characterizes solvable subgroup lattices, we find it somewhat unsatisfactory. We don’t know how to use the implicit lattice conditions to calculate Möbius numbers. And, as the following example will show, lattices satisfying the implicit conditions need not be shellable. Example 5.2. Consider the lattice whose Hasse diagram is pictured in Figure 5.1. The element M is easily verified to be modular in this lattice, and the interval [0̂, M] is a Boolean lattice, hence modular. But since the interval [C, 1̂] is disconnected, the lattice is not shellable or Cohen-Macaulay. It is our opinion that a good definition of “solvable lattice” should be equivalent to solvability on subgroup lattices, and obey as many of the useful properties possessed by supersolvable A BROAD CLASS OF SHELLABLE LATTICES 19 1 M C 0 Figure 5.1. A non-shellable lattice satisfying Conditions (2) and (3) of Proposition 5.1. lattices as possible. Among these are: EL-shellability, efficient computation of homotopy type and/or homology bases and/or Möbius numbers, and self-duality of the property. Perhaps even more importantly, such a definition should have many combinatorial examples. We believe comodernistic lattices to be an important step towards understanding a definition or definitions of “solvable lattice”. As Theorem 1.7 states, comodernism is equivalent to solvability on subgroup lattices. While we do not know if a comodernistic lattice is always EL-shellable, we have shown such a lattice to be CL-shellable. The CL-labeling allows efficient calculation of homotopy type, and consequences thereof. Perhaps most importantly, there are many natural examples of comodernistic lattices. Unfortunately, comodernism is not a self-dual property, as is made clear by Example 4.7. 5.3. Proof of Theorem 1.7. We begin the proof by reviewing a few elementary facts about groups and subgroup lattices, all of which can be found in [31], or easily verified by the reader. We say that H permutes with K if HK = KH. Lemma 5.3. Let H, K, and L be subgroups of a group G. (1) If H permutes with K, then HK = KH is the join in L(G) of H and K. (2) If N ⊳ G, then N permutes with every subgroup of G. (3) (Dedekind Identity) If H ⊆ K, then H(K ∩ L) = K ∩ HL and (K ∩ L)H = K ∩ LH. Corollary 5.4. If K ⊃ H permutes with every subgroup on the interval [H, G], then K is a modular element of this interval. By Lemma 2.15, the elements of a sub-M-chain of a comodernistic lattice satisfy the modularity condition as in part (3) of Proposition 5.1. It follows immediately by the same proposition that G is solvable if L(G) is comodernistic. For the other direction, every subgroup of a solvable group is solvable. Thus, it suffices to find a modular coatom in the interval [H, G] over any subgroup H. Let 1 = N0 ( N1 ( A BROAD CLASS OF SHELLABLE LATTICES 20 · · · ( Nk = G be a chief series. It follows that each HNi is a subgroup for each i. Let ℓ be the maximal index such that HNℓ < G, and let K be any coatom of the interval [HNℓ , G]. We will show that K permutes with every subgroup L on [H, G], hence is modular on the same interval. For any such L, we have H(L ∩ Nℓ+1 ) = L ∩ HNℓ+1 = L ∩ G = L, and similarly (L ∩ Nℓ+1 )H = L ∩ Nℓ+1 H = L ∩ G = L, so that H permutes with L ∩ Nℓ+1 . Moreover, Nℓ ⊆ K ∩ Nℓ+1 ⊆ Nℓ+1 , and since Nℓ+1 /Nℓ is abelian, the Correspondence Theorem gives that K ∩ Nℓ+1 ⊳ Nℓ+1 . Thus, K ∩ Nℓ+1 permutes with L ∩ Nℓ+1 . Now KL = H(K ∩ Nℓ+1 )H(L ∩ Nℓ+1 ) = H(L ∩ Nℓ+1 )H(K ∩ Nℓ+1 ) = LK, as desired. 5.4. Homotopy type of the subgroup lattice of a solvable group. It is immediate from Lemma 5.3 that if N ⊳ G then HN permutes with all subgroups on the interval [H, G]. Thus, a chief series {Ni } lifts to a chain of left-modular elements {HNi } on the interval [H, G]. In a solvable group we can (by the proof of Theorem 1.7) complete the chain {HNi } to a sub-M-chain. Let λ be constructed according to this choice of sub-M-chain in all applicable intervals, and consider the decreasing chains of λ. It is immediate by basic facts about left-modular elements that a decreasing maximal chain on L(G) contains as a subset a chain of complements to the chief series {Ni }. Since a chain of complements to {Ni } in a solvable group is a maximal chain [12, Lemma 9.10], such chains are exactly the decreasing maximal chains. The order complex of a CL-shellable poset is a bouquet of spheres, where the spheres are in bijective correspondence with the decreasing chains of the poset. Thus, we recover the homotopy-type calculation of [41] (see also [43, 44]). 6. k-equal partition and related lattices In this section, we will show that the k-equal partition lattices are comodernistic. We’ll also show two related families of lattices to be comodernistic. 6.1. k-equal partition lattices. Recall that the k-equal partition lattice Πn,k is the subposet of the partition lattice Πn consisting of all partitions whose non-singleton blocks have size at least k. Theorem 6.1. For any 1 ≤ k ≤ n, the k-equal partition lattice Πn,k is comodernistic. Proof. Consider an interval [π ′ , π] in Πn,k . Let π be C1 | C2 | . . . | Cm , and assume without loss of generality that C1 is not a block in π ′ . Then C1 is formed by merging blocks B1 , . . . , Bℓ of π ′ . Suppose that the Bi ’s are ordered by increasing size, so that |B1 | ≤ |B2 | ≤ · · · . Consider the element m = B1 | B2 ∪ · · · ∪ Bℓ | C2 . . . | Cm . A BROAD CLASS OF SHELLABLE LATTICES 21 Suppose that σ is some partition on [π ′ , π], and D is the block of σ containing B1 . If D = B1 , then σ ∧ m = σ. Otherwise, there are two cases: Case 1: |B1 | > 1. Then σ ∧ m is formed by splitting D into blocks B1 , D \ B1 . Notice that |D \ B1 | ≥ |B1 | ≥ k by the ordering of the Bi ’s. Case 2: |B1 | = 1. Then if |D| > k, then σ ∧ m is formed by splitting D into smaller blocks B1 , D \ B1 . Otherwise, we have |D| = k, and σ ∧ m is formed by splitting D into k singletons. In either situation, we have σ ∧ m ⋖ σ, so Lemma 2.12 gives m to be left-modular on the desired interval.  We recover from Theorem 6.1 a weaker form of the result [7, Theorem 6.1] that k-equal partition lattices are EL-shellable. Repeated application of Lemma 3.7 recovers the same set of decreasing chains for the comodernistic labeling as in [7, Corollary 6.2]. See also [9]. 6.2. k, h-equal partition lattices in type B. By a sign pattern of a set S, we refer to an assignment of + or − to each element of S, considered up to reversing the signs of every element of S. Thus, if S has an order, an equivalent notion is to assign a + to the first element of S and either + or − to each remaining element. A signed partition of {0, 1, . . . , n} then consists of a partition of the set, together with a sign pattern assignment for each block not containing 0. The block containing 0 is called the zero block, and other blocks are called signed blocks. The signed partition lattice ΠB n consists of all signed partitions of {0, 1, . . . , n}. The cover relations in ΠB are of two types: n merging two signed blocks, and selecting one of the two possible patterns of the merged set; or merging a signed block with the zero block (thereby ‘forgetting’ the sign pattern on the signed block). The signed partition lattice is well-known to be supersolvable. Indeed, if π is a signed partition where every signed block is a singleton, then π is left-modular in ΠB n. Björner and Sagan [5] considered the signed k, h-equal partition lattice, where 1 ≤ h < k ≤ n. This is the subposet ΠB n,k,h consisting of all signed partitions whose non-singleton signed blocks have size at least k, and whose zero block is either a singleton or has size at least h + 1. Theorem 6.2. For any 1 ≤ h < k ≤ n, the signed k, h-equal partition lattice ΠB n,k,h is comodernistic. Proof. We proceed similarly to the proof of Theorem 6.1. Let [π ′ , π] be an interval in ΠB n,k,h , and π be C1 | C2 | . . . | Cm. Assume without loss of generality that C1 is not a block in π ′ . Then C1 is formed by merging blocks B1 , . . . , Bℓ of π ′ . Let B1 be the smallest signed block in this list, and consider the element m = B1 | B2 ∪ · · · ∪ Bℓ | C2 . . . | Cm . Now let σ be some partition in the interval [π ′ , π], and D be the block of σ containing B1 . If D = B1 , then σ ∧ m = σ. Otherwise, there are two cases: A BROAD CLASS OF SHELLABLE LATTICES 22 Case 1: |B1 | > 1. Since σ is on [π ′ , π], we see that D ⊆ C1 . If D is a signed block, then |D \ B1 | ≥ |B1 | ≥ k by the ordering of the blocks. Otherwise, by choice of B1 , no part of σ ∧ m is a signed singleton block contained in C1 . It follows that |D \ B1 | ≥ h + 1. In either situation, it follows that σ ∧ m is formed by splitting D into blocks B1 , D \ B1 . Case 2: |B1 | = 1. If |D| > k, then σ ∧ m is formed by splitting D into smaller blocks B1 , D \ B1 . Similarly if 0 ∈ D and |D| > h + 1. Otherwise, we have |D| = k or |D| = h (depending on whether 0 ∈ D), and σ ∧ m is formed by splitting D into singletons. In either case, we have σ ∧ m ⋖ σ, hence that m is left-modular on the desired interval by Lemma 2.12.  Björner and Sagan [5, Theorem 4.4] showed ΠB n,k,h to be EL-shellable. We recover from Theorem 6.2 the weaker result of CL-shellability. However, we remark that our proof is significantly simpler, and still allows easy computation of a cohomology basis, etc. There is also a “type D analogue” of Πn,k and ΠB n,k,h . Björner and Sagan considered this lattice in [5], but left the question of shellability open. Feichtner and Kozlov gave a partial answer to the type D shellability question in [16]. Our basic technique in the proofs of Theorems 6.1 and 6.2 is to show that left-modularity of B coatoms in Πn and ΠB n is sometimes inherited in the join subsemilattices Πn,k and Πn,k,h . It is easy to verify from the (here omitted) definition that the type D analogue of Πn and ΠB n has no left-modular coatoms, see also [19]. For this reason, the straightforward translation of our techniques will not work in type D. We leave open the question of under what circumstances the type D analogue of Πn,k has left-modular coatoms, or is comodernistic. 6.3. Partition lattices with restricted element-block size incidences. The k-equal partition lattices admit generalizations in several directions. One such generalization, examined in [7], is that of the subposet of partitions where the size of every block is in some set T . Further generalizations in similar directions are studied in [14, 15]. We consider here a different direction. Motivated by the signed k, h-equal partition lattices, Gottlieb [17] examined a related sublattice of the (unsigned) partition lattice. In Gottlieb’s lattice, the size of a block is restricted to be at least k or at least h, depending on whether or not the block contains a distinguished element. We further generalize to allow each element to have a different block-size restriction associated to it. More formally, we consider a map aff : [n] → [n], which we consider as providing an affinity to each element x of [n]. We consider two subposets of the partition lattice Πn : Π∀aff , {π ∈ Πn : every nonsingleton block B has |B| ≥ aff(x) for every x ∈ B} , and Π∃aff , {π ∈ Πn : every nonsingleton block B contains some x such that |B| ≥ aff(x)} . It is clear that both subposets are join subsemilattices of Πn , hence lattices. Theorem 6.3. For any selection of affinity map aff, the lattices Π∃aff and Π∀aff are comodernistic. A BROAD CLASS OF SHELLABLE LATTICES 23 Proof. As in the proof of Theorem 6.1, consider an interval [π ′ , π]. Let some block of π split nontrivially into blocks B1 , B2 , . . . , Bℓ of π ′ . As in Theorem 6.1, assume that the blocks are sorted by increasing size, and in particular that |B1 | ≤ |Bi | for all i. Now, if there are multiple singleton blocks B1 = {x1 }, B2 = {x2 }, . . . Bj = {xj }, then sort these by affinity. In the case of Π∃aff , let aff(x1 ) ≥ aff(x2 ) ≥ . . . ; while for Π∀aff reverse to require aff(x1 ) ≤ aff(x2 ) ≤ . . . . The remainder of the proof now goes through entirely similarly to that of Theorems 6.1 and 6.2.  Theorem 6.3 has applications to lower bounds of the complexity of a certain computational problem, directly analogous to the work in [3, 4] with Πn,k . 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Medical Concept Representation Learning from Electronic Health Records and its Application on Heart Failure Prediction Edward Choi*,MS; Andy Schuetz**,PhD; Walter F. Stewart**,PhD; Jimeng Sun*,PhD *Georgia Institute of Technology, Atlanta, USA **Research Development & Dissemination, Sutter Health, Walnut Creek, USA Corresponding Author: Jimeng Sun Georgia Institute of Technology 266 Ferst Drive, Atlanta, GA 30313 Tel: 404.894.0482 E-mail: [email protected] Keywords: Neural Networks, Representation learning, Predictive modeling, Heart Failure prediction. Word Count: 3600 ABSTRACT Objective: To transform heterogeneous clinical data from electronic health records into clinically meaningful constructed features using data driven method that rely, in part, on temporal relations among data. Materials and Methods: The clinically meaningful representations of medical concepts and patients are the key for health analytic applications. Most of existing approaches directly construct features mapped to raw data (e.g., ICD or CPT codes), or utilize some ontology mapping such as SNOMED codes. However, none of the existing approaches leverage EHR data directly for learning such concept representation. We propose a new way to represent heterogeneous medical concepts (e.g., diagnoses, medications and procedures) based on co-occurrence patterns in longitudinal electronic health records. The intuition behind the method is to map medical concepts that are co-occuring closely in time to similar concept vectors so that their distance will be small. We also derive a simple method to construct patient vectors from the related medical concept vectors. Results: For qualitative evaluation, we study similar medical concepts across diagnosis, medication and procedure. In quantitative evaluation, our proposed representation significantly improves the predictive modeling performance for onset of heart failure (HF), where classification methods (e.g. logistic regression, neural network, support vector machine and K-nearest neighbors) achieve up to 23% improvement in area under the ROC curve (AUC) using this proposed representation. Conclusion: We proposed an effective method for patient and medical concept representation learning. The resulting representation can map relevant concepts together and also improves predictive modeling performance. Introduction Growth in use of electronic health records (EHR) in health care delivery is opening unprecedented opportunities to predict patient risk, understand what works best for a given patient, and to personalize clinical decision-making. But, raw EHR data, represented by a heterogeneous mix of elements (e.g., clinical measures, diagnoses, medications, procedures) and voluminous unstructured content, may not be optimal for analytic uses or even for clinical care. While higher order clinical features (e.g., disease phenotypes) are intuitively more meaningful and can reduce data volume, they may fail to capture meaningful information inherent to patient data. We explored whether novel data driven methods that rely on the temporal occurrence of EHR data elements could yield higher order intuitively interpretable features that both capture pathophysiologic relations inherent to data and improve performance of predictive models. Growth in use of EHRs is raising fundamental questions on optimal ways to represent structured and unstructured data. Medical ontologies such as SNOMED, RxNorm and LOINC offer structured hierarchical means of compressing data and of understand relations among data from different domains (e.g., disease diagnosis, labs, prescriptions). But, these ontologies do not offer the means of extracting meaningful relations inherent to longitudinal patient data. Scalable methods that can detect pathophysiologic relations inherent to longitudinal EHR data and construct intuitive features may accelerate more effective use of EHR data in clinical care and advances in performance of predictive analytics. The abstract concepts inherent to existing ontologies does not provide a means to connect elements in different domains to a common underlying pathophysiologic constructs that are represented by how data elements co-occur in time. The data driven approach we developed logically organizes data into higher order constructs. Heterogeneous medical data were mapped to a low-dimensional space that accounted for temporal clustering of similar concepts (e.g., A1c lab test, ICD-9 code for diabetes, prescription for metformin). Co-occurring clusters (e.g., diabetes and peripheral neuropathy) were then identified and formed into higher order pathophysiologic feature sets organized by prevalence. We propose to learn such a medical concept representation on longitudinal EHR data based on a state-of-the-art neural network model. We also propose an efficient way to derive patient representation based on the medical concept representation (or medical concept vectors). We calculated for a set of diseases their closest diseases, medications and procedures to demonstrate the clinical knowledge captured by the medical concept representations. We use those learned representation for heart failure prediction tasks, where significant performance improvement up to 23% in AUC can be obtained on many classification models (logistic regression: AUC 0.766 to 0.791, SVM: AUC 0.736 to 0.791, neural network: AUC 0.779 to 0.814, KNN: AUC 0.637 to 0.785) BACKGROUND Representation Learning in Natural Language Processing Recently, neural network based representation learning has shown success in many fields such as computer vision [1] [2] [3], audio processing [4] [5] and natural language processing (NLP) [6] [7] [8]. We discuss representation learning in NLP, in particular, as our proposed method is based on Skip-gram [6] [9], a popular method for learning word representations. Mikolov et al. [6] [9] proposed Skip-gram, a simple model based on neural network that can learn real-valued multi-dimensional vectors that capture relations between words by training on massive amount of text. The trained real-valued vectors will have similar values for syntactically and semantically close words such as dog and cat or would and could, but distinct values for words that are not. Pennington et al. [10] proposed GloVe, a word representation algorithm based on the global co-occurrence matrix. While GloVe and Skip-gram essentially achieve the same goal by taking different approaches, GloVe is computationally faster than Skip-gram as it precomputes co-occurrence information before the actual learning. Skip-gram, however, requires less number of hyper-parameters to tune than GloVe, and generally shows better performance [11]. As natural language text can be seen as a sequence of codes, medical records such as diagnoses, medications and procedures can also be seen as a sequence of codes over time. In this work, we propose a framework for mapping raw medical concepts (e.g., ICD9, CPT) into related concept vectors using Skip-gram and validating the utility of the resulting medical concept vectors. Representation Learning in the Clinical Field A few researchers applied representation learning in the clinical field recently. Minnaro-Gimenez et al. [12] learned the representations of medical terms by applying Skip-gram to various medical text. They collected the medical text from PubMed, Merck Manuals, Medscape and Wikipedia. De Vine et al. [13] learned the representations of UMLS concepts from free-text patient records and medical journal abstracts. The first preprocessed the text to map the words to UMLS concepts, then applied Skip-gram to learn the representations of the concepts. More recently, Choi et al. [14] applied Skip-gram to structured dataset from a health insurance company, where the dataset consisted of patient visit records along with diagnosis codes(ICD9), lab test results(LOINC), and drug usage(NDC). Their goal, to learn efficient representations of medical concepts, partially overlaps with our goal. Our study however, is focused on learning the representations of medical concepts and using them to generate patient representations, apply them to a real-world prediction problem to demonstrate improved performance provided by the efficient representation learning. MATERIALS AND METHODS Training Phase Medical Concept Representation Learning EHR Intensity Kidney failure Medical Mapping Concept Vectors Patient Representation Construction Aggregation Time Patient Vectors Heart Failure Prediction Model Training Input Model Output Heart Failure Risk Score Prediction Phase Figure 1. Flowchart of the proposed method Intensity Kidney failure In Figure 1, we give a Time high-level overview of the steps we take to perform HF prediction. In the training phase, we first train medical concept vectors from the EHR dataset using Skip-gram. Then, we construct patient representation using the medical concept vectors. The patient representation is then used to train heart failure prediction models using various classifiers, namely logistic regression, support vector machine (SVM), multi-layer perceptron with one hidden layer (MLP) and K-nearest neighbors classifier (KNN). In the prediction phase, we map the medical record of a patient to medical concept vectors and generate patient vectors by aggregating the concept vectors. Then we plug the patient vectors into the trained model, which in turn will generate the risk score for heart failure. In the following sections, we will describe medical concept representation learning and patient representation construction in more detail. Medical Concept Representation Learning N-dimensional vector D-dimensional vector Bronchitis: [1, 0, 0, 0, 0, …, 0, 0, 0] Pneumonia: [0, 1, 0, 0, 0, …, 0, 0, 0] Obesity: [0, 0, 1, 0, 0, …, 0, 0, 0] Bronchitis: [0.4, -0.2, …, 0.2] Pneumonia: [0.3, -0.3, …, 0.1] Obesity: [-0.7, 1.4, …, 1.2] Cataract: Cataract: N diagnoses [0, 0, 0, 0, 0, …, 0, 0, 1] (a) One-hot encoding for diagnoses [1.2, 0.8, …, 1.5] (b) A better representation of diagnoses Figure 2. Two different representation of diagnoses. Typically, raw data dimensionality N(~10,000) is much larger than concept dimensionality D(50~1,000) Figure 2 depicts a straightforward motivation for using a better representation for medical concepts. Figure 2(a) shows one-hot encoding of N unique diagnoses using Ndimensional vectors. It is easy to see that this is not an effective representation in that the difference between Bronchitis and Pneumonia are the same as the difference between Pneumonia and Obesity. Figure 2(b) shows a better representation in that Bronchitis and Pneumonia share similar values compared to other diagnoses. By using Skip-gram, we will be able to better represent not only diagnoses but also medications and procedures as multidimensional real-valued vectors that will capture the latent relations between them. (a) Patient medical records on a timeline Pneumonia Benzonatate Amoxicillin Time Cough Cough Benzonatate Fever Pneumonia Chest X-ray Chest X-ray Fever (b) Predicting neighboring medical concepts given Fever Bezonanate Fever Chest X-ray Amoxicillin Pneumonia (c) Predicting neighboring medical concepts given Pneumonia D-dimensional v(ct-2) v(ct-1) v(ct+1) v(ct+2) v(ct) mapping ct (d) Model architecture of Skip-gram Figure 3. Training examples and the model architecture of Skip-gram Figure 3(a) is an example of a patient medical record in a temporal order. Skip-gram assumes the meaning of a concept is determined by its context(or neighbors). Therefore, given a sequence of concepts, Skip-gram picks a target concept and tries to predict its neighbors, as shown by Figure 3(b). Then we slide the context window, pick the next target and do the same context prediction, which is shown by Figure 3(c). Since the goal of Skipgram is to learn the vector representation of concepts, we need to convert medical concepts to D-dimensional vectors, where D is a user-chosen value typically between 50 and 1000. Therefore the actual prediction is conducted with vectors, as shown by Figure 3(d), where c" is the concept at the t-th timestep, 𝐯 𝑐" the vector that represents c" . The goal of Skipgram is to maximize the following average log probability, 1 𝑇 4 log 𝑝 𝑐"+, |𝑐" , where "56 ./0,0/,,23 𝑝 𝑐"+, |𝑐" = 𝑒𝑥𝑝 𝐯 𝑐"+, ? @56 𝑒𝑥𝑝 > 𝐯 𝑐" 𝐯 𝑐 > 𝐯 𝑐" where T is the length of the sequence of medical concepts, w the size of the context window, 𝑐" the target medical concept at timestep t, 𝑐"+, the neighboring medical concept at timestep t+j, 𝐯 𝑐 the vector that represents the medical concept c, N the total number of medical concepts. The size of the context window is typically set to 5, giving us 10 concepts surrounding the target concept. Note that the conditional probability is expressed as a softmax function. Simply put, by maximizing the softmax score of the inner product of the neighboring concepts, Skip-gram learns real-valued vectors that efficiently capture the fine-grained relations between concepts. It needs to be mentioned that our formulation of Skip-gram is different from the original Skip-gram. In Mikolov et al. [9], they distinguish the vectors for the target concept and the vectors for the neighboring concept. In our formulation, we force the two sets of vectors to hold the same values as suggested by [15]. This simpler formulation allowed faster training and impressive results. Patient Representation Construction In this section, we describe a simple derivation of patient representation using the learned medical concept vectors. One of the impressive features of Skip-gram in Mikolov et al. [9] was that the word vectors supported syntactically and semantically meaningful linear operations that enabled word analogy calculations such that the resulting vector of King – Man + Woman is closest to Queen vector. We expect that the medical concept representations learned by Skip-gram will show similar properties so that the concept vectors will support clinically meaningful vector additions. Then, an efficient representation of a patient will be as simple as converting all medical concepts in his medical history to medical concept vectors, then summing all those vectors to obtain a single representation vector, as shown in Figure 4. In the experiments, we show examples of clinically meaningful concept vector additions. Benzonatate Pneumonia Amoxicillin (a) Time Cough Fever Chest X-ray (b) (c) Medical concept vectors + + + + + = Patient representation Figure 4. Patient representation construction. (a) represents a medical record of a patient on a timeline. (b) The medical concepts are represented as vectors using the trained medical concept vectors. (c) The patient is represented as a vector by summing all medical concept vectors. EXPERIMENTS AND RESULTS Population and Source of Data Data were from Sutter Palo Alto Medical Foundation (Sutter-PAMF) primary care patients. Sutter-PAMF is a large primary care and multispecialty group practice that has used an EHR for more than a decade. The study dataset was extracted with cases and controls identified within the interval from 05/16/2000 to 05/23/2013. The EHR data included demographics, smoking and alcohol consumptions, clinical and laboratory values, International Classification of Disease version 9 (ICD-9) codes associated with encounters, order, and referrals, procedure information in Current Procedural Terminology (CPT) codes and medication prescription information in medical names. The dataset contained 265,336 patients with 555,609 unique clinical events in total. Configuration for Medical Concept Representation Learning To apply Skip-gram, we scanned through encounter, medication order, procedure order and problem list records of all 265,336 patients, and extracted diagnosis, medication and procedure codes assigned to each patient in temporal order. If a patient received multiple diagnoses, medications or procedures at a single visit, then those medical codes were given the same timestamp. The respective number of unique diagnoses, medications and procedures was 11,460, 17,769 and 9,370 totaling to 38,599 unique medical concepts. We used 100-dimensional vectors to represent medical concepts(i.e. D=100 in Figure 2(b)), considering 300 was sufficient to effectively represent 692,000 vocabularies in NLP. [9] We used Theano [16], a Python library for evaluating mathematical expression to implement Skip-gram. Theano can also take advantage of GPUs to greatly improve the speed of calculations involving large matrices. For optimization, we used Adadelta [17], which employs adaptive learning rate. Unlike stochastic gradient descent (SGD), which is widely used for training neural networks, Adadelta does not depend very strongly on the Diagnosis Codes using SkipGram ion reducec by t-SNE ntation of Diagnosis Codes using SkipGram 15by t-SNE of Diagnosis Codes using SkipGram Dimension reducec Vector Representation Dimension reducec by t-SNE 10 des using SkipGram setting of the learning rate, and shows good performance. Using Theano 0.7 and CUDA 7 t-SNE Dimension2 5 Vector Representation of Diagnosis Codes using SkipGram Dimension reducec by t-SNE is Codes using SkipGram on an Ubuntu machine E5-2697 and 0 Vector Representation of Diagnosis Codeswith using Xeon SkipGram by t-SNE Nvidia Tesla K80, it took approximately Dimension reducec by t-SNE 43 hours to run 10 epochs of Adadelta with the batch size of 100. -5 Evaluation of Medical Concept Representation Learning -10 Infectious And Parasitic Diseases Diseases Of The Nervous System And Sense Organs Neoplasms Diseases Of The Circulatory System Endocrine, Nutritional And Metabolic Diseases, And Immunity Disorders Diseases Of The Respiratory System Diseases Of The Blood And Blood-Forming Organs Diseases Of The Digestive System Vector Representation of Diagnosis Codes using SkipGram Mental Disorders Diseases Of The Genitourinary System -15 -20 Dimension reducec by t-SNE 30 -25 25 -30 -30 -10 20 -5 -25 -20 0 -15 5 -10 10 -5 15 0 20 5 10 15 20 25 Highcharts.com t-SNE Dimension1 Highcharts.com 15 5 -20 -15 -10 10 -5 0 5 10 15 20 25 30 t-SNE Dimension1 Highcharts.com 5 5 -5 0 t-SNE Dimension1 -15 -10 -5 t-SNE Dimension2 0 t-SNE Dimension1 10 15 5 0 20 10 0 25 15 30 Highcharts.com 20 5 25 10 15 30 Highcharts.com 20 25 10 15 30 t-SNE Dimension1 -5 5 Highcharts.com -10 20 25 30 Highcharts.com -15 -20 25 ension1 -15 5 -20 -10 10 -15 -5 -20 -25 15 0 t-SNE Dimension1 20 25 -5 -10 5 0 10 30 15 5 20 10 25 15 20 Benign Skin Neoplasms 30 Highcharts.com Malignant Skin Neoplasms t-SNE Dimension1 Highcharts.com 25 30 Highcharts.com -30 -30 30 25 Dimension1 30 t-SNE -25 -20 -15 -10 -5 0 5 10 15 20 25 30 t-SNE Dimension1 Highcharts.com 053.29: Herpes zoster with other ophthalmic complications 053.21: Herpes zoster keratoconjunctivitis 053.19: Herpes zoster with other nervous system complications 364.04: Secondary iridocyclitis, noninfectious 364.00: Acute and subacute iridocyclitis, unspecified 053.22: Herpes zoster iridocyclitis 364.3: Unspecified iridocyclitis Figure 5. Diagnosis vectors projected to a 2D space by t-SNE Figure 5 shows the trained diagnosis vectors plotted in a 2D space, where we used t-SNE [18] to reduce the dimensions from 100 to 2. t-SNE is a dimensionality reduction algorithm that was specifically developed for plotting high-dimensional data into a two or three dimensional space. We randomly chose 1,000 diagnoses from 10 uppermost categories of ICD-9, which are displayed at the top of the figure. It is readily visible that diagnoses are generally well grouped by their corresponding categories. However, if diagnoses from the same category are in fact quite different, they should be apart. This is shown by the red box and the blue box in Figure 5. Even though they are from the same neoplasms category, red box indicates a group of malignant skin neoplasms (172.X, 173.X) while blue box indicates a group of benign skin neoplasms (216.X). Detailed figure of the red and blue boxes are in the supplementary section. What is more, as the black box shows, diagnoses from different groups are located close to one another if they are actually related. In the black box, iridocyclitis and eye infections related to herpes zoster are closely located, which corresponds to the fact that approximately 43% herpes zoster ophthalmicus (HZO) patients develop iridocyclitis. [19] In order to see how well the representation learning captured the relations between medications and procedures as well as diagnoses, we conduct the following study. We chose 100 diagnoses that occurred most frequently in the data, obtained for each diagnosis 50 closest vectors in terms of cosine similarity, picked 5 diagnosis, medication and procedure vectors among the 50 vectors. Table 2 depicts a portion of the entire lis. Note that some cells contain less than 5 items, which is because there was less than 5 items in the 50 closest vectors. The entire list is provided in the supplementary section. Table 1. Examples of diagnoses and their closest medical concepts. Diagnoses -Bronchitis, not specified as acute or chronic Acute upper (490) respiratory -Cough (786.2) infections -Acute sinusitis, unspecified (461.9) (465.9) -Acute bronchitis (466.0) -Acute pharyngitis (462) Medications Procedures -Azithromycin 250 mg po tabs -Pulse oximetry single -Promethazine-Codeine 6.25-10 -Serv prov during reg sched mg/5ml po syrp eve/wkend/hol hrs -Amoxicillin 500 mg po caps -Chest PA & lateral -Fluticasone Propionate 50 mcg/act -Gyn cytology (pap) pa na susp -Influenza vac (flu clinic -Flonase 50 mcg/act na susp only) 3+yo pa -Diabetes mellitus (250.00) -Metformin hcl 500 mg po tabs Diabetes -Mixed hyperlipidemia (272.2) -Metformin hcl 1000 mg po tabs mellitus -Other abnormal glucose (790.29) -Glucose blood vi strp (250.02) -Obesity, unspecified (278.00) -Lisinopril 10 mg po tabs -Pure hypercholesterolemia (272.0) -Lisinopril 20 mg po tabs -Anemia, unspecified (285.9) -Furosemide 20 mg po tabs -Congestive heart failure, unspecified (428.0) -Hydrochlorothiazide 25 mg po tabs -Diabetic eye exam (no bill) -Diabetes education, int -Ophthalmology, int -Diabetic foot exam (no bill) -Influenza vac 3+yr (v04.81) im -Debridement of nails, 6 or more Edema -Unspecified essential hypertension (401.9) -Hydrocodone-Acetaminophen 5-500 -OV est pt min serv (782.3) -Atrial fibrillation (427.31) mg po tabs -Chronic kidney disease, Stage III (moderate) -Cephalexin 500 mg po caps -ECG and interpretation (585.3) -Chest PA & lateral -Furosemide 40 mg po tabs -Blepharitis, unspecified (373.00) Tear film -Senile cataract, unspecified (366.10) insufficiency, -Presbyopia (367.4) unspecified -Preglaucoma, unspecified (365.00) (375.15) -Other chronic allergic conjunctivitis -Refraction -Glasses -Erythromycin 5 mg/gm op oint -Patanol 0.1 % op soln (372.14) Benign essential hypertension (401.1) -EKG -Visual field exam extended -Visual field exam limited -Referral to ophthalmology, int -Ophthalmology, int -Hyperlipidemia (272.4) -Hydrochlorothiazide 25 mg po tabs -Essential hypertension (401.9) -Atenonol 50 mg po tabs -Pure hypercholesterolemia (272.0) -Lisinopril 10 mg po tabs -Mixed hyperlipidemia (272.2) -Lisinopril 40 mg po tabs -Diabetes mellitus (250.00) -Lisinopril 20 mg po tabs -ECG and interpretation -Influenza vac 3+yr im -Immun admin im/sq/id/perc 1st vac only -GI, int -OV est pt lev 3 Evaluation of Medical Concept Vector Additions Table 2. Vector operations of medical vectors trained by Skip-gram. Diagnoses Hypertension (401.9) + Obesity (278.0) -Hyperlipidemia (272.4) -Hydrochlorothiazide -Diabetes (250.00) -Valsartan -Coronary atherosclerosis (414.00) -Nifedipine -Hypertension (401.1) -Lisinopril -Chronic kidney disease (585.3) -Losartan potassium -Pneumonia (486) Fever (780.60) + Cough (786.2) Medications -Acute bronchitis (466.0) -Acute upper respiratory infections (465.9) -Bronchitis (490) -Acute sinusitis (461.9) + Pain in/around Eye (379.91) -X-ray chest -Chest PA & Lateral -Promethazine-codeine -Pulse oximetry -Guaifenesin-codeine -Serv prov during reg sched -Proair HFA eve/wkend/hol hrs -Levofloxacin -Inhalation Rx for obstruction MDI/NEB -Ophthalmology -Visual discomfort (368.13) -Glasses -Regular astigmatism (367.21) -Erythromycin ointment -Referral to ophthalmology -Presbyopia (367.4) -Patanol -Blepharitis (373.00) Loss of Weight (783.21) + Anxiety State (300.00) + Speech Disturbance (784.59) -Peripheral refraction -Visual field exam -Diabetic eye exam -Depressive disorder (311) -Lorazepam -Referral to GI -Malais & fatigue (780.79) -Zolpidem tartrate -ECG & Interpretation -Insomnia (780.52) -Omeprazole -GI -Generalized anxiety disorder (300.02) -Alprazolam -EKG -Esophageal reflux (530.81) -Trazodone HCL -Chest PA & Lateral -Dysarthria (784.51) Hallucination (780.1) N/A -Azithromycin -Tear film insufficiency (375.15) Visual Disturbance (368.8) Procedures -Secondary parkinsonism (332.1) -Senile dementia with delirium (290.3) -Mental disorder (294.9) -Paranoid state (297.9) -Referral to geriatrics -Midorine HCL -Mental status exam -Risperdal -Referral to -Rivastigmine Tartrate neuropsychology -Rivastigmine -Referral to speech therapy Home visit est pt lev 2 Due to the difficulty of generating medically interesting examples, we chose 5 intuitive examples as shown by the first column of Table 3 to give a simple demonstration of the medical concept vector additions. We again generated 50 closest vectors to the sum of two medical concept vectors and picked 5 from each diagnosis, medication and procedure category. Setup for Heart Failure Prediction Evaluation In this section, we first describe why we chose heart failure (HF) prediction task as an application. Then we briefly mention the models to use, followed by the description of the data processing steps to create the training data for all models. Lastly, the evaluation strategy will be followed by implementation details. Heart failure prediction task: Onset of HF is associated with a high level of disability, health care costs and mortality (roughly ~50% risk of mortality within 5 years of diagnosis). [20] [21] There has been relatively little progress in slowing the progression of HF severity, largely because it is difficult to detect before actual diagnosis. As a consequence, intervention has primarily been confined to the time period after diagnosis, with little or no impact on disease progression. Earlier detection of HF could lead to improved outcomes through patient engagement and more assertive treatment with angiotensin converting enzyme (ACE)-inhibitors or Angiotensin II receptor blockers (ARBs), mild exercise, and reduced salt intake, and possibly other options [22] [23] [24] [25]. Models for performance comparison: We aim to emphasize the effectiveness of the medical concept representation and the patient representation derived from it. Therefore we trained four popular classifiers, namely logistic regression, MLP, SVM, and KNN using both one-hot vectors and medical concept vectors. Definition of Cases and Controls: Criteria for incident onset of HF, are described in [26] and were adopted from [27]. The criteria are defined as: 1) Qualifying ICD-9 codes for HF appeared as a diagnosis code in either the encounter, the problem list, or the medication order fields. Qualifying ICD-9 codes are listed in the supplementary section. Qualifying ICD-9 codes with image and other related orders were excluded because these orders often represent a suspicion of HF, where the results are often negative; 2) a minimum of three clinical encounters with qualifying ICD-9 codes had to occur within 12 months of each other, where the date of diagnosis was assigned to the earliest of the three dates. If the time span between the first and second appearances of the HF diagnostic code was greater than 12 months, the date of the second encounter was used as the first qualifying encounter; 3) ages 50 or greater to less than 85 at the time of HF diagnosis. Up to ten (nine on average) eligible primary care clinic-, sex-, and age-matched (in 5-year age intervals) controls were selected for each incident HF case. Primary care patients were eligible as controls if they had no HF diagnosis in the 12-month period before diagnosis of the incident HF case. Control subjects were required to have their first office encounter within one year of the matching HF case patient’s first office visit, and have at least one office encounter 30 days before or any time after the case’s HF diagnosis date to ensure similar duration of observations among cases and controls. From 265,336 Sutter-PAMF patients, 3,884 incident HF cases and 28,903 control patients were identified. Data processing: To train the four models, we generated the dataset again from the encounter, medication order, procedure order and problem list records of 3,884 cases and 28,903 controls. Based on the HF diagnosis date (HFDx) of each patient, we extracted all records from the 18-month period before the HFDx. To train the models with medical concept vectors, we converted the medical records to patient vectors as shown in Figure 4. To train the models with one-hot encoding, we converted the medical records to aggregated one-hot vectors in the same fashion as Figure 4, using one-hot vectors instead of medical concept vectors. In order to study the relation between the medical concept vectors trained with different sizes of data and their influence on the models’ prediction performance, we used three kinds of medical concept vectors: 1) The one trained with only HF cases (3,884 patients), 2) The one trained with HF cases and controls (32,787 patients), 3) The one trained with the full sample (265,336 patients). Note that medical concept vectors trained with smaller number of patients cover less number of medical concepts. Therefore, when converting patient records to patient vectors as Figure 4, we excluded all medical codes that did not have matching medical concept vectors. All input vectors were normalized to zero mean and unit variance. Evaluation strategy: We used six-fold cross validation to train and evaluate all models, and to estimate how well the models will generalize to independent datasets. Prediction performance was measured using area under the ROC curve (AUC), on data not used in the training. We used the confidence score to calculate its AUC for SVM. Detailed explanation of the cross validation is given in the supplementary section. Implementation details: Logistic regression and MLP were implemented with Theano and trained with Adadelta. SVM and KNN were implemented with Python Scikit-Learn. All models were trained by the same machine used for medical concept representation learning. Hyper-parameters used for training each model are described in the supplementary section. Evaluation of Heart Failure Prediction One-hot encoding Medical concept vectors trained on 4K patients Medical concept vectors trained on 33K patients Medical concept vecotrs trained on 265K patients 1 0.95 0.9 0.85 AUC 0.8 0.75 0.7 0.65 0.6 0.55 0.5 Logistic Regression SVM MLP KNN Figure 6. Heart failure prediction performance of various models and input vectors. Figure 6 shows the average AUC of 6-fold cross validations of various models and input vectors. The colors represent different training input vectors. The error bars indicate the standard deviation derived from the 6-fold cross evaluation. The power of medical concept representation learning is evident as all models show significant improvement in the HF prediction performance. Logistic regression and SVM, both being linear models, show similar performance when trained with medical concept vectors, although SVM benefits slightly more from using the better representation of medical concepts. MLP also benefits from using medical concept vectors, and, being a non-linear model, shows better performance compared to logistic regression and SVM. It is interesting that KNN benefits the most from using the medical concept vectors, even the ones trained on the smallest dataset. Considering the fact that KNN classification is based on the distances between data points, this is a clear indication that proper medical concept representation can alleviate the sparsity problem induced by the simple one-hot encoding. Figure 6 also tells us that medical concept representation is best learned with a large dataset as shown by Mikolov et al. [6] However, in most models, especially KNN, even the medical concept vectors trained with the smallest number of patients improves the prediction performance. It is quite surprising given the fact that we used less amount of information by excluding unmatched medical codes when using medical concept vectors trained with a small number of patients, the models still show better prediction performance. This again is a clear proof that medical concept representation learning provides more effective way to represent medical concepts than one-hot encoding. Table 3. Training speed improvement. (*Since KNN does not require training, we display classification time instead) One-hot encoding Medical concept vectors Speed -up Logistic regression SVM MLP KNN* 81.3 20.5 85.7 2900.82 5.3 1.9 5.6 36.66 x15.3 x10.8 x15.3 x79.1 Table 4 depicts the training time for each model when using one-hot encoding and medical concept vectors. Considering the high-dimensionality of one-hot encoding, training the models with medical concept vectors should provide significant speed-up, as shown by the last row of Table 4. This shows that medical concept vectors not only improve performance, but also significantly reduce the training time. Before discussing future work, we would like to emphasize the fact that our entire experiments were conducted completely without expert knowledge such as medical ontologies or features designed by medical experts. Using only the medical order records, we were able to produce clinically meaningful representation of medical concepts. This is an inspiring discovery that can be extended for numerous other medical problems. Future Work Although medical concept vectors have shown impressive results, it would be even more effective if deeper medical information could be embedded such as lab results or patient demographic information. This would enable us to represent the medical state of patients more accurately. Using expert knowledge is another thing we should try. Even though we have shown impressive performance only by using medical records, this does not mean we cannot benefit from well-established expert medical knowledge, such as specific features or medical ontologies. Another natural extension of our work is to address other medical problems. Although this work focused on the early detection of heart failure, our approach is very general that it could be applied to any kind of disease prediction problem. And the medical concept vectors can also be used in numerous medical applications as well. CONCLUSION We proposed a new way of representing heterogeneous medical concepts as realvalued vectors and constructing efficient patient representation using the state-of-the-art Deep Learning method. We have qualitatively shown that the trained medical concept vectors indeed captured medical insights compatible with our medical knowledge and experience. For the heart failure prediction task, medical concept vectors improved the performance of many classifiers, thus quantitatively proving its effectiveness. We discussed the limitation of our method and possible future works, which include deeper utilization of medical information, combining expert knowledge into our framework, and expanding our approach to various medical applications. References [1] Alex Krizhevsky, Ilya Sutskever, and Geoffrey Hinton, "Imagenet classification with deep convolutional neural networks," in Advances in Neural Information Processing Systems, 2012, pp. 1097-1105. [2] Pascal Vincent, Hugo Larochelle, Yoshua Bengio, and Pierre-Antoine Manzagol, "Extracting and composing robust features with denoising autoencoders," in International Conference on Machine learning, 2008, pp. 1096-1103. [3] Quoc Le et al., "uilding high-level features using large scale unsupervised learning," in arXiv:1112.6209 , 2012. 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[8] Richard Socher, Jeffrey Pennington, Eric Huang, Andrew Ng, and Christopher Manning, "Semi-supervised recursive autoencoders for predicting sentiment distributions," in Empirical Methods in Natural Language Processing, 2011, pp. 151161. [9] Tomas Mikolov, Ilya Sutskever, Kai Chen, Greg Corrado, and Jeff Dean, "Distributed representations of words and phrases and their compositionality," in Advances in Neural Information Processing Systems, 2013, pp. 3111-3119. [10] Jeffrey Pennington, Richard Socher, and Christopher Manning, "Glove: Global vectors for word representation," in Empirical Methods on Natural Language Processing, 2014, pp. 1532-1543. [11] Radim Rehurek. (2014, Dec.) Rare Technologies. [Online]. http://rare- technologies.com/making-sense-of-word2vec/ [12] Jose Minarro-Gimenez, Oscar Marin-Alonso, and Matthias Samwald, "Exploring the application of deep learning techniques on medical text corpora," Studies in health technology and informatics, vol. 205, pp. 584-588, 2013. 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[19] Janice Thean, Anthony Hall, and Richard Stawell, "Uveitis in herpes zoster ophthalmicus," Clinical & Experimental Ophthalmology, vol. 29, no. 6, pp. 406-410, 2001. [20] Veronique L. Roger et al., "Trends in heart failure incidence and survival in a community-based population," JAMA, vol. 292, no. 3, pp. 344-350, July 2004. [21] Sherry L. Murphy, Jiaquan Xu, and Kenneth D. Kochanek, "Deaths: final data for 2010," National Vital Stat Rep, vol. 61, no. 4, pp. 1-117, May 2010. [22] SOLVD Investigators, "Effect of enalapril on mortality and the development of heart failure in asymptomatic patients with reduced left ventricular ejection fractions," N Engl j Med, vol. 327, pp. 685-691, 1992. [23] J Arnold et al., "Prevention of heart failure in patients in the Heart Outcomes Prevention Evaluation (HOPE) study," Circulation, vol. 107, no. 9, pp. 1284-1290, 2003. [24] Sebastiano Sciarretta, Francesca Palano, Giuliano Tocci, Rossella Baldini, and Massimo Volpe, "Antihypertensive treatment and development of heart failure in hypertension: a Bayesian network meta-analysis of studies in patients with hypertension and high cardiovascular risk," Archives of internal medicine, vol. 171, no. 5, pp. 384-394, 2011. [25] Chao-Hung Wang, Richard Weisel, Peter Liu, Paul Fedak, and Subodh Verma, "Glitazones and heart failure critical appraisal for the clinician," Circulation, vol. 107, no. 10, pp. 1350-1354, 2003. [26] Rajakrishnan Vijayakrishnan et al., "Prevalence of heart failure signs and symptoms in a large primary care population identified through the use of text and data mining of the electronic health record," Journal of Cardiac Failure, vol. 20, no. 7, pp. 459464, 2014. [27] Jerry Gurwitz et al., "Contemporary prevalence and correlates of incident heart failure with preserved ejection fraction," The American journal of medicine, vol. 126, no. 5, pp. 393-400, 2013. SUPPLEMENTARY Table 4. Qualifying ICD-9 codes for heart failure ICD-9 Code 398.91 402.01 402.11 402.91 404.01 404.03 404.11 404.13 404.91 404.93 428.0 428.1 428.20 428.21 428.22 428.23 428.30 428.31 428.32 428.33 428.40 428.41 428.42 428.43 428.9 Description Rheumatic heart failure (congestive) Malignant hypertensive heart disease with heart failure Benign hypertensive heart disease with heart failure Unspecified hypertensive heart disease with heart failure Hypertensive heart and chronic kidney disease, malignant, with heart failure and with chronic kidney disease stage I through stage IV, or unspecified Hypertensive heart and chronic kidney disease, malignant, with heart failure and with chronic kidney disease stage V or end stage renal disease Hypertensive heart and chronic kidney disease, benign, with heart failure and with chronic kidney disease stage I through stage IV, or unspecified Hypertensive heart and chronic kidney disease, benign, with heart failure and chronic kidney disease stage V or end stage renal disease Hypertensive heart and chronic kidney disease, unspecified, with heart failure and with chronic kidney disease stage I through stage IV, or unspecified Hypertensive heart and chronic kidney disease, unspecified, with heart failure and chronic kidney disease stage V or end stage renal disease Congestive heart failure, unspecified Left heart failure Systolic heart failure, unspecified Acute systolic heart failure Chronic systolic heart failure Acute on chronic systolic heart failure Diastolic heart failure, unspecified Acute diastolic heart failure Chronic diastolic heart failure Acute on chronic diastolic heart failure Combined systolic and diastolic heart failure, unspecified Acute combined systolic and diastolic heart failure Chronic combined systolic and diastolic heart failure Acute on chronic combined systolic and diastolic heart failure Heart failure, unspecified Red and Blue Box of Figure 5 173.71 173.31 172.5 173.41 173.51 173.50 173.91 172.4 173.6 172.3 173.4 173.0 173.7 238.9 173.2 173.3 173.5 173.1 173.9 Figure 7. Detailed version of the red box of Figure 5 Table 5. List of ICD-9 codes that appear in Figure 7, and their descriptions ICD-9 Code 172.3 172.4 172.5 173.0 173.1 173.2 173.3 173.31 173.4 173.41 173.5 173.50 173.51 173.6 173.7 173.71 173.9 173.91 Description Malignant melanoma of skin of other and unspecified parts of face Malignant melanoma of skin of scalp and neck Malignant melanoma of skin of trunk, except scrotum Other and unspecified malignant neoplasm of skin of lip Other and unspecified malignant neoplasm of skin of eyelid, including canthus Other and unspecified malignant neoplasm of skin of ear and external auditory canal Other and unspecified malignant neoplasm of skin of other and unspecified parts of face Basal cell carcinoma of skin of other and unspecified parts of face Other and unspecified malignant neoplasm of scalp and skin of neck Basal cell carcinoma of scalp and skin of neck Other and unspecified malignant neoplasm of skin of trunk, except scrotum Unspecified malignant neoplasm of skin of trunk, except scrotum Basal cell carcinoma of skin of trunk, except scrotum Other and unspecified malignant neoplasm of skin of upper limb, including shoulder Other and unspecified malignant neoplasm of skin of lower limb, including hip Basal cell carcinoma of skin of lower limb, including hip Other and unspecified malignant neoplasm of skin, site unspecified Basal cell carcinoma of skin, site unspecified 238.9 Neoplasm of uncertain behavior, site unspecified 448.9 228.00 216.5 216.7 216.6 216.9 216.3 078.10 216.8 238.2 448.1 Figure 8. Detailed version of the blue box of Figure 5 Table 6. List of ICD-9 codes that appear in Figure 8, and their descriptions ICD-9 Code 078.10 216.3 216.5 216.6 216.7 216.8 216.9 228.00 238.2 448.1 448.9 Description Viral warts, unspecified Benign neoplasm of skin of other and unspecified parts of face Benign neoplasm of skin of trunk, except scrotum Benign neoplasm of skin of upper limb, including shoulder Benign neoplasm of skin of lower limb, including hip Benign neoplasm of other specified sites of skin Benign neoplasm of skin, site unspecified Hemangioma of unspecified site Neoplasm of uncertain behavior of skin Nevus, non-neoplastic Other and unspecified capillary diseases 6-fold Cross Validation Scheme Training Set Validation Set Test Set 32,787 Patients Fold 1 Fold 2 Fold 6 Figure 9. Diagram of 6-fold cross validation Figure 6 depicts the 6-fold cross validation we performed for HF prediction. As explained earlier, the entire cohort is divided into 7 chunks, and two chunks take turn to play as the validation set and the test set. Hyper-parameters used for training models After experimenting with various values, the following hyper-parameter setting produced the best performance. We used Theano 0.7 and CUDA 7 for training logistic regression, MLP, and GRU models. SVM was implemented with Scikit-Learn Linear SVC. KNN was implemented with Scikit-Learn KNeighborsClassifier. Table 7. Hyper-parameter settings for training the models Model Logistic regression, one-hot vectors Logistic regression, medical concept vectors SVM, one-hot vectors SVM, medical concept vectors MLP, one-hot vectors MLP, medical concept vectors KNN, one-hot vectors KNN, medical concept vectors Hyper-parameter L2 regularization: 0.1, Max epoch: 100 L2 regularization: 0.01, Max epoch: 100 L2 regularization: 0.000001, Dual: False L2 regularization: 0.001, Dual: False L2 regularization: 0.01, Hidden layer size: 15, Max epoch: 100 L2 regularization: 0.001, Hidden layer size: 100, Max epoch: 100 Number of neighbors: 15 Number of neighbors: 100 Complete table of 100 frequent diagnoses and their closest diagnoses, medications and procedures Diagnoses Medications Procedures Myalgia and myositis, unspecified (729.1) -Lumbago (724.2) -Thoracic or lumbosacral neuritis or radiculitis, unspecified (724.4) -Degeneration of lumbar or lumbosacral intervertebral disc (722.52) -Other malaise and fatigue (780.79) -PR INJ TRIGGER POINT(S) 1 OR 2 MUSCLES -PHYSICAL MEDICINE, INT -TRIAMCINOLONE ACETONIDE *KENALOG INJ -ASP/INJ MAJOR JOINT/BURS/CYST -PR METHLYPRED ACET (BU 21 - 40MG) *DEPOMEDROL INJ Pain in limb (729.5) -Keratoderma, acquired (701.1) -Plantar fascial fibromatosis (728.71) -Other specified diseases of nail (703.8) -Disturbance of skin sensation (782.0) -Senile cataract, unspecified (366.10) -Presbyopia (367.4) -Preglaucoma, unspecified (365.00) -Other chronic allergic conjunctivitis (372.14) -HYDROCODONEACETAMINOPHEN 5-500 MG PO TABS -ZOLPIDEM TARTRATE 10 MG PO TABS -CYCLOBENZAPRINE HCL 10 MG PO TABS -HYDROCODONEACETAMINOPHEN 10-325 MG PO TABS -TRAMADOL HCL 50 MG PO TABS -HYDROCODONEACETAMINOPHEN 5-500 MG PO TABS -GLASSES -ERYTHROMYCIN 5 MG/GM OP OINT -PATANOL 0.1 % OP SOLN -PR REFRACTION -PR VISUAL FIELD EXAM EXTENDED -PR VISUAL FIELD EXAM LIMITED -REFERRAL TO OPHTHALMOLOGY, INT -OPHTHALMOLOGY, INT -REFERRAL TO NEUROLOGY, INT -NEUROLOGY, INT -SERV PROV DURING REG SCHED EVE/WKEND/HOL HRS -PR OV EST PT LEV 3 -PR ECG AND INTERPRETATION Tear film insufficiency, unspecified (375.15) Headache (784.0) -Dizziness and giddiness (780.4) -Cervicalgia (723.1) -Other malaise and fatigue (780.79) -Acute sinusitis, unspecified (461.9) Disturbance of skin sensation (782.0) -Brachial neuritis or radiculitis NOS (723.4) -Cervicalgia (723.1) -Pain in limb (729.5) -Headache (784.0) Edema (782.3) -Congestive heart failure, unspecified (428.0) -Unspecified essential hypertension (401.9) -Atrial fibrillation (427.31) -Chronic kidney disease, Stage III (moderate) (585.3) -HYDROCODONEACETAMINOPHEN 5-500 MG PO TABS -FLUTICASONE PROPIONATE 50 MCG/ACT NA SUSP -CYCLOBENZAPRINE HCL 10 MG PO TABS -FLONASE 50 MCG/ACT NA SUSP -AZITHROMYCIN 250 MG PO TABS -HYDROCODONEACETAMINOPHEN 5-500 MG PO TABS -PREDNISONE 20 MG PO TABS -FUROSEMIDE 20 MG PO TABS -HYDROCHLOROTHIAZIDE 25 MG PO TABS -HYDROCODONEACETAMINOPHEN 5-500 MG PO TABS -CEPHALEXIN 500 MG PO CAPS -DEBRIDEMENT OF NAILS, 6 OR MORE -PODIATRY, INT -REFERRAL TO PODIATRIC MEDICINE/SURGERY, INT -PR OV EST PT LEV 3 -FOOT COMPLETE -NEUROLOGY, INT -SENSE/MIXED N CONDUCTION TST -REFERRAL TO NEUROLOGY, INT -PR NERVE COND STUDY MOTOR W F-WAVE EA -PR EMG NEEDLE ONE EXTREMITY -DEBRIDEMENT OF NAILS, 6 OR MORE -PR OV EST PT MIN SERV -EKG -PR ECG AND INTERPRETATION -CHEST PA & LATERAL Asthma, unspecified type, unspecified (493.90) -Allergic rhinitis, cause unspecified (477.9) -Acute bronchitis (466.0) -Bronchitis, not specified as acute or chronic (490) -Acute upper respiratory infections of unspecified site (465.9) Routine gynecological examination (V72.31) -Other screening mammogram (V76.12) -Screening for lipoid disorders (V77.91) -Routine general medical examination at a health care facility (V70.0) -Screening for unspecified condition (V82.9) -Mixed hyperlipidemia (272.2) -Other abnormal glucose (790.29) -Obesity, unspecified (278.00) -Pure hypercholesterolemia (272.0) Diabetes mellitus without mention of complication, type II or unspecified type, uncontrolled (250.02) Need for prophylactic vaccination and inoculation against diphtheria-tetanuspertussis, combined [DTP] [DTaP] (V06.1) Diabetes mellitus without mention of complication, type II or unspecified type, not stated as uncontrolled (250.00) Personal history of colonic polyps (V12.72) -Laboratory examination ordered as part of a routine general medical examination (V72.62) -Need for prophylactic vaccination and inoculation against other combinations of diseases (V06.8) -Screening for lipoid disorders (V77.91) -Need for prophylactic vaccination and inoculation against influenza (V04.81) -Unspecified essential hypertension (401.9) -Diabetes mellitus without mention of complication, type II or unspecified type, uncontrolled (250.02) -Obesity, unspecified (278.00) -Benign essential hypertension (401.1) -Special screening for malignant neoplasms of colon (V76.51) -Family history of malignant neoplasm of gastrointestinal tract (V16.0) -Diverticulosis of colon (without mention of hemorrhage) (562.10) -Routine general medical examination at a health care facility (V70.0) -FUROSEMIDE 40 MG PO TABS -AZITHROMYCIN 250 MG PO TABS -ALBUTEROL SULFATE HFA 108 (90 BASE) MCG/ACT IN AERS -ALBUTEROL SULFATE HFA 108 MCG/ACT IN AERS -PROMETHAZINE-CODEINE 6.25-10 MG/5ML PO SYRP -ALBUTEROL 90 MCG/ACT IN AERS -OBTAINING SCREEN PAP SMEAR -GYN CYTOLOGY (PAP) PA -MAMMOGRAM SCREENING -MAMMO SCREENING BILATERAL DIGITAL -SCRN MAMMO DIR DIG IMAGE BIL (V76.12) -METFORMIN HCL 500 MG PO TABS -METFORMIN HCL 1000 MG PO TABS -GLUCOSE BLOOD VI STRP -LISINOPRIL 10 MG PO TABS -LISINOPRIL 20 MG PO TABS -FLUTICASONE PROPIONATE 50 MCG/ACT NA SUSP -AZITHROMYCIN 250 MG PO TABS -ZOLPIDEM TARTRATE 10 MG PO TABS -SIMVASTATIN 20 MG PO TABS -METFORMIN HCL 500 MG PO TABS -METFORMIN HCL 1000 MG PO TABS -GLUCOSE BLOOD VI STRP -HYDROCHLOROTHIAZIDE 25 MG PO TABS -LISINOPRIL 10 MG PO TABS -PEG-KCL-NACL-NASULFNA ASC-C 100 G PO SOLR -MOVIPREP 100 G PO SOLR -SIMVASTATIN 20 MG PO TABS -OMEPRAZOLE 20 MG PO CPDR -CHEST PA & LATERAL -PULSE OXIMETRY SINGLE -PR INHALATION RX FOR OBSTRUCTION MDI/NEB -XR CHEST 2 VIEWS PA LATERAL -IMMUN ADMIN IM/SQ/ID/PERC 1ST VAC ONLY -DIABETIC EYE EXAM (NO BILL) -DIABETES EDUCATION, INT -OPHTHALMOLOGY, INT -DIABETIC FOOT EXAM (NO BILL) -INFLUENZA VAC 3+YR (V04.81) IM -TDAP VAC 11-64 YO *ADACEL (V06.8) IM -MAMMO SCREENING BILATERAL DIGITAL -PR INFLUENZA VACC PRES FREE 3+ YO 0.5ML IM -REFERRAL TO GI, INT -PNEUMOCOCCAL VAC (V03.82) IM/SQ -DIABETIC EYE EXAM (NO BILL) -PR DIABETIC EYE EXAM (NO BILL) -INFLUENZA VAC 3+YR (V04.81) IM -IMMUN ADMIN IM/SQ/ID/PERC 1ST VAC ONLY -OPHTHALMOLOGY, INT -PR COLONOSCOPY W/BIOPSY(S) -PR COLONOSCOPY W/RMVL LESN/TUM/POLYP SNARE -REFERRAL TO GI, INT -GI, INT -PR COLONOSCOPY DIAG Acute upper respiratory infections of unspecified site (465.9) -Cough (786.2) -Acute sinusitis, unspecified (461.9) -Acute bronchitis (466.0) -Acute pharyngitis (462) Need for prophylactic vaccination and inoculation against tetanus-diphtheria [Td] (DT) (V06.5) -Routine general medical examination at a health care facility (V70.0) -Other general medical examination for administrative purposes (V70.3) -Need for prophylactic vaccination and inoculation against other combinations of diseases (V06.8) -Multiphasic screening (V82.6) -Unspecified essential hypertension (401.9) -Allergic rhinitis, cause unspecified (477.9) -Cough (786.2) -Routine general medical examination at a health care facility (V70.0) -Lens replaced by other means (V43.1) -Senile cataract, unspecified (366.10) -Unspecified aftercare (V58.9) -Follow-up examination, following surgery, unspecified (V67.00) -Irritable bowel syndrome (564.1) -Nausea with vomiting (787.01) -Hemorrhage of rectum and anus (569.3) -Nausea alone (787.02) Esophageal reflux (530.81) Other specified aftercare following surgery (V58.49) Diarrhea (787.91) Coronary atherosclerosis of unspecified type of vessel, native or graft (414.00) -Unspecified essential hypertension (401.9) -Atrial fibrillation (427.31) -Benign essential hypertension (401.1) -Congestive heart failure, unspecified (428.0) Dizziness and giddiness (780.4) -Benign paroxysmal positional vertigo (386.11) -Other malaise and fatigue (780.79) -Unspecified hearing loss (389.9) -Chest pain, unspecified (786.50) -Unspecified hearing loss (389.9) -Infective otitis externa, Impacted cerumen (380.4) -AZITHROMYCIN 250 MG PO TABS -PROMETHAZINE-CODEINE 6.25-10 MG/5ML PO SYRP -AMOXICILLIN 500 MG PO CAPS -FLUTICASONE PROPIONATE 50 MCG/ACT NA SUSP -FLONASE 50 MCG/ACT NA SUSP -TD VAC 7+ YO (V06.5) IM -IMMUN ADMIN IM/SQ/ID/PERC 1ST VAC ONLY -CHART ABSTRACTION SIMP (V70.3) -MAMMOGRAM SCREENING -IMMUN ADMIN, EACH ADD -PULSE OXIMETRY SINGLE -SERV PROV DURING REG SCHED EVE/WKEND/HOL HRS -CHEST PA & LATERAL -GYN CYTOLOGY (PAP) PA -INFLUENZA VAC (FLU CLINIC ONLY) 3+YO PA -OMEPRAZOLE 20 MG PO CPDR -ACIPHEX 20 MG PO TBEC -PROTONIX 40 MG PO TBEC -PRILOSEC 20 MG PO CPDR -FLUTICASONE PROPIONATE 50 MCG/ACT NA SUSP -GLASSES -HYDROCODONEACETAMINOPHEN 5-500 MG PO TABS -VICODIN 5-500 MG PO TABS -CEPHALEXIN 500 MG PO CAPS -IBUPROFEN 600 MG PO TABS -METRONIDAZOLE 500 MG PO TABS -CIPRO 500 MG PO TABS -OMEPRAZOLE 20 MG PO CPDR -CIPROFLOXACIN HCL 500 MG PO TABS -PROMETHAZINE HCL 25 MG PO TABS -SIMVASTATIN 40 MG PO TABS -NITROGLYCERIN 0.4 MG SL SUBL -SIMVASTATIN 20 MG PO TABS -LISINOPRIL 10 MG PO TABS -LISINOPRIL 20 MG PO TABS -LORAZEPAM 0.5 MG PO TABS -MECLIZINE HCL 25 MG PO TABS -FLUTICASONE PROPIONATE 50 MCG/ACT NA SUSP -GI, INT -PR ECG AND INTERPRETATION -PR LARYNGOSCOPY FLEX DIAG -PR OV EST PT LEV 3 -CHEST PA & LATERAL -NEOMYCIN-POLYMYXINHC 1 % OT SOLN -AMOXICILLIN 500 MG PO -PR ECG AND INTERPRETATION -PR OPHTHALMIC BIOMETRY -FOOT COMPLETE -EKG -PR REFRACTION -GI, INT -REFERRAL TO GI, INT -SERV PROV DURING REG SCHED EVE/WKEND/HOL HRS -PR COLONOSCOPY W/BIOPSY(S) -US ABDOMEN COMPLETE -PR ECG AND INTERPRETATION -EKG -STRESS ECHO -TTE W/O DOPPLER CMPLT -PR CARDIAC STRESS COMPLTE -PR ECG AND INTERPRETATION -PR TYMPANOMETRY (IMPEDANCE TESTING) -EKG -PR COMPREHENSIVE HEARING TEST -ENT, INT -PR REMVL EAR WAX UNI/BILAT (380.4) -ENT, INT unspecified (380.10) -Dysfunction of Eustachian tube (381.81) -Dizziness and giddiness (780.4) CAPS -FLONASE 50 MCG/ACT NA SUSP -FLUTICASONE PROPIONATE 50 MCG/ACT NA SUSP -TRIAMCINOLONE ACETONIDE 0.1 % EX CREA -FLUOCINONIDE 0.05 % EX CREA Other chronic dermatitis due to solar radiation (692.74) -Personal history of other malignant neoplasm of skin (V10.83) -Other seborrheic keratosis (702.19) -Benign neoplasm of other specified sites of skin (216.8) -Inflamed seborrheic keratosis (702.11) Chronic kidney disease, Stage III (moderate) (585.3) -Anemia, unspecified (285.9) -Diabetes with renal manifestations, type II or unspecified type, not stated as uncontrolled (250.40) -Proteinuria (791.0) -Unspecified essential hypertension (401.9) -Unspecified essential hypertension (401.9) -Other and unspecified hyperlipidemia (272.4) -Long-term (current) use of other medications (V58.69) -Screening for malignant neoplasms of prostate (V76.44) -Keratoderma, acquired (701.1) -Peripheral vascular disease, unspecified (443.9) -Ingrowing nail (703.0) -Diabetes with neurological manifestations, type II or unspecified type, not stated as uncontrolled (250.60) -Tear film insufficiency, unspecified (375.15) -Glaucomatous atrophy [cupping] of optic disc (377.14) -Vitreous degeneration (379.21) -Lens replaced by other means (V43.1) -AMLODIPINE BESYLATE 10 MG PO TABS -FUROSEMIDE 20 MG PO TABS -AMLODIPINE BESYLATE 5 MG PO TABS -LISINOPRIL 40 MG PO TABS -HYDROCHLOROTHIAZIDE 25 MG PO TABS -COLCHICINE 0.6 MG PO TABS -ALLOPURINOL 100 MG PO TABS -ALLOPURINOL 300 MG PO TABS -INDOMETHACIN 50 MG PO CAPS -ATENOLOL 50 MG PO TABS -ECONAZOLE NITRATE 1 % EX CREA -Pre-operative cardiovascular examination (V72.81) -Other specified preoperative examination (V72.83) -Unspecified aftercare (V58.9) -Senile cataract, unspecified (366.10) -Acute upper respiratory infections of unspecified site (465.9) -Bronchitis, not specified as acute or chronic (490) -HYDROCODONEACETAMINOPHEN 5-500 MG PO TABS -VICODIN 5-500 MG PO TABS -IBUPROFEN 600 MG PO TABS Gout, unspecified (274.9) Other specified diseases of nail (703.8) Preglaucoma, unspecified (365.00) Pre-operative examination, unspecified (V72.84) Cough (786.2) -GLASSES -AZITHROMYCIN 250 MG PO TABS -PROMETHAZINE-CODEINE 6.25-10 MG/5ML PO SYRP -ALBUTEROL SULFATE HFA -PR COMPREHENSIVE HEARING TEST -PR TYMPANOMETRY (IMPEDANCE TESTING) -CERUMEN RMVL BY NURSE/MA (NO BILL) -PR DESTR PRE-MALIG LESN 1ST -PR DESTR PRE-MALIG LESN 2-14 -PR BIOPSY OF SKIN/SQ/MUC MEMBR LESN SINGLE -PR DESTRUCT BENIGN LESION UP TO 14 LESIONS (LIQUID NITROGEN) -PR OV EST PT LEV 3 -PR PROTHROMBIN TIME -PR TX/ PROPHY/ DX INJ SQ/ IM -PR OV EST PT MIN SERV -EKG -DEBRIDEMENT OF NAILS, 6 OR MORE -DEBRIDEMENT OF NAILS, 6 OR MORE -TRIAMCINOLONE ACETONIDE *KENALOG INJ -DEBRIDEMENT OF NAILS, 6 OR MORE -TRIM SKIN LESIONS, 2 TO 4 -PARE CORN/CALLUS, SINGLE LESION -PODIATRY, INT -PR DESTR PRE-MALIG LESN 1ST -PR VISUAL FIELD EXAM EXTENDED -PR FUNDAL PHOTOGRAPHY -PR OPHTHALMIC DX IMAGING -PR VISUAL FIELD EXAM LIMITED -PR CMPTR OPHTH IMG OPTIC NERVE -PR ECG AND INTERPRETATION -EKG -CHEST PA & LATERAL -PR OV EST PT LEV 3 -ECG AND INTERPRETATION (ORDERS ONLY) SZ -CHEST PA & LATERAL -XR CHEST 2 VIEWS PA LATERAL -PULSE OXIMETRY SINGLE -SERV PROV DURING REG -Asthma, unspecified type, unspecified (493.90) -Acute sinusitis, unspecified (461.9) Palpitations (785.1) Need for prophylactic vaccination and inoculation against influenza (V04.81) Benign essential hypertension (401.1) Unspecified essential hypertension (401.9) Other screening mammogram (V76.12) Elevated blood pressure reading without diagnosis of hypertension (796.2) -Cardiac dysrhythmia, unspecified (427.9) -Syncope and collapse (780.2) -Dizziness and giddiness (780.4) -Shortness of breath (786.05) -Other and unspecified hyperlipidemia (272.4) -Need for prophylactic vaccination and inoculation against diphtheria-tetanuspertussis, combined [DTP] [DTaP] (V06.1) -Unspecified essential hypertension (401.9) -Screening for unspecified condition (V82.9) -Unspecified essential hypertension (401.9) -Pure hypercholesterolemia (272.0) -Mixed hyperlipidemia (272.2) -Diabetes mellitus without mention of complication, type II or unspecified type, not stated as uncontrolled (250.00) -Diabetes mellitus without mention of complication, type II or unspecified type, not stated as uncontrolled (250.00) -Benign essential hypertension (401.1) -Coronary atherosclerosis of unspecified type of vessel, native or graft (414.00) -Obesity, unspecified (278.00) -Screening for malignant neoplasms of cervix (V76.2) -Routine general medical examination at a health care facility (V70.0) -Screening for unspecified condition (V82.9) -Special screening for malignant neoplasms of colon (V76.51) -Screening for unspecified condition (V82.9) -Impaired fasting glucose (790.21) -Family history of ischemic heart disease (V17.3) -Obesity, unspecified (278.00) 108 (90 BASE) MCG/ACT IN AERS -FLUTICASONE PROPIONATE 50 MCG/ACT NA SUSP -GUAIFENESIN-CODEINE 100-10 MG/5ML PO SYRP -ATENOLOL 25 MG PO TABS -METOPROLOL SUCCINATE ER 25 MG PO TB24 -LORAZEPAM 0.5 MG PO TABS SCHED EVE/WKEND/HOL HRS -PR ECG AND INTERPRETATION -AZITHROMYCIN 250 MG PO TABS -FLUTICASONE PROPIONATE 50 MCG/ACT NA SUSP -SIMVASTATIN 20 MG PO TABS -HYDROCHLOROTHIAZIDE 25 MG PO TABS -PR INFLUENZA VACC PRES FREE 3+ YO 0.5ML IM -INFLUENZA VAC 3+YR (V04.81) IM -INFLUENZA VAC (FLU CLINIC ONLY) 3+YO PA -IMMUN ADMIN IM/SQ/ID/PERC 1ST VAC ONLY -TDAP VAC 11-64 YO *ADACEL (V06.8) IM -PR ECG AND INTERPRETATION -INFLUENZA VAC 3+YR (V04.81) IM -IMMUN ADMIN IM/SQ/ID/PERC 1ST VAC ONLY -GI, INT -PR OV EST PT LEV 3 -HYDROCHLOROTHIAZIDE 25 MG PO TABS -ATENOLOL 50 MG PO TABS -LISINOPRIL 10 MG PO TABS -LISINOPRIL 40 MG PO TABS -LISINOPRIL 20 MG PO TABS -HOLTER 24 HR ECG -PR ECG AND INTERPRETATION -PR CARDIAC STRESS COMPLTE -EKG -TTE W/O DOPPLER CMPLT -HYDROCHLOROTHIAZIDE 25 MG PO TABS -LISINOPRIL 10 MG PO TABS -ATENOLOL 50 MG PO TABS -LISINOPRIL 20 MG PO TABS -SIMVASTATIN 20 MG PO TABS -PR ECG AND INTERPRETATION -EKG -INFLUENZA VAC 3+YR (V04.81) IM -IMMUN ADMIN IM/SQ/ID/PERC 1ST VAC ONLY -FLUTICASONE PROPIONATE 50 MCG/ACT NA SUSP -MAMMO SCREENING BILATERAL DIGITAL -MAMMOGRAM SCREENING -GYN CYTOLOGY (PAP) PA -OBTAINING SCREEN PAP SMEAR -PR OV EST PT LEV 3 -AZITHROMYCIN 250 MG PO TABS -PR OV EST PT LEV 3 -PR OV EST PT LEV 2 -CHART ABSTRACTION SIMP (V70.3) -PR ECG AND INTERPRETATION -TD VAC 7+ YO (V06.5) IM Symptomatic menopausal or female climacteric states (627.2) Congestive heart failure, unspecified (428.0) Other and unspecified hyperlipidemia (272.4) Pure hypercholesterolemia (272.0) Mixed hyperlipidemia (272.2) Unspecified acquired hypothyroidism (244.9) Depressive disorder, not elsewhere classified (311) -Other screening mammogram (V76.12) -Routine gynecological examination (V72.31) -Screening for malignant neoplasms of cervix (V76.2) -Disorder of bone and cartilage, unspecified (733.90) -Edema (782.3) -Coronary atherosclerosis of unspecified type of vessel, native or graft (414.00) -Aortic valve disorders (424.1) -Mitral valve disorders (424.0) -Diabetes mellitus without mention of complication, type II or unspecified type, not stated as uncontrolled (250.00) -Benign essential hypertension (401.1) -Coronary atherosclerosis of unspecified type of vessel, native or graft (414.00) -Need for prophylactic vaccination and inoculation against influenza (V04.81) -Benign essential hypertension (401.1) -Other and unspecified hyperlipidemia (272.4) -Impaired fasting glucose (790.21) -Diabetes mellitus without mention of complication, type II or unspecified type, uncontrolled (250.02) -Benign essential hypertension (401.1) -Diabetes mellitus without mention of complication, type II or unspecified type, uncontrolled (250.02) -Impaired fasting glucose (790.21) -Obesity, unspecified (278.00) -Unspecified essential hypertension (401.9) -Benign essential hypertension (401.1) -Other screening mammogram (V76.12) -Other malaise and fatigue (780.79) -Insomnia, unspecified (780.52) -Other malaise and fatigue (780.79) -Unspecified essential hypertension (401.9) -ZOLPIDEM TARTRATE 10 MG PO TABS -MAMMOGRAM SCREENING -MAMMO SCREENING BILATERAL DIGITAL -GYN CYTOLOGY (PAP) PA -PR OV EST PT LEV 3 -OBTAINING SCREEN PAP SMEAR -FUROSEMIDE 20 MG PO TABS -FUROSEMIDE 40 MG PO TABS -LASIX 20 MG PO TABS -LASIX 40 MG PO TABS -LISINOPRIL 5 MG PO TABS -TTE W/O DOPPLER CMPLT -PR OV EST PT MIN SERV -TRANSTHORACIC COMPLETE, ECHO -CHEST PA & LATERAL -PR PROTHROMBIN TIME -SIMVASTATIN 20 MG PO TABS -SIMVASTATIN 40 MG PO TABS -HYDROCHLOROTHIAZIDE 25 MG PO TABS -LISINOPRIL 10 MG PO TABS -LIPITOR 10 MG PO TABS -PR ECG AND INTERPRETATION -IMMUN ADMIN IM/SQ/ID/PERC 1ST VAC ONLY -PR OV EST PT LEV 3 -INFLUENZA VAC 3+YR (V04.81) IM -PR INFLUENZA VACC PRES FREE 3+ YO 0.5ML IM -SIMVASTATIN 20 MG PO TABS -SIMVASTATIN 40 MG PO TABS -LISINOPRIL 10 MG PO TABS -ATENOLOL 50 MG PO TABS -HYDROCHLOROTHIAZIDE 25 MG PO TABS -EKG -PR ECG AND INTERPRETATION -TDAP VAC 11-64 YO *ADACEL (V06.8) IM -MAMMO SCREENING BILATERAL DIGITAL -SIMVASTATIN 40 MG PO TABS -SIMVASTATIN 20 MG PO TABS -LISINOPRIL 10 MG PO TABS -SIMVASTATIN 10 MG PO TABS -METFORMIN HCL 500 MG PO TABS -TDAP VAC 11-64 YO *ADACEL (V06.8) IM -INFLUENZA VAC 3+YR (V04.81) IM -EKG -PR CARDIAC STRESS COMPLTE -LEVOTHYROXINE SODIUM 75 MCG PO TABS -HYDROCHLOROTHIAZIDE 25 MG PO TABS -LEVOTHYROXINE SODIUM 100 MCG PO TABS -SIMVASTATIN 20 MG PO TABS -SIMVASTATIN 40 MG PO TABS -ZOLPIDEM TARTRATE 10 MG PO TABS -LORAZEPAM 0.5 MG PO TABS -HYDROCHLOROTHIAZIDE 25 MG PO TABS -TRAZODONE HCL 50 MG PO -MAMMO SCREENING BILATERAL DIGITAL -IMMUN ADMIN IM/SQ/ID/PERC 1ST VAC ONLY -PR OV EST PT LEV 3 -MAMMOGRAM SCREENING -PR INFLUENZA VACC PRES FREE 3+ YO 0.5ML IM -IMMUN ADMIN IM/SQ/ID/PERC 1ST VAC ONLY -PR OV EST PT LEV 3 -INFLUENZA VAC 3+YR (V04.81) IM -Other and unspecified hyperlipidemia (272.4) Personal history of other malignant neoplasm of skin (V10.83) Bronchitis, not specified as acute or chronic (490) -Other chronic dermatitis due to solar radiation (692.74) -Benign neoplasm of other specified sites of skin (216.8) -Other seborrheic keratosis (702.19) -Neoplasm of uncertain behavior of skin (238.2) -Acute upper respiratory infections of unspecified site (465.9) -Cough (786.2) -Acute sinusitis, unspecified (461.9) -Acute nasopharyngitis [common cold] (460) Obstructive sleep apnea (adult)(pediatric) (327.23) -Sleep disturbance, unspecified (780.50) -Obesity, unspecified (278.00) -Other respiratory abnormalities (786.09) -Other malaise and fatigue (780.79) Anemia, unspecified (285.9) -Chronic kidney disease, Stage III (moderate) (585.3) -Edema (782.3) -Unspecified essential hypertension (401.9) -Iron deficiency anemia, unspecified (280.9) -Tear film insufficiency, unspecified (375.15) -Vitreous degeneration (379.21) -Regular astigmatism (367.21) -Other specified aftercare following surgery (V58.49) Lens replaced by other means (V43.1) Chest pain, unspecified (786.50) -Palpitations (785.1) -Other chest pain (786.59) -Other respiratory abnormalities (786.09) -Other malaise and fatigue (780.79) Acute sinusitis, unspecified (461.9) -Cough (786.2) -Bronchitis, not specified as acute or chronic (490) -Allergic rhinitis, cause unspecified (477.9) -Acute pharyngitis (462) TABS -HYDROCODONEACETAMINOPHEN 5-500 MG PO TABS -FLUOCINONIDE 0.05 % EX CREA -TRIAMCINOLONE ACETONIDE 0.1 % EX CREA -PROMETHAZINE-CODEINE 6.25-10 MG/5ML PO SYRP -AZITHROMYCIN 250 MG PO TABS -ZITHROMAX Z-PAK 250 MG PO TABS -ALBUTEROL 90 MCG/ACT IN AERS -ALBUTEROL SULFATE HFA 108 MCG/ACT IN AERS -ZOLPIDEM TARTRATE 10 MG PO TABS -FLUTICASONE PROPIONATE 50 MCG/ACT NA SUSP -SIMVASTATIN 20 MG PO TABS -ZOLPIDEM TARTRATE 5 MG PO TABS -AZITHROMYCIN 250 MG PO TABS -HYDROCHLOROTHIAZIDE 25 MG PO TABS -GLASSES -AZITHROMYCIN 250 MG PO TABS -NITROGLYCERIN 0.4 MG SL SUBL -HYDROCODONEACETAMINOPHEN 5-500 MG PO TABS -OMEPRAZOLE 20 MG PO CPDR -AMOXICILLIN 500 MG PO CAPS -AZITHROMYCIN 250 MG PO TABS -FLONASE 50 MCG/ACT NA SUSP -PROMETHAZINE-CODEINE 6.25-10 MG/5ML PO SYRP -MAMMO SCREENING BILATERAL DIGITAL -PR DESTR PRE-MALIG LESN 1ST -PR DESTR PRE-MALIG LESN 2-14 -PR BIOPSY OF SKIN/SQ/MUC MEMBR LESN SINGLE -SURGICAL PATHOLOGY PA -PR OV EST PT LEV 3 -PULSE OXIMETRY SINGLE -CHEST PA & LATERAL -PR INHALATION RX FOR OBSTRUCTION MDI/NEB -SERV PROV DURING REG SCHED EVE/WKEND/HOL HRS -XR CHEST 2 VIEWS PA LATERAL -REFERRAL TO SLEEP DISORDERS, INT -PR POLYSOMNOGRAPHY 4 OR MORE -PR POLYSOMNOGRAPHY W/C P -SLEEP STUDY ATTENDED -CPAP EQUIPMENT DME -INFLUENZA VAC 3+YR (V04.81) IM -PR ECG AND INTERPRETATION -GI, INT -EKG -CHEST PA & LATERAL -PR VISUAL FIELD EXAM EXTENDED -PR REFRACTION -PR OPHTHALMIC DX IMAGING -DIABETIC EYE EXAM (NO BILL) -PR DIABETIC EYE EXAM (NO BILL) -PR ECG AND INTERPRETATION -PR CARDIAC STRESS COMPLTE -EKG -STRESS ECHO -CHEST PA & LATERAL -PULSE OXIMETRY SINGLE -SERV PROV DURING REG SCHED EVE/WKEND/HOL HRS -CHEST PA & LATERAL -IMMUN ADMIN IM/SQ/ID/PERC 1ST VAC Atrial fibrillation (427.31) Urinary tract infection, site not specified (599.0) -Encounter for therapeutic drug monitoring (V58.83) -Coronary atherosclerosis of unspecified type of vessel, native or graft (414.00) -Congestive heart failure, unspecified (428.0) -Unspecified essential hypertension (401.9) -Dysuria (788.1) -Urinary frequency (788.41) -Other malaise and fatigue (780.79) -Vaginitis and vulvovaginitis, unspecified (616.10) Pain in joint, shoulder region (719.41) -Other affections of shoulder region, not elsewhere classified (726.2) -Rotator cuff (capsule) sprain (840.4) -Cervicalgia (723.1) -Pain in joint, lower leg (719.46) Encounter for therapeutic drug monitoring (V58.83) -Atrial fibrillation (427.31) -Congestive heart failure, unspecified (428.0) -Coronary atherosclerosis of unspecified type of vessel, native or graft (414.00) Malignant neoplasm of breast (female), unspecified (174.9) -Lump or mass in breast (611.72) -Unspecified aftercare (V58.9) -Disorder of bone and cartilage, unspecified (733.90) -Pre-operative examination, unspecified (V72.84) Contact dermatitis and other eczema, unspecified cause (692.9) -Unspecified disorder of skin and subcutaneous tissue (709.9) -Other atopic dermatitis and related conditions (691.8) -Other seborrheic keratosis (702.19) -Other chronic dermatitis due to solar radiation (692.74) -Special screening for malignant neoplasms of colon (V76.51) -Diverticulosis of colon (without mention of Benign neoplasm of colon (211.3) -FLUTICASONE PROPIONATE 50 MCG/ACT NA SUSP -WARFARIN SODIUM 5 MG PO TABS -ATENOLOL 50 MG PO TABS -ATENOLOL 25 MG PO TABS -METOPROLOL SUCCINATE ER 50 MG PO TB24 -SIMVASTATIN 40 MG PO TABS ONLY -ENT, INT -NITROFURANTOIN MONOHYD MACRO 100 MG PO CAPS -CIPROFLOXACIN HCL 250 MG PO TABS -SULFAMETHOXAZOLETMP DS 800-160 MG PO TABS -MACROBID 100 MG PO CAPS -CIPRO 500 MG PO TABS -HYDROCODONEACETAMINOPHEN 5-500 MG PO TABS -VICODIN 5-500 MG PO TABS -NAPROXEN 500 MG PO TABS -SERV PROV DURING REG SCHED EVE/WKEND/HOL HRS -PR URINALYSIS AUTO W/O SCOPE -PR ECG AND INTERPRETATION -CYSTOURETHROSCOPY -PR URINALYSIS DIP W/O SCOPE -WARFARIN SODIUM 5 MG PO TABS -COUMADIN 5 MG PO TABS -WARFARIN SODIUM 2.5 MG PO TABS -ATENOLOL 50 MG PO TABS -WARFARIN SODIUM 2 MG PO TABS -LORAZEPAM 1 MG PO TABS -LORAZEPAM 0.5 MG PO TABS -TAMOXIFEN CITRATE 20 MG PO TABS -VICODIN 5-500 MG PO TABS -ANASTROZOLE 1 MG PO TABS -TRIAMCINOLONE ACETONIDE 0.1 % EX CREA -FLUOCINONIDE 0.05 % EX CREA -TRIAMCINOLONE ACETONIDE 0.1 % EX OINT -DESONIDE 0.05 % EX CREA -FLUTICASONE PROPIONATE 50 MCG/ACT NA SUSP -PEG-KCL-NACL-NASULFNA ASC-C 100 G PO SOLR -MOVIPREP 100 G PO SOLR -OMEPRAZOLE 20 MG PO CPDR -PR OV EST PT MIN SERV -PR PROTHROMBIN TIME -EKG -PR ECG AND INTERPRETATION -TTE W/O DOPPLER CMPLT -SHOULDER 2+ VIEWS -ASP/INJ MAJOR JOINT/BURS/CYST -TRIAMCINOLONE ACETONIDE *KENALOG INJ -UPPER EXTREM JOINT MRI -PR THERAPEUTIC EXERCISES -PR OV EST PT MIN SERV -PR PROTHROMBIN TIME -DEBRIDEMENT OF NAILS, 6 OR MORE -PR DESTR PRE-MALIG LESN 2-14 -PR ECG AND INTERPRETATION -MAMMOGRAM BILAT DIAGNOSTIC -DEXA BONE DENSITY AXIAL SKELETON HIPS/PELV/SPINE -MAMMO DIAGNOSTIC BILATERAL DIGITAL -VENIPUNC FNGR,HEEL,EAR -REFERRAL TO ONCOLOGY, INT -DERMATOLOGY, INT -PR BIOPSY OF SKIN/SQ/MUC MEMBR LESN SINGLE -PR DESTR PRE-MALIG LESN 1ST -REFERRAL TO DERMATOLOGY, INT -PR OV EST PT LEV 3 -PR COLONOSCOPY W/BIOPSY(S) -PR COLONOSCOPY W/RMVL LESN/TUM/POLYP SNARE hemorrhage) (562.10) -Family history of malignant neoplasm of gastrointestinal tract (V16.0) -Hemorrhage of rectum and anus (569.3) -Other screening mammogram (V76.12) -Symptomatic menopausal or female climacteric states (627.2) -Routine general medical examination at a health care facility (V70.0) -Special screening for osteoporosis (V82.81) -Myopia (367.1) -Hypermetropia (367.0) -Astigmatism, unspecified (367.20) -Other specified visual disturbances (368.8) -SIMVASTATIN 20 MG PO TABS -REFERRAL TO GI, INT -GI, INT -PR COLONOSCOPY DIAG -FOSAMAX 70 MG PO TABS -DEXA BONE DENSITY AXIAL SKELETON HIPS/PELV/SPINE -BONE DENSITY (DEXA) -MAMMO SCREENING BILATERAL DIGITAL -DEXA BONE DENSITY STUDY 1+ SITES AXIAL SKEL (HIP/PELVIS/SPINE) -PR OV EST PT LEV 3 -PR REFRACTION -DIABETIC EYE EXAM (NO BILL) -PR VISUAL FIELD EXAM EXTENDED -INFLUENZA VAC (FLU CLINIC ONLY) 3+YO PA -PR VISUAL FIELD EXAM LIMITED -Screening for malignant neoplasms of cervix (V76.2) -Routine general medical examination at a health care facility (V70.0) -Routine gynecological examination (V72.31) -Screening for unspecified condition (V82.9) -Regular astigmatism (367.21) -Astigmatism, unspecified (367.20) -Other specified visual disturbances (368.8) -Hypermetropia (367.0) -CHART ABSTRACTION SIMP (V70.3) -PR OV EST PT LEV 2 -TDAP VAC 11-64 YO *ADACEL (V06.8) IM -GYN CYTOLOGY (PAP) PA -PR OV EST PT LEV 3 Acute pharyngitis (462) -Acute sinusitis, unspecified (461.9) -Acute nasopharyngitis [common cold] (460) -Bronchitis, not specified as acute or chronic (490) -Conjunctivitis, unspecified (372.30) Screening for unspecified condition (V82.9) -Other screening mammogram (V76.12) -Other general medical examination for administrative purposes (V70.3) -Need for prophylactic vaccination and inoculation against tetanus-diphtheria [Td] (DT) (V06.5) -Screening for malignant neoplasms of cervix (V76.2) -AZITHROMYCIN 250 MG PO TABS -AMOXICILLIN 500 MG PO CAPS -FLONASE 50 MCG/ACT NA SUSP -PROMETHAZINE-CODEINE 6.25-10 MG/5ML PO SYRP -AUGMENTIN 875-125 MG PO TABS -PR OV EST PT LEV 3 -PR OV EST PT LEV 2 -IMMUN ADMIN IM/SQ/ID/PERC 1ST VAC ONLY -MAMMOGRAM SCREENING -GYN CYTOLOGY (PAP) PA Disorder of bone and cartilage, unspecified (733.90) Presbyopia (367.4) Screening for lipoid disorders (V77.91) Myopia (367.1) -GLASSES -GLASSES -PR REFRACTION -CONTACT LENS MISC PA -INFLUENZA VAC (FLU CLINIC ONLY) 3+YO PA -PR VISUAL FIELD EXAM EXTENDED -DIABETIC EYE EXAM (NO BILL) -PR STREP A RAPID ASSAY W/OPTIC -SERV PROV DURING REG SCHED EVE/WKEND/HOL HRS -PULSE OXIMETRY SINGLE -ENT, INT -CHART ABSTRACTION SIMP (V70.3) Screening for malignant neoplasms of cervix (V76.2) Impotence of organic origin (607.84) Unspecified sinusitis (chronic) (473.9) Need for prophylactic vaccination and inoculation against viral hepatitis (V05.3) Senile cataract, unspecified (366.10) -Other screening mammogram (V76.12) -Screening for lipoid disorders (V77.91) -Routine general medical examination at a health care facility (V70.0) -Screening for unspecified condition (V82.9) -Hypertrophy (benign) of prostate without urinary obstruction and other lower urinary tract symptom (LUTS) (600.00) -Other and unspecified hyperlipidemia (272.4) -Hypertrophy (benign) of prostate with urinary obstruction and other lower urinary tract symptoms (LUTS) (600.01) -Unspecified essential hypertension (401.9) -Acute sinusitis, unspecified (461.9) -Allergic rhinitis, cause unspecified (477.9) -Dysfunction of Eustachian tube (381.81) -Headache (784.0) -Other specified counseling (V65.49) -Need for prophylactic vaccination and inoculation against other combinations of diseases (V06.8) -Need for prophylactic vaccination and inoculation against poliomyelitis (V04.0) -Need for prophylactic vaccination and inoculation against tetanus-diphtheria [Td] (DT) (V06.5) -Presbyopia (367.4) -Regular astigmatism (367.21) -Preglaucoma, unspecified (365.00) -Tear film insufficiency, unspecified (375.15) -GYN CYTOLOGY (PAP) PA -OBTAINING SCREEN PAP SMEAR -MAMMOGRAM SCREENING -PR OV EST PT LEV 3 -PR OV EST PT LEV 2 -VIAGRA 100 MG PO TABS -SILDENAFIL CITRATE 100 MG PO TABS -VIAGRA 50 MG PO TABS -TADALAFIL 20 MG PO TABS -SIMVASTATIN 20 MG PO TABS -PR OV EST PT LEV 3 -IMMUN ADMIN IM/SQ/ID/PERC 1ST VAC ONLY -PR DESTR PRE-MALIG LESN 1ST -PR OV EST PT LEV 2 -INFLUENZA VAC 3+YR (V04.81) IM -AMOXICILLIN-POT CLAVULANATE 875-125 MG PO TABS -FLUTICASONE PROPIONATE 50 MCG/ACT NA SUSP -MOMETASONE FUROATE 50 MCG/ACT NA SUSP -FLONASE 50 MCG/ACT NA SUSP -AUGMENTIN 875-125 MG PO TABS -CIPROFLOXACIN HCL 500 MG PO TABS -MEFLOQUINE HCL 250 MG PO TABS -PR NASAL ENDO DIAG -ENT, INT -CT MAXILLOFACIAL W/O CONTRAST -PR LARYNGOSCOPY FLEX DIAG -REFERRAL TO ENT, INT -GLASSES -PR REFRACTION -PR VISUAL FIELD EXAM EXTENDED -DIABETIC EYE EXAM (NO BILL) -PR VISUAL FIELD EXAM LIMITED -PR OPHTHALMIC DX IMAGING -PR INFLUENZA VACC PRES FREE 3+ YO 0.5ML IM -MAMMO SCREENING BILATERAL DIGITAL -PR OV EST PT LEV 3 -IMMUN ADMIN IM/SQ/ID/PERC 1ST VAC ONLY -TDAP VAC 11-64 YO *ADACEL (V06.8) IM -ASP/INJ MAJOR JOINT/BURS/CYST Insomnia, unspecified (780.52) -Depressive disorder, not elsewhere classified (311) -Other malaise and fatigue (780.79) -Sleep disturbance, unspecified (780.50) -Need for prophylactic vaccination and inoculation against influenza (V04.81) -ZOLPIDEM TARTRATE 10 MG PO TABS -AMBIEN 10 MG PO TABS -LORAZEPAM 0.5 MG PO TABS -ZOLPIDEM TARTRATE 5 MG PO TABS -TRAZODONE HCL 50 MG PO TABS Osteoarthrosis, localized, primary, lower leg (715.16) -Osteoarthrosis, localized, not specified whether -HYDROCODONEACETAMINOPHEN 10-325 -HEP A VAC ADULT (V05.3) IM -IMMUN ADMIN, EACH ADD -HEP B VAC ADULT (V05.3) IM -IMMUN ADMIN IM/SQ/ID/PERC 1ST VAC ONLY -TDAP VAC 11-64 YO *ADACEL (V06.8) IM Obesity, unspecified (278.00) Screening for malignant neoplasms of prostate (V76.44) Unspecified vitamin D deficiency (268.9) Lumbago (724.2) primary or secondary, lower leg (715.36) -Chondromalacia of patella (717.7) -Pain in joint, pelvic region and thigh (719.45) -Enthesopathy of hip region (726.5) MG PO TABS -HYDROCODONEACETAMINOPHEN 5-500 MG PO TABS -Routine general medical examination at a health care facility (V70.0) -Other and unspecified hyperlipidemia (272.4) -Unspecified sleep apnea (780.57) -Unspecified essential hypertension (401.9) -Special screening for malignant neoplasms of colon (V76.51) -Screening for unspecified condition (V82.9) -Impaired fasting glucose (790.21) -Impotence of organic origin (607.84) -Laboratory examination ordered as part of a routine general medical examination (V72.62) -Other malaise and fatigue (780.79) -Need for prophylactic vaccination and inoculation against diphtheria-tetanuspertussis, combined [DTP] [DTaP] (V06.1) -Anemia, unspecified (285.9) -Sciatica (724.3) -Cervicalgia (723.1) -Thoracic or lumbosacral neuritis or radiculitis, unspecified (724.4) -Degeneration of lumbar or lumbosacral intervertebral disc (722.52) -METFORMIN HCL 500 MG PO TABS -HYDROCHLOROTHIAZIDE 25 MG PO TABS Backache, unspecified (724.5) -Sciatica (724.3) -Cervicalgia (723.1) -Thoracic or lumbosacral neuritis or radiculitis, unspecified (724.4) -Pain in joint, pelvic region and thigh (719.45) Thoracic or lumbosacral neuritis or radiculitis, unspecified (724.4) -Spinal stenosis, lumbar region, without neurogenic claudication (724.02) -Displacement of lumbar intervertebral disc without myelopathy (722.10) -Sciatica (724.3) -Lumbago (724.2) -TRIAMCINOLONE ACETONIDE *KENALOG INJ -REFERRAL TO ORTHOPEDICS, INT -KNEE 3 VIEW -BETAMETH ACET 3MG & NA PHOS 3MG *CELESTONE INJ -PR OV EST PT LEV 3 -NUTRITION, INT -PR ECG AND INTERPRETATION -IMMUN ADMIN IM/SQ/ID/PERC 1ST VAC ONLY -PR OV EST PT LEV 2 -SIMVASTATIN 20 MG PO TABS -IMMUN ADMIN IM/SQ/ID/PERC 1ST VAC ONLY -PR OV EST PT LEV 3 -PR OV EST PT LEV 2 -GI, INT -TDAP VAC 11-64 YO *ADACEL (V06.8) IM -ERGOCALCIFEROL 50000 UNITS PO CAPS -FLUTICASONE PROPIONATE 50 MCG/ACT NA SUSP -SIMVASTATIN 20 MG PO TABS -OMEPRAZOLE 20 MG PO CPDR -ZOLPIDEM TARTRATE 10 MG PO TABS -MAMMO SCREENING BILATERAL DIGITAL -TDAP VAC 11-64 YO *ADACEL (V06.8) IM -DEXA BONE DENSITY AXIAL SKELETON HIPS/PELV/SPINE -EKG -REFERRAL TO GI, INT -HYDROCODONEACETAMINOPHEN 5-500 MG PO TABS -CYCLOBENZAPRINE HCL 10 MG PO TABS -FLEXERIL 10 MG PO TABS -VICODIN 5-500 MG PO TABS -CARISOPRODOL 350 MG PO TABS -HYDROCODONEACETAMINOPHEN 5-500 MG PO TABS -CYCLOBENZAPRINE HCL 10 MG PO TABS -NAPROXEN 500 MG PO TABS -FLEXERIL 10 MG PO TABS -VICODIN 5-500 MG PO TABS -HYDROCODONEACETAMINOPHEN 10-325 MG PO TABS -HYDROCODONEACETAMINOPHEN 5-500 MG PO TABS -GABAPENTIN 300 MG PO CAPS -CYCLOBENZAPRINE HCL 10 MG PO TABS -PREDNISONE 20 MG PO TABS -LS SPINE 2 VIEWS -LUMBAR SPINE MRI W/O CONTR -PHYSICAL MEDICINE, INT -PHYSICAL THERAPY, INT -PHYSICAL MEDICINE, INT -LS SPINE 2 VIEWS -PHYSICAL MEDICINE, INT -PR ECG AND INTERPRETATION -LUMBAR SPINE MRI W/O CONTR -PR OV EST PT LEV 3 -LUMBAR SPINE MRI W/O CONTR -MRI LUMBAR SPINE WO CONTRAST -REFERRAL TO PHYSICAL MEDICINE, INT -PHYSICAL MEDICINE, INT -PHYSICAL MEDICINE, INT Inflamed seborrheic keratosis (702.11) -Other chronic dermatitis due to solar radiation (692.74) -Other dyschromia (709.09) -Viral warts, unspecified (078.10) -Nevus, non-neoplastic (448.1) -TRIAMCINOLONE ACETONIDE 0.1 % EX CREA -ECONAZOLE NITRATE 1 % EX CREA Other seborrheic keratosis (702.19) -Actinic keratosis (702.0) -Inflamed seborrheic keratosis (702.11) -Other dyschromia (709.09) -Personal history of other malignant neoplasm of skin (V10.83) -TRIAMCINOLONE ACETONIDE 0.1 % EX CREA -FLUOCINONIDE 0.05 % EX CREA Allergic rhinitis due to other allergen (477.8) -Allergic rhinitis due to pollen (477.0) -Asthma, unspecified type, unspecified (493.90) -Extrinsic asthma, unspecified (493.00) -Acute sinusitis, unspecified (461.9) Allergic rhinitis, cause unspecified (477.9) -Acute sinusitis, unspecified (461.9) -Routine general medical examination at a health care facility (V70.0) -Acute upper respiratory infections of unspecified site (465.9) -Cough (786.2) Abdominal pain, unspecified site (789.00) -Abdominal pain, epigastric (789.06) -Abdominal pain, other specified site (789.09) -Esophageal reflux (530.81) -Constipation, unspecified (564.00) (V72.3) -Other general medical examination for administrative purposes (V70.3) -Need for prophylactic vaccination and inoculation against tetanus-diphtheria [Td] (DT) (V06.5) -Screening for unspecified condition (V82.9) -Symptomatic menopausal or female climacteric states (627.2) -Screening for malignant neoplasms of prostate (V76.44) -Routine general medical examination at a health care -FLUTICASONE PROPIONATE 50 MCG/ACT NA SUSP -FLONASE 50 MCG/ACT NA SUSP -MOMETASONE FUROATE 50 MCG/ACT NA SUSP -ALLEGRA 180 MG PO TABS -NASONEX 50 MCG/ACT NA SUSP -FLUTICASONE PROPIONATE 50 MCG/ACT NA SUSP -FLONASE 50 MCG/ACT NA SUSP -MOMETASONE FUROATE 50 MCG/ACT NA SUSP -AZITHROMYCIN 250 MG PO TABS -NASONEX 50 MCG/ACT NA SUSP -CIPRO 500 MG PO TABS -METRONIDAZOLE 500 MG PO TABS -OMEPRAZOLE 20 MG PO CPDR -HYDROCODONEACETAMINOPHEN 5-500 MG PO TABS -ACIPHEX 20 MG PO TBEC -ALBUTEROL 90 MCG/ACT IN AERS Other specified vaccinations against streptococcus pneumoniae [pneumococcus] (V03.82) -ALBUTEROL SULFATE HFA 108 (90 BASE) MCG/ACT IN AERS -PR DESTRUCT BENIGN LESION UP TO 14 LESIONS (LIQUID NITROGEN) -PR DESTR PRE-MALIG LESN 1ST -PR DESTR PRE-MALIG LESN 2-14 -PR BIOPSY OF SKIN/SQ/MUC MEMBR LESN SINGLE -DERMATOLOGY, INT -PR DESTR PRE-MALIG LESN 1ST -PR DESTR PRE-MALIG LESN 2-14 -PR BIOPSY OF SKIN/SQ/MUC MEMBR LESN SINGLE -PR DESTRUCT BENIGN LESION UP TO 14 LESIONS (LIQUID NITROGEN) -PR OV EST PT LEV 3 -IMMUNOTHERAPY MULTIPLE -ANTIGEN ALLERGY MAT FOR INJ (06041) PA -PR IMMUNOTX W EXTRACT 2+ INJ -INFLUENZA VAC (FLU CLINIC ONLY) 3+YO PA -ENT, INT -PR OV EST PT LEV 3 -IMMUN ADMIN IM/SQ/ID/PERC 1ST VAC ONLY -IMMUNOTHERAPY MULTIPLE -MAMMOGRAM SCREENING -INFLUENZA VAC (FLU CLINIC ONLY) 3+YO PA -US ABDOMEN COMPLETE -GI, INT -ABDOMEN/PELVIS CT PA -PR ECG AND INTERPRETATION -SERV PROV DURING REG SCHED EVE/WKEND/HOL HRS -GYN CYTOLOGY (PAP) PA -MAMMOGRAM SCREENING -CHART ABSTRACTION SIMP (V70.3) -PR OV EST PT LEV 2 -TD VAC 7+ YO (V06.5) IM -PNEUMOCOCCAL VAC (V03.82) IM/SQ -PNEUMOCOCCAL POLYSACC VAC 2+ YO (V03.82) IM/SQ Other psoriasis (696.1) facility (V70.0) -Need for prophylactic vaccination and inoculation against tetanus-diphtheria [Td] (DT) (V06.5) -Need for prophylactic vaccination and inoculation against other specified disease (V05.8) -Other dyschromia (709.09) -Other atopic dermatitis and related conditions (691.8) -Seborrheic dermatitis, unspecified (690.10) -Unspecified pruritic disorder (698.9) Other general medical examination for administrative purposes (V70.3) -Routine general medical examination at a health care facility (V70.0) -Need for prophylactic vaccination and inoculation against tetanus-diphtheria [Td] (DT) (V06.5) -Screening for lipoid disorders (V77.91) Migraine, unspecified, without mention of intractable migraine without mention of status migrainosus (346.90) -Anxiety state, unspecified (300.00) -Cervicalgia (723.1) -Other screening mammogram (V76.12) -Myalgia and myositis, unspecified (729.1) Routine general medical examination at a health care facility (V70.0) -Other screening mammogram (V76.12) -Special screening for malignant neoplasms of colon (V76.51) -Screening for malignant neoplasms of prostate (V76.44) -Need for prophylactic vaccination and inoculation against diphtheria-tetanuspertussis, combined [DTP] [DTaP] (V06.1) -Routine general medical examination at a health care facility (V70.0) -Benign neoplasm of colon (211.3) -Screening for unspecified condition (V82.9) -Other screening mammogram (V76.12) -Pain in joint, shoulder region (719.41) -Pain in joint, lower leg (719.46) -Lumbago (724.2) -Pain in joint, pelvic region and thigh (719.45) Special screening for malignant neoplasms of colon (V76.51) Care involving other physical therapy (V57.1) -IMMUN ADMIN IM/SQ/ID/PERC 1ST VAC ONLY -INFLUENZA VAC 3+YR (V04.81) IM -TDAP VAC 11-64 YO *ADACEL (V06.8) IM -CLOBETASOL PROPIONATE 0.05 % EX OINT -CLOBETASOL PROPIONATE 0.05 % EX CREA -FLUOCINONIDE 0.05 % EX CREA -DESONIDE 0.05 % EX CREA -TRIAMCINOLONE ACETONIDE 0.1 % EX CREA -ALBUTEROL 90 MCG/ACT IN AERS -FLONASE 50 MCG/ACT NA SUSP -SUMATRIPTAN SUCCINATE 50 MG PO TABS -HYDROCODONEACETAMINOPHEN 5-500 MG PO TABS -PROMETHAZINE HCL 25 MG PO TABS -IMITREX 100 MG PO TABS -ZOLPIDEM TARTRATE 10 MG PO TABS -PR OV EST PT LEV 3 -PR OV EST PT LEV 2 -IMMUN ADMIN IM/SQ/ID/PERC 1ST VAC ONLY -TDAP VAC 11-64 YO *ADACEL (V06.8) IM -GYN CYTOLOGY (PAP) PA -PR ULTRAVIOLET THERAPY -DERMATOLOGY, INT -PR BIOPSY OF SKIN/SQ/MUC MEMBR LESN SINGLE -PR DESTR PRE-MALIG LESN 1ST -DERMATOLOGY, INT -CHART ABSTRACTION SIMP (V70.3) -CHART ABSTRACTION INT (V70.3) -PAMFONLINE ENROLLMENT -MAMMOGRAM SCREENING -PR CHART ABSTRACTION COMP (V70.3) -MAMMO SCREENING BILATERAL DIGITAL -GYN CYTOLOGY (PAP) PA -PR OV EST PT LEV 3 -MAMMOGRAM SCREENING -NEUROLOGY, INT -PEG-KCL-NACL-NASULFNA ASC-C 100 G PO SOLR -GI, INT -REFERRAL TO GI, INT -PR OV EST PT LEV 3 -PR OV EST PT LEV 2 -IMMUN ADMIN IM/SQ/ID/PERC 1ST VAC ONLY -HYDROCODONEACETAMINOPHEN 5-500 MG PO TABS -HYDROCODONEACETAMINOPHEN 10-325 MG PO TABS -CYCLOBENZAPRINE HCL 10 MG PO TABS -HYDROCODONE- -PR PHYS THERAPY EVALUATION -PR THERAPEUTIC EXERCISES -PR MANUAL THER TECH 1+REGIONS EA 15 MIN -PR ELECTRIC STIMULATION THERAPY -REFERRAL TO PHYSICAL THERAPY, INT ACETAMINOPHEN 5-325 MG PO TABS -FOLIC ACID 1 MG PO TABS -METHOTREXATE 2.5 MG PO TABS -PREDNISONE 5 MG PO TABS -PREDNISONE 1 MG PO TABS -HYDROXYCHLOROQUINE SULFATE 200 MG PO TABS Rheumatoid arthritis (714.0) -Unspecified inflammatory polyarthropathy (714.9) -Arthropathy, unspecified, site unspecified (716.90) -Sicca syndrome (710.2) Osteoporosis, unspecified (733.00) -Unspecified acquired hypothyroidism (244.9) -Osteoarthrosis, unspecified whether generalized or localized, site unspecified (715.90) -Benign essential hypertension (401.1) -Unspecified essential hypertension (401.9) -FOSAMAX 70 MG PO TABS -ALENDRONATE SODIUM 70 MG PO TABS -HYDROCHLOROTHIAZIDE 25 MG PO TABS -ACTONEL 35 MG PO TABS Other malaise and fatigue (780.79) -Routine general medical examination at a health care facility (V70.0) -Dizziness and giddiness (780.4) -Other respiratory abnormalities (786.09) -Screening for unspecified condition (V82.9) -Atrial fibrillation (427.31) -Congestive heart failure, unspecified (428.0) -Coronary atherosclerosis of unspecified type of vessel, native or graft (414.00) -AZITHROMYCIN 250 MG PO TABS -FLUTICASONE PROPIONATE 50 MCG/ACT NA SUSP Long-term (current) use of anticoagulants (V58.61) Spinal stenosis, lumbar region, without neurogenic claudication (724.02) -Degeneration of lumbar or lumbosacral intervertebral disc (722.52) -Lumbosacral spondylosis without myelopathy (721.3) -Displacement of lumbar intervertebral disc without myelopathy (722.10) -Lumbago (724.2) Unspecified hemorrhoids without mention of complication (455.6) -Internal hemorrhoids without mention of complication (455.0) -Benign neoplasm of colon (211.3) -Anal or rectal pain (569.42) -Personal history of colonic polyps (V12.72) -Dermatophytosis of foot (110.4) -Ingrowing nail (703.0) -Keratoderma, acquired (701.1) -Peripheral vascular disease, unspecified (443.9) Dermatophytosis of nail (110.1) -RHEUMATOLOGY, INT -METHYLPRED ACET 80MG *DEPO-MEDROL INJ -TRIAMCINOLONE ACETONIDE *KENALOG INJ -ASP/INJ INT JOINT/BURS/CYST -PR METHLYPRED ACET (BU 21 - 40MG) *DEPOMEDROL INJ -DEXA BONE DENSITY AXIAL SKELETON HIPS/PELV/SPINE -BONE DENSITY (DEXA) -DEXA BONE DENSITY STUDY 1+ SITES AXIAL SKEL (HIP/PELVIS/SPINE) -INFLUENZA VAC 3+YR (V04.81) IM -MAMMOGRAM SCREENING -PR ECG AND INTERPRETATION -CHEST PA & LATERAL -PR OV EST PT LEV 3 -EKG -TDAP VAC 11-64 YO *ADACEL (V06.8) IM -WARFARIN SODIUM 5 MG PO TABS -COUMADIN 5 MG PO TABS -WARFARIN SODIUM 2.5 MG PO TABS -ATENOLOL 50 MG PO TABS -WARFARIN SODIUM 2 MG PO TABS -HYDROCODONEACETAMINOPHEN 10-325 MG PO TABS -GABAPENTIN 300 MG PO CAPS -HYDROCODONEACETAMINOPHEN 5-500 MG PO TABS -TRAMADOL HCL 50 MG PO TABS -PREDNISONE 20 MG PO TABS -HYDROCORTISONE ACETATE 25 MG PR SUPP -ANUSOL-HC 25 MG PR SUPP -HYDROCORTISONE 2.5 % PR CREA -ANUSOL-HC 2.5 % PR CREA -PEG-KCL-NACL-NASULFNA ASC-C 100 G PO SOLR -PR OV EST PT MIN SERV -PR PROTHROMBIN TIME -DEBRIDEMENT OF NAILS, 6 OR MORE -PR ECG AND INTERPRETATION -EKG -ECONAZOLE NITRATE 1 % EX CREA -TERBINAFINE HCL 250 MG PO TABS -TRIAMCINOLONE ACETONIDE 0.1 % EX CREA -DEBRIDEMENT OF NAILS, 6 OR MORE -TRIM SKIN LESIONS, 2 TO 4 -PARE CORN/CALLUS, SINGLE LESION -PODIATRY, INT -LUMBAR SPINE MRI W/O CONTR -LS SPINE 2 VIEWS -PHYSICAL MEDICINE, INT -MRI LUMBAR SPINE WO CONTRAST -PHYSICAL MEDICINE, INT -ANOSCOPY -SURGERY, INT -PR COLONOSCOPY W/BIOPSY(S) -LIGATION OF HEMORRHOID(S) -GI, INT -CEPHALEXIN 500 MG PO CAPS -HYDROCODONEACETAMINOPHEN 5-500 MG PO TABS -VICODIN 5-500 MG PO TABS -IBUPROFEN 600 MG PO TABS Pain in joint, lower leg (719.46) -Chondromalacia of patella (717.7) -Tear of medial cartilage or meniscus of knee, current (836.0) -Pain in limb (729.5) -Pain in joint, pelvic region and thigh (719.45) Pain in joint, pelvic region and thigh (719.45) -Lumbago (724.2) -Sciatica (724.3) -Backache, unspecified (724.5) -Pain in joint, lower leg (719.46) -HYDROCODONEACETAMINOPHEN 5-500 MG PO TABS -VICODIN 5-500 MG PO TABS Anxiety state, unspecified (300.00) -Insomnia, unspecified (780.52) -Generalized anxiety disorder (300.02) -Other malaise and fatigue (780.79) -Headache (784.0) Generalized anxiety disorder (300.02) -Depressive disorder, not elsewhere classified (311) -Insomnia, unspecified (780.52) -Irritable bowel syndrome (564.1) -Sleep disturbance, unspecified (780.50) Cervicalgia (723.1) -Lumbago (724.2) -Backache, unspecified (724.5) -Headache (784.0) -Brachial neuritis or radiculitis NOS (723.4) Impaired fasting glucose (790.21) -Routine general medical examination at a health care facility (V70.0) -Screening for malignant neoplasms of prostate (V76.44) -Other and unspecified hyperlipidemia (272.4) -Overweight (278.02) -Other chronic dermatitis due to solar radiation (692.74) -Other seborrheic keratosis (702.19) -Benign neoplasm of other specified sites of skin (216.8) -Neoplasm of uncertain behavior of skin (238.2) -LORAZEPAM 0.5 MG PO TABS -ZOLPIDEM TARTRATE 10 MG PO TABS -LORAZEPAM 1 MG PO TABS -ALPRAZOLAM 0.25 MG PO TABS -CLONAZEPAM 0.5 MG PO TABS -LORAZEPAM 0.5 MG PO TABS -CLONAZEPAM 0.5 MG PO TABS -TRAZODONE HCL 50 MG PO TABS -LORAZEPAM 1 MG PO TABS -AMBIEN 10 MG PO TABS -CYCLOBENZAPRINE HCL 10 MG PO TABS -HYDROCODONEACETAMINOPHEN 5-500 MG PO TABS -FLEXERIL 10 MG PO TABS -NAPROXEN 500 MG PO TABS -CARISOPRODOL 350 MG PO TABS -SIMVASTATIN 20 MG PO TABS -HYDROCHLOROTHIAZIDE 25 MG PO TABS -SIMVASTATIN 40 MG PO TABS Actinic keratosis (702.0) -TRIAMCINOLONE ACETONIDE 0.1 % EX CREA -FLUOCINONIDE 0.05 % EX CREA -GLASSES -PR DESTR PRE-MALIG LESN 1ST -ASP/INJ MAJOR JOINT/BURS/CYST -REFERRAL TO ORTHOPEDICS, INT -TRIAMCINOLONE ACETONIDE *KENALOG INJ -LOWER EXTREMITY JOINT MRI W/O CONTR -ORTHOPEDICS, INT -PELVIS LIMITED -LS SPINE 2 VIEWS -XR PELVIS 1 OR 2 VIEWS -REFERRAL TO ORTHOPEDICS, INT -ASP/INJ MAJOR JOINT/BURS/CYST -MAMMO SCREENING BILATERAL DIGITAL -EKG -PR ECG AND INTERPRETATION -PR INFLUENZA VACC PRES FREE 3+ YO 0.5ML IM -TDAP VAC 11-64 YO *ADACEL (V06.8) IM -MAMMOGRAM SCREENING -GI, INT -CHART ABSTRACTION INT (V70.3) -PHYSICAL MEDICINE, INT -SPINE CERVICAL COMPLETE -PHYSICAL THERAPY, INT -PHYSICAL THERAPY, EXT -PR THERAPEUTIC EXERCISES -PR OV EST PT LEV 3 -TDAP VAC 11-64 YO *ADACEL (V06.8) IM -PR OV EST PT LEV 2 -NUTRITION, INT -PR INFLUENZA VACC PRES FREE 3+ YO 0.5ML IM -PR DESTR PRE-MALIG LESN 1ST -PR DESTR PRE-MALIG LESN 2-14 -PR BIOPSY OF SKIN/SQ/MUC MEMBR LESN SINGLE -SURGICAL PATHOLOGY PA -PR OV EST PT LEV 3 

1 Maximizing Indoor Wireless Coverage Using UAVs Equipped with Directional Antennas arXiv:1705.09772v1 [] 27 May 2017 Hazim Shakhatreh and Abdallah Khreishah Abstract Unmanned aerial vehicles (UAVs) can be used to provide wireless coverage during emergency cases where each UAV serves as an aerial wireless base station when the cellular network goes down. They can also be used to supplement the ground base station in order to provide better coverage and higher data rates for the users. In this paper, we aim to maximize the indoor wireless coverage using UAVs equipped with directional antennas. We study the case that the UAVs are using one channel, thus in order to maximize the total indoor wireless coverage, we avoid any overlapping in their coverage volumes. We present two methods to place the UAVs; providing wireless coverage from one building side and from two building sides. In the first method, we utilize circle packing theory to determine the 3-D locations of the UAVs in a way that the total coverage area is maximized. In the second method, we place the UAVs in front of two building sides and efficiently arrange the UAVs in alternating upsidedown arrangements. We show that the upside-down arrangements problem can be transformed from 3D to 2D and based on that we present an efficient algorithm to solve the problem. Our results show that the upside-down arrangements of UAVs, can improve the maximum total coverage by 100% compared to providing wireless coverage from one building side. Index Terms Unmanned aerial vehicles, coverage, circle packing theory. I. I NTRODUCTION Cells on wheels (COW), are used to provide expanded wireless coverage for short-term demands, when cellular coverage is either minimal, never present or compromised by the disaster [1]. UAVs can also be used to provide wireless coverage during emergency cases and Hazim Shakhatreh and Abdallah Khreishah are with the Department of Electrical and Computer Engineering, New Jersey Institute of Technology (email: {hms35,abdallah}@njit.edu) 2 special events (such as concerts, indoor sporting events, etc.), when the cellular network service is not available or it is unable to serve users [2]–[5]. Compared to the COW, the advantage of using UAV-based aerial base stations is their ability to quickly and easily move [6]. The main disadvantage of using UAVs as aerial base stations is their energy capacity, the UAVs need to return periodically to a charging station for recharging, due to their limited battery capacity. In [7], the authors integrate the recharging requirements into the coverage problem and examine the minimum number of required UAVs for enabling continuous coverage under that setting. Directional antennas are used to improve the received signal at their associated users, and also reduce interference since other- aerial base stations are targeting/serving other users in other directions [8]. The authors in [9] study the optimal deployment of UAVs equipped with directional antennas, using circle packing theory. The 3D locations of the UAVs are determined in a way that the total coverage area is maximized. In [10], the authors investigate the problem by characterizing the coverage area for a target outage probability, they show that for the case of Rician fading there exists a unique optimum height that maximizes the coverage area. In [11], the authors propose a heuristic algorithm to find the positions of aerial base stations in an area with different user densities, the goal is to find the minimum number of UAVs and their 3D placement so that all the users are served. However, it is assumed that all users are outdoor and the location of each user represented by an outdoor 2D point. In [12], the authors use multiple UAVs to design efficient UAV relay networks to support military operations. They describe the tradeoff between connectivity among the UAVs and maximizing the covered area. However, they use the UAVs as wireless relays and do not take into account their mutual interference in downlink channels. In [13], the authors propose a computational method for positioning aerial base stations with the goal of minimizing their number, while fully providing the required bandwidth over the disaster area. It is assumed that overlapping aerial base stations coverage areas are allowed and they use the Inter-Cell Interference Coordination (ICIC) methods to schedule radio resources to avoid inter-cell interference. The authors in [4], [5] use a single UAV equipped with omnidirectional antenna to provide wireless coverage for indoor users inside a high-rise building, where the objective is to find the 3D location of a UAV that minimizes the total transmit power required to cover the entire high-rise building. In [14], the authors use UAVs equipped with omnidirectional antennas to minimize the number of UAVs required to cover the indoor users. We summarize our main contributions as follows: • In order to maximize the indoor wireless coverage, we present two methods to place the 3 UAVs, providing wireless coverage from one building side and from two building sides. In this paper, we study the case that the UAVs are using one channel, thus we avoid any overlapping in their coverage volumes (to avoid interference). In the first method, we utilize circle packing theory to determine the 3-D locations of the UAVs in a way that the total coverage area is maximized. In the second method, we place the UAVs in front of two building sides and efficiently arrange the UAVs in alternating upside-down arrangements. • We show that the upside-down arrangements problem can be transformed from 3D to 2D and based on that we present an efficient algorithm to solve the problem. • We demonstrate through simulation results that the upside-down arrangements of UAVs, can improve the maximum total coverage by 100% compared to providing wireless coverage from one building side. The rest of this paper is organized as follows. In Section II, we describe the system model. In Section III, we show the appropriate placement of UAVs that maximizes the total indoor wireless coverage. Finally, we present our numerical results in Section IV and make concluding remarks in Section V. II. S YSTEM M ODEL A. System Settings Consider a 3D building, as shown in Figure 1, where N UAVs must be deployed to maximize wireless coverage to indoor users located within the building. The dimensions of the high-rise building, in the shape of a rectangular prism, be [0, xb ] × [0, yb] × [0, zb ]. Let (xk , yk , zk ) denote the 3D location of UAV k∈ N, and let (Xi , Yi , Zi ) denote the location of user i. Also, let dout,i be the distance between the UAV and indoor user i, and let din,i be the distance between the building wall and indoor user i. Each UAV uses a directional antenna to provide wireless coverage where the antenna half power beamwidth is θB . The authors in [15] use an outdoor directional antenna to provide wireless coverage for indoor users. They show that the highest RSRP (Reference Signal Received Power) and throughput values are measured along the main beam direction, thus the radiation pattern of a directional antenna is a cone and the indoor volume covered by a UAV is a truncated cone, as shown in Figure 2. Here, ri is the radius of the circle that is located at yz-rectangular side ((0,0,0), (0,0,zb ) , (0,yb ,zb ), (0,yb ,0))), rj is the 4 Fig. 1: System model radius of the circle that is located at yz-rectangular side ((xb ,0,0), (xb ,0,zb ) , (xb ,yb ,zb ), (xb ,yb ,0)) and xb is the horizontal width of the building. The volume of a truncated cone is given by: 1 V = πxb (ri2 + rj2 + ri rj ) 3 B. UAV Power Consumption In [16], the authors show that significant power gains are attainable for indoor users even in rich indoor scattering conditions, if the indoor users use directional antennas. Now, consider a transmission between k-th UAV located at (xk , yk , zk ) and i-th indoor user located at (Xi , Yi , Zi ). The received signal power at i-th indoor user location can be given by: Pr,ik (dB) = Pt + Gt + Gr − Li where Pr,ik is the received signal power, Pt is the transmit power of UAV, Gt is the antenna gain of the UAV. It can be approximated by Gt ≈ 29000 2 θB with θB in degrees [17], [18] and Gr is the antenna gain of indoor user i, which is given by [16]: Gr (dB) = Gr,dir + Gr,omni − GRF 5 Fig. 2: 3D Dimensions of the truncated cone Fig. 3: Building sides where Gr,dir and Gr,omni are free-space antenna gains of a directive and an omnidirectional antenna respectively and GRF is the decrease in gain advantage of a directive over an omnidirectional antenna, due to the presence of clutter. Also, Li is the path loss which for the Outdoor-Indoor communication is: Li = LF + LB + LI = (w log10 d3D,i + w log10 fGhz + g1 ) +(g2 + g3 (1 − cos θi )2 ) + (g4 d2D,i ) where LF is the free space path loss, LB is the building penetration loss, and LI is the indoor loss. In the path loss model, we also have w=20, g1 =32.4, g2 =14, g3 =15, g4 =0.5 [19] and fGhz is the carrier frequency. C. Placement of UAVs Choosing the appropriate placement of UAVs will be a critical issue when we aim to maximize the indoor wireless coverage. In this paper, we assume that we can place the UAVs in front of building sides A, B and above the building C as shown Figure 3. We also assume that the UAVs 6 Fig. 4: Placing two UAVs in front of building side A Fig. 5: Placing two UAVs in front of two building sides A and B are using one channel. In this section, we demonstrate why avoiding the overlapping between UAV’s coverage volumes will strengthen the total indoor wireless coverage. 1) Overlapping between UAV’s coverage volumes is allowed: Now, when we place two UAVs in front of building sides A as shown in Figure 4 (the UAVs have different z-coordinates and same x- and y- coordinates), the indoor users located in G1 ’s and G2 ’s locations will have high SINR. On the other hand, the indoor users located in G3 ’s location will have low SINR. This is because the dependency of SINR on the location of indoor user. Similarly, when we place two UAVs in front of two building sides A and B as shown in Figure 5 (the UAVs have different xcoordinates and same y- and z- coordinates), the indoor users located in G1 ’s and G2 ’s locations will have high SINR. On the other hand, the indoor users located in G3 ’s location will have low SINR. In Figure 6 (the UAVs have same y-coordinates and same x- and z- coordinates), 7 Fig. 6: Placing one UAV in front of building side A and one UAV above the building C Fig. 7: UAVs with small antenna half power beamwidth θB when we place one UAV in front of building side A and one UAV above the building C, the indoor users located in G1 ’s and G2 ’s locations will have high SINR. On the other hand, the indoor users located in G3 ’s location will have low SINR. From the previous examples, we can conclude that allowing the UAVs coverage volumes to overlap will result in that some users are not satisfied. In the next section, we place the UAVs in a way that maximizes the total coverage, and avoids any overlapping in their coverage volumes. 2) Overlapping between UAV’s coverage volumes is not allowed: In Figure 7, we avoid the overlapping between UAV’s coverage volumes by using UAVs with small antenna half power beamwidths θB . Actually, this is impractical way to cover the building, due to the high number of UAVs required to cover the building. In Figure 8, we place the UAVs in front of two building sides and efficiently arrange the UAVs in alternating upside-down arrangements. We can notice 8 that this method will maximize the indoor wireless coverage where the uncovered holes are minimized and the overlapping between UAV’s coverage volumes is not allowed. III. M AXIMIZING I NDOOR W IRELESS C OVERAGE In this section, the UAVs are assumed to be symmetric having the same transmit power, the same horizontal location xk , the same channel and the same antenna half power beamwidth θB . We show two methods to place the UAVs in a way that tries to maximize the total coverage, and avoids any overlapping in their coverage volumes. A. Providing Wireless Coverage from one building side In this method, we place all UAVs in front of one building side (side A, side B or side C). The objective is to determine the three-dimensional location of each UAV k∈ N in a way that the total covered volume is maximized. Now, consider that we place the UAVs in front of building side A, then the projection of UAV’s coverage on the building side B is a circle as shown in Figure 9. Our problem can be formulated as: max |N| ⋆ 1 ⋆ π ⋆ xb ⋆ (ri2 + rj2 + ri rj ) 3 subject to q (yk − yq )2 + (zk − zq )2 ≥ 2rj , k 6= q ∈ N zb − (zk + rj ) ≥ 0, k ∈ N (zk − rj ) ≥ 0, k ∈ N yb − (yk + rj ) ≥ 0, k ∈ N (yk − rj ) ≥ 0, k ∈ N The objective is to maximize the indoor wireless coverage (covered volume). Constraint set (1) guarantees that truncated cones cannot overlap each other. Constraint sets (2-5) ensure that UAV k should not cover outside the 3D building, see Figure 9. We model this problem by utilizing the well-known problem, circle packing problem. In this problem, N circles should be packed inside a given surface such that the packing density is maximized and no overlapping occurs [20], note that the surface in our problem is a rectangle. The authors of [20] tackle this problem by solving a number of decision problems. The decision problem is: 9 Fig. 8: UAVs in alternating upside-down arrangements Given N circles of radius rj and a rectangle of dimension d1 × d2 , whether is it possible to locate all the circles into the rectangle or not. In [20], the authors introduce a nonlinear model for this problem. Finding the answer for the decision problem will depend on finding the global minimizer of a nonconvex and nonlinear optimization problem. In each decision problem, they investigate the feasibility of packing N identical circles. If this is feasible, N is incremented by one and the decision problem is solved again. The algorithm will stop when the decision problem yields an infeasible packing [21]. The pseudo code of the algorithm is shown in Algorithm 1. In the next section, we utilize the two building sides to maximize the indoor wireless coverage. This will allow us to extend the indoor wireless coverage compared with providing wireless coverage from one building side, because the holes induced by the cones of the UAVs in one side can be filled by the cones induced by the UAVs in the other side without causing overlap among the two sets of cones. B. Providing Wireless Coverage from two building sides In this method, we place the UAVs in front of two building sides (side A and side B) and efficiently arrange the UAVs in alternating upside-down arrangements (see Figures 10 and 11). In Theorem 1, we find the horizontal location of the UAV xU AV that guarantees the upside-down arrangements of the truncated cones. In Theorem 2, we prove that if the truncated cones do not intersect in 3D, then the circles do not intersect in building sides (A and B), and vice versa. In 10 Fig. 9: Circle packing in a rectangle Theorem 3, we prove that if we maximize the percentage of covered area of building sides (A and B), then we maximize the percentage of covered volume of building, and vice versa. These theorems help us to transform the geometric problem from 3D to 2D and present an efficient algorithm that maximizes the indoor wireless coverage. Theorem 1. The horizontal location of the UAV xU AV that guarantees the upside-down arrangements of the truncated cones will be equal to 0.7071xb regardless of the antenna half power beamwidth angle θB . Proof. The radius of the smaller circular face ri is given by: ri = rj xU AV xb + xU AV (1) Now, we divide the building sides A and B to square cells (as shown in Figures 10 and 11), 11 Algorithm 1 Circle packing in a rectangle 1: N ←− 1 2: Solve the decision problem for N circles 3: If Answer = YES 4: Then N ←− N + 1 5: Return to step 2 6: If Answer = NO 7: n ←− N − 1 8: End 9: Output n the large circle in Figure 10 and the small circle in Figure 11 will represent the projections of UAV’s coverage on building sides A and B when the UAV is placed in front of building side B. Similarly, the four small circles quarters in Figure 10 and the four large circles quarters in Figure 11 will represent the projections of UAVs coverage on building sides A and B when the UAV is placed in front of building side A. From Figures 10 and 11, the diagonal of the square cell is: D = 2rj + 2ri where rj is the radius of the larger circular face and ri is the radius of the smaller circular face. By applying the pythagoreans theorem, we get: √ 4rj2 + 4rj2 = (2rj + 2ri )2 =⇒ 8rj = 2rj + 2ri =⇒ √ 8−2 ri = rj = γrj 2 (2) From equations √ (1) and (2), we get: √ √ 8−2 xU AV = =⇒ 2xU AV = xb ( 8 − 2) + xU AV ( 8 − 2) =⇒ xb + xU AV √ 2 ( 8 − 2) √ = 0.7071xb  xU AV = xb (4 − 8) Thus, to guarantee the upside-down arrangements of the truncated cones, we must place the UAVs at horizontal distance equals to 0.7071xb . Theorems 2 and 3 help us to transform the geometric problem from 3D to 2D and present an efficient algorithm that maximizes the indoor wireless coverage. 12 Fig. 10: The square cell in side A Fig. 11: The square cell in side B Fig. 12: Four circles (with Fig. 13: Four circles (with radius rj ) in building side A radius ri ) in building side B Theorem 2. The truncated cones do not intersect in 3D iff The circles do not intersect in building sides (A and B). Proof. First, we prove that if the truncated cones do not intersect in 3D, then the circles do not intersect in building sides (A and B). Assume that we have a set of truncated cones G = {1, 2, ..., N} and they do not intersect in 3D space. Each truncated cone n ∈ G can be represented by a number of 2D circles {c1n , c2n , ..., c|h|n}, where |h| is the height of the truncated cone, c1n is the smaller circular face and c|h|n is the larger circular face. It is obvious that if the |G| truncated cones do not intersect in 3D space then the smaller and larger circular faces do not intersect in building sides (A and B). 13 Second, we prove that if the circles do not intersect in building sides (A and B), then the truncated cones do not intersect in 3D. Assume that four circles (with large radius rj ) not intersect in building side A (see Figure 12), then the circles (with small radius ri ) in building side B will appear as shown Figure 13. Now, we need to do two steps: 1) Connect the lines between these points (A|h| with A1 , B|h| with B1 , C|h| with C1 and D|h| with D1 ). 2) Draw circles that pass through four points Ak , Bk , Ck and Dk where k ∈ h. After these two steps, the circles that have been drawn in step two will represent a truncated cone that his circular bases do not intersect with the four circles in building sides (A and B). Also, the truncated cones do not intersect in 3D space.  Theorem 3. We maximize the percentage of covered area of building sides (A and B) iff We maximize the percentage of covered volume of building Proof. First, we divide the building sides A and B to square cells (as shown in Figures 10 and 11). The percentage of covered volume is given by: (yb ∗ zb ) ⌋ ∗ 2 ∗ ( π3 ∗ xb ∗ (ri2 + ri rj + rj2 )) 2 4rj V = (xb ∗ yb ∗ zb ) ⌊ (3) Where: (yb ∗ zb ) ⌊ ⌋: the number of square cells in the building side. 4rj2 2: the number of truncated cones in the square cell (see Figures 7 and 8). π 3 ∗ xb ∗ (ri2 + ri rj + rj2 ): the volume of truncated cone. (xb ∗ yb ∗ zb ): the volume of the building. Now, from equations (2) and (3), we get: (yb ∗ zb ) ) ∗ (γ 2 + γ + 1)rj2 ⌋ ∗ ( 2π 3 4rj2 (yb ∗ zb ) 2 ⌋rj = K1 ⌊ V = (yb ∗ zb ) 4rj2 ⌊ (4) Where: ( 2π )(γ 2 + γ + 1) K1 = 3 (yb ∗ zb ) The percentage of covered area of building sides (A and B) is given by: (yb ∗ zb ) (yb ∗ zb ) (yb ∗ zb ) ⌋ ∗ (πri2 + πrj2 ) ⌊ ⌋ ∗ (πri2 + πrj2 ) ⌋ ∗ 2π(ri2 + rj2 ) ⌊ 2 2 4rj 4rj 4rj2 W = + = (yb ∗ zb ) (yb ∗ zb ) (yb ∗ zb ) ⌊ (5) 14 Now, from equations (2) and (5), we get: (yb ∗ zb ) ⌋ ∗ 2π(γ 2 + 1)rj2 2 4rj (yb ∗ zb ) 2 ⌋rj = K2 ⌊ W = (yb ∗ zb ) 4rj2 ⌊ (6) Where: (2π)(γ 2 + 1) K2 = (yb ∗ zb ) To prove that maximizing the percentage of covered volume of building is equivalent to maximizing the percentage of covered area of building sides (A and B). From equations (4) and (6), (yb ∗ zb ) 2 (yb ∗ zb ) 2 ⌋rj is equivalent to maximizing K2 ⌊ ⌋rj where K1 and maximizing V = K1 ⌊ 2 4rj 4rj2 K2 are constants. To prove that maximizing the percentage of covered area of building sides (A and B) is equivalent to maximizing the percentage of covered volume of building. From equations (4) and (yb ∗ zb ) 2 (yb ∗ zb ) 2 ⌋rj is equivalent to maximizing K1 ⌊ ⌋rj where K1 (6), maximizing W = K2 ⌊ 2 4rj 4rj2 and K2 are constants.  In Algorithm 2, we maximize the covered volume by placing the UAVs in alternating upsidedown arrangements. First, we find the horizontal distance between the building and the UAVs xU AV = 0.7071xb (see Theorem 1) that guarantees the alternating upside-down arrangements. Then, we divide the building sides A and B to square cells and place one UAV in front of the square cell. In steps (8-16), we find the 3D locations of UAVs that cover the building from side B. On the other hand, steps (17-25) find the 3D locations of UAVs that cover the building from side A. Finally, the algorithm will output total number of UAVs and the total covered volume. IV. SIMULATION RESULTS Let the dimensions of the building, in the shape of a rectangular prism, be [0, xb = 30]×[0, yb = 40] × [0, zb = 60]. We use three methods to cover the building using UAVs. In the first method, we place all UAVs in front of one building side (A or B) (FOBS). In the second method, we place all UAVs above the building (C) (ABS). In the third method, we arrange the UAVs in alternating upside-down arrangements (AUDA). For the first and second methods, we utilize the circle packing in a rectangle approach [22] to maximize the covered volume. For the third method, we apply Algorithm 2 to maximize the covered volume. In Figure 14, we find the maximum total coverage for different antenna half power beamwidth angles θB . As can be seen from the simulation results, the maximum total coverage is less than half for the FOBS and 15 Algorithm 2 Maximizing Indoor Wireless Coverage Using UAVs 1: Input: 2: The dimensions of building xb , yb and zb 3: The radius of the larger circular face rj 4: Initialization: √ 8−2 5: ri = rj 2 6: xU AV = 0.7071xb 7: u = q = 0 8: The 3D locations of UAVs that cover the building from side B are given by: yb 9: For k1 = 1 : ⌊ ⌋ 2rj zb 10: For s1 = 1 : ⌊ ⌋ 2rj 11: u=u+1 12: xq = xU AV + xb 13: yu = (2k1 − 1)rj 14: zu = (2s1 − 1)rj 15: End 16: End 17: The 3D locations of UAVs that cover the building from side A are given by: yb 18: For k2 = 1 : ⌊ ⌋ 3rj zb ⌋ 19: For s2 = 1 : ⌊ 3rj 20: q =q+1 21: xq = −xU AV 22: yq = (2k2 )rj 23: zq = (2s2 )rj 24: End 25: End 26: Output: 27: The number of UAVs= u + q + 2k1 + 2s1 28: The covered volume=(u)(2)( π3 ∗ xb ∗ (ri2 + ri rj + rj2 )) 16 Total coverage (Normalized) 1 FOBS ABS AUDA 0.8 0.6 0.4 0.2 0 5 10 15 The antenna half power beamwidth ( 20 θ ) B Fig. 14: Total coverage vs. θB 100 FOBS ABS AUDA Number of UAVs 80 60 40 20 0 8 10 12 14 16 18 The antenna half power beamwidth ( 20 θ ) 22 B Fig. 15: Number of UAVs vs. θB ABS methods, this is because providing wireless coverage from one building side will only maximize the covered area of the building side. On the other hand, we improve the maximum total coverage by applying the AUDA, this is because AUDA will allow us to use a higher number of UAVs to provide wireless coverage compared with providing wireless coverage from one building side, as shown in Figure 15. In order to provide full wireless coverage for the building, we use UAVs with different channels to cover the holes in the building. In Figure 16, we find the total number of UAVs required to 17 300 FOBS ABS AUDA Number of UAVs 250 200 150 100 50 0 8 10 12 14 16 18 The antenna half power beamwidth ( 20 θ ) 22 B Fig. 16: Number of UAVs vs. θB provide full coverage. As can be seen from the figure, FOBS and ABS need high number of UAVs to guarantee full wireless coverage for the building, due to the irregular shapes of the holes in the building. Here, we can easily specify the number of UAVs required to cover each hole in the building, due to the small projections of the holes in the building side. On the other hand, AUDA needs fewer number of UAVs to provide full wireless coverage, due to the small-regular shapes of the uncovered spaces inside the building. Here, we need only one UAV to cover each hole. In Figure 17, we find the total transmit power consumed by UAVs when the building is fully covered. Here, we assume that the threshold SNR equals 25dB, the noise power equals -120dBm, the frequency of the channel is 2GHz and the antenna gain of each indoor user is 14.4 dB [16]. As can be seen from the figure, the total transmit power in all methods is very small, due to the high gain of the directional antennas. Also, we can notice that the total power consumed in FOBS and ABS is higher than that of AUDA. This is because the number of UAVs required to fully cover the building in AUDA is fewer than that for FOBS and ABS. V. C ONCLUSION Choosing the appropriate placement of UAVs will be a critical issue when we aim to maximize the indoor wireless coverage. In this paper, we study the case that the UAVs are using one channel, thus in order to maximize the total indoor wireless coverage, we avoid any overlapping in their coverage volumes. We present two methods to place the UAVs; providing wireless coverage from 18 Total transmit power consumed by UAVs (Watts) ×10 -4 1.2 1 0.8 0.6 0.4 FOBS ABS AUDA 0.2 0 8 10 12 14 16 18 The antenna half power beamwidth ( 20 θ ) 22 B Fig. 17: Total transmit power vs. θB one building side and from two building sides. In the first method, we utilize circle packing theory to determine the 3-D locations of the UAVs in a way that the total coverage area is maximized. In the second method, we place the UAVs in front of two building sides and efficiently arrange the UAVs in alternating upside-down arrangements. We show that the upside-down arrangements problem can be transformed from 3D to 2D and based on that we present an efficient algorithm to solve the problem. Our results show that the upside-down arrangements, can improve the maximum total coverage by 100% compared to providing wireless coverage from one building side. ACKNOWLEDGMENT This work was supported in part by the NSF under Grant CNS-1647170. R EFERENCES [1] wikipedia, “Mobile cell sites,” (Accessed on April 2017). [Online]. Available: https://en.wikipedia.org/wiki/Mobile{ }cell{ }sites [2] P. Bupe, R. Haddad, and F. Rios-Gutierrez, “Relief and emergency communication network based on an autonomous decentralized uav clustering network,” in SoutheastCon 2015. IEEE, 2015, pp. 1–8. [3] R. I. Bor-Yaliniz, A. El-Keyi, and H. Yanikomeroglu, “Efficient 3-d placement of an aerial base station in next generation cellular networks,” in Communications (ICC), 2016 IEEE International Conference on. IEEE, 2016, pp. 1–5. [4] H. Shakhatreh, A. Khreishah, and B. 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arXiv:1705.06456v1 [] 18 May 2017 Groups whose Chermak-Delgado lattice is a quasi-antichain∗ Lijian An Department of Mathematics, Shanxi Normal University Linfen, Shanxi 041004, P. R. China April 1, 2018 Abstract A quasiantichain is a lattice consisting of a maximum, a minimum, and the atoms of the lattice. The width of a quasiantichian is the number of atoms. For a positive integer w (≥ 3), a quasiantichain of width w is denoted by Mw . In [3], it is proved that Mw can be as a Chermak-Delgado lattice of a finite group if and only if w = 1 + pa for some positive integer a. Let t be the number of abelian atoms in CD(G). If t > 2, then, according to [3], there exists a positive integer b such that t = pb + 1. The converse is still an open question. In this paper, we proved that a = b or a = 2b. Keywords finite p-groups generated torsion module Chermak-Delgado lattice quasi-antichain finitely 2000 Mathematics subject classification: 20D15. Chermak and Delgado [5] defined a family of functions from the set of subgroups of a finite group into the set of positive integers. They then used these functions to obtain a variety of results, including a proof that every finite group G has a characteristic abelian subgroup N such that |G : N | ≤ |G : A|2 for any abelian A ≤ G. ChermakDelgado measures are values of one of these functions. For any subgroup H of G, the Chermak-Delgado measure of H (in G) is denoted by mG (H), and defined as mG (H) = |H||CG (H)|. The maximal Chermak-Delgado measure of G is denoted by m∗ (G). That is , m∗ (G) = max{mG (H) | H ≤ G}. In [6], Isaacs showed the following theorem. Theorem 1. ([6]) Given a finite group G, let CD(G) = {H | mG (H) = m∗ (G)}. Then (1) CD(G) is a lattice of subgroups of G. (2) If H, K ∈ CD(G), then hH, Ki = HK. (3) If H ∈ CD(G), then CG (H) ∈ CD(G) and CG (CG (H)) = H. By Theorem 1, Chermak-Delgado lattice of a finite groups is always a self-dual lattice. It is natural to ask a question: which types of self-dual lattices can be used as Chermak-Delgado lattices of finite groups. Some special cases of the above question ∗ This work was supported by NSFC (No. 11471198) 1 are proposed and solved. In [2], it is proved that, for any integer n, a chain of length n can be a Chermak-Delgado lattice of a finite p-group. In [1], general conclusions are given. Theorem 2. ([1]) If L is a Chermak-Delgado lattice of a finite p-group G such that both G/Z(G) and G′ are elementary abelian, then are L+ and L++ , where L+ is a mixed 3-string with center component isomorphic to L and the remaining components being m-diamonds (a lattice with subgroups in the configuration of an m-dimensional cube), L++ is a mixed 3-string with center component isomorphic to L and the remaining components being lattice isomorphic to Mp+1 (a quasiantichain of width p + 1, see the following definition). A quasiantichain is a lattice consisting of a maximum, a minimum, and the atoms of the lattice. The width of a quasiantichian is the number of atoms. For a positive integer w (≥ 3), a quasiantichain of width w is denoted by Mw . In [3], it is proved that Mw can be as a Chermak-Delgado lattice of a finite group if and only if w = 1 + pa for some positive integer a. According to [3], if Mw is a Chermak-Delgado lattice of a finite group G, then G is nilpotent of class 2. Moreover, we may choose G to be a p-group of class 2 without loss of generality. Let M1 , M2 , . . . , Mw be all atoms of CD(G). Then G has the following properties: (P1) Both G/Z(G) and G′ are elementary abelian; (P2) Mi /Z(G) ∼ = Mj /Z(G) and G/Z(G) = Mi /Z(G) × Mj /Z(G) for i 6= j; (P3) Let M1 = hx1 , x2 , . . . , xn iZ(G) and M2 = hy1 , y2 , . . . , yn iZ(G) such that M1 /Z(G) and M2 /Z(G) are elementary abelian groups of order pn . Then, for k ≥ 3, Mk = hx1 y1′ , x2 y2′ , . . . , xn yn′ iZ(G) where M2 = hy1′ , y2′ , . . . , yn′ iZ(G). Moreover, there is (k) an invertible matrix Ck = (cij )n×n over Fp such that yj′ ≡ n Y c (k) yi ij (mod Z(G)), j = 1, 2, . . . , n. i=1 In this paper, Ck is called the characteristic matrix of Mk relative to (x1 , x2 , . . . , xn ) and (y1 , y2 , . . . , yn ). Let G be a p-group with CD(G) a quasi-antichain of width w ≥ 3. Let t be the number of abelian atoms in CD(G). If t > 2, then, according to [3], there exists a positive integer b such that t = pb + 1. The converse is still an open question. At the end of [3], such questions are proposed definitely: (Q1) Which values of t are possible in quasi-antichain Chermak-Delgado lattices of width w = pa + 1 when a > 1? (Q2) Are there examples of groups G with G ∈ CD(G) and CD(G) a quasi-antichain where t = 0 and p ≡ 1 modulo 4? This paper answer above questions completely. It is amazing that there are only two possible relations: a = b or a = 2b. We also prove that a | n, which is another appli2 cation about the decomposition of a finitely generated torsion module over a principal ideal domain. Examples related to (Q1) and (Q2) are given. Throughout the article, let p be a prime and G be a finite p-group with properties (P1), (P2) and (P3). Theorem 3. Suppose that there are totally t > 2 abelian atoms in CD(G), which is a quasi-antichain of width w. Then there are positive integers a and b such that w = pa +1 and t = pb + 1, where a = b or a = 2b. If |G/Z(G)| = p2n , then a | n. Proof Without loss of generality, we may assume that M1 , M2 and M3 are abelian atoms. For convenience, the operation of the group G is replaced to be addition. According to (P3), M3 = hx1 + y1′ , x2 + y2′ , . . . , xn + yn′ iZ(G) such that yj′ ≡ n Y (3) cij yi (mod Z(G)), j = 1, 2, . . . , n. i=1 (3) where C3 = (cij )n×n is the characteristic matrix of Mk relative to (x1 , x2 , . . . , xn ) and (y1 , y2 , . . . , yn ). Replacing y1 , y2 , . . . , yn with y1′ , y2′ , . . . , yn′ respectively, we may assume that C3 = In . Let zij = [xi , yj ] and Z = (zij )n×n . Then G′ = hzij | i, j = 1, 2, . . . , ni. We have the following formalized calculation: [((x1 , x2 , . . . , xn ) + (y1 , y2 , . . . , yn )Ck )T , (x1 , x2 , . . . , xn ) + (y1 , y2 , . . . , yn )Ck′ ] = [(x1 , x2 , . . . , xn )T + CkT (y1 , y2 , . . . , yn )T , (x1 , x2 , . . . , xn ) + (y1 , y2 , . . . , yn )Ck′ ] = [(x1 , x2 , . . . , xn )T , (y1 , y2 , . . . , yn )Ck′ ] + [CkT (y1 , y2 , . . . , yn )T , (x1 , x2 , . . . , xn )] = ZCk′ + CkT (−Z T ) = ZCk′ − (ZCk )T . Hence CG (Mk ) = Mk′ if and only if (ZCk )T = ZCk′ . In particular, Mk is abelian if and only if ZCk is symmetric. Since M3 is abelian, Z T = Z. Let V = {C ∈ Mn (Fp ) | ∃ C ′ ∈ Mn (Fp ) s.t. C T Z = ZC ′ }. It is straightforward to prove that V is a linear space over Fp . Hence |V| = pa for some positive integer a. In the following, we prove that V is also a field. Assume that On 6= C ∈ V and C T Z = ZC ′ . Let (s1 , s2 , . . . , sn ) = (x1 , x2 , . . . , xn ) + (y1 , y2 , . . . , yn )C and (t1 , t2 , . . . , tn ) = (x1 , x2 , . . . , xn ) + (y1 , y2 , . . . , yn )C ′ . Then, by the above formalized calculation, [(s1 , s2 , . . . , sn )T , (t1 , t2 , . . . , tn )] = On . Let N1 = hs1 , s2 , . . . , sn iZ(G) and N2 = ht1 , t2 , . . . , tn iZ(G). Then N2 ≤ CG (N1 ). Hence |N1 ||CG (N1 )| ≥ |N1 ||N2 | = |M1 |2 = m∗ (G). It implies that N1 ∈ CD(G), and, there is some k ≥ 3 such that N1 = Mk . Hence V \ {On } is the set of Ck where 3 ≤ k ≤ w. 3 Since Ck is invertible, V is a division algebra. By Wedderburn’s little theorem, V is also a finite field. Now we know that w − 2 = pa − 1. hence w = pa + 1. Let A be a a primitive element of V. Then V = {On , A, A2 , . . . , Ap −1 = In }. Next we study A applying decomposition of a finitely generated torsion module over a principal ideal domain. M2 /Z(G) can be regarded as a n-dimensional vector space V2 over Fp , in which (y1 Z(G), y2 Z(G), . . . , yn Z(G)) is a base for V2 . At this time, A = (aij ) is a matrix of a single transformation A relative to the given base, where n Y A(yj Z(G)) = ( aij yi )Z(G), j = 1, 2, . . . , n. i=1 We can make V2 an Fp [λ]-module by defining the action of any polynomial g(λ) = b0 + b1 λ + · · · + bm λm on any vector y ∈ V2 as g(λ)y = b0 + b1 (Ay) + · · · + bm (Am y). Let mA (λ) be the minimal polynomial of A. Then V2 is a torsion Fp [λ]-module of exponent mA (λ). Obviously, mA (x) is irreducible with degree a. By fundamental structure theorem for finitely generated torsion modules over a principle ideal domain, V2 is a direct sum of cyclic module: V2 = Fp [λ]v1 ⊕ Fp [λ]v2 ⊕ · · · ⊕ Fp [λ]vr It is easy to see that annvi = mA (λ) where i = 1, 2, . . . , r. Let mA (λ) = k0 + k1 λ + · · · + ka−1 λa−1 − ka λa . Then n = ra and v1 , Av1 , . . . , Aa−1 v1 , v2 , Av2 , . . . , Aa−1 v2 , . . . , vr , Avr , . . . , Aa−1 vr is a set of generators for M2 . Assume that (v1 , v2 , . . . , vn ) = (y1 , y2 , . . . , yn )S. Let (u1 , u2 , . . . un ) = (x1 , x2 , . . . , xn )S where S = (sij ) is an invertible matrix. Then the characteristic matrix of Mk relative to (u1 , u2 , . . . , un ) and (v1 , v2 , . . . , vn ) is S −1 Ck S. Matrices related in this way are said to be similar. Without loss of generality, we may assume that A is the characteristic matrix relative to (v1 , Av1 , . . . , Aa−1 vr ). It is clear that A has the form   A1 Oa . . . Oa  Oa A2 . . . Oa    A= . .. ..  , ..  .. . . .  Oa 0 ... Ar where Ai is the companion matrix of mA (λ). That is,  0 0 ... k0  1 0 ... k1  Ai = B =  . . .. . . . .  . . . . 0 ... 1 ka−1 4    ,  and A = Diag(B, B, . . . , B). If AT Z = ZA′ , then A′ T Z = ZA since Z T = Z. there are 1 ≤ k ≤ pa − 1 such that AT Z = ZAk . Let  Z11 Z12 . . . Z1r  Z21 Z22 . . . Z2r  Z= . .. .. ..  .. . . . Zr1 Zr2 . . . Zrr It follows that A′ ∈ V. Hence    ,  T for 1 ≤ i ≤ j ≤ r. It follows where Zij are a × a matrices. Since Z T = Z, Zji = Zij from AT Z = ZAk that  (1) Zij  (2)  Zij Let Zij =   ..  . (a) Zij B T Zij = Zij B k (1) B T Zji = Zji B k (2)    . Then   (2)    B Zij =     Zij (3) Zij .. . T (1) (2) (a) k0 Zij + k1 Zij + · · · + ka−1 Zij      (3) By (1) and (3), (2) (1) (3) (2) (a) (a−1) Zij = Zij B k , Zij = Zij B k , . . . , Zij = Zij (1) (2) (a) Bk (a) k0 Zij + k1 Zij + · · · + ka−1 Zij = Zij B k (4) (5) Hence (1) Zij (k0 + k1 B k + · · · + ka−1 B (a−1)k − B ak ) = 0 (6) (1) That is, Zij mA (B k ) = 0. We claim that mA (B k ) = 0. Otherwise, mA (Ak ) 6= 0. Since mA (Ak ) ∈ V and V is a field, mA (Ak ) is invertible. Hence mA (B k ) is also invertible. (1) Notice that if Zij = 0 then Zij = Oa by (4) and (5). Hence we may choose Zij such (1) (1) that Zij 6= 0. In this case, Zij mA (B k ) 6= 0, a contradiction. Thus mA (B k ) = 0 and a 2 mA (Ak ) = 0. Since Ap , Ap , . . . , Ap = A are all zero points of mA (λ), there exists a 1 ≤ e ≤ a such that k = pe . e It is easy to see that (Am )T Z = ZAmp if and only if (pa − 1) | m(pe − 1). Let W = {C ∈ V | C T Z = ZC}. Then W \ {On } = {Am ∈ V | (pa − 1) | m(pe − 1)}. Hence 5 pa −1 W \ {On } is a cyclic group generated by A (pa −1,pe −1) of order (pa − 1, pe − 1) = p(a,e) − 1. Let b = (a, e). Notice that |W| = t − 1. It is easy to see that t = pb + 1. By (1), e (B p )T Zij = Zij B p 2e (7) By (2), e (B p )T Zij = Zij B (8) Hence (pa − 1) | (p2e − 1). Thus a | 2e. It follows that a | 2b. Hence a = e = b or a = 2e = 2b.  Remark 4. Let F be a field containing pa elements and F ∗ be the multiply group of F . Then F ∗ is cyclic with order pa − 1. Let F ∗ = hbi and B : f 7→ bf be linear transformation over F , where F is regarded as a linear space over field Fp . Of course, the order of B is pa −1. Let p(x) be the minimal polynomial of B. It follows from CayleyHamilton theorem that deg(p(x)) = r ≤ a. Let W = {f (B) f (x) ∈ Fp [x]} = {f1 (B) | f1 (x) ∈ Fp [x], deg(f1 ) < r}. Then dimW = deg(p(x)) = r and hence |W | = pr . On the a other hand, {1, B, B 2 , . . . , B p −1 } ⊆ W and hence |W | ≥ pa . So we get r = a. Let the minimal polynomial of B is p(x) = xa − ka−1 xa−1 − · · · − k1 x − k0 . Then a matrix of B is Frobenius form  0 0  1 0  B= . . ..  .. 0 ... ... ... .. . k0 k1 .. . 1 ka−1    .  Lemma 5. Let B be the matrix introduced in Remark 4. Z = (zij ) is a a × a matrix. If B T Z = ZB, Then Z is symmetric. Proof By calculation, the (u, v)th entry of B T Z is zu+1,v , while the (u, v)th entry of ZB is zu,v+1 , where 1 ≤ u, v ≤ a − 1. Hence zu+1,v = zu,v+1 where 1 ≤ u, v ≤ a − 1. Then, for u ≤ v, we have zu,v+1 = zu+1,v = zu+2,v−1 = · · · = zv,u+1 = zv+1,u . Hence Z is symmetric. Theorem 6. Suppose that ??? Proof Let G be generated by {x1 , x2 , . . . , xn , y1 , y2 , . . . , yn } with defining relationships xpi = yip = 1 and [xi , xj ] = [yi , yj ] = 1 for all i, j such that 1 ≤ i, j ≤ n, [x(u−1)a+i , y(v−1)a+j ] = z(u−1)a+i,(v−1)a+j for 1 ≤ u, v ≤ r and 1 ≤ i, j ≤ a, z(u−1)a+1,(v−1)a+j ∈ Z(G) for 1 ≤ u ≤ v ≤ r and 1 ≤ j ≤ a. For convenience, 6 we use addition operation to replace mutiplication operation of G. We also use the following notations (where 1 ≤ u, v ≤ r and 1 ≤ i, j, k ≤ a): (k) Zuv := (z(u−1)a+k,(v−1)a+1 , z(u−1)a+k,(v−1)a+2 , . . . , z(u−1)a+k,(v−1)a+a ),  Zuv   = (z(u−1)a+i,(v−1)a+j ) =    (1) Zuv (2) Zuv .. . (3) Zuv       and Z =      Z11 Z12 . . . Z21 Z22 . . . .. .. .. . . . Zr1 Zr2 . . . Z1r Z2r .. . Zrr (k)    .  (1) Using above notations, we continue to give defining relationships Zuv = Zuv B k−1 for T for 1 ≤ v < u ≤ r. It is easy to see that 1 ≤ u ≤ v ≤ r and 2 ≤ k ≤ a, Zuv = Zvu G′ = Z(G) = hz(u−1)a+1,(v−1)a+j | 1 ≤ u ≤ v ≤ r, 1 ≤ j ≤ ai is elementary abelian of r+1 r+5 order p 2 n . Hence |G| = p 2 n . (k) (1) Since Zuv = Zuv B k−1 for 1 ≤ u ≤ v ≤ r and 2 ≤ k ≤ a, B T Zuv = Zuv B for 1 ≤ u ≤ v ≤ r. By Lemma 5, Zuv is symmetric for 1 ≤ u ≤ v ≤ r. Hence T T T = Z Zuv = Zvu vu for 1 ≤ v < u ≤ r. Moreover Zuv = Zvu = Zuv = Zvu for all 1 ≤ u, v ≤ r and Z T = Z. Let X = hx1 , x2 , . . . , xn iZ(G) and Y = hy1 , y2 , . . . , yn iZ(G). Assertion 1: CG (x) = X for all x ∈ X \ Z(G). Q Let x = ni=1 ci xi +z where z ∈ Z(G). Write Ck = (c(k−1)a+1 , c(k−1)a+2 , . . . , cka ) for 1 ≤ k ≤ a. Since x 6∈ Z(G), there exists a k0 such that Ck0 6= (0, 0, . . . , 0). Exchanging x(k0 −1)a+1 , y(k0 −1)a+1 , . . . , xk0 a , yk0 a and x(r−1)a+1 , y(r−1)a+1 , . . . , xra , yra , we have Cr 6= (0, 0, . . . , 0). Let Hi = h[x, y(i−1)a+j ] | 1 ≤ j ≤ ai for 1 ≤ i ≤ r. We will prove that H1 +H2 +· · ·+Hv is of order pva . Use induction, we may assume that H1 +H2 +· · ·+Hv−1 is of order p(v−1)a . By calculation, ([x, y(v−1)a+1 ], . . . , [x, yva ]) n X = ck ([xk , y(v−1)a+1 ], . . . , [xk , yva ]) k=1 = = = = = r X a X u=1 i=1 r X a X u=1 i=1 r X a X c(u−1)a+i ([x(u−1)a+i , y(v−1)a+1 ], . . . , [x(u−1)a+i , yva ]) (i) c(u−1)a+i Zuv (1) i−1 c(u−1)a+i Zuv B u=1 i=1 a r X X (1) c(u−1)a+i B i−1 ) Zuv ( u=1 i=1 a r−1 X X (1) c(u−1)a+i B i−1 ) + Zuv ( u=1 i=1 7 a X (1) Zrv ( i=1 c(r−1)a+i B i−1 ) P P Since Cr 6= (0, 0, . . . , 0), ( ai=1 c(r−1)a+i B i−1 ) 6= Oa . By Remark 4, ( ai=1 c(r−1)a+i B i−1 ) (1) P is invertible. Hence Zrv ( ai=1 c(r−1)a+i B i−1 ) is of rank a. It follows that (H1 + H2 + · · · + Hv−1 ) ∩ Hv = 0. Thus H1 + H2 + · · · + Hv is of order pva . By above discussion, H1 +H2 +· · ·+Hv is of order pn . Hence |[x, G]| = pn . It follows r+3 r+3 that |CG (x)| = |G|/pn = p 2 n . Since |X| = p 2 n and X ≤ CG (x), CG (x) = X. Similarly, we have CG (y) = Y for all y ∈ Y \ Z(G). Then CG (X) = X and CG (Y ) = Y , yielding mG (G) = mG (X) = mG (Y ) = p(r+3)n . Assertion 2: m∗ (G) = p(r+3)n . Otherwise, by the dual-property of CD-lattice, there exists H ∈ CD(G) such that (r+3) (r+1) n 2 H < G and |H| > p 2 n . Since |H ∩ X| = |H||X| = |Z(G)|, there exists |HX| > p x ∈ H ∩ X \ Z(G). Hence CG (H) ≤ CG (x) = X. Similarly, we have CG (H) ≤ Y . Hence CG (H) = Z(G) and mG (H) < mG (G), a contradiction. Above discussion also gives that CD(G) is a quasi-antichain, in which every atom (r+3) is of order p 2 n . Assertion 3: w = t = 1 + pa . Now we have G, Z(G), X, Y ∈ CD(G). Let M be an atom different from X and Y . Then, by the same reason given in Theorem 3, we may let M = hw1 , w2 , . . . , wn i and N = CG (M ) = hv1 , v2 , . . . , vn i where (w1 , w2 , . . . , wn ) = (x1 , x2 , . . . , xn ) + (y1 , y2 , . . . , yn )C, (v1 , v2 , . . . , vn ) = (x1 , x2 , . . . , x3n ) + (y1 , y2 , . . . , yn )D. Since [M, N ] = 0, C T Z = ZD. Let  C11 C12 . . . C1r  C21 C22 . . . C2r  C= . .. .. ..  .. . . . Cr1 Cr2 . . . Crr Then r X k=1 T Cki Zkj =        and D =    r X D11 D12 . . . D21 D22 . . . .. .. .. . . . Dr1 Dr2 . . . Zik Dkj for 1 ≤ i, j ≤ r. D1r D2r .. . Drr    .  (9) k=1 Taking i 6= j in Equation (9) and compiling two sides, we get Cik = Oa for i 6= k, Dkj = Oa for k 6= j, and CiiT Zij = Zij Djj for 1 ≤ i, j ≤ r and i 6= j. (10) Taking i = j in Equation (9) and compiling two sides, we get CiiT Zii = Zii Dii for 1 ≤ i ≤ r. (11) Notice that Zij and Z11 have no essential difference. Thus Cii = Cjj and Dii = Djj for i 6= j. 8 Let V = {U ∈ Ma (Fp ) | ∃ V ∈ Ma (Fp ) s.t. U T Z11 = Z11 V }. Similar to the proof in Theorem 3, V is a division algebra. Hence V has at most pa elements (otherwise, there exists a non-zero matrix in which the elements of the first row are all zero, a contradiction). Notice that B T Z11 = Z11 B. Hence B k ∈ V where 1 ≤ k ≤ pa − 1. it a follows that V = {On , B, B 2 , . . . , B p −1 = In }. Therefore C = D = Diag(B k , B k , . . . , B k ) for some 1 ≤ k ≤ pa − 1. Hence w = t = pa + 1.  Theorem 7. Suppose that a, b and r are positive integers such that a = 2b and r ≥ 3. Let n = ar. Then there exists a group G such that |G/Z(G)| = p2n , and CD(G) is a quasi-antichain of width w = pa + 1, in which the number of abelian atoms is t = pb + 1. Proof Let G be generated by {x1 , x2 , . . . , xn , y1 , y2 , . . . , yn } with defining relationships similar to that in Theorem 6. The followings are different defining relationships from (k) that in Theorem 6: [x(u−1)a+i , y(u−1)a+j ] = 1 for 1 ≤ u ≤ r and 1 ≤ i, j ≤ a, Zuv = b (1) Zuv B (k−1)p for 1 ≤ u < v ≤ r and 2 ≤ k ≤ a. In this case, G′ = Z(G) = hz(u−1)a+1,(v−1)a+j | 1 ≤ u < v ≤ r, 1 ≤ j ≤ ai is r+3 r−1 elementary abelian of order p 2 n . Hence |G| = p 2 n . b b (k) (1) Since Zuv = Zuv B (k−1)p for 1 ≤ u < v ≤ r and 2 ≤ k ≤ a, B T Zuv = Zuv B p for 1 ≤ u < v ≤ r. In this case, Zuv is not symmetric and Zuv 6= Zvu for all 1 ≤ v < u ≤ r. Z T = Z is still hold. Let X = hx1 , x2 , . . . , xn iZ(G) and Y = hy1 , y2 , . . . , yn iZ(G). Similar to Assertion 1 in the proof of Theorem 6, we have |[x, Y ]| ≥ p(r−1)a for all x ∈ X \ Z(G). It follows that |CY (x)| ≤ |Y |/p(r−1)a = pa |Z(G)|. Similarly, |CX (y)| ≤ pa |Z(G)| for all y ∈ Y \ Z(G). It is easy to check that CG (X) = X and CG (Y ) = Y , yielding mG (G) = mG (X) = mG (Y ) = p(r+1)n . Assertion (1): m∗ (G) = p(r+1)n . Otherwise, by the dual-property of CD-lattice, there exists H ∈ CD(G) such that (r−1) (r+1) n 2 = |Z(G)|, there exists H < G and |H| > p 2 n . Since |H ∩ X| = |H||X| |HX| > p x ∈ H ∩ X \ Z(G). Hence CG (H) ≤ CG (x) = XCY (x). Similarly, there exists y ∈ H ∩ Y \ Z(G). Hence CG (H) ≤ CG (y) = Y CX (y). It follows that CG (H) ≤ CX (y)CY (x). Hence |CG (H)| ≤ p2a |Z(G)|. Obviously CG (H) > Z(G). Hence there exists 1 6= x′ y ′ ∈ CG (H) \ Z(G), where x′ ∈ CX (y), y ′ ∈ CY (x). Whatever, we have |H| ≤ r+1 |CG (x′ y ′ )| ≤ |G : [G, x′ y ′ ]| ≤ p 2 n+a . Hence mG (H) = |H||CG (H)| ≤ prn+3a ≤ p(r+1)n , a contradiction. Assertion (2): CD(G) is a quasi-antichain. Otherwise, by the dual-property of CD-lattice, there exists H ∈ CD(G) such that (r+1) H < G and |H| > p 2 n . By Assertion (1), n = 3a and there exists x ∈ H ∩ X \ Z(G) and y ∈ H ∩ Y \ Z(G) such that CG (H) = CX (y)CY (x), where |CX (y)| = |CY (x)| = pa |Z(G)|. Take x′ ∈ CX (y) \ Z(G) and y ′ ∈ CY (x) \ Z(G). Then H ≤ CX (y ′ )CX (x′ ) ≤ (r+1) p2a |Z(G)| < p 2 n , a contradiction. 9 Assertion (3): w = pa + 1 and t = pb + 1. Let M be an atom different from X and Y . Then, by the same reason given in Theorem 3, we may let M = hw1 , w2 , . . . , wn i and N = CG (M ) = hv1 , v2 , . . . , v3n i where (w1 , w2 , . . . , wn ) = (x1 , x2 , . . . , xn ) + (y1 , y2 , . . . , yn )C, (v1 , v2 , . . . , v3 ) = (x1 , x2 , . . . , x3n ) + (y1 , y2 , . . . , y3 )D. We also have C T Z = ZD.  C11 C12  C21 C22  C= . ..  .. . Then Let ... ... .. . C1r C2r .. . Cr1 Cr2 . . . Crr r X k=1 T Zkj = Cki        and D =    r X D11 D12 . . . D21 D22 . . . .. .. .. . . . Dr1 Dr2 . . . Zik Dkj for 1 ≤ i, j ≤ r. D1r D2r .. . Drr    .  (12) k=1 Taking i 6= j in Equation (12) and compiling two sides, we get Cik = Oa for i 6= k, Dkj = Oa for k 6= j, and CiiT Zij = Zij Djj for 1 ≤ i, j ≤ r and i 6= j. (13) Notice that Zij , Zik and Zki have no essential difference for k 6= i, j. Thus Cii = Ckk and Djj = Dkk for k 6= i, j. Since r ≥ 3, C11 = C22 = · · · = Crr and D11 = D22 = · · · = Drr . Let V = {U ∈ Ma (Fp ) | ∃ V ∈ Ma (Fp ) s.t. U T Z12 = Z12 V }. Similar to the proof in Theorem 3, V is a division algebra. Hence V has at most pa elements (otherwise, there exists a non-zero matrix in which the elements of the first row are all zero, a b contradiction). Notice that B T Z12 = Z12 B p . Hence B k ∈ V where 1 ≤ k ≤ pa − 1. It a follows that V = {On , B, B 2 , . . . , B p −1 = In }. b b b Therefore C = Diag(B k , B k , . . . , B k ) and D = Diag(B kp , B kp , . . . , B kp ) for some 1 ≤ k ≤ pa − 1. Hence w = pa + 1. It is easy to see that M = N if and only if C = D if and only if (pb + 1) | k. Hence the number of abelian atoms t = pb + 1.  Theorem 8. Suppose that a and r are positive integers such that r ≥ 3. Let n = ar. Then there exists a group G such that |G/Z(G)| = p2n , and CD(G) is a quasi-antichain of width w = pa + 1, in which the number of abelian atoms is 1 or 2 for p = 2 or p 6= 2 respectively. Proof Let G be generated by {x1 , x2 , . . . , xn , y1 , y2 , . . . , yn } with defining relationships xpi = yip = 1 and [xi , yj ] = 1 for all i, j such that 1 ≤ i, j ≤ n, [x(u−1)a+i , x(u−1)a+j ] = 1 for 1 ≤ u ≤ r and 1 ≤ i < j ≤ a, [x(u−1)a+i , x(v−1)a+j ] = z(u−1)a+i,(v−1)a+j for 1 ≤ u < v ≤ r and 1 ≤ i, j ≤ a, [yi , yj ] = [xj , xi ] for every i, j with 1 ≤ i < j ≤ n, z(u−1)a+1,(v−1)a+j ∈ Z(G) for 1 ≤ u < v ≤ r and 1 ≤ j ≤ a. For convenience, we use 10 addition operation to replace mutiplication operation of G. We also use the following notations (where 1 ≤ u, v ≤ r and 1 ≤ i, j, k ≤ a): (k) Zuv := (z(u−1)a+k,(v−1)a+1 , z(u−1)a+k,(v−1)a+2 , . . . , z(u−1)a+k,(v−1)a+a ),  (1)    Zuv Z11 Z12 . . . Z1r  (2)   Z21 Z22 . . . Z2r   Zuv     Zuv = (z(u−1)a+i,(v−1)a+j ) =  and Z =  .. .. ..  . ..  ..   . . . .   .  (3) Zr1 Zr2 . . . Zuv (k) Zrr (1) Using above notations, we continue to give defining relationships Zuv = Zuv B k−1 for T for 1 ≤ v < u ≤ r, 1 ≤ u < v ≤ r and 2 ≤ k ≤ a. It is easy to see that Zuv = −Zvu G′ = Z(G) = hz(u−1)a+1,(v−1)a+j | 1 ≤ u < v ≤ r, 1 ≤ j ≤ ai is elementary abelian of r+3 r−1 order p 2 n . Hence |G| = p 2 n . Since [x(u−1)a+i , x(u−1)a+j ] = 1 for 1 ≤ u ≤ r and 1 ≤ i < j ≤ a, Zuu = Oa for (k) (1) 1 ≤ u ≤ r. Since Zuv = Zuv B k−1 for 1 ≤ u < v ≤ r and 2 ≤ k ≤ a, B T Zuv = Zuv B for 1 ≤ u < v ≤ r. By Lemma 5, Zuv is symmetric for 1 ≤ u < v ≤ r. Hence T T = −Z Zuv = −Zvu vu for 1 ≤ v < u ≤ r. Moreover B Zuv = Zuv B for all 1 ≤ u, v ≤ r and Z T = −Z. Let X = hx1 , x2 , . . . , xn iZ(G) and Y = hy1 , y2 , . . . , yn iZ(G). Similar to Assertion 1 in the proof of Theorem 6, we have |[x, X]| ≥ p(r−1)a for all x ∈ X \ Z(G). It follows that |CX (x)| ≤ |X|/p(r−1)a = pa |Z(G)|. Similarly, |CY (y)| ≤ pa |Z(G)| for all y ∈ Y \Z(G). It is easy to check that CG (X) = Y and CG (Y ) = X, yielding mG (G) = mG (X) = mG (Y ) = p(r+1)n . Similar to Assertion (1) and Assertion (2) in the proof of Theorem 7, we have = p(r+1)n and CD(G) is a quasi-antichain. Let M be an atom different from X and Y . Then, by the same reason given in Theorem 3, we may let M = hw1 , w2 , . . . , wn i and N = CG (M ) = hv1 , v2 , . . . , v3n i where (w1 , w2 , . . . , wn ) = (x1 , x2 , . . . , xn ) + (y1 , y2 , . . . , yn )C, m∗ (G) (v1 , v2 , . . . , vn ) = (x1 , x2 , . . . , xn )D + (y1 , y2 , . . . , yn ). Since [M, N ] = 0, C T Z = ZD. Let  C11 C12 . . . C1r  C21 C22 . . . C2r  C= . .. .. ..  .. . . . Cr1 Cr2 . . . Crr        and D =    D11 D12 . . . D21 D22 . . . .. .. .. . . . Dr1 Dr2 . . . D1r D2r .. . Drr    .  Similar to Assertion 3 in the proof of Theorem 6, we have C = D = Diag(B k , B k , . . . , B k ) for some 1 ≤ k ≤ pa − 1. Hence w = pa + 1. It is easy to see that M = N if and only if CD = In if and only if (pa − 1) | 2k. Hence the number of abelian atoms is 1 or 2 for p = 2 and p 6= 2 respectively.  11 Theorem 9. Suppose that p is and odd prime, a is odd, and r is positive integers such that r ≥ 3. Let n = ar. Then there exists a group G such that |G/Z(G)| = p2n , and CD(G) is a quasi-antichain of width w = pa + 1, in which the number of abelian atoms is t = 0. Proof Let G be generated by {x1 , x2 , . . . , xn , y1 , y2 , . . . , yn } with defining relationships similar to that in Theorem 8. The unique different defining relationship is [yi , yj ] = [xj , xi ]ν for every i, j with 1 ≤ i < j ≤ n, where ν is a fixed quadratic non-residue module p. Similar to the proof of Theorem 8, we have m∗ (G) = p(r+1)n , CD(G) is a quasiantichain, w = pa + 1, νC = D = Diag(B k , B k , . . . , B k ) for some 1 ≤ k ≤ pa − 1, and M = N if and only if νB 2k = Ia . By the definition of B, hB pa −1 p−1 i = hlIa | 1 ≤ l ≤ p − 1i. Since ν is not a square, a −1 m pp−1 there exists an odd m such that B = νIa . a −1 pa −1 2k a If νB = Ia , then (p −1) | 2k +m p−1 . Notice that m and pp−1 = 1+p+· · ·+pa−1 are all odd. There is no integer k such that νB 2k = Ia . Hence the number of abelian atoms is t = 0.  References [1] AN, L., Brenna, J., Qu, H., Wilcox, E., Chermak-Delgado lattice extension theorems, Algebra comm., 2015, 43(5): 2201–2213. [2] Brewster, B., Hauck, P., Wilcox, E., Groups whose Chermak-Delgado lattice is a chain, Journal of Group Theory, 2014, 17(2): 253–265. [3] Brewster, B., Hauck, P., Wilcox, E., Quasi-antichain Chermak-Delgado lattice of finite groups, Arch. Math., 2014, 103: 301–311. [4] Brewster, B., Wilcox, E., Some groups with computable Chermak-Delgado lattices, Bulletin of the Australian Mathematical Society, 2012, 86(1): 29–40. [5] Chermak, A., Delgado, A., A measuring argument for finite groups, Proceedings of the American Mathematical Society, 1989, 107(4): 907–914. [6] Isaacs, I. M., Finite Group Theory, American Mathematical Society, 2008. 12 

1 Investigations of a Robotic Testbed with Viscoelastic Liquid Cooled Actuators arXiv:1711.01649v2 [] 8 Mar 2018 Donghyun Kim, Junhyeok Ahn, Orion Campbell, Nicholas Paine, and Luis Sentis, Abstract—We design, build, and thoroughly test a new type of actuator dubbed viscoelastic liquid cooled actuator (VLCA) for robotic applications. VLCAs excel in the following five critical axes of performance: energy efficiency, torque density, impact resistence, joint position and force controllability. We first study the design objectives and choices of the VLCA to enhance the performance on the needed criteria. We follow by an investigation on viscoelastic materials in terms of their damping, viscous and hysteresis properties as well as parameters related to the longterm performance. As part of the actuator design, we configure a disturbance observer to provide high-fidelity force control to enable a wide range of impedance control capabilities. We proceed to design a robotic system capable to lift payloads of 32.5 kg, which is three times larger than its own weight. In addition, we experiment with Cartesian trajectory control up to 2 Hz with a vertical range of motion of 32 cm while carrying a payload of 10 kg. Finally, we perform experiments on impedance control and mechanical robustness by studying the response of the robotics testbed to hammering impacts and external force interactions. Index Terms—Viscoelastic liquid cooled actuator, Torque feedback control, Impedance control. I. I NTRODUCTION ERIES elastic actuators (SEAs) [1] have been extensively used in robotics [2], [3] due to their impact resistance and high-fidelity torque controllability. One drawback of SEAs is the difficulty that arises when using a joint position controller due to the presence of the elastic element in the drivetrain. To remedy this problem the addition of dampers has been previously considered [4]–[6]. However, incorporating mechanical dampers makes actuators bulky and increases their mechanical complexity. One way to avoid this complexity is to employ elastomers instead of metal springs. Using a viscoelastic material instead of combined spring-damper systems enables compactness [7] and simplified drivetrains [8]. However, it is difficult to achieve high bandwidth torque control due to the nonlinear behavior of elastomers. To address this difficulty, [9] models the force-displacement curve of elastomer using a “standard linear model.” The estimated elastomer force is employed in a closed-loop force controller. Unfortunately, the hysteresis in the urethane elastomer destabilized the system at frequencies above 2 Hz. In contrast our controllers achieve a bandwidth of 70 Hz. The study on [10] accomplishes reasonably good torque control performance, but the range of torques is small to ensure that the elastomer operates in the linear region; our design and control methods described here achieve more than an order of magnitude higher range of torques with high fidelity tracking. To sufficiently address the nonlinear behavior of elastomers, which severely reduce force control performance, we empiri- S cally analyze various viscoelastic materials with a custom-built elastomer testbed. We measure each material’s linearity, creep, compression set, and damping under preloaded conditions, which is a study under-documented in the academic literature. To achieve stable and accurate force control, we study various feedback control schemes. In a previous work, we showed that the active passivity obtained from motor velocity feedback [11] and model-based control such as disturbance observer (DOB) [12] play an essential role in achieving high-fidelity force feedback control. Here, we analyze the phase margins of various feedback controllers and empirically show their operation in the new actuators. We verify the stability and accuracy of our controllers by studying impedance control and impact tests. To test our new actuator, we have designed a two degree-offreedom (DOF), robotic testbed, shown in Fig. 5. It integrates two of our new actuators, one in the ankle, and another in the knee, while restricting motions to the sagittal plane. With the foot bolted to the floor for initial tests, weight plates can be loaded on the hip joint to serve as an end-effector payload. We test operational space control to show stable and accurate operational space impedance behaviors. We perform dynamic motions with high payloads to showcase another important aspect of our system, which is its cooling system aimed at significantly increasing the power of the robot. The torque density of electric motors is often limited by sustainable core temperature. For this reason, the maximum continuous torque achieved by these motors can be significantly enhanced using an effective cooling system. Our previous study [13] analyzed the improvements on achievable power based on thermal data of electric motors and proposed metrics for design of cooling systems. Based on the metrics from that study, we chose a 120 W Maxon EC-max 40, which is expected to exert 3.59 times larger continuous torque when using the proposed liquid cooling system. We demonstrate the effectiveness of liquid cooling by exerting 860N continous force during 5 min and 4500N peak force during 0.5s while keeping the core temperatures below 115◦C, which is much smaller than the maximum, 155◦C. We accurately track fast motions of 2 Hz while carrying a 10 kg payload for endurance tests. In addition we perform heavy lift tests with a payload of 32.5 kg keeping the motor temperatures under 80◦C. The main contribution of this paper is the introduction of a new viscoelastic liquid cooled actuator and a thorough study of its performance and its use on a multidof testbed. We demonstrate that the use of liquid cooling and the elastomer significantly improve joint position controllability and power density over traditional SEAs. More concretely, we 1) design 2 a new actuator, dubbed the VLCA, 2) extensively study viscoelastic materials, 3) extensively analyze torque feedback controllers for VLCAs, and 4) examine the performance in a multidof prototype. II. BACKGROUND Existing actuators can be characterized using four criteria: power source (electric or hydraulic), cooling type (air or liquid), elasticity of the drivetrain (rigid or elastic), and drivetrain type (direct, harmonic drive, ball screw, etc.) [14], [15]. One of the most powerful and common solutions is the combination of hydraulic, liquid-cooling, rigid and direct drive actuation. This achieves high power-to-weight and torque-to-weight ratios, joint position controllability, and shock tolerance. Existing robots that use this type of actuators include Atlas, Spot, Big Dog, and Wildcat of Boston Dynamics, BLEEX of Berkeley [16], and HyQ of IIT [17]. However, hydraulics are less energy efficient primarily because they require more energy transformations [18]. Typically, a gasoline engine or electric motor spins a pump, which compresses hydraulic fluid, which is modulated by a hydraulic servo valve, which finally causes a hydraulic piston to apply a force. Each stage in this process incurs some efficiency loss, and the total losses can be very significant. The combination of electric, air-cooled, rigid, and harmonic drive actuators are other widely used actuation types. Some robots utilizing these actuator types include Asimo of Honda, HRP2,3,4 of AIST [19], HUBO of KAIST [20], REEM-C of PAL Robotics, JOHNNIE and LOLA of Tech. Univ. of Munich [21], [22], CHIMP of CMU [23], Robosimian of NASA JPL [24], and more. These actuators have precise position control and high torque density. For example, LOLA’s theoretical knee peak torque-density (129N m/kg) is comparable to ours (107N m/kg), although they did not validate their number experimentally and their max speed is roughly 2/3 of our max speed [22]. Compared to us, low shock tolerance, low fidelity force sensing, and low efficiency gearboxes are common drawbacks of these type of actuators. According to Harmonic Drive AGs catalog, the efficiency of harmonic drives may be as poor as 25% and only increases above 80% when optimal combinations of input shaft speed, ambient temperature, gear ratio, and lubrication are present. Conversely, the efficiency of our VLCA is consistently above 80% due to the use of a ball screw mechanism. [25] used liquid cooling for electric, rigid, harmonic drive actuators to enhance continuous power-to-weight ratio. The robots using this type of actuation include SCHAFT and Jaxon [26]. These actuators share the advantages and disadvantages of electric, rigid, harmonic drive actuators, but have a significant increase of the continuous power output and torque density. One of our studies [13], indicates a 2x increase in sustained power output by retrofitting an electric motor with liquid cooling. Other published results indicate a 6x increase in torque density through liquid cooling [14], [27], though such performance required custom-designing a motor specifically for liquid cooling. In our case we use an off-theshelf electric motor. In contrast with our design, these actuators do not employ viscoelastic materials reducing their mechanical robustness and high quality force sensing and control. Although the increased power density achieved via liquid cooling amplifies an electric actuator’s power, the rigid drivetrain is still vulnerable to external impacts. To increase impact tolerance, many robots (e.g. Walkman and COMAN of IIT [28], Valkyrie of NASA [29], MABEL and MARLO in UMich [30], [31], and StarlETH of ETH [32]) adopt electric, air-cooled, elastic, harmonic drive actuators. This type of actuation provides high quality force sensing, force control, impact resistance, and energy efficiency. However, precise joint position control is difficult because of the elasticity in the drivetrain and the coupled effect of force feedback control and realtime latencies [33]. Low efficiency originating from the harmonic drives is another drawback. As an alternative to harmonic drives, ball screws are great drives for mechanical power transmission. SAFFiR, THOR, and ESCHER of Virginia Tech [34]–[36], M2V2 of IHMC [37], Spring Flamingo of MIT [38], Hume of UT Austin [11], and the X1 Mina exoskeleton of NASA [39] use electric, air-cooled, elastic, ball-screw drives. These actuators show energy efficiency, good power and force density, low noise force sensing, high fidelity force controllability, and low backlash. Compared to these actuators our design significantly reduces the bulk of the actuator and increases its joint position controllability. There are some other actuators that have special features such as the electric actuators used in MIT’s cheetah [40], which allow for shock resistance through a transparent but backlash-prone drivetrain. However, the lack of passive damping limits the joint position controllability of these type of actuators compared to us. III. VISCOELASTIC MATERIAL CHARACTERIZATION The primary driver for using elastomers instead of metal springs is to benefit from their intrinsic damping properties. However, the mechanical properties of viscoelastic materials can be difficult to predict, thus making the design of an actuator based on these materials a challenging endeavor. The most challenging aspect of incorporating elastomers into the structural path of an actuator is in estimating or modeling their complex mechanical properties. Elastomers possess both hysteresis and strain-dependent stress, which result in nonlinear force displacement characteristics. Additionally, elastomers also exhibit time-varying stress-relaxation effects when exposed to a constant load. The result of this effect is a gradual reduction of restoration forces when operating under a load. A third challenge when using elastomers in compression is compression set. This phenomenon occurs when elastomers are subjected to compressive loads over long periods of time. An elastomer that has been compressed will exhibit a shorter free-length than an uncompressed elastomer. Compression set is a common failure mode for o-rings, and in our application, it could lead to actuator backlash if not accounted for properly. To address these various engineering challenges we designed experiments to empirically measure the following four properties of our viscoelastic springs: 1) force versus displacement, 2) stress relaxation, 3) compression set, and 4) 3 E-stop Load cell Displacement sensor Elastic material EtherCAT-based embedded control system Belt drive BLDC motor 2500 2000 1500 Force (N) 1000 Ball screw drive (a) Viscoelastic material Testbed 0 -500 Spring steel Viton 75A Polyurethane 80A Polyurethane 90A EPDM 80A Reinforced Silicon 70A Buna- N 90A Silicone 90A -1000 Reinforced silicone 70A Viton 75A Buna-N 90A Polyurethane 80A Polyurethane 90A Spring steel 500 EPDM 80A -1500 -2000 0 1 2 3 4 5 -2500 6 -5 Compression set (%) -4 -3 -2 -1 0 1 2 3 Dispacement (m) 5 10 -4 (c) Force vs Displacement curve Phase (deg) Force (N) Magnitude (dB) (b) Complession set 4 time (sec) (d) Stess relaxation Increasing system bandwidth Spring steel Viton 75A Buna-N 90A Polyurethane 90A Frequency (Hz) (e) Dynamic response of four elastomer Fig. 1. Viscoelastic material test. (a) The elastomer testbed is designed and constructed to study various material properties of candidate viscoelastic materials. (b) We measured each elastomers free length both before and after they were placed in the preloaded testbed. (c) A strong correlation between material hardness and the materials stiffness can be observed. An exception to this correlation is the fabric reinforced silicone which we hypothesize had increased stiffness due to the inelastic nature of its reinforcing fabric. Nonlinear effects such as hysteresis can also be observed in this plot. (d) We command a rapid change in material displacements and then measured the materials force change versus time for 300 seconds. Note that the test of reinforced silicone 70A is omitted due to its excessive stiffness. (d) Although the bandwidths of the four responses are different, their damping ratios (signal peak value) are relatively constant, which implies different damping. frequency response, which will be used to characterize each material’s effective viscous damping. We built a viscoelastic material testbed, depicted in Fig. 1(a), to measure each of these properties. We selected and tested the seven candidate materials that are listed in Table I. The dimension of the tested materials are fairly regular, with 46mm diameter and 27mm thickness. A. Compression set Compression set is the reduction in length of an elastomer after prolonged compression. The drawback of using materials with compression set in compliant actuation is that the materials must be installed with larger amounts of preload forces to avoid the material sliding out of place during usage. To measure this property, we measured each elastomers free length both before and after the elastomer was placed in the preloaded testbed. The result of our compression set experiments are summarized in Table I. B. Force versus displacement In the design of compliant actuation, it is essential to know how much a spring will compress given an applied force. This displacement determines the required sensitivity of a spring-deflection sensor and also affects mechanical aspects of the actuator such as usable actuator range of motion and clearance to other components due to Poisson ratio expansion. In this experiment, we identify the force versus displacement curves for the various elastomer springs. Experimental data for all eight springs as shown in Fig 1(b). Note that there is a disagreement between our empirical measurements and the analytic model relating stiffness to hardness, i.e. the Gent’s relation shown in [41]. This mismatch arises because in our experiments the materials are preloaded whereas the analytical models assume unloaded materials. 4 Materials Compression Linearity set (%) (R-square) Linear stiffness (N/mm) Spring steel 0 0.996 860.8 Polyurethane 90A 2 0.992 8109 Preloaded elastic modulus (N/mm) Material damping (N s/m) Creep (%) Material Cost ($) 0 0 - 112.5 16000 15.3 19.40 Reinforced silicone 70A 2.7 0.978 57570 798.7 242000 - 29.08 Buna-N 90A 2.8 0.975 11270 156.4 29000 25 51.47 Viton 75A 4 0.963 2430 33.7 9000 30.14 105.62 Polyurethane 80A 4.5 0.993 2266 31.4 4000 16.8 19.40 EPDM 80A 6.48 0.939 6499 90.2 16000 23.4 35.28 Silicone 90A - 0.983 12460 172.9 37000 10.7 29.41 TABLE I S UMMARY OF VISCOELASTIC MATERIALS C. Stress relaxation E. Selection of Polyurethane 90A Stress-relaxation is an undesirable property in compliant actuators for two reasons. First, the time-varying force degrades the quality of the compliant material as a force sensor. When a material with significant stress-relaxation properties is used, the only way to accurately estimate actuator force based on deflection data is to model the effect and then pass deflection data through this model to obtain a force estimate. This model introduces complexity and more room for error. The second reason stress-relaxation can be problematic is that it can lead to the loss of contact forces in compression-based spring structures. The experiment for stress relaxation is conducted as follow: 1) enforce a desired displacement to a material, 2) record the force data over time from the load cell, 3) subtract the initially measured force from all of the force data. Empirically measured stress-relaxation properties for each of the materials are shown in Fig. 1 (c), which represents force offsets as time goes under the same displacement enforced. Note that each material shows different initial force due to the different stiffness and each initial force data is subtracted in the plot. A variety of other experiments were conducted to strengthen our analysis and are summarized in Table I. Based on these results, Polyurethane 90A appears to be a strong candidate for viscoelastic actuators based on its high linearity (0.992), low compression set (2%), low creep (15%), and reasonably high damping (16000 N s/m). It is also the cheapest of the materials and comes in the largest variety of hardnesses and sizes. D. Dynamic response In regards to compliant actuation, the primary benefit of using an elastomer spring is its viscous properties, which can characterize the dynamic response of an actuator in series with such a component. To perform this experiment, we generate motor current to track an exponential chirp signal, testing frequencies between 0.001Hz and 200Hz. Given the inputoutput relation of the system, we can fit a second order transfer function to the experimental data to obtain an estimate of the system’s viscous properties. However, this measure also includes the viscoelastic testbed’s ballscrew drive train friction (Fig. 1(a)). To quantify the elastomer spring damping independently of the damping of the testbed drive train, the latter (8000 N s/m) was first characterized using a metal spring, and then subtracted from subsequent tests of the elastomer springs to obtain estimates for the viscous properties of the elastomer materials. Fig. 1(d) shows the frequency response results for current input and force output of three different springs, while controlling the damping ratio. The elastomers have higher stiffness than the metal spring, hence their natural frequencies are higher. IV. V ISCOELASTIC L IQUID C OOLED ACTUATION The design objectives of the VLCA are 1) power density, 2) efficiency, 3) impact tolerance, 4) joint position controllability, and 5) force controllability. Compactness of actuators is also one of the critical design parameters, which encourage us to use elastomers instead of metal springs and mechanical dampers. Our previous work [13] shows a significant improvement in motor current, torque, output power and system efficiency for liquid cooled commercial off-the-shelf (COTS) electric motors and studied several Maxon motors for comparison. As an extension of this previous work, in this new study we studied COTS motors and their thermal behavior models and selected the Maxon EC-max 40 brushless 120 W (Fig. 2(e)), with a custom housing designed for the liquid cooling system (Fig. 2(h)). The limit of continuous current increases by a factor of 3.59 when liquid convection is used for cooling the motor. Therefore, a continuous motor torque of 0.701 N · m is theoretically achievable. Energetically, this actuator is designed to achieve 366 W continuous power and 1098W short-term power output with an 85% ball screw efficiency (Fig. 2(b)) since short-term power is generally three time larger than continuous power. With the total actuator mass of 1.692 kg, this translates into a continuous power of 216W/kg and a short-term power of 650W/kg. The liquid pump, radiator, and reservoir are products of Swiftech which weight approximately 1kg. By combining convection liquid cooling, high power brushless DC (BLDC) motors, and a high-efficiency ball screw, we aim to surpass existing electric actuation technologies with COTS motors in terms of power density. In terms of controls, a common problem with conventional SEAs is their lack of physical damping at their mechanical output. As a result, active damping must be provided from torque produced by the motor [42]. However, the presence of 5 (e) BLDC Motor (a) Timing belt transmission (b) Ball screw drive (c) Load cell (d) Actuator output (f) Quadrature encoder (g) Temperature sensor (h) Liquid cooling jacket Opposite side Motor part Rubber part Load part (i) Tube connector (j) Polyuretane elastomer (k) Compliance deflection sensor (l) Mechanical ground pivot (m) Quadrature encoder (deflection) ground Fig. 2. Viscoelastic Liquid Cooled Actuator. The labels are explanatory. In addition, the actuator contains five sensors: a load cell, a quadrature encoder for the electric motor, a temperature sensor, and two elastomer deflection sensors. One of the elastomer deflection sensors is absolute and the other one is a quadrature encoder. The quadrature encoder gives high quality velocity data of the elastomer deflection. signal latency and derivative signal filtering limit the amount by which this active damping can be increased, resulting in SEA driven robots achieving only relatively low output impedances [33] and thus operating with limited joint position control accuracy and bandwidth. Our VLCA design incorporates damping directly into the compliant element itself, reducing the requirements placed on active damping efforts from the controller. The incorporation of passive damping aims to increase the output impedance while retaining compliance properties, resulting in higher joint position control bandwidth. The material properties we took into consideration will be introduced in Section III. The retention of a compliant element in the VLCA drive enables the measurement of actuator forces based on deflection. The inclusion of a load cell (Fig. 2(c)) on the actuators output serves as a redundant force sensor and is used to calibrate the force displacement characteristics of the viscoelastic element. Mechanical power is transmitted when the motor turns a ball nut via a low-loss timing belt and pulley (Fig. 2 (a)), which causes a ball screw to apply a force to the actuator’s output (Fig. 2(d)). The rigid assembly consisting of the motor, ball screw, and ball nut connects in series to a compliant viscoelastic element (Fig. 2(j)), which connects to the mechanical ground of the actuator (Fig. 2(k)). When the actuator applies a force, the reaction force compresses the viscoelastic element. The viscoelastic element enables the actuator to be more shock tolerant than rigid actuators yet also maintain high output impedance due to the inherent damping in the elastomer. V. ACTUATOR F ORCE F EEDBACK C ONTROL To demonstrate various impedance behaviors in operational space, robots must have a stable force controller. Stable and accurate operational space control (OSC) is not trivial to achieve because of the bandwidth interference between outer position feedback control (OSC) and inner torque feedback control [11]. Since stable torque control is a critical component for a successful OSC implementation, we extensively study various force feedback controls. Jm (kg m2 ) bm (N m s) mr (kg) br (N s/m) kr (N/m) 3.8e−5 2.0e−4 1.3 2.0e4 5.5e6 TABLE II ACTUATOR PARAMETERS The first step in this analysis is to identify the actuator dynamics. The transfer functions of the reaction force sensed in the series elastic actuators (elastomer deflection) are well explained in [43]. When the actuator output is fixed, the transfer function from the motor current input to the elastomer deflection is given by Px = xr ηkτ Nm = , (1) 2 + m )s2 + (b N 2 + b )s + k im (Jm Nm r m m r r where η, kτ , Nm , and im are the ball screw efficiency, the torque constant of a motor, the speed reduction ratio of the motor to the ball screw, and the current input for the motor, respectively. The equations follow the nomenclature in Fig. 3(a). We can find η, kτ , and Nm in data sheets, which are 0.9, 0.0448 N · m/A, and 3316 respectively. The gear ratio of the drivetrain is computed by dividing the speed reduction of pulleys (2.111) with lead length of the ball screw (0.004m) using the equation 2π × 2.111/0.004. However, we need to experimentally identify kr , br , Jm , and bm . We infer kr by dividing the force measurement from the load cell by the elastomer deflection. The other parameters are estimated by comparing the frequency response of the model and experimental data. The frequency response test is done with the ankle actuator while prohibiting joint movement with a load and an offset force command. The results are presented in Fig. 3 with solid gray lines. Note that the dotted gray lines are the estimated response from the transfer function (measured elastomer force/ input motor force) using the parameters of Table. II. The estimated response and experimental result match closely with one another, implying that the parameters we found are close to the actual values. We also study the frequency response for different load masses to understand how the dynamics changes as the joint moves. When 10kg is attached to the end of link, the reflected 6 (a) VCLA -149 o 27.5 Hz Fig. 3. Frequency response of VLCA. Gray solid lines are experimental data and the other lines are estimated response with the model using empirically parameters. (b) Open-loop PDm PIDm PDf PDm + DOB 60 Magnitude (dB) Experiment result 40 20 0 -20 -40 0 Phase margin -45 47.6 -90 41.0 1) Proportional (P) + Derivative (Df ) using velocity signal obtained by a low-pass derivative filtered elastomer deflection 2) Proportional (P) + Derivative (Dm ) using motor velocity signal measured by a quadrature encoder connected to a motor axis The second controller (PDm ) has benefits over the first one (PDf ) with respect to sensor signal quality. The velocity of motor is directly measured by a quadrature encoder rather than low-pass filtered elastomer deflection data, which is relatively noisy and lagged. In addition, Fig. 4 shows that the phase margin of the second controller (47.6) is larger than the first one (17.1). To remove the force tracking error at low frequencies, we consider two options: augmenting the controller either with integral control or with a DOB on the PDm controller. To compare the two controllers, we analyzed the phase margins of all the mentioned controllers. First, we chose to focus on the location where the sensor data returns in order to address the time delay of digital controllers (Fig. 4 (a) and (c)). Next, we have to compute the open-loop transfer function for each closed loop system. For example, the PDf controller’s closed loop transfer function is
kr Px kp (Fr − e−T s Fk ) + Fr − kd,f Qd e−T s Fk , N (2) where Fk , Fr , T , and Qd are the measured force from a elastomer deflection, a reference force, a time delay, a low pass derivative filter, respectively.For convenience, we use N instead of the multiplication of three terms, ηkτ Nm . When 34.0 -225 10 (c) 42.8 17.7 -135 -180 mass to the actuator varies from 1500kg to 2500kg because the length of the effective moment arm changes depending on joint position. In Fig. 3(b), the bode plots are presented and the response is not significantly different than the fixed output case. Therefore, we design and analyze the feedback controller based on the fixed output dynamics. For the force feedback controller, we first compare two options, which we have used in our previous studies [11], [12]: Fk = kr -3dB Phase (deg) Inf. mass 2500 kg 2000 kg 1500 kg 0 10 1 10 2 Frequency (Hz) VCLA kr s Fig. 4. Stability analysis of controllers. Phase margins of each controllers and open-loop system are presented. gathering the term with e−T s of Eq. (2), we obtain Fk kr Px (Kp + 1)/N = . Fr 1 + e−T s kr Px (Kp + Kd,f Qd )/N (3) Then, the open-loop transfer function of the closed system with the time delay is open PPD = kr Px (Kp + Kd,f Qd )/N. f (4) We can apply the same method for the PIDm and PDm +DOB controllers. The transfer function of PIDm , which is presented in Fig. 4(c), is 1 kr Px  (Fr − e−T s Fk )(Kp + Ki ) + Fr Fk = N s (5) Fk  −T s − Kd,m e sNm . kr Then it becomes Fk kr Px (Kp + Ki /s + 1)/N = . Fr 1 + e−T s Px (kr (Kp + Ki /s) + Kd sNm )/N (6) When we apply a DOB instead of integral control, we need the inverse of the plant. In our case, the plant of the DOB is PDm , which is similar to Eq. (6) except that Ki and e−T s are omitted: kr Px (Kp + 1) PPDm (= Pc ) = . (7) N + Px (kr Kp + Kd,m sNm ) 7 Fig. 5. Robotic testbed. Our testbed consists of two VLCAs at the ankle and the knee. The foot of the testbed is fixed on the ground. The linkages are designed to vary the maximum peak torques and velocities depending on postures. As the joint positions change, the ratios between ball screw velocities (L̇0,1 ) and joint velocities (q̇0,1 ) also change because of effective lengths of moment arms vary. The linkages are designed to exert more torque when the robot crouches, which is the posture that the gravitational loads on the joints are large. The formulation of PDm including the DOB, which is shown in Fig. 4(c), is Fk = kr Px (Kp + 1)(Fd − e−T s Pc−1 Qτ d Fk ) , (8) (N + e−T s Px (kr Kp + Kd,m sNm )) (1 − Qτ d ) the hip, and the foot is fixed on the ground. With this testbed, we intended to demonstrate coordinated position control with two VLCAs, the viability of liquid cooling on an articulated platform, cartesian position control of a weighted end effector, and verification of a linkage design. The two joints each have a different linkage structure that was carefully designed so that the moment arm accommodates the expected torques and joint velocities as the robot posture changes (Fig. 5). For example, each joint can exert a peak torque of approximately 270 N m and the maximum joint velocity ranges between 7.5 rad/s and 20+ rad/s depending on the mechanical advantage of the linkage along the configurations. The joints can exert a maximum continuous torque of 91 N m at the point of highest mechanical advantage. This posture dependent ratio of torque and velocity is a unique benefit of prismatic actuators. Given cartesian motion trajectories, which are 2nd order Bspline or sinusoidal functions, the centralized controller computes the torque commands with operational space position and velocity, which are updated by the sensed joint position and velocity. The OSC formulation that we use is −1 τ = AJhip (ẍdes + Kp e + Kd ė − J˙hip q̇) + b + g, (11) where A, b, and g represent inertia, coriolis, and gravity joint torque, respectively. ẍdes , e, and ė are desired trajectory acceleration, position and velocity error, respectively. q̇ ∈ R2 is the joint velocity of the robot and τ is the joint torque. Jhip is a jacobian of the hip, which is a 2 × 2 square matrix and assumed to be full-rank. where Qτ d is a second order low-pass filter. Then the transfer function is VII. R ESULTS Fk kr Px (Kp + 1) . = We first conducted various single actuator tests to show Fd N (1 − Qτ d ) + e−T s (N Qτ d + Px (kr Kp + Kd,m sNm )) (9) basic performance such as torque and joint position controllability, continuous and peak torque, and impact resistance. SubThe open-loop transfer function is sequently, we focused on the performance of OSC using the N Qτ d + Px (kr Kp + Kd,m sNm ) open robotic testbed integrated with DOB based torque controllers PPD = (10) m +DOB N (1 − Qτ d ) to demonstrate actuator efficiency and high power motions. open open open open The bode plots of PPD , P , P , and P PDm PIDm PDm +DOB f are presented in Fig. 4(b). The gains (Kp , Kd,m , Ki ) are the A. Single Actuator Tests same as the values that we use in the experiments presented in Fig. 6(a) shows the experimental results of our frequency Section VII-A, which are 4, 15, and 300, respectively. The PDf response testing as well as the estimated response based on controller uses Kd Nm /kr for Kd,f to normalize the derivative the transfer functions. We compare three types of controllers: gain. The cutoff frequency of the DOB is set to 15Hz because PD , PID , and PD + DOB. As we predicted in the m m m this is where the PDm +DOB shows a magnitude trend similar analysis of Section V, the PD + DOB controller shows less m to the integral controller (PIDm ). The results imply that the phase drop and overshoot than PID . The integral control m PDm +DOB controller is more stable than PIDm with respect feedback gain used in the experiment is 300 and the cutoff to phase margin and maximum phase lag. This analysis is also frequency of the DOB’s Q filter is 60Hz, which shows τd experimentally verified in Section VII-A. similar error to the PID controller (Fig. 6(b)). Another test m VI. ROBOTIC T ESTBED We built a robotic testbed shown in Fig. 5. To demonstrate dynamic motion, we implemented an operational space controller (OSC) incorporating the multi-body dynamics of the robot. We designed and built a robotic testbed (Fig. 5) consisting of two VLCAs - one for the ankle (q0 ) and one for the knee (q1 ). The design constrains motion to the sagittal plane, the robot carries 10kg, 23kg, or 32.5kg of weight at presented in Fig. 6(c) also supports the stability and accuracy of torque control. In the test, we command a ramp in joint torque from 1 to 25N m in 0.1s. The sensed torque (blue solid line) almost overlaps the commanded torque (red dashed line). Fig. 6(d) is the result of a joint position control test designed to show that VLCAs have better joint position controllability than SEAs using springs. In the experiment, we use a joint encoder for position control and a motor quadrature encoder for velocity feedback. To compare the VLCAs performance 8 20 10 1.8 2 2.2 2.4 2.6 2.8 -1.8 JPos (rad) Phase (deg) (c) Torque fast response Open (estimated) (estimated) (estimated) -2 -2.4 1.6 1.8 2 2.2 2.4 2.6 70 1 60 0 50 40 -2 -3 2.8 30 0 1 2 3 time (sec) 20 (f) peak force (d) Position fast response Actuator force (N) 80 1000 150 500 100 0 -500 time (sec) 50 actuator force core temperature (w/ liquid) core temperature (w/o liquid) 0 50 100 150 200 250 0 350 300 Motor temperature (oC) Error (dB) (a) Frequency responses of different controllers Frequency (Hz) 90 2 time (sec) Reference 100 -1 (sim.) metal spring (sim.) elastomer (exp.) command (exp.) sensed -2.2 Frequency (Hz) 110 3 0 1.6 Open x 103 4 command sensed Actuator force (N) Magnitude (dB) 5 Motor temperature (oC) Torque (Nm) 30 time (sec) (e) Continous force and core temperature (b) Error magintude and chirp test trajectories Fig. 6. Torque Feedback Control Test. (a) Experimental data and estimated response based on the transfer functions are presented. Estimated response of PD controller is identical to the PD+DOB since DOB theoretically does not change the transfer function. The plot show PD+DOB shows better performance in terms of less overshoot and smaller phase drop near to the natural frequency. (b) We choose integral controller feedback gain that shows similar accuracy of PD+DOB’s. The left is error magnitude of three controllers. PD controller has larger error than the other two controller in the low frequency region. The right is torque trajectories in the time domain. Load cell (solid holding) Load cell (w/ elastomer) Rubber deflection (solid holding) Rubber deflection (w/ elastomer) 1000 Actuator force (N) with that of spring-based SEAs, we present simulation results for a spring-based SEA on the same plot as the experiment result for the VLCA. The green dashed line is the simulated step response of our actuator and the yellow dotted line is the result of the simulation model using the same parameters except the spring stiffness and damping. The spring stiffness was selected to be 11% of the elastomer’s, based on the results of our tests in Section III, and the damping for the spring case was set to 8000 N s/m which only includes the drivetrain friction. The results show a notable improvement in joint position control when using an elastomer instead of a steel spring. Fig. 6(e) shows the continous force and the motor core temperature trend with and without liquid cooling. The observed continous force is 860N and the motor core temperature settles at 115◦C with liquid cooling. Fig. 6(f) is the the result of shortterm torque test. In the experiment, we fix the output of the actuator and command a 31A current for 0.5s. The observed force measured by a loadcell (Fig. 2(c)) is 4500N, which is a little smaller than the theoretically expected value, 5900N. Considering that the estimated core temperature surpassed 107◦C (< 155◦C limit), we expect that the theoretical value is reasonable. Thus, we conclude that the maximum force density of our actuator is larger than 2700N/kg and potentially 3500N/kg. Fig. 7 shows loadcell and elastomer force data from the impact tests. In the tests, we hit the loadcell connected to the ball screw (Fig. 2(c)) with a hammer falling from a constant height while fixing the actuator in two different places to 500 0 95% interval -500 -1000 -2 0 2 4 6 8 10 12 14 time (ms) Fig. 7. Impact test. 83 trials are plotted and estimated with gaussian process. We can see the deflections of the elastomer, which imply that the elastic element absorbes the external impact force. compare the rigid actuator to viscoelastic actuator response. In the rigid scenario, outer case of ballnut, a blue part in Fig. 2, is fixed to exclude the elastomer from the external impact force path. In the second case, we fixed the ground pin of the actuator, which is depicted by a gray part in Fig. 2(l), to see how the elastomers react to the impact. The impact experiment is challenging because the number of data points we can obtain is very small with a 1ms update rate. To overcome the lack of data points, we estimate the mean and variance of 83 trials by gaussian process regression. The results presented in Fig. 7 imply that there is no significant difference in the forces measured by the loadcell in both cases, which is predictable because the elastic element is placed behind the drivetrain. However, the elastomer does Ankle command sensed 20 0 power supply 14 16 18 20 22 24 14 16 18 20 22 24 40 Wm Wb 0.2 Hip position (m) joint electric motor 50 30 0 -0.1 14 16 18 20 22 24 1 Wk Average efficiency 1.5 0.1 Wk / W m Compliant in horizontal direction servo drive -20 Knee Stiff in vertical direction Torque (Nm) 9 1 0.5 0.8 14 16 (a) Impedance control 22 0 0.6 0.4 -20 12 14 16 18 40 12 14 16 18 Hip position (m) 0.1 10 12 10 12 14 16 18 14 16 18 1 time (sec) (b) Operational space impact test 0.9 0.1 0.85 0 0.8 -0.1 2 command 2.5 sensed 0.9 0.75 0.7 0.8 0.65 0.7 0.6 0.6 1.5 time (sec) 1.5 2 2.5 3 3.5 5 (sec) 3 1 0.5 0.3 0.2 0 -0.2 0.5 1 1.5 2 2.5 3 3.5 0 -0.1 0.8 1 1 time (sec) 0.2 1.5 0.5 -0.4 30 10 1 0 24 Wk / W b Ankle 20 -10 10 Knee Torque (Nm) Hitting down 18 time (sec) 2 2.5 -0.05 0 0.05 (c) Fast up and down (1.7 Hz) Fig. 8. Operational Space Impedance Control Test. (a) The robot demonstrates different impedance: stiff in the vertical direction and compliant in the horizontal direction. The high tracking performance of force feedback control results in the overlapped commanded and sensed torques. (b) To show the stability, we hit the weight with a hammer while operating the impedance control. Even under the impact, force control show stable and accurate tracking. (c) The robot demonstrates a 1.7Hz up and down motion while carrying 10kg weight at the hip, and shows a position error of less than 2.5cm. play a significant role in absorbing energy from the impact which is evident from large elastomer deflection in the second case. Thus, the presence of the elastic element mitigates the propagation of an impulse to the link where the actuator grounds. B. Operational Space Impedance Control Fig. 8 shows our OSC experimental tests (Section VI) carrying a 10kg weight. In the first test presented in Fig. 8(a), the commanded behavior is to be compliant in the horizontal direction (x) and to be stiff in the vertical direction (y). When pushing the hip with a sponge in the x direction, the robot smoothly moves back to comply with the push, but it strongly resists the given vertical disturbance to maintain the Fig. 9. Efficiency analysis of the ankle actuator. Efficiencies of mechanical system using electrical power has 3 steps from a power supply to robot joints. The graph shows the ratio of the mechanical power of the ankle joint and the motor power and the ratio of the joint power and power supply’s input power. commanded height. To show the stability of our controller, we also test the response to impacts by hitting the weight with a hammer (Fig. 8(b)). Even when there are sudden disturbances, the torque controllers rapidly respond to maintain good torque tracking performance as shown in Fig. 6(d). Fig. 8(c) shows the tracking performance of our system while following a fast vertical hip trajectory. While traveling 0.3m with 1.7Hz frequency, the hip position errors are bounded by 0.025m. This result demonstrates that our system is capable of stable and accurate OSC, which is challenging because of the bandwidth conflict induced by its cascaded structure. C. Efficiency Analysis Fig. 9 explains the power flow from the power supply to the robot joint. Input current (Ib ) and voltage (Vb ) are measured in the micro-controllers and the product of those two yields the input power from the power supply. θ̇m is measured by the quadrature encoder connected to the motor’s axis (Fig. 2(f)) and τm is computed from kτ im with im measured in the micro-controller. Joint velocity is low-pass derivative filtered joint positions measured at the absolute joint encoders. The torque (τk ) is computed from projecting the load cell data across the linkage’s effective moment arm. In this test, the robot lifts a 23kg load using five different durations to observe efficiency over a range of different speeds and torques. The results are presented in Fig. 9 with the description of three different power measures. The sensed torque data measured by a load cell is noisy; therefore, we compute the average of the drivetrain efficiency for a clearer comparison. The averages are the integrations of efficiency divided by the time durations. Here we only integrate efficiency while the mechanical power is positive, to prevent confounding our results by incorporating the work done by gravity. 7 7.5 8 200 100 0 -100 0 7 joint torque 7.5 mechanical power 8 400 200 0 -200 -400 -600 200 100 0 6.5 7 7.5 Mechanical power (W) Ankle 7 200 -200 6.5 Knee 8 5 4 6.5 Joint torque (Nm) 7.5 8 (a) 2Hz up and down motion Ankle 0 45 40 -1000 35 -2000 -3000 0 0.5 1 1.5 2 2.5 actuator force 3 3.5 4 motor temperature 30 4.5 80 Knee -1000 60 -2000 -3000 40 -4000 0 Power (W) Ankle Knee 6.5 6 1000 Actuator force (N) command joint encoder motor encoder 2.6 2.4 2.2 2 1.8 0.5 1 1.5 2 2.5 3 3.5 4 Motor temperature (oC) Joint position (rad) 10 4.5 total ankle knee 1500 1000 500 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 time (sec) (b) Heavy weight lift Fig. 10. High power motion experiment. (a) Joint position data from joint encoder and motor encoder are shown. In this experiment, the maximum observed torque of the ankle joint is 250 N m and the maximum observed mechanical power of the knee joint is 310W. (b) The robot lifts by 0.3m a 32.5 kg load during 0.4s. There is still a safety margin with respect to the limits equal to 5900N and 155◦C. The experimental results show that the drivetrain efficiency is approximately 0.89, which means that we lose only a small amount of power in the drivetrain and most of the torque from the motor is delivered to the joint. This high efficiency indicates only minor drivetrain friction, which is beneficial for dynamics-based motion controllers. D. High Power Motion Experiment To demonstrate high power motions such as fast vertical trajectories and heavy payload lifts, we use the motor position control mode, which uses the quadrature encoders attached directly to the motor for feedback. Fig. 10(a) presents the results of a test comprised of 2Hz vertical motion with 0.32 m of travel while carrying a load of 10 kg at the hip. With respect to mechanical power, the knee joint repeatedly exerts 305W, which is close to the predicted constant power (360W). Although the limited range of motion makes it hard to demonstrate continuous mechanical power, these results convincingly support our claim of enhanced continuous power enabled through liquid cooling. Fig. 10(b) presents another test in which the robot lifts a 32.5kg weight. We can see that the robot operates in the safe region (≤ 5900N and ≤ 155◦C) while demonstrating high power motion. 6(e). As we can see, when turning off liquid cooling the temperature rises quickly above safety limits whereas when turning on the cooling we can sustain large payload torques for long periods of time. The use of elastomers versus steel springs has demonstrated a clear improvement on joint position performance as shown in Fig. 6(d). This capability is important to achieve a large range of output joint or Cartesian space impedances. In the future we will explore further reducing the size of our viscoelastic liquid cooled actuators. Maintaining the current compact design structure we can still reduce another significant percentage the bulk of the actuator by exploring new types of bearings, ballnut sizes and piston bearings at the front end of the actuator. We will also explore using different material for the liquid cooling actuator jacket. The current polyoxymethylene material is easily breakable and develops cracks due to the vibrations and impacts of this kind of robotic applications. In the future we will switch to sealed metal chambers for instance. Further in the future we will consider designing our own motor stators and rotors for improved performance. We expect this kind of actuators to make their way into full humanoid robots and high performance exoskeleton devices and we look forward to participate in such interesting future studies. ACKNOWLEDGMENT VIII. C ONCLUDING R EMARKS Overall our main contribution has been on the design and extensive testing of a new viscoelastic liquid cooled actuator for robotics. One of the tests addressed is impedance control in the operational space instead of joint impedance control. It is often the case that humanoid robots require impedance control in the operational space. For instance, controlling the operational space impedance can enable improved locomotion behaviors such as running. Our controllers demonstrate that we can control the impedance in the Cartesian operational space as a potential functionality for future robotic systems. 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arXiv:1703.03372v3 [] 15 Mar 2017 LesionSeg: Semantic segmentation of skin lesions using Deep Convolutional Neural Network Dhanesh Ramachandram School of Engineering University of Guelph Guelph, ON N1G 2W1, Canada [email protected] Terrance DeVries School of Engineering University of Guelph Guelph, ON N1G 2W1, Canada [email protected] Executive Summary We present a method for skin lesion segmentation for the ISIC 2017 Skin Lesion Segmentation Challenge. Our approach is based on a Fully Convolutional Neural Network architecture which is trained end to end, from scratch, on a small dataset. Our semantic segmentation architecture utilizes several recent innovations in deep learning particularly in the combined use of (i) atrous convolutions to increase the effective field of view of the network’s receptive field without increasing the number of parameters, (ii) network-in-network 1 × 1 convolution layers to increase network capacity and (iii) state-of-art super-resolution upsampling of predictions using subpixel CNN layers. We achieved a IOU score of 0.642 on the validation set provided by the organizers. 1 Background One of the fundamental and challenging tasks in digital image analysis is segmentation, which is the process of assigning pixel-wise labels to regions in an image that share some high-level semantics, hence the term “semantic segmentation”. In skin lesion segmentation, the goal is to assign pixel-wise labels to regions in dermoscopy images that represents skin lesions, such as melanoma, seborrhoeic keratosis or benign nevus. Skin lesion segmentation is challenging due to a variety of factors, such as variations in skin tone, uneven illumination, partial obstruction due to the presence of hair, low contrast between lesion and surrounding skin, and the presence of freckles or gauze in the image frame, which may be mistaken for lesions. A successful lesion segmentation technique should be robust enough to accommodate these variability. Skin lesion segmentation is a widely researched topic in medical image analysis[1–3]. Until recently, most skin lesion segmentation approaches were based on hand crafted algorithms[4–6]. Such approaches require carefully designed pre-processing and post-processing approaches such as hair removal, edge-preserving smoothing and morphological operations. The robustness of the such approaches can be somewhat limited however, as each new scenario may require custom tuning. An alternative approach to manually crafting segmentation algorithms is to instead leverage machine learning techniques to learn a model capable of successfully dealing with the numerous factors of variation. Specifically applicable to this application are artificial neural networks, which have made an impressive resurgence in recent years. The active research area, now known as deep learning, is currently enjoying interest and support not only from the academic community, but also from industry. The advances in hardware performance and low costs involved have made it viable to analyse very large data sizes using very deep neural network architectures in a reasonable amount of time. Deep learning-based techniques have resulted in state-of-the-art performance for many practical problems, especially in areas involving high-dimensional unstructured data such as in computer vision, speech and natural language processing. Medical imaging problems have also been given much attention by 2017 ISIC Skin Lesion Segmentation Challenge deep learning researchers[7] and has seen tremendous success for the related skin lesion classification problem[8]. The notion of being able to train in an end-to end manner, without requiring any manual feature engineering or complicated hand-crafted algorithms, is very attractive indeed. 2 Semantic Segmentation The deep learning approach to image segmentation is known as semantic segmentation. In contrast to low-level image segmentation, which operate purely on local image characteristics such as colour, shape and texture, deep semantic segmentation algorithms are trained using thousands of examples to recognize and delineate regions in an image corresponding to some high-level semantics. A convolutional neural network can be adapted to perform perform semantic segmentation by replacing the top layer1 of a classification network into a convolutional layer. As FCNN use downsampling implemented via the max-pooling or strided convolutions to capture context, these architectures often employ a single or several progressive upsampling layers which are used to upscale lower resolution pixel-wise predictions to match the dimensions of the input image. Ground truth segmentation masks provide pixel-wise labels for the segmentation task, now cast as a pixel-wise classification problem. The fully convolutional neural network architecture was first proposed by Long et al.[9]. Subsequently, a number of similar architectures have been reported in the literature[10, 11]. For the ISIC 2017 challenge, the skin lesion segmentation task is a binary segmentation task - the goal is to produce accurate segmentation of various skin lesions, benign and malignant, against a variety of background which may consist of skin, colored markers, or dark-vignetted region produced by the dermoscope. Fig.?? shows an example lesion and its corresponding binary mask. The training dataset consists of 2000 dermoscopy images and corresponding binary masks. The images consists of 3 types of skin lesions: nevus, seborrhoeic keratosis and melanoma; the latter lesion being malignant. The images are also of various dimensions. Figure 1: Left: An example of a skin lesion image with a blue marker in the background. Right: The corresponding ground truth binary segmentation mask. 3 Our Approach Our approach for lesion segmentation is primarily a fully convolutional neural network, trained from scratch, in an end-to-end manner. 3.1 Architecture The inputs to the network are images resized to 448 × 448. The first convolution layer uses a stride of 2 to downsample the image by a factor of 2. This is followed by series of 2D convolution layers interspersed with 1 × 1 convolutions. We also apply batch-normalization for the convolutional layers and use ReLu activations. The 1 × 1 convolution layers add capacity to the network without increasing the number of parameters.The last two convolutions are dilated convolutional layers or 1 typically the softmax layer 2 Layer Filter Size Stride/Rate Padding Non-linearity Conv1-1 Conv1-2 Conv1-3 Conv2-1 Conv2-2 Conv2-3 Conv3-1 Conv3-2 Conv3-3 Subpixel 5x5x64 3x3x96 1x1x96 3x3x128 3x3x256 1x1x256 3x3x256 3x3x256 3x3x128 3x3x32 2 1 1 2 1 1 2 2 2 1 Mirror Mirror Same Mirror Mirror Same Mirror Mirror Mirror Mirror ReLu ReLu ReLu ReLu ReLu ReLu ReLu ReLu None None Type 2D Convolution 2D Convolution 2D Convolution 2D Convolution 2D Convolution 2D Convolution Atrous Convolution Atrous Convolution Atrous Convolution Subpixel CNN Layer Table 1: LesionSeg Architecture atrous convolutions[12] with a rate of 2. These two layers effectively enlarge the field of view of filters to incorporate larger context without increasing the number of parameters or the amount of computation. Before upsampling is performed, the final subpixel convolution layer is added. This layer has number of filters equal to the upsampling factor and is applied with a stride of 1 and without any non-linearities. The upsampling layer is a subpixel convolutional layer introduced by [13] which produced state-of-art super-resolution reconstruction accuracy superior to the commonly used bilinear upsampling as used by [9]. 3.2 Preprocessing The input images and the corresponding ground-truth masks are resized to 448 by 448 pixels. We perform data augmentation on-the-fly by randomly rotating both the image and its mask by 90-degree increments as well as flipping the images. In addition, we also perform per-image-standardization of the input image. 3.3 Training We trained the network using Adam[14] optimization using a per-pixel cross-entropy loss function. During training, we performed randomly sampled images using a batch size of 32. The network is trained until no improvement in the mean IOU is observed. 3.4 Post-processing As the test and validation sets have images of different resolutions up to 6688 × 4439 pixels, we resorted to upscaling the network output back to its original image dimensions using bicubic interpolation and then binarizing the upsampled output mask using a threshold of 128. We also apply morphological opening to eliminate small, spurious errors made by the semantic segmentation using a 3 × 3 disk-shaped kernel. 4 Results and Discussion We achieved a IOU score of 0.642 for the validation set using our approach. Example segmentation outputs from our lesion segmentation architecture is shown in Fig.2. 3 (a) Sample 1 (b) Sample 2 (c) Sample 3 (d) Sample 4 Figure 2: Examples of segmentation output using our approach. References [1] M Emre Celebi, Quan Wen, Hitoshi Iyatomi, Kouhei Shimizu, Huiyu Zhou, and Gerald Schaefer. A state-of-the-art survey on lesion border detection in dermoscopy images, 2015. [2] Konstantin Korotkov and Rafael Garcia. Computerized analysis of pigmented skin lesions: a review. Artificial intelligence in medicine, 56(2):69–90, 2012. [3] M Emre Celebi, Hitoshi Iyatomi, Gerald Schaefer, and William V Stoecker. Lesion border detection in dermoscopy images. Computerized medical imaging and graphics, 33(2):148–153, 2009. [4] Huiyu Zhou, Gerald Schaefer, M Emre Celebi, Faquan Lin, and Tangwei Liu. Gradient vector flow with mean shift for skin lesion segmentation. Computerized Medical Imaging and Graphics, 35(2):121–127, 2011. [5] Xiaojing Yuan, Ning Situ, and George Zouridakis. A narrow band graph partitioning method for skin lesion segmentation. Pattern Recognition, 42(6):1017–1028, 2009. [6] Gerald Schaefer, Maher I Rajab, M Emre Celebi, and Hitoshi Iyatomi. Colour and contrast enhancement for improved skin lesion segmentation. Computerized Medical Imaging and Graphics, 35(2):99–104, 2011. [7] Ge Wang. A perspective on deep imaging. IEEE Access, 4:8914–8924, 2016. [8] Andre Esteva, Brett Kuprel, Roberto A Novoa, Justin Ko, Susan M Swetter, Helen M Blau, and Sebastian Thrun. Dermatologist-level classification of skin cancer with deep neural networks. Nature, 542(7639):115–118, 2017. [9] Jonathan Long, Evan Shelhamer, and Trevor Darrell. Fully convolutional networks for semantic segmentation. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 3431–3440, 2015. [10] Liang-Chieh Chen, George Papandreou, Iasonas Kokkinos, Kevin Murphy, and Alan L Yuille. Semantic image segmentation with deep convolutional nets and fully connected crfs. arXiv preprint arXiv:1412.7062, 2014. [11] Vijay Badrinarayanan, Alex Kendall, and Roberto Cipolla. Segnet: A deep convolutional encoder-decoder architecture for image segmentation. arXiv preprint arXiv:1511.00561, 2015. 4 [12] Liang-Chieh Chen, George Papandreou, Iasonas Kokkinos, Kevin Murphy, and Alan L Yuille. Deeplab: Semantic image segmentation with deep convolutional nets, atrous convolution, and fully connected crfs. arXiv preprint arXiv:1606.00915, 2016. [13] Wenzhe Shi, Jose Caballero, Ferenc Huszár, Johannes Totz, Andrew P Aitken, Rob Bishop, Daniel Rueckert, and Zehan Wang. Real-time single image and video super-resolution using an efficient sub-pixel convolutional neural network. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 1874–1883, 2016. [14] Diederik Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014. 5 

Submitted to the Annals of Statistics arXiv: arXiv:0000.0000 arXiv:1710.11268v3 [] 11 Dec 2017 THEORETICAL AND COMPUTATIONAL GUARANTEES OF MEAN FIELD VARIATIONAL INFERENCE FOR COMMUNITY DETECTION By Anderson Y. Zhang, and Harrison H. Zhou Yale University The mean field variational Bayes method is becoming increasingly popular in statistics and machine learning. Its iterative Coordinate Ascent Variational Inference algorithm has been widely applied to large scale Bayesian inference. See Blei et al. (2017) for a recent comprehensive review. Despite the popularity of the mean field method there exist remarkably little fundamental theoretical justifications. To the best of our knowledge, the iterative algorithm has never been investigated for any high dimensional and complex model. In this paper, we study the mean field method for community detection under the Stochastic Block Model. For an iterative Batch Coordinate Ascent Variational Inference algorithm, we show that it has a linear convergence rate and converges to the minimax rate within log n iterations. This complements the results of Bickel et al. (2013) which studied the global minimum of the mean field variational Bayes and obtained asymptotic normal estimation of global model parameters. In addition, we obtain similar optimality results for Gibbs sampling and an iterative procedure to calculate maximum likelihood estimation, which can be of independent interest. 1. Introduction. A major challenge of large scale Bayesian inference is the calculation of posterior distribution. For high dimensional and complex models, the exact calculation of posterior distribution is often computationally intractable. To address this challenge, the mean field variational method [2, 19, 30] is used to approximate posterior distributions in a wide range of applications in many fields including natural language processing [6, 22], computational neuroscience [14, 26], and network science [1, 8, 17]. This method is different from Markov chain Monte Carlo (MCMC) [13, 28], another popular approximation algorithm. The variational inference approximation is deterministic for each iterative update, while MCMC is a randomized sampling algorithm, so that for large-scale data analysis, the mean field variational Bayes usually converges faster than MCMC [7], which is particularly attractive in the big data era. MSC 2010 subject classifications: Primary 60G05 Keywords and phrases: mean field, variational inference, Bayesian, community detection, stochastic block model 1 2 ZHANG, ZHOU In spite of a wide range of successful applications of the mean field variational Bayes, its fundamental theoretical properties are rarely investigated. The existing literature [3, 8, 31, 33, 34] is mostly on low dimensional parameter estimation and on the global minimum of the variational Bayes method. For example, in a recent inspiring paper, Wang and Blei [32] studied the frequentist consistency of the variational method for a general class of latent variable models. They obtained consistency for low dimensional global parameters and further showed asymptotic normality, assuming the global minimum of the variational Bayes method can be achieved. However, it is often computationally infeasible to attain the global minimum when the model is high-dimensional or complex. This motivates us to investigate the statistical properties of the mean field in high dimensional settings, and more importantly, to understand the statistical and computational guarantees of the iterative variational inference algorithms. The success and the popularity of the mean field method in Bayesian inference mainly lies in the success of its iterative algorithm: Coordinate Ascent Variational Inference (CAVI) [7], which provides a computationally efficient way to approximate the posterior distribution. It is important to understand what statistical properties CAVI has and how do they compare to the optimal statistical accuracy. In addition, we want to investigate how fast CAVI converges for the purpose of implementation. With the ambition of establishing a universal theory of the mean field iterative algorithm for general models in mind, in this paper, we consider the community detection problem [1, 4, 12, 24, 25, 35] under the Stochastic Block Model (SBM) [4, 18, 21, 29] as our first step. Community detection has been an active research area in recent years, with the SBM as a popular choice of model. The Bayesian framework and the variational inference for community detection are considered in [1, 3, 8, 11, 17, 27]. For high dimensional settings, Celisse et al. [8] and Bickel et al. [3] are arguably the first to study the statistical properties of the mean field for SBMs. The authors built an interesting connection between full likelihood and variational likelihood, and then studied the closeness of maximum likelihood and maximum variational likelihood, from which they obtained consistency and asymptotic normality for global parameter estimation. From a personal communication with the authors of Bickel et al. [3], an implication of their results is that the variational method achieves exact community recovery under a strong signal-to-noise (SNR) ratio. Their analysis idea is fascinating, but it is not clear whether it is possible to extend the analysis to other SNR conditions under which exact recovery may never be possible. More importantly, it may not be computationally feasible to maximize the MEAN FIELD FOR COMMUNITY DETECTION 3 variational likelihood for the SBM, as seen from Theorem 2.1. In this paper, we consider the statistical and computational guarantees of the iterative variational inference algorithm for community detection. The primary goal of community detection problem is to recover the community membership in a network. We measure the performance of the iterative variational inference algorithm by comparing its output with the ground truth. Denote the underlying ground truth by Z ∗ . For a network of n nodes and k communities, Z ∗ is an n × k matrix with each row a standard Euclidean ∗ }n basis in Rk . The index of non-zero coordinate of each row {Zi,· i=1 gives the community assignment information for the corresponding node. We propose an iterative algorithm called Batch Coordinate Ascent Variational Inference (BCAVI), a slight modification of CAVI with batch updates, to make parallel and distributed computing possible. Let π (s) denote the output of the s-th iteration, an n × k matrix with nonnegative entries. The summation (s) of each row {πi,· }ni=1 is equal to 1, which is interpreted as an approximate posterior probability of assigning the corresponding node of each row into k communities. The performance of π (s) is measured by an `1 loss `(·, ·) compared with Z ∗ . An Informal Statement of Main Result: Let π (s) be the estimation of community membership from the iterative algorithm BCAVI after s iterations. Under weak regularity condition, for some cn = on (1), with high probability, we have for all s ≥ 0, (1) `(π (s+1) , Z ∗ ) ≤ minimax rate + cn `(π (s) , Z ∗ ). The main contribution of this paper is Equation (1). The coefficient cn is on (1) and is independent of s, which implies `(π (s) , Z ∗ ) decreases at a fast linear rate. In addition, we show that BCAVI converges to the statistical optimality [35]. It is worth mentioning that after log n iterations BCAVI attains the minimax rate, up to an error on (n−a ) for any constant a > 0. The conditions required for the analysis of BCAVI are relatively mild. We allow the number of communities to grow. The sizes of the communities are not assumed to be of the same order. The separation condition on global parameters covers a wide range of settings from consistent community detection to exact recovery. To the best of our knowledge this provides arguably the first theoretical justification for the iterative algorithm of the mean field variational method in a high-dimensional and complex setting. Though we focus on the problem of community detection in this paper, we hope the analysis would shed some 4 ZHANG, ZHOU light on analyzing other models, which may eventually lead to a general framework of understanding the mean field theory. The techniques of analyzing the mean field can be extended to providing theoretical guarantees for other iterative algorithms, including Gibbs sampling and an iterative procedure for maximum likelihood estimation, which can be of independent interest. Results similar to Equation (1) are obtained for both methods under the SBM. Organization. The paper is organized as follows. In Section 2 we introduce the mean field theory and the implementation of BCAVI algorithm for community detection. All the theoretical justifications for the mean field method are in Section 3. Discussions on the convergence of the global minimizer and other iterative algorithms are presented in Section 4. The proofs of theorems are in Section 5. We include all the auxiliary lemmas and propositions and their corresponding proofs in the supplemental material. Notation. Throughout this paper, for any matrix X ∈ Rn×m P , its `1 norm is defined in analogous to that of a vector. That is, kXk1 = i,j |Xi,j |. We use the notation Xi,· and X·,i to indicate its i-th row and column respectively. For matrices P X, Y of the same dimension, their inner product is defined as hX, Y i = i,j Xi,j Yi,j . For any set D, we use |D| for its cardinality. We denote Ber(p) for a Bernoulli random variable with success probability p. For two positive sequences xn and yn , xn . yn means xn ≤ cyn for some constant c not depending on n. We adopt the notation xn  yn if xn . yn and yn . xn . To distinguish from the probabilities p, q, we use bold p and q to indicate distributions. The Kullback-Leibler divergence between two distributions is defined as KL(pkq) = Eq log(p(x)/q(x)). We use ψ(·) for the digamma function, which is defined as the logarithmic derivative of Gamma d function, i.e., ψ(x) = dx [log Γ(x)]. In any Rd , we denote {ea }da=1 to be the standard Euclidean basis with e1 = (1, 0, 0, . . .), e2 = (0, 1, 0, . . . , 0), . . . , ed = (0, 0, 0, . . . , 1). We let 1d be a vector of length d whose entries are all 1. We use [d] to indicate the set {1, 2, . . . , d}. Throughout this paper, the superscript “pri” (e.g., π pri ) indicates that this is a hyperparameters of priors. 2. Mean Field Method for Community Detection. In this section, we first give a brief introduction to the variational inference method in Section 2.1. Then we introduce the community detection problem and the Stochastic Block Model in Section 2.2. The Bayesian framework is presented in Section 2.3. Its mean field approximation and CAVI updates are given in Section 2.4 and Section 2.5 respectively. The BCAVI algorithm is introduced in Section 2.6. MEAN FIELD FOR COMMUNITY DETECTION 5 2.1. Mean Field Variational Inference. We first present the mean field method in a general setting and then consider its application to the community detection problem. Let p(x|y) be an arbitrary posterior distribution for x, given observation y. Here x can be a vector of latent variables, with coordinates {xi }. It may be difficult to compute the posterior p(x|y) exactly. The variational Bayes ignores Q the dependence among {xi }, by simply taking a product measure q(x) = i qi (xi ) to approximate it. Usually each qi (xi ) is simple and easy to compute. The best approximation is obtained by minimizing the Kullback-Leibler divergence between q(x) and p(x|y): (2) q̂MF = arg min KL(qkp). q∈Q Despite the fact that every measure q has a simple product structure, the global minimizer q̂MF remains computationally intractable. To address this issue, an iterative Coordinate Ascent Variational Inference (CAVI) is widely used to approximate the global minimum. It is a greedy algorithm. The value of KL(qkp) decreases in each coordinate update:   Y q̂i = min KL qi qj p , ∀i. (3) qi ∈Qi j6=i The coordinate update has an explicit formula   (4) q̂i (xi ) ∝ exp Eq−i [log p(xi |x−i , y)] , where x−i indicates all the coordinates in x except xi , and the expectation Q is over q−i = j6=i qj (xj ). Equation (4) is usually easy to compute, which makes CAVI computationally attractive, although CAVI only guarantees to achieve a local minimum. In summary, the mean field variational inference via CAVI can be represented in the following diagram: approx. approx. p(x|y) ⇐= q̂MF (x) ⇐= q̂CAVI (x), where q̂MF (x), the global minimum, serves mainly as an intermediate step in the mean field methodology. What is implemented in practice to approximate global minimum is an iterative algorithm like CAVI. This motivates us to consider directly the theoretical guarantees of the iterative algorithm in this paper. We refer the readers to a nice review and tutorial by Blei et al. [7] for more detail on the variational inference and CAVI. The derivation from Equation (3) to Equation (4) can be found in many variational inference literatures [5, 7]. We include it in Appendix D in the supplemental material for completeness. 6 ZHANG, ZHOU 2.2. Community Detection and Stochastic Block Model. The Stochastic Block Model (SBM) has been a popular model for community detection. Consider an n-node network with its adjacency matrix denoted by A. It is an unweighted and undirected network without self-loops, with A ∈ {0, 1}n×n , A = AT and Ai,i = 0, ∀i ∈ [n]. Each edge is an independent Bernoulli random variable with EAi,j = Pi,j , ∀i < j. In the SBM, the value of connectivity probability Pi,j depends on the communities the two endpoints i and j belong to. We assume Pi,j = p if both nodes come from the same community and Pi,j = q otherwise. There are k communities in the network. We denote z ∈ [k]n , as the assignment vector, with zi indicating the index of community the i-th node belongs to. Thus, the connectivity probability matrix P can be written as Pi,j = Bzi ,zj , where B ∈ [0, 1]k×k with diagonal entries as p and off-diagonal entries as q. That is, B = q1k 1Tk + (p − q)Ik . Let Z ∈ Π0 be the assignment matrix where Π0 = {π ∈ {0, 1}n×k : kπi,· k0 = 1, ∀i ∈ [n]}. In each row {Zi,· }ni=1 there is only one 1 with all the other coordinates as 0, indicating the assignment of community for the corresponding node. Then T , ∀i < j, or in a matrix form P can be equivalently written as Pi,j = Zi,· BZj,· Pi,j = (ZBZ T )i,j , ∀i < j. The goal of community detection is to recover the assignment vector z, or equivalently, the assignment matrix Z. The equivalence can be seen by observing that there is a bijection r between z ∈ [k]n and Z ∈ Π0 which is defined as follows, (5) r(z) = Z, where Zi,a = I{a = zi }, ∀i ∈ [n], a ∈ [k]. Since they are uniquely determined by each other, in our paper we may use z directly without explicitly defining z = r−1 (Z) (or vice versa) when there is no ambiguity. 2.3. A Bayesian Framework. Throughout the whole paper, we assume k, the number of communities, is known. We observe the adjacency matrix A. The global parameters p and q and the community assignment Z are unknown. From the description of the model in Section 2.2, we can write down the distribution of A as follows: Y Ai,j p(A|Z, p, q) = Bzi ,zj (1 − Bzi ,zj )1−Ai,j , (6) i<j 7 MEAN FIELD FOR COMMUNITY DETECTION with B = q1k 1Tk + (p − q)Ik and z = r−1 (Z). We are interested in Bayesian inference for estimating Z, with prior to be given on both p, q and Z. We assume that {zi }ni=1 have independent categorical (a.k.a. multinomial P pri n pri with size one) priors with hyperparameters {πi,· }i=1 , where ka=1 πi,a = n 1, ∀i ∈ [n]. In other words, {Zi,· }i=1 are independently distributed by pri P(Zi,· = eTa ) = πi,a , ∀a = 1, 2, . . . , k, where {ea }ka=1 are the coordinate vectors. Here we allow the priors for Zi,· to be different for different i. If additionally πi,· = πj,· for all i 6= j is assumed, and then this is reduced to the usual case of i.i.d. priors. Since {Ai,j }i<j are Bernoulli, it is natural to consider a conjugate Beta prior for p and q. Let p ∼ Beta(αppri , βppri ) and q ∼ Beta(αqpri , βqpri ). Then the joint distribution is (7) p(A, Z, p, q) = " Y i × "  # Y A 1−Ai,j  π pri  Bzii,j ,zj (1 − Bzi ,zj ) i,zi i<j Γ(αppri + βppri ) αpri p p −1 (1 pri pri Γ(αp )Γ(βp ) − p) βppri −1 #" Γ(αqpri + βqpri ) Γ(αqpri )Γ(βqpri ) q αpri q −1 (1 − q) Our main interest is to infer Z, from the posterior distribution p(Z, p, q|A). However, the exact calculation of p(Z, p, q|A) is computationally intractable. 2.4. Mean Field Approximation. Since the posterior distribution p(Z, p, q|A) is computationally intractable, we apply the mean field approximation to approximate it by a product measure, qπ,αp ,βp ,αq ,βq (Z, p, q) = qπ (Z)qαp ,βp (p)qαq ,βq (q) where {r−1 (Zi,· )}ni=1 areQindependent categorical variables with parameters {πi,· }ni=1 , i.e., qπ (Z) = ni=1 qπi,· (Zi,· ) with qπi,· (Zi,· = ea ) = πi,a , ∀i ∈ [n], a ∈ [k], and qαp ,βp (p) and qαq ,βq (q) are Beta with parameters αp , βp , αq , βq due to conjugacy. See Figure 1 for the graphical presentation of qπ,αp ,βp ,αq ,βq (Z, p, q). Note that the distribution class of q is fully captured by the parameters (π, αp , βp , αq , βq ), and then the optimization in Equation (2) is equivalent βqpri −1 # . n×kn,i,jwhere i,jhyperparameter i,j ! ! 1 1]prior 1 1] with πi,a ∈[n]. [0, ,i,jwhere n with prior hyperparameter π (a.k.a. ∈k [0, πi,a =a=1 1, ∀i ∈ It is k a=1 n×k We assume {z }multinomial have categorical (a.k.a. multinomi n×k We {z }its have categorical with size one) n T n{Z n iand iand Tp, estimating with prior given on both q Z. i=1 imating Z, with given on assume both p, qbe Z. i=1 equivalently to state that, } are ind with hyperparameter ∈ [0, 1] , where π = 1, ∀i priorprior with hyperparameter π ∈ [0, 1] , where π = 1, ∀i ∈ [n]. It is equivalently to state that, {Z } are independently distributed with n T i,· P(Z = e ) = π , ∀u = 1 set D, we denote |D|Z, toprior cardinality. set D, we We denote denote |D| Ber(p) to be its to cardinality. be a Bernoulli We denote Ber(p) to be aB i,a T i,· i,a ! i=1 ! P(Z = e ) = π , ∀u = 1, 2, . . . , k, i=1 i,· i,a equivalently to state that, {Z } are independ a=1 a=1 equivalently to state that, {Zi,· }ai=1 are)i,a independently with a= i,· i,· P(Z k πi,a , ∀u i=1ea ) = k πi,a , ∀u = distributed P(Z = e = 1, 2, . . . , k, n×k i,· n×k i,· a n n n prior with hyperparameter π ∈ [0, 1] , where π n prior with hyperparameter π ∈ [0, 1] , where π = 1, ∀i ∈ [n]. It is i,a i,a We assume {z } have categorical (a.k.a. multinomial with size one) a=1 T We assume equivalently {zi }i=1random have to categorical (a.k.a. multinomial with size one) a=1 variable with equivalently success probability random variable p. 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T a=1 i,· na if {e ni,· n{Z ni,· n nP(Z n j,· n If a=1 i,· j,· equivalently state }notation uivalently to state that, distributed with = e ) = π , = 1, 2, . . . , k, P(Z = e ) π , ∀u = 1, 2, . . . , k, i,·= 6 ZHANG, 6 ZHOU ZHANG, ZHOU i,· }i=1 are independently i,· i,a i,· i,a i=1 a n a eters on Z varies. If we assume π = π eters on Z varies. If we assume π = π for all i = ̸ j, then this can be i,· i,· j eters Z varies. Ifusual we assume πj,· sam for i,·and j,·ip into the case with pria onnation Zi,·into varies. 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It is Due to the fact that p, q ∈ (0, 1) and all the {A } are Bernoulli random α , β , α , β . Thus the full likelihood function is rs on Zi,· varies. Ifwith we assume π = π for all i = ̸ j, then this can be α , β , α , β . Thus the full likelihood function is α , β , α , β . Thus the full likelihood funct p p q q α , β , α , β . Thus the full likelihood function is α , β , α , β . Thus the full likelihood function is α β , α , β . Thus the full likelihood function is p p q q i,a i,a i,j the i<j{Ai,j }i<j are parameter i,· fact j,· pthat i,j i<j pq are q a=1 q1) andrandom Due toi,· the fact p, qpq∈ Bernoulli (0, all p j,·pthat p p p q pp, Due toi,·the and all the {A }pi<j q q1) q qqpqq∈ q(0, a=1 i,jstudied (SBM)tohas been the most studied (SBM) model has fornbeen community the most detection. model Consider for community detection. 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Conside but it is computationally Consider called mean filed variational method mean isto filed one popular to tackle method is distribution onep(Z, popular to tackle +n isBayes MFdisthere multiple variables involved in posterior distribut there multiple variables involved in posterior distribution p(x), it. The philosophy of mean filed theory isBayes to the posterior MF munity. The optimization inare Equation (8)the can benintractable. shown to be equivalent to The philosophy of are mean filed theory tocalled approximate posterior dis(4) q̂ = min KL (4) q̂ = min KL(q∥p). est but it is computationally intractable. C est but it is computationally intractable. Consider distribution MF q(x) = q (x ), a way to reduce computational com q(x) = q (x ), a way to reduce computational complexity. The soMF i theory i disi theory i The i=1 i=1 q∈Q ′ va it. The philosophy of mean filed it. is philosophy to approximate of mean the filed posterior is to approximate posterior dis(4) q̂ = min KL(q∥p q∈Q (4) q̂ = min KL(q∥p). a more explicit optimization as follows. Recall ψ(·) is the digamma function ′independence ′ ′,βstructure. ′ ′,α ′ ,β ′the ′ (Z)q q (Z, p, q) = q ′ ′ ′ ′ ′ ′ ′ ′ q (Z, p, q) = q (Z)q (p)q (q) mean field theory usually assumes the of these mean field theory usually assumes the independence of these variables in π ,α π α tribution p(x|y) by some distribution q(x) with simpler When π ,αp ,βpstructure. ,αq ,βq π αp ,βp α bution p(x|y) by some distribution q(x) with simpler When p q ,β pq q q p ,β q∈Q + mean field theory to search over some distribut mean field theory isp(x|y) tocalled search over someq∈Q distribution class such d distribution tribution+ p(x|y) by some tribution q(x) simpler by When simpler structure. When ′Q, ′ ,α ′ ,β ′ (Z, p, q) = qπ′ (Z nwith ′ ,βstructure. ′ distribution ′ ,β ′ (Z, p, ′q ′ ′(p)q ′ (q) n called qπ′some q) q(x) = qis (Z)q with ψ(x) = Γ(x)]. ,α π ′with α α′q ,β p ,βp qqq p a p ,αway q in q the p ,βπ p ,α dxa[log q(x) = q (x ), to reduce computational complexity q(x) = q (x ), way to reduce computational complexity. The soBack to our case, the posterior distribu there are multiple variables involved posterior distribution p(x), Back to our case, the posterior distribution p(Z, p, q|A) is the main in re are multiple variables involved in the posterior distribution p(x), i i i i i=1 i=1 that the Kullback-Leibler divergence is minimized. That i that the Kullback-Leibler divergence is minimized. That is there are multiple variables involved there in are the multiple posterior variables distribution involved p(x), in the posterior distribution p(x), Back to our case, the posterior distribution Back to our case, the posterior distribution p(Z, p, q|A) is the main interest but it is computationally intractable. est but it is computationally intractable. Consider distribution called mean field theory is to search over some distribution clas called mean field theory is to search over some distribution class Q, such mean field theory usually assumes the independence of these variables in an field theory usually assumes the independence of these variables in MF MF MF MF MF mean field theory usually assumes mean the field independence theory usually of these assumes variables the in independence of these variables in 2.1. The mean computationally fieldMF estimator (π̂ intractable. , α̂but α̂computationally , β̂q )distribution + + est intractable. Con p , β̂it p is,Consider q q̂MF +n Theorem n est but it is + nq̂ (4)= min KL(q∥p). = min KL(q∥p). (4) that the Kullback-Leibler divergence is minimized. That is divergence is minimized. That is q(x) =a Kullback-Leibler q (x ), a way to reduce q(x) = computational q (x ), complexity. a way to reduce The socomputational complexity. q(x) = q (x ), a way to reduce computational complexity. TheThe x) = ni=1 qthat way to reduce computational complexity. The soi i i i i i defined in Equation (8) is equivalent to i (x i ),the i=1 i=1 q ′ q∈Q q∈Q ′ ,α ′ ′,β ′ ,α′ ,β′′ (Z, i=1 q p,soq)so= qπ ′ ( ′ ′ ′ ′ ′ ′ (Z, p, q) = q (Z)q (p)q π π ,αp ,βp ,αq ,βq π αp ,βpp p q αq ,βq′ (q) called mean field theory is to search called over mean some field distribution theory is to class search Q, such over some distribution class Q, such ′ ′ ′ ′ ′ ′ (Z)qα q (Z, p, q) = q ′ ,α ′ ,βsearch ′ ,α′ ,β ′ (Z, ′distribution ′,α ′,β ′q,β ′ (q) q p, q) = q (Z)q (p)q mean field theory is to over some class Q, such π ,α ,β π led mean field theory is called to search over some distribution class Q, such π π α ,β α p p q MF MF p p q q p p q q MF to MFthe MF (4) q̂ = min KL(q∥p). Back to our case, the posterior distribution p(Z, p, q|A) (4) q̂ = min KL(q∥p). Back our case, posterior distribution p(Z, p, q|A) is the main interthat the Kullback-Leibler is the minimized. Kullback-Leibler That is divergence is minimized. That is (π̂ , α̂pMFdivergence , β̂pMF , α̂that , β̂ ) = arg min f (π, α , β , α , β ; A), p p q q q q that the Kullback-Leibler divergence is minimized. That is q∈Q q∈Q at the Kullback-Leibler divergence is minimized. That is π∈Π1 (4) est but it is Consider computationally intractable. Consider distrib est but it is computationally intractable. distribution MF MF α p ,βp ,αq ,βq >0 q̂ =(4) min KL(q∥p). q̂ = min KL(q∥p). MF q∈Q distribution q∈Qis distribution Back to our case, the KL(q∥p). posterior p(Z, p, q|A) is the m Back to our case, theKL(q∥p). posterior p,qq|A) the main inter(4) q̂MF = p(Z, min q̂ = min ′ ,α ′ ,β ′ ′ α′′ ,β ′ ,α′ ,β ′ ,α′ ,β ′ (Z, p, q) = qπ ′ (Z)q ′ ,β ′ (p)q ′ (q) (Z, p, q) = qπ′ (Z)qα′p ,βp′ (p)qα′q ,β q π π α where q∈Q p ,αq ,β p p q q p pp qq q q∈Q estdistribution but computationally intractable. est but intractable. Consider distribution Back it to is ourcomputationally case, the posterior Backittoisour p(Z, case, p, q|A) the posterior is the main distribution interp(Z,Consider p, q|A) is thedistribution main inter1 est but is computationally intractable. estp(Z, but Consider itposterior isq|A) computationally distribution intractable. Consider distribution T main toβour case, the distribution p,q )]q|A) is the main interBack to our case, theitposterior distribution interf (π, αBack = thA − λ1p, 1Tn + λI′isn ,′the ππ i′ + [ψ(α )′ − ψ(β kAk q′; A) q p(Z, 1 ′(p)q ′ (Z, ′ (Z)q ′ (Z, p, q) =n q ′q ′ ′,β ′ ,β p, q) = q qpπ,′β,αp ,′pα,βqp′, ,α (Z)q (q) 2 π ,α ,β ,α ,β π α′p ,βp′ (p)qα′q ,βq′ (q) ,β π α α p pp pq q q q q q est but computationally Consider distribution ′ ,α ′ ,β ′ ,αit ′ ,β is ′ (Z, ′ (Z)q ′ ,β ′ intractable. ′(p)q ′ ,βα ′ ′,α ′ ′,β ′ (Z, ′ (Z)q ′ ,β ′ (p)qα′ ,β ′ (q) but it is computationally Consider distribution qπintractable. p, q) = q q (q) p, q) = q n π α π ,α ,β π α h i p p q q p p p p q qq q X p p q q qπ′ ,α′p ,βp′ ,α′q ,βq′ (Z, p, q) = n pri [ψ(βq ) − ψ(αq + βq )] − KL Categorical(πi,· )kCategorical(πi,· ) 2 ′ ,α ′ ,β ′ (Z, ′ ′ ′ ′ ′ qπ′ ,αα′p ,β p, q) = q (Z)q (p)q (q) ′ ,β ′ (p)q ′ ′ qπ′ (Z)q (q) π α ,β αq ,βq i=1 pp p q q αq ,βq  p p  − KL Beta(αp , βp )kBeta(αppri , βppri ) − KL Beta(αq , βq )kBeta(αqpri , βqpri ) , + 9 MEAN FIELD FOR COMMUNITY DETECTION and t = [[ψ(αp ) − ψ(βp )] − [ψ(αq ) − ψ(βq )]] /2 (9) λ = [[ψ(βq ) − ψ(αq + βq )] − [ψ(βp ) − ψ(αp + βp )]] /(2t). (10) The explicit formulation in Theorem 2.1 is helpful to understand the global minimizer of the mean field method. However, the global minimizer π̂ MF remains computationally infeasible as the objective function is not convex. Fortunately, there is a practically useful algorithm to approximate it. 2.5. Coordinate Ascent Variational Inference. CAVI is possibly the most popular algorithm to approximate the global minimum of the mean field variational Bayes. It is an iterative algorithm. In Equation (8), there are latent variables {Zi,· }ni=1 , p, q. CAVI updates them one by one. Since the distribution class of q is uniquely determined by the parameters {πi,· }ni=1 , αp , βp , αq , βq , equivalently we are updating those parameters iteratively. Theorem 2.2 gives explicit formulas for the coordinate updates. Theorem 2.2. Starts with some π, αp , βp , αq , βq , the CAVI update for each coordinate (i.e., Equation (3) and Equation (4)) has an explicit expression as follows: • Update on p: αp0 = αppri + k XX πi,a πj,a Ai,j , and i<j a=1 βp0 = βppri + k XX i<j a=1 πi,a πj,a (1 − Ai,j ). • Update on q: XX XX αq0 = αqpri + πi,a πj,b Ai,j , and βq0 = βqpri + πi,a πj,b (1 − Ai,j ). i<j a6=b i<j a6=b • Update on Zi,· , ∀i = 1, 2, . . . , n:   X pri 0 πi,a ∝ πi,a exp 2t πj,a (Ai,j − λ) , ∀a = 1, 2, . . . , k, j6=i where t and λ are defined in EquationP (9) and Equation (10) respec0 = 1. tively, and the normalization satisfies ka=1 πi,a All coordinate updates in Theorem 2.2 have explicit formulas, which makes CAVI a computationally attractive way to approximate the global optimum q̂MF for the community detection problem. 10 ZHANG, ZHOU 2.6. Batch Coordinate Ascent Variational Inference. The Batch Coordinate Ascent Variational Inference (BCAVI) is a batch version of CAVI. The difference lies in that CAVI updates the rows of π sequentially one by one, 0 } according to Thewhile BCAVI uses the value of π to update all rows {πi,· orem 2.2. This makes BCAVI especially suitable for parallel and distributed computing, a nice feature for large scale network analysis. We define a mapping h : Π1 → Π1 as follows. For any π ∈ Π1 , we have   X pri (11) [ht,λ (π)]i,a ∝ πi,a exp 2t πja (Ai,j − λ) , j6=i with parameters t and λ. For BCAVI, we update π by π 0 = ht,λ (π) in each batch iteration, with t, λ defined in Equations (14) and (15). See Algorithm 1 for the detailed implementation of BCAVI algorithm. Algorithm 1: Batch Coordinate Ascent Variational Inference (BCAVI) 1 Input: Adjacency matrix A, number of communities k, hyperparameters π pri , αppri , βppri , αqpri , βqpri , initializer π (0) , number of iterations S. Output: Mean variational Bayes approximation π̂, α̂p , β̂p , α̂q , β̂q . for s = 1, 2, . . . , S do (s) (s) (s) (s) Update αp , βp , αq , βq by (12) αp(s) = αppri + k X X (s−1) Ai,j πi,a (s−1) πj,a , βp(s) = βppri + a=1 i<j k X X (s−1) (s−1) (1 − Ai,j )πi,a πj,a , a=1 i<j (13) αq(s) = αqpri + XX (s−1) Ai,j πi,a (s−1) πj,b , βq(s) = βqpri + a6=b i<j a6=b i<j 2 XX (s−1) (s−1) (1 − Ai,j )πi,a πj,b . Define i h ii 1 hh ψ(αp(s) ) − ψ(βp(s) ) − ψ(αq(s) ) − ψ(βq(s) ) 2 i h ii 1 hh = (s) ψ(βq(s) ) − ψ(αq(s) + βq(s) ) − ψ(βp(s) ) − ψ(αp(s) + βp(s) ) , 2t (14) t(s) = (15) λ(s) where ψ(·) is the digamma function. Then update π (s) with π (s) = ht(s) ,λ(s) (π (s−1) ), 3 where the mapping h(·) is defined as in Equation (11). end (S) (S) (S) (S) We have π̂ = π (S) , α̂p = αp , β̂p = βp , α̂q = αq , β̂q = βq . 11 MEAN FIELD FOR COMMUNITY DETECTION Remark 2.1. The definitions of t(s) and λ(s) in Equations (14) and (15) involve the digamma function, which costs a non-negligible computational resources each time called. Note that we have ψ(x) ∈ (log(x− 21 ), log x) for all x > 1/2. For the computational purpose, we propose to use the logarithmic function instead of digamma function in Algorithm 1, i.e., Equations (14) and (15) are replaced by (s) (s) (s) (16) t(s) = 1 αp βq log (s) (s) , 2 β p αq (s) (s) and λ(s) = (s) (s) (s) βq (αp + βp ) 1 log (s) . (s) (s) (s) 2t (αq + βq )βp (s) Later we show that αp , βp , αq , βq are all at least in the order of np, which goes to infinity, and thus the error caused by using the logarithmic function to replace the digamma function is negligible. All theoretical guarantees obtained in Section 3 for Algorithm 1 (i.e., Theorem 3.1, Theorem 3.2) still hold if we use Equation (16) to replace Equations (14) and (15). 3. Theoretical Justifications. In this section, we establish theoretical justifications for BCAVI for community detection under the Stochastic Block Model. Though Z, p and q are all unknown, the main interest of community detection is on the recovery of the assignment matrix Z, while p and q are nuisance parameters. As a result, our main focus is on developing convergence rate of BCAVI for π. 3.1. Loss Function. We use `1 norm to measure the performance of recovering Z. Let Φ be the set of all the bijections from [k] to [k]. Then for any Z, Z ∗ ∈ Π1 , the loss function is defined as X ∗ `(Z, Z ∗ ) = inf kZ − φ ◦ Z ∗ k1 = inf (17) |Zi,a − Zi,φ(a) |. φ∈Φ φ∈Φ i,a Note that the infimum over Φ addresses the issue of identifiability over the labels. For instance, in the case of n = 4, k = 2, the assignment vector z = (1, 1, 2, 2) and z 0 = (2, 2, 1, 1) give the same partition. In Equation (17) two equivalent assignments give the same loss. There are a few reasons for the choise of the `1 norm. When both Z, Z 0 ∈ Π0 , the `1 distance between Z and Z 0 is equal to the `0 norm, i.e., the Hamming distance between the corresponding assignment vectors r−1 (Z) and r−1 (Z 0 ), which is the default metric used in community detection literature [12, 35]. The other reason is related to the interpretation of Π1 . Since each row of Π1 corresponds to a categorical distribution, it is natural to use the `1 norm, the total variation distance, to measure their diffidence. 12 ZHANG, ZHOU 3.2. Ground Truth. We use the superscript asterisk (∗ ) to indicate the ground truth. The ground truth of connectivity matrix B ∗ is B ∗ = q ∗ 1k 1Tk + (p∗ − q ∗ )Ik , where p∗ is the within community connection probability and q ∗ is the between community connection probability. Throughout the paper, we assume p∗ > q ∗ such that the network satisfies the so-called “assortative” property, with the within-community connectivity probability larger than the between-community connectivity probability. We further assume the network is generated by the true assignment matrix Z ∗ in the sense that Pi,j = (Z ∗ B ∗ Z ∗T )i,j for all i 6= j. We are interested in deriving a statistical guarantee of `(π̂ (s) , Z ∗ ). Throughout this section we (ρ,ρ0 ) (ρ,ρ0 ) consider cases Z ∗ ∈ Π0 or Z ∗ ∈ Π0 , where Π0 is defined to be a subset of Π0 with all the community sizes bounded between ρn/k and ρ0 n/k. That is, (ρ,ρ0 ) Π0 = {π ∈ Π0 : ρn/k ≤ |{i ∈ [n] : πi,a = 1}| ≤ ρ0 n/k, ∀a ∈ [k]}. It is worth mentioning that ρ, ρ0 are not necessarily constants. We allow the community sizes not to be of the same order in the theoretical analysis. 3.3. Theoretical Justifications for BCAVI. In Theorem 3.1, we present theoretic guarantees of the convergence rate of BCAVI when initialized properly. Define pri pri , and n̄min = min[na + nb ]/2. w = max max πi,a /πi,b i∈[n] a,b∈[k] a6=b When w = 1, the priors for {r−1 (Zi,· )}ni=1 are i.i.d. Categorical(1/k, 1/k, . . . , 1/k) and n̄min = n/2 when there exist only two communities. The following quantity I plays a key role in the minimax theory [35] hp i p I = −2 log p∗ q ∗ + (1 − p∗ )(1 − q ∗ ) , which is the Rényi divergence of order 1/2 between two Bernoulli distributions: Ber(p∗ ) and Ber(q ∗ ). The proof of Theorem 3.1 is deferred to Section 5.3. Theorem 3.1. Let Z ∗ ∈ Π0 . Let 0 < c0 < 1 be any constant. Assume 0 < c0 p∗ < q ∗ < p∗ = on (1), (18) nI/[wk[n/n̄min ]2 ] → ∞, and αppri , βppri , αqpri , βqpri = on ((p∗ − q ∗ )n2 /k). MEAN FIELD FOR COMMUNITY DETECTION 13 Under the assumption that the initializer π (0) satisfies `(π (0) , Z ∗ ) ≤ cinit n̄min for some sufficiently small constant cinit with probability at least 1 − , there exist some constant c > 0 and some η = on (1) such that in each iteration for the BCAVI algorithm, we have `(π (s) , Z ∗ ) `(π (s+1) , Z ∗ ) ≤ n exp(−(1 − η)n̄min I) + p , ∀s ≥ 0, nI/[wk[n/n̄min ]2 ] 1 holds uniformly with probability at least 1 − exp[−(n̄min I) 2 ] − n−c − . Theorem 3.1 establishes a linear convergence rate for BCAVI algorithm. The coefficient [nI/[wk[n/n̄min ]2 ]]−1/2 is independence of s, and goes to 0 when n grows. The following theorem is an immediate consequence of Theorem 3.1. Theorem 3.2. Under the same condition as in Theorem 3.1, for any s ≥ s0 , [nI/k]/ log[nI/[wk[n/n̄min ]2 ]], we have ( n exp(−(1 − o(1))ρnI/k), k ≥ 3; (s) ∗ `(π̂ , Z ) ≤ n exp(−(1 − 2η)n̄min I) ≤ n exp(−(1 − o(1))nI/2), k = 2, 1 with probability at least 1 − exp[−(n̄min I) 2 ] − n−c − . Theorem 3.2 shows that BCAVI provably attains the statistical optimality from the minimax lower bound in Theorem 3.3 after at most s0 iterations. When the network is sparse, i.e., p∗ and q ∗ are at most in an order of (log n)/n, the quantity s0 can be shown to be o(log n), and then BCAVI converges to be minimax rate within log n iterations. When the network is dense, i.e., p∗ and q ∗ are far bigger than (log n)/n, log n iterations are not enough to attain the minimax rate. However, `(π (s) , Z ∗ ) = o(n−a ) for any a > 0 when s ≥ log n, and thus all the nodes can be correctly clustered with high probability by clustering each note to a community with the highest assignment probability. Therefore, it is enough to pick the number of iterations to be log n in implementing BCAVI. Under the assumption nI/(k log k) → ∞, we have ( n exp(−(1 − o(1))ρnI/k), k ≥ 3; inf sup E`(π̂, Z ∗ ) ≥ . π̂ n exp(−(1 − o(1))nI/2), k = 2, (ρ,ρ0 ) Z ∗ ∈Π Theorem 3.3. 0 14 ZHANG, ZHOU Theorem 3.3 gives the minimax lower bound for community detection problems with respect to the `(·, ·) loss. In Theorem 3.2, under the addi(ρ,ρ0 ) tional assumption that Z ∗ ∈ Π0 , it immediately reveals that BCAVI converges to the minimax rate after s0 iterations. As a consequence, BCAVI is not only computationally efficient, but also achieves statistical optimality. The minimax lower bound in Theorem 3.3 is almost identical to the minimaxity established in [35]. The only difference is that [35] consider a `0 loss function. The proof of Theorem 3.3 is just a routine extension of that in [35]. Therefore, we omit the proof. To help understand Theorem 3.1, we add a remark on conditions on model parameters and priors, and a remark on initialization. Remark 1 (Conditions on model parameters and priors). The community sizes are not necessarily of the same order in Theorem 3.1. If we further pri assume ρ, ρ0 are constants, and the prior πi,a  1/k, ∀i ∈ [n], a ∈ [k] (for example, uniform prior), and then the first condition in Equation (18) is equivalent to nI/k 3 → ∞, noting that n/n̄min  k and w  1. This condition is necessary for consistent community detection [35] when k is finite. The assumptions in Equation (18) is slightly stronger than the assumption in [23], which is essentially nI ≥ Ck 2 log k for a sufficient large constant C. Under the assumption nI/k 3 → ∞, since we have I  (p∗ − q ∗ )2 /p∗ , it can be shown that p∗ , q ∗ are far bigger than n−1 , and then the second part of Equation (18) can also be easily satisfied. For instance, we can simply set αppri , βppri , αqpri , βqpri all equals to 1, i.e., consider non-informative priors. Remark 2 (Initialization). The requirement on the initializers for BCAVI in Theorem 3.1 is relatively weak. When k is a constant and the community sizes are of the same order, the condition needed is `(π (0) , Z ∗ ) ≤ cn for some small constant c. Many existing methodologies in community detection literature can be used. One popular choice is spectral clustering. Established in [9, 12, 21], the spectral clustering has a mis-clustering error bound as O(k 2 /I). From Equation (18), the error is o(n̄min ), and then the condition that Theorem 3.1 requires for initialization is satisfied. The semidefinite programming (SDP), another popular method for community detection, also enjoys satisfactory theoretical guarantees [10, 16], and is suitable as an initializer. 4. Discussion. MEAN FIELD FOR COMMUNITY DETECTION 15 4.1. Statistical Guarantee of Global Minimizer. Though it is often challenging to obtain the global minimizer of the mean field method, it is still interesting to understand the statistical property of the global minimizer π̂ MF . Assume that both p∗ and q ∗ are known, the optimization problem stated in Theorem 2.1 can be further simplified. The posterior distribution becomes Q p(Z|A). We use a product measure qπ (Z) = i qi (πi,· ) for approximation, and then π̂ MF = arg minπ∈Π1 KL[qπ (Z)kp(Z|A)]. Theorem 4.1 reveals that π̂ MF is rate-optimal, not surprisingly given the theoretical results obtained for BCAVI, an approximation of π̂ MF . Theorem 4.1. Assume p∗ and q ∗ are known. Under the assumption ρnI/[wk 2 [n/n̄min ]2 ] → ∞, there exist some constant c > 0 and η = on (1) such that `(π̂ MF , Z ∗ ) ≤ n exp(−(1 − η)n̄min I) 1 with probability at least 1 − exp[−(n̄min I) 2 ] − n−c . 4.2. Gibbs Sampling. In Section 3.3 we analyze an iterative algorithm, BCAVI, and establish its linear convergence towards statistical optimality. The framework and methodology we establish is not limited to BCAVI, but can be extended to other iterative algorithms, including Gibbs sampling. As a popular Markov chain Monte Carlo (MCMC) algorithm, Gibbs sampling has been widely used in practice to approximate the posterior distribution. There is a strong tie between Gibbs sampling and the mean field variational inference: both implement coordinate updates using conditional distributions. Using the general notation introduced in Section 2.1, to approximate p(x|y), Gibbs sampling obtains the update on xi by a random generation from the conditional distribution p(xi |x−i ,y), while the variational  inference updates in a deterministic way with exp Eq−i log p(xi |x−i , y) . We present a batched version of Gibbs sampling for community detection. It involves iterative updates with • Generate p(s) by sampling from p(p|q (s−1) , Z (s−1) , A); • Generate q (s) by sampling from p(q|p(s−1) , Z (s−1) , A); (s) (s−1) (s−1) • Generate Zi,· independently by sampling from p(Zi,· |Z−i,· , p(s) , q (s) , A), for i ∈ [n]. We include the detailed implementation as Algorithm 2 in the supplemental material (Section A.1). The similarity between Algorithm 1 and Algorithm 2 makes it possible for us to analyze the output of Gibbs sampling in a similar way as we did for the variational inference. 16 ZHANG, ZHOU Theorem 4.2. Assume the initializer Z (0) satisfies `(Z (0) , Z ∗ ) ≤ cinit n̄min for some sufficiently small constant cinit with probability at least 1−. Under the same condition as in Theorem 3.1, there exist some constant c > 0 and some η, η 0 = on (1) that go to 0 slowly, such that for all s ≥ 0 of the batched Gibbs sampling (Algorithm 2), we have h i EZ (s+1) `(Z (s+1) , Z ∗ ) A, Z (0) ≤ n exp(−(1 − η)n̄min I) + csn `(Z (0) , Z ∗ ) + (s + 1)nbn 1 −c holds probability 1 − exp[−(n̄min I) p2 )] − n − , where bn =  with  at least 02 2 02 2 exp −η n̄min + exp −η n I and cn = 1/ nI/[wk[n/n̄min ]2 ]. Consequently, for s = [nI/k]/ log[nI/[wk[n/n̄min ]2 ]], we have h i EZ (s+1) `(Z (s+1) , Z ∗ ) A, Z (0) ≤ exp(−(1 − 2η)n̄min I), 1 with probability at least 1 − exp[−(n̄min I) 2 ] − n−c − . Theorem 4.2 establishes theoretical justification for batched Gibbs sampling for community detection. Despite that we have the same cn and similar convergence as Theorem 3.1, some extra efforts are needed due to the existence of randomness in each iterative update. The additional term of bn is necessary to handle the extreme events due to random generation. Note that (s + 1)nbn is dominated by n exp(−(1 − η)n̄min I) as long as s ≤ en . Thus, when s ≤ en , we have similar “linear convergence” results as in Theorem 3.1. 4.3. An Iterative Algorithm for Maximum Likelihood Estimation. Maximum likelihood estimator (MLE) usually yields statistical optimality. However, the maximization of the likelihood p(A|Z, p, q) over Z, p, q is computationally infeasible. Inspired by the procedures proposed in Algorithm 1 and Algorithm 2, we may approach max p(A|Z, p, q) by alternating maximization. We use a batched coordinate maximization: • Maximize p(A|p, q (s−1) , Z (s−1) ) over p to obtain p(s) ; • Maximize p(A|p(s−1) , q, Z (s−1) ) over q to obtain q (s) ; (s−1) (s) • Maximize p(A|p(s−1) , q (s−1) , Zi,· , Z−i,· ) over Zi,· to obtain Zi,· , for each i ∈ [n]. We include its detailed implementation in Algorithm 3 in the supplemental material (Section A.2). We have the following theoretical guarantee of this iterative algorithm to approximate the MLE. 17 MEAN FIELD FOR COMMUNITY DETECTION Theorem 4.3. Assume the initializer Z (0) satisfies `(Z (0) , Z ∗ ) ≤ cinit n̄min for some sufficiently small constant cinit with probability at least 1−. Under the same condition as in Theorem 3.1, there exist some constant c > 0 and some η = on (1), such that in each iteration of the BCAVI algorithm, `(Z (s) , Z ∗ ) `(Z (s+1) , Z ∗ ) ≤ n exp(−(1 − η)n̄min I) + p , ∀s ≥ 0, nI/[wk[n/n̄min ]2 ] 1 holds with probability at least 1 − exp[−(n̄min I) 2 ] − n−c − . Algorithm 3 is essentially the same with the procedure proposed in [12]. However, [12] can only analyze the performance of one single iteration from Z (0) (i.e., `(Z (1) , Z ∗ )), and it requires extra data splitting steps. Theorem 4.3 provides a stronger and cleaner result compared with that of [12]. 5. Proofs of Main Theorems. In this section, we give proofs of the theorems in Section 2 and Section 3. We first present the proof of Theorem 2.1 in Section 5.1. Then we give the proof Theorem 2.2 in Section 5.2. The proof of Theorem 3.1 is given in Section 5.3. 5.1. Proof of Theorem 2.1. From Equation (8), by some algebra (see Equation (53) in Appendix D for detailed derivation) we have (19) (π̂ MF , α̂pMF , β̂pMF , α̂qMF , β̂qMF ) = arg min π∈Π1 αp ,βp ,αq ,βq >0 Eq [log p(A|Z, p, q)] − KL(q(Z, p, q)kp(Z, p, q)), where we use q instead of qπ,αp ,βp ,αq ,βq for simplicity. From the conditional distribution in Equation (6), the log-likelihood function can be simplified as   XX Bab log p(A|Z, p, q) = Zia Zjb Ai,j log + log(1 − Bab ) . 1 − Bab a,b i<j Due to the independence of Z and p, q under q, we have    XX   Eq [log p(A|Z, p, q)] = Eq(p,q) Eq(Z) Zi,a Zj,b Ai,j log a,b i<j   XX = Eq(p,q)  πi,a πj,b Ai,j log a,b i<j Bab 1 − Bab   + log(1 − Bab )    Bab + log(1 − Bab )  . 1 − Bab 18 ZHANG, ZHOU Since Ba,a = p, ∀a ∈ [k] and Ba,b = q, ∀a 6= b, we have (20)     XX 1−p  p(1 − q) + log Eq [log p(A|Z, p, q)] = Eq(p,q)  πi,a πj,a Ai,j log q(1 − p) 1−q a i<j   i h XX q + Eq(p,q)  + log(1 − q)  . πi,a πj,b Ai,j log 1−q a,b i<j By properties of Beta distribution, we obtain Eq(p,q) log p(1 − q) = Eq(p) [log p − log(1 − p)] − Eq(q) [log q − log(1 − q)] q(1 − p) = [ψ(αp ) − ψ(βp )] − [ψ(αq ) − ψ(βq )] , and Eq(p,q) log 1−q = Eq(q) log(1 − q) − Eq(p) log(1 − p) 1−p = [ψ(βq ) − ψ(αq + βq )] − [ψ(βp ) − ψ(αp + βp )] . This leads to (21)       X X XX 1−p  p(1 − q) + log = 2t  Eq(p,q)  πi,a πj,a (Ai,j − λ) πi,a πj,a Ai,j log q(1 − p) 1 − q a a i<j i<j = thA − λ1n 1Tn + λIn , ππ T i. Similarly we can obtain (22)  h XX Eq(p,q)  πi,a πj,b Ai,j log  a,b i<j q 1−q  i + log(1 − q)  X X  XX q = Eq(q) log Ai,j πi,a πj,b + Eq(q) log(1 − q) πi,a πj,b 1−q i<j a,b 1 n = [ψ(αq ) − ψ(βq )] kAk1 + [ψ(βq ) − ψ(αq + βq )] , 2 2 i<j a,b 19 MEAN FIELD FOR COMMUNITY DETECTION where we use the fact that kπi,· k1 = 1, ∀i ∈ [n]. Now consider the KullbackLeibler divergence between q(Z, p, q) and p(Z, p, q). Due to the independence of p, q and {Zi,· }ni=1 in both distributions, we have (23) KL(q(Z, p, q)kp(Z, p, q)) = KL(q(Z)kp(Z)) + KL(q(p)kp(p)) + KL(q(q)kp(q)) n h i X pri = KL Categorical(πi,· )kCategorical(πi,· ) i=1     + KL Beta(αp , βp )kBeta(αppri , βppri ) + KL Beta(αq , βq )kBeta(αqpri , βqpri ) . By Equations (19) - (23), we conclude with the desired result. 5.2. Proof of Theorem 2.2. Note that   " k # X X Bzi ,zj = Zi,a Zj,a p +  Zi,a Zj,b  q. a=1 a6=b We rewrite the joint distribution p(p, q, z, A) in Equation (7) as follows, (24) p(p, q, Z, A)   " n # Pk Pk Y pri Y Y    Z Z Z Z i,a j,b i,a j,a   q Ai,j (1 − q)1−Ai,j a6=b = πi,zi  pAi,j (1 − p)1−Ai,j a=1 i=1 × " i<j Γ(αppri + βppri ) Γ(αppri )Γ(βppri ) p αpri p −1 (1 − p) βppri −1 #" i<j Γ(αqpri + βqpri ) Γ(αqpri )Γ(βqpri ) q αpri q −1 (1 − q) βqpri −1 # . Updates on p and q. From Equation (24), p has conditional probability as  " # P pri pri Y k  pri pri Γ(α + β ) Z Z p p i,a j,a  pαp −1 (1 − p)βp −1 . p(p|q, Z, A) ∝  pAi,j (1 − p)1−Ai,j a=1 pri pri Γ(α )Γ(β ) p p i<j Then the CAVI update in Equation (4) leads to   q̂(p) ∝ exp Eq(q,Z) log p(p|q, Z, A)  " # k pri pri XX  A  pri pri Γ(α + β ) p p ∝ exp Eq(Z) Zi,a Zj,a log p i,j (1 − p)1−Ai,j  pαp −1 (1 − p)βp −1 pri pri Γ(α )Γ(β ) p p i<j a=1  " # k pri pri XX  A  pri pri Γ(α + β ) p p = exp  πi,a πj,a log p i,j (1 − p)1−Ai,j  pαp −1 (1 − p)βp −1 . pri pri Γ(α )Γ(β ) p p i<j a=1 20 ZHANG, ZHOU It can be written as " # h P Pk i Γ(αpri + β pri ) pri P Pk pri p p q̂(p) ∝ p i<j a=1 πi,a πj,a Ai,j (1 − p) i<j a=1 πi,a πj,a (1−Ai,j ) pαp −1 (1 − p)βp −1 . Γ(αppri )Γ(βppri ) The distribution of p is still Beta p ∼ Beta(αp0 , βp0 ), with αp0 = αppri + k XX πi,a πj,a Ai,j , and βp0 i<j a=1 = βppri + k XX i<j a=1 πi,a πj,a (1 − Ai,j ). Similar analysis on q yields updates on αq0 and βq0 . Hence, its proof is omitted. Updates on {Zi,· }ni=1 . Zi,· is From Equation (24), the conditional distribution on pri p(Zi,· |Z−i,· , p, q, A) ∝ πi,z i   Y Ai,j  Bzi ,zj (1 − Bzi ,zj )1−Ai,j  . j6=i Consequently, up to a constant not depending on i, we have log P(Zi,a = 1|Z−i,· , p, q, A)   X pri = log πi,a + log  Zj,a Ai,j log   XX   q p + log(1 − p) + Zj,b Ai,j log + log(1 − q)  1−p 1−q j6=i b6=a j6=i       X X 1 − q q p(1 − q) pri = log πi,a − log + Ai,j log + log(1 − q)  . + log  Zj,a Ai,j log q(1 − p) 1−p 1−q j6=i j6=i Then the CAVI update from Equation (4) leads to 0 πi,a = q̂Zi,· (Zi,a = 1)   ∝ exp Eq(p,q,z−i ) log P(Zi,a = 1|Z−i,· , p, q, A) h i = exp Eq(p) Eq(q) Eq(Z−i,· ) log P(Zi, = 1|Z−i,· , p, q, A)     X 1 − q p(1 − q) pri , (25) − log ∝ πi,a exp Eq(p) Eq(q) πj,a Ai,j log q(1 − p) 1−p j6=i where we use the property that p, q, Z are all independent of each other under q. Recall that p ∼ Beta(αp , βp ) and q ∼ Beta(αq , βq ). It can be shown that p Eq(p) log = ψ(αp ) − ψ(βp ), and Eq(p) log(1 − p) = ψ(βp ) − ψ(αp + βp ), 1−p MEAN FIELD FOR COMMUNITY DETECTION 21 where ψ(·) is digamma function. Similar results hold for Eq(q) log(q/(1 − q)) and Eq(q) log(1 − q). Plug in these expectations to Equation (25), we have   X pri 0 πi,a ∝ πi,a exp 2t πj,a (Ai,j − λ) . j6=i 5.3. Proof of Theorem 3.1. Theorem 3.1 gives a theoretical justification for all iterations in the BCAVI algorithm. Due to the limit of pages, in this section we assume `(π (0) , Z ∗ ) = o(n̄min ). The proof of the case `(π (0) , Z ∗ ) in a constant order of n̄min is essentially the same with slight modification, and we defer it to Section B.1 in the supplemental material. To prove the theorem, it is sufficient if we are able to show the loss `(·, Z ∗ ) decreases in a desired way for one BCAVI iteration, when the community assignment is in an appropriate neighborhood of the truth. Let γ = o(1) be any sequence that goes to zero when n grows. Define t∗ and λ∗ as the true counterparts of t and λ, by t∗ = 1 p∗ (1 − q ∗ ) 1 1 − q∗ ∗ log ∗ , and λ = log . 2 q (1 − p∗ ) 2t∗ 1 − p∗ The proof of Theorem 3.1 involves three parts as follows. Part One: One Iteration. Consider any π ∈ Π1 such that kπ − Z ∗ k1 ≤ γ n̄min . Let η 0 be any sequence such that η 0 = o(1). Consider any t and λ with |t−t∗ | ≤ η 0 (p∗ −q ∗ )/p∗ and |λ−λ∗ | ≤ η 0 (p∗ −q ∗ ). We define F to be the event, that after applying the mapping ht,λ (·), there exists some η = o(1) such that kπ − Z ∗ k1 kht,λ (π) − Z ∗ k1 ≤ n exp(−(1 − η)n̄min I) + p , nI/[wk[n/n̄min ]2 ] holds uniformly over all the eligible π, t and λ. We have 1 P(F) ≥ 1 − exp[−(n̄min I) 2 )] − n−r , for some constant r > 0. We defer its proof to the later part of this section. Part Two: Consistency of Model Parameters. Consider any π ∈ Π1 such that kπ − Z ∗ k1 ≤ γ n̄min . Define (26) αp = αppri + k X X a=1 i<j Ai,j πi,a πj,a , βp = βppri + k X X (1 − Ai,j )πi,a πj,a , a=1 i<j 22 ZHANG, ZHOU and (27) αq = αqpri + XX Ai,j πi,a πj,b , a6=b i<j βq = βqpri + XX (1 − Ai,j )πi,a πj,b , a6=b i<j and consequently, (28) (29) 1 [[ψ(αp ) − ψ(βp )] − [ψ(αq ) − ψ(βq )]] 2 1 λ= [[ψ(βq ) − ψ(αq + βq )] − [ψ(βp ) − ψ(αp + βp )]] . 2t t= From Lemma C.1, we have a concentration of t, λ towards t∗ , λ∗ . That is, there exists some η 0 = o(1), such that with probability at least 1 − e3 5−n , the following inequalities hold |t − t∗ | ≤ η 0 (p∗ − q ∗ )/p∗ , and |λ − λ∗ | ≤ η 0 (p∗ − q ∗ ), uniformly over all the eligible π. Part Three: Multiple Iterations. Consider any π ∈ Π1 such that kπ − Z ∗ k1 ≤ γ n̄min . Define αp , βp , αq , βq , t, λ as Equations (26) - (29). A combination of results from Part One and Part Two immediately implies that (30) kπ − Z ∗ k1 , kht,λ (π) − Z ∗ k1 ≤ n exp(−(1 − η)n̄min I) + p nI/[wk[n/n̄min ]2 ] 1 holds uniformly over all the eligible π with probability at least 1−exp[−(n̄min I) 2 )]− n−r . This is sufficient to show Theorem 3.1. The only thing left to be proved, the most critical part towards the proof of Theorem 3.1, is the claim we made in Part One. We are going to prove the claim as follow. Proof Sketch of Part One. The error associated with the [ht,λ (π)]i,· is a function of π and Ai,· . It can be decomposed into a summation of two terms, one only involves the ground truth Z ∗ and the other involves the deviation π − Z ∗ . That is, ∗ [ht,λ (π)]i,· − Zi,· 1 ≤ fi,1 (Z ∗ , Ai,· ) + fi,2 (π − Z ∗ , Ai,· ). 23 MEAN FIELD FOR COMMUNITY DETECTION Consequently, (31) ∗ kht,λ (π) − Z k1 ≤ n X |i=1 ∗ fi,1 (Z , Ai,· ) + {z involves Z∗ } n X |i=1 fi,2 (π − Z ∗ , Ai,· ) . {z involves π−Z ∗ } With a proper choice of f·,1 and f·,2 , the first term on the RHS of Equation (31) leads to the minimax rate n exp(−(1 − η)n̄min I). Up to a constant not dependent on π, Z ∗ or A, the second term can be written as n X i=1 fi,2 (π − Z ∗ , Ai,· ) . X a ∗ T ∗ (π·,a − Z·,a ) (A − EA)(A − EA)T (π·,a − Z·,a ). In this way it is all about the random EA and P there exist∗ sharp P matrix A − ∗ 2 ≤ bounds on kA − EAkop . Note that a π·,a − Z·,a a π·,a − Z·,a 1 ≤ ∗ kπ − Z k1 . The second term ends up being upper bounded by kπ − π ∗ k1 multiplied by a coefficient factor. Proof of Part One. Denote z = r−1 (Z ∗ ). By the definition of ht,λ (·) in Equation (11), we have h P i P pri 2 a6=zi πi,a exp 2t j6=i πj,a (Ai,j − λ) ∗ i h P ≤ P [ht,λ (π)]i,· − Zi,· 1 pri π π (A − λ) exp 2t j,a i,j a i,a j6=i   X X ≤ 2w 1 ∧ exp 2t (πj,a − πj,zi )(Ai,j − λ) . a6=zi j6=i Define f (x) = 1 ∧ exp(−x). It can be shown P that for any x0 < 0 and any integer m ≥ 1 we have f (x) ≤ exp(x0 ) + m−1 l=0 exp(lx0 /m)I{x ≥ (l + 1)x0 /m}, which can be seen as a stepwise approximation of theP continuous function f (x). By taking x0 = −(na +nzi )I/2 and letting x = 2t j6=i (πj,a − πj,zi )(Ai,j − λ), we have "     m−1 X X (n + n )I l(na + nzi )I a zi ∗ [ht,λ (π)]i,· − Zi,· 1 ≤ 2w exp − + 2w exp − 2 2m a6=zi l=0 # X  X (l + 1)(na + nzi )I × I 2t (πj,a − πj,zi )(Ai,j − λ) ≥ − . 2m a6=zi j6=i We choose some m → ∞ slowly such that (32) m = o(n̄min I) and m = o([wnI/[k[n/n̄min ]2 ]1/4 ). 24 ZHANG, ZHOU Thus, we have ∗ kht,λ (π) − Z k1 ≤ 2wnk exp(−n̄min I) + 2w m−1 k X XX l=0 a=1 b6=a "  l(na + nb )I exp − 2m  # X X (l + 1)(na + nb )I × I (πj,a − πj,b )(Ai,j − λ) ≥ − 4mt (33) i:zi =b j6=i where we use the fact that mina6=b (na + bb )/2 ≥ n̄min . The key to the rest of the analysis is to understand Equation (33) through P the decomposition of the critical quantity j6=i (πj,a − πj,b )(Ai,j − λ). We will show for any pair of a, b ∈ [k] such that a 6= b, and any i ∈ [n] such that zi = b, it is equal to a summation of two terms: one only involves the ground truth Z ∗ , and the other involves the deviation π − Z ∗ . The former remains steady along iterations and contributes to the minimax rate, while the latter needs to be connected with the error kπ − Z ∗ k1 . ∗ + Z∗ − Let θa,b be a vector of length n such that [θa,b ]j = πj,a − Zj,a j,b πj,b , ∀j ∈ [n]. Then we have (34) X X X ∗ ∗ ∗ ∗ − πj,b )(Ai,j − λ) )(Ai,j − λ) + (πj,a − Zj,a + Zj,b (πj,a − πj,b )(Ai,j − λ) = (Zj,a − Zj,b j6=i j6=i j6=i = X j6=i = X j6=i | ∗ ∗ )(Ai,j (Zj,a − Zj,b ∗ (Zj,a − ∗ Zj,b )(Ai,j X − λ) + (Ai,j − λ)[θa,b ]j j6=i − λ) + (Ai,· − EAi,· )θa,b + {z } involves Z ∗ | With the help of Equation (34), Equation (33) can be written as kht,λ (π) − Z ∗ k1 ≤ 2wnk exp(−n̄min I) + 2w m−1 k X XX l=0 a=1 b6=a "   l(na + nb )I exp − 2m X (EAi,j − λ)[θa,b ]j . j6=i {z involves π−Z ∗ # X X X (l + 3/2)(n + n )I a b ∗ ∗ × I (Zj,a − Zj,b )(Ai,j − λ) ≥ − − (EAi,j − λ)[θa,b ]j 4mt i:zi =b j6=i j6=i "" m−1  # # k X X X X  l(na + nb )I n̄min I + 2w exp − × I (Ai,· − EAi,· )θa,b ≥ . 2m 4mt a=1 b6=a l=0 i:zi =b } MEAN FIELD FOR COMMUNITY DETECTION Equations (18) and (32) imply we have Pm−1 l=0 exp [−l(na + nb )I/(2m)] ≤ 2. Thus, kht,λ (π) − Z ∗ k1 ≤ 2wnk exp(−n̄min I) + 2wLsum + 4wLsum , | {z1 } | {z2 } involves Z ∗ where 25 Lsum 1 , m−1 k X XX l=0 a=1 b6=a  l(na + nb )I exp − 2m  X involves π−Z ∗ L1,i (a, b, l), i:zi =b P ∗ ∗ with L (a, b, l) , I[ 1,i j6=i (Zj,a −Zj,b )(Ai,j −λ) ≥ −(l+3/2)(na +nb )I/(4mt)− P j6=i (EAi,j − λ)[θa,b ]j ], and Lsum 2 , k X X X a=1 b6=a i:zi =b  I (Ai,· − EAi,· )θa,b  n̄min I ≥ . 4mt In this way we turn kht,λ (π) − Z ∗ k1 into calculations on Lsum and Lsum 1 2 , where the former only involves the ground truth Z ∗ and the latter only involves the deviation π − Z ∗ . and Lsum as follows. Their proofs We can obtain upper bounds on Lsum 1 2 are deferred to the end of this section. 00 • For Lsum 1 , there exists a sequence η = o(1) such that with probability 1 at least 1 − exp[−2(n̄min I) 2 ], we have   (35) Lsum ≤ nmk exp −(1 − 2η 00 )n̄min I . 1 • For Lsum 2 , there exist constants c and r such that with probability at least 1 − n−r − exp(−5np∗ ), we have (36) Lsum ≤ 2 cknp∗ kπ − Z ∗ k1 cn2 kp∗ exp(−5np∗ ) + . (n̄min I/(mt∗ ))2 n̄min I/(mt∗ ) Thus, we have   kht,λ (π) − Z ∗ k1 ≤ 2wnk exp(−n̄min I) + 2wnmk exp −(1 − 2η 00 )n̄min I + 4cwknp∗ kπ − Z ∗ k1 4cwkn2 p∗ exp(−5np∗ ) , + (n̄min I/(mt∗ ))2 n̄min I/(mt∗ ) 1 with probability at least 1−exp[−2(n̄min I) 2 ]−n−r −exp(−5np∗ ). By Propositions C.2 and C.3, we have p∗ t∗2  I. Then due to Equation (32), we have " #   wknp∗ 1 n 2 k 2  wm =o p , (n̄min I/(mt∗ ))2 n̄min nI nI/[wk[n/n̄min ]2 ] 26 ZHANG, ZHOU and  √ ∗ n wkn2 p∗ exp(−5np∗ ) np n exp(−5np∗ ) ≤ n exp(−5n̄min I).  wmk √ n̄min I/(mt∗ ) n̄ nI min 1 Thus, with probability at least 1 − exp[−(n̄min I) 2 ] − n−r , there exists some η = o(1), such that kπ − Z ∗ k1 kht,λ (π) − Z ∗ k1 ≤ n exp(−(1 − η)n̄min I) + p . nI/[wk[n/n̄min ]2 ] The proof for Part One is complete. The very last thing remained to be obtained is upper bounds on Lsum and Lsum 1 2 , i.e., Equations (35) and (36). Recall the definition of θa,b . We have some properties on θa,b which will be useful in the analysis for Lsum and Lsum 1 2 : kθa,b k∞ ≤ 2 and (37) ∗ kθa,b k1 ≤ π·,a − Z·,a 1 ∗ + π·,b − Z·,b 1 ≤ kπ − Z ∗ k1 ≤ γ n̄min , 1 ≤ 2k kπ − Z ∗ k1 . and (38) k X X a=1 b6=a kθa,b k1 ≤ 2k 1. Bounds on Lsum 1 . X a ∗ π·,a − Z·,a By applying Markov inequality, we have EL1,i (a, b, l)   ∗ X X t (l + 3/2)(na + nb )I ∗ ∗ − t∗ (EAi,j − λ)[θa,b ]j  = P t∗ (Zj,a − Zj,b )(Ai,j − λ) ≥ − 4mt j6=i j6=i    ∗  X t (l + 3/2)(na + nb )I ∗ ∗ ≤ exp + t∗ (EAi,j − λ1Tn )θa,b E exp t∗ (Zj,a − Zj,b )(Ai,j − λ) . 4mt j6=i With the help of Proposition C.1, we have   X ∗ ∗ )(Ai,j − λ) E exp t∗ (Zj,a − Zj,b j6=i = exp(−t∗ (λ − λ∗ )(na − nb )) exp(−t∗ λ∗ (na − nb )) Y j6=i ∗ ∗ E exp(t∗ (Zj,a − Zj,b )Ai,j ) b   na −n tX 2 b  tX −tY  na +n Ee ∗ ∗ −tλ 2 = exp(−t (λ − λ )(na − nb )) e Ee Ee Ee−tY   (na + nb )I = exp(−t∗ (λ − λ∗ )(na − nb )) exp − . 2 27 MEAN FIELD FOR COMMUNITY DETECTION Hence (39) ELsum 1 m−1 k X XX "     X t∗ (l + 3/2)(na + nb )I + t∗ (EAi,j − λ)[θa,b ]j  4mt l=0 a=1 b6=a j6=i  # (na + nb )I × exp(−t∗ (λ − λ∗ )(na − nb )) exp − 2   m−1 k X t∗ (l+3/2) l XX X (1 + m − 2mt )(na + nb )I ≤ exp − (EAi,j − λ)[θa,b ]j  . − t∗ (λ − λ∗ )(na − nb ) + t∗ 2 = exp − l(na + nb )I exp  2m l=0 a=1 b6=a j6=i We are going to show −(1−η 00 )n̄min I upper bounds terms in the exponent of RHS of Equation (39) by some η 00 = o(1). We first present some properties of λ∗ , t∗ and I that will be helpful: (40) (41) (42) I  (p∗ − q ∗ )2 /p∗ , λ∗ ∈ (q ∗ , p∗ ), and t∗  (p∗ − q ∗ )/p∗ . Here Equations (40) and (41) are proved by Propositions C.2 and C.3 respectively. Equation (42) is due to t∗  log(1 + (p∗ − q ∗ )/q ∗ )  (p∗ − q ∗ )/p∗ under the assumption that p∗ , q ∗ = o(1), p∗  q ∗ . The first term in the exponent of Equation (39) is upper bounded by −(1 − 7/(8m))n̄min I by the assumption t∗ /t = 1 + o(1). Since |t∗ (λ − λ∗ )| ≤ η 0 t∗ (p∗ − q ∗ ), by Equations (40) and (42) the second term is upper bounded by η 0 n̄min I up to a constant factor. For the last term in the exponent of Equation (39), since |λ − λ∗ | ≤ η 0 (p∗ − q ∗ ) we have t∗ X X X (λ∗ − λ)[θa,b ]i (EAi,j − λ)[θa,b ]i ≤ t∗ (EAi,j − λ∗ )[θa,b ]i + t∗ j6=i j6=i j6=i 0 ∗ ∗ ∗ ≤ (1 + η )t (p − q ) kθa,b k1 ≤ (1 + η 0 )t∗ (p∗ − q ∗ )γ n̄min . γ n̄min I, where we use Equations (37) and (40) - (42). As a consequence, there exists a sequence η 00 = o(1) that goes to zero slower than m−1 , γ, η 0 , such that the summation of three terms in the exponent of the RHS of Equation (39) is upper bounded by −(1 − η 00 )n̄min I. 28 ZHANG, ZHOU Thus, Equation (39) can be written as   ELsum ≤ nmk exp −(1 − η 00 )n̄min I . 1 1 Since η 00 goes to 0 slower than m−1 , we have η 00 ≥ m−1 ≥ (n̄min I) 4 by Equation (32). Then by applying Markov inequality, we have h i      1 P Lsum ≥ nmk exp −(1 − 2η 00 )n̄min I ≤ exp −η 00 n̄min I ≤ exp −2(n̄min I) 2 . 1 1 That is, with probability at least 1 − exp[−2(n̄min I) 2 ], Equation (35) holds. 2. Bounds on Lsum 2 . Depending on whether the network is dense or sparse, we consider two scenarios. (1) Dense Scenario: q ∗ ≥ (log n)/n. In this scenario, we have a sharp bound on kA − EAkop . First we observe that X X T [(Ai,· − EAi,· )θa,b ]2 = θa,b [(Ai,· − EAi,· )T (Ai,· − EAi,· )]θa,b i:zi =b i:zi =b ≤ = T θa,b X i T θa,b [(A [(Ai,· − EAi,· )T (Ai,· − EAi,· )]θa,b − EA)T (A − EA)]θa,b . By applying Markov inequality, we have Lsum ≤ 2 k X T X θa,b [(A − EA)T (A − EA)]θa,b a=1 b6=a (n̄min I/(4mt))2 . Since kθa,b k∞ ≤ 2, we have kθa,b k2 ≤ 2 kθa,b k1 . Lemma C.3 shows kA − √ EAkop ≤ c1 np holds with probability at least 1 − n−r for some constants c1 , r > 0. Together with Equation (38), we have k X X a=1 b6=a T θa,b [(A T − EA) (A − EA)]θa,b ≤ ≤ k X X a=1 b6=a k X X a=1 b6=a kA − EAk2op kθa,b k2 2c1 np kθa,b k1 ≤ 4c1 knp kπ − Z ∗ k1 . Thus, with probability at least 1 − n−r , Lsum ≤ 2 4c1 knp kπ − Z ∗ k1 . (n̄min I/(4mt))2 29 MEAN FIELD FOR COMMUNITY DETECTION (2) Sparse Scenario: q ∗ < (log n)/n. When the network is sparse, the previous upper bound on kA − EAkop no longer holds. Instead, removing nodes with large degrees is required to yield provably sharp bound on kA − P EAkop . Define S = {i ∈ [n], j Ai,j ≥ 20np∗ }. We define Ã, P̃ such that Ãi,j = Ai,j I{i, j ∈ / S} and P̃i,j = (EAi,j )I{i, j ∈ / S}. Then we have the decomposition as  X  n̄min I L2 (a, b) , I (Ai,· − EAi,· )θa,b ≥ 4mt i:zi =b   X n̄min I ≤ I (Ãi,· − P̃i,· )θa,b ≥ 8mt i:zi =b   X X n̄min I  + I  (Ai,j − EAi,j )[θa,b ]i,j I{i ∈ S or j ∈ S} ≥ 8mt i:zi =b j6=i , L2,1 (a, b) + L2,2 (a, b). Pk P Define Lsum 2,1 , a=1 b6=a L2,1 (a, b). We have Lsum 2,1 ≤ k X T X θa,b [(à − P̃ )T (à − P̃ )]θa,b (n̄min I/(8mt))2 a=1 b6=a ≤ k X X 2kà − P̃ k2op kθa,b k a=1 b6=a (n̄min I/(8mt))2 1 . √ Lemma C.4 shows kÃ− P̃ kop ≤ c2 np holds with probability at least 1−n−1 for some constant c2 > 0. Then we have Lsum 2,1 ≤ 4c2 knp kπ − Z ∗ k1 . (n̄min I/(8mt))2 P Lemma C.5 shows i,j |Ai,j − EAi,j |I{i ∈ S} ≤ 20n2 p∗ exp(−5np∗ ) holds with probability at least 1 − exp(−5np∗ ). Then by applying Markov inequality, we have   k X X  Lsum L2,2 (a, b) 2,2 , a=1 ≤ ≤ ≤ k X b6=a n X a=1 i,j=1 k X 4 a=1 |Ai,j − EAi,j ||[θa,b ]i,j |I{i ∈ S or j ∈ S} n̄min I/(8mt) P i,j |Ai,j − EAi,j |I{i ∈ S} n̄min I/(8mt) 80n2 kp∗ exp(−5np∗ ) . n̄min I/(8mt) 30 ZHANG, ZHOU As a consequence, we have sum Lsum ≤ Lsum 2 2,1 + L2,2 ≤ 4c2 knp∗ kπ − Z ∗ k1 80n2 kp∗ exp(−5np∗ ) + , (n̄min I/(8mt))2 n̄min I/(8mt) with probability at least 1 − n−1 − exp(−5np∗ ). By the bounds on Lsum and 1 ∗ = 1 + o(1), we obtain Equation (36). Lsum , and due to t/t 2 SUPPLEMENTARY MATERIAL Supplement A: Supplement to “Theoretical and Computational Guarantees of Mean Field Variational Inference for Community Detection” (url to be specified). In the supplement [36], we provide the detailed implementations of the batched Gibbs sampling and an iterative algorithm for MLE in Algorithm 2 and Algorithm 3 respectively. We include proof of Theorem 4.1, Theorem 4.2 and Theorem 4.3. 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Output: Gibbs sampling Ẑ, p̂, q̂. for s = 1, 2, . . . , S do (s) (s) (s) (s) Update αp , βp , αq , βq by αp(s) = αppri + k X X (s−1) , βp(s) = βppri + k X X (s−1) (s−1) (1 − Ai,j )Zi,a Zj,a , (s−1) , βq(s) = βqpri + XX (s−1) (s−1) (1 − Ai,j )Zi,a Zj,b . (s−1) Zj,a (s−1) Zj,b Ai,j Zi,a a=1 i<j a=1 i<j αq(s) = αqpri + XX Ai,j Zi,a a6=b i<j a6=b i<j (s) 2 (s) (s) (s) Then generate p(s) ∼ Beta(αp , βp ) and q (s) ∼ Beta(αq , βq ) independently. Define t(s) = p(s) (1 − q (s) ) 1 log , 2 (1 − p(s) )q (s) and λ(s) = 1 1 − q (s) log . (s) 2t 1 − p(s) Then update π (s) with π (s) = ht(s) ,λ(s) (Z (s−1) ), where ht,λ (·) is defined as in Equation (11). Independently generate each row of Z (s) from distributions (s) (s) P(Zi,· = ea ) = πi,a , ∀a ∈ [k], ∀i ∈ [n]. 3 end We have ẑ = z (S) , p̂ = p(S) and q̂ = q (S) . 2 ZHANG, ZHOU A.2. An Iterative Algorithm for Maximum Likelihood Estimation. We first define a mapping h0 : Π0 → Π0 as follows   X [h0λ (Z)]i,a = I a = arg max Zi,b (Ai,j − λ) . (43) b j6=i Here if the maximizer is not unique, we simply pick the smallest index. Algorithm 3: An Iterative Algorithm for MLE 1 Input: Adjacency matrix A, number of communities k, some initializers z (0) , number of iterations S. Output: Estimation Ẑ, p̂, q̂. for s = 1, 2, . . . , S do Update p(s) , q (s) by Pk p (s) P a=1 = Pk a=1 (s−1) Zj,a (s−1) (s−1) Ai,j )Zi,a Zj,a (s−1) i<j P Ai,j Zi,a i<j (1 − P Ai,j Zi,a and P q (s) = P a6=b a6=b 2 (s−1) i<j P i<j (1 (s−1) Zj,b (s−1) − Ai,j )Zi,a (s−1) . Zj,b Define t(s) = p(s) (1 − q (s) ) 1 log , 2 (1 − p(s) )q (s) and λ(s) = 1 1 − q (s) log . (s) 2t 1 − p(s) Then update π (s) with Z (s) = h0λ(s) (Z (s−1) ), where h0λ (·) is defined as in Equation (43). 3 end We have ẑ = z (S) , p̂ = p(S) and q̂ = q (S) . APPENDIX B: PROOFS OF OTHER THEOREMS In this section, we first validate Theorem 3.1 when `(π (0) , π ∗ ) is in a constant order of n̄min , which complements the proof presented in Section 5.3. The we give proofs of theorems stated in Section 4, including Theorem 4.1, Theorem 4.2 and Theorem 4.3. B.1. Proof of Theorem 3.1 for the case `(π (0) , π ∗ ) in a constant order of n̄min . For any π such that `(π, π ∗ ) ≤ cinit n̄min , we are going to 3 MEAN FIELD FOR COMMUNITY DETECTION show when cinit is sufficiently small (44) `(π, Z ∗ ) `(ht,λ (π), Z ∗ ) ≤ n exp(−n̄min I/25) + p , 2 nI/[wk[n/n̄min ]2 ] with probability at least 1−exp(−n̄min I/10)−n−r for some constant r > 0. If it holds, for any π (0) such that `(π (0) , Z ∗ ) = cn̄min for some pconstant c ≤ cinit , (0) ∗ the term n exp(−n̄min I/25) is dominated by `(π , Z )/ nI/[wk[n/n̄min ]2 ] which implies `(π (1) , Z ∗ ) ≤ n exp(−(1 − η)/n̄min I) + p `(π (0) , Z ∗ ) nI/[wk[n/n̄min ]2 ] . It also implies `(π (1) , Z ∗ ) = o(n̄min ), which means after the first iteration, the results in Section 5.3 can be directly applied and the proof is complete. The proof of Equation (44) mainly follows the proof of Part One in Section 5.3. We have   X X ∗ [ht,λ (π)]i,· − Zi,· ≤ 2w 1 ∧ exp 2t (πj,a − πj,zi )(Ai,j − λ) . 1 a6=zi j6=i Note that the inequality 1 ∧ exp(−x) ≤ f (x0 ) + I{x ≥ x0 } holds for any x0 ≥ 0. By taking x0 = (na + nzi )I/4, we have      X X (n + n )I (n + n )I zi a zi ∗ exp − a  , ≤ 2w + I  (πj,a − πj,zi )(Ai,j − λ) ≥ − [ht,λ (π)]i,· − Zi,· 1 4 8t a6=zi j6=i and consequently, kht,λ (π) − Z ∗ k1 ≤ 2wnk exp(−n̄min I/2) # k X X X X (na + nzi )I + 2w . I (πj,a − πj,b )(Ai,j − λ) ≥ − 8t a=1 b6=a i:zi =b j6=i Define θa,b the same way as in Section 5.3, and by the same argument, we have  k X X  X n̄min I ∗ kht,λ (π) − Z k1 ≤ 2wnk exp(−n̄min I/2) + 2w I (Ai,· − EAi,· )θa,b ≥ 8t a=1 b6=a i:zi =b  k X X X X (na + nb )I X ∗ ∗ + 2w I (Zj,a − Zj,b )(Ai,j − λ) ≥ − − (EAi,j − λ)[θa,b ]j . 4t a=1 b6=a i:zi =b j6=i j6=i 4 ZHANG, ZHOU From Lemma C.1, when cinit is sufficiently small, with probability at least 1 − e3 5−n we have   |λ − λ∗ | |t − t∗ | (45) , ≤ 24c0 cinit . max (p∗ − q ∗ )/p∗ (p∗ − q ∗ ) Proposition C.3 shows that λ∗ ∈ (q ∗ + c(p∗ − q ∗ ), q ∗ + (1 − c)(p∗ − q ∗ )) for some positive constant 0 < c < 1/2. Therefore, when cinit is sufficiently small, we have λ ∈ (q ∗ , p∗ ). Thus, X j6=i (EAi,j − λ)[θa,b ]j ≤ (p∗ − q ∗ ) kθa,b k1 ≤ (p∗ − q ∗ ) kπ − Z ∗ k1 ≤ cinit (p∗ − q ∗ )n̄min , where we use Equation (37). By Equations (40) - (42), it is smaller than (na + nzi )/(8t) when cinit is sufficiently small. As a consequence, we have ∗ kht,λ (π) − Z k1 ≤ 2wnk exp(−n̄min I/2) + 2w + 2w k X X a=1 b6=a i:zi =b  I (Ai,· − EAi,· )θa,b n̄min I ≥ 8t  X X (na + nb )I ∗ ∗ )(Ai,j − λ) ≥ − (Zj,a − Zj,b I . 8t a=1 b6=a i:zi =b Define Lsum 1 sum and L2 = k X X X Pk = Pk j6=i a=1 P P b6=a P P i:zi =b I hP b6=a i:zi =b I [(Ai,· ∗ j6=i (Zj,a − ∗ )(A Zj,b i,j i − λ) ≥ −(na + nb )I/(8t) − EAi,· )θa,b ≥ n̄min I/(8t)]. Our analysis on them is quite similar to that in Section 5.3. By Markov inequality,   k X X X X ∗ ∗ )(Ai,j − λ) ≥ −t∗ (na + nb )I/(8t) (Zj,a − Zj,b P t∗ ELsum = 1 a=1 a=1 b6=a i:zi =b ≤ ≤ k X X X a=1 b6=a i:zi =b k X X  X a=1 b6=a i:zi =b j6=i exp   t∗ (n    X a + nb )I ∗ ∗ − t∗ (λ − λ∗ )(na − nb ) E exp t∗ (Zj,a − Zj,b )(Ai,j − λ∗ ) 8t j6=i  t∗ (na + nb )I (na + nb )I ∗ ∗ − t (λ − λ )(na − nb ) − . exp 8t 2 By Equations (40) - (42) and (45), when cinit is small enough, t∗ /t ≤ 2 and t∗ |λ − λ∗ | ≤ I/6. Thus ELsum ≤ nk exp(−n̄min I/12). 1 5 MEAN FIELD FOR COMMUNITY DETECTION Hence, with probability at least 1 − exp(−n̄min I/24), Lsum ≤ nk exp(−n̄min I/24). 1 For Lsum we use the same argument as in Section 5.3 and obtain 2 Lsum ≤ 2 4c2 knp∗ kπ − Z ∗ k1 80n2 kp∗ exp(−5np∗ ) + , (n̄min I/(8t))2 n̄min I/(8t) with probability at least 1−n−r −exp(−5np∗ ) for some constants r, c1 , c2 > 0. Recall that kht,λ (π) − Z ∗ k1 ≤ 2wnk exp(−n̄min I/2) + 2wLsum + 2wLsum 1 2 . Using the same argument as in Section 5.3, we conclude with 1 kπ − Z ∗ k1 , kht,λ (π) − Z ∗ k1 ≤ n exp(−n̄min I/25) + p 2 nI/[wk[n/n̄min ]2 ] with probability at least 1 − exp(−n̄min I/10) − n−r . B.2. Proof of Theorem 4.1. Define t∗ = 1 2t∗ ∗ 1 2 ∗ ∗ (1−q ) ∗ = log qp∗ (1−p ∗ ) and λ 1−q log 1−p ∗ . By the same simplification we derive in Theorem 2.1, we have π̂ MF = arg max f 0 (π; A), π∈Π1 where f 0 (π; A) = hA + λ∗ In − λ∗ 1n 1Tn , ππ T i − n 1 X pri KL(Categorical(πi,· )kCategorical(πi,· )). t∗ i=1 Recall the definition of ht,λ (·) as in Equation (11). A key observation is that π̂ MF = ht∗ ,λ∗ (π̂ MF ), otherwise if there exists some i ∈ [n] such that MF . This indicates the implementation of CAVI [ht∗ ,λ∗ (π̂ MF )]i,· not equal to π̂i,· update on the i-th row of π will make change, leading to the decrease of f 0 (·; A). This contradicts with the fact that π̂ MF is the global minimizer. The fixed-point property of π̂ MF is the key to our analysis. It involves three steps. • Step One. For any π such that `(π, Z ∗ ) = o(n̄min ), by the same analysis as in the proof of Theorem 3.1, we are able to show that there exist constant r > 0 and sequence η = o(1) such that kht∗ ,λ∗ (π) − Z ∗ k1 ≤ n exp(−(1 − η)n̄min I) + p 1 kπ − Z ∗ k1 nI/[wk[n/n̄min ]2 ] with probability at least 1 − exp[−(n̄min I) 2 ] − n−r . , 6 ZHANG, ZHOU • Step Two. Lemma C.6 presents some loose upper bound for `(π̂ MF , Z ∗ ). That is, under the assumption ρnI/[wk 2 [n/n̄min ]2 ] → ∞, with probability at least 1 − e3 5−n , we have `(π̂ MF , Z ∗ ) ≤ o(n̄min ). • Step Three. Using the property that ht∗ ,λ∗ (π̂ MF ) = π̂ MF , we have π̂ MF − Z ∗ 1 ≤ n exp(−(1 − η)n̄min I) + p π̂ MF − Z ∗ 1 nI/[wk[n/n̄min ]2 ] 1 holds with probability at least 1 − exp[−(n̄min I) 2 ] − n−r . Then we obtain the desired result by simple algebra. B.3. Proof of Theorem 4.2. By law of total expectation, we have (46) EZ (s+1) h Z (s+1) −Z ∗ 1 A, Z (0) i   h i (0) (s+1) ∗ (s+1) (0) = Eπ(s+1) EZ (s+1) Z A, Z −Z π , A, Z 1 h i = Eπ(s+1) π (s+1) − Z ∗ A, Z (0) , 1 where the first equation is due to that the conditional expectation of Z (s+1) is π (s+1) . We are going to build the connection between π (s) and π (s+1) . In Algorithm 2, there are intermediate steps between π (s) and π (s+1) as follows: π (s) Z (s) (p(s+1) , q (s+1) ) → (t(s+1) , λ(s+1) ) → π (s+1) , where we use the plain right arrow (→) to indicate deterministic generation and the curved right arrow ( ) to indicate random generation. Despite a slight abuse of notation, we define π (0) = Z (0) . Analogous to the proof of Theorem 3.1 in Section 5.3, we assume `(Z (0) , Z ∗ ) = o(n̄min ). The proof for the case `(Z (0) , Z ∗ ) in the same order of n̄min is similar and thus is omitted. Let γ = o(1) be any sequence goes to 0 when n grows. We define a series of events as follows: • global event F: We define F exactly the same way as we define in the proof of Theorem 3.1 in Section 5.3 with respect to sequences γ and 1 η 0 , and we have P(F) ≥ 1 − exp[−(n̄min I) 2 )] − n−r for some constant r > 0. We have η 0 = o(1) whose value will be determined later. 7 MEAN FIELD FOR COMMUNITY DETECTION • global event G: Consider any Z ∈ Π1 such that kZ − Z ∗ k1 ≤ γ n̄min . Define αp = αppri + k X X Ai,j Zi,a Zj,a , βp = βppri + a=1 i<j αq = αqpri + XX k X X (1 − Ai,j )Zi,a Zj,a , a=1 i<j Ai,j Zi,a Zj,b , βq = βqpri + a6=b i<j XX a6=b i<j (1 − Ai,j )Zi,a Zj,b . Define G be the event that   αq αp ∗ ∗ −p , −q ≤ η 00 (p∗ − q ∗ ) max αp + βp αq + β q holds uniformly over all the eligible Z for some sequence η 00 = o(1). Then by the same analysis as in Lemma C.1, we have P(G) ≥ 1−e3 5−n . (s) (s) • local events {H1 }Ss=1 : We define H1 = { π (s) − Z ∗ 1 ≥ γ n̄min /2}. (s) (s) • local events {H2 }Ss=1 : We define H2 = { Z (s) − Z ∗ the conditional probability, we have (s) ≤P ≥ γ n̄min }. For (s) P(H2 = 1|H1 = 0) " n X h (s) ∗ ≤P Zi,· − Zi,· " 1 i=1 n h X (s) Zi,· i=1 Since π (s) − Z ∗ we have (s) P(H2 = − ∗ Zi,· 1 − ∗ − Zi,· − (s) πi,· ∗ Zi,· − 1 1 i i ≥ γ n̄min − π (s) − Z ∗ ≥ γ n̄min /2 (s) H1 # 1 (s) H1 =0 ≤ γ n̄min /2 given H1 = 0 by Bernstein inequality, " (γ n̄min )2 /8 = 0) ≤ exp − (s) π − Z ∗ 1 + γ n̄min /6   ≤ exp −3(γ n̄min )2 /16 . (s) # =0 (s) 1 (s) 1|H1 (s) 1 (s) πi,· # • local events {H3 }Ss=1 : We define H3 = {|t(s) −t∗ | ≥ η 0 (p∗ −q ∗ )/p∗ , or |λ(s) − (s) λ∗ | ≥ η 0 (p∗ − q ∗ )}. If the global event G holds and the local event H2 does not hold, we have ( ) (s+1) (s+1) αp αq ∗ ∗ max − p , (s+1) −q ≤ η 00 (p∗ − q ∗ ). (s+1) (s+1) (s+1) αp + βp αq + βq 8 ZHANG, ZHOU P P (s) (s) (s+1) (s+1) Note that αp + βp = αppri + βppri + ka=1 i<j Zi,a Zj,a ≥ n2 /k. Using the tail bound of Beta distribution (Lemma C.7) we are able to show " # (s+1) α p (s) P p(s+1) − (s+1) ≥ η 00 (p∗ − q ∗ ) H2 = 0, G = 1 (s+1) αp + βp   ∗ ∗ 2 002 2 (p − q ) ≤ exp −η n 2p∗  002 2  ≤ exp −η n I/2 , where the last inequality is due to Proposition C.2. This leads to h i   (s) P p(s+1) − p∗ ≥ 2η 00 (p∗ − q ∗ ) H2 = 0, G = 1 ≤ exp −η 002 n2 I/2 . And similar result holds for q (s+1) . Then by the same analysis as in the proof of Lemma C.1, max{|p(s+1) − p∗ |, |q (s+1) − q ∗ |} ≤ 2η 00 (p∗ − q ∗ ) leads to ( ) ∗ |t(s+1) − t∗ | |λ(s+1)−λ | max , ≤ 16c0 η 00 . (p∗ − q ∗ )/p∗ p∗ − q ∗ By taking η 0 = 16c0 η 00 , we obtain (s+1) P(H3   (s) = 1|H2 = 0, G = 1) ≤ 2 exp −η 002 n2 I/2 . Note that events F and G are about the adjacency matrix A. The events (s+1) and H3 are for π (x) , Z (s) and (p(s+1) , q (s+1) ) respectively. With all the above events defined, we can continue our analysis for Equation (46). (s) (s) (s+1) C Under the event F ∩ G ∩ (H1 ∪ H2 ∪ H3 ) we have (s) (s) H1 , H2 π (s+1) − Z ∗ (47) 1 ≤ n exp(−(1 − η)n̄min I) + cn π (s) − Z ∗ 1 , where cn = [nI/[wk[n/n̄min ]2 ]]−1/2 . As a consequence, under the event F ∩ Q (v) (v) (v+1) C ) , we have G ∩ ( sv=0 H1 ∪ H2 ∪ H3 π (s+1) − Z ∗ 1 ≤ n exp(−(1 − 2η)n̄min I) + csn π (0) − Z ∗ 1 . Therefore, we have (48) Eπ(s+1) h i (0) H1 = 0, F = 1, G = 1 ≤ n exp(−(1 − 2η)n̄min I) 1 # " s Y (v) (0) (v) (v+1) H1 = 0, F = 1, G = 1 . + nP H1 ∪ H 2 ∪ H 3 π (s+1) − Z ∗ + csn π (0) − Z ∗ 1 v=1 MEAN FIELD FOR COMMUNITY DETECTION 9 Due to the small value of cn , if π (s) − Z ∗ 1 ≤ γ n̄min , Equation (47) immediately implies π (s+1) − Z ∗ 1 ≤ γ n̄min . This implies that under the event F ∪ G we have (s+1) H1 (s) (s) (s+1) ⊂ H1 ∪ H2 ∪ H3 , ∀s ≥ 0, and consequently, s Y v=0 (v) (v) (v+1) H1 ∪ H 2 ∪ H 3 (0) ⊂ H1 s Y v=0 (v) (v+1) H2 ∪ H3 , ∀s ≥ 1. Thus, (49) P " s Y v=0 ≤P ≤ " (v) H1 s Y v=0 s X ∪ (v) H2 (v) H2 ∪ (v) ∪ (v+1) H3 (v+1) H3 (v) (0) H1 (0) H1 = 0, F = 1, G = 1 # = 0, F = 1, G = 1 P(H2 = 1|H1 = 0) + v=0 n X (v+1) P(H3 v=0 (v) = 1|H2 = 0, G = 1)     ≤ (s + 1) exp −3(γ n̄min )2 /16 + 2 exp −η 002 n2 I/2 . (0)  # 1 Note that P(H1 = 0, F = 1, G = 1) ≥ 1−exp[−(n̄min I) 2 )]−n−r −e3 5−n −. Recall we define π (0) = Z (0) . By Equations (46), (48) and (49), we have h i EZ (s+1) Z (s+1) − Z ∗ A, Z (0) ≤ n exp(−(1 − 2η)n̄min I) + csn Z (0) − Z ∗ + (s + 1)nbn , 1 1 1 −r 3 −n 2 with probability at least 1 − exp[−(n̄   002 2min I)  )] − n − e 5 − , where bn = 2 exp −3(γ n̄min ) /16 + 2 exp −η n I/2 . B.4. Proof of Theorem 4.3. Note the similarity between Algorithm 3 and Algorithm 1. We can prove Theorem 4.3 with almost the identical argument used in the proof of Theorem 3.1, thus omitted. APPENDIX C: STATEMENTS AND PROOFS OF AUXILIARY LEMMAS AND PROPOSITIONS We include all the auxiliary propositions and lemmas in this section. 10 ZHANG, ZHOU C.1. Statements and Proofs of Lemmas and Propositions for Theorem 3.1. Lemma C.1. Let cinit be some sufficiently small constant. Consider any π ∈ Π1 such that kπ − Z ∗ k1 ≤ cinit n/k. Let αp , βp , αq , βq , t, λ be the outputs after one step CAVI iteration from π described in Algorithm 1. That is, they are defined as Equations (26) - (29). Define P P P Pk i<j a6=b πi,a πj,b Ai,j i<j a=1 πi,a πj,a Ai,j , and q̂ = P P p̂ = P Pk . i<j a6=b πi,a πj,b a=1 πi,a πj,a i<j Under the same assumption as in Theorem 3.1, there exists some sequence  = o(1) such that with probability at least 1 − e3 5−n , the following inequality holds   kπ − Z ∗ k1 |p̂ − p∗ | |q̂ − q ∗ | |t − t∗ | |λ − λ∗ | max , , , ≤  + 24c , 0 p∗ − q ∗ p∗ − q ∗ (p∗ − q ∗ )/p∗ p∗ − q ∗ n/k uniformly over all the eligible π. In addition if we further assume cinit goes to 0, the LHS of the above inequality will be simply upper bounded by . Proof. We are going to obtain tight bounds on |p̂ − p∗ | and |q̂ − q ∗ | first. Note that we have the “variance-bias” decomposition as in P Pk P P | i<j ka=1 πi,a πj,a (Ai,j − EAi,j )| i<j a=1 πi,a πj,a EAi,j ∗ + − p∗ . |p̂ − p | ≤ P Pk P Pk a=1 πi,a πj,a a=1 πi,a πj,a i<j i<j We have concentration inequality holds for the numerator in the first term by Lemma C.2. That is, with probability at least 1 − e3 5−n , we have k XX i<j a=1 πi,a πj,a (Ai,j − EAi,j ) = p 1 hA − EA, ππ T i ≤ 3n np∗ 2 holds uniformly over all π ∈ Π1 . For the denominator, we have k k n2 X X 1X n2 ≥ πi,a πj,a = kπ·,a k21 ≥ , 2 2 2k i<j a=1 a=1 P since ka=1 kπ·,a k1 = n. Thus, we are able to obtain an upper bound on the first term as r P P | i<j ka=1 πi,a πj,a (Ai,j − EAi,j )| k 2 p∗ . ≤6 P Pk n π π i,a j,a i<j a=1 11 MEAN FIELD FOR COMMUNITY DETECTION For the second term, since EAi,j = p∗ we have P Pk a=1 πi,a πj,a EAi,j Pk i<j a=1 πi,a πj,a i<j P Pk ∗ ∗ ∗ a=1 Zi,a Zj,a + q (1 − − p∗ = (p∗ − q ∗ ) hP P ∗ ∗ a=1 Zi,a Zj,a ), k a=1 πi,a πj,a i<j P hππ T , 11T − = (p∗ − q ∗ ) P Pk = (p∗ − q ∗ ) Pk i hP Pk i<j a=1 πi,a πj,a Z ∗ Z ∗T i i<j a=1 πi,a πj,a hππ T − Z ∗ Z ∗T , 11T − P i<j k a=1 1 Pk Z ∗ Z ∗T i a=1 πi,a πj,a ∗ Z∗ − Zi,a j,a i , where in the last inequality we use the orthogonality between Z ∗ Z ∗T and 11T − Z ∗ Z ∗T . For its numerator, we have hππ T − Z ∗ Z ∗T , 11T − Z ∗ Z ∗T i ≤ ππ T − Z ∗ Z ∗T 1 ∗ ≤ kπ − Z k1 (kπk1 + kZ ∗ k1 ) ≤ kπ − Z ∗ k1 (2 kZ ∗ k1 + kπ − Z ∗ k1 ) ≤ 3n kπ − Z ∗ k1 . This leads to P Pk 3n kπ − Z ∗ k1 (p∗ − q ∗ ) i<j a=1 πi,a πj,a EAi,j ≤ 3kn−1 (p∗ − q ∗ ) kπ − Z ∗ k1 . − p∗ ≤ P Pk 2 /k n π π a=1 i,a j,a i<j Thus, |p̂ − p∗ | ≤ 6 r k 2 p∗ + 3kn−1 (p∗ − q ∗ ) kπ − Z ∗ k1 ≤ n Similar result holds for |q̂ − q ∗ |. Denote η0 = q "s # 3 kπ − Z ∗ k1 k 2 p∗ + (p∗ − q ∗ ). n(p∗ − q ∗ )2 n/k k 2 p∗ n(p∗ −q ∗ )2 + 3kπ−Z ∗ k1 , n/k thus max{|p̂ − p∗ |, |q̂ − q ∗ |} ≤ η0 (p∗ − q ∗ ). By the assumption of nI in Equation (18) and Proposition C.2, we have n(p∗ − q ∗ )2 /(k 2 p∗ )  nI/k 2 → ∞. Therefore, the first term in η0 goes to 0. The second term in η0 is at most 3cinit which implies η0 ≤ 4cinit . By the fact that the digamma function satisfies ψ(x) ∈ (log(x−1/2), log x), ∀x ≥ 12 ZHANG, ZHOU 1/2, we have αp − 1/2 βp i  hP P i  h pri P P k αp − 1/2 + i<j ka=1 πi,a πj,a Ai,j π π i<j a=1 i,a j,a h i  hP P i  = log  P P k 1 + βppri − i<j ka=1 πi,a πj,a Ai,j π π i<j a=1 i,a j,a hP P i   k p̂ + (αppri − 1/2) i<j a=1 πi,a πj,a  h i . = log Pk pri  P 1 − p̂ + βp i<j a=1 πi,a πj,a ψ(αp ) − ψ(βp ) ≥ log P P Recall that we have shown i<j ka=1 πi,a πj,a lies in the interval of (n2 /(2k), n2 /2). By Equation (18), there exists a sequence η 0 = o(1) such that αp , βp ≤ η 0 (p∗ − q ∗ )n2 /k. Then we have ψ(αp ) − ψ(βp ) ≥ log p∗ − |p∗ − p̂| − η 0 (p∗ − q ∗ ) . 1 − p∗ + |p∗ − p̂| + η 0 (p∗ − q ∗ ) Similar analysis leads to ψ(αq ) − ψ(βq ) ≤ log q ∗ + |q ∗ − q̂| + η 0 (p∗ − q ∗ ) . 1 − q ∗ − |q ∗ − q̂| − η 0 (p∗ − q ∗ ) Together we have   ∗ p − |p∗ − p̂| − η 0 (p∗ − q ∗ ) 1 − q ∗ − |q ∗ − q̂| − η 0 (p∗ − q ∗ ) ∗ − t∗ t − t ≥ log 1 − p∗ + |p∗ − p̂| + η 0 (p∗ − q ∗ ) q ∗ + |q ∗ − q̂| + η 0 (p∗ − q ∗ ) " #  |p∗ − p̂| + η 0 (p∗ − q ∗ ) 4 p∗ (1 − q ∗ ) ≥ log 1 − − t∗ q∗ q ∗ (1 − p∗ )   ∗ ∗ 0 p −q = 4 log 1 − (η0 + η ) . q∗ Recall that we assume c0 p∗ < q ∗ < p∗ . Thus (η0 + η 0 )(p∗ − q ∗ )/p∗ ≤ 5cinit c0 . When cinit is sufficiently small, we have (η0 + η 0 )(p∗ − q ∗ )/p∗ ≤ 1/2. Then using the fact −x ≥ log(1 − x) ≥ −2x, ∀x ∈ (0, 1/2). We have t − t∗ ≥ −8(η0 + η 0 )(p∗ − q ∗ )/q ∗ . Analogously we can obtain the same upper bound on t̂ − t∗ , and then |t − t∗ | ≤ 8c0 (η0 + η 0 ) p∗ − q ∗ . p∗ MEAN FIELD FOR COMMUNITY DETECTION 13 Identical analysis can be applied towards bounds on |λ̂ − λ∗ |. Note that h i   Pk pri  P 1 − p̂ + β π π p i,a j,a i<j a=1 βp i , = log  log  hP Pk αp + βp 1 + (αppri + βppri ) π π i<j a=1 i,a j,a similarly for αq , βq . Omitting the immediate steps, we end up with |λ − λ∗ | = | [ψ(βq ) − ψ(αq + βq )] − [ψ(βp ) − ψ(αp + βp )] − λ∗ | ≤ 8(η0 + η 0 )(p∗ − q ∗ ). The proof is complete after we unify and rephrase all the aforementioned results. Lemma C.2. Let A ∈ [0, 1]n×n such that A = AT and Ai,i = 0, ∀i ∈ [n]. Assume {Ai,j }i<j are independent random variable, and there exists p ≤ 1 P 2 −1 such that 9n ≤ n(n−1) i<j Var(Ai,j ) ≤ p, and then we have √ sup hA − EA, ππ T i ≤ 6n np, π∈Π1 with probability at least 1 − e3 5−n . Proof. This result is a direct consequence of Grothendieck inequality [15] (see also Theorem 3.1 of [16] for a rephrased statement) on the matrix A−EA. The Lemma 4.1 of [16] proves that with probability at least 1−e3 5−n , X √ sup (Ai,j − EAi,j )si tj ≤ 3n np. s,t∈{−1,1}n i,j Then by applying Grothendieck inequality we obtain X √ sup (Ai,j − EAi,j )XiT Xj ≤ 3cn np, kXi k2 ≤1,∀i∈[n] i,j where c is a positive constant smaller than 2. This concludes with √ sup hA − EA, ππ T i ≤ 6n np, π∈Π1 Proposition C.1. Assume 0 < q < p < 1. Let X ∼ Ber(q) and Y ∼ p(1−q) 1−q 1 Ber(p). Recall the definition λ = log 1−p / log p(1−q) q(1−p) , t = 2 log q(1−p) and p √ I = −2 log[ pq + (1 − p)(1 − q)]. Then the following two equations hold (50) e tλ =  EetX Ee−tY  12 , and EetX Ee−tY = exp(−I). 14 ZHANG, ZHOU Proof. The proof is straightforward and all by calculation. Note that E exp(tX) = pet + 1 − p and E exp(tY ) = qet + 1 − q. We can easily obtain p √ EetX Ee−tY = (pet + 1 − p)(qe−t + 1 − q) = ( pq + (1 − p)(1 − q))2 = exp(−I). We can justify the first part of Equation (50) in a similar way. Lemma C.3. [Theorem 5.2 of [21]] Let A ∈ {0, 1}n×n be a symmetric binary matrix with Ai,i = 0, ∀i ∈ [n], and {Ai,j }i<j are independent Bernoulli random variable. If p , maxi,j EAi,j ≥ log n/n. Then there exist constants c, r > 0 such that √ kA − EAkop ≤ c np, with probability at least 1 − n−r . The following lemma on the operator norm of sparse networks is from [9]. In the original statement of Lemma 12 in [9], “with probability 1 − o(1)” is stated. However, its proof in [9] gives explicit form of the probability that the statement holds, which is at least 1 − n−1 . Lemma C.4. [Lemma 12 of [9]] Suppose M is random symmetric matrix with zero on the diagonal whose entries above the diagonal are independent with the following distribution ( 1 − pi,j , w.p. pi,j ; Mi,j = −pi,j , w.p. 1 − pi,j . Let p , maxi,j pi,j and M̃ be the matrix obtained from M by zeroing out all the rows and columns having more than 20np positive entries. Then there exists some constant c > 0 such that √ kM̃ kop ≤ c np, holds with probability at least 1 − n−1 . Lemma C.5. Let A ∈ {0, 1}n×n be a symmetric binary matrix with Ai,i = 0, ∀i ∈ [n], and {Ai,j }i<j are independent Bernoulli random variable. Let P P p ≥ maxi,j EAi,j . Define S = {i ∈ [n], j Ai,j ≥ 20np} and Zi = j |Ai,j − EAi,j |I{i ∈ S}. Then with probability at least 1 − exp(−5np), we have X Zi ≤ 20n2 p exp(−5np). i MEAN FIELD FOR COMMUNITY DETECTION 15 P Proof. Note that E j |Ai,j − EAi,j | ≤ 2np(1 − p) ≤ 2np. For any s ≥ 20np, we have   X X P(Zi > s) ≤ P  |Ai,j − EAi,j | − E |Ai,j − EAi,j | > s − 2np j " ≤ exp − 1 2 (s − 2np)2 np + 13 (s − 2np) # j ≤ exp(−s/2), by implementing Bernstein inequality. Applying Bernstein inequality again we have   X P(Zi > 0) = P  Ai,j ≥ 20np j   X X ≤ P Ai,j − E Ai,j ≥ 18np j  ≤ exp − j (18np)2 /2 np + 18np/3 ≤ exp(−21np/2).  Thus, we are able to bound EZi with Z 20np Z ∞ EZi ≤ P(Zi > 0) ds + P(Zi > s) ds 0 20np Z ∞ ≤ 20np exp(−21np/2) + exp(−s/2) 20np ≤ 20np exp(−10np). By Markov inequality, we have   " # X X 2 2 P |Ai,j − EAi,j |I{i ∈ S} ≥ 20n p exp(−5np) = P Zi ≥ 20n p exp(−5np) i,j i ≤ nEZ1 2 20n p exp(−5np) ≤ exp(−5np). 16 ZHANG, ZHOU Proposition C.2. Under the iassumption that 0 < q < p = o(1). For h√ p I = −2 log pq + (1 − p)(1 − q) we have √ √ I = (1 + o(1))( p − q)2 . Consequently, (p − q)2 /(4p) ≤ I ≤ (p − q)2 /p. Proof. It is a partial result of Lemma B.1 in [35]. p(1−q) 1−q Proposition C.3. Define λ = log 1−p . For any p, q > 0 such / log q(1−p) that p, q = o(1) and p  q, there exists a constant 0 < c < 1/2 such that λ−q ∈ (c, 1 − c). p−q Proof. First we are going to establish the lower bound. Let x = p − q, and then we can rewrite λ as λ= 1 1+ log(1+x/q) log(1+x/(1−q−x)) . Case I: x ≥ q/10. Define s = (p − q)/q. Since p  q we have s ≥ 1/10 and also upper bounded by some constant. We have   λ−q 1 1 1 = − 1 log(1+s) p−q s q1+ log(1+sq/(1−(s+1)q))   1 (1 − q) log(1 + sq/(1 − (s + 1)q)) − q log(1 + s) = s q log(1 + sq/(1 − (s + 1)q)) + q log(1 + s) sq 1 (1 − q) 1−(s+1)q − q log(1 + s) ≥ s 2q log(1 + s) 1 1−q ≥ , 8 log(1 + s) which is lower bounded by some constant c > 0. Case II: x < q/10. By Taylor theorem, there exist constants 0 ≤ 1 , 2 ≤ 1/10 such that     x x 1 − 1 x 2 log 1 + , = − q q 2 q    2 x x 1 − 2 x and log 1 + = − . 1−q−x 1−q−x 2 1−q−x MEAN FIELD FOR COMMUNITY DETECTION 17 Thus, we have log(1 + xq ) log(1 + x 1−q−x ) =  q(1 − q)2 − 2q(1 − q) + q 2 (1 − 1−1 2 2 (1 − q) 2 2 q) − 3− 2 q x  x + c1 x2 + c2 x3 , where c1 = (1 − 1 )(1 − q) + q and c2 = −(1 − 1 )/2. Thus, " # 2 2 q 2 (1 − q) − 3− q x 1 λ−q 2   = −q 3−2 2 1 2 2 3 p−q x q(1 − q) − 2q(1 − q) + 1− 2 (1 − q) + 2 q x + c1 x + c2 x 1  2 2 1 2 2 2 q(1 − q) + 2 q (1 − q) − 2 (1 − q) q + c1 qx + c2 qx = 3−2 2 1 2 2 3 q(1 − q) − 2q(1 − q) + 1− 2 (1 − q) + 2 q x + c1 x + c2 x Note that |c1 |, |c2 | ≤ 1. We have 1 q(1 − q) λ−q ≥ 4 ≥ 1/8. p−q 2q(1 − q) By using exactly the same discussion, we can show (p − λ)/(p − q) > c. Thus, we proved the desired bound stated in the proposition. C.2. Statements and Proofs of Lemmas and Propositions for Theorem 4.1. Lemma C.6. Let Z ∗ ∈ Π0 . Assume p∗ , q ∗ = o(1) and p∗  q ∗ . Define t∗ , λ∗ and π̂ MF the same way as in Theorem 4.1. If nI/[k log kw] → ∞, we have with probability at least 1 − e3 5−n , √ Z ∗ Z ∗T − π̂ MF (π̂ MF )T 1 . n2 / nI. (ρ,ρ0 ) If we further assume Z ∗ ∈ Π0 probability at least 1 − e3 5−n , with arbitrary ρ, ρ0 , and then we have with `(π̂ MF , Z ∗ ) . ρ−1 n p k 2 /(nI). Proof. Form Lemma C.2, with probability at least 1 − e3 5−n , we have uniformly for all π ∈ Π1 p (51) |hA − EA, ππ T i| ≤ 6n np∗ . In the remaining part of the proof, we always assume holds. P the above event pri Denote f 0 (π) = hA+λ∗ In −λ∗ 1n 1Tn , ππ T i−(t∗ )−1 ni=1 KL(πi,· kπi,· ) for any 18 ZHANG, ZHOU pri pri π ∈ Π1 . Here we adopt the notation KL(πi,· kπi,· ) short for KL(Categorical(πi,· )kCategorical(πi,· )), and we do it in the same way in the rest part of the proof. Thus, p hEA + λ∗ In − λ∗ 1n 1Tn , π̂ MF (π̂ MF )T i ≥ hA + λ∗ In − λ∗ 1n 1Tn , π̂ MF (π̂ MF )T i − 6n np∗ n X p pri 0 MF ∗ −1 MF ∗ = f (π̂ ) − 6n np + (t ) KL(π̂i,· kπi,· ) i=1 n X p pri ∗ −1 MF 0 ∗ ∗ KL(π̂i,· kπi,· ) ≥ f (Z ) − 6n np + (t ) i=1 T ∗ ∗T 1n 1n , Z Z i p ≥ hEA + λ In − λ − 12n np∗ n n X X pri pri ∗ −1 MF ∗ −1 ∗ + (t ) KL(π̂i,· kπi,· ) − (t ) KL(Zi,· kπi,· ), ∗ ∗ i=1 i=1 where we use Equation (51) twice in the first and last inequality. Note that for any π ∈ Π1 , we have X X pri pri |KL(πi,· kπi,· )| ≤ | πi,j log πi,j | + | πi,j log πi,j | ≤ log k + log w, j j P where the second inequality is due to 0 ≥ j πi,j log πi,j = KL(πi,· kk −1 1k )− log k ≥ − log k, where k −1 1k can be explicitly written as a length-k vector (1/k, 1/k, . . . , 1/k). Then we have n X i=1 pri MF KL(π̂i,· kπi,· ) − n X i=1 pri ∗ KL(Zi,· kπi,· ) ≤ 2n log kw. Thus, hEA + λ∗ In − λ∗ 1n 1Tn , Z ∗ Z ∗T − π̂ MF (π̂ MF )T i ≤ 12n By Proposition C.4, we have ∗ hEA + λ In − λ ∗ 1n 1Tn , Z ∗ Z ∗T − π̂ MF (π̂ p np∗ + 2(t∗ )−1 n log kw.    λ∗ − q ∗ λ∗ − q ∗ α+ ∗ γ , ) i ≥ 2(p − q ) 1 − ∗ p − q∗ p − q∗ MF T ∗ ∗ where α = hZ ∗ Z ∗T − π̂ MF (π̂ MF )T , Z ∗ Z ∗T − In i/2 and γ = hπ̂ MF (π̂ MF )T − Z ∗ Z ∗T , 1n 1Tn − Z ∗ Z ∗T i/2. By Proposition C.3, there exists a constant c > 0 such that (52) hEA + λ∗ In − λ∗ 1n 1Tn , Z ∗ Z ∗T − π̂ MF (π̂ MF )T i ≥ 2c(p∗ − q ∗ )(α + γ). MEAN FIELD FOR COMMUNITY DETECTION 19 Note that the following inequality holds 2(α + γ) = Z ∗ Z ∗T − π̂ MF (π̂ MF )T ≥ Z ∗ Z ∗T − π̂ MF (π̂ MF )T 1 1 − hZ ∗ Z ∗T − π̂ MF (π̂ MF )T , In i/2 − n/2. These together lead to Z ∗ Z ∗T − π̂ MF (π̂ MF )T 1 ≤ i h p 1 ∗ + 2(t∗ )−1 n log kw + c(p∗ − q ∗ )n/2 . np 12n c(p∗ − q ∗ ) Note that t∗  (p∗ − q ∗ )/p∗ when p∗  q ∗ . Together by Proposition C.2, as long as nI/[k log kw] → ∞, the last two terms in the RHS of the above formula is dominated by the first term. Thus, Z ∗ Z ∗T − π̂ MF (π̂ MF )T (ρ,ρ0 ) If we further assume Z ∗ ∈ Π0 to 1 n2 .√ . nI , Proposition C.5 and Equation (52) lead hEA + λ∗ In − λ∗ 1n 1Tn , Z ∗ Z ∗T − π̂ MF (π̂ MF )T i ≥ ρcn(p∗ − q ∗ ) `(π̂ MF , Z ∗ ). 8k So we have p 8k (12n np∗ + 2(t∗ )−1 n log kw) ∗ ∗ ρcn(p − q ) s 192k np∗ ≤ . ∗ ρc (p − q ∗ )2 `(π̂ MF , Z ∗ ) ≤ Before we state the remaining lemmas and propositions used in the Proof of Lemma C.6, we first introduce two definitions. For any π, π 0 ∈ [0, 1]n×k , 0 0 0 0 0 0 define α(π; π 0 ) = hπ π T −ππ T , π π T −In i/2 and γ(π; π 0 ) = hππ T −π π T , 1n 1Tn − 0 0T π π i/2. Proposition C.4. Define P = Z ∗ BZ ∗T − pIn , with B = q1k 1Tk + (p − q)Ik . We have the equation    λ−q λ−q α(π; Z ∗ ) + γ(π; Z ∗ ) . hP + λIn − λ1n 1Tn , Z ∗ Z ∗T − ππ T i = 2(p − q) 1 − p−q p−q 20 ZHANG, ZHOU Proof. Note that Z ∗ BZ ∗T − pIn = (p − q)Z ∗ Z ∗T + q1n 1Tn . We have λ−q λ−p 1n 1Tn + In , Z ∗ Z ∗T − ππ T i p−q p−q − In , Z ∗ Z ∗T − ππ T i hP + λIn − λ1n 1Tn , Z ∗ Z ∗T − ππ T i = (p − q)hZ ∗ Z ∗T − = (p − q)hZ ∗ Z ∗T + (λ − q)hIn − 1n 1Tn , Z ∗ Z ∗T − ππ T i = (p − λ)hZ ∗ Z ∗T − In , Z ∗ Z ∗T − ππ T i + (λ − q)hZ ∗ Z ∗T − 1n 1Tn , Z ∗ Z ∗T − ππ T i = 2(p − q)α(π; Z ∗ ) + 2(λ − q)γ(π; Z ∗ ). Consequently, we obtain the desired bound. (ρ,ρ0 ) If Z ∗ ∈ Π0 Proposition C.5. , π ∈ Π1 , we have α(π; Z ∗ ) + γ(π; Z ∗ ) ≥ ρn `(π, Z ∗ ). 16k Proof. We use α, γ instead of α(π; Z ∗ ), γ(π; Z ∗ ) for simplicity. Without ∗ ∗ = 1} loss of generality Z ∗ ). Define Cu = {i : Zi,u P we assume kπ − Z k1 = `(π, P and Lu,v = i∈Cu πi,v . We have the equality v Lu,v = |Cu | and also α= and γ = 1X 2 u 1X 2  |Cu |2 − X X X i,j∈Cu w X u6=v i∈Cu ,j∈Cv w  " # X X 1 1X X πi,w πj,w  = |Cu |2 − L2u,w = Lu,w Lu,w0 2 u 2 u 0 w πi,w πj,w = 1 XX 2 w6=w Lu,w Lv,w . u6=v w We define [k] into two disjoint subsets S1 and S2 where o n 3 S1 = u ∈ [k] : ∀v 6= u, Lu,v ≤ |Cu | , 4 n o 3 and S2 = i ∈ [k] : ∃v 6= u, Lu,v > |Cu | . 4 P Define L = v6=u Lu,v . For any u ∈ S1 , if Lu,u ≥ |Cu |/4, we have |Cu |2 − P 2 P 2 u 1 2 w Lu,w ≥ w Lu,w ≥ Lu,u Lu ≥ |Cu |Lu /4. If Lu,u < 4 |Cu | we have |Cu | − 3 2 8 |Cu | ≥ |Cu |Lu /4 as well. This leads to " # X 1 X 1 X α≥ |Cu |2 − L2u,w ≥ |Cu |Lu . 2 8 w u∈S1 u∈S1 MEAN FIELD FOR COMMUNITY DETECTION 21 For any u ∈ S2 there exists a v 6= u such that Lu,v > 43 |Cu |. We must have Lu,u + Lv,v ≥ Lu,v + Lv,u otherwise kπ − Z ∗ k1 = `(π, Z ∗ ) does not hold since we can switch the u-th and v-th columns of π to make − Z ∗ k1 smaller. P kπP Consequently, we have Lv,v ≥ Lu /2. So we have u0 6=u w Lu,w Lu0 ,w ≥ Lu,v Lv,v ≥ 3|Cu |Lu /8. Then we have γ≥ 1 X XX 3 X Lu,w Lu0 ,w ≥ |Cu |Lu . 2 8 0 w u∈S2 u 6=u u∈S2 Thus, α+γ ≥ ρn X ρn ρn 1 X |Cu |Lu ≥ Lu ≥ kπ − Z ∗ k1 = `(π, Z ∗ ). 16 u 16k u 16k 16k C.3. Statements and Proofs of Lemmas and Propositions for Theorem 4.2. Lemma C.7. Let X ∼ Beta(α, β) where α = n2 p and β = n2 (1 − p) with p = o(1). Let η = o(1). Then we have P(|X − p| ≥ ηp) ≤ exp(−η 2 n2 p/2). Proof. Note X has the same distribution as Y /(Y + Z) where Y and Z are independent χ2 random variables with Y ∼ χ2 (2α) and Z ∼ χ2 (2β). Then by using tail bound of χ2 distribution (i.e., Proposition C.6) P(|X − p| ≥ ηp) ≤ P(|Y − 2n2 p| ≥ 2ηn2 p) + P(|Y + Z − 2n2 | ≥ ηn2 ) ≤ 2 exp(−η 2 n2 p/4) + 2 exp(−η 2 n2 /16) ≤ exp(−η 2 n2 p/2). Proposition C.6. Let X ∼ χ2 (k) we have   P |X − k| ≥ kt ≤ 2 exp(−kt2 /8), ∀t ∈ (0, 1). Proof. See Lemma 1 of [20]. 22 ZHANG, ZHOU APPENDIX D: GENERAL DERIVATIONS OF CAVI FOR VARIATIONAL INFERENCE In this section, we provide the derivation from Equation (3) to Equation (4). First we have (53) h q(x) i KL(q(x)kp(x|y)) = Eq(x) log p(x|y) = Eq(x) [log q(x)] − Eq(x) [log p(x|y)] = Eq(x) [log q(x)] − Eq(x) [log p(x, y)] + log p(y) = −(Eq(x) [log p(x, y)] − Eq(x) [log q(x)]) + log p(y)   = − Eq(x) [log p(y|x)] − KL(q(x)kp(x)) + log p(y). Thus, to minimize KL(q(x)kp(x|y)) w.r.t. q(x) is equivalent to maximize Eq(x) [log p(y|x)] − KL(q(x)kp(x)). n Recall we have independence under both p and Q q for {xi }i=1 . For simplicity, denote x−i to be {xj }j6=i and q−i to be j6=i qj . We have the decomposition bi (qi ) , Eq(x) [log p(x, y)] − Eq(x) [log q(x)]     = Eqi Eq−i [log p(xi , x−i , y)] − Eqi Eq−i [log q(xi , x−i )]   = Eqi Eq−i [log p(xi |x−i , y)] − Eqi [log qi (xi )] + const log qi (xi )   + const, = −Eqi log −1 c exp Eq−i [log p(xi |x−i , y)]   P where the constant includes all terms not depending on xi and c = xi exp Eq−i [log p(xi |x−i , y)] which is also independent of xi . It is obvious that to solve Equation (3) is equivalent to q̂i = arg max bi (qi ) qi    = arg min KL qi kc−1 exp Eq−i [log p(xi |x−i , y)] . qi   Immediately we have q̂i (xi ) = c−1 exp Eq−i [log p(xi |x−i , y)] . Or we may write it as   q̂i (xi ) ∝ exp Eq−i [log p(xi |x−i , y)] . Department of Statistics Yale University New Haven, CT 06511 E-mail: [email protected] E-mail: [email protected] URL: http://www.stat.yale.edu/˜hz68/ 

A smooth transition from Wishart to GOE Miklós Z. Rácz ∗ Jacob Richey † arXiv:1611.05838v1 [] 17 Nov 2016 November 18, 2016 Abstract It is well known that an n × n Wishart matrix with d degrees of freedom is close to the appropriately centered and scaled Gaussian Orthogonal Ensemble (GOE) if d is large enough. Recent work of Bubeck, Ding, Eldan, and Racz, and independently Jiang and Li, shows that the transition happens when d = Θ(n3 ). Here we consider this critical window and explicitly compute the total variation distance between the Wishart and GOE matrices when d/n3 → c ∈ (0, ∞). This shows, in particular, that the phase transition from Wishart to GOE is smooth. 1 Introduction The Wishart distribution is a fundamental object appearing in many domains, such as statistics, geometry, quantum physics, and wireless communications, among others. In statistics it arises as the distribution of the sample covariance matrix of a sample from a multivariate normal distribution. In geometry it is known as the Gram matrix of inner products of n points in Rd , and it is also the starting point for canonical models of random geometric graphs [5, 3, 8]. It is well known that an n × n Wishart matrix with d degrees of freedom is close to the appropriately centered and scaled Gaussian Orthogonal Ensemble (GOE) if d is
large enough (see, e.g., [5]). Recent work [3, 9] shows that the transition happens when d = Θ n3 and in this paper we study this critical window. In Theorem 1.2 below we explicitly compute the total variation distance between the Wishart and GOE matrices when d/n3 → c ∈ (0, ∞), showing, in particular, that the phase transition from Wishart to GOE is smooth. 1.1 Main result Let X be an n × d matrix where the entries are i.i.d. standard normal random variables, and let W ≡ W (n, d) = XX T be the corresponding n × n Wishart matrix with d degrees of freedom.1 Let M (n) be an n × n matrix drawn from the Gaussian Orthogonal Ensemble, i.e., a symmetric n × n random matrix where the diagonal entries are i.i.d. normal random variables with mean zero and variance 2, and the entries above the diagonal are i.i.d. standard normal random variables, with the entries on and above the diagonal all independent. In order to match the first moment √ and the scale of the Wishart matrix, we center and scale M (n) appropriately: let M (n, d) := dM (n) + dIn , where In is the n × n identity matrix. ∗ Microsoft Research; [email protected]. University of Washington; [email protected]. 1 In statistics the number of samples is usually denoted by n and the number of parameters is usually denoted by p, resulting in a p × p Wishart matrix with n degrees of freedom. Here our notation is taken with the geometric perspective in mind, following [5, 3, 4]. † 1 If d is large enough compared to n, then the Wishart matrix becomes approximately like the GOE. Recent work of Bubeck, Ding, Eldan,
and Racz [3], and independently Jiang and Li [9], shows that the transition happens when d = Θ n3 . Specifically, they proved the following theorem, where we write TV for total variation distance. Theorem 1.1. Define the random matrix ensembles W (n, d) and M (n, d) as above. (a) (Bubeck, Ding, Eldan, and Racz [3]) If d/n3 → 0 then TV (W (n, d), M (n, d)) → 1. (b) (Bubeck, Ding, Eldan, and Racz [3]; Jiang and Li [9]) If d/n3 → ∞ then TV (W (n, d), M (n, d)) → 0. Our focus is on the critical window and our main result is the explicit computation of the limiting total variation distance between W (n, d) and M (n, d) when d/n3 → c ∈ (0, ∞). Theorem 1.2. Define the random matrix ensembles W (n, d) and M (n, d) as above and let d = d(n) be such that d/n3 → c ∈ (0, ∞). Then   1 √ √ , (1.1) lim TV (W (n, d), M (n, d)) = Erf n→∞ 4 3 c where recall that the error function is defined as 2 Erf (x) = √ π Z x 2 e−t dt. 0 From this result we can immediately read off that, as c → 0 and c → ∞, the total variation distance goes to 1 and 0, respectively, recovering the previous results described in Theorem 1.1. Since Erf (x) = √2π x (1 + o (1)) as x → 0, the limiting total variation distance decays as   1 1 √ √ Erf ∼ √ √ 4 3 c 2 3π c as c → ∞. The behavior of the limit when c is small is plotted in Figure 1. Figure 1: The limiting total variation distance as a function of c, when c is close to 0. From the proof we shall see that the limit in (1.1) is the expected value of an explicit function of a two-dimensional Gaussian, which comes from the central limit theorem for the first and third moments of the empirical spectral distribution of a GOE matrix. 2 1.2 Further related work and open problems Several recent works have explored extensions of Theorem 1.1, and Theorem 1.2 raises further questions. Robustness. Bubeck and Ganguly [4] showed that the critical dimension is universal in the following sense: Theorem 1.1 holds (up to logarithmic factors) if the entries of X are i.i.d. from a sufficiently smooth distribution. What can be said about the transition in the critical regime? Are there other distributions for which the limiting total variation distance can be computed explicitly? If not, can one prove similar qualitative behavior? Anisotropy. Eldan and Mikulincer [8] studied the effect of anisotropy on the power of detecting geometry in random geometric graphs. This is directly related to studying Wishart matrices where each row of X is a multivariate normal with a diagonal covariance matrix. The authors introduce new notions of dimensionality and prove a theorem similar to Theorem 1.1 with appropriate upper and lower bounds on the “effective critical dimension”. While the primary open problem is to close the gap between these bounds, one may also ask about the nature of the transition at the effective critical dimension: can anisotropy cause qualitatively different behavior? Other regimes. Theorems 1.1 and 1.2 state that as d/n3 → ∞, all statistics of the Wishart W (n, d) and the GOE M (n, d) have asymptotically the same distribution, but this is not the case if d/n3 remains bounded. In the random matrix literature there has been lots of work showing that particular statistics of these ensembles have asymptotically the same distribution even when d  n3 . For instance, when d = Θ (n), then the limiting empirical spectral distribution of the Wishart is the Marchenko-Pastur law, which shows the difference between the Wishart and GOE, but the largest eigenvalue of the Wishart already behaves like that of the GOE [10, 6, 7]. This naturally raises the question of whether there are other regimes of d and n where there are interesting phase transitions. 2 Proof of Theorem 1.2 The main reason that allows for an explicit computation of the limiting total variation distance in Theorem 1.2 is that both W (n, d) and M (n, d) have explicit densities. The proof of Theorem 1.2 is similar to the case of d/n3 → ∞ presented in [3] and proceeds by a Taylor expansion of the ratio of the densities of the two random matrix ensembles. The difference compared to the case of d/n3 → ∞ is that here the Taylor expansion has to be done to one degree higher. As we shall see, taking the limit of the total variation distance as d/n3 → c ∈ (0, ∞) then requires using the central limit theorem for the moments of the empirical spectral distribution of a GOE matrix. n(n+1) Step 1: Writing out the total variation distance. Let P ⊂ R 2 denote the cone of positive semidefinite matrices. It is well known (see, e.g., [11]) that when d ≥ n, W (n, d) has the following density with respect to the Lebesgue measure on P:
1 (det (A)) 2 (d−n−1) exp − 21 Tr (A) fn,d (A) := 1
, Q 1 2 2 dn π 4 n(n−1) ni=1 Γ 12 (d + 1 − i) where Tr (A) denotes the trace of the matrix A. The density of a GOE random matrix with respect

n(n+1) 1 n to the Lebesgue measure on R 2 is A 7→ (2π)− 4 n(n+1) 2− 2 exp − 14 Tr A2 and so the density 3 n(n+1) of M (n, d) with respect to the Lebesgue measure on R 2 is    1 Tr (A − dIn )2 exp − 4d . gn,d (A) := 1 n (2πd) 4 n(n+1) 2 2 n(n+1) Denote the measure given by this density by µn,d , let λ denote the Lebesgue measure on R 2 and write A  0 if A is positive semidefinite. We can then write Z
TV (W (n, d), M (n, d)) = n(n+1) gn,d (A) − fn,d (A) 1{A0} + dλ (A) R 2   Z fn,d (A) 1{A0} = n(n+1) 1 − dµn,d (A) , (2.1) gn,d (A) R 2 + where x+ := max {x, Let Q denote the set of symmetric matrices for which all of the eigenvalues h 0}. √ √ i are in the interval d − 3 dn, d + 3 dn . Since d/n3 → c > 0, we have that Q ⊂ P for all n large enough. It is known (see, e.g., [1]) that, with probability 1 − o (1), all the eigenvalues of M (n) √ √ are in the interval [−3 n, 3 n], which implies that M (n, d) ∈ Q. Since the integrand in (2.1) is bounded, we can then write  Z  fn,d (A) TV (W (n, d), M (n, d)) = 1− dµn,d (A) + o (1) (2.2) gn,d (A) + Q and so we may restrict our attention to Q. Define αn,d (A) := log (fn,d (A) /gn,d (A)). Denote the eigenvalues of an n × n matrix A by λ1 (A) ≤ · · · ≤ λn (A); when P from the context, we omit the dependence on Q the matrix is obvious A. Recall that det (A) = ni=1 λi and Tr (A) = ni=1 λi . We then have that  n  1X 1 2 αn,d (A) = (λi − d) (d − n − 1) log λi − λi + 2 2d i=1     n X n (n + 3) dn n n (n + 1) 1 + − log 2 + log π + log d − log Γ (d + 1 − i) . 4 2 2 4 2 i=1

By Stirling’s formula we know that log Γ (z) = z − 12 log z − z + 12 log (2π) + O z1 as z → ∞, so  n  1X 1 2 αn,d (A) = (d − n − 1) log λi − λi + (λi − d) 2 2d i=1 n n n n (n + 1) 1X 1X log d − (d − i) log (d + 1 − i) + (d + 1 − i) + O . 4 2 2 d i=1 i=1  
(i−1)2 (i−1)3 i−1 Now writing log (d + 1 − i) = log d + log 1 − i−1 = log d − − + O we get that 2 3 d d 2d d +  n  1X 1 n3 n 2 αn,d (A) = (d − n − 1) log λi − λi + (λi − d) − {(d − n − 1) log d − d} − + o (1) . 2 2d 2 12d i=1 n o 1 Defining h (x) := 21 (d − n − 1) log (x/d) − (x − d) + 2d (x − d)2 , we have that αn,d (A) = n X h (λi ) − i=1 4 n3 + o (1) . 12d (2.3) Step 2: Taylor expansion and taking the limit. The derivatives of h at d are h (d) = 0, n+1 00 (3) (d) = d−n−1 , h(4) (d) = − 3(d−n−1) , and also h(5) (x) = 12(d−n−1) . = − n+1 2d , h (d) = 2d2 , h d3 d4 x5 Approximating h with its fourth order Taylor polynomial around d we get that h0 (d) h(x) = − d−n−1 d−n−1 d−n−1 n+1 n+1 (x − d)2 + (x − d)3 − (x − d)4 + (x − d)5 , (x − d)+ 2d 4d2 6d3 8d4 10ξ 5 where ξ is some real number between x and d. From 2.3 we see that to compute αn,d (A), we need to compute the sum over the eigenvalues {λi }ni=1 of each term in the expansion. First, we argue the contribution from the remainder term Recall that A ∈ Q, h that h is negligible. √ √ i √ √ i and hence λi ∈ d − 3 dn, d + 3 dn for every i ∈ [n]. If x ∈ d − 3 dn, d + 3 dn , then d−n−1 (x − d)5 ≤ 10ξ 5    √ 5 d−n−1 1  √ 5 3 dn = O n2 , 10 d − 3 dn
where we used that d = Θ n3 . Summing n such terms gives a term of order O(1/n), which is negligible in the limit. Turning to the four terms that matter, defining n n n+1X S1 (n, d) := − (λi − d) , 2d n+1X S2 (n, d) := (λi − d)2 , 4d2 i=1 S3 (n, d) := n d−n−1X 6d3 i=1 n d−n−1X (λi − d)4 , S4 (n, d) := − 8d4 3 (λi − d) , i=1 i=1 and also letting S0 (n, d) := −n3 /(12d), we thus have that αn,d (A) = S0 (n, d) + S1 (n, d) + S2 (n, d) + S3 (n, d) + S4 (n, d) + o (1) . (2.4) (λi − d). If {λi }ni=1 are the eigenvalues of M (n, d), then {µi }ni=1 are P the eigenvalues of √1n M (n). Recall that the empirical spectral distribution n1 ni=1 δµi converges √ 1 weakly to the semicircle distribution with density ρsc (x) = 2π 4 − x2 1{|x|≤2} (see, e.g., [1]). With this notation we can rewrite the four quantities above as follows: √ n n n (n + 1) X n2 (n + 1) 1 X 2 √ S1 (n, d) = − × µi , S2 (n, d) = × µi , 4d n 2 d i=1 i=1 r n n d − n − 1 n3 1X 4 d−n−1 n3 X 3 S4 (n, d) = − × × µi . × × µi , S3 (n, d) = 8d d n 6d d For i ∈ [n] define µi := √1 dn i=1 i=1 Let U be a random variable distributed according to the semicircle law. We know that for any fixed k ∈ N, the k th moment of the empirical spectral distribution converges in probability to the k th moment of the semicircle law (see [1, Lemmas 2.1.6 and 2.1.7]), i.e., n h i 1X k µi → E U k . n i=1     Since E U 2 = 1 and E U 4 = 2, we have that, as d/n3 → c ∈ (0, ∞), S2 (n, d) → 1/(4c) and S4 (n, d) → −1/(4c), so put together we have that S2 (n, d) + S4 (n, d) → 0. 5 The central limit theorem for the moments of the empirical spectral distribution of a GOE matrix (see [1, Theorem 2.1.31 and Exercise 2.1.35] and [2]) shows that ! n n X X µi , µ3i =⇒ (N1 , N3 ) , i=1 i=1 6 ). The entries where (N1 , N3 ) are jointly normal with mean zero and covariance matrix C = ( 26 24 of the covariance matrix C are special cases of the more general formulas found in [1, 2], so we do not describe the computations here, but one can verify these numbers by using the identity n X µki = Tr  √1 M n n  k  X k  1 √ (n) = M (n) n i=1 i=1 i,i and computing appropriate moments of normal random variables. Putting everything together we see that, as d/n3 → c ∈ (0, ∞), we have that   1 1 1 1 1 √ √ ,− . N1 , , N3 , − (S0 , S1 , S2 , S3 , S4 ) =⇒ − 12c 2 c 4c 6 c 4c (2.5) Therefore, since the function (s0 , s1 , s2 , s3 , s4 ) 7→ (1 − exp (s0 + s1 + s2 + s3 + s4 ))+ is continuous and bounded, we have by (2.2), (2.4), and (2.5) that     1 1 1 TV (W (n, d), M (n, d)) → E 1 − exp − − √ N1 + √ N3 . (2.6) 12c 2 c 6 c + Step 3: Evaluating the limit. What remains is to evaluate the expectation on the right hand side of (2.6). Let Y and Z be independent normal random variables with mean zero and variances 2 and 6, respectively. Then (N1 , N3 ) and (Y, 3Y + Z) have the same distribution, since both are Gaussian with the same mean vector and covariance matrix. Notice that 1 1 1 − √ Y + √ (3Y + Z) = √ Z 2 c 6 c 6 c and hence the right hand side of (2.6) is equal to     1 1 E 1 − exp − + √ Z . 12c 6 c + √ √ Since 1 − exp (−1/(12c) + z/(6 c)) ≥ 0 if and only if z ≤ 1/(2 c), we have that     Z √1   1 2 c z2 1 1 1 z − 1 + √ E 1 − exp − + √ Z = 1 − e 12c 6 c · √ e− 12 dz 12c 6 c 12π −∞ + Z √1 Z √1 √ 2 (z−1/ c)2 2 c 2 c 1 1 − z12 − 12 √ √ = e dz − e dz 12π −∞ 12π −∞       1 1 1 √ √ , =P Z< √ −P Z <− √ = Erf 2 c 2 c 4 3 c which concludes the proof. 6 References [1] G. W. Anderson, A. Guionnet, and O. Zeitouni. An Introduction to Random Matrices. Cambridge University Press, 2010. [2] G. W. Anderson and O. Zeitouni. A CLT for a band matrix model. Probability Theory and Related Fields, 134(2):283–338, 2006. [3] S. Bubeck, J. Ding, R. Eldan, and M. Z. Rácz. Testing for high-dimensional geometry in random graphs. Random Structures & Algorithms, 49(3):503–532, 2016. [4] S. Bubeck and S. Ganguly. Entropic CLT and phase transition in high-dimensional Wishart matrices. Preprint available at http://arxiv.org/abs/1509.03258, 2015. [5] L. Devroye, A. György, G. Lugosi, and F. Udina. High-dimensional random geometric graphs and their clique number. Electronic Journal of Probability, 16:2481–2508, 2011. [6] N. El Karoui. On the largest eigenvalue of Wishart matrices with identity covariance when n, p and p/n → ∞. arXiv preprint math/0309355, 2003. [7] N. El Karoui. Tracy-Widom limit for the largest eigenvalue of a large class of complex sample covariance matrices. The Annals of Probability, pages 663–714, 2007. [8] R. Eldan and D. Mikulincer. Information and dimensionality of anisotropic random geometric graphs. Preprint available at https://arxiv.org/abs/1609.02490, 2016. [9] T. Jiang and D. Li. Approximation of Rectangular Beta-Laguerre Ensembles and Large Deviations. Journal of Theoretical Probability, 28:804–847, 2015. [10] I. M. Johnstone. On the distribution of the largest eigenvalue in principal components analysis. Annals of Statistics, 29(2):295–327, 2001. [11] J. Wishart. The Generalised Product Moment Distribution in Samples from a Normal Multivariate Population. Biometrika, 20A(1/2):32–52, 1928. 7 

arXiv:1408.6821v3 [math.CO] 19 May 2016 Looking for vertex number one Alan Frieze∗, Wesley Pegden† Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh PA 15213, U.S.A. May 20, 2016 Abstract Given an instance of the preferential attachment graph Gn = ([n], En ), we would like to find vertex 1, using only ‘local’ information about the graph; that is, by exploring the neighborhoods of small sets of vertices. Borgs et al. gave an an algorithm which runs in time O(log4 n), which is local in the sense that at each step, it needs only to search the neighborhood of a set of vertices of size O(log4 n). We give an algorithm to find vertex 1, which w.h.p. runs in time O(ω log n) and which is local in the strongest sense of operating only on neighborhoods of single vertices. Here ω = ω(n) is any function that goes to infinity with n. 1 Introduction The Preferential Attachment Graph Gn was first discussed by Barabási and Albert [2] and then rigorously analysed by Bollobás, Riordan, Spencer and Tusnády [3]. It is perhaps the simplest model of a natural process that produces a graph with a power law degree sequence. The Preferential Attachment Graph can be viewed as a sequence of random graphs G1 , G2 , . . . , Gn where Gt+1 is obtained from Gt as follows: Given Gt , we add vertex t + 1 and m random edges {ei = (t + 1, ui ) : 1 ≤ i ≤ m} incident with vertex t + 1. Here the constant m is a parameter of the model. The vertices ui are not chosen uniformly from Vt , instead they are chosen with probabilities proportional to their degrees. This tends to generate some very high degree vertices, ∗ † Research supported in part by NSF grant CCF1013110 Research supported in part by NSF grant DMS 1 compared with what one would expect in Erdős-Rényi models with the same edge-density. We refer to u1 , u2 , . . . , um as the left choices of vertex t + 1. We also say that t + 1 is a right neighbor of ui for i = 1, 2, . . . , m. We consider the problem of searching through the preferential attachment graph looking for vertex number 1, using only local information. This was addressed by Borgs, Brautbar, Chayes, Khanna and Lucier [5] in the context of the Preferential Attachment Graph Gn = (Vn , En ). Here Vn = [n] = {1, 2, . . . , n}. They present the following local algorithm that searches for vertex 1, in a graph which may be too large to hold in memory in its entirety. 1: Initialize a list L to contain an arbitrary node u in the graph. 2: while L does not contain node 1 do 3: Add a node of maximum degree in N(L) to L od; 4: return L Here for vertex set L, we let N(L) = {w ∈ / L : ∃v ∈ L s.t. {v, w} ∈ En }. They show that w.h.p. the algorithm succeeds in reaching vertex 1 in O(log4 n) steps. (We assume that an algorithm can recognize vertex 1 when it is reached.) In [5], they also show how a local algorithm to find vertex 1 can be used to give local algorithms for some other problems. We also note that Brautbar and Kearns [6] considered local algorithms in a more general context. There the algorithm is allowed to jump to random vertices as well as crawl around the graph in the search for vertices of high degree and high clustering coefficient. We should note that, as the maximum degree in Gn is n1/2−o(1) w.h.p., one cannot hope to have a polylog(n) time algorithm if we have to check the degrees of the neighbors as we progress. Thus the algorithm above operates on the assumption that we can find the highest-degree neighbor of a vertex in O(1) time. This would be the case, for example, if the neighborhood of a vertex is stored as a linked-list which is sorted by degrees. In the same situation, we can also determine the K highest degree neighbors of a vertex in constant time for any constant K, and in the present manuscript we assume such a constant-time step is possible. In particular, in this setting, each of steps 2-7 of the following Degree Climbing Algorithm takes constant time. We let dn (v) denote the degree of vertex v ∈ Vn . Algorithm DCA: The algorithm generates a sequence of vertices v1 , v2 , . . . , until vertex 1 is reached. Step 1 Carry out a random walk on G until it is mixed; i.e., until the variation distance between the current vertex and the steady state is o(1). We let v1 be the terminal vertex of the walk. (See Remark 1.1 for comments on this step.) Step 2 t ← 1. 2 Step 3 repeat Step 4  Let Ct = w1 , w2, . . . , wm/2 be the m/2 neighbors of vt of largest degree. (In the case of ties for the m/2th largest degree, vertices will be placed randomly in Ct in order to make |Ct | = m/2. Also m is large here and we could replace m/2 by ⌊m/2⌋ if m is odd without affecting the analysis by very much.) Step 5 Choose vt+1 randomly from Ct . Step 6 t ← t + 1. Step 7 until dn (vt ) ≥ arbitrary. n1/2 log1/100 n (SUCCESS) or t > 2ω log4/3 n (FAILURE), where ω → ∞ is Step 8 Assuming Success, starting from vT , where T is the value of t at this point, do a random 1/2 walk on the vertices of degree at least logn1/20 n until vertex 1 is reached. Remark 1.1. It is known that w.h.p. the mixing time of a random walk on Gn is O(log n), see Mihail, Papadimitriou and Saberi [11]. So we can assume that the distribution of v1 is close to n (v) the steady state πv = d2mn . Note that Algorithm DCA is a local algorithm in a strong sense: the algorithm only requires access to the current vertex and its neighborhood. (Unlike the algorithm from [5], it does not need access to the neighborhood of the entire set Pt = {v1 , . . . vt } of vertices visited so far.) Our main result is the following: Theorem 1.2. If m is sufficiently large then then w.h.p. Algorithm DCA finds vertex 1 in Gn in O(ω log n) time. DCA is thus currently the fastest as well as the “most local” algorithm to find vertex 1. We conjecture that the factor ω in the running time is unnecessary. Conjecture 1.3. Algorithm finds vertex 1 in Gn in O(log n) time, w.h.p. We note that w.h.p. the diameter of Gn is ∼ execution time much below O(log n). log n log log n and so we cannot expect to improve the The bulk of our proof consists of showing that the execution of Steps 2–7 requires only time O(ω log n) w.h.p. for any ω = ω(n) → ∞. This analysis requires a careful accounting of conditional probabilities. This is facilitated by the conditional model of the preferential attachment graph due to Bollobás and Riordan [4]. One contribution of our paper is to recast their model in terms of sums of independent copies of the rate one exponential random variables; this will be essential to our analysis. 3 Outline of the paper In Section 2 we reformulate the construction of Bollobás and Riordan [4] in terms of sums of independent copies of the exponential random variable of rate one. Section 3 is the heart of the paper. The aim is to show that if vt is not too small, then the ratio vt+1 /vt is bounded above by 3/4 in expectation. We deduce from this that w.h.p. the main loop, Steps 2–7, only takes O(ω log n) rounds. The idea is to determine a degree bound ∆ such that many of vt ’s left neighbors have degree at least ∆, while only few of vt ’s right neighbors have degree at least ∆. In this way, vt+1 is likely to be significantly smaller than vt . Once we find a vertex vT of high enough degree, then we know that w.h.p. vT is not very large and lies in a small connected subgraph of vertices of high degree that contains vertex one. Then a simple argument based on the worst-case covertime of a graph suffices to show that only o(log n) more steps are required. Our proofs will use various parameters. For convenience, we collect here in table form a dictionary of some notations, giving a brief (and imprecise) description of the role each plays in our proof, for later reference. ω := λ0 := Definition Role in proof O(log log n) An arbitrarily chosen slowly growing fucntion. 1 log 40/m n A (usually valid) lower bound on random variables ηi (cf. Section 2.1). n1 := log1/100 n W.h.p. the main loop never visits v ≤ n1 . Pt := {v1 , . . . vt } The set of vertices visited up to time t. Ψ:= (log log n)10 Vertices v > Ψvt will not be important in the search for vt+1 . L:= m1/5 A large constant, significantly smaller than m. Notation: We write An ∼ Bn if An = (1 + o(1))Bn as n → ∞. We write α . β in place of α ≤ o(1) + (1 + o(1))β. 4 2 2.1 Preliminaries A different model of the preferential attachement graph Bollobás and Riordan [4] gave an ingenious construction equivalent to the preferential attachment graph model. We choose x1 , x2 , . . . , x2mn independently and uniformly from [0, 1]. We then let {ℓi , ri } = {x2i−1 , x2i } where ℓi < ri for i = 1, 2, . . . , mn. We then sort the ri in increasing order R1 < R2 < · · · < Rmn and let R0 = 0. We then let Wj = Rmj and wj = Wj − Wj−1 and Ij = (Wj−1 , Wj ] for j = 1, 2, . . . , n. Given this we can define Gn as follows: It has vertex set Vn = [n] and an edge {x, y} , x ≤ y for each pair ℓi , ri , where ℓi ∈ Ix and ri ∈ Iy . We recast the construction of Bollobás and Riordan as follows: we can generate the sequence R1 , R2 , . . . , Rmn by letting 1/2  Υi Ri = , (1) Υmn+1 where Υ0 = 0 and ΥN = ξ1 + ξ2 + · · · + ξN for N ≥ 1 and where ξ1 , ξ2, . . . , ξmn+1 are independent exponential rate one random variables i.e. Pr(ξi ≥ 2 are independent and uniform in [0, 1] (as they x) = e−x for all i. This is because r12 , r22 , . . . , rmn are each chosen as the maximum of two uniform points) and the order statistics of N independent uniform [0, 1] random variables can be expressed as the ratios Υi /ΥN +1 for 1 ≤ i ≤ N. We refer to the distribution of ΥN as ERL(N), as it is known in the literature as the Erlang distribution. 2.2 Important properties The advantage of our modification of the variant of the Bollobàs and Riordan construction is that if we define ηi := ξ(i−1)m+1 + ξ(i−1)m+2 + · · · + ξim , then ηi is closely related to the size of Ii . It can then be used to estimate the degree of vertex i. This will simplify the analysis since ηi is simply a sum of exponentials. In this section, we make this claim (along with other more obscure asymptotic properties of this model) precise. In particular, we let E denote the event that the following properties hold for Gn . In the appendix, we prove that Gn has all these properties w.h.p. 5 (P1) For Υk,ℓ = Υk − Υℓ , we have Υk,ℓ " 1/2 Lθk,ℓ ∈ (k − ℓ) 1 ± 3(k − ℓ)1/2 # for (k, ℓ) = (mn + 1, 0) or   l=0 1 k−ℓ 2 ∈ {ω, ω + 1, . . . , n} and k − l ≥ log n k ≥ log30 n, l > 0  m  1/300 log n 0 < l < k < log30 n. Here, where n0 = λ20 n , ω log2 n λ0 = θk,ℓ 1 , log20/m n   log k    1/2   k = (k − ℓ)1/2   (k−ℓ)3/2 log n    n1/2   n ω 3/2 log2 n Similarly define θk =  1/2  k3/2 k log n 1/2 n   n ω 3/2 log2 n ω ≤ l < k ≤ log30 n, ω ≤ k ≤ n2/5 , l = 0, log30 n < k ≤ n2/5 , n2/5 < k ≤ n0 , n0 < k. k ≤ n2/5 , n2/5 < k ≤ n0 , n0 < k. #    1/2 " 1/2 1/2 Lθi i i 1 ± 1/2 ∼ for ω ≤ i ≤ n. (P2) Wi ∈ n i n " # 1/2 ηi 2Lθi ηi (P3) wi ∈ 1 ± 1/2 1/2 ∼ for ω ≤ i ≤ n. 1/2 2m(in) m i 2m(in)1/2 (P4) λ0 ≤ ηi ≤ 40m log log n for i ∈ [log30 n]. (P5) ηi ≤ log n for i ∈ [n]. Some properties give asymptotics for intermediate quantities in the Bollobas/Riordan model (e.g., (P2), (P3)), while the rest give worst-case bounds on parameters in various ranges for i. The very technical (P1) is just giving constraints on the gaps between the points Υk in the Bollobas/Riordan model. 2.3 Inequalities We will use the following inequalities from Hoeffding [9] at several points in the paper. Let Z = Z1 + Z2 + . . . + ZN be the sum of independent [0, 1] random variables and suppose that 6 µ = E(Z). Then if α > 1 we have n 2 o  (  exp − α3µ , α ≤ 1. α2 µ ≤ Pr(Z ≥ (1 + α)µ) ≤ exp −  2 + α/3 exp − αµ , α > 1. 3  βµ e Pr(Z ≥ βµ) ≤ e−µ , β>1 β  2  α µ , 0 ≤ α ≤ 1. Pr(Z ≤ (1 − α)µ) ≤ exp − 2 (2) (3) (4) Our main use for these inequalities is to get a bound on vertex degrees, see Section 2.4. In addition to these concentration inequalities, we use various inequalities bounding the tails of the random variable η. We note that the probability density φ(x) of the sum η of m independent exponential rate one random variables is given by φ(x) := xm−1 e−x . (m − 1)! That is, Pr(a ≤ η ≤ b) = Z b φ(y)dy. (5) a The equation (5) is a standard result, which can be verified by induction on m (for example, see exercise 4.14.10 of Grimmett and Stirzaker [8]). Although we will frequently need to bound the probability (5), this integral cannot be evaluated exactly in general, and thus we will often use simple bounds on φ(η). We summarise what we need in the following lemma: Lemma 2.1. (a) Pr(η ≤ xm) ≤ m(xe1−x )m for x ≤ 1 − 1 . m (6) (b) Pr(η ≤ x) ≤ (1 − e−x )m ≤ xm . (c) Pr(η ≥ βm) ≤  eβ eβ m ≤ e−3m/10 for β ≥ 2. (d) 2 m/3 Pr(η ≥ (1 + α)m) ≤ e−α for 0 < α < 1. (e) 2 m/2 Pr(η ≤ (1 − α)m) ≤ e−α 7 for 0 < α < 1. (7) Proof. (a) φ(η) is maximized at η = m − 1. Taking φ(mx) (x ≤ 1 − 1/m) as an upper bound on φ(y) for y ∈ [0, mx] and m! ≥ (m/e)m in (5) gives us (6). Q (b) Writing η = ξ1 + ξ2 + · · · + ξm we have Pr(η ≤ x) ≤ m i=1 Pr(ξi ≤ x). (c) If η = ξ1 + ξ2 + · · · + ξN , then with λ = (β − 1)/β, Pr(η ≥ βm) = Pr(eλη ≥ eλβm ) ≤ e−λβm E(eλη ) = e−λβm m Y E(eλξi ) = i=1 −λβm e (1 − λ)−m = (βe−(β−1) )m . (8) (d) Putting β = 1 + α into (8) we see that 2 m/3 Pr(η ≥ (1 + α)m) ≤ ((1 + α)e−α )m ≤ e−α . (e) With λ = α/(1 − α) we now have Pr(η ≤ (1 − α)m) = Pr(e−λη ≥ e−λ(1−α)m ) ≤ eλ(1−α)m E(e−λη ) = eλ(1−α)m m Y E(e−λξi ) = i=1 λ(1−α)m e 2.4 (1 + λ) −m 2 m/2 = ((1 − α)eα )m ≤ e−α . Properties of the degree sequence We will use the following properties of the degree sequence throughout: let !  1/2  n 1/2 i 5L log log n ζ(i) = 1− − 3/4 i n ω log n !  1/2  n 1/2 i 5L log log n + . 1− + 3/4 ζ (i) = i n ω log n (9) (10) Note that + ζ(i) ∼ ζ (i) ζ(i) ∼  n 1/2 i   2 log log n . if i ≤ n 1 − log n if i = o(n). Also, let d¯n (i) denote the expected value of dn (i) in Gn . Lemma 2.2. 8 (11) (12) (a) If E occurs then d¯n − m ∈ [ηi ζ(i), ηi ζ + (i)]. 2 η ζ(i)/2 i (b) Pr(dn (i) − m ≤ (1 − α)ηi ζ(i)) ≤ e−α for 0 ≤ α ≤ 1. 2 η ζ + (i)/3 i (c) Pr(dn (i) − m ≥ (1 + α)ηi ζ + (i)) ≤ e−α (d) Pr(dn (i) − m ≥ βηi ζ + (i)) ≤ (e/β)βηi ζ + (i) for 0 ≤ α ≤ 1. for β ≥ 2. (e) W.h.p. ηi ≥ λ0 and ω ≤ i ≤ n1/2 implies that dn (i) ∼ ηi (f ) W.h.p. ω ≤ i ≤ log30 n implies that dn (i) ∼ ηi
n 1/2 . i (g) W.h.p. ω ≤ i ≤ n1/2 implies dn (i) . max {1, ηi } (h) W.h.p. n1/2 ≤ i ≤ n implies dn (i) ≤ n1/3 . (i) W.h.p. 1 ≤ i ≤ log1/49 n implies that dn (i) ≥ (j) W.h.p. dn (i) ≥ n log1/20 n
n 1/2 i
n 1/2 . i . n1/2 . log1/20 n implies i ≤ log1/9 n. Proof. We defer the proof, which is straightforward but tedious, to the appendix. Remark 2.3. We will for the rest of the paper condition on the occurrence of E. All probabilities include this conditioning. We will omit the conditioning in the text in order to simplify expressions. 3 Analysis of the main loop Since the variation distance after Step 1 is o(1), it suffices to prove Theorem 1.2 under the assumption that we begin Step 2, with v1 chosen randomly, exactly according to the stationary distribution. The main loop consists of Steps 2–7. Let v0 = 1 and v1 , v2 , . . . , vs for s ≥ 1 be the sequence of vertices followed by the algorithm up to time s. Let ρt = vt+1 /vt , and define T1 , T2 by   T1 = min t : vt ≤ log30 n and T2 = T1 + 30ω log4/3 log n and T0 = min 2ω log4/3 n, T2 . (13) We will prove, see Lemma 3.2, that E(ρt ) ≤ 3 for 1 ≤ t ≤ T0 . 4 (14) Recalling that T is the time when Step 8 begins, we note that if T < t ≤ T0 then this statement is meaningless. So, we will keep to the following notational convention: if Xt is some quantity that depends on t ≤ T and t > T then Xt = XT . 9 Now, roughly speaking, if r = 2 log4/3 n and µ is the number of steps in the main loop, then we would hope to have   1 1 Pr(µ ≥ r) ≤ Pr ρ0 ρ1 · · · ρr ≥ ≤ n E(ρ0 ρ1 · · · ρr ) ≤ n n and so w.h.p. the algorithm will complete the main loop within 2 log4/3 n steps. Unfortunately, we cannot justify Q the last inequality, seeing as the ρt are not independent. I.e. we cannot replace E(ρ0 ρ1 · · · ρr ) by ri=0 E(ρi ). We proceed instead as in the next lemma. Lemma 3.1. Assuming (14) we have the w.h.p. DCA completes the main loop in at most T0 steps with SUCCESS. Proof. We let s0 denote the number of vertices visited by the main loop, and then define Zs = ρ0 ρ1 · · · ρs for s ≤ s0 , and Zs = ρ0 ρ1 · · · ρs0 ( 43 )s−s0 for s > s0 . Suppose first that T1 > ω log4/3 n. Now (14) and Jensen’s inequality implies that for s ≥ 1, min(s,s0 ) E(log(Zs )) = X E(log(ρi )) + i=0 s X log 43 min(s,s0 )+1 ≤ min(s,s0 ) X log E(ρi ) + i=0 s X log 43 ≤ s log(3/4). (15) min(s,s0 )+1 Now log(Zs ) ≥ (s − s0 ) log(3/4) − log n ≥ s log(3/4) − log n (16) since ρ1 ρ2 . . . ρs0 ≥ 1/n. Now let α = Pr(log(Zs ) ≤ (1 − β)s log(3/4)) where α, β are to be determined. Then, (15), (16) imply that (1 − α)(1 − β)s log(3/4) + α(s log(3/4) − log n) ≤ E(log(Zs )) ≤ s log(3/4). (17) Equation (17) then implies that α≥ βs log(4/3) . βs log(4/3) + log n Now putting s = ω log4/3 n and β = 1/2 we see that (18) becomes α ≥1− 2 = 1 − o(1). ω+2 So w.h.p. after at most ω log4/3 n steps, we will have exited the main loop with SUCCESS. 10 (18) Suppose now that T1 ≤ ω log4/3 n. Using the argument that gave us (18) we obtain T − T1 ≤ ω log4/3 log30 n w.h.p. To prove Lemma 3.2, we will use a method of deferred decisions, exposing various parameters of Gn as we proceed. At time t, we will consider all random variables in the model from Section 2.1 as being exposed if they have affected the algorithm’s trajectory thus far, and condition on their particular evaluation. To reduce the conditioning necessary, we will actually analyze a modified algorithm, NARROW-DCA(τ ), and then later show that the trajectory of NARROW-DCA(τ ) is the same as that of the DCA algorithm, w.h.p., when identical sources of randomness are used. NARROW-DCA(τ ) is the same as the DCA algorithm, except that for the first τ rounds of the algorithm, a modified version of Step 4 is used: Modified Step 4 Let  Ct = w1 , w2 , . . . , wm/2 be the m/2 neighbors of vt of largest degree from {1, . . . , Ψvt } where Ψ := (log log n)10 . For rounds τ + 1, τ + 2, . . . , the behavior of NARROW-DCA(τ ) is the same as for DCA. Notice that NARROW-DCA “cheats” by using the indices of the vertices, which we do not actually expect to be able to use. Nevertheless, we will see later that w.h.p., for τ = 2ω log4/3 n, the path of this algorithm is the same as for the DCA algorithm, justifying its role in our analysis. 3.1 Analyzing one step Our analysis of one step of the main loop consists of the following lemma: Lemma 3.2. Let ρt be the ratio of vt+1 /vt which appears in a run of the algorithm NARROWDCA(t). Then for all t ≤ T0 (see (13)), we have that E(ρt ) ≤ 3 4 and Pr (ρt ≥ Ψ) ≤ 1 . log2 n (19) The first statement ensures that NARROW-DCA(t) makes progress in expectation in the tth jump. The second part of this statement implies by induction that for any t ≤ ω log n, the behavior of NARROW-DCA(t) is identical to the behavior of the DCA algorithm for the first t steps. Thus together these statements give (14). To prove Lemma 3.2, we will prove a stronger statement which is conditioned on the history of the algorithm at time t. The history Ht of the process at the end of step t consists of 11 (H1) The sequence v1 , v2 , . . . , vt . (H2) The left-choices λ(vs , 1), λ(vs , 2), . . . , λ(vs , m), 1 ≤ s < t and the corresponding left neighbors NL (vs ) = {u1,s , u2,s , . . . , um,s }. These are the m ℓ′i s that correspond to the m ri ’s associated with vs as defined at the beginning of Section 2.1. (H3) The lists u′1,s , u′2,s , . . . , u′r,s of all vertices u′k,s which have the property that (i) vs ∈ NL (u′k,s) and (ii) u′k,s ≤ Ψvs for 1 ≤ s < t. (It is important to notice that s < t here.) (H4) The values ηvi and the intervals Ivi for i = 1, 2, . . . , t. (H5) The values ηw and the intervals Iw and the degrees deg(w), for w ∈ Here, St i=1 N(vi ). N(v) = NL (v) ∪ NR (v) where NR (v) = {w ≤ Ψv : v ∈ NL (w)} . We note that at any step t, and for a fixed random sequence used in the NARROW-DCA(t) algorithm, Ht contains all random variables which have determined the behavior of the algorithm so far, in the sense that if we modify any random variables from the random graph model described in Section 2.1 while preserving all values in the history, then the trajectory of the algorithm will not change. We write Ht to refer to a particular evaluation of the history (so that we will be conditioning on events of the form Ht = Ht ). Structure of the proof The essential structure of our proof of Lemma 3.2 is as follows: Part 1 We will define the notion of a typical history Ht . Part 2 We will prove that for t ≤ T0 and any typical history Ht , random variables ηv which are not explicitly exposed in Ht are essentially unconditioned by the event Ht = Ht (Lemma 3.3). Part 3 We will prove by induction that Ht is typical w.h.p., for t ≤ T0 . Part 4 We will use Part 2 and Part 3 to prove that for t ≤ T0 , E(ρt | Ht ) ≤ L3 2 21ηvt + + 2 3 mL m and Pr (ρt ≥ Ψ | Ht ) ≤ 1 log2 n (20) by using using nearly unconditioned distributions of random variables which are not revealed in Ht to estimate the probabilities of various events. Here E(ρt | Ht ) is short for E(ρt | Ht = Ht ). (Note that ηvt in (20) is simply a real number determined by Ht .) In this context, we always work under the assumption that Ht is typical. 12 Part 5 We will also prove for t ≤ T0 that E(ηvt+1 ) ≤ 4m. (21) Now the expected value statement in (19) follows from (21) and the first part of (20), by removing the conditioning on Ht . Part 1 Let Pt denote the sequence of vertices v1 , v2 , . . . , vt determined by the history Ht . We now define the notion of a typical history Ht . For this purpose, we consider the reordered values (t) (t) (t) 0 ≤ λ1 < λ2 < · · · < λN (t) where n o (t) (t) (t) (t) Λ0 = λ1 , λ2 , . . . , λN (t) = {λ(vs , i) : 1 ≤ s ≤ t, 1 ≤ i ≤ m} . (t) (t) Given this we define v = vj to be the index such that λj ∈ Iv and then let n o (t) (t) VL = vj : 1 ≤ j ≤ N(t) . We also define (t) VR = {v : v ∈ NR (Pt )} Now let us reorder n o (t) (t) (t) (t) (t) V (t) = x1 < x2 < · · · < xM (t) = VL ∪ VR . (t) (t) We define the extreme points x0 = 0 and xM (t)+1 = n + 1 and define M (t)+1 (t) Xj = (t) [xj−1 + (t) 1, xj − 1] and X (t) = (t) [ Xj = [n] \ V (t) (t) (t) and Nj = |Xj |, j=1 (t) Uj = [Wx(t) j−1 +1 , Wx(t) −1 ] and U (t) = j N [ (t) Uj (t) (t) and Lj = |Uj |. j=1 A typical history Ht , t ≤ T0 is now one with the following properties: (S1) There do not exist s1 , s2 ≤ t such that either (i) s1 ≤ t − 2 and vs1 and vs2 are neighbors or (ii) s1 ≤ t − 3 and there exists a vertex w such that w ∈ N(vs1 ) ∩ N(vs2 ). (We say that the path is self-avoiding.) (S2) The points of Λ(t) are well-separated, in the following sense: ( (t) log2 n xj−1 ≥ log30 n. (t) (t) |xj − xj−1 | ≥ log1/400 n Otherwise. 13 (22) We observe that (T1) If Ht is typical then vj+1 is chosen from X (j) for all j < t. (t) (t) (T2) Each Uj is the union of intervals Iv , v ∈ Xj . Part 2 We prove the following: Lemma 3.3. For any vertex v ∈ X (t) , any interval R ⊆ R, and any typical history Ht , we have that v ∈ / Pt ∪ N(Pt ) implies Pr (ηv ∈ R | Ht ) ∼ Pr (ERL(m) ∈ R) . (23) The following lemma is the starting point for the proof of Lemma 3.3. (t) Lemma 3.4. Let j ∈ [M(t)+1], let Ht be any typical history, and let X ′ be the value of Xj in Ht . Then the distribution of the random variables ηv , v ∈ X ′ conditioned on Ht = Ht is equivalent to of the random variables ηv , v ∈ X ′ conditioned only on the relationship P the distribution 2 2 v∈X ′ ηv = A1 − A0 , where A1 , A0 are the values of Wx(t) −1 and Wx(t) +1 , respectively, in Ht . j j−1 Proof. Suppose we fix everything except for ηv , v ∈ X ′ . By everything we mean every other ηw and all of the λ(v, i) and the random bits we use to make our choices in Step 5 of DCA; we ′ ′ ′ let Ht be the corresponding Phistory. Suppose now that we replace ηv , v ∈ X with ηv , v ∈ X without changing the sum v∈X ′ ηv . Then Wx(t) +1 remains the same, as it depends only on ηv j−1 for v ∈ / X ′ , and thus Wx(t) −1 remains the same as well, since the difference A21 − A20 is unchanged. j In particular, this implies that Ht remains a valid history. We confirm this by induction. Suppose that Hs , s < t remains valid. We first note that because the λ(vs , i) are unchanged, none of vs′ s (t) left neighbors are in Xj . Also, NR (vs ) and the vertex degrees for w ∈ NR (vs ) will not be affected (t) by the change, even if vs < min Xj . So Hs+1 will be unchanged, completing the induction. We are now ready to prove Lemma 3.3. (t) (t) Proof of Lemma 3.3. Suppose that v ∈ X ′ = Xj , then M = Nj Lemma 3.4 to write Pr(ηv ≤ x | Ht ) = Pr ηv ≤ x X w∈X ′ 14 ηw = A21 − ≥ ζn → ∞. We now use A20 , ! where A1 and A0 are the values of Wx(t) −1 and Wx(t) j j−1)+1 , respectively, in Ht , so that A1 − A0 is (t) the value of Lj in Ht . Now from (P1) we have that A := A21 − A22 ∈ [(1 − ε)mM, (1 + ε)mM] for M = |X ′ | w.h.p., for any ε > 0. Thus we fix any µ ∈ [(1 − ε)mM, (1 + ε)mM] and show that ! X ηw = µ = (1 + O(ε)) Pr (ERL(m) ≤ x) . Pr ηv ≤ x w∈X ′ The lemma follows since ε is arbitrary. We write Pr ηv ≤ x X w∈X ′ x ηw = µ ! η m−1 e−η (µ − η)(M −1)m−1 e−(µ−η) (Mm − 1)! · · M m−1 −µ dη ((M − 1)m − 1)! µ e η=0 (m − 1)!  (M −1)m−1 Q Z x m−1 −η 1 − µη eη m i=1 (Mm − i) η e · dη = µm η=0 (m − 1)!    2   Z x m−1 −η  m  η η η e dη · 1 + O · exp η − ((M − 1)m − 1) +O = µ µ2 M η=0 (m − 1)! Z x m−1 −η η e = (1 + O(ε)) dη. η=0 (m − 1)! = Z Here we used that Ht typical implies that M ≥ log1/400 n → ∞. Part 3 In the next section we will need a lower bound on vt+1 . Let ( 1 v ≥ log30 n log3 n φv = 1 . v < log30 n. (log log n)3 Lemma 3.5. W.h.p. ρt ≥ φvt for 1 ≤ t ≤ T0 . Proof. The values of λ(vt , i), i = 1, 2, . . . , m are unconditioned by Ht , see (H2). It then follows from (P2) that if vt ≥ log30 n then Pr(vt+1 ≤ φvt vt | Ht ) . m Wφvt m . mφ1/2 . vt = Wvt log3/2 n There are O(ω log n) choices for t and so this deals with vt ≥ log30 n. 15 (24) Now there are O(log log n) choices of t ∈ [T1 , T0 ] for which vt ≤ log30 n. In this case we can replace the RHS of (24) by 1/(log log n)3/2 . We will also need to bound the size of NR (vt ) for all t. Lemma 3.6. W.h.p., for all t ≤ T0 , |NR (vt )| ≤ ( log3 n (log log n)20 vt ≥ log30 n. vt ≤ log30 n. Proof. The size of NR (v), v = vt is stochastically bounded by Bin(Ψv, ηv /v). This is because if w ∈ NR (v) then w ≤ Ψv. Also, for any such w, the probability that it has v as a left neighbor is at most mwv /Ww . ηv /(vw)1/2 ≤ ηv /v. This uses property (S1) to see that the values of λ(w, i), i = 1, 2, . . . , m are unconditioned by Ht . Thus, if θv = log3 n if v ≥ log30 n and equal to (log log n)20 otherwise,     θ ηv θv Ψv eΨηv v Pr(|NR (v)| ≥ θv | Ht ) ≤ ≤ . (25) θv v θv 3 If v ≥ log30 n then the RHS of (25) is at most (e/ log n)log n which is clearly small enough to han20 dle T possible values for t. If v < log30 n then the RHS of (25) is at most (40e/(log log n)9 )(log log n) which is small enough to handle O(ω log log n) possible values for t such that v < log30 n. Continuing Part 3, we now show that the DCA walk doesn’t contain cycles. Lemma 3.7. W.h.p. the path Pt , t ≤ T0 is self avoiding. Proof. We proceed by induction and assume that the claim of the lemma is valid up to time t − 1. Now consider the choice of vt . Case 1: There is an edge vs vt where s ≤ t − 2: (a): vt ∈ NL (vs ) ∩ NL (vt−1 ). We bound the probability of this (conditional on E, Ht ) asymptotically by X X mwv X X ηv . . Wvt−1 2(vvt−1 )1/2 s∈[t−2] v∈NL (vs ) (26) s∈[t−2] v∈NL (vs ) Here, and throughout the proof of Case 1, v denotes a possibility for vt and mwv /Wvt−1 bounds the probability that vt−1 chooses v. Remember that these choices are still uniform, given the history. We split the sum in (26) as X X s∈[t−2] v∈NL (vs ) vs >log30 n ηv + 2(vvt−1 )1/2 X X s∈[t−2] v∈NL (vs ) vs ≤log30 n 16 ηv . 2(vvt−1 )1/2 Consider the first sum. There are less than t choices for s; m choices for v and ηv ≤ log n. Now v ∈ NL (vs ) and Lemma 3.5 implies that v ≥ log27 n. So we can bound the first sum by   1 1 1 . =o (# of s) · (# of v) · (max ηv ) · 1/2 ≤ T0 · m · log n · v log11 n log27/2 n Summing this estimate over t ≤ T0 gives o(1). For the second sum, we bound the number of choices of s by O(ω log log n) and ηv by O(log log n), since v ≤ vs . We use the fact (see Section 3.2) that vt−1 ≥ log1/100 n. So we can therefore bound the second sum by   1 1 1 (# of s) · (# of v) · (max ηv ) · 1/2 ≤b ω log log n · m · log log n · =o . log1/200 n log1/300 n vt−1 (We use A ≤b B in place of A = O(B).) There are O(ω log log n) choices for T0 ≥ t > s ≥ T1 and so we can sum this estimate over choices of t. (b): vt ∈ NL (vs ) ∩ NR (vt−1 ). Using vt ∈ NR (vt−1 ), we bound the probability of this asymptotically by X X s∈[t−2] v∈NL (vs ) vs >log30 n ηvt−1 + 2(vvt−1 )1/2 X X s∈[t−2] v∈NL (vs ) vs ≤log30 n ηvt−1 . 2(vvt−1 )1/2 For the first sum we use the argument of Case (a) without any change, except for bounding ηvt−1 by log n as opposed to bounding ηv by the same. This gives a bound   1 1 1 (# of s) · (# of v) · (max ηvt−1 ) · 1/2 ≤b T0 · m · log n · . =o v log11 n log27/2 n This is small enough to inflate by the number of choices for t. For the second sum we split into two cases: (i) vt−1 ≥ log30 n and (ii) vt−1 < log30 n. This enables us to control ηvt−1 . For the first case we obtain   1 1 1 (# of s) · (# of v) · (max ηvt−1 ) · 1/2 ≤ ω log log n · m · log n · =o . log15 n log13 n vt−1 The RHS is small enough to handle the O(ω log n) choices for t. For the second case we obtain (# of s) · (# of v) · (max ηvt−1 ) · 1 1/2 vt−1 ≤ ω log log n · m · log log n · 1 log1/200 n The RHS is small enough to handle the O(ω log log n) choices for t. 17 =o  1 log1/300 n  . (c): vt ∈ NR (vs ) ∩ NL (vt−1 ). Using vt ∈ NL (vt−1 ), we bound the probability of this asymptotically by X X X X ηv ηv + . 1/2 2(vvt−1 ) 2(vvt−1 )1/2 s∈[t−2] v∈NR (vs ) vs ≤log29 n s∈[t−2] v∈NR (vs ) vs >log29 n For the first sum we use v ≥ vs and the argument of Case (a) without change, but notice we split over vs > log29 n or not here. This gives a bound of   1 1 1 (# of s) · (# of v) · (max ηv ) · 1/2 ≤ T0 · m · log n · =o . v log12 n log29/2 n For the second sum we use v ≤ Ψvs to bound v by log30 n. We also use Lemma 3.6 to bound the number of choices of v by (log log n)20 . This gives a bound of   1 1 1 20 . (# of s)·(# of v)·(max ηv )· 1/2 ≤b ω log log n·(log log n) ·log log n· 1/200 = o log n log1/300 n vt−1 (d): vt ∈ NR (vs ) ∩ NR (vt−1 ). Using vt ∈ NR (vt−1 ), we bound the probability of this asymptotically by X X s∈[t−2] v∈NR (vs ) vs >log30 n ηvt−1 + 2(vvt−1 )1/2 X X s∈[t−2] v∈NR (vs ) vs ≤log30 n ηvt−1 . 2(vvt−1 )1/2 For the first sum we use v ≥ vs and Lemma 3.6 to bound the number of choices for v and then we have a bound of   1 1 1 3 . =o (# of s) · (# of v) · (max ηvt−1 ) · 1/2 ≤ T0 · log n · log n · v log15 n log9 n For the second sum we split into two cases: (i) vt−1 ≥ log30 n and (ii) vt−1 < log30 n. This enables us to control ηvt−1 . We also use Lemma 3.6 to bound the number of choices for v in each case. Thus in the first case we have the bound   1 1 1 3 (# of s) · (# of v) · (max ηvt−1 ) · 1/2 ≤b ω log log n · log n · log n · =o . log15 n log10 n vt−1 In the second case we have (# of s)·(# of v)·(max ηvt−1 )· 1 1/2 vt−1 20 ≤b ω log log n·(log log n) ·log log n· 1 log1/200 n =o  1 log1/300 n  Case 2: There is a path vs , v, vt where s < t. The calculations that we have done for Case 1 carry through unchanged. We just replace vt−1 by vt throughout the calculation and treat v as an arbitrary vertex as opposed to a choice of vt . 18 . (t) The xj are separated We now prove that w.h.p. points λi are well-separated. Let  J1 = j : vj ≥ log30 n . Lemma 3.8. Equation (22) holds w.h.p. for all t ≤ T0 . Proof. We consider cases. (t) (t) (t) Case 1: xj−1 , xj ∈ VR . For this we write ζv,w = ( log2 n min {v, w} ≥ log30 n. . log1/300 n Otherwise. Pr(∃1 ≤ s ≤ t, v ∈ NR (vs ), w ∈ NR (vt ) : |v − w| ≤ ζv,w | E, Ht) ≤ X X ηvs ηvt . (vs vt vw)1/2 1≤s≤t≤T 0 ≤ ≤ X v∈NR (vs ),w∈NR (vt ) |v−w|≤ζv,w X ζv,w ηvs ηvt (v − ζv,w )(vs vt )1/2 (27) 1≤s≤t≤T0 v∈NR (vs ) ∗ X ζs,t ηvs ηvt |NR (vs )| 2 . (vs vt )1/2 1≤s≤t≤T0 (28) ∗ Here ζs,t will be a bound on the possible value of ζv,w in (27). Case 1a: max {vs , vt } ≥ log29 n: ∗ In this case ζs,t ≤ log2 n and we can bound the summand of (28) by ∗ ζs,t · log2 n · log3 n · 1 log 29/2 n = 1 log 15/2 n . Multiplying by a bound T02 on the number of summands gives a bound of o(1). Here, and in the next case, we use Lemma 3.6 to bound |NR (vs )|. Case 1b: max {vs , vt } < log29 n: Here we have max {v, w} ≤ Ψ log29 n ≤ log30 n. In this case we can bound the summand of (28) by   1 (log log n)5 1/300 2 20 log n · (40m log log n) · (log log n) · . =o log1/100 n log1/200 We only have to inflate this by (T0 − T1 )2 = O((ω log log n)2 ). This completes the case where (t) (t) xj−1 , xj ∈ R(t) . 19 (t) (t) (t) Case 2: xj−1 , xj ∈ VL We first show that the gaps λj − λj−1 are large. Define β1 = and log1/300 n log15/2 n and β = . 2 n1/2 n1/2 ( β1 εj = β2 and σ1 = λj = λ(vt , i), vt ∈ J1 . otherwise. 1 log 15/2 n and σ2 = 1 log 1/200 n . We drop the superscript t for the rest of the lemma. Claim 3.9. Pr(∃λj ∈ Λ0 : λj−1 > λj − εj | Ht ) = o(1). Proof of Claim 3.9. This follows from the fact that Pr(∃j : λj−1 > λj − εj ) ≤ o(1) + (1 + o(1))(m2 T12 σ1 + m2 (T0 − T1 )(T1 σ1 + (T0 − T1 )σ2 )). We have fewer than m2 T12 choices for s = τ (j − 1), t = τ (j) ∈ J1 . Assume first that s < t. Given such a choice, we have that w.h.p. Wvt & log15 n/n1/2 by (P2). Now λj will have been chosen uniformly from 0 to ≈ Wvt and so the probability it lies in [λj−1, λj−1 + εj ] is at most ≈ β1 /Wvs , which explains the term m2 T12 σ1 . If s > t then we repeat the above argument with [λj−1 , λj−1 + ε1 ] replaced by [λj − ε1 , λj ] The term m2 (T0 − T1 )T1 σ1 arises in the same way with j − 1 ∈ J1 , j ∈ / J1 or vice-versa. The term m2 (T0 − T1 )2 σ2 arises from the case where j − 1, j ∈ / J1 . Here we can only assume 1/200 1/2 that Wj & log n/n . This follows from (P2), (P4) and Lemma 2.2 and the fact that we exit the main loop with SUCCESS when we see a vertex of degree at least n1/2 / log1/100 n. Assuming that s < t we see that the probability that λj lies in [λj−1 , λj−1 + ε2 ] is at most β2 /Wvt ∼ β2 /(log1/200 n/n1/2 ) = o(1). Given the Claim and (P4), (P5) we have that w.h.p. ( n 1 β1 − log 1/2 ≥ 2 β1 n Wv(t) −1 − Wv(t) +1 ≥ log n j j−1 ≥ 21 β2 β2 − 40mnlog 1/2 j ∈ J1 . j∈ / J1 (29) Now, Wv(t) −1 − Wv(t) j j−1 +1 =  Υm(τ (j)−1) Υmn+1 1/2 −  Υm(τ (j−1)+1) Υmn+1 20 1/2 = Υm(τ (j)−1) − Υm(τ (j−1)+1) 1/2 1/2 1/2 Υmn+1 (Υm(τ (j)−1) + Υm(τ (j−1)+1) ) . Or, ( β1 n1/2 1/2 1/2 1/2 ηu = (Wv(t) −1 − Wv(t) +1 )Υmn+1 (Υm(τ (j)−1) + Υm(τ (j−1)+1) ) ≥ j j−1 β2 n1/2 (t) j ∈ J1 . . j∈ / J1 X u∈Xj It follows that w.h.p. (t) |xj−1 (t) (t) (t) − (t) xj | = (t) |Xj | ≥ ( β1 n1/2 log n β2 n1/2 40m log log n j ∈ J1 . j∈ / J1 . (t) Case 3: xj ∈ VL , xj−1 ∈ VR Let θv = β1 , v ≥ log30 n and θv = β2 otherwise. We write ηvs wv + 2θv · 1/2 (vvs ) Wvt s,t,v,k  X  ηv ηv 2n1/2 θv ηvs s + .(30) . v(vs vt )1/2 v(vs vt )1/2 s,t,v,k Pr(∃s < t, v, k : v ∈ NR (vs ), λ(vt , k) ∈ Iv ± θv | E, Ht ) ≤ X We bound the sum in the RHS of (30) as follows: If max {v, vs } ≥ log30 n then we bound the first sum by n X 1 1 1 2 ≤b (ω 2 log2 n) · log2 n · log n · = o(1). (#s, t, k) · log n · · 15 v log n log15 n v=1 We bound the second sum by n X 1 1 1 15/2 (#s, t, k) · 2 log n · log n · · ≤ (ω 2 log2 n) · log15/2 n · log n · log n · = o(1). 15 15 v log n log n v=1 When max {v, vs , vt } < log30 n we bound the first sum by log30 n (#s, t, k) · (40 log log n)2 · X 1 1 ≤b · 1/200 v log n v=1 (ω log log n)2 · (log log n)2 · log log n · 1 log 1/200 log 1/200 n = o(1). (31) We bound the second sum by log30 n (#s, t, k) · log 1/300 n · 40 log log n · X 1 1 ≤b · 1/200 v log n v=1 (ω log log n)2 · log1/300 n · log log n · 1 n = o(1). (32) Finally, if max {v, vs } < log30 n ≤ log30 n then we have to replace (#s, t, k) in (31), (32) by 1/2 O(ω 2 log n log log n). But this is compensated by a factor 1/vt ≤ 1/ log15 n. It follows that (29) holds w.h.p. and the proof continues as for Case 2. 21 Part 4 We now assume t ≤ T0 . We begin by showing that DCA only uncovers a small part of the distribution of the η’s. Let Ξt = Pt ∪ N(Pt ) and St,j = X wv . v∈Ξt Lemma 3.10. W.h.p., St,j = o(Wj ) for log1/100 n ≤ j and 1 ≤ t ≤ T0 . Proof. Assume first that j ≥ log30 n. It follows from (P2), (P3), (P5) and Lemma 3.6 that w.h.p.   max ηvs T0 log n(m + log3 n) ω log5 n St,j ≤ T0 × × (m + max |NR (vs )|) . =O . n1/2 2mn1/2 n1/2  10   1/2 log n j . =Ω Wj ≥ (1 − o(1)) n n1/2 This completes this case. Now assume that j ≤ log30 n. (P2), (P3), (P4) and Lemma 3.6 that w.h.p.  1/2 ω log4/3 log n j 20 St,j . 40m log log n × × (m + (log log n) ) ≪ Wj ∼ 1/2 2mn n for log1/100 n ≤ j ≤ log30 n. Dealing with left neighbors The calculation of the ratio ρt takes contributions from two cases: where vt+1 is a left-neighbor of vt , and where vt+1 is a right-neighbor of vt . Lemma 3.11. 2 E(ρt 1vt+1 <vt | Ht ) ≤ . 3 Proof. Let D denote the (m/2)th largest degree of a vertex in NR (vt ). We write X E(ρt 1vt+1 <vt | Ht | D = d) Pr(D = d) E(ρt 1vt+1 <vt | Ht ) = d  ζd ≤ E vt d   ζD , =E vt X  22 Pr(D = d) where ζd is the index of the smallest degree left neighbor of vt that has degree at least d. We let ζd = 0 if there are no such left neighbors. We now couple ζ with a random variable that is independent of the algorithm and can be used in its place. Going back to Section 2.1 let us associate ℓk for k ≥ ω with an index µk chosen uniformly from [⌊k/m⌋]. In this way, vertex i ≥ ω is associated with m uniformly chosen vertices ai,1 , ai,2 , . . . , ai,m in [i − 1]. Furthermore, we can couple these choices so that if NL (i) = {bi,1 , bi,2 , . . . , bi,m } then given we have (i) Pr(bi,j ≤ ai,j ) ≥ 1 − o(1) and (ii) bi,j ≤ 2ai,j for all i, j. This because Pr(bi,j ≤ k) ∼ Wk /Wi ∼ (k/i)1/2 (giving (i)) and (k/i)1/2 ≥ k/i and (1 − o(1))(k/i)1/2 ≥ k/2i (giving (ii)). So now let µ be the index of the uniform choice associated with the largest degree left neighbor of vt that has degree at least D. Thus     ζD µ 1 2 .E = + o(1) ≤ . E vt vt 2 3 Dealing with right neighbors It will be more difficult to consider the contribution of right-neighbors. In preparation, for λ0 ≤ γ ≤ 1 − 1/m we define ∆iγ := m + γmζ(i) where ζ(i), ζ + (i) are defined in (9), (10) respectively. We note that ηi ζ(i) is a lower bound for the expected degree of vertex i, i ≥ ω, see Lemma 2.2(a). Note also that ηi ζ + (i) is an upper bound for the expected degree of vertex i, i ≥ ω. The parameter ∆iγ is a degree threshold. For a suitable parameter γ, we wish it to be the case that there should be many left-neighbors but few right-neighbors which have degree greater than ∆iγ . We define   γi∗ = max γ : j ∈ NL (i) : dn (j) ≥ ∆iγ ≥ m/2 . ∆vγtv∗ is a lower bound on the degree needed for vertex j > vt to be considered by DCA as the t next vertex; thus we proceed by analyzing the distribution of γv∗t . We first derive upper bounds for Pr(γv∗t ≤ γ | Ht ). Lemma 3.12. There exists c1 > 0 such that 1/2 )c1 m + me−c1 γ Pr(γv∗t ≤ γ | Ht ) . (γ 1/2 e1−γ 1/2 )c 1 m Pr(γv∗t ≤ 45 | Ht ) . e−c1 m . Pr(γv∗t ≥ γ | Ht ) . γ −c1 m , 2 1 , 0≤γ≤ . 8 1 −c1 /γ 1/2 + me , 0≤γ≤ . 8 Pr(γv∗t ≤ γ | Ht ) . (γ 1/2 e1−γ 2 5 γ ≥ 10 . 23 1/2 m (33) (34) (35) (36)   Proof. For j < vt , we define events Aj = ηj ≤ γ 1/2 m and Dj = dn (j) ≤ ∆vγt . We need to T  estimate Pr j∈S Dj for subsets S ⊆ NL (vt ) of size m/2. We write \ j∈S Dj ⊆ \ (Aj ∪ (Āj ∩ Dj )) ⊆ j∈S \ Aj ∪ j∈S [ (Āj ∩ Dj ). (37) j∈S Now, using inequality (6) and equation (23), we see that if 0 ≤ γ ≤ 1/8 then for j < vt , Pr(ηj ≤ γ 1/2 m | Ht ) . m(γ 1/2 e1−γ 1/2 )m . (38) The RHS of (38) includes a factor of 1 + o(1) due to conditioning on E, Ht . So, Pr \ Aj j∈S Ht ! . (m(γ 1/2 e1−γ 1/2 )m )|S| . (39) Furthermore, because j ∈ NL (vt ) implies that i ≥ j and hence ζ(vt ) ≤ ζ(j), Pr((dn (j) ≤ ∆iγ ) ∧ Āj | Ht ) ≤ Pr (dn (j) − m ≤ γmζ(j) | Ht ) Pr(ηj > γ 1/2 m|Ht )   (1 − γ 1/2 )2 1/2 γ mζ(j) . . exp − 2 (40) Explanation of (40): We remark first that the conditioning on E, Ht only adds a (1+o(1)) factor to the upper bound on our probability estimate. We now apply Lemma 2.2(b) with 1 − α = γ and ηj ≥ γ 1/2 m. From (37) (summing over all m/2 subsets of NL (vt )) and (40) (summing over NL (vt )) we obtain  Pr(γv∗t ≤ γ | Ht ) = Pr(| j ∈ NL (vt ) : dn (j) ≤ ∆iγ | ≥ m/2| | Ht ) .    m/2 (1 − γ 1/2 )2 1/2 m 1/2 1−γ 1/2 m 2 (m(γ e ) ) + m exp − γ mζ(vt ) . (41) 2 We observe that j ∈ NL (vt ) implies that dn (j) ≥ m + 1. So, m + 1 < ∆iγ implies ζ(vt ) > 1 . mγ (42)  m/2 1/2 2 1/2 1−γ 1/2 m Using (42) in (41) verifies (34), after bounding 2 (m(γ e ) ) by (γ 1/2 e1−γ )c1 m . m From (37) and (40), Pr(γv∗t ≤ γ | Ht ) . (m(γ 1/2 e1−γ 1/2 )m )m/2 + (1 − γ 1/2 )2 1/2 γ mζ + Pr(A1 | Ht ) + Pr(Ā1 | Ht ) m exp − 2 −1 24   9n 10  (43) Here A1 =   9n NL (vt ) ∩ 10  m ≥ 4  . Explanation of (43): The first term is from (39). If Ā1 holds then vt has at least one left neighbor j ≤ 9n/10. The final term comes from using (40) and ζ(j) ≥ ζ(9n/10). The factor Pr(Ā1 )−1 handles the conditioning on Ā1 . The factor m is the union bound for choices of j.   Now NL (vt ) ∩ 9n is dominated by the binomial Bin(m, n/10) and so Pr(A1 | Ht ) ≤ e−d1 m . 10 Now ζ(9n/10) ≥ 1/20 and plugging these facts into (43) yields (33). Here we have absorbed the 1/2 e−d1 m term into me−c1 γ m and we will do so again below. We continue with the proof of (35). For j ∈ NL (vt ), we observe that if dn (j) ≤ ∆vγt and γ ≤ 54 then 5m dn (j) − m ≤ ζ(vt ). 4 We now estimate the probability that a uniform random choice of j ∈ NL (vt ) (for fixed Ht , which determines vt ) has certain properties. We first observe that     W3i/5 3i Pr j ≥ Ht . 1 − ∼ 5 Wvt  1/2 ! 2 3 < . 1− 5 5 (44) (For this we used (P2).) Now (6) implies that Pr(ηj ≤ 0.99m | Ht ) ≤ e−d2 m . (45) Moreover, for ηj > 0.99m and j < 3vt /5, we have ζ(j) = ζ(vt )  1/2 vt j where ε= Thus we have 1− 1−
j 1/2 −ε n
1/2 vt −ε n !  1/2 vt ≥ j 5L log log n . ω 3/4 log n (46)  1/2 5 E(dn (j) − m | Ht ) ≥ ηj ζ(j) & 0.99m ζ(vt). 3 Now 0.99 × (5/3)1/2 = 1.278.. > 1.01 × 5/4 and so    ζ(j)ηj 5ζ(vt ) Ht ≤ Pr dn (j) − m ≤ Pr dn (j) − m ≤ 4 1.01 25 Ht  ≤ e−d3 ηj ζ(j) ≤ e−d4 m (47) using Lemma 2.2(b). It follows from (44) and (45) and (47) that       5 m 2 ∗ −d2 m −d4 m Pr γvt < ≥ ≤ e−d3 m . Ht ≤ Pr Bin m, e +e + 4 5 2 This completes the proof of (35). To deal with (36) we observe that if dn (j) ≥ ∆vγt and γ ≥ 105 then vt j ∈ NL (vt ) and j ≤ 1/2 γ But or ηj ≥ γ 1/2  1/2 j m ≥ γ 1/4 m or dn (j) − m ≥ γ 3/4 ηj ζ(j). vt  vt Pr j ∈ NL (vt ) and j ≤ 1/2 γ Ht And, using (P3) and γ ≥ 105 ,  Pr ηj ≥ γ 1/4 m Ht  . vt X l=2vt /γ  wl Wvt . Z Wvt /γ 1/2 1 . 1/4 . Wvt γ ∞ ηl =γ 1/4 m (48) ηlm e−ηl dηl (m − 1)! 1/4 vt X e−γ m . 2(vt l)1/2 l=2vt /γ . e−γ 1/4 m . (49) Lastly, using (44), (45) and Lemma 2.2(d) and ζ + (j) . ζ(j) for j ≤ 3n/5 we have
2 3/4 Pr dn (j) − m ≥ γ 3/4 ηj ζ(j) | Ht ≤ + e−d2 m + e−γ η ≤ 0.41. 5 (50) It follows from (48), (49) and (50) that     
1 m ∗ −γ 1/4 m Pr γvt ≥ γ | Ht . Pr Bin m, (1 + o(1)) . e−d3 m . +e + 0.41 ≥ γ 1/4 2 This completes the proof of the lemma. Corollary 3.13. W.h.p. γv∗s ≥ 1/(log log n)2 for s = 1, 2, . . . , T = O(log n). Proof. The value of γv∗s is determined when vs is first visited and in this case we can apply Lemma 3.12. In which case the result follows directly from (34). We now have a handle on the distribution of γv∗t . We now put bounds on the expected number of j > vt that can be considered to be a candidate for vt+1 , conditioned on the value of γv∗t . In particular, we let  Dγvt = j > vt : dn (j) ≥ ∆vγt . 26 We will bound the size of Dγi by dividing Dγi into many parts bounding each part; in particular, κ ∈ N we let h i  i, i2 ∩ D i κ = 0. γ γ i (51) Jγi,κ = h

 i2 1 + κ−1 , i2 1 + κ ∩ D i 1 ≤ κ ≤ 2nγ 2 L . γ γ L γ L i Note that Jγi,0 = ∅ if γ ≥ 1. Finally, we let rγi,κ := |Jγi,κ | rγi and := X rγi,κ and siγ := κ≥0   i X κ + 1 i,κ j≤ 2 rγ . 1+ γ κ≥0 L i,κ X X (52) κ≥0 j∈Jγ   Remark 3.14. We have that E m2 v1t svγt | Ht is an upper bound on the expectation of the ratio , conditioned on the event that vt+1 > vt , since each right neighbor whose index is ρt = vt+1 vt included in the sum svγt has probability of at most m2 of being chosen by the algorithm.   3L3 Lemma 3.15. If vt ≤ n 1 − m then E(rγvt | Ht ) ≤ svγt E | Ht vt Moreover,   ≤
ηvt L max {0, (1 − γ)} + 7 + 10Le−c2 γL . γL (53)
ηvt L max {0, (1 − γ)} + 13 + 100Le−c2 γL . 3 γ L If vt ≤ n/5 and κ ≥ (log log n)4 and γ ≥ 1/(log log n)2 then Pr(rγvt ,κ > 0) ≤ (54) 1 . log2 n (55) Note that (55) implies the second inequality in (19). Proof. Recall from Lemma 3.10 that w.h.p., ST,j = o(Wj ) for j ≥ log1/100 n. (56) We write E(rγvt ,κ | Ht ) Z ∞ m−1 −ηj X
ηj e mwvt . Pr dn (j) ≥ ∆vγt | Ht , ηj dηj ηj =0 (m − 1)! vt ,κ Wj − St,j (57) j∈Jγ . ηvt X v ,κ j∈Jγ t 1 2(vt j)1/2 Z ∞ ηj =0
ηjm−1 e−ηj Pr dn (j) ≥ ∆vγt | Ht , ηj dηj . (m − 1)! 27 (58) Explanation of (57) and (58): We sum over the relevant j and fix ηj . We multiply by the density of ηj and integrate. Using (56) we see that mwvt ηvt mwvt ∼ ∼ . Wj − St,j Wj 2(vt j)1/2 This is asymptotically equal to the expected number of times j chooses vt as a neighbor. Thus E(rγvt ,κ | Ht ) . X v ,κ j∈Jγ t where (59)
ηjm−1 e−ηj Ij = Pr dn (j) ≥ ∆vγt | Ht , ηj dηj ≤ 1. ηj =0 (m − 1)! Z If m is large then ηvt Ij 2(vt j)1/2 ∞ E rγvt ,0 | Ht Continuing, for κ ≥ 1, we write:
( 0 . P ηvt v j∈J0 t (γ) 2(vt j)1/2 . ηvt 1−γ γ γ ≥ 1. γ < 1. Ij . A1 + A2 + A3 , (60) (61) where A1 = Z A2 = Z A3 = Z (1−1/L)γm ηj =0 (1+1/L)m
ηjm−1 e−ηj Pr dn (j) ≥ ∆vγt | ηj dηj . (m − 1)! ηj =(1−1/L)m ∞ ηj =(1+1/L)m
ηjm−1 e−ηj Pr dn (j) ≥ ∆vγt | ηj dηj (m − 1)!
ηjm−1 e−ηj Pr dn (j) ≥ ∆vγt | ηj dηj . (m − 1)! and then write rγvt ,κ = rγvt ,κ,1 + rγvt ,κ,2 + rγvt ,κ,3. Here rγvt ,κ,l is equal to the RHS of (59) with Ij replaced by Al . The implicit (1 + o(1)) factor in
(61) arises from replacing
Pr dn (j) ≥ ∆vγt | ηj , Ht by Pr dn (j) ≥ ∆vγt | ηj in the integrals, i.e., ignoring the conditioning due to Ht . Since j > vt , the only effect  of Ht is on Wj through wvt . Here we have that w.h.p.
ηvt log n vt 1/2 Wj ∼ n = o(Wvt ). and wvt ∼ 2m(vt n)1/2 = O 2m(v 1/2 t n) Case 1: n1 ≤ vt < n/5: Note that in this case 1 ζ(vt ) ≥ 2  1/2 n ≥ 1. vt In the following we use Lemma 2.1 to estimate the integrals over ηj . We observe that E(rγvt ,κ,1 | Ht ) 28 (62) . X ηvt 2(vt j)1/2 X ηvt 2(vt j)1/2 v ,κ j∈Jγ t . v ,κ j∈Jγ t Z (1− 1 )m m−1 −ηj L
ηj e Pr dn (j) ≥ ∆vγt | Ht , ηj dηj , (m − 1)! ηj =0 ( ! Z (1− 1 )m m−1 −ηj L ηj e 1 1 ≤ κ ≤ 10L, · dηj , (63) (m − 1)! (e(L/κ)1/2 )d0 γmζ(vt ) κ > 10L ηj =0 Explanation of (63): We remark first that the conditioning on Ht only adds a (1 + o(1)) factor to the upper bound on our probability estimate. We will use Lemma 2.2 to bound the probability that degrees are large. Now with our bound on vt and within the range of integration, the ratio of ∆vγt − m to the mean of dn (j) − m is ∆vγt − m ζ + (j)  1/2  1− γm vnt =  1/2  ηj nj 1− 
vt 1/2 n  1/2 − o(1) γm j ≥  &
ηj vt j 1/2 + o(1) n  1/2   κ−1 κ 1/2 L 1+ ≥ when κ ≥ 10L. (64) L−1 L L We then use (11) and Lemma 2.2(d) with β = (κ/L)1/2 . Continuing, we observe that 1/2   κ 1/2 κ−1 1 1+ − 1+ ≤ L L 2L and so E(rγvt ,κ,1 ηv e−m/(2L | Ht ) ≤ t γvt 1/2 2) ηv e−m/(2L ≤ t γL 2)  1/2   1/2 ! κ+1 κ−1 vt vt × 1+ − 1+ L L ( ! 1 1 ≤ κ ≤ 10L, (e(L/κ)1/2 )d0 γmζ(vt ) κ > 10L, ( ! 1 1 ≤ κ ≤ 10L, (e(L/κ)1/2 )d0 γmζ(vt ) κ > 10L. (65)  · (66) Continuing, it follows from (65) that X v ,κ j∈Jγ t 1 j 1/2 ≤ vt 1/2 . γL E(rγvt ,κ,2 | Ht ) Z (1+1/L)m X
ηjm−1 e−ηj ηvt vt Pr d (j) ≥ ∆ | H , η dηj ≤ n t j γ 1/2 ηj =(1−1/L)m (m − 1)! vt ,κ 2(vt j) j∈Jγ 29 (67) ≤ X v ,κ j∈Jγ t ηvt 2(vt j)1/2    1 n o d1 γmζ(vt ) × exp − L2   (e(L/κ)1/2 )d0 γmζ(vt )  1 ηvt  −d1 γmζ(vt )/L2 × e ≤ γL   (e(L/κ)1/2 )d0 γmζ(vt ) κ ≤ 3, 4 ≤ κ ≤ 10L, (68) κ > 10L, 1 ≤ κ ≤ 3, 4 ≤ κ ≤ 10L, κ > 10L. (69) where we have used (67). Explanation for (68): We proceed in a similar manner to (64) and use   1/2 
vt 1/2 n   vt 1 − − o(1) γm ∆γ − m vt n 1 1 m. =  ≥ 1 + if κ ≥ 4, ηj ≥ 1 −  1/2 
ζ + (j) L L j 1/2 n 1− n +ε ηj j Then we use Lemma 2.2(c),(d). Continuing, E(rγvt ,κ,3 | Ht ) ≤ X v ,κ j∈Jγ t ηvt 2(vt j)1/2 Z ∞ ηj =(1+1/L)m
ηjm−1 e−ηj Pr dn (j) ≥ ∆vγt | Ht , ηj dηj (m − 1)! (70) We bound the integral in (70) by something independent of j and then as above, there is a factor ηvt /γL arising from the sum over j. For all 1 ≤ κ ≤ 80L + 1, we simply use the bound   Z ∞ ηjm−1 e−ηj d2 m dηj ≤ exp − 2 . 1 L ηj =m(1+ L ) (m − 1)! (71) For κ ≥ 80L + 2, we split the integral from (70) into pieces B1κ , B2κ (whose definition depends on κ), which we will bound individually. In particular, we use B1κ = Z ≤ Z ∞ ηj =m(1+ κ−1 L ) 1/4 ∞ ηj =m(1+ κ−1 L ) 1/4
ηjm−1 e−ηj Pr dn (j) ≥ ∆vγt | Ht , ηj dηj (m − 1)! ηjm−1 e−ηj dηj (m − 1)! 1/4 ≤ e−d3 m(κ/L) (72) and 30 B2κ m(1+ κ−1 L ) 1/4
ηjm−1 e−ηj Pr dn (j) ≥ ∆vγt | Ht , ηj dηj = 1 ηj =m(1+ L ) (m − 1)!  1/4 !  1/4 d4 γmζ(vt ) κ − 1 eL ≤ Pr dn (j) ≥ ∆vγt Ht , ηj ≤ m 1 + ≤ (73) L κ1/4 Z to bound the integral in (70) by B1κ + B2κ for all κ ≥ 80L + 2. Therefore, gathering the many terms together (and using that κ ≤ on m large to allow crude upper bounding, we see that 2nLγ 2 vt from (51)) and relying γL 2 E(rγvt | Ht ) . L max {0, (1 − γ)} [from (60)] + 10Le−m/(2L ) [from (66)] ηvt 2  2nLγ  X/i  eL1/2 d0 γmζ(vt ) −d1 γmζ(vt )/L2 −m/(2L2 ) + 10Le [from (69)] + 2 + e [from (66) and (69)] 1/2 κ κ=10L   d2 m + 6[from (69)] + 100L exp − 2 [from (71)] L 2  1/4 d4 γmζ(vt ) ! 2nLγ /i X eL 1/4 + [from (72) and (73)]. (74) e−d3 m(κ/L) + 1/4 κ κ=80L+2 We first observe that if vnt < γ102 then the summations κ = 10L, . . . , 2nLγ 2 /vt etc. above are empty. For larger n/vt we can therefore assume that γm(n/vt )1/2 ≥ m which implies (see (62)) that γmζ(vt ) ≥ m/2 and then we can assume that 2nLγ 2 /vt X  eL1/2 d0 γmζ(vt ) 1 ≤ and 1/2 κ 1000 κ=10L 2nLγ 2 /vt X  eL1/4 d4 γmζ(vt ) 1 ≤ . 1/4 κ 1000 κ=80L+1 (75) Plugging these estimates into (74) and making some simplifications, we obtain (53). Going back to (52) we have   vt 2sγ γ 3L Ht ≤ L max {0, (1 − γ)} E ηvt mi −m/(2L2 ) + 200Le −d1 γmζ(vt )/L2 + 100Le  2  2nLγ X/vt 2κ  eL1/2 d0 γmζ(vt ) L κ1/2 κ=10L  1/4 d4 γmζ(vt ) ! eL 1/4 e−d3 m(κ/L) + . κ1/4 −m/(2L2 ) + 2+e 2  2nLγ  X/vt 2κ d m 2 + 12 + 104 L exp − 2 + L L κ=80L+2 Making similar estimates to what we did for (75) gives us (54). We obtain (55) from (P5), (66), (69), (72) and (73). Indeed, if Jγvt ,κ 6= ∅ then from its definition 2n 1/2 we must have vt ≤ 2Lγ . Together with vt ≤ n/5 we obtain that ζ(vt ) ≥ 2Lκ1/2 γ . Thus, in this κ−1 31 case,  eL1/2 κ1/2 d0 γmζ(vt )  eL1/2 (log log n)2 d0 m(log log n)2 /2L1/2   1 ≤ =o . (76) log10 n This deals with the probabilities in (66) and (69). For (69) we rely m large to to show that 1/4 e−d3 m(κ/L) = o(1/ log10 n). Equation (72) is dealt with in a similar manner to (66). Here we  1/4 d4 γmζ(vt ) which is the square root of (76). have eL κ1/4   3 Case 2: n/5 ≤ vt ≤ n 1 − 3L : m The upper bound on vt implies that mζ(vt ) ≥ L3 . Using the same definitions of rγvt ,κ,l , l = 1, 2, 3 as above: X E(rγvt ,κ,1 | Ht ) ≤ X X ≤ X X κ≥1 j∈Jγvt ,κ κ≥1 κ≥1 j∈Jγvt ,κ ηvt 2(vt j)1/2 Z (1− 1 )m m−1 −ηj L ηj e dηj , (m − 1)! ηj =0 ηvt 2 e−m/(2L ) , 1/2 2(vt j) from Lemma 2.1(e),  1/2 ηvt n 2 ≤ e−m/(2L ) γ vt 51/2 ηvt −m/(2L2 ) e . ≤ γ X E(rγvt ,κ,2 | Ht ) κ≥1 . X X ηvt 2(vt j)1/2 Z X X ηvt 2(vt j)1/2 Z κ≥1 j∈Jγvt ,κ . κ≥1 j∈Jγvt ,κ (1+1/L)m ηj =(1−1/L)m (1+1/L)m ηj =(1−1/L)m   2 X ηv  −d5 γL t 4e ≤  d γL3  γL  κ≥1  eL1/2 6 κ1/2 X E(rγvt ,κ,3 | Ht ) ≤ κ ≤ 3, 4 ≤ κ ≤ 10L, κ > 10L, X X κ≥1 j∈Jγvt ,κ κ≥1 ≤
ηjm−1 e−ηj Pr dn (j) ≥ ∆vγt | ηj dηj (m − 1)!   κ ≤ 3, 1 n o ηjm−1 e−ηj  d5 γmζ(vt ) 4 ≤ κ ≤ 10L, · exp − L2 (m − 1)!   (e(L/κ)1/2 )d6 γmζ(vt ) κ > 10L, X X κ≥1 v ,κ j∈Jγ t ηvt 2(vt j)1/2 Z ∞ ηj =(1+1/L)m ηvt −m/(3L2 ) e , 2(vt j)1/2 32 ηjm−1 e−ηj (m − 1)! from Lemma 2.1(d),  1/2 ηvt −m/(3L2 ) n e ≤ γ vt 1/2 5 ηvt −m/(3L2 ) ≤ e . γ The above upper bounds are small enough to give the lemma in this case, without trouble. We are now in a position to prove (20). We confirmed the second part of the statement (20) above, using (55), so only the first part remains. The first part follows immediately from Lemma 3.11 and the following, by addition: Lemma 3.16. E(ρt 1vt+1 ≥vt | Ht ) ≤ Proof. We consider cases.  Case 1: n1 ≤ vt ≤ n 1 − 3L3 m 21ηvt L3 + 2. mL m  : Then, E(ρt 1vt+1 ≥vt | Ht ) ≤ I1 + I2 + I3 + I4 , where 5/4 I1 = Z ≤ Z γ=1/8 5/4 γ=1/8 ≤ E(ρt 1vt+1 ≥vt | γv∗t = γ)d Pr(γv∗t ≤ γ),  1000ηvt m −c1 m 2 × 83 × ηvt × mL Z , ≤ ηvt e Z 10000  I2 = γ=5/4 ≤ I3 = 7L + 13 + 100Le−c2 γL 8  d Pr(γv∗t ≤ γ), by Remark 3.14, Lemma 3.15, 5/4 γ=1/8 d Pr(γv∗t ≤ γ), from (35). (77)  2 × ηvt × (13 + 100Le−c2 γL ) d Pr(γv∗t ≤ γ), mγ 3 L 20ηvt .  ZmL ∞ γ=10000  (78)  2 × ηvt × (13 + 100Le−c2 γL ) d Pr(γv∗t ≤ γ) mγ 3 L Z ∞ 27ηvt γ −cm dγ, from (36), ≤ 15 10 Lm γ=100 27ηv 1 = 15 t × 2(cm−1) . 10 Lm 10 (cm − 1) Z 1/8 I4 = E(ρt 1vt+1 ≥vt | γv∗t = γ)d Pr(γv∗t ≤ γ) γ=0 33 (79) 1/2 ≤ e−d0 m , (80) 1/2 To obtain the term e−d0 m in (80) we use (33) and (34) to obtain  max γ ∈ [0, 1/8] : γ −2 Pr(γ ∗ ≤ γ) ≤ o n 2 1/2 2 max γ ∈ [0, 1/8] : (γ 1/2−4/(c1 m ) e1−γ )c1 m + oo n n −2 −c1 mγ 1/2 −c1 /γ 1/2 max γ ∈ [0, 1/8] : mγ min e ,e 2 )−1/2 ≤ (84/(c1 m 1/2 e1−1/8 2 1/2 )c1 m + m2 e−c1 m . The first case of the lemma now follows from (77), (78), (79) and (80).   3 : Case 2: vt > n 1 − 3L m   3 We observe first that n ≤ vt 1 + 4L . Then we let Z = dn (vt ) − m be the number of right m neighbors of vt . Furthermore, n X wi . E(Z | Ht ) . W j j=v +1 t   3 vt 1+ 4L m X j=vt +1 ηvt . ηvt 2(vt j)1/2 !  1/2 4L3 4L3 1+ − 1 ≤ ηvt . m m Case 2a: ηvt ≥ 1/L1/2 . We use (81) and Lemma 2.2(d) to prove   ηvt 1/2 Pr Z ≥ Ht ≤ e−d1 ηvt L ≤ e−d1 L . L (81) (82) Then we can write   4L3 2ηvt 3ηvt 1/2 E(ρt 1vt+1 ≥vt | Ht ) ≤ 1 + × e−d1 L + ≤ . m Lm Lm   3 Explanation: ρt will be at most 1 + 4L if the unlikely event in (82) occurs. Failing this, the m chance that ρt > 1 is at most 2Z m ≤ 2ηvt . Lm Case 2b: ηvt < 1/L1/2 . It follows from (81) that E(Z | Ht ) . 4L5/2 /m. It then follows from Lemma 2.2(d) that   L3 1/2 Ht ≤ e−d2 L . Pr Z ≥ 3m We then have E(ρt 1vt+1 ≥vt | Ht ) ≤  4L3 1+ m  34 × e−d2 L 1/2 + 2L3 L3 ≤ . 3m2 3m2 Part 5 We now prove (21). To do this, we will obtain a recurrence for E(ηvt+1 | Ht ), and, at the end, obtain the bound 4m by averaging over the possible histories Ht . We begin by writing E(ηvt+1 | Ht ) = E(ηvt+1 1vt+1 <vt | Ht ) + E(ηvt+1 1vt+1 >vt | Ht ) (83) We consider each term in (83) separately. For the first term, since ηvt+1 1vt+1 <vt ≤ max {ηl : 1 ≤ l < vt , l ∈ NL (vt )} 1vt+1 <vt ≤ max {ηl : 1 ≤ l < vt , l ∈ NL (vt )} , we have that E(ηvt+1 1vt+1 <vt | Ht ) ≤ E (max {ηl : 1 ≤ l < vt , l ∈ NL (vt )} | Ht ) Z ∞ = Pr(max {ηl : 1 ≤ l < vt , l ∈ NL (vt )} ≥ η | Ht )dη η=0 Z ∞ = Pr(∃1 ≤ l < vt , l ∈ NL (vt ) : ηl ≥ η | Ht )dη η=0 ≤ . Z t −1 ∞ vX η=0 l=1 Z ∞ vX t −1 η=0 l=1 vX t −1 Z ∞ Pr(l ∈ NL (vt ) and ηl ≥ η | Ht )dη wl Wvt Z ∞ Z ∞ ηl =η ηlm−1 e−ηl dηl dη (m − 1)! ηlm−1 e−ηl ηl dηl dη 1/2 (m − 1)! 2m(lv t) η=0 η =η l l=1 Z ∞ Z ∞ ηlm e−ηl dηl dη . 2m + (1 + o(1)) η=2m ηl =η (m − 1)! Z ∞ . 2m + 4e−3η/10 dη, from Lemma 2.1(c), . η=2m −3m/5 . 2m + 20e ≤ 3m. (84) We now bound the second term of (83). We consider two cases, according to properties of the history Ht (which determines vt and ηvt ).
1 Case 1: Ht is such that vt ≤ 1 − ω1/2 n. In this case, we have that E(ηvt+1 1vt+1 >vt | Ht )  o  n ≤ E max ηl : vt < l ≤ n, vt+1 ∈ NL (l), dn (l) ≥ ∆vγtv∗ 1vt+1 >vt | Ht t 35 o   n ≤ E max ηl : vt < l ≤ n, vt+1 ∈ NL (l), dn (l) ≥ ∆vγtv∗ | Ht . t So we have that E(ηvt+1 1vt+1 >vt | Ht ) o   n vt ≤ E max ηl : vt < l ≤ n, vt+1 ∈ NL (l), dn (l) ≥ ∆γv∗ | Ht t Z ∞ o n vt = Pr(max ηl : vt < l ≤ n, vt ∈ NL (l), dn (l) ≥ ∆γv∗ ≥ η | Ht )dη t η=0 Z ∞   = Pr ∃vt < l ≤ n, vt ∈ NL (l) : ηl ≥ η, dn (l) ≥ ∆vγtv∗ | Ht dη ≤ η=0 n X l=vt +1 n X t Z ∞ η=0   Pr (vt ∈ NL (l)) ∧ (ηl ≥ η) ∧ (dn (l) ≥ ∆vγtv∗ ) | Ht , ηl dη ηvt . 2(lvt )1/2 l=v +1 t t Z ∞ η=0 Z ∞ ηl =η   ηlm−1 e−ηl Pr dn (l) ≥ ∆vγtv∗ | Ht , ηl dηl dη. t (m − 1)! (85) Recall that in the final two lines, vt and ηvt are not random variables, but are the actual values of these random variables in the history Ht , so this is a deterministic upper bound on E(ηvt+1 1vt+1 >vt | Ht ). We split the sum in the RHS of (85) into E1 + E2 + E3 + E4 according to the ranges of l and η, and bound each separately. The first part consists of Z Z n   X ηvt 1l≤4m2 vt /(γv∗t )2 2m ∞ ηlm−1 e−ηl vt | H , η Pr d (l) ≥ ∆ E1 = ∗ t l dηl dη. n γvt 1/2 2(lv (m − 1)! t) η=0 η =η l l=v +1 t Even though vt and ηvt are constants (determined by Ht ), we caution that γv∗t and so also E1 are random variables. Observe that we have that Z ∞ m−1 −ηl Z ∞ m−1 −ηl   ηl e ηl e vt Pr dn (l) ≥ ∆γv∗ | Ht , ηl dηl ≤ dηl ≤ 1 t ηl =η (m − 1)! ηl =η (m − 1)! which allows us to write E1 1γv∗t ≤5/4 ≤ 1γv∗t ≤5/4 n X 2mηvt 1l≤4m2 vt /(γv∗t )2 l=vt +1 2(lvt )1/2 ≤ 5m2 ηvt 1γv∗t ≤5/4 . γv∗t (86) We will use this expression when we take the expectation over γv∗t ≤ 5/4. We also have that Z ∞ m−1 −ηl
ηl e Pr (dn (l) ≥ ∆vγt∗ ) ∧ (γv∗t > 5/4) | Ht , ηl dηl ≤ I1 + I2 + I3 , ηl =η (m − 1)! 36 (87) where I1 = Z 7m/8 ηl =η Z ∞ ηlm−1 e−ηl dηl ≤ m (m − 1)!  7 1/8 e 8 m ≤ e−d0 m , from Lemma 2.1(a), ηlm−1 e−ηl dηl ≤ e−d1 m , from Lemma 2.1(d). I2 = ηl =9m/8 (m − 1)!   Z 9m/8
ηlm−1 e−ηl ∗ ∗ Pr dn (l) − m ≥ γvt mζ(vt ) ∧ (γvt > 5/4) Ht , ηl dηl , I3 = ηl =7m/8 (m − 1)!   Z 9m/8 ηlm−1 e−ηl 5 ≤ Pr dn (l) − m ≥ mζ(vt ) Ht , ηl dηl . 4 ηl =7m/8 (m − 1)! (88) We bound I3 with two subcases: Subcase 1a: ζ(l) > 0.   Z 9m/8 8 10ζ(vt) ηlm−1 e−ηl I3 ≤ Pr dn (l) − m ≥ ηl ζ(l) Ht , ηl dηl [since m ≥ ηl ], 9ζ(l) 9 ηl =7m/8 (m − 1)!  n o 2 (10ζ(vt )−9ζ(l)) ηl Z 9m/8 10ζ(vt) ≤ 18ζ(l) ηlm−1 e−ηl exp n− 81ζ(l) o [from (2) and ℓ ≥ vt + 1], ≤ (10ζ(vt )−9ζ(l))ηl 10ζ(vt) > 18ζ(l) ηl =7m/8 (m − 1)! exp − 27 ≤ Z 9m/8 ηl =7m/8  n o exp − 7mζ(vt ) 10ζ(vt ) ≤ 18ζ(l) 648 n o (m − 1)! exp − 7mζ(vt ) 10ζ(vt ) > 18ζ(l) 216 ηlm−1 e−ηl ≤ e−mζ(vt )/100 . (89) Subcase 1b: ζ(l) ≤ 0. In this case, we go back to (88) and use ζ + (l) in place of ζ(l), see (10).   Z 9m/8 10ζ(vt) + ηlm−1 e−ηl Pr dn (l) − m ≥ ηl ζ (l) Ht , ηl dηl , I3 ≤ 9ζ + (l) ηl =7m/8 (m − 1)! (90) For ε as in (46) we see that ζ(l) ≤ 0 implies that l ≥ n(1 − ε)2 . In which case ζ + (l) ≤ On the other hand, vt ≤ 1 − 1 ω 1/2 2ε ≤ 3ε. 1−ε (91) implies that ζ(vt ) ≥ 1 ω 1/2 − 2ε ≥ 1 . 2ω 1/2 t) ≥ Comparing (91) and (92), we see that ζ(vt ) ≫ ζ + (l). From this and (3) with β = 10ζ(v 9ζ + (l) we deduce that   10ζ(vt ) + ηl ζ (l) Ht , ηl ≤ (6εω 1/2)10ηl ζ(vt ) ≤ (6εω 1/2 )35mζ(vt )/36 . Pr dn (l) − m ≥ 9ζ + (l) 37 (92) 1 6εω 1/2 Plugging this estimate into (90) we obtain something stronger than (89), finishing Subcase 1b and giving that I3 ≤ e−mζ(vt )/100 in all cases. Having bounded the three terms in (87), we then have that E1 1γv∗t >5/4 ≤ n X ηvt 1l≤4m2 vt /(γv∗t )2 (lvt )1/2 l=vt +1 5m e−d2 m ∗ γvt ≤ ηvt  −d2 m e−d2 m + e−mζ(vt )/100 n e−mζ(vt )/100 X 1 + vt 1/2 l1/2 l=v +1 t −mζ(vt )/100 ≤ ηvt 4me +e !
(n + 1)1/2 − (vt + 1)1/2 · vt 1/2 ≤ ηvt (4me−d2 m + 2ζ(vt)e−mζ(vt )/100 )   200 −d2 m . + ≤ ηvt 4me m  (93) It follows from (86) and (93) that    2 200 5m −d2 m . 1γ ∗ ≤5/4 + 4me + E1 ≤ ηvt γv∗t vt m We continue with the other parts of the RHS of (85): Z Z ∞ m−1 −ηl n X ηvt 1l≤4m2 vt /(γv∗t )2 ∞ ηl e E2 = Pr(dn (l) ≥ ∆vγt∗ | Ht , ηl )dηl dη 1/2 2(lv ) (m − 1)! t η=2m ηl =η l=v +1 t 4m2 vt /(γv∗t )2 ≤ X ηvt 2(lvt )1/2 Z ∞ X ηvt 2(lvt )1/2 Z ∞ X ηvt ×m 2(lvt )1/2 l=vt +1 4m2 vt /(γv∗t )2 ≤ l=vt +1 4m2 vt /(γv∗t )2 ≤ l=vt +1 4m2 vt /(γv∗t )2 = ≤ X l=vt +1 −d3 m e ηvt . γv∗t η=2m Z ∞  eη/m eη/m ηl =η η=2m Z ηlm−1 e−ηl dηl dη (m − 1)! m dη from Lemma 2.1(c), ∞ e−3mx/10 dx x=2 10ηvt e−3m/5 6(lvt )1/2 (94) Note that we aborbed an O(m) factor into the expression in (94). This is valid because m is large. We continue to do this where possible. Z Z n X ηvt 1l>4m2 vt /(γv∗t )2 γv∗t (l/vt )1/2 ∞ ηlm−1 e−ηl Pr(dn (l) ≥ ∆vγtv∗ | Ht )dηl dη E3 = 1/2 t 2(lv ) (m − 1)! t η=0 ηl =η l=v +1 t 38 n X γv∗t (l/vt )1/2 ∞ ηlm−1 e−ηl dηl dη ∗ (l/v )1/2 (m − 1)! η=0 η =γ t l 2 ∗ 2 v t l=4m vt /(γvt ) ( ) Z γv∗ (l/vt )1/2 n X t 3γv∗t (l/vt )1/2 ηvt exp − dη from Lemma 2.1(c), ≤ 2(lvt )1/2 η=0 10 2 ∗ 2 l=4m vt /(γvt ) ) ( n X 3γv∗t (l/vt )1/2 ηvt ≤ , exp − 2(lvt )1/2 10 2 ∗ 2 l=4m vt /(γvt ) ) ( Z ηvt γv∗t n 3γv∗t (x/vt )1/2 ≤ dx, exp − vt 10 x=4m2 vt /(γv∗t )2 Z ∞ ηvt γv∗t 8vt ≤ × ∗ 2 ye−3y/5 dy, vt (γvt ) y=m ≤ ηvt 2(lvt )1/2 Z Z ηvt e−d4 m = . γv∗t Z Z ∞ m−1 −ηl n X ηvt 1l>4m2 vt /(γv∗t )2 ∞ ηl e Pr(dn (l) ≥ ∆vγtv∗ | Ht , ηl )dηl dη E4 = 1/2 t 2(lvt ) η=γv∗t (l/vt )1/2 ηl =η (m − 1)! l=i Z ∞ Z ∞ m−1 −ηl n X ηl e ηvt dηl dη ≤ 1/2 2(lvt ) η=γv∗ (l/vt )1/2 ηl =η (m − 1)! 2 ∗ 2 l=4m vt /(γvt ) ≤ n X l=4m2 vt /(γv∗t )2 t ηvt 2(lvt )1/2 Z ∞ e−3η/10 dη η=γv∗t (l/vt )1/2 ) ( 3γv∗t (l/vt )1/2 5ηvt , ≤ exp − 3(lvt )1/2 10 2 2 ∗ l=4m vt /(γvt ) ) ( Z 1/2 ∗ (x/j) 3γ 2ηvt ∞ ≤ 1/2 dx, x−1/2 exp − vt i 10 2 ∗ 2 x=4m vt /(γvt ) Z 4ηvt ∞ −3y/10 e dy, = ∗ γvt y=2m n X ≤ ηvt e−d5 m . γv∗t Thus, E(ηvt+1 1vt+1 >vt    5m∗ 2 + e−d∗6 m ηvt ≤ γvt γvt  | Ht ) ≤ E1 + E2 + E3 + E4 ≤  −d m 7 200 e  ηvt + ∗ γvt m 7m2 η γv∗t vt γv∗t ≤ 5/4 γv∗t > 5/4. We now integrate with respect to the value of γv∗t . (Note that γv∗t is actually a discrete random variable, so that Pr(γv∗t ≤ γ | Ht ) is discontinuous, but one can view this as a Riemann-Stieltjes integral. We write Pr† (γv∗t ≤ γ) below in place of Pr(γv∗t ≤ γ | Ht ).) Using Lemma 3.12 we see 39 that if m is large then integrating over γ, E(ηvt+1 1vt+1 >vt | Ht ) ! Z ∞ Z 5/4 −d7 m e 200 7m2 d Pr† (γv∗t ≤ γ) + d Pr† (γv∗t ≤ γ) + ≤ ηvt γ γ m γ=5/4 γ=0 ! Z 5/4 7m2 200 ≤ ηvt d Pr† (γv∗t ≤ γ) + e−d8 m + γ m γ=0 ! Z 1/m Z 5/4 2 7m2 200 7m = ηvt d Pr† (γv∗t ≤ γ) + d Pr† (γv∗t ≤ γ) + e−d8 m + γ γ m 1/2 γ=0 γ=1/m 1/m Z 1/m1/2  2 7m2 7m † ∗ + Pr (γvt ≤ γ) Pr† (γv∗t ≤ γ)dγ ≤ ηvt 2 γ γ γ=0 0 ! Z 5/4 7m2 200 + d Pr† (γv∗t ≤ γ) + e−d8 m + γ m γ=1/m Z 1/m 7m2 −d9 m1/2 Pr† (γv∗t ≤ γ)dγ+ + ≤ ηvt e 2 γ γ=0 ! Z 5/4 7m3 200 , from (34), d Pr† (γv∗t ≤ γ) + e−d8 m + m γ=1/m γ     200 1 † ∗ −d8 m −d9 m1/2 −d10 m1/2 4 , ≤ γvt ≤ 5/4 + e + +e + 7m Pr ≤ ηvt e m m   200 −d9 m1/4 −d10 m1/2 4 −c1 m −d8 m +e + 7m e +e + ≤ ηvt e , from (35) m   200 −d12 m1/4 + ≤ ηvt e . m (95) Combining (84) and (95) via (83), we have that E(ηvt+1   200 −cm1/4 ηvt . | Ht ) ≤ 3m + e + m This completes Case 1. Case 2 is much shorter.
1 Case 2: Ht is such that vt > 1 − ω1/2 n. E(ηvt+1 | Ht ) . ∼ ∼ n X l=vt +1 n X l=vt +1 n X ηvt 2(lvt )1/2 Z ηvt 2(lvt )1/2 Z mηvt 2(lvt )1/2 l=v +1 t 40 ∞ η=0 Z ∞ ηl =η ∞ −η e η=0 ηlm−1 e−ηl dηl dη (m − 1)! m X i=1 η m−i dη (m − i)! (96) . mηvt ≤ (n + 1)1/2 − (vt + 1)1/2 (vt + 1)1/2 mηvt . ω 1/2 This completes Case 2. In particular, for sufficiently large n we see that for any typical Ht (i.e., in both Case 1 and Case 2), the bound from (96) is valid. Putting Et = E ∩ {Ht is typical} we deduce from (96) that E(ηvt+1   200 −cm1/4 | Et ) ≤ 3m + e + E(ηvt | Et ) m   200 −cm1/4 E(ηvt | Et−1 ). . 3m + e + m (97) (98) We obtain (98) from (97) because Et ⊆ Et−1 and so E(ηvt | Et−1 ) ≥ E(ηvt | Et ) Pr(Et | Et−1 ) ∼ E(ηvt | Et ). Because m is large, (21) will follow by induction once we have shown that E(ηv1 ) ≤ 3m. (99) Here we will use the assumption that v1 is chosen exacty according to the stationary distribution for a simple random walk on Gn . In particular, we have ! n X dn (i) Pr(ηv1 ≥ η) ≤ E 1η ≥η , 2mn v1 i=1 and Lemma 2.2 implies that if η ≥ 2m E((dn (i) − m)1ηv1 ≥η ) .  n 1/2 Z i ∞ η=2m  n 1/2 η m e−η dη . × 4me−η/2 . (m − 1)! i Furthermore, E(m1ηvt ≥η ) = m Pr(ηv1 ≥ η) ≤ 5me−η/2 . So, if η ≥ 2m, then Pr(ηv1 Therefore, E(ηv1 ) ≤ 2m + Z n 9e−η/2 X 1 ≤ 10e−η/2 . ≥ η) . 2n1/2 i=1 i1/2 ∞ Pr(ηv1 ≥ η)dη ≤ 2m + 10 η=2m Z ∞ η=2m and (99) follows. 41 e−η/2 dη 3.2 Exiting the main loop with success In summary, it follows that w.h.p. DCA reaches Step 7 in O(ω log n) time. Also, at this time vT ≤ log1/49 n. This follows from Lemma 2.2(g), 2.2(h) and (P4). Furthermore, this justifies using n1 as a lower bound on vertices visited during the main loop. The random walk of Step 8 will w.h.p. take place on [log1/9 n]. This follows from Lemma 2.2(j). Vertex 1 will be in the 1/2 same component as vt in the subgraph of Gn induced by vertices of degree at least logn1/20 n . This is because there is a path from vT to vertex 1 through vertices in [vT ] only and furthermore it 1/2 follows from Lemma 2.2(i) that w.h.p. every vertex on this path has degree at least logn1/20 n . The expected time to visit all vertices of a graph with ν vertices is O(ν 3 ), see for example Aleliunas, Karp, Lipton, Lovász and Rackoff [1]. Consequently, vertex 1 will be reached in a further O((log1/9 n)3 ) = o(log n) steps w.h.p, completing the proof of Theorem 1.2. 4 Concluding remarks We have described an algorithm that finds a distinguished vertex quickly and which is local in a strong sense. There are some natural questions that are left unanswered: • Can the running time be improved from O(ω log n) to O(log n)? • Can we get polylog expected running time for DCA if m = 2? • Can we extend the analysis to other more general models of web graphs e.g. Cooper and Frieze [7]. In this case, we would not be able to use the model described in Section 2. As a final observation, the algorithm DCA could be used to find the vertex of largest degree: if we replace Step 8 by “Do the random walk for log n steps and output the vertex of largest degree encountered” then w.h.p. this will produce a vertex of highest degree. This is because log n will be enough time to visit all vertices v ≤ log1/39 n, where the maximum degree vertex lies. References [1] R. Aleliunas, R.M. Karp, R.J. Lipton, L. Lovász and C. Rackoff, Random Walks, Universal Traversal Sequences, and the Complexity of Maze Problems. Proceedings of the 20th Annual IEEE Symposium on Foundations of Computer Science (1979) 218-223. [2] A. Barabási and R. Albert, Emergence of scaling in random networks, Science 286 (1999) 509-512. [3] B. Bollobás, O. Riordan, J. Spencer and G. Tusnády, The degree sequence of a scale-free random graph process, Random Structures and Algorithms 18 (2001) 279-290. 42 [4] B. Bollobás and O. Riordan, The Diameter of a Scale-Free Random Graph, Combinatorica 24 (2004) 5-34. [5] C. Borgs, M. Brautbar, J. Chayes, S. Khanna and B. Lucier, The Power of Local Information in Social Networks, http://arxiv.org/abs/1212.0884. [6] M. Brautbar and M. Kearns, Local Algorithms for Finding Intersting Individuals in Large Networks, in Innovations in Computer Science (ITCS) (2010) 188-199. [7] C. Cooper and A.M. Frieze, On a general model of web graphs, Random Structures and Algorithms 22 (2003) 311-335. [8] G. Grimmett and D. Stirzaker, Probability and Random Processes, Third Edition, Oxford University Press, Oxford UK, 2001. [9] W. Hoeffding, Probability inequalities for sums of bounded random variables, Journal of the American Statistical Association 58 (1963) 13-30. [10] M. Jerrum and A. Sinclair. The Markov chain Monte Carlo method: an approach to approximate counting and integration. In Approximation Algorithms for NP-hard Problems. (D. Hochbaum ed.) PWS (1996) 482-520. [11] M. Mihail, C. Papadimitriou and A. Sabeeri, On Certain Connectivity Properties of the Internet Topology, Journal of Computer and System Sciences 72 (2006) 239251. A Proofs of properties (P1)–(P5) In this section we give proofs of (P1)–(P5), which we list here for convenience. (P1) For Υk,ℓ = Υk − Υℓ , we have Υk,ℓ " 1/2 Lθk,ℓ ∈ (k − ℓ) 1 ± 3(k − ℓ)1/2 # for (k, ℓ) = (mn + 1, 0) or   l=0 1 k−ℓ 2 ∈ {ω, ω + 1, . . . , n} and k − l ≥ log n k ≥ log30 n, l > 0  m  1/300 log n 0 < l < k < log30 n. 43 Here n0 = λ20 n , ω log2 n λ0 = 1 , log20/m n   log k    1/2   k = (k − ℓ)1/2   (k−ℓ)3/2 log n    n1/2   n θk,ℓ ω ≤ l < k ≤ log30 n, l > 1 ω ≤ k ≤ n2/5 , l = 0 log30 n < k ≤ n2/5 n2/5 < k ≤ n0 . n0 < k. ω 3/2 log2 n #    1/2 " 1/2 1/2 i Lθi i (P2) Wi ∈ 1 ± 1/2 ∼ for ω ≤ i ≤ n. n i n (P3) wi ∼ ηi for ω ≤ i ≤ n. 2m(in)1/2 (P4) λ0 ≤ ηi ≤ 40m log log n for i ∈ [log30 n]. (P5) ηi ≤ log n for i ∈ [n]. Proof of (P1) Applying Lemma 2.1(d),(e) to (1) for i ≥ 1 we see that Pr(¬(P1))   n X L2 θk,0 ≤2 exp − +2 27 k=ω n2/5 L2 k 1/2 exp − =2 27 k=ω X  30  n X k−ℓ=log1/300 n +2 n0 X k=n2/5 +1 L2 θk,ℓ exp − 27   L2 θmn+1,0 + 2 exp − 27 L2 k 3/2 log n exp − 27n1/2    +2 n+1 X k=n0 L2 n exp − 27ω 3/2 log2 n +1     n2/5 X L2 log k L2 (k − ℓ)1/2 exp − +2 +2 exp − + 27 27 30 1/300 k−ℓ=log n k−ℓ=log n     n n 0 X X L2 n L2 (k − ℓ)3/2 log n +2 exp − 2 exp − 27n1/2 27ω 3/2 log2 n 2/5 k−ℓ=n +1 log k−ℓ=n Xn  +1 0 = o(1). Proof of (P2) For this we use Wi =  Υmi Υmn+1 44 1/2 .   Then, #  1/2 " 1/2 Lθi i Wi ∈ / 1 ± 1/2 n i implies that either Υmn+1 " 1/2 Lθi ∈ / (mn + 1) 1 ± 3(mn + 1)1/2 # or Υmi " 1/2 Lθi ∈ / mi 1 ± 1/2 3i # . These events are ruled out w.h.p. by (P1). Proof of (P3) We use (1 + x)1/2 ≤ 1 + wi =  x 2 for 0 ≤ |x| ≤ 1. Then, Υmi Υmn+1 1/2 −  Υm(i−1) Υmn+1 1/2 !   1/2 Υm(i−1) 1/2 ηi = 1+ −1 Υmn+1 Υm(i−1) 1/2   1/2 Lθi m(i − 1) 1 + 3m1/2 (i−1)1/2 η  i ≤  1/2  1/2 Lθi 2m(i − 1) 1 − (mn + 1) 1 − 1/2  3(mn+1) ηi ≤ 2m(in)1/2 1/2 2Lθ 1 + 1/2i 1/2 m i ! 1/2 Lθi 3m1/2 (i−1)1/2  . A similar calculation gives 1/2 ηi wi ≥ 2m(in)1/2 2Lθ 1 − 1/2i 1/2 m i ! . Proof of (P4) The upper bound follows from Lemma 2.1(c). For the lower bound, we observe by (7) that the expected number of i ≤ log30 n with ηi ≤ λ0 is at most log30 n × λm 0 = o(1). Proof of (P5) This follows from Lemma 2.1(c). 45 B Proof of Lemma 2.2 We restate the lemma for convenience. Lemma B.1. (a) If E occurs then d¯n − m ∈ [ηi ζ(i), ηi ζ + (i)]. 2 η ζ(i)/2 i (b) Pr(dn (i) − m ≤ (1 − α)mζ(j)) ≤ e−α for 0 ≤ α ≤ 1. 2 η ζ + (i)/3 i (c) Pr(dn (i) − m ≥ (1 + α)mζ + (j)) ≤ e−α (d) Pr(dn (i) − m ≥ βmζ + (j)) ≤ (e/β)βηi ζ + (i) for 0 ≤ α ≤ 1. for β ≥ 2. (e) W.h.p. ηi ≥ λ0 and ω ≤ i ≤ n1/2 implies that dn (i) ∼ ηi (f ) W.h.p. ω ≤ i ≤ log30 n implies that dn (i) ∼ ηi
n 1/2 . i (g) W.h.p. ω ≤ i ≤ n1/2 implies that dn (i) . max {1, ηi } (h) W.h.p. n1/2 ≤ i ≤ n implies dn (i) ≤ n1/3 . (i) W.h.p. 1 ≤ i ≤ log1/49 n implies that dn (i) ≥ (j) W.h.p. dn (i) ≥ n log1/20 n
n 1/2 . i
n 1/2 i . n1/2 . log1/20 n implies i ≤ log1/9 n. Proof. (a) Suppose that we fix the values for W1 , W2 , . . . , Wn . Then the degree dn (i) of vertex i can be expressed n X m X dn (i) = m + ζj,k j=i k=1 where the ζj,k are independent Bernouilli random variables such that   wi wi Pr(ζj,k = 1) ∈ , . Wj Wj−1 So, putting d¯n (i) = E(dn (i)) we have mwi n n X X 1 1 ≤ d¯n (i) − m ≤ mwi . W W j j j=i−1 j=i Now assuming that (P2), we have for ω ≤ i ≤ n, n  1/2 n X X n 1 ≥ Wj j j=i j=i 46 1/2 2Lθj 1 − 1/2 j ! . But 1/2 n X θj j=ω j 2/5 n X 1 ≤ + 3/4 j j=ω n0 X j=n2/5 +1 n X n1/2 log1/2 n + n1/4 j 1/4 j=n +1 jω 3/4 log n 0 3n1/2 log log n 4n1/2 + 3ω 3/4 log n ω 3/4 log n 4n1/2 log log n ≤ . ω 3/4 log n ≤ 4n1/10 + It follows that   9Ln1/2 log log n 1/2 1/2 1/2 ¯ 2(n − (i + 1) ) − dn (i) ≥ m + mwi n ω 3/4 log n !  1/2  n 1/2 i 9L log log n 1 1− , − 1/2 1/2 − ≥ m + ηi i n n i 2ω 3/4 log n after using (P3). A similar calculation gives a similar upper bound for d¯n (i) and this proves that " #  1/2  n 1/2 i 5L log log n i ≥ ω implies that d¯n (i) ∈ m + ηi 1− . ± 3/4 i n ω log n It follows from (2) and (4) that     L2 ηi n1/2 Pr dn (i) − m ≤ (1 − α)ηi ζ(i) ηi ≤ exp − 1/2 1/2 . 4i ω     L2 ηi n1/2 Pr dn (i) − m ≥ (1 + α)ηi ζ(i) ηi ≤ exp − 1/2 1/2 . 4i ω (a) For ηi ≥ λ0 and ω ≤ i ≤ n0 we have   L2 ηi n1/2 2 exp − 1/2 1/2 ≤ e−L log n/4 . 4i ω (b) This follows from (a) and (4). (c) This follows from (a) and (2). (d) This follows from (a) and (3). (e) This follows from (a), (b), (c) and (12). (f) This follows from (e) and (P4). 47 (g) This follows from (c) and (12). (h) The degree of i ≥ n1/2 is stochastically dominated by the degree of n1/2 . Also, the probability that dn (n1/2 ) exceeds the stated upper bound is o(1/n). So (h) follows from (g). (i) For ω ≤ i ≤ log1/49 n, this follows from (f) and (P4). For 1 ≤ i < ω we can use (b) with ηi ≥ λ0 and α = n−1/10 . (j) This follows from (e), (f) and (g) and (P4). 48 

Morphological Error Detection in 3D Segmentations David Rolnick1∗, Yaron Meirovitch1 , Toufiq Parag2 , Hanspeter Pfister2 Viren Jain3 , Jeff W. Lichtman2 , Edward S. Boyden1 , Nir Shavit1 arXiv:1705.10882v1 [] 30 May 2017 1 Massachusetts Institute of Technology, 2 Harvard University, 3 Google, Inc. Abstract Deep learning algorithms for connectomics rely upon localized classification, rather than overall morphology. This leads to a high incidence of erroneously merged objects. Humans, by contrast, can easily detect such errors by acquiring intuition for the correct morphology of objects. Biological neurons have complicated and variable shapes, which are challenging to learn, and merge errors take a multitude of different forms. We present an algorithm, MergeNet, that shows 3D ConvNets can, in fact, detect merge errors from high-level neuronal morphology. MergeNet follows unsupervised training and operates across datasets. We demonstrate the performance of MergeNet both on a variety of connectomics data and on a dataset created from merged MNIST images. 1 Introduction The neural network of the brain remains a mystery, even as engineers have succeeded in building artificial neural networks that can solve a wide variety of problems. Understanding the brain at a deeper level could significantly impact both biology and artificial intelligence [3, 8, 19, 23, 35]. Perhaps appropriately, artificial neural networks are now being used to map biological neural networks. However, humans still outperform computer vision algorithms in segmenting brain tissue. Deep learning has not yet attained the intuition that allows humans to recognize and trace the fine, intermingled branches of neurons. The field of connectomics aims to reconstruct three-dimensional networks of biological neurons from high-resolution microscope images. Automated segmentation is a necessity due to the quantities of data involved. In one recent study [9], the brain of a larval zebrafish was annotated by hand, requiring more than a year of human labor. It is estimated that mapping a single human brain would require a zettabyte (one billion terabytes) of image data [17], clearly more than can be manually segmented. State-of-the-art algorithms apply a convolutional neural network (ConvNet) to predict, for each voxel of an image, whether it is on the boundary (cell membrane) of a neuron. The predicted membranes are then filled in by subsequent algorithms [10]. Such methods are prone both to split errors, in which true objects are subdivided, and to merge errors, in which objects are fused together. The latter pose a particular challenge. Neurons are highly variable, unpredictably sprouting thousands of branches, so their correct shapes cannot be catalogued. Erroneously merged neurons are obvious to trained humans because they simply don’t look right, but it has hitherto been impossible to make such determinations automatically. ∗ Correspondence should be addressed to: [email protected]. 1 Figure 2: A single, relatively simple merge error detected and localized by MergeNet. This object, within the Kasthuri dataset [12], occurred within the training data of the algorithm, but was not labeled as a merge error. MergeNet was nonetheless able to correct the label. This capability allows MergeNet to be trained on an uncertain segmentation, then used to correct errors within the same segmentation, without requiring any manual annotation. Figure 1: A probability map localizing merge errors, as predicted by MergeNet, for an object within the ECS dataset. Orange indicates a high probability of merge error, blue the absence of error. Location (A) illustrates a merge between two neurons running in parallel, (B) a merge between three neurons simultaneously (the two parallel neurons, plus a third perpendicular to them), and (C) a merge between a large neuron segment and a small branch from another neuron. MergeNet is able to learn that all of these diverse morphologies (and others not illustrated in this example) represent merge errors, but that locations such as (D) and (E) are normally occurring branch points within a single neuron. We introduce a deep learning approach for detecting merge errors that leverages the morphological intuition of human annotators. Instead of relying upon voxelwise membrane predictions or microscope images, we zoom out and capture as much context as possible. Using only threedimensional binary masks, our algorithm is able to learn to distinguish the shapes of plausible neurons from those that have been erroneously fused together. We test our network, MergeNet, both on connectomics datasets and on an illustrative dataset derived from MNIST [15]. The key contributions of this approach include: • Localization of merge errors. MergeNet is able to detect merge errors with high accuracy within a three-dimensional segmentation and to pinpoint their locations for correction (see Figures 1 and 2). • Unsupervised training. The algorithm can be trained using any reasonably accurate segmentation, without the need for any additional annotation. It is even able to correct errors within its own training data. • Generalizability and scalability across datasets. MergeNet can be applied irrespective of the segmentation algorithm or imaging method. It can be trained on one dataset and run 2 on another with high performance. By downsampling volumetric data, our ConvNet is able to process three million voxels a second, faster than most membrane prediction systems. 2 Related work There have been numerous recent advances in using neural networks to recognize general threedimensional objects. Methods include taking 2D projections of the input [32], combined 2D-3D approaches [6, 16], and purely 3D networks [20, 36]. Accelerated implementation techniques for 3D networks have been introduced by Budden, et al. [4] and Zlateski, Lee, and Seung [38]. Within the field of connectomics, Maitin-Shepard et al. [18] describe CELIS, a neural network approach for optimizing local features of a segmented image. Januszewski et al. [11] and Meirovitch et al. [22] present approaches for directly segmenting individual neurons from microscope images, without recourse to membrane prediction and agglomeration algorithms. Deep learning techniques have likewise been used to detect synapses between neurons [29, 31] and to localize voltage measurements in neural circuits [2] (progress towards a functional connectome). New forms of data are also being leveraged for connectomics [27, 34], thanks to advances in biochemical engineering. Many authors cite the frequent problems posed by merge errors (see e.g. [26]); however, almost no approaches have been proposed for detecting them automatically. Meirovitch et al. [22] suggest a hard-coded heuristic to find “X-junctions”, one variety of merge error, by analyzing graph theoretical representations of neurons as skeletons (see also [37]). Recent work including [13, 24] has considered the problem of deep learning on graphs, and Farhoodi, Ramkumar, and Kording [7] use Generative Adversarial Networks (GANs) to generate neuron skeletons. However, such methods have not to date been brought to bear on connectomic reconstruction of neural circuits. 3 Methods Our algorithm, MergeNet, operates on an image segmentation to correct errors within it. Given an object within the proposed segmentation, MergeNet determines whether points chosen within the object are the location of erroneous merges. If no such points exist, then the object is determined to be free from merge errors. Input and architecture. The input to our network is a three-dimensional window of the object in question, representing a 51 × 51 × 51 section of the object, centered at the chosen point. (These dimensions are chosen as a tradeoff between enhancing speed and capturing more information, as we discuss further in sections §4 and 5.) Crucially, the window is given as a binary mask : that is, each voxel is 0 or 1 depending on whether it is assigned to the object. MergeNet is not given data from the original image, inducing the network to learn general morphological features present in the binary mask. The network follows a simple convolutional architecture, containing six convolutional layers with rectified linear unit (ReLU) activation, and three max-pooling layers, followed by a densely connected layer and softmax output. The desired output is a 1-hot vector for the two classes “merge” and “no merge”, and is trained with cross-entropy loss. 2D MergeNet. We also constructed a simpler 2D version of MergeNet to illustrate the identification of merge errors within two-dimensional images. In this case, the input to the network is a square binary mask, which passes through four convolutional layers and two max-pooling layers. This network was trained to recognize merges between binarized digits from the MNIST dataset [15]. Random digits were drawn 3 from the training dataset and chained together, with merge errors given by pixels at the points of contact between neighboring digits. Testing was performed on similar merges created from the testing dataset. The size of the input window to the network was varied to compare accuracy across a variety of contextual scales. Downsampling. To apply MergeNet to connectomics data, we begin by downsampling all objects. Segmentations of neural data are typically performed at very high resolution, approximately 5 nm. The finest morphological details of neurons, however, are on the order of 100 nm. Commonly, data is anisotropic, with resolution in the z direction being significantly lower than that in x and y. We tested MergeNet with downsampling ratios of 10 × 10 × 2 and 25 × 25 × 5 to compensate for anisotropy. Downsampling an object is performed by a max-pooling procedure. That is, every voxel within the downsampled image represents the intersection of the object with a corresponding subvolume of the original image with dimensions e.g. 25 × 25 × 5. Training. The network was trained on artificially induced merge errors between objects within segmentations of neural tissue. Merges consisted of identifying immediately adjacent objects and designating points of overlap as the locations of merge errors. Negative examples consisted of windows centered at random points of objects within the segmentation. Artificial merge errors are used owing to the impracticality of manually annotating data over large enough volumes to determine sufficient merge errors for training. As we demonstrate, such training suffices for effective detection of real merge errors and has numerous other advantages (detailed in section §5). Training was performed on various segmentations of the Kasthuri dataset [12], and the algorithm was evaluated both on this dataset and on segmented objects of the ECS dataset, a 20 × 20 × 20 micron cube of rat cortex data to which we were given access. Output. To run a trained instance of MergeNet on objects within a segmentation, it is not necessary to apply the network to every voxel since the predictions at nearby points may be interpolated. Sample points are therefore taken within each downsampled object, and MergeNet is run on windows centered at these points. The real-valued predictions at sample points are then interpolated over the entire object to give a heatmap of probabilities for merge errors. As the distribution of training examples is balanced between positive and negative examples, which is not true when the network is applied in practice, the output must be normalized or thresholded after the softmax layer. We find it effective to classify as a merge error any voxel at which the prediction exceeds 0.9. 4 Results We first consider the illustrative example of the merged MNIST dataset. After training on five million examples within this dataset, we obtained a maximum pixelwise accuracy of 96.8 percent on a test set constructed from held out digits and equally distributed between positive and negative examples. This corresponds to almost perfect identification of individual merge error regions, as shown in Figure 3. (Ambiguous pixels on the edges of merge error regions were the most likely pixels to be misclassified.) Accuracy increases with morphological information. In Figure 4, we show the dependence of accuracy on the window size used. Intuitively, a larger window gives the network more morphological context to work from, and plateaus in this case at approximately the dimensions of a pair of fused MNIST digits (40-50 pixels across), which represents the maximum scale at which morphology is useful to the network. Figure 3 provides a qualitative 4 Figure 3: Predictions of MergeNet on merged MNIST digits, shown on a sample of the test set. The top image shows predictions of the network with 12 × 12 input windows, with predicted merges shown in yellow. The middle image shows predictions with 24 × 24 input windows. The final image shows the actual merges, in red. Note that both networks are quite accurate, but that for 12 × 12 input, the algorithm makes several erroneous predictions of merge errors (shown with blue arrows), which are not made for 24 × 24 input. This illustrates how greater morphological context leads to qualitatively better predictions. A quantitative assessment is shown in Figure 4. comparison of performance between a smaller and a larger window size. The smaller window size erroneously predicts merges within digits, while the large window size allows the network to recognize the shapes of these digits and identify only merge errors between digits. There is, of course, a tradeoff between accuracy and the time required to train and run the network. Slowdown resulting from larger window size is considerable for three-dimensional ConvNets. We have attempted to choose the parameters of MergeNet with this tradeoff in mind. MergeNet detects merge errors across datasets. We trained MergeNet on two segmentations of the Kasthuri dataset [21] and tested performance on artificially merged objects omitted from the training set. Segmentation A was relatively poor, and training on it yielded performance of only 77.3 percent. Segmentation B was more accurate, yielding performance of 90.0 percent. Training on both segmentations together yielded the best performance on both test sets, showing that MergeNet is able to leverage contextual information from one segmentation to improve performance on another. We also note that after training on the low-quality Segmentation A, MergeNet was able to detect errors within its own training set, as shown in Figure 2. Testing on Seg. A Testing on Seg. B Training on Seg. A 77.3% 70.7% Training on Seg. B Training on both 82.6% 84.5% 90.0% 91.9% MergeNet generalizes broadly across datasets as well as segmentations of the same dataset, and applies to both artificial and natural merge errors, though the latter are harder to quantify owing to the paucity of large-scale annotation. After training on the Kasthuri dataset, MergeNet was able to detect naturally occurring merge errors within segmentations of the ECS dataset, obtained from a state-of-the-art U-Net segmentation algorithm [30]. Example output is shown in Figures 1 and 5. 5 Figure 4: Performance of MergeNet on merged MNIST digits, shown as a function of input window width (with example windows shown). Note that increasing window size increases performance, but only up to a size of 40-50 pixels. We see that performance plateaus at the point at which morphological context captures all of the information about the neighboring digits. For the case of neurons, which are larger and more complicated, morphological context does not plateau; however, there is a tradeoff between more context and greater speed, since the time required to run the 3D ConvNet depends strongly upon input size. 5 Discussion We will now consider the capabilities of the MergeNet algorithm and discuss opportunities that it offers within the field of connectomics. Detection and localization of merge errors. MergeNet is a powerful tool for detecting and pinpointing merge errors. Once a merge location has been flagged with high spatial precision, other algorithms can be used to create a more accurate local segmentation, thereby correcting any errors that occured. Flood-filling networks [11] and MaskExtend [22] are two examples of algorithms that have high accuracy, but are extremely time-consuming to run over large volumes, making them ideally suited to segment at the merge locations flagged by MergeNet. Alternatively, the agglomeration algorithm NeuroProof [25], used in transforming membrane probabilities to segmentations, can be tuned to be more or less sensitive to merge errors. A more merge-sensitive setting could be applied at those locations flagged by our algorithm. If the thresholding step is omitted, then the output of MergeNet may be thought of as a probability distribution of merge errors over objects within a segmentation. This distribution may be treated as a Bayesian prior and updated if other information is available; multiple proofreading algorithms can work together. Thus, for example, synapse detection algorithms [29, 31] may provide additional evidence for a merge error if synapses of two kinds are found on the same segmented object, but are normally found on different types of neurons. In such a scenario, the probability at the relevant location would be increased from its value computed by MergeNet, according to the confidence of the synapse detection algorithm. We envision MergeNet being the first step towards fully automated proofreading of connectomics data, which will become increasingly necessary as such data is processed at ever greater scale. 6 Unsupervised training. MergeNet is trained on merge errors created by fusing adjacent objects within a segmentation. This allows training to proceed without any direct human annotation. In testing MergeNet, we performed training on several automatically generated segmentations of EM data and obtained good results, even though the training data was not free of merge errors and other mistakes. It is highly advantageous to eliminate the need for further data annotation, since this is the step in connectomics that has traditionally consumed by far the most human effort. The ability to run on any (reasonably accurate) segmentation also means that MergeNet can be trained on far larger datasets than those for which manual annotation could reasonably be obtained. Automated segmentations already exist for volumes of neural tissue as large as 232,000 cubic microns [28]. Comparison of segmentations. Automatic detection of merge errors allows us to compare the performance of alternative segmentation algorithms in the absence of ground truth annotation. We ran MergeNet with the same parameter settings on two segmentations within the ECS dataset, after training on other data. These alternative segmentations were produced by two different versions of a state-of-the-art UNet segmentation algorithm [30]. For the simpler algorithm, 33 of the 300 largest objects within the segmentation (those most likely to have merge errors) were flagged as unlikely, while, for the more advanced algorithm, only 15 of the largest 300 objects were flagged. This indicates that the latter pipeline produces more plausible objects, making fewer merge errors. The size of the objects was comparable in each case, so there is no indication that this improvement came with a greater propensity for erroneous splits within single objects. Thus, MergeNet was able to perform a fully automatic comparison of two segmentation algorithms and confirm that one outperforms the other. Correction of the training set. We have already observed that MergeNet can be trained on any reasonably correct segmentation. In fact, it is possible to leverage artificial merge errors within the training set to detect real merge errors that may occur there (as shown in Figure 2). This is remarkable, since the predictions MergeNet makes in this case should conflict with the labels it was itself trained on - namely, when objects from the training set that have not been artificially merged are nonetheless the result of real merge errors. Our observations align with results showing that neural networks are capable of learning from data even when the labels given are unreliable [33]. Independence from lower-order errors. Since MergeNet makes use of the global morphology of neurons, it is not reliant on earlier stages of the connectomics pipeline, such as microscope images or membrane predictions. Thus, it is able to correct errors that arise at early stages of the pipeline, including those at the experimental stage. EM images are prone to various catastrophic errors; most notably, individual tissue slices can tear before imaging, leading to distortion or gaps in the predicted membranes. Algorithms that stitch together adjacent microscope images also sometimes fail, leading to pieces of neurons in one image being erroneously aligned with pieces from the neighboring image. Typically, such errors are propagated or magnified by later stages of the connectomics pipeline, since algorithms such as watershed and NeuroProof [25] assume that their input is mostly true. By contrast, MergeNet can look at the broader picture and use “common sense” as would a human proofreader. The output of a membrane-detection algorithm also can induce errors in object morphology. Common sources of error at this stage are ambiguously stained tissue slices and intracellular membranes, such as those from mitochondria, which can be confused with the external cell membrane. Figure 5 shows an error in predictions from the U-Net algorithm, where a large gap in the predicted membrane has allowed two objects to be fused together into one (shown in blue). By utilizing the overall 3D context, MergeNet is able to detect and localize the error (shown in red). 7 Figure 5: The output of MergeNet at a merge error, superimposed over the erroneously predicted membranes that led to the merge error. MergeNet output at individual pixels has been thresholded above 0.9, with red denoting predicted merge error and blue the absence of error. Observe that the predicted membranes have a wide gap at the region MergeNet has flagged; this gap is incorrect and the membrane should extend between the two objects, separating them. Note that MergeNet used only threedimensional morphological information to detect this error, and did not make use of the (erroneous) membrane predictions that are shown, or the underlying microscope images. Figure 6: Glial cell, flagged as a merge error by MergeNet. While glia are not merge errors, they are also not neurons and did not occur in the training set for MergeNet. As the algorithm recognizes, the morphology of glia is markedly different from that of neurons. Specifically training MergeNet to recognize glia could be useful in segmenting these cells, which occur along with neurons in brain tissue. Generalizability to different datasets. One of the challenges of traditional connectomics algorithms is that there are numerous different imaging techniques, which can each be applied to the nervous systems of various organisms, and in some cases also to structurally distinct regions of the nervous system within a single organism. For networks used in connectomics for image segmentation, it is often necessary to obtain ground truth annotations on each new dataset, which consumes considerable time and effort. MergeNet, by contrast, is highly transferrable between datasets. Not only can the algorithm be trained on an unverified segmentation and can correct it, but it can also be trained on one dataset and then run on a segmentation of a different dataset, without any retraining. Figures 1 and 5 show images obtained by training on a segmentation of the Kasthuri dataset [12] and then running on the ECS dataset. Applicability to anisotropic data. The microscope data underlying connectomics segmentation is often anisotropic, where the particular dimensions in the x, y, z directions depend upon the particular imaging procedure used. For example, the Kasthuri dataset has resolution of 6 × 6 × 30 nanometers, while our ECS dataset is even more anisotropic, with resolution of 4 × 4 × 30 nanometers. Some imaging technologies do yield isotropic data, such as expansion microscopy (ExM) [5] and focused ion beam scanning elec8 tron microscopy (FIB-SEM) [14]. Various techniques have been proposed to work with anisotropic data, including 2D ConvNets feeding into 3D ConvNets [16] and a combination of convolutional and recurrent networks [6]. MergeNet cancels the effect of anisotropy, as necessary, by downsampling differentially along the x, y, z directions. Thus, the network is able to transform any segmentation into one in which morphology is approximately isotropic, making learning much easier. We also anticipate that it may be possible to train on data from one imaging modality, then to apply a different downsampling ratio to run on data with different anisotropy. For example, MergeNet could be trained on an EM segmentation, then run on an ExM segmentation. Scalability. The MergeNet algorithm is designed to be scalable, so that it can be used to proofread segmentations of extremely large datasets. The network is applied only once objects have been downsampled by a large factor in each dimension, and is applied then only to sampled points within the downsampled object. These two reductions in cost allow the 3D ConvNet to be run at scale, even though 3D kernels are slower to implement than 2D kernels. We tested the speed of MergeNet on an object with 36,874,153 original voxels, downsampled to 18,777 voxels, from which we sampled 1,024, allowing us to generate a dense probability map across the entire object. The ConvNet ran in 11.3 seconds on an Nvidia Tesla K20m GPU. This corresponds to a speed of over three million voxels per second within the original image. Thus, the network could be applied to a volume of 1 billion voxels in a minute using five GPUs. By comparison, the fastest membrane-prediction algorithm can process 1 billion voxels within 2 minutes on a 72core machine [21], demonstrating that our algorithm can be integrated into a scalable connectomics pipeline. Note that our experiments were performed using TensorFlow [1]; we have not attempted to optimize time for training or running the network, though recent work indicates that significant further speedup may be possible [4, 38]. Detection of non-neuronal objects. While MergeNet is trained only on merge errors, it also seems to be able to detect non-neuronal objects, as a byproduct of learning plausible shapes for neurons. In particular, we observe that MergeNet often detects glia (nonneuronal cells that occur in neural tissue), the morphologies of which are distinctively different from those of neurons. Figure 6 shows an example of a glial object from the ECS dataset; notice that MergeNet finds the morphology implausible, even though it has been trained on neither positive nor negative examples of glia. Quantifying the accuracy of glia detection is challenging, however, since little ground truth has been annotated for this task, and most connectomics algorithms are unable to distinguish glia. Finally, let us consider when (and why) MergeNet succeeds and fails on different inputs. The algorithm does not simply label all branch points within a neuron as merge errors, or else it would be effectively useless. However, the network can be confused by examples such as two branches that diverge from a main segment at approximately the same point, resembling the cross of two distinct objects. MergeNet also misses some merge errors. For example, when two neuronal segments run closely in parallel, there may at some points along the boundary be no morphological clues that two objects are present. It is worth noting, however, that parallel neuronal segments can in fact be detected by MergeNet, as shown at point (A) of Figure 1. 6 Conclusion Though merge errors occur universally in automated segmentations of neural tissue, they have never been addressed in generality, as they are difficult to detect using existing connectomics methods. 9 We have shown that a 3D ConvNet can proofread a segmented image for merge errors by “zooming out”, ignoring the image itself, and instead leveraging the general morphological characteristics of neurons. We have demonstrated that our algorithm, MergeNet, is able to generalize without retraining to detect errors within a range of segmentations and across a range of datasets. Relying solely upon unsupervised training, it can nonetheless detect errors within its own training set. Our algorithm enables automatic comparison of segmentation methods, and can be integrated at scale into existing pipelines without the requirement of additional annotation, opening up the possibility of fully automated error detection and correction within neural circuits. While MergeNet can detect and localize merge errors, it cannot, by itself, correct them. One could conceive a variation on the MergeNet algorithm that is used to mark the exact boundary of a merge error, allowing a cut to be made automatically along the boundary so as to correct the merge without additional effort. However, in practice this is a much more challenging task. Often it is impossible to determine the exact division of objects at a merge error purely from morphology. For instance, when two largely parallel objects touch, it may not be evident which is the continuation of which past the point of contact, even if the erroneous merge itself is obvious. Likewise, some merge errors consist of three or more objects that have been confused in some complex way, e.g. by virtue of poor image quality at that location. In such cases, any merge-correction algorithm must have recourse to the underlying microscope images or membrane probabilities, rather than relying purely upon morphological cues. Deep learning approaches leveraging morphology have the potential to transform biological image analysis. It may, for instance, become possible to classify types of neurons automatically, or to identify anomalies such as cancer cells. We anticipate a growth in such algorithms as the scale of biological data grows and as progress in connectomics leads to a deeper understanding of the brain. 7 Acknowledgments We are grateful for support from the National Science Foundation (NSF) under grants IIS-1447786, CCF- 1563880, and 1122374 and from the Intelligence Advanced Research Projects Activity (IARPA) under grant 138076-5093555. 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Geometric Adaptive Control of Attitude Dynamics on SO(3) with State Inequality Constraints arXiv:1602.04286v1 [math.OC] 13 Feb 2016 Shankar Kulumani, Christopher Poole, and Taeyoung Lee Abstract— This paper presents a new geometric adaptive control system with state inequality constraints for the attitude dynamics of a rigid body. The control system is designed such that the desired attitude is asymptotically stabilized, while the controlled attitude trajectory avoids undesired regions defined by an inequality constraint. In addition, we develop an adaptive update law that enables attitude stabilization in the presence of unknown disturbances. The attitude dynamics and the proposed control systems are developed on the special orthogonal group such that singularities and ambiguities of other attitude parameterizations, such as Euler angles and quaternions are completely avoided. The effectiveness of the proposed control system is demonstrated through numerical simulations and experimental results. I. I NTRODUCTION Rigid body attitude control is an important problem for aerospace vehicles, ground and underwater vehicles, as well as robotic systems [1], [2]. One distinctive feature of the attitude dynamics of rigid bodies is that it evolves on a nonlinear manifold. The three-dimensional special orthogonal group, or SO(3), is the set of 3 × 3 orthogonal matrices whose determinant is one. This configuration space is nonEuclidean and yields unique stability properties which are not observable on a linear space. For example, it is impossible to achieve global attitude stabilization using continuous timeinvariant feedback [3]. Attitude control is typically studied using a variety of attitude parameterizations, such as Euler angles or quaternions [4]. All attitude parameterizations fail to represent the nonlinear configuration space both globally and uniquely [5]. For example, minimal attitude representations, such as Euler angle sequences or modified Rodriguez parameters, suffer from singularities. These attitude representations are not suitable for large angular slews. Quaternions do not have singularities but they double cover the special orthogonal group. As a result, any physical attitude is represented by a pair of antipodal quaternions on the three-sphere. During implementation, the designer must carefully resolve this nonunique representation in quaternion based attitude control systems to avoid undesirable unwinding behavior [3]. Many physical rigid body systems must perform large angular slews in the presence of state constraints. For example, autonomous spacecraft or aerial systems are typically equipped with sensitive optical payloads, such as infrared Shankar Kulumani, Christopher Poole, Taeyoung Lee, Mechanical and Aerospace Engineering, George Washington University, Washington DC 20052 {skulumani,poolec,tylee}@gwu.edu This research has been supported in part by NSF under the grants CMMI1243000, CMMI-1335008, and CNS-1337722. or interferometric sensors. These systems require retargeting while avoiding direct exposure to sunlight or other bright objects. The removal of constrained regions from the rotational configuration space results in a nonconvex region. The attitude control problem in the absence of constraints has been extensively studied [6], [7], [8]. However, the attitude control problem in the presence of constraints has received much less attention. Several approaches have been developed to treat the attitude control problem in the presence of constraints. A conceptually straightforward approach is used in [9] to determine feasible attitude trajectories prior to implementation. The algorithm determines an intermediate point such that an unconstrained maneuver can be calculated for each subsegment. Typically, an optimal or easily implementable on-board control scheme for attitude maneuvers is applied to maneuver the vehicle along these segments. In this manner it is possible to accomplish constraint avoidance by linking several intermediary unconstrained maneuvers. While this method is conceptually simple, it is difficult to generalize for an arbitrary number of constraints. In addition, this approach is only applicable to problems where the selection of intermediate points are computationally feasible. The approach in [10] involves the use of randomized motion planning algorithms to solve the constrained attitude control problem. A graph is generated consisting of vertices from an initial attitude to a desired attitude. A random iterative search is conducted to determine a path through a directed graph such that a given cost functional is minimized. The random search approach can only stochastically guarantee attitude convergence as it can be shown that as the number of vertices in the graph grow, the probability of nonconvergence goes to zero. However, the computational demand grows as the size of the graph is increased. As a result, random search approaches are ill-suited to on-board implementation or in scenarios that require agile maneuvers. Model predictive control for spacecraft attitude dynamics is studied in [11], [12], [13]. These methods rely on linear or non-linear state dynamics to repeatedly solve a finitetime optimal control problem. As a result, model predictive control methods are also computational expensive and apply direct optimization methods to solve the necessary conditions for optimality. Therefore these methods are complicated to implement and not applicable for real-time control applications. Artificial potential functions are commonly used to handle kinematic constraints for a wide range of problems in robotics [14]. The goal is the design of attractive and repulsive terms which drive the system toward or away from a certain state, respectively. The superposition of the these functions allows one to apply standard feedback control schemes for stabilization and tracking. More specifically, artificial potential functions have previously been applied to the spacecraft attitude control problem in [15], [16]. However, both of these approaches were developed using attitude parameterizations, namely Euler angles and quaternions, and as such, they are limited by the singularities of minimal representations or the ambiguity of quaternions. This paper is focused on developing an adaptive attitude control scheme in the presence of attitude inequality constraints on SO(3). We apply a potential function based approach developed directly on the nonlinear manifold SO(3). By characterizing the attitude both globally and uniquely on SO(3), our approach avoids the issues of attitude parameterizations, such as kinematic singularities and ambiguities, and is geometrically exact. A configuration error function on SO(3) with a logarithmic barrier function is proposed to avoid constrained regions. Instead of calculating a priori trajectories, as in the geometric and randomized approaches, our approach results in a closed-loop attitude control system. This makes it ideal for on-board implementation on UAV or spacecraft systems. In addition, unlike previous approaches our control system can handle an arbitrary number of constrained regions without modification. Furthermore, we formulate an adaptive update law to enable attitude convergence in the presence of uncertain disturbances. The stability of the proposed control systems is verified via mathematically rigorous Lyapunov analysis on SO(3). In short, the proposed attitude control system in the presence of inequality constraints is computationally efficient and able to handle uncertain disturbances. The effectiveness of this approach is illustrated via numerical simulation and experimental results. II. P ROBLEM F ORMULATION A. Attitude Dynamics We assume that the external disturbance is expressed by W (R, Ω)∆, where W (R, Ω) : SO(3) × R3 → R3×p is a known function of the attitude and the angular velocity. The disturbance is represented by ∆ ∈ Rp and is an unknown, but fixed uncertain parameter. In addition, we assume that a bound on W (R, Ω) and ∆ is known and given by kW k ≤ BW , for x = [x1 , x2 , x3 ]T ∈ R3 . The inverse of the hat map is denoted by the vee map ∨ : so(3) → R3 . Several properties of the hat map are summarized as x · ŷz = y · ẑx, x̂ŷz = (x · z)y − (x · y)z, x[ × y = x̂ŷ − ŷx̂ = yxT − xy T ,  1  tr[Ax̂] = tr x̂(A − AT ) = −xT (A − AT )∨ , 2 x̂A + AT x̂ = ({tr[A] I3×3 − A} x)∧ , ∧ Rx̂R = (Rx) , SO(3) = {R ∈ R3×3 | RT R = I, det[R] = 1}, where a rotation matrix R ∈ SO(3) represents the transformation of the representation of a vector from the bodyfixed frame to the inertial reference frame. The equations of motion are given by J Ω̇ + Ω × JΩ = u + W (R, Ω)∆, (1) Ṙ = RΩ̂, (2) where J ∈ R3×3 is the inertia matrix, and Ω ∈ R3 is the angular velocity represented with respect to the bodyfixed frame. The control moment is denoted by u ∈ R3 , and it is expressed with respect to the body-fixed frame. (3) This form of uncertainty enters the system dynamics through the input channel and as a result is referred to as a matched uncertainty. While this form of uncertainty is easier than the unmatched variety many physically realizable disturbances may be modeled in this manner. For example, orbital spacecraft are subject to gravity gradient torques caused by the non-spherical distribution of mass of both the spacecraft and central gravitational body. This form of disturbance may be represented as a body fixed torque on the vehicle. In addition, for typical scenarios, where the spacecraft is significantly smaller than the orbital radius, the disturbance torque may be assumed constant over short time intervals. In (2), the hat map ∧ : R3 → so(3) represents the transformation of a vector in R3 to a 3 × 3 skew-symmetric matrix such that x̂y = x × y for any x, y ∈ R3 [6]. More explicitly,   0 −x3 x2 0 −x1  , x̂ =  x3 −x2 x1 0 T Consider the attitude dynamics of a rigid body. We define an inertial reference frame and a body frame whose origin is at the center of mass and aligned with the principle directions of the body. The configuration manifold of the attitude dynamics is the special orthogonal group: k∆k ≤ B∆ . R(x × y) = Rx × Ry (4) (5) (6) (7) (8) for any x, y, z ∈ R3 , A ∈ R3×3 and R ∈ SO(3). Throughout this paper, the dot product of two vectors is denoted by x · y = xT y for any x, y ∈ Rn and the maximum eigenvalue and the minimum eigenvalue of J are denoted by λM and λm , respectively. The 2-norm of a matrix A is denoted by kAk, norm is denoted by kAk ≤ kAkF = p and its Frobenius p tr[AT A] ≤ rank(A) kAk. B. State Inequality Constraint The two-sphere is the manifold of unit-vectors in R3 such that S2 = {q ∈ R3 | kqk = 1}. We define r ∈ S2 to be a unit vector from the mass center of the rigid body along a certain direction and it is represented with respect to the body-fixed frame. For example, r may represent the pointing direction of an on-board optical sensor. We define v ∈ S2 to be a unit vector from the mass center of the rigid body toward an undesired pointing direction and represented in the inertial reference frame. For example, v may represent the inertial direction of a bright celestial object or the incoming direction of particles or other debris. It is further assumed that optical sensor has a strict non-exposure constraint with respect to the celestial object. We formulate this hard constraint as rT RT v ≤ cos θ, ◦ (9) ◦ where we assume 0 ≤ θ ≤ 90 is the required minimum angular separation between r and RT v. The objective is to a determine a control input u that stabilizes the system from an initial attitude R0 to a desired attitude Rd while ensuring that (9) is always satisfied. III. ATTITUDE C ONTROL ON SO(3) WITH I NEQUALITY C ONSTRAINTS The first step in designing a control system on a nonlinear manifold Q is the selection of a proper configuration error function. This configuration error function, Ψ : Q × Q → R, is a smooth and proper positive definite function that measures the error between the current configuration and a desired configuration. Once an appropriate configuration error function is chosen, one can then define a configuration error vector and a velocity error vector in the tangent space Tq Q through the derivatives of Ψ [6]. With the configuration error function and vectors the remaining procedure is analogous to nonlinear control design on Euclidean vector spaces. One chooses control inputs as functions of the state through a Lyapunov analysis on Q. To handle the attitude inequality constraint, we propose a new attitude configuration error function. More explicitly, we extend the trace form used in [6], [17] for attitude control on SO(3) with the addition of a logarithmic barrier function. Based on the proposed configuration error function, nonlinear geometric attitude controllers are constructed. A smooth control system is first developed assuming that there is no disturbance, and then it is extended to include an adaptive update law for stabilization in the presence of unknown disturbances. The proposed attitude configuration error function and several properties are summarized as follows. Proposition 1 (Attitude Error Function) Define an attitude error function Ψ : SO(3) → R, an attitude error vector eR ∈ R3 , and an angular velocity error vector eΩ ∈ R3 as follows: Ψ(R) = A(R)B(R), (10) eR = eRA B(R) + A(R)eRB , (11) eΩ = Ω, (12) with
 1  tr G I − RdT R , 2   1 cos θ − rT RT v B(R) = 1 − ln . α 1 + cos θ
∨ 1 eRA = GRdT R − RT Rd G , 2 A(R) = (13) (14) (15) eRB
∨ RT v r = . α (rT RT v − cos θ) (16) where α ∈ R is defined as a positive constant and the matrix G ∈ R3×3 is defined as a diagonal matrix matrix for distinct, positive constants g1 , g2 , g3 ∈ R. Then, the following properties hold (i) Ψ is positive definite about R = Rd (ii) The variation of A(R) with respect to a variation of δR = Rη̂ for η ∈ R3 is given by DR A · δR = η · eRA . (17) (iii) The variation of B(R) with respect to a variation of δR = Rη̂ for η ∈ R3 is given by DR B · δR = η · eRB . (18) (iv) The critical points of Ψ are Rd , and Rd exp(πŝ) for s ∈ {e1 , e2 , e3 } satisfying RT v = ±r. (v) An upper bound of keRA k is given as: 2 keRA k ≤ A(R) , b1 where the constant b1 is given by b1 = (19) h1 h2 +h3 for h1 = min {g1 + g2 , g2 + g3 , g3 + g1 } , n o 2 2 2 h2 = min (g1 − g2 ) , (g2 − g3 ) , (g3 − g1 ) , o n 2 2 2 h3 = min (g1 + g2 ) , (g2 + g3 ) , (g3 + g1 ) . Proof: See Appendix A. Equation (10) is composed of an attractive term, A(R) toward the desired attitude, and a repulsive term, B(R) away from the undesired direction RT v. In order to visualize the attitude error function on SO(3) we utilize a spherical coordinate representation. Recall that the spherical coordinate system represents the position of a point relative to an origin in terms of a radial distance, azimuth, and elevation. This coordinate system is commonly used to define locations on the Earth in terms of a latitude and longitude. Similarly, the positions of celestial objects are defined on the celestial sphere in terms of right ascension and declination. We apply this concept and parametrize the rotation matrix R ∈ SO(3) in terms of the spherical angles −180◦ ≤ λ ≤ 180◦ and −90◦ ≤ β ≤ 90◦ . Using the elementary Euler rotations the rotation matrix is now defined as R = exp(λê2 ) exp(βê3 ). We iterate over the domains of λ and β in order to rotate the body-fixed vector r throughout the two-sphere S2 . Applying this method, Fig. 1 allows us to visualize the error function on SO(3). The attractive error function, given by (13), has been previously used for attitude control on SO(3). The potential well of A(R) is illustrated in Fig. 1(a), where the desired attitude lies at the minimum of A(R). To incorporate the state inequality constraints we apply a logarithmic barrier term. Barrier functions are typically used in optimal control and motion planning applications. A visualization of the configuration error function is presented in Fig. 1(b) which shows that as the boundary of the Attract Avoid 3 3 2.5 2.5 2 2 1.5 1.5 1 1 0.5 0.5 0 where the matrices E(R, Rd ), F (R) ∈ R3×3 are given by 
1  T E(R, Rd ) = tr R Rd G I − RT Rd G , (27) 2 
1 F (R) = v T Rr I − RT vrT + α (rT RT v − cos θ)  RT v̂Rrv T Rr̂ . (28) (rT RT v − cos θ) 0 50 50 100 0 100 0 0 -50 Latitude (β) 0 -50 -100 Latitude (β) Longitude (λ) -100 Longitude (λ) (a) Attractive A(R) (b) Repulsive B(R) Total Proof: See Appendix B. 3 2.5 A. Attitude Control without Disturbance 2 1.5 We introduce a nonlinear geometric controller for the attitude stabilization of a rigid body. We first assume that there is no disturbance, i.e., ∆ = 0. 1 0.5 0 50 100 0 0 -50 Latitude (β) -100 Longitude (λ) (c) Configuration Ψ Proposition 3 (Attitude Control) Given a desired attitude command (Rd , Ωd = 0), which satisfies the constraint (9), and positive constants kR , kΩ ∈ R we define a control input u ∈ R3 as follows Fig. 1: Configuration error function visualization constraint is neared, or rT RT v → cos θ, the barrier term 1 increases, B → ∞. We use the scale factor 1+cos θ to ensure that Ψ remains positive definite. The logarithmic function is popular as it quickly decays away from the constraint boundary. The positive constant α serves to shape the barrier function. As α is increased the impact of B(R) is reduced away from the constraint boundary. The superposition of the attractive and repulsive functions is shown in Fig. 1(c). The control system is defined such that the attitude trajectory follows the negative gradient of Ψ toward the minimum at R = Rd , while avoiding the constrained region. While (14) represents a single inequality constraint given as (9), it is readily generalized to multiple constraints of an arbitrary form. For example, the configuration error function P can be formulated as Ψ = A[1 + i Ci ], where Ci has the form of Ci = B − 1 for the i-th constraint. In this manner, one may enforce multiple state inequality constraints, and we later demonstrate this through numerical simulation. u = −kR eR − kΩ eΩ + Ω × JΩ. (29) Then the zero equilibrium of the attitude error is asymptotically stable, and the inequality constraint is satisfied. Proof: See Appendix C. This proposition only guarantees that the attitude error vector eR asymptotically converges to zero. However, this does not necessarily imply that R → Rd as t → ∞, since there are at most three additional critical points of Ψ where eR = 0 and RT v = ±r. At an undesired equilibrium R = exp (πêi )Rd and eΩ = 0. However, we can show that these undesired equilibrium points are unstable in the sense of Lyapunov [17]. As a result, we can claim that the desired equilibrium R = Rd and eΩ = 0 is almost globally asymptotically stable, which means that the set of initial conditions that do not converge to the desired attitude has zero Lebesgue measure. B. Adaptive Control Proposition 2 (Error Dynamics) The attitude error dynamics for Ψ, eR , eΩ satisfy d (Ψ) = eR · eΩ , dt (20) d (eR ) = ėRA B(R) + eRA Ḃ(R) + Ȧ(R)eRB + AėRB , dt (21) d (eRA ) = E(R, Rd )eΩ , (22) dt d (eRB ) = F (R)eΩ , (23) dt d (A(R)) = eRA · eΩ , (24) dt d (B(R)) = eRB · eΩ , (25) dt d (eΩ ) = J −1 (−Ω × JΩ + u + W (R, Ω)∆) , (26) dt We extend the results of the previous section with the addition of a fixed but unknown disturbance ∆. This scenario is typical of many mechanical systems and represents unmodeled dynamics or external moments acting on the system. For example, Earth orbiting spacecraft typically experience a torque due to a gravitational gradient. Aerial vehicles will similarly experience external torques due to air currents or turbulence. An adaptive control system is introduced to asymptotically stabilize the system to a desired attitude while ensuring that state constraints are satisfied. Proposition 4 (Bound on ėR ) Consider a domain D about the desired attitude defined as  D = R ∈ SO(3)|Ψ < ψ < h1 , rT RT v < β < cos θ . (30) Then the following statements hold: TABLE I: Constraint Parameters (i) Upper bounds of A(R) and B(R) are given by kAk < cA , kBk < cB . (31) Constraint Vector (v) [0.174, −0.934, −0.034]T [0, 0.7071, 0.7071]T [−0.853, 0.436, −0.286]T [−0.122, −0.140, −0.983]T (ii) Upper bounds of E(R, Rd ) and F (R) are given by These results are combined to yield a maximum upper bound of the time derivative of the attitude error vector ėR as kėR k ≤ H keΩ k , (37) Proof: See Appendix D. Proposition 5 (Adaptive Attitude Control) Given a desired attitude command (Rd , Ωd = 0) and positive constants kR , kΩ , k∆ , c ∈ R, we define a control input u ∈ R3 and ¯ as an adaptive update law for the estimated uncertainty ∆ follows: ¯ u = −kR eR − kΩ eΩ + Ω × JΩ − W ∆, T ˙ ¯ = k∆ W (eΩ + ceR ) . ∆ (38) (39) If c is chosen such that 0<c< eR1 2 4 6 8 2 10 t(sec) 2 0 −2 0 1.5 Ψ 1 0 −1 0 2 4 6 8 10 6 8 10 1 t(sec) 0.5 2 4 t(sec) 0 0 2 4 6 8 10 t(sec) (a) Attitude error vector eR (b) Configuration error Ψ Fig. 2: Attitude stabilization without adaptive update law (36) where H ∈ R is defined as H = kBk kEk + 2 keRA k keRB k + kAk kF k . 0 0 eR2 (iii) Upper bounds of the attitude error vectors eRA and eRB are given by r ψ keRA k ≤ , (34) b1 sin θ . (35) keRB k ≤ α (cos θ − β) 2.5 0.5 eR3 1 kEk ≤ √ tr[G] , (32) 2

2 β 2 + 1 (β − cos θ) + 1 + β 2 β 2 − 2 kF k ≤ . 4 α2 (β − cos θ) (33) Angle (θ) 40◦ 40◦ 40◦ 20◦ inequality constraints for the first time on SO(3). The presented control system is computed in real-time and offers significant computational advantages over previous iterative methods. In addition, the riguous mathematical proof guarantees stability. IV. N UMERICAL E XAMPLES We demonstrate the performance of the proposed control system via numerical simulation. The inertia tensor of a rigid body is given as   5.57 × 10−3 6.17 × 10−5 −2.50 × 10−5 J =  6.17 × 10−5 5.57 × 10−3 1.00 × 10−5  kg m2 . −2.50 × 10−5 1.00 × 10−5 1.05 × 10−2 The control system parameters are chosen as 4kR kΩ 2 + 4k − Rλ H , kΩ M (40) the zero equilibrium of the error vectors is stable in the sense ¯ is of Lyapunov. Furthermore, eR , eΩ → 0 as t → ∞, and ∆ uniformly bounded. Proof: See Appendix E. Nonlinear adaptive controllers have been developed for attitude stabilization in terms of modified Rodriguez parameters and quaternions, as well as attitude tracking in terms of Euler angles. The proposed control system is developed on SO(3) and avoids the singularities of Euler angles and Rodriguez parameters while incorporating state inequality constraints. In addition, the unwinding and double coverage ambiguity of quaternions are also completely avoided. The control system handles uncertain disturbances while avoiding constrained regions. Compared to the previous work on constrained attitude control, we present a geometrically exact control system without parameterizations. In addition, we incorporate state G = diag[0.9, 1.1, 1.0], c = 1.0, kR = 0.4, k∆ = 0.5, kΩ = 0.296, α = 15. A body fixed sensor is defined as r = [1, 0, 0], while multiple inequality constraints are defined in Table I. The simulation parameters are chosen to be similar to those found in [15], however we increase the size of the constraint regions to create a more challenging scenario for the control system. The initial state is defined as R0 = exp(225◦ × π 180 ê3 ), Ω0 = 0. The desired state is Rd = I, Ωd = 0. We show simulation results for the system stabilizing about the desired attitude with and without the adaptive update law from We assume a fixed disturbance of  Proposition5. T ∆ = 0.2 0.2 0.2 , with the function W (R, Ω) = I. This form is equivalent to an integral control term which penalizes deviations from the desired configuration. The first term of (39) has the effect of increasing the proportional gain of the control system, since the time derivative of the attitude error vector, ėR , is linear with respect to the angular velocity error vector eΩ . 2.5 180 160 ar ccos(r T R T v i ) 2 Ψ 1.5 1 140 120 100 80 0.5 60 0 0 2 4 6 8 10 40 0 2 4 6 8 10 t(sec) t(sec) (a) Configuration error Ψ (b) Angle to constraints Fig. 4: Attitude control testbed 0.3 ¯ ∆ 0.2 0.1 0 −0.1 0 2 4 6 8 10 t(sec) ¯ (c) Disturbance estimate ∆ (d) Attitude trajectory Fig. 3: Attitude stabilization with adaptive update law Simulation results without the adaptive update law are shown in Fig. 2. Without the update law, the system does not achieve zero steady state error. Fig. 2(b) shows that the configuration error function does not converge to zero and there exist steady state errors. Fig. 3 shows the results with the addition of the adaptive update law. The addition of the adaptive update law allows the system to converge to the desired attitude in the presence of constraints. The path of the body fixed sensor in the inertial frame, namely Rr, is illustrated in Fig. 3(d). The initial attitude is represented with the green circle while the final attitude is marked with a green ×. The inequality constraints from Table I are depicted as red cones, where the cone half angle is θ. The control system is able to asymptotically converge to zero attitude error. Fig. 3(b) shows that the angle arccos(rT RT vi ) between the body fixed sensor and each constraint is satisfied for the entire maneuver. In addition, the estimate of the disturbance converges to the the true value as shown in Fig. 3(c). Both control system are able to automatically avoid the constrained regions. In addition, these results show that it is straightforward to incorporate an arbitrary amount of large constraints. In spite of this challenging configuration space the proposed control system offers a simple method of avoiding constrained regions. These closed-loop feedback results are computed in real time and offer a significant advantage over typical open-loop planning methods. These results show that the proposed geometric adaptive approach is critical to attitude stabilization in the presence of state constraints and disturbances. V. E XPERIMENT ON H EXROTOR UAV A hexrotor unmanned aerial vehicle (UAV) has been developed at the Flight Dynamics and Controls Laboratory (FDCL) at the George Washington University [18]. The UAV is composed of three pairs of counter-rotating propellers. The propeller pairs of the hexrotor are angled relative to one another to allow for a fully actuated rigid body. The hexrotor UAV, shown in Fig. 4, is composed of the following hardware: • Onboard ODROID XU3 computer module. • VectorNav VN100 IMU operating via TTL serial • BLDC motors with BL-Ctrl-2.0 ESC via I2C. • Position and attitude over WiFi (TCP) communication from Vicon motion capture system. • Commands sent over WiFi to onboard controller. In order to constrain the motion and test only the attitude dynamics we attach the hexrotor to a spherical joint. The center of rotation is below the center of gravity of the hexrotor. As a result, there is a destabilizing gravitational moment and the resulting attitude dynamics are similar to an inverted pendulum model. We augment the control input in (38) with an additional term to negate the effect of the gravitational moment. A sensor pointing direction is defined in the body frame of the hexrotor as r = [1, 0, 0]T . We define an obstacle in the inertial frame as v = [ √12 , √12 , 0]T with θ = 12◦ . An initial state is defined as R(0) = exp( π2 ê3 ), while the desired state is Rd = I. This results in the UAV performing a 90◦ yaw rotation about the vertical axis of the spherical joint and the constrained region is on the shortest path connecting R0 and Rd . The attitude control system is identical to the one presented in Proposition 5 with the exception of a gravity moment term and the following parameters: kR = 0.4, kΩ = 0.7, c = 0.1, α = 8 and k∆ = 0.05. The experimental results are shown in Fig. 5. In order to maneuver the system “close” to the constrained zone we utilize several intermediary set points on either side of the obstacle. From the initial attitude the hexrotor rotates to the first set point, pauses, and then continues around the obstacle to the second set point before continuing toward the desired attitude. As a result this creates the stepped behavior of the configuration error history as shown in Fig. 5(b). The brushless motors of the hexrotor allow for large control inputs which are critical to enable aggressive maneuvers. When constrained to the spherical joint the hexrotor is capable of performing responsive attitude changes with high angular velocities. In addition, The on-board control and motion capture system operate at a discrete interval of approximately 100 Hz. It is possible for the system to violate the constraint between these discrete steps and cause eR1 0.5 0 −0.5 0 1.5 5 10 15 1 Ψ eR2 t(sec) 1 0 −1 0 5 10 15 0.5 eR3 t(sec) 2 1 0 0 5 10 15 t(sec) 0 0 5 10 15 t(sec) u1 u2 (a) Attitude error vector eR 0.2 0 −0.2 0 0.5 0 −0.5 0 (b) Configuration error Ψ A PPENDIX 5 10 15 10 15 10 15 A. Proof of Proposition 1 t(sec) 5 To prove (i) we note that (13) is a positive definite function about R = Rd [6]. The constraint angle is assumed 0◦ ≤ θ ≤ 90◦ such that 0 ≤ cos θ. The term rT RT v represents the cosine of the angle between the body fixed vector r and the inertial vector v. It follows that u3 t(sec) 0 −0.2 −0.4 0 analysis. In addition, we have demonstrated the control system via numerical simulation and hardware experiments on a hexrotor UAV. A novel feature of this control is that it is computed autonomously on-board the UAV. This is in contrast to many state constrained attitude control systems which require an a priori attitude trajectory to be calculated. The presented method is simple, efficient and ideal for hardware implementation on embedded systems. 5 t(sec) (c) Control input u (d) Attitude Trajectory Fig. 5: Constrained Attitude stabilization experiment numerical exceptions within the embedded software. As a result, conservative control gains are chosen to ensure the hexrotor operates in a sedate manner and to allow sufficient time for the measurement and control software to operate. There exist several sources of error in the experimental setup. The motion capture system uses a series of optical sensors to determine the relative position of several tracking markers. These markers as well as the cameras must remain fixed to ensure accurate attitude measurement. In addition, the spherical joint is not fixed at the center of mass but is instead offset due to the physical structure of the hexrotor. As a result a disturbance moment is induced on the resultant motion. This results in a small steady state error in the vicinity of the desired attitude. Over time (39) will remain non-zero while eR 6= 0. This will cause an increase in control input until the steady-state error is reduced. Further tuning of the control gains would enable a faster response and a reduced settling time. The hexrotor avoids the constrained region illustrated by the circular cone in Fig. 5(d), by rotating around the boundary of the constraint. This verifies that the proposed control system exhibits the desired performance in the experimental setting as well. A video clip showing the attitude maneuver is available https://youtu.be/dsmAbwQram4. 0≤ cos θ − rT RT v ≤ 1, 1 + cos θ for all R ∈ SO(3). As a result, its negative logarithm is always positive and from (14), 1 < B. The error function Ψ = AB is composed of two positive terms and is therefore also positive definite, and it is minimized at R = Rd . Next, we consider (ii). The variation of (13) is taken with respect to δR = Rη̂ as
∨ 1 DR A · δR = η · GRdT R − RT Rd G , 2 where we used (6). A straightforward application of the chain and product rules of differentiation allows us to show (iii) as
∨ − RT v r DR B · δR = η · , α (cos θ − rT RT v) where the scalar triple product (4) was used. The critical points of eRA are derived in [6]. There are four critical points of eRA , the desired attitude Rd as well as rotations about each body fixed axis by 180◦ . The repulsive
∧ error vector eRB is zero only when the numerator RT v r = 0. This condition only occurs if the desired attitude results in the body fixed vector r becoming aligned with RT v while simultaneously satisfying {Rd } ∪ {Rd exp(πŝ} for s ∈ {e1 , e2 , e3 }. Since we assume the system will not operate in violation of the constraints, the addition of the barrier function does not add additional critical points to the control system. The desired equilibrium is eR = 0 and A = 0. The proof of keRA k given by (v) is available in [17]. VI. C ONCLUSIONS We have developed a geometric adaptive control system which incorporates state inequality constraints on SO(3). The presented control system is developed directly on SO(3) and it avoids singularities and ambiguities that are inherent to attitude parameterizations. The attitude configuration error is augmented with a barrier function to avoid the constrained region, and an adaptive control law is proposed to cancel the effects of uncertainties. We show the stability of the proposed control system through a rigorous mathematical B. Proof of Proposition 2 From the kinematics (2) and noting that Ṙd = 0 the time derivative of RdT R is given as
d RdT R = RdT RêΩ . dt Applying this to the time derivative of (13) gives  d 1  (A) = − tr GRdT RêΩ . dt 2 Applying (6) into this shows (24). Next, the time derivative of the repulsive error function is given by   T T r Ω̂R v d (B) = . dt α (rT RT v − cos θ) Using the scalar triple product, given by (4), one can reduce this to (25). The time derivative of the attractive attitude error vector, eRA , is given by
∨ d 1 (eRA ) = êΩ RT Rd G + (RT Rd G)T êΩ . dt 2 Using the hat map property given in (7) this is further reduced to (22) and (27). We take the time derivative of the repulsive attitude error vector, eRB , as d (eRB ) = aΩv T Rr − aRT vΩT r + bRT v̂Rr, dt with a ∈ R and b ∈ R given by 
−1 a = α rT RT v − cos θ ,b= rT Ω̂RT v α (rT RT v − cos θ) 2.
Using
the scalar triple product from (4) as r · Ω × RT v = RT v · r × Ω gives (23) and (28). We show the time derivative of the configuration error function as d (Ψ) = ȦB + AḂ. dt A straightforward substitution of (13), (14), (24) and (25) into this and appplying (11) shows (20). We show (26) by rearranging (1) as d eΩ = Ω̇ = J −1 (u − Ω × JΩ + W (R, Ω)∆) . dt C. Proof of Proposition 3 Consider the following Lyapunov function: 1 eΩ · JeΩ + kR Ψ(R, Rd ). (41) 2 From (i) of Proposition 1, V ≥ 0. Using (20) and (26) with ∆ = 0, the time derivative of V is given by V= 2 V̇ = −kΩ keΩ k . (42) Since V is positive definite and V̇ is negative semi-definite, the zero equilibrium point eR , eΩ is stable in the sense of Lyapunov. This also implies limt→∞ keΩ k = 0 and keR k is uniformly bounded, as the Lyapunov function is nonincreasing. From (22) and (23), limt→∞ ėR = 0. One can show that këR k is bounded. From Barbalat’s Lemma, it follows limt→∞ kėR k = 0 [19, Lemma 8.2]. Therefore, the equilibrium is asymptotically stable. Furthermore, since V̇ ≤ 0 the Lyapunov function is uniformly bounded which implies Ψ(R(t)) ≤ V(t) ≤ V(0). In addition, the logarithmic term in (14) ensures Ψ(R) → ∞ as rT RT v → cos θ. Therefore, the inequality constraint is always satisfied given that the desired equilibrium lies in the feasible set. D. Proof of Proposition 4 The selected domain ensures that the configuration error function is bounded Ψ < ψ. This implies that that both A(R) and B(R) are bounded by constants cA cB < ψ < h1 . Furthermore, since kBk > 1 this ensures that cA , cB < ψ and shows (31). Next, we show (32) and (33) using the Frobenius norm. The Frobenius norm kEkF is given in [17] as q q 1 2 tr[G2 ] + tr[RT Rd G] . kEkF = tr[E T E] = 2 Applying Rodrigues’ formula and the Matlab symbolic toolbox, this is simplified to  1 1   2 2 2 2 tr G + tr[G] ≤ tr[G] , kEkF ≤ 4 2 which shows (32), since kEk ≤ kEkF . To show (33), we apply the Frobenius norm kF kF :   T   T  1 kF kF = 2 tr a a − 2tr a b 2 T T α (r R v − cos θ)  T        +2tr a c + tr bT b − 2tr bT c + tr cT c . where the terms a, b, and c are given by a = rT RrI, b = RT vrT , c= RT v̂Rrv T Rr̂ . rT RT v − cos θ A straightforward computation of aT a shows that  
2 tr aT a = v T Rr tr[I] ≤ 3β 2 , where we used the fact that v T Rr = rT RT v <  β from  our given domain. Similarly, one can show that tr aT b is equivalent to    
2 tr aT b = v T Rrtr RT vrT = v T Rr ≤ β 2 ,  T where = xT y. The product  T we used the fact that tr xy tr a c is given by h  
∨
i v T Rr tr aT c = T T tr RT v rv T R r̂ , r R v − cos θ where we used the hat map property (8). One can show that T T tr[a  T c] ≤ 0 over the range −1 ≤ v Rr ≤ cos θ. Next, tr b b is equivalent to     tr bT b = tr rv T RRT vrT = 1,   since r, v ∈ S2 . Finally, tr cT c is reduced to       tr cT c = tr r̂RT vrT −I + RT vv T R rv T Rr̂ , 2 where we used the fact that x̂2 = − kxk I +xxT . Expanding and collecting like terms gives
2
4  T  1 − 2 v T Rr + v T Rr tr c c = . 2 (rT RT v − cos θ) Using the the given domain rT RT v ≤ β gives the upper bound (33). The bound on eRA is given in (19) while eRB arises from the definition of the cross product ka × bk = kak kbk sin θ. Finally, we can find the upper bound (21) as kėR k ≤ (kBk kEk + 2 keRA k keRB k + kAk kF k) keΩ k . Using (31–35) one can define H in terms of known values. E. Proof of Proposition 5 Consider the Lyapunov function V given by V= 1 1 eΩ · JeΩ + kR Ψ + cJeΩ · eR + e∆ · e∆ , (43) 2 2k∆ over the domain D in (30). From Proposition 4, the Lyapunov function is bounded in D by V ≤ z T W z, (44) ¯ z = [keR k, keΩ k, ke∆ k] ∈ R and the where e∆ = ∆ − ∆, matrix W ∈ R3×3 is given by   0 kR ψ 21 cλM 0 . W =  12 cλM 21 λM 1 0 0 2k∆ T 3 The time derivative of V with the control inputs (38) is given by T V̇ = − kΩ eTΩ eΩ + (eΩ + ceR ) W e∆ − kR ceTR eR 1 T ¯˙ e ∆, (45) − kΩ ceTR eΩ + cJeTΩ ėR − k∆ ∆ ¯˙ The terms linearly dependent on where we used ė = −∆. ∆ e∆ are combined with (39) to yield   1 ¯˙ T T e∆ W (eΩ + ceR ) − ∆ = 0. k∆ Using Proposition 4 an upper bound on V̇ is written as V̇ ≤ −ζ T M ζ, where ζ = [keR k, keΩ k] ∈ R2 , and the matrix M ∈ R2×2 is   kΩ c kR c 2 . (46) M = kΩ c kΩ − cλM H 2 If c is chosen such that (40) is satisfied the matrix M is positive definite. This implies that V̇ is negative semidefinite and limt→∞ ζ = 0. As the Lyapunov function is nonincreasing z is uniformly bounded. R EFERENCES [1] P. Hughes, Spacecraft Attitude Dynamics. Dover Publications, 2004. [2] J. R. Wertz, Spacecraft Attitude Determination and Control. Springer, 1978, vol. 73. [3] S. P. Bhat and D. S. Bernstein, “A topological obstruction to continuous global stabilization of rotational motion and the unwinding phenomenon,” Systems & Control Letters, 2000. [4] M. D. Shuster, “A survey of attitude representations,” Navigation, vol. 8, no. 9, 1993. [5] N. Chaturvedi, A. K. Sanyal, N. H. McClamroch, et al., “Rigid-body attitude control,” Control Systems, IEEE, vol. 31, no. 3, pp. 30–51, 2011. [6] F. Bullo and A. D. Lewis, Geometric Control of Mechanical Systems, ser. Texts in Applied Mathematics. New York-Heidelberg-Berlin: Springer Verlag, 2004, vol. 49. [7] C. Mayhew and A. Teel, “Synergistic potential functions for hybrid control of rigid-body attitude,” in Proceedings of the American Control Conference, 2011, pp. 875–880. [8] T. Lee, “Global exponential attitude tracking controls on SO(3),” IEEE Transactions on Automatic Control, vol. 60, no. 10, pp. 2837– 2842, 2015. [9] H. B. Hablani, “Attitude commands avoiding bright objects and maintaining communication with ground station,” Journal of Guidance, Control, and Dynamics, vol. 22, no. 6, pp. 759–767, 2015/09/19 1999. [Online]. Available: http://dx.doi.org/10.2514/2. 4469 [10] E. Frazzoli, M. Dahleh, E. Feron, and R. Kornfeld, “A randomized attitude slew planning algorithm for autonomous spacecraft,” in AIAA Guidance, Navigation, and Control Conference and Exhibit, Montreal, Canada, 2001. [11] A. Guiggiani, I. Kolmanovsky, P. Patrinos, and A. Bemporad, “Fixedpoint constrained model predictive control of spacecraft attitude,” arXiv:1411.0479, 2014. [Online]. Available: http://arxiv.org/abs/1411. 0479 [12] U. Kalabic, R. Gupta, S. Di Cairano, A. Bloch, and I. Kolmanovsky, “Constrained spacecraft attitude control on SO(3) using fast nonlinear model predictive control using reference governors and nonlinear model predictive control,” in American Control Conference (ACC), 2014, June 2014, pp. 5586–5593. [13] R. Gupta, U. Kalabic, S. Di Cairano, A. Bloch, and I. Kolmanovsky, “Constrained spacecraft attitude control on SO(3) using fast nonlinear model predictive control,” in American Control Conference (ACC), 2015, July 2015, pp. 2980–2986. [14] E. Rimon and D. E. Koditschek, “Exact robot navigation using artificial potential functions,” Robotics and Automation, IEEE Transactions on, vol. 8, no. 5, pp. 501–518, 1992. [15] U. Lee and M. Mesbahi, “Spacecraft Reorientation in Presence of Attitude Constraints via Logarithmic Barrier Potentials,” in Proceedings of the American Control Conference, 2011, pp. 450–455. [16] C. R. McInnes, “Large angle slew maneuvers with autonomous sun vector avoidance,” Journal of Guidance, Control, and Dynamics, vol. 17, no. 4, pp. 875–877, 2015/07/10 1994. [Online]. Available: http://dx.doi.org/10.2514/3.21283 [17] T. Lee, “Robust adaptive tracking on SO(3) with an application to the attitude dynamics of a quadrotor UAV,” IEEE Transactions on Control Systems Technology, vol. 21, no. 5, pp. 1924–1930, September 2013. [18] E. Kaufman, K. Caldwell, D. Lee, and T. Lee, “Design and development of a free-floating hexrotor UAV for 6-dof maneuvers,” in Proceedings of the IEEE Aerospace Conference, 2014. [19] H. K. Khalil, Nonlinear Systems, 3rd ed. Prentice Hall New Jersey, 2002. 

INSPECTRE: Privately Estimating the Unseen arXiv:1803.00008v1 [] 28 Feb 2018 Jayadev Acharya∗ ECE, Cornell University [email protected] Gautam Kamath† EECS & CSAIL, MIT [email protected] Ziteng Sun‡ ECE, Cornell University [email protected] Huanyu Zhang§ ECE, Cornell University [email protected] March 2, 2018 Abstract We develop differentially private methods for estimating various distributional properties. Given a sample from a discrete distribution p, some functional f , and accuracy and privacy parameters α and ε, the goal is to estimate f (p) up to accuracy α, while maintaining ε-differential privacy of the sample. We prove almost-tight bounds on the sample size required for this problem for several functionals of interest, including support size, support coverage, and entropy. We show that the cost of privacy is negligible in a variety of settings, both theoretically and experimentally. Our methods are based on a sensitivity analysis of several state-of-the-art methods for estimating these properties with sublinear sample complexities. 1 Introduction How can we infer a distribution given a sample from it? If data is in abundance, the solution may be simple – the empirical distribution will approximate the true distribution. However, challenges arise when data is scarce in comparison to the size of the domain, and especially when we wish to quantify “rare events.” This is frequently the case: for example, it has recently been observed that there are several very rare genetic mutations which occur in humans, and we wish to know how many such mutations exist [KC12, TBO+ 12, NWE+ 12]. Many of these mutations have only been seen once, and we can infer that there are many which have not been seen at all. Over the last decade, a large body of work has focused on developing theoretically sound and effective tools for such settings [OSW16] and references therein, including the problem of estimating the frequency distribution of rare genetic variations [ZVV+ 16]. However, in many settings where one wishes to perform statistical inference, data may contain sensitive information about individuals. For example, in medical studies, where the data may contain individuals’ health records and whether they carry some disease which bears a social stigma. Alternatively, one can consider a map application which suggests routes based on aggregate positions of individuals, which contains delicate information including users’ residence data. In these settings, it is critical that our methods protect sensitive information contained in the dataset. This does not preclude our overall goals of statistical analysis, as we are trying to infer properties of the population p, and not the samples which are drawn from said population. That said, without careful experimental design, published statistical findings may be prone to leaking sensitive information about the sample. As a notable example, it was recently shown that one can determine the identity of some individuals who participated in genome-wide association studies [HSR+ 08]. This ∗ Supported by NSF CCF-1657471 and a Cornell University startup grant. by ONR N00014-12-1-0999, NSF CCF-1617730, CCF-1650733, and CCF-1741137. Work partially done while author was an intern at Microsoft Research, New England. ‡ Supported by NSF CCF-1657471 and a Cornell University startup grant. § Supported by NSF CCF-1657471 and a Cornell University startup grant. † Supported 1 realization has motivated a surge of interest in developing data sharing techniques with an explicit focus on maintaining privacy of the data [JS13, USF13, YFSU14, SSB16]. Privacy-preserving computation has enjoyed significant study in a number of fields, including statistics and almost every branch of computer science, including cryptography, machine learning, algorithms, and database theory – see, e.g., [Dal77, AW89, AA01, DN03, Dwo08, DR14] and references therein. Perhaps the most celebrated notion of privacy, proposed by theoretical computer scientists, is differential privacy [DMNS06]. Informally, an algorithm is differentially private if its outputs on neighboring datasets (differing in a single element) are statistically close (for a more precise definition, see Section 2). Differential privacy has become the standard for theoretically-sound data privacy, leading to its adoption by several large technology companies, including Google and Apple [EPK14, Dif17]. Our focus in this paper is to develop tools for privately performing several distribution property estimation tasks. In particular, we study the tradeoff between statistical accuracy, privacy, and error rate in the sample size. Our model is that we are given sample access to some unknown discrete distribution p, over a domain of size k, which is possibly unknown in some tasks. We wish to estimate the following properties: • Support Coverage: If we take m samples from the distribution, what is the expected number of unique elements we expect to see? • Support Size: How many elements of the support have non-zero probability? • Entropy: What is the Shannon entropy of the distribution? For more formal statements of these problems, see Section 2.1. We require that our output is α-accurate, satisfies (ε, 0)-differential privacy, and is correct with probability 1 − β. The goal is to give an algorithm with minimal sample complexity n, while simultaneously being computationally efficient. Theoretical Results. Our main results show that privacy can be achieved for all these problems at a very low cost. For example, if one wishes to privately estimate entropy, this incurs an additional additive cost in the sample complexity which is very close to linear in 1/αε. We draw attention to two features of this bound. First, this is independent of k. All the problems we consider have complexity Θ(k/ log k), so in the primary regime of study where k  1/αε, this small additive cost is dwarfed by the inherent sample complexity of the non-private problem. Second, the bound is almost linear in 1/αε. We note that performing even the most basic statistical task privately, estimating the bias of a coin, incurs this linear dependence. Surprisingly, we show that much more sophisticated inference tasks can be privatized at almost no cost. In particular, these properties imply that the additive cost of privacy is o(1) in the most studied regime where the support size is large. In general, this is not true – for many other problems, including distribution estimation and hypothesis testing, the additional cost of privacy depends significantly on the support size or dimension [DHS15, CDK17, ASZ17, ADR17]. We also provide lower bounds, showing that our upper bounds are almost tight. A more formal statement of our results appears in Section 3. Experimental Results. We demonstrate the efficacy of our method with experimental evaluations. As a baseline, we compare with the non-private algorithms of [OSW16] and [WY18]. Overall, we find that our algorithms’ performance is nearly identical, showing that, in many cases, privacy comes (essentially) for free. We begin with an evaluation on synthetic data. Then, inspired by [VV13, OSW16], we analyze text corpus consisting of words from Hamlet, in order to estimate the number of unique words which occur. Finally, we investigate name frequencies in the US census data. This setting has been previously considered by [OSW16], but we emphasize that this is an application where private statistical analysis is critical. This is proven by efforts of the US Census Bureau to incorporate differential privacy into the 2020 US census [DLS+ 17]. Techniques. Our approach works by choosing statistics for these tasks which possess bounded sensitivity, which is well-known to imply privacy under the Laplace or Gaussian mechanism. We note that bounded sensitivity of statistics is not always something that can be taken for granted. Indeed, for many fundamental tasks, optimal algorithms for the non-private setting may be highly sensitive, thus necessitating crucial modifications to obtain differential privacy [ADK15, CDK17]. Thus, careful choice and design of statistics must be a priority when performing inference with privacy considerations. To this end, we leverage recent results of [ADOS17], which studies estimators for non-private versions of the problems we consider. The main technical work in their paper exploits bounded sensitivity to show sharp cutoff-style concentration bounds for certain estimators, which operate using the principle of best-polynomial approximation. They use these results to show that a single algorithm, the Profile Maximum Likelihood (PML), can estimate all these properties simultaneously. On the other hand, we consider the sensitivity of these estimators for purposes of privacy – the same property is utilized by both works for very different 2 purposes, a connection which may be of independent interest. We note that bounded sensitivity of a statistic may be exploited for purposes other than privacy. For instance, by McDiarmid’s inequality, any such statistic also enjoys very sharp concentration of measure, implying that one can boost the success probability of the test at an additive cost which is logarithmic in the inverse of the failure probability. One may naturally conjecture that, if a statistical task is based on a primitive which concentrates in this sense, then it may also be privatized at a low cost. However, this is not true – estimating a discrete distribution in `1 distance is such a task, but the cost of privatization depends significantly on the support size [DHS15]. One can observe that, algorithmically, our method is quite simple: compute the non-private statistic, and add a relatively small amount of Laplace noise. The non-private statistics have recently been demonstrated to be practical [OSW16, WY18], and the additional cost of the Laplace mechanism is minimal. This is in contrast to several differentially private algorithms which invoke significant overhead in the quest for privacy. Our algorithms attain almost-optimal rates (which are optimal up to constant factors for most parameter regimes of interest), while simultaneously operating effectively in practice, as demonstrated in our experimental results. Related Work. Over the last decade, there have been a flurry of works on the problems we study in this paper by the computer science and information theory communities, including Shannon and Rényi entropy estimation [Pan03, VV17, JVHW17, AOST17, OS17, WY18], support coverage and support size estimation [OSW16, WY18]. A recent paper studies the general problem of estimating functionals of discrete distribution from samples in terms of the smoothness of the functional [FS17]. These have culminated in a nearly-complete understanding of the sample complexity of these properties, with optimal sample complexities (up to constant factors) for most parameter regimes. Recently, there has been significant interest in performing statistical tasks under differential privacy constraints. Perhaps most relevant to this work are [CDK17, ASZ17, ADR17], which study the sample complexity of differentialy privately performing classical distribution testing problems, including identity and closeness testing. Other works investigating private hypothesis testing include [WLK15, GLRV16, KR17, KSF17, Rog17, GR17], which focus less on characterizing the finite-sample guarantees of such tests, and more on understanding their asymptotic properties and applications to computing p-values. There has also been study on private distribution learning [DHS15, DJW17, KV18], in which we wish to estimate parameters of the distribution, rather than just a particular property of interest. A number of other problems have been studied with privacy requirements, including clustering [WWS15, BDL+ 17], principal component analysis [CSS13, KT13, HP14], ordinary least squares [She17], and much more. 2 Preliminaries We will start with some definitions. Pk def Let ∆ = {(p(1), . . . , p(k)) : p(i) ≥ 0, i=1 p(i) = 1, 1 ≤ k ≤ ∞} be the set of discrete distributions over a countable support. Let ∆k be the set of distributions in ∆ with at most k non-zero probability values. A property f (p) is a mapping from ∆ → R. We now describe the classical distribution property estimation problem, and then state the problem under differential privacy. Property Estimation Problem. Given α, β, f , and independent samples X1n from an unknown distribution p, design an estimator fˆ : X1n → R such that with probability at least 1 − β, fˆ(X1n ) − f (p) < α.   def The sample complexity of fˆ, is Cfˆ(f, α, β) = min{n : Pr fˆ(X1n ) − f (p) > α < β} is the smallest number of samples to estimate f to accuracy α, and error β. We study the problem for β = 1/3, and by the median trick, we can boost the error probability to β with an additional multiplicative log(1/β) more samples: def Cfˆ(f, α) = Cfˆ(f, α, 1/3). The sample complexity of estimating a property f (p) is the minimum sample complexity over all estimators: C(f, α) = minfˆ Cfˆ(f, α). An estimator fˆ is ε-differentially private (DP) [DMNS06] if for any X1n and Y1n , with dham (X1n , Y1n ) ≤ 1, Pr (f (X1n )∈S) ≤ eε , for all i, and measurable S. Pr (f (Y1n )∈S ) 3 Private Property Estimation. Given α, ε, β, f , and independent samples X1n from an unknown distribution p, design an ε-differentially private estimator fˆ : X1n → R such that with probability at least 1 − β, fˆ(X1n ) − f (p) < α. Similar to the non-private setting, the sample complexity of ε-differentially private estimation problem is C(f, α, ε) = minfˆ:fˆis ε-DP Cfˆ(f, α, 1/3), the smallest number of samples n for which there exists such a ±α estimator with error probability at most 1/3. In their original paper [DMNS06] provides a scheme for differential privacy, known as the Laplace mechanism. This method adds Laplace noise to a non-private scheme in order to make it private. We first define the sensitivity of an estimator, and then state their result in our setting. def Definition 1. The sensitivity of an estimator fˆ : [k]n → R is ∆n,fˆ = maxdham (X1n ,Y1n )≤1 fˆ(X1n ) − fˆ(Y1n ) . Let Dfˆ(α, ε) = min{n : ∆n,fˆ ≤ αε}. Lemma 1.  n  α o ,ε . C(f, α, ε) = O min Cfˆ(f, α/2) + Dfˆ 4 fˆ Proof. [DMNS06] showed that for a function with sensitivity ∆n,fˆ, adding Laplace noise X ∼ Lap(∆n,fˆ/ε) makes the output ε-differentially private. By the definition of Dfˆ( α4 , ε), the Laplace noise we add has |x| 1 − b , hence we have e parameter at most α4 . Recall that the probability density function of Lap(b) is 2b 1 α Pr (|X| > α/2) < e2 . By the union bound, we get an additive error less than α = 2 + α2 with probability at most 1/3 + e12 < 0.5. Hence, with the median trick, we can boost the error probability to 1/3, at the cost of a constant factor in the number of samples. To prove sample complexity lower bounds for differentially private estimators, we observe that the estimator can be used to test between two distributions with distinct property values, hence is a harder problem. For lower bounds on differentially private testing, [ASZ17] gives the following argument based on coupling: Lemma 2. Suppose there is a coupling between distributions p and q over X n , such that E [dham (X1n , Y1n )] ≤ D. Then, any ε-differentially private algorithm that distinguishes between p and q with error probability at
most 1/3 must satisfy D = Ω 1ε . 2.1 Problems of Interest Support Size. The support size of a distribution p is S(p) = |{x : p(x) > 0}|, the number of symbols with non-zero probability values. However, notice that estimating S(p) from samples can be hard due to the presence of symbols with negligible, yet non-zero probabilities. To circumvent this issue, [RRSS09] proposed def to study the problem when the smallest probability is bounded. Let ∆≥ k1 = {p ∈ ∆ : p(x) ∈ {0} ∪ [1/k, 1]} be the set of all distributions where all non-zero probabilities have value at least 1/k. For p ∈ ∆≥ k1 , our goal is to estimate S(p) up to ±αk with the least number of samples from p. P Support Coverage. For a distribution p, and an integer m, let Sm (p) = x (1 − (1 − p(x))m ), be the expected number of symbols that appear when we obtain m independent samples from the distribution p. The objective is to find the least number of samples n in order to estimate Sm (p) to an additive ±αm. Support coverage arises in many ecological and biological studies [CCG+ 12] to quantify the number of new elements (gene mutations, species, words, etc) that can be expected to be seen in the future. Good and Toulmin [GT56] proposed an estimator that for any constant α, requires m/2 samples to estimate Sm (p). P 1 Entropy. The Shannon entropy of a distribution p is H(p) = x p(x) log p(x) , H(p) is a central object in information theory [CT06], and also arises in many fields such as machine learning [Now12], neuroscience [BWM97, NBdRvS04], and others. Estimating H(p) is hard with any finite number of samples due to the possibility of infinite support. To circumvent this, a natural approach is to consider distributions in ∆k . The goal is to estimate the entropy of a distribution in ∆k to an additive ±α, where ∆k is all discrete distributions over at most k symbols. 4 3 Statement of Results Our theoretical results for estimating support coverage, support size, and entropy are given below. Algorithms for these problems and proofs of these statements are provided in Section 4. Our experimental results are described and discussed in Section 5. Theorem 1. For any ε = Ω(1/m), the sample complexity of support coverage is   m log(1/α) m log(1/α) C(Sm , α, ε) = O + . log m log(εm) Furthermore,   m log(1/α) 1 C(Sm , α, ε) = Ω + . log m αε Theorem 2. For any ε = Ω(1/k), the sample complexity of support size estimation is  C(S, α, ε) = O  k log2 (1/α) k log2 (1/α) + . log k log(εk) Furthermore,   k log2 (1/α) 1 C(S, α, ε) = Ω + . log k αε Theorem 3. Let λ > 0 be any small fixed constant. For instance, λ can be chosen to be any constant between 0.01 and 1. We have the following upper bounds on the sample complexity of entropy estimation:    log2 (min{k, n}) 1 1 k + + log C(H, α, ε) = O α α2 αε αε and k log2 (min{k, n}) C(H, α, ε) = O + + λ2 α log k α2  1 αε 1+λ ! . Furthermore,  C(H, α, ε) = Ω  k log2 (min{k, n}) log k + + . α log k α2 αε We provide some discussion of our results. At a high level, we wish to emphasize the following two points: 1. Our upper bounds show that the cost of privacy in these settings is often negligible compared to the sample complexity of the non-private statistical task, especially when we are dealing with distributions over a large support. Furthermore, our upper bounds are almost tight in all parameters. 2. The algorithmic complexity introduced by the requirement of privacy is minimal, consisting only of a single step which noises the output of an estimator. In other words, our methods are realizable in practice, and we demonstrate the effectiveness on several synthetic and real-data examples. First, we examine our results on support size and support coverage estimation. We note that we focus on the regime where ε is not exceptionally small, as the privacy requirement becomes somewhat unusual. For instance, non-privately, if we have m samples for the problem of support coverage, then the empirical plug-in estimator is the best we can do. However, if ε = O(1/m), then group privacy [DR14] implies that the algorithm’s output distribution on any dataset of m samples must be very similar – however, these samples may have an arbitrary value of support coverage ∈ [m], which precludes hopes for a highly accurate estimator. To avoid degeneracies of this nature, we restrict our attention to ε = Ω(1/m). In this regime, if ε = Ω(mγ /m) for any constant γ > 0, then up to constant factors, our upper bound is within a constant factor of the optimal sample complexity without privacy constratints. In other words, for most meaningful values of ε, privacy comes for free. 5 Next, we turn our attention to entropy estimation. We note that the second upper bound in Theorem 3 has a parameter λ that indicates a tradeoff between the sample complexity incurred in the first and third term. This parameter determines the degree of a polynomial to be used for entropy estimation. As the degree becomes smaller (corresponding to a large λ), accuracy of the polynomial estimator decreases, however, at the same time, low-degree polynomials have a small sensitivity, allowing us to privatize the outcome. In terms of our theoretical results, one can think of λ = 0.01. With this parameter setting, it can be observed that our upper bounds are almost tight. For example, one can see that the upper and lower bounds match to either logarithmic factors (when looking at the first upper bound), or a very small polynomial factor in 1/αε (when looking at the second upper bound). For our experimental results, we experimentally determined an effective value for the parameter λ on a single synthetic instance. We then show that this choice of parameter generalizes, giving highly-accurate private estimation in other instances, on both synthetic on real-world data. 4 Algorithms and Analysis In this section, we prove our results for support coverage in Section 4.1, support size in Section 4.2, and entropy in Section 4.3. In each section, we first describe and analyze our algorithms for the relevant problem. We then go on to describe and analyze a lower bound construction, showing that our upper bounds are almost tight. All our algorithms fall into the following simple framework: 1. Compute a non-private estimate of the property; 2. Privatize this estimate by adding Laplace noise, where the parameter is determined through analysis of the estimator and potentially computation of the estimator’s sensitivity. 4.1 Support Coverage Estimation In this section, we prove Theorem 1, about support coverage estimation: Theorem 1. For any ε = Ω(1/m), the sample complexity of support coverage is   m log(1/α) m log(1/α) + . C(Sm , α, ε) = O log m log(εm) Furthermore,   1 m log(1/α) + . C(Sm , α, ε) = Ω log m αε Our upper bound is described and analyzed in Section 4.1.1, while our lower bound appears in Section 4.1.2. 4.1.1 Upper Bound for Support Coverage Estimation Let ϕi be the number of symbols that appear i times in X1n . We will use the following non-private support coverage estimator from [OSW16]: Ŝm (X1n ) = n X
ϕi 1 + (−t)i · Pr (Z ≥ i) , i=1 where Z is a Poisson random variable with mean r (which is a parameter to be instantiated later), and t = (m − n)/n. Our private estimator of support coverage is derived by adding Laplace noise to this non-private estimator with the appropriate noise parameter, and thus the performance of our private estimator, is analyzed by bounding the sensitivity and the bias of this non-private estimator according to Lemma 1. The sensitivity and bias of this estimator is bounded in the following lemmas. Lemma 3. Suppose m > 2n, then the maximum coefficient of ϕi in Ŝm (p) is at most 1 + er(t−1) . 6 Proof. By the definition of Z, we know Pr (Z ≥ i) = P∞ k=i k e−r rk! , hence we have: |1 + (−t)i · Pr (Z ≥ i)| ≤ 1 + ti ≤1+e ∞ X e−r k=i ∞ X −r k=i ≤ 1 + e−r =1+e rk k! (rt)k k! ∞ X (rt)k k=0 r(t−1) k! The bias of the estimator is bounded in Lemma 4 of [ADOS17]: Lemma 4. Suppose m > 2n, then h i E Ŝm (X1n ) − Sm (p) ≤ 2 + 2er(t−1) + min(m, S(p)) · e−r . Using these results, letting r = log(1/α), [OSW16] showed that there is a constant C, such that with n = C logmm log(1/α) samples, with probability at least 0.9, Ŝm (X1n ) Sm (p) − ≤ α. m m Ŝ (X n ) Our upper bound in Theorem 1 is derived by the following analysis of the sensitivity of m m 1 . If we change one sample in X1n , at most two of the ϕj ’s change. Hence by Lemma 3, the sensitivity of the estimator satisfies !  2  Ŝm (X1n ) ≤ · 1 + er(t−1) . (1) ∆ m m By Lemma 1, there is a private algorithm for support coverage estimation as long as ! Ŝm (X1n ) ≤ αε, ∆ m which by (1) holds if 2(1 + exp(r(t − 1))) ≤ αεm. Let r = log(3/α), note that t − 1 = m n − 2. Suppose αεm > 2, then, the condition above reduces to       1 3 m · − 2 ≤ log αεm − 1 . log α n 2 This is equivalent to m log(3/α) log( 21 αεm − 1) + 2 log(3/α) m log(3/α) = log( 23 εm − 3/α) + log(3/α) n≥ Suppose αεm > 2, then the condition above reduces to the requirement that   m log(1/α) n=O . log(εm) 7 (2) 4.1.2 Lower Bound for Support Coverage Estimation We now prove the lower bound described in Theorem 1. Note that the first term in the lower bound is the sample complexity of non-private support coverage estimation, shown in [OSW16]. Therefore, we turn our attention to prove the latter term in the sample complexity. Consider the following two distributions. u1 is uniform over [m(1 + α)]. u2 is distributed over m + 1 α 1 ∀i ∈ [m] and u2 [4] = 1+α . Moreover, 4 ∈ / [m(1 + α)]. Then, elements [m] ∪ {4} where u2 [i] = m(1+α)   m  1 Sm (u1 ) = m(1 + α) · 1 − 1 − , m(1 + α) and Sm (u2 ) =   m· 1− 1− 1 m(1 + α) m    + 1− 1− α 1+α m  hence, Sm (u2 ) − Sm (u1 )   = mα · 1 − 1 − 1 m(1 + α) m    − 1− 1− α 1+α m  = Ω(αm) α Hence we know there support coverage differs by Ω(αm). Moreover, their total variation distance is 1+α . The following lemma is folklore, based on the coupling interpretation of total variation distance, and the fact that total variation distance is subadditive for product measures. Lemma 5. For any two distributions p, and q, there is a coupling between n iid samples from the two distributions with an expected Hamming distance of dTV (p, q) · n. Using Lemma 5 and dTV (u1 , u2 ) = α 1+α , we have Lemma 6. Suppose u1 and u2 are as defined before, there is a coupling between un1 and un2 with expected α Hamming distance equal to 1+α n. Moreover, given n samples, we must be able to privately distinguish between u1 and u2 given an α accurate estimator of support coverage with privacy considerations. Thus, according to Lemma 2 and 6, we have:   α 1 1 n≥ ⇒n=Ω . 1+α ε εα 4.2 Support Size Estimation In this section, we prove our main theorem about support size estimation, Theorem 2: Theorem 2. For any ε = Ω(1/k), the sample complexity of support size estimation is  C(S, α, ε) = O  k log2 (1/α) k log2 (1/α) + . log k log(εk) Furthermore,   1 k log2 (1/α) + . C(S, α, ε) = Ω log k αε Our upper bound is described and analyzed in Section 4.2.1, while our lower bound appears in Section 4.2.2. 8 4.2.1 Upper Bound for Support Size Estimation In [OSW16], it is shown that the support coverage estimator can be used to obtain optimal results for estimating the support size of a distribution. In this fashion, taking m = k log(3/α), we we may use an estimate of the support coverage Sm (p) as an estimator of S(p). In particular, their result is based on the following observation. Lemma 7. Suppose m ≥ k log(3/α), then for any p ∈ ∆≥ k1 , |Sm (p) − S(p)| ≤ αk . 3 Proof. From the definition of Sm (p), we have Sm (p) ≤ S(p). For the other side, X X m S(p) − Sm (p) = (1 − p(x)) ≤ e−mp(x) x x ≤ k · e− log(3/α) = kα . 3 Therefore, estimating Sm (p) for m = k log(3/α), up to ±αk/3, also estimates S(p) up to ±αk. Therefore, the goal is to estimate the smallest value of n to solve the support coverage problem. Suppose r = log(3/α), and m = k log(3/α) = k · r in the support coverage problem. Then, we have t= k log(3/α) m −1= − 1. n n (3) Then, by Lemma 4 in the previous section, we have h i E Ŝm (X1n ) − S(p) h i ≤ E Ŝm (X1n ) − Sm (p) + |Sm (p) − S(p)| ≤ 2 + 2er(t−1) + min{m, k} · e−r + ≤ 2 + 2er(t−1) + k · e− log(3/α) + ≤ 2 + 2er(t−1) + 2 kα 3 kα 3 kα . 3 We will find conditions on n such that the middle
term above is at most kα. Toward this end, note that 2er(t−1) ≤ αk holds if and only if r(t − 1) ≤ log αk 2 . Plugging in (3), this holds when     k log(3/α) αk log(3/α) · − 2 ≤ log , n 2 which is equivalent to n≥ k log2 (3/α) log αk 2 + 2 log  3 α =O k log2 (1/α) log k  where we have assumed without loss of generality that α > k1 . The computations for sensitivity are very similar. From Lemma 1 1, we need to find the value of n such that 2 + 2er(t−1) ≤ αεk, where we assume that n ≤ 12 k log(3/α), else we just add noise to the true number of observed distinct elements By computations similar to the expectation case, this reduces to n≥ k log2 (3/α) . 3 log αεk 2 + log α 9 Therefore, this gives us a sample complexity of k log2 (1/α) n=O log (εk)   (4) for the sensitivity result to hold. We note that the bound above blows up when ε ≤ k1 . However, we note that our lower bound implies that we need at least Ω(1/ε) = Ω(k) samples in this case, which is not in the sub-linear regime that we are interested in. We therefore consider only the regime where the privacy parameter ε is at least 1/k. 4.2.2 Lower Bound for Support Size Estimation In this section, we prove a lower bound for support size estimation, as described in Theorem 2. The techniques are similar to those for support coverage in Section 4.1.2. The first term of the complexity is the lower bound for non-private setting. This follows by combining the lower bound of [OSW16] for support coverage, with the equivalence between estimation of support size and coverage as implied by Lemma 7. We focus on the second term in the sequel. Consider the following two distributions: u1 is a uniform distribution over [k] and u2 is a uniform distribution over [(1−α)k]. Then the support size of these two distribution differs by αk, and dTV (u1 , u2 ) = α. Hence by Lemma 5, we know the following: Lemma 8. Suppose u1 ∼ U [k] and u2 ∼ U [(1 − α)k], there is a coupling between un1 and un2 with expected Hamming distance equal to αn. Moreover, given n samples, we must be able to privately distinguish between u1 and u2 given an α accurate estimator of entropy with privacy considerations. Thus, according to Lemma 2 and Lemma 8, we have:   1 1 . αn ≥ ⇒ n = Ω ε εα 4.3 Entropy Estimation In this section, we prove our main theorem about entropy estimation, Theorem 3: Theorem 3. Let λ > 0 be any small fixed constant. For instance, λ can be chosen to be any constant between 0.01 and 1. We have the following upper bounds on the sample complexity of entropy estimation:    log2 (min{k, n}) 1 1 k + + log C(H, α, ε) = O α α2 αε αε and k log2 (min{k, n}) C(H, α, ε) = O + + λ2 α log k α2  1 αε 1+λ ! . Furthermore,  C(H, α, ε) = Ω  k log2 (min{k, n}) log k + + . α log k α2 αε We describe and analyze two upper bounds. The first is based on the empirical entropy estimator, and is described and analyzed in Section 4.3.1. The second is based on the method of best-polynomial approximation, and appears in Section 4.3.2. Finally, our lower bound is in Section 4.3.3. 4.3.1 Upper Bound for Entropy Estimation: The Empirical Estimator Our first private entropy estimator is derived by adding Laplace noise into the empirical estimator. The parameter of the Laplace distribution is ∆(H(ε p̂n )) , where ∆(H(p̂n )) denotes the sensitivity of the empirical estimator. By analyzing its sensitivity and bias, we prove an upper bound on the sample complexity for private entropy estimation and get the first upper bound in Theorem 3. 10 Let p̂n be the empirical distribution, and let H(p̂n ) be the entropy of the empirical distribution. The theorem is based on the following three facts:   log n ∆(H(p̂n )) = O . (5) n   k |H(p) − E [H(p̂n )]| = O , (6) n  2  log (min{k, n}) V ar (H(p̂n )) = O , (7) n With these three facts in hand, the sample complexity of the empirical
estimator can be bounded as follows. 1 1 log( αε ) . We also need |H(p) − E [H(p̂n )]| = By Lemma 1, we need ∆(H(p̂n )) ≤ αε, which gives  n = O αε
2 log (min{k,n}) k 2 O(α) and V ar (H(p̂n )) = O α , which gives n = O n + . α2 Proof of (5). The largest change in any Nx when we change one symbol is one. Moreover, at most two Nx change. Therefore, n j n j+1 log − log n j+1 n j j j 1 n = 2 · max log + log j=1...n−1 n j+1 n j+1   j 1 n j log , log ≤ 2 · max max j=1...n−1 n j+1 n j+1   1 log n ≤ 2 · max , , n n log n . =2· n ∆(H(p̂n )) ≤ 2 · Proof of (6). max j=1...n−1 (8) (9) (10) By the concavity of entropy function, we know that E [H(p̂n )] ≤ H(p). Therefore, E [|H(p) − H(p̂n )|] = H(p) − E [H(p̂n )] " # X =E (p̂n (x) log p̂n (x) − p(x) log p(x)) x " =E X x # " # X p̂n (x) p̂n (x) log +E (p̂n (x) − p(x)) log p(x) p(x) x = E [D(p̂n kp)]   ≤ E dχ2 (p̂n || p) " # X (p̂n (x) − p(x))2 =E p(x) x X (p(x)/n) ≤ p(x) x = (11) (12) (13) k . n (14) 11 2 Proof of (7). The variance bound of logn k is given precisely in Lemma 15 of [JVHW17]. To obtain the other half of the bound of, we apply the bounded differences inequality in the form stated in Corollary 3.2 of [BLM13]. Lemma 9. Let f : Ωn → R be a function. Suppose further that max 0 0 z1 ,...,zn ,zi f (z1 , . . . , zn ) − f (z1 , . . . , zi−1 , zi , . . . , zn ) ≤ ci . Then for independent variables Z1 , . . . , Zn , n Var(f (Z1 , . . . , Zn )) ≤ 1X 2 c . 4 i=1 i Therefore, using Lemma 9 and Equation (5)  V ar (H(p̂n )) ≤ n · 4.3.2 4 log2 n n2  = 4 log2 n . n Upper Bound for Entropy Estimation: Best-Polynomial Approximation We prove an upper bound on the sample complexity for private entropy estimation if one adds Laplace noise into best-polynomial estimator.This will give us the second upper bound in Theorem 3. In the non-private setting the optimal sample complexity of estimating H(p) over ∆k is given by Theorem 1 of [WY16]   k log2 (min{k, n}) Θ + . α log k α2 However, this estimator can have a large sensitivity. [ADOS17] designed an estimator that has the same sample complexity but a smaller sensitivity. We restate Lemma 6 of [ADOS17] here: Lemma 10. Let λ > 0 be a fixed small constant, which may be taken to be any value between 0.01 and 1. Then there is an entropy estimator with sample complexity   1 k log2 (min{k, n}) + Θ 2· , λ α log k α2 and has sensitivity nλ /n. We can now invoke Lemma 1 on the estimator in this lemma to obtain the upper bound on private entropy estimation. 4.3.3 Lower Bound for Entropy Estimation We now prove the lower bound for entropy estimation. Note that any lower bound on privately testing two distributions p, and q such that H(p) − H(q) = Θ(α) is a lower bound on estimating entropy. We analyze the following construction for Proposition 2 of [WY16]. The two distributions p, and q over [k] are defined as: 1 − p(1) 2 ,p(i) = , for i = 2, . . . , k, 3 k−1 2−η 1 − q(1) q(1) = ,q(i) = , for i = 2, . . . , k. 3 k−1 p(1) = Then, by the grouping property of entropy, H(p) = h(2/3) + 1 1+η · log(k − 1), and H(q) = h((2 − η)/3) + · log(k − 1), 3 3 12 (15) (16) which gives H(p) − H(q) = Ω(η log k). For η = α/ log k, the entropy difference becomes Θ(α). The total variation distance between p and q is η/3. By Lemma 5 in the paper, there is a coupling over X1n , and Y1n generated from p and q with expected Hamming distance at most dTV (p, q) · n. This along with Lemma 2 in the paper gives a lower bound of Ω(log k/αε) on the sample complexity. 5 Experiments We evaluated our methods for entropy estimation and support coverage on both synthetic and real data. Overall, we found that privacy is quite cheap: private estimators achieve accuracy which is comparable or near-indistinguishable to non-private estimators in many settings. Our results on entropy estimation and support coverage appear in Sections 5.1 and 5.2, respectively. Code of our implementation is available at https://github.com/HuanyuZhang/INSPECTRE. 5.1 Entropy We compare the performance of our entropy estimator with a number of alternatives, both private and non-private. Non-private algorithms considered include the plug-in estimator (plug-in), the Miller-Madow Estimator (MM) [Mil55], the sample optimal polynomial approximation estimator (poly) of [WY16]. We analyze the privatized versions of plug-in, and poly in Sections 4.3.1 and 4.3.2, respectively. The implementation of the latter is based on code from the authors of [WY16]1 . We compare performance on different distributions including uniform, a distribution with two steps, Zipf(1/2), a distribution with Dirichlet-1 prior, and a distribution with Dirichlet-1/2 prior, and over varying support sizes. While plug-in, and MM are parameter free, poly (and its private counterpart) have to choose the degree L of the polynomial to use, which manifests in the parameter λ in the statement of Theorem 3. [WY16] suggest the value of L = 1.6 log k in their experiments. However, since we add further noise, we choose a single L as follows: (i) Run privatized poly for different L values and distributions for k = 2000, ε = 1, (b) Choose the value of L that performs well across different distributions (See Figure 1). We choose L = 1.2·log k from this, and use it for all other experiments. To evaluate the sensitivity of poly, we computed the estimator’s value at all possible input values, computed the sensitivity, (namely, ∆ = maxdham (X1n ,Y1n )≤1 |poly(X1n )−poly(Y1n )|),
and added noise distributed as Lap 0, ∆ ε . 0.6 0.4 0.8 0.6 500 750 1000 1250 1500 Number of samples 1750 2000 0.0 1.0 0.8 500 750 1000 1250 1500 Number of samples 1750 2000 0.0 1.50 1.25 1.00 0.75 L=0.3log(k) L=0.6log(k) L=0.9log(k) L=1.2log(k) L=1.5log(k) L=1.8log(k) 2.0 1.5 1.0 0.5 0.25 0.2 250 L=0.3log(k) L=0.6log(k) L=0.9log(k) L=1.2log(k) L=1.5log(k) L=1.8log(k) 1.75 0.50 0.4 0.2 250 1.2 0.6 0.4 0.2 0.0 1.0 1.4 Dirichlet-1/2 prior Dirichlet-1 prior L=0.3log(k) L=0.6log(k) L=0.9log(k) L=1.2log(k) L=1.5log(k) L=1.8log(k) 1.6 RMSE 0.8 1.2 RMSE RMSE 1.0 Zipf 1/2 L=0.3log(k) L=0.6log(k) L=0.9log(k) L=1.2log(k) L=1.5log(k) L=1.8log(k) 1.4 RMSE Two steps L=0.3log(k) L=0.6log(k) L=0.9log(k) L=1.2log(k) L=1.5log(k) L=1.8log(k) RMSE Uniform 1.2 250 500 750 1000 1250 1500 Number of samples 1750 2000 0.00 250 500 750 1000 1250 1500 Number of samples 1750 2000 0.0 250 500 750 1000 1250 1500 Number of samples 1750 2000 Figure 1: RMSE comparison for private Polynomial Approximation Estimator with various values for degree L, k = 2000, ε = 1. The RMSE of various estimators for k = 1000, and ε = 1 for various distributions are illustrated in Figure 2. The RMSE is averaged over 100 iterations in the plots. We observe that the performance of our private-poly is near-indistinguishable from the non-private poly, particularly as the number of samples increases. It also performs significantly better than all other alternatives, including the non-private Miller-Madow and the plug-in estimator. The cost of privacy is minimal for several other settings of k and ε, for which results appear in Section A. 1 See https://github.com/Albuso0/entropy for their code for entropy estimation. 13 2.5 1.5 2.5 2.0 1.5 2.0 2.0 1.5 1.0 1.0 0.5 0.5 0.5 0.5 400 600 800 0.0 1000 Number of samples 200 400 600 800 0.0 1000 Number of samples 200 400 600 800 Number of samples 2.0 1.5 1.0 0.5 0.0 1000 Plug-in Miller Poly Poly-Laplace Plug-in-Laplace 2.5 1.5 1.0 200 Dirichlet-1/2 prior Plug-in Miller Poly Poly-Laplace Plug-in-Laplace 2.5 1.0 0.0 Dirichlet-1 prior 3.0 Plug-in Miller Poly Poly-Laplace Plug-in-Laplace 3.0 RMSE 2.0 RMSE RMSE 2.5 Zipf 1/2 Plug-in Miller Poly Poly-Laplace Plug-in-Laplace 3.0 RMSE Two steps Plug-in Miller Poly Poly-Laplace Plug-in-Laplace 3.0 RMSE Uniform 3.5 200 400 600 800 0.0 1000 Number of samples 200 400 600 800 1000 Number of samples Figure 2: Comparison of various estimators for the entropy, k = 1000, ε = 1. 5.2 Support Coverage We investigate the cost of privacy for the problem of support coverage. We provide a comparison between the Smoothed Good-Toulmin estimator (SGT) of [OSW16] and our algorithm, which is a privatized version of their statistic (see Section 4.1.1). Our implementation is based on code provided by the authors of [OSW16]. As shown in our theoretical results, the sensitivity of SGT is at most 2(1 + er (t − 1)), necessitating the addition of Laplace noise with parameter 2(1 + er(t−1) )/ε. Note that while the theory suggests we select the 2 1 loge n(t+1) parameter r = log(1/α), α is unknown. We instead set r = 2t t−1 , as previously done in [OSW16]. 5.2.1 Evaluation on Synthetic Data In our synthetic experiments, we consider different distributions over different support sizes k. We generate n = k/2 samples, and then estimate the support coverage at m = n · t. For large t, estimation is harder. Some results of our evaluation on synthetic are displayed in Figure 3. We compare the performance of SGT, and privatized versions of SGT with parameters ε = 1, 2, and 10. For this instance, we fixed the domain size k = 20000. We ran the methods described above with n = k/2 samples, and estimated the support coverage at m = nt, for t ranging from 1 to 10. The performance of the estimators is measured in terms of RMSE over 1000 iterations. 1500 Uniform 1000 Non-private Private eps=10 Private eps=2 Private eps=1 800 Two steps 1200 Non-private Private eps=10 Private eps=2 Private eps=1 1000 Zipf 1/2 1400 Non-private Private eps=10 Private eps=2 Private eps=1 1200 1000 800 400 RMSE RMSE RMSE RMSE 600 1000 600 400 500 0 1 200 2 3 4 5 t 6 7 8 9 10 0 1 3 4 5 t 6 7 8 9 10 0 1 1800 1600 1400 600 3 4 5 t 6 7 8 9 10 0 1 1000 800 600 400 200 2 Dirichlet-1/2 prior Non-private Private eps=10 Private eps=2 Private eps=1 1200 800 400 200 2 Dirichlet-1 prior Non-private Private eps=10 Private eps=2 Private eps=1 RMSE 2000 200 2 3 4 5 t 6 7 8 9 10 0 1 2 3 4 5 t 6 7 8 9 10 Figure 3: Comparison between the private estimator with the non-private SGT when k = 20000 We observe that, in this setting, the cost of privacy is relatively small for reasonable values of ε. This is as predicted by our theoretical results, where unless ε is extremely small (less than 1/k) the non-private sample complexity dominates the privacy requirement. However, we found that for smaller support sizes (as shown in Section A.2), the cost of privacy can be significant. We provide an intuitive explanation for why no private estimator can perform well on such instances. To minimize the number of parameters, we instead argue about the related problem of support-size estimation. Suppose we are trying to distinguish between distributions which are uniform over supports of size 100 and 200. We note that, if we draw n = 50 samples, the “profile” of the samples (i.e., the histogram of the histogram) will be very similar for the two distributions. In particular, if one modifies only a few samples (say, five or six), one could convert one profile into the other. In other words, these two profiles are almost-neighboring datasets, but simultaneously correspond to very different support sizes. This pits the two goals of privacy and accuracy at odds with each other, thus resulting in a degradation in accuracy. 5.2.2 Evaluation on Census Data and Hamlet We conclude with experiments for support coverage on two real-world datasets, the 2000 US Census data and the text of Shakespeare’s play Hamlet, inspired by investigations in [OSW16] and [VV17]. Our investigation 14 on US Census data is also inspired by the fact that this is a setting where privacy is of practical importance, evidenced by the proposed adoption of differential privacy in the 2020 US Census [DLS+ 17]. The Census dataset contains a list of last names that appear at least 100 times. Since the dataset is so oversampled, even a small fraction of the data is likely to contain almost all the names. As such, we make the task non-trivial by subsampling mtotal = 86080 individuals from the data, obtaining 20412 distinct last names. We then sample n of the mtotal individuals without replacement and attempt to estimate the total number of last names. Figure 4 displays the RMSE over 100 iterations of this process. We observe that even an exceptionally stringent privacy budget of ε = 0.5, the performance is almost indistinguishable from the non-private SGT estimator. 6000 Non-private Private eps=2 Private eps=1 Private eps=0.5 5000 RMSE 4000 3000 2000 1000 0 0.1 0.2 0.3 0.4 0.5 0.6 Fraction of seen names 0.7 0.8 0.9 1.0 Figure 4: Comparison between our private estimator with the SGT on Census Data The Hamlet dataset has mtotal = 31, 999 words, of which 4804 are distinct. Since the distribution is not as oversampled as the Census data, we do not need to subsample the data. Besides this difference, the experimental setup is identical to that of the Census dataset. Once again, as we can see in Figure 5, we get near-indistinguishable performance between the non-private and private estimators, even for very small values of ε. Our experimental results demonstrate that privacy is realizable in practice, with particularly accurate performance on real-world datasets. 1800 Non-private Private eps=2 Private eps=1 Private eps=0.5 1600 1400 RMSE 1200 1000 800 600 400 200 0 0.1 0.2 0.3 0.4 0.5 0.6 Fraction of seen words 0.7 0.8 0.9 1.0 Figure 5: Comparison between our private estimator with the SGT on Hamlet References [AA01] Dakshi Agrawal and Charu C. Aggarwal. On the design and quantification of privacy preserving data mining algorithms. In Proceedings of the 20th ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems, PODS ’01, pages 247–255, New York, NY, USA, 2001. ACM. 15 [ADK15] Jayadev Acharya, Constantinos Daskalakis, and Gautam Kamath. Optimal testing for properties of distributions. 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Journal of Biomedical Informatics, 50:133–141, 2014. [ZVV+ 16] James Zou, Gregory Valiant, Paul Valiant, Konrad Karczewski, Siu On Chan, Kaitlin Samocha, Monkol Lek, Shamil Sunyaev, Mark Daly, and Daniel G. MacArthur. Quantifying unobserved protein-coding variants in human populations provides a roadmap for large-scale sequencing projects. Nature Communications, 7, 2016. A Additional Experimental Results This section contains additional plots of our synthetic experimental results. Section A.1 contains experiments on entropy estimation, while Section A.2 contains experiments on estimation of support coverage. A.1 Entropy Estimation We present four more plots of our synthetic experimental results for entropy estimation. Figures 6 and 7 are on a smaller support of k = 100, with ε = 1 and 2, respectively. Figures 8 and 9 are on a support of k = 1000, with ε = 0.5 and 2. 19 Uniform 3.0 2.0 2.5 1.5 2.0 1.5 2.5 1.5 1.0 0.5 0.5 0.5 40 60 Number of samples 80 100 20 Zipf 1 2.0 40 60 Number of samples 80 0.0 100 2.5 RMSE RMSE 1.5 0.5 0.5 0.5 80 100 20 40 60 Number of samples (a) 100 1.5 1.0 60 80 80 Plug-in Miller Poly Poly-Laplace Plug-in-Laplace 2.0 1.0 40 60 2.5 1.0 Number of samples 40 Number of samples Dirichlet-1/2 prior Plug-in Miller Poly Poly-Laplace Plug-in-Laplace 2.0 1.5 20 20 Dirichlet-1 prior 3.0 Plug-in Miller Poly Poly-Laplace Plug-in-Laplace 2.5 RMSE 2.0 1.0 20 Plug-in Miller Poly Poly-Laplace Plug-in-Laplace 3.0 1.0 0.0 Zipf 1/2 3.5 Plug-in Miller Poly Poly-Laplace Plug-in-Laplace 3.0 RMSE RMSE 2.5 Two steps 3.5 Plug-in Miller Poly Poly-Laplace Plug-in-Laplace RMSE 3.5 100 20 40 60 Number of samples (b) 80 100 (c) Figure 6: Comparison of various estimators for the entropy, k = 100, ε = 1. Uniform 3.0 2.0 1.5 Zipf 1/2 Plug-in Miller Poly Poly-Laplace Plug-in-Laplace 3.0 2.5 RMSE RMSE 2.5 Two steps 3.5 Plug-in Miller Poly Poly-Laplace Plug-in-Laplace 2.0 1.5 2.5 2.0 1.5 1.0 1.0 1.0 0.5 0.5 0.5 20 40 60 Number of samples 80 0.0 100 Zipf 1 2.5 40 60 Number of samples 80 100 20 Dirichlet-1 prior 3.0 Plug-in Miller Poly Poly-Laplace Plug-in-Laplace 2.0 20 Plug-in Miller Poly Poly-Laplace Plug-in-Laplace 3.0 RMSE 3.5 60 80 100 Dirichlet-1/2 prior Plug-in Miller Poly Poly-Laplace Plug-in-Laplace 2.5 40 Number of samples Plug-in Miller Poly Poly-Laplace Plug-in-Laplace 2.5 2.0 1.5 1.0 1.0 0.5 0.5 20 40 60 Number of samples (a) 80 100 RMSE 1.5 RMSE RMSE 2.0 1.5 1.0 0.5 20 40 60 Number of samples (b) 80 100 20 40 60 Number of samples (c) Figure 7: Comparison of various estimators for the entropy, k = 100, ε = 2. 20 80 100 Two steps Uniform Plug-in Miller Poly Poly-Laplace Plug-in-Laplace 3.0 2.0 3.0 2.5 RMSE 1.5 2.5 1.5 2.0 1.5 1.0 1.0 0.5 0.5 0.5 0.0 0.0 1.0 200 400 600 800 Number of samples 1000 Zipf 1 1.75 1.50 1.00 400 600 800 Number of samples 0.0 1000 200 Dirichlet-1 prior 400 600 800 Number of samples 1000 Dirichlet-1/2 prior Plug-in Miller Poly Poly-Laplace Plug-in-Laplace 2.5 Plug-in Miller Poly Poly-Laplace Plug-in-Laplace 2.5 2.0 2.0 RMSE 1.25 200 3.0 Plug-in Miller Poly Poly-Laplace Plug-in-Laplace 2.00 RMSE 2.0 Plug-in Miller Poly Poly-Laplace Plug-in-Laplace 3.0 RMSE RMSE 2.5 Zipf 1/2 Plug-in Miller Poly Poly-Laplace Plug-in-Laplace RMSE 3.5 1.5 0.75 1.0 0.50 0.5 1.5 1.0 0.5 0.25 200 400 600 800 Number of samples 0.0 1000 200 400 600 800 Number of samples (a) 0.0 1000 200 400 600 800 Number of samples (b) 1000 (c) Figure 8: Comparison of various estimators for the entropy, k = 1000, ε = 0.5. Two steps Uniform Plug-in Miller Poly Poly-Laplace Plug-in-Laplace 3.0 2.0 3.0 2.5 RMSE 1.5 2.5 1.5 2.0 1.5 1.0 1.0 0.5 0.5 0.5 0.0 0.0 1.0 200 400 600 800 Number of samples 1000 Zipf 1 2.00 1.50 1.00 0.75 400 600 800 Number of samples 200 400 600 800 Number of samples 1000 Dirichlet-1/2 prior Plug-in Miller Poly Poly-Laplace Plug-in-Laplace 2.0 1.5 1.0 0.50 0.0 1000 Dirichlet-1 prior 2.5 RMSE 1.25 200 3.0 Plug-in Miller Poly Poly-Laplace Plug-in-Laplace 1.75 RMSE 2.0 Plug-in Miller Poly Poly-Laplace Plug-in-Laplace 3.0 Plug-in Miller Poly Poly-Laplace Plug-in-Laplace 2.5 2.0 RMSE RMSE 2.5 Zipf 1/2 Plug-in Miller Poly Poly-Laplace Plug-in-Laplace RMSE 3.5 1.5 1.0 0.5 0.5 0.25 0.00 200 400 600 800 Number of samples (a) 1000 0.0 200 400 600 800 Number of samples (b) 1000 0.0 200 400 600 (c) Figure 9: Comparison of various estimators for the entropy, k = 1000, ε = 2. 21 800 Number of samples 1000 200 150 100 50 0 1 300 250 Two steps 200 Non-private Private eps=10 Private eps=2 Private eps=1 150 RMSE RMSE 150 Uniform Non-private Private eps=10 Private eps=2 Private eps=1 RMSE 200 100 50 2 3 4 5 6 t Zipf 1 7 8 9 0 1 10 200 Non-private Private eps=10 Private eps=2 Private eps=1 150 Zipf 1/2 Non-private Private eps=10 Private eps=2 Private eps=1 100 50 2 3 4 5 6 t Dirichlet-1 prior 7 8 9 0 1 10 200 Non-private Private eps=10 Private eps=2 Private eps=1 150 2 3 4 5 6 7 8 9 10 6 7 8 9 10 t Dirichlet-1/2 prior Non-private Private eps=10 Private eps=2 Private eps=1 150 RMSE RMSE RMSE 200 100 100 100 50 50 50 0 1 2 3 4 5 t 6 7 8 9 (a) 10 0 1 2 3 4 5 t (b) 6 7 8 9 10 0 1 2 3 4 5 t (c) Figure 10: Comparison between the private estimator with the non-private SGT when k = 1000. A.2 Support Coverage We present three additional plots of our synthetic experimental results for support coverage estimation. In particular, Figures 10, 11, and 12 show support coverage for k = 1000, 5000, 100000. 22 450 Non-private Private eps=10 Private eps=2 Private eps=1 500 400 350 RMSE RMSE 400 300 200 100 Two steps 450 Non-private Private eps=10 Private eps=2 Private eps=1 400 350 300 300 250 250 RMSE Uniform 600 200 0 1 1000 800 2 3 4 5 6 t Zipf 1 7 8 9 150 100 100 0 1 10 600 Non-private Private eps=10 Private eps=2 Private eps=1 500 50 2 3 4 5 6 t Dirichlet-1 prior 7 8 9 0 1 10 600 Non-private Private eps=10 Private eps=2 Private eps=1 500 RMSE RMSE 400 200 0 1 2 3 4 5 t 6 7 8 9 300 3 4 5 6 7 8 9 10 6 7 8 9 10 t Dirichlet-1/2 prior Non-private Private eps=10 Private eps=2 Private eps=1 300 200 200 100 100 0 1 10 2 400 RMSE 400 600 200 150 50 Zipf 1/2 Non-private Private eps=10 Private eps=2 Private eps=1 2 3 4 5 (a) t 6 7 8 9 0 1 10 2 3 4 5 (b) t (c) Figure 11: Comparison between the private estimator with the non-private SGT when k = 5000. 3000 2500 RMSE RMSE 4000 12000 2 3 4 5 6 t Zipf 1 7 8 9 1500 0 1 10 4500 Non-private Private eps=10 Private eps=2 Private eps=1 4000 3500 6000 2 3 4 5 6 t Dirichlet-1 prior 7 8 9 6000 Non-private Private eps=10 Private eps=2 Private eps=1 5000 2000 3 4 5 (a) t 6 7 8 9 10 3 4 5 6 7 8 9 10 6 7 8 9 10 t Dirichlet-1/2 prior Non-private Private eps=10 Private eps=2 Private eps=1 3000 2000 1000 500 2 2 4000 2500 0 1 1500 0 1 10 1000 2000 2000 500 1500 4000 Zipf 1/2 Non-private Private eps=10 Private eps=2 Private eps=1 1000 3000 8000 0 1 2500 500 RMSE RMSE 10000 3000 1000 2000 14000 3500 2000 6000 0 1 Two steps Non-private Private eps=10 Private eps=2 Private eps=1 RMSE 8000 Uniform Non-private Private eps=10 Private eps=2 Private eps=1 RMSE 10000 2 3 4 5 (b) t 6 7 8 9 10 0 1 2 3 4 5 t (c) Figure 12: Comparison between the private estimator with the non-private SGT when k = 100000. 23 

A Report of a Significant Error On a Frequently Used Pseudo Random Number Generator arXiv:1408.1900v1 [cs.MS] 7 Aug 2014 Ayşe Ferhan Yeşil, M. Cemal Yalabik Department of Physics, Bilkent University, 06800 Ankara, Turkey Abstract Emergence of stochastic simulations as an extensively used computational tool for scientific purposes intensified the need for more accurate ways of generating sufficiently long sequences of uncorrelated random numbers. Even though several different methods have been proposed for this end, deterministic algorithms known as pseudo-random number generators (PRNGs) emerged to be the most widely used tool as a replicable, portable and easy to use method to generate such random number sequences. Here, we introduce a simple Poisson process whose simulation gives systematic errors when the very commonly used random number generator of the GNU C Library (Glibc) is utilised. The PRNG of Glibc is an additive lagged Fibonacci generator, the family of such PRNGs are accepted as relatively safe among other PRNGs. The systematic errors indicate complex correlation relations among random numbers which requires a further explanation. Keywords: pseudo-random numbers, pseudo-random number generator, PACS: 02.70.-c, 05.10.-a 1. Introduction Random numbers are at the core of stochastic simulations for scientific purposes. Nearly in all of these cases, but especially in Monte Carlo simulations, long sequences of uncorrelated, uniformly distributed random numbers are required. Several different methods are used to generate such random number sequences, but as a replicable, portable and easy to use method, deterministic algorithms, namely pseudo-random number generators (PRNGs), are preferred extensively. However, to be confident of such a number sequence being “sufficiently” random, i.e. any combination of random numbers within Preprint submitted to Journal Of Computational Physics August 15, 2014 the sequence is uncorrelated, is not straightforward since there is no complete and universal statistical test of analysis. In other words, even though one cannot be confident that a sequence is totally random by any analysis known so far, it can still be decided that the sequence is prone to correlations in the case of empirical or theoretical tests indicating in that direction [1]. As a result, correlation tests for PRNGs have become necessary tools for physicists [3, 5, 6, 4, 2, 12, 7, 9, 8, 10, 11]. Exemplifying the significance of these tests, it is important to note that most of the tests are direct results of PRNGs that go wrong. One of the first such paradigmatic examples, dated back to 1993, is encountered on the simulation of 2-D Ising Model which shows that results of various algorithms are different from the exact result and they also differ from algorithm to algorithm depending on the PRNGs used. This analysis showed that Generalized Feedback Shift Register (GFSR) PRNGs (a type of lagged Fibonacci generators (LFGs) which utilise exclusive-or operations) with smaller lags produce dramatically wrong results, up to hundred times of the standard deviation, when used with Wolff algorithm [3]. The depth first nature of the algorithms are suspected to interfere with the intrinsic triple correlations of the GFSRs. However, Schmid et al showed that even though the algorithm is randomly updating the system one spin at a time, if the model has triple structure in itself such as Blume-Capel model, use of random numbers produced by GFSR in the simulations lead to dramatic errors [4]. On the other hand, GFSRs are not the only faulty PRNGs. LFGs with addition and subtraction operations also cause systematic errors if they have small lags [5]. The default random number generator of GlibC, the library of the commonly used C-compiler unix based environments, is an LFG with addition operation LFG(31,3,+). It’s lag is 31 which is generally considered as small. Grassberger reported of correlations in Lagged Fibonacci Generators with correlation lengths proportional to their lags. His method resembles Vattulainen’s n-block method[10]. Grassberger proposed that it might be related to the triple correlations intrinsic to the random number sequence produced by any LFG. He also showed while simulating 3-D self-avoiding walks on a cubic lattice, that even an LFG with lag 250, LFG(250,103,+), gives poor results. This led to the suspicion that LFG(55,24,+) and LFG(17,5,+) may operate poorly, and was demonstrated to be so [6]. In light of these, the default PRNG of GlibC should be considered as unreliable since it has a small lag as well. In this paper, a simple test model, which shows strong correlations when utilised with the default random number generator provided with the C com2 piler of Linux distributions, is introduced. C is a very commonly used programming language and is also very popular among scientific programming communities. The GNU C Library (Glibc) is used as the C library in the GNU systems and most systems with the Linux kernel. The PRNG of this library is constructed using a linear additive feedback method. We demonstrate the existence of relatively strong correlations through the use of a simple algorithm which is related to a one dimensional diffusive gas problem. To inform the unspecialised reader we provide here further information about the pseudo-random number generators: There are basically two common types of PRNGs: Linear Congruential Generators (LCGs) and Lagged Fibonacci Generators (LFGs) [5]. LCGs are associated with the general equation Xi = (AXi−1 + B) mod M (1) which can be represented in short as LCG(A, B, M), with a maximum period of M that can be reached through a suitable choice of A and B. Here A, B and M are integers, and Xi is the ith generated random number. LFGs have the general equation: Xi = Xi−P ⊙ Xi−Q (2) where P and Q are the lags, conventionally chosen as P > Q. LFGs can be represented as LF G(P, Q, ⊙) [5]. The symbol ⊙ represents any binary arithmetic operation such as addition, subtraction, multiplication or exclusive-or (xor) operation. If this operation is addition or subtraction, the maximum period that can be reached for b-bit precision is (2P −1)2b−1 . If the arithmetic operation is multiplication, the maximum period is (2P − 1)2b−3 [5]. Moreover, if the operator is xor the maximum period becomes 2P −1P . The random number generator of Glibc uses an additive lagged Fibonacci generator, which is a combination of a LCG and LFG utilizing 32-bit integer operations. The lags are produced by a linear congruential generator LCG(16807, 0, 231 − 1) and then by using those lags a LF G(31, 3, +) produces the random number sequence. (The first 344 members of the sequence are omitted.) Newest versions of Glibc allow initialization of lags to be done by any seed (X0 ), but here we stick to the default case in which the initial seed is equal to 1. In order to apply Eqn 2., an initial set of Xi for i < P must be defined. This is accomplished by X0 = 1 3 Xi = 16807Xi−1 mod (231 − 1) for 1 ≤ i < 31 Xi = Xi−31 for i = 31, ..., 33. (3) The generation part is, Xi = (Xi−3 + Xi−31 ) mod 232 for i ≥ 34. (4) All Xi ’s for 0 ≤ i ≤ 343 are discarded and in the end ith output of the RNG becomes Ri = Xi+344 ÷ 2 [13]. The integer division by 2 is carried out to convert the number to a 31 bit number, a leading zero bit is necessary to obtain a positive number. In order to obtain real pseudo random numbers, the result is divided by the biggest 31 bit integer: ri = Ri . −1 231 (5) This result then provides a pseudo-random real number linearly distributed between 0 and 1 with 31 bit precession. 2. Results The model we study is simple: There are N boxes numbered 1 to N in a chain structure, all boxes are equally likely to receive a ball at a random time driven by a Poisson process (Fig. 1). Each time when a ball is received, a random time increment of δt has elapsed, where δt = − log(r)/N and r is a uniformly distributed random number between 0 and 1. This then defines a Poisson process in which an event occurs with rate N per unit time. With this choice of δt, on the average N balls are placed in unit time. The box that will receive the ball is chosen randomly as well, such that ith box is chosen if i − 1 < rN < i (Fig. 2). In between these two steps (choosing the random time and choosing the random box), n samples are chosen from the random number sequence in vain, i.e., they are discarded from the used sequence. Furthermore, the simulation is stopped at multiples of a time period T , so that at each time step when the interval crosses a multiple of T . The time variable is set to that time (that is, no balls are placed to any box) and then the simulation continues as described in Fig. 1. The pseudocode for the model can be seen in the Table 1. 4 Figure 1: a) Each box is equally likely to receive a ball at a random time. b) Time t is incremented by δt at each step. The small dots on the axis indicate n draws in vain from the PRNG, and the larger dots indicate a ball is placed to a random box at that point in time. Whenever δt crosses a multiple of T , the process is paused, i.e., no ball is placed to any box and time is set to that multiple of T . The process involving the period (T ) dependence is necessary to observe the correlation. Indeed Fig. 3 shows that the process is sensitive on the period. In our research, we were analysing the effect of a periodic perturbation on a system, and that is how was the problem discovered. We then tested the process for time increment derived from a simpler form δt = r/N, and still found correlations. It is expected that over a large period of time each site will be equally likely to be occupied. However, for n = 0, boxes in the middle have a tendency to be occupied more than the sites near the both ends (Fig. 2). Similarly, for n = 1 and n = 2, correlations are observed. However if n > 2, correlations vanish, in other words there are no preferences left among the boxes. Note that the induced correlations are significant, i.e., they exceed 5 1.01 n=2 1.005 n=0 n=3 1 0.995 n=1 0.99 0.985 0 2 4 6 8 10 12 14 16 18 20 Figure 2: n = 0, 1, 2 and 3, T = 0.25, N = 20. n is the samples drawn in vain from the PRNG, T is the period and N is the number of boxes. Dependence on n is related to the correlations among consecutive numbers. 1% in our example. Also note that, simulations are done for 109 trials, which is well below the period of the PRNG. Note that, the abscissa of the graphs in Figures 2 and 3 correspond to the size of the random number used for selecting a box. Different shapes of graphs in these plots correspond to a non-uniform distribution for the size of the random number. Dependences on n and T are then related to the correlations among the random numbers. Since we wanted a measure of the systematic error in the probability distribution P , we looked P at the first coefficient of the Fourier expansion of this distribution: c = N j=1 exp(iπj/N)(P (j) − c0 ) where c0 is the average value of P . Figure 4 shows magnitude of c, which represents the size of the systematic error as a function of the period. 6 Pseudocode for the process 1. SET N to lattice size 2. SET period to T 3. SET totalT ime to 0 4. SET n to number of draws in vain 5. SET a to 1 6. SET occupation of k th site o(k) to 0 for 1 ≤ k ≤ N 7. DRAW a random number r 8. SET incrementalT ime to −log(r)/N 9. IF totalT ime plus incrementalT ime is smaller than a × T THEN 10. INCREASE totalT ime → totalT ime + incrementalT ime 11. DRAW a random number r 12. DRAW n random numbers in vain 13. CHOOSE k th box as k = ⌈rN ⌉ 14. INCREMENT o(k) → o(k) + 1 15. ELSE 16. INCREMENT a → a + 1 17. SET totalT ime as a × T 18. REPEAT 6 − 17 steps for 1 billion times Table 1: The pseudocode which is neccesary to produce the results in this paper. 3. Conclusions Given the common usage of C as a scientific programming language, we report of correlations in the default random number generator of Glibc. Even though it can pass some tests, PRNG of Glibc shows significant correlations under this very simple test. Systematic errors of this magnitude will cause problems in simulations which demand random numbers with low correlations, such as Monte Carlo analysis. Scientific community should beware of accepting default random number generators even for well established, commonly used programming languages as C. We could not identify the precise mechanism which leads to this behavior. A mathematical analysis of this mechanism represents an interesting challenge for further analysis. 4. Acknowledgements This work was supported by Turkish Academy of Sciences. 7 1.015 T=0.3 1.01 T=0.2 1.005 T=0.4 1 T=0.5 0.995 0.99 T=0.1 0.985 0 5 10 15 20 25 Figure 3: n = 0, T = 0.1, 0.2, 0.3, 0.4 and 0.5. For different values of periods we see different shapes of graphs, however for higher periods correlations cease to exist. 5. References References [1] I. Vattulainen, arXiv:cond-mat/9411062 [2] R. M. Ziff, Comput. Phys. 12, 385 (1998) [3] A. M. Ferrenberg, D. P. Landau and Y. J. Wong, Phys. Rev. Lett. 69, 3382 (1992). [4] F. Schmid and N. Wilding, Int. J. Mod. Phys. C 6, 781 (1995) [5] P. D. Coddington, Northeast Parallel Architecture Center, Paper 14 (1994). [6] P. Grassberger, Physics Letters A, 181 (1993) [7] S. Mertens and H. Bauke, Phys. Rev. E 69, 055702(R) (2004) 8 0.07 0.06 0.05 c 0.04 0.03 0.02 0.01 0 0 0.5 1 1.5 2 2.5 τ Figure 4: For different values of period the magnitude of first Fourier coefficient c. It represents the size of the systematic error as a function of the period. [8] D. E. Knuth, The Art of Computer Programming, 3rd ed. (AddisonWesley, New York, 1998), Vol. 2 [9] H. G. Katzgraber, arXiv:1005.4117 [10] I. Vattulainen, T. Ala-Nissila and K. Kankaala, Phys. Rev. E 52, 3205 (1995) [11] I. Vattulainen, K. Kankaala, J. Saarinena, T. Ala-Nissilaa, Comp. Phys. Comm., 86(1995) 209-226 [12] I. Vattulainen, T. Ala-Nissila and K. Kankaala, Phys. Rev. Lett. 73, 2513 (1994) [13] This webpage also contains code to generate the random number sequence discussed in this paper. Peter Selinger: The GLIBC Pseudorandom Number Generator. N.p., n.d. Web. 24 Feb. 2014 9 

Control refinement for discrete-time descriptor systems: a behavioural approach via simulation relations F. Chen ∗ , S. Haesaert ∗ , A. Abate ∗∗ , and S. Weiland ∗ ∗ arXiv:1704.01672v1 [] 6 Apr 2017 Department of Electrical Engineering Eindhoven University of Technology, Eindhoven, The Netherlands ∗∗ Department of Computer Science University of Oxford, Oxford, United Kingdom Abstract: The analysis of industrial processes, modelled as descriptor systems, is often computationally hard due to the presence of both algebraic couplings and difference equations of high order. In this paper, we introduce a control refinement notion for these descriptor systems that enables analysis and control design over related reduced-order systems. Utilising the behavioural framework, we extend upon the standard hierarchical control refinement for ordinary systems and allow for algebraic couplings inherent to descriptor systems. Keywords: Descriptor systems, simulation relations, control refinement, behavioural theory. 1. INTRODUCTION Complex industrial processes generally contain algebraic couplings in addition to differential (or difference) equations of high order. These systems, referred to as descriptor systems (Kunkel and Mehrmann, 2006; Dai, 1989), are commonly used in the modelling of mechanical systems. The presence of algebraic equations, or couplings, together with large state dimensions renders numerical simulation and controller design challenging. Instead model reduction methods (Antoulas, 2005) can be applied to replace the systems with reduced order ones. Even though most methods have been developed for systems with only ordinary difference equations, recent research also targets descriptor systems (Cao et al., 2015). In this paper, we newly target the use of descriptor systems of reduced order for the verifiable design of controllers. A rich body of literature on verification and formal controller synthesis exists for systems solely composed of difference equations. This includes the algorithmic design of certifiable (hybrid) controllers and the verification of pre-specified requirements (Tabuada, 2009; Kloetzer and Belta, 2008). Usually, these methods first reduce the original, concrete systems to abstract systems with finite or smaller dimensional state spaces over which the verification or controller synthesis can be run. A such controller obtained for the abstract system can be refined over the concrete system leveraging the existence of a similarity relation, e.g., an (approximate) simulation relation, between the two systems (Tabuada, 2009; Girard and Pappas, 2011). For the application of these relations in control problems, a hierarchical control framework is presented by (Girard and Pappas, 2009). Currently, the control synthesis over descriptor systems cannot be dealt with in this fashion due to the presence of algebraic equations. The presence of similarity relations between descriptor systems has also been a topic under investigation in (Megawati and Van der Schaft, 2015). This work on similarity relations deals with continuous-time descriptor systems that are unconstrained and non-deterministic, and focuses on the conditions for bisimilarity and on the construction of similarity relations. Instead in this work, we specifically consider the control refinement problem for discrete-time descriptor systems via simulation relations within a behavioural framework, such that properties verified over the future behaviour of the abstract system are also verified over the concrete controlled system. Within the behavioural theory (Willems and Polderman, 2013), a formal distinction is made between a system (its behaviour) and its representations, enabling us to investigate descriptor systems and refinement control problems without having to directly deal with their inherent anticausality. In the next section, we define the notion of dynamical systems and control within a behavioural framework and use it to formalise the control refinement problem. Subsequently, Section 3 is dedicated to the exact control refinement for descriptor systems and contains the main results of the paper. The last section closes with the conclusions. 2. THE BEHAVIOURAL FRAMEWORK 2.1 Discrete-time descriptor systems As introduced by (Willems and Polderman, 2013), we define dynamical systems as follows. Definition 1. A dynamical system Σ is defined as a triple Σ = (T, W, B) with the time axis T, the signal space W, and the behaviour B ⊂ WT . ✷ In this definition, WT denotes the collection of all timedependent functions w : T → W. The set of trajectories or time-dependent functions given by B represents the trajectories that are compatible with the system. This set is referred to as the behaviour of the system (Willems and Polderman, 2013). Generally, the representation of the behaviour of a dynamical system by equations, such as a set of ordinary differential equations, state space equations and transfer functions, is non-unique. Hence we distinguish a dynamical system (its behaviour) from the mathematical equations used to represent its governing laws. We consider dynamical systems evolving over discrete-time (T := N = {0, 1, 2, . . .}) that can be represented by a combination of linear difference and algebraic equations. The dynamics of such a linear discrete-time descriptor system (DS) are defined by the tuple (E, A, B, C) as Ex(t + 1) = Ax(t) + Bu(t), (1) y(t) = Cx(t), with the state x(t) ∈ X = Rn , the input u(t) ∈ U = Rp , and the output y(t) ∈ Y = Rk and t ∈ N. Further, E, A ∈ Rn×n , B ∈ Rn×p and C ∈ Rk×n are constant matrices and we presume that rank(B) = p and rank(C) = k. We say that a trajectory w = (u, x, y), with w : N → (U × X × Y), satisfies (1) if for all t ∈ N the equations in (1) evaluated at u(t), x(t), x(t + 1), y(t) hold. Then the collection of all trajectories w defines the full behaviour, or equivalently the input-state-output behaviour as Bi/s/o := {(u, x, y) ∈ (U × X × Y)N | (1) is satisfied}. (2) Σ := (T, W, B) = (N, U × X × Y, Binit i/s/o ) with • the time axis T := N = {0, 1, 2, . . .}, • the full signal space W := U × X × Y, and • the initialised behaviour 1 N Binit i/s/o = {w ∈ W |w = (u, x, y) s.t. (1) and s.t. x(0) = x0 ∈ X0 }. 2.2 Control of descriptor systems Controller synthesis amounts to synthesising a system Σc , called a controller, which, after interconnection with Σ, restricts the behaviour B of Σ to desirable (or controlled) trajectories. Thus, in the behavioural framework, control is defined through interconnections (or via variable sharing as specified next), rather than based on the causal transmission of signals or information, as in classical system theory. Let Σ1 = (T, C1 ×W, B1 ) and Σ2 = (T, C2 ×W, B2 ) be two dynamical systems. Then, as depicted in Fig. 1a and defined in (Willems and Polderman, 2013), the interconnection of Σ1 and Σ2 over W, denoted by Σ = Σ1 ×w Σ2 with the shared variable w ∈ W, yields the dynamical system Σ = (T, C1 × C2 × W, B) with B = {(c1 , c2 , w) : T → C1 × C2 × W | (c1 , w) ∈ B1 , (c2 , w) ∈ B2 }. The variable x is considered as a latent variable, therefore the manifest, or equivalently the input-output behaviour associated with (1) is defined by Bi/o:= {(u, y)|∃x ∈ XN s.t. (u, x, y) ∈ Bi/s/o }. If E is non-singular, we refer to the corresponding dynamical system as a non-singular DS. In that case, we can transform (1) into standard state space equations, as x(t + 1) = Ãx(t) + B̃u(t), y(t) = Cx(t), with à = E −1 A, B̃ = E −1 (3) B. Further Bi/s/o as in (2) is {(u, x, y) ∈ (U × X × Y)N | (u, x, y) s.t. (3) holds}. Similarly, if E is non-singular, Bi/o can be defined by (3). The tuple with dynamics (1) defines a dynamical system Σ evolving over the combined signal space W = U × X × Y with behaviour B := Bi/s/o given in (2). Similarly, for W restricted to input-output space, the tuple (N, U×Y, Bi/o ) defines the manifest or induced dynamical system. We are specifically interested in the behaviour initialised at t = 0 with a given set of initial states X0 ⊂ X. For this, we say that a trajectory w : N → (U × X × Y) is initialised with X0 if (1) holds and x(0) = x0 ∈ X0 . Such a trajectory, initialised with x0 ∈ X0 , is also called the continuation of x0 . We refer to the collection of initialised trajectories related to X0 as the initialised behaviour Binit i/s/o . This allows us to formalise our definition of the descriptor system evolving over N. Definition 2. (Discrete-time descriptor systems (DS)). A (discrete-time) descriptor system is defined as a dynamical system Σ initialised with X0 , whose behaviour can be represented by the combination of algebraic equations and difference equations given in (1), that is (4) c1 c2 w Σ1 Σ2 (a) The interconnected system Σ obtained via the shared variables w in W between dynamical systems Σ1 and Σ2 with signal spaces C1 × W and C2 × W. BΣ BΣ×Σc BΣ c (b) The controlled behaviour BΣ×Σc = BΣ ∩ BΣc is given as the intersection of the behaviours of the dynamical system Σ and its controller Σc . Fig. 1. The left figure (a) portrays the general interconnection of two dynamical systems. In figure (b), the more specific case of behavioural intersection for a system and its controller is depicted. Observe that w ∈ WT contains the signals shared by both Σ1 and Σ2 , while c1 ∈ CT1 only belongs to Σ1 and c2 ∈ CT2 only belongs to Σ2 . So, in the interconnected system, the shared variable w satisfies the laws of both B1 and B2 . Note that it is always possible to trivially extend the signal spaces of Σ1 and Σ2 (and the associated behaviour) such that a full interconnection structure is obtained, that is, such that both C1 and C2 are empty and the behaviour of the interconnected system is B = B1 ∩ B2 . Hence, a full interconnection of Σ = (T, W, BΣ ) and Σc = (T, W, BΣc ) is simply Σ ×w Σc = (T, W, BΣ ∩ BΣc ), with the intersection of the behaviours, denoted by BΣ×Σc , as portrayed in Fig. 1b. That is, interconnection and intersection are equivalent in full interconnections. Further, we define a well-posed controller Σc for Σ as follows. Definition 3. Consider a dynamical system Σ = (T, W, B), with initialised behaviour as defined in (4). We say that a system Σc = (T, W, Bc ) is a well-posed controller for Σ if the following conditions are satisfied: 1 In the sequel the indexes init and i/s/o will be dropped. (1) BΣ×Σc := BΣ ∩ BΣc 6= {∅}; (2) For every initial state x0 ∈ X0 , there exists a unique continuation in BΣ×Σc . Denote with C(Σ) the collection of all well-posed controllers for Σ. We want a controller that accepts any initial state of the system. This is formalised in the second condition by requiring that for any initial state of Σ, there exists a unique continuation in BΣ×Σc . We elucidate the properties of a well-posed linear controller as follows. Example 1. For a system Σ as in (1), consider a controller Σc , which is a DS, and has dynamics given as Ec x(t + 1) = Ac x(t) + Bc u(t), (5) with Ec , Ac ∈ Rnc ×n and Bc ∈ Rnc ×p . Suppose that the controller shares the variables u and x with the system Σ. That is, w = (u, x). The interconnected system Σ ×w Σc yields the state evolutions of the combined system as       B A E u(t), (6) x(t) + x(t + 1) = Bc Ac Ec and can be rewritten     to A E −B x(t + 1) = x(t). Ac Ec −Bc u(t) (7) If for any x(t) ∈ X, there exists a pair (x(t + 1), u(t)) such that (7) holds, then this implies that for any initial state x0 ∈ X0 of Σ there exists a continuation in the controlled behaviour. In addition, if the pair (x(t + 1), u(t)) is unique for any x(t) ∈ X, then this continuation is unique and we say that Σc ∈ C(Σ). This existence and uniqueness of the pairs (x(t+ 1), u(t)) depends on the solutions of the matrix equality (7). We use the classical results on the solutions of matrix equalities (cf. (Abadir and Magnus, 2005)) to conclude that the first well-posedness condition is satisfied if and only if     E B E B A rank = rank . (8) Ec Bc Ec Bc Ac If in addition, rank  E B A Ec Bc Ac  = n + p, (9) then the second condition is also satisfied and Σc ∈ C(Σ). Of interest is the design of well-posed controllers subject to specifications over the future output behaviour of the controlled system. We thus consider specifications defined over the output space. In order to analyse the output behaviour, we introduce a projection map. For B ⊂ (W1 × W2 )T we denote with ΠW2 a projection given as ΠW2 (B) := {w2 ∈ WT2 |∃w1 ∈ WT1 (E, A, B, C) as in (1) and initialised with X0 . A well-posed controller for Σ is referred to Σc ∈ C(Σ). The controlled concrete system is the interconnected system Σ×w Σc with the shared variables w = (u, x). Now, we consider a simpler DS Σa , related to the concrete DS Σ, with dynamics given as (Ea , Aa , Ba , Ca ) and initialised with Xa0 . We assume that the synthesis of a wellposed controller Σca for Σa is substantially easier than for Σ. We refer to this simpler system Σa as the abstract DS, and we note that its signals take values ua (t), xa (t), ya (t) with xa (t) ∈ Xa = Rm , ua (t) ∈ Ua = Rq , ya (t) ∈ Ya = Y = Rk and t ∈ N. With respect to the concrete system, the abstract DS is generally a reduced-order system. The controlled abstract system Σa ×wa Σca is the interconnected system with the shared variables wa = (ua , xa ). If we assume that we can compute a well-posed controller for the abstract system, then the control synthesis problem reduces to a control refinement problem. Definition 4. (Exact control refinement). Let Σa and Σ be the abstract and concrete DS, respectively. We say that controller Σc ∈ C(Σ) refines the controller Σca ∈ C(Σa ) if ΠY (BΣ×Σc ) ⊆ ΠY (BΣa ×Σca ). Then we formalise the exact control refinement problem. Problem 1. Let Σa and Σ be the abstract and concrete DS, respectively. For any Σca ∈ C(Σa ), refine Σca to Σc , s.t. Σc ∈ C(Σ) and ΠY (BΣ×Σc ) ⊆ ΠY (BΣa ×Σca ). In the next section, we will show that the existence of a solution to this problem hinges on certain conditions involving similarity relations between the concrete and abstract DS. For this, we will first introduce simulation relations to formally characterise this similarity. 3. EXACT CONTROL REFINEMENT 3.1 Similarity relations between DS We give the notion of simulation relation as defined in (Tabuada, 2009) for transition systems and applied to pairs of DS Σ1 and Σ2 that share the same output space Y1 = Y2 = Y. Definition 5. Let Σ1 and Σ2 be two DS with respective dynamics (E1 , A1 , B1 , C1 ) and (E2 , A2 , B2 , C2 ) over state spaces X1 and X2 . A relation R ⊆ X1 × X2 is called a simulation relation from Σ1 to Σ2 , if ∀(x1 , x2 ) ∈ R, (1) for all (u1 , x+ 1 ) ∈ U1 × X1 subject to E1 x+ 1 = A1 x1 + B1 u1 there exists (u2 , x+ 2 ) ∈ U2 × X2 subject to s.t. (w1 , w2 ) ∈ B}. We focus here on finding a controller Σc for a given dynamical system Σ such that the output behaviour ΠY (BΣ×Σc ) of the interconnected system satisfies some specifications. 2.3 Exact control refinement & problem statement Let us refer to the original DS that represents the real physical system as the concrete DS. It is for this system that we would like to develop a well-posed controller. Recall that the DS is a dynamical system Σ with dynamics such that E2 x+ 2 = A2 x2 + + (x1 , x2 ) ∈ R, and + B2 u 2 (2) we have C1 x1 = C2 x2 . We say that Σ1 is simulated by Σ2 , denoted by Σ1  Σ2 , if there exists a simulation relation R from Σ1 to Σ2 and if in addition ∀x10 ∈ X10 , ∃x20 ∈ X20 such that (x10 , x20 ) ∈ R. We call R ⊆ X1 × X2 a bisimulation relation between Σ1 and Σ2 , if R is a simulation relation from Σ1 to Σ2 and its inverse R−1 ⊆ X2 × X1 is a simulation relation from Σ2 to Σ1 . We say that Σ1 and Σ2 are bisimilar, denoted by Σ1 ∼ = Σ2 , if Σ1  Σ2 w.r.t. R and Σ2  Σ1 w.r.t. R−1 . Simulation relations as defined above are transitive. Let R12 and R23 be simulation relations respectively, from Σ1 to Σ2 and from Σ2 to Σ3 . Then a simulation relation from Σ1 to Σ3 is given as a composition of R12 and R23 , namely R12 ◦R23 = {(x1 , x3 ) | ∃x2 : (x1 , x2 ) ∈ R12 ∧(x2 , x3 ) ∈ R23 }. We also have that Σ1  Σ2 and Σ2  Σ3 implies Σ1  Σ3 and, in addition, Σ1 ∼ = Σ3 . = Σ3 implies Σ1 ∼ = Σ2 and Σ2 ∼ Simulation relations have also implications on the properties of the output behaviours of the two systems. More precisely, if a system is simulated by another system then this implies output behaviour inclusion. This follows from Proposition 4.9 in (Tabuada, 2009) and is formalised next. Proposition 6. Let Σ1 and Σ2 be two DS with simulation relations as defined in Definition 5. Then, Σ1  Σ2 =⇒ ΠY (BΣ1 ) ⊆ ΠY (BΣ2 ), Σ1 ∼ = Σ2 =⇒ ΠY (BΣ1 ) = ΠY (BΣ2 ). Simulation relations can also be used for the controller design for deterministic systems such as nonsingular DS (Tabuada, 2009; Fainekos et al., 2007; Girard and Pappas, 2009). This will be used in the next subsection, where we consider the exact control refinement for non-singular DS. After that, we introduce a transformation of a singular DS to an auxiliary nonsingular DS representation, referred to as a driving variable (DV) system. The exact control refinement problem is then solved based on the introduced notions. 3.2 Control refinement for non-singular DS Let us consider the simple case where the concrete and abstract systems of interest are given with non-singular dynamics. For these systems, the existence of a simulation relation also implies the existence of an interface function (Girard and Pappas, 2009), which is formulated as follows. Definition 7. (Interface). Let Σ1 and Σ2 be two nonsingular DS defined over the same output space Y with a simulation relation R from Σ1 to Σ2 . A mapping F : U1 × X1 × X2 7→ U2 is an interface related to R, if ∀(x1 , x2 ) ∈ R and for all u1 ∈ U1 , u2 := F (u1 , x1 , x2 ) ∈ U2 is such that + (x+ 1 , x2 ) ∈ R with + x+ 1 = A1 x1 + B1 u1 and x2 = A2 x2 + B2 u2 . It follows from Definition 5 that there exists at least one interface related to R if two deterministic, or non-singular systems are in a simulation relation. As such we can solve the exact refinement problem as follows. Theorem 8. Let Σ1 and Σ2 be two non-singular DS defined over the same output space Y with dynamics (I, A1 , B1 , C1 ) and (I, A2 , B2 , C2 ), which are initialised with X10 and X20 , respectively. If there exists a relation R ⊆ X1 × X2 such that (1) R is a simulation relation from Σ1 to Σ2 , and (2) ∀x20 ∈ X20 , ∃x10 ∈ X10 s.t. (x10 , x20 ) ∈ R, then for any controller Σc1 ∈ C(Σ1 ), there exists a controller Σc2 ∈ C(Σ2 ) that is an exact control refinement for Σc1 and thus achieves with ΠY (BΣ2 ×Σc2 ) ⊆ ΠY (BΣ1 ×Σc1 ). Proof. Since R is a simulation relation from Σ1 to Σ2 , there exists an interface function F : U1 × X1 × X2 → U2 as given in Definition 7, cf (Tabuada, 2009; Girard and Pappas, 2009). Additionally, due to (2) there exists a map, F0 : X20 → X10 such that for all x20 ∈ X20 it holds that (F0 (x20 ), x20 ) ∈ R. Next, we construct the controller Σc2 that achieves exact control refinement for Σc1 as Σc2 := (Σ1 ×w1 Σc1 ) ×w1 ΣF , where w1 = (u1 , x1 ) and where ΣF := (N, W, BF ) is a dynamical system taking values in the combined signal space with BF := {(x1 , u1 , x2 , u2 ) ∈ W|x10 = F0 (x20 ) and u2 = F (x1 , u1 , x2 )}. The dynamical system Σc2 is a well-posed controller for Σ2 with Σ2 ×w2 Σc2 sharing w2 = (u2 , x2 ). Denote with BΣ2 ×Σc2 the behaviour of the controlled system, then due to the construction of ΣF it follows that BΣ2 ×Σc2 is nonempty and ∀x20 ∈ X20 , ∃x10 ∈ X10 such that (x10 , x20 ) has a unique continuation in BΣ2 ×Σc2 . Furthermore it holds that ΠY (BΣ2 ×Σc2 ) ⊆ ΠY (BΣ1 ×Σc1 ). ✷ The design of the controller Σc2 that achieves exact control refinement for Σc1 is similar to that in (Tabuada, 2009), which also holds in the behavioural framework. 3.3 Driving variable systems Since it is difficult to control and analyse a DS directly, we develop a transformation to a system representation that is in non-singular DS form and is driven by an auxiliary input. We refer to this non-singular DS as the driving variable (DV) system (Weiland, 1991). We investigate whether the DS and the obtained DV system are bisimilar and behaviourally equivalent. Let us first introduce with a simple example the apparent non-determinism or anticausality in the DS. Later-on, we show the connections between a DS and its related DV system. Example 2. Consider the DS with dynamics (E, A, B, C) defined as h1i h 0 iT h1 0 0i h −1 0 0 i E = 0 0 1 , A = 0 1 0 , B = 1 , C = 0.2 , (10) 0 0 0 1 0 0 1 0.5 T and x(t) = [x1 (t) x2 (t) x3 (t)] . In this case, the input u(t) = −x3 (t) is constrained by the third state component. Now the state trajectories of (10) can be found as follows: • for a given input sequence u : N → U, we have x2 (t) = −u(t)−u(t+1), and thus we can use this anticausal relation of the DS to find the corresponding state trajectories; • alternatively, we can allow the next state x2 (t + 1) to be freely chosen, and for arbitrary state x2 (t), the equations (10) impose constraints on the input sequence that is, therefore, no longer free as u(t) = −x3 (t). We embrace the latter, non-deterministic interpretation of the DS. This non-determinism can be characterised by introducing an auxiliary driving input of a so-called DV system. We reorganise the state evolution of (1). For simplicity we omit the time index in x(t) and u(t) and denote x(t + 1) as x+  + x = Ax, (11) M u where M = [E −B]. For any x, we notice that the pairs (u, x+ ) are non-unique due to the non-determinism related to x+ . If M has full row rank, then it has a right inverse. This always holds when the DS is reachable (cf. Definition 2-1.1 (Dai, 1989)). In that case we can characterise the non-determinism as follows. Let M + be a right inverse of M such that M M + = I and N be a matrix such that im N = ker M and N T N = I. Then all pairs (u, x+ ) that are compatible with state x in (11) are parametrised as  + x = M + Ax + N s, (12) u where s is a free variable. We now claim that all transitions (x, u, x+ ) in (12) for some variable s satisfy (11). To see this, multiply M on both sides of (12) to regain (11). Now assume that there exists a tuple (x, u, x+ ) satisfying (11) that does not satisfy (12). Then there exists an s and a vector z 6= 0 that is not an element of the kernel of M and such that the right side of (12) becomes M + Ax + N s + z. Multiplying again with M , we infer that there is an additional non-zero term M z and that (11) cannot hold. In conclusion any transition of (11) is also a transition of (12) and vice versa. Example 3. [Example 2: cont’d] For the DS of Example 2, the related DV system ΣDV is developed as h −1 0 −1 i h 0 i x(t + 1) = 0 0 0 x(t) + −1 s(t) 0 1 −1 0 (13) u(t) = [ 0 0 −1 ]x(t) y(t) = [ 0 0.2 0.5 ]x(t). As indicated by (13), the input u(t) is a function of the state trajectory. The non-determinism of x2 (t + 1) is characterised by −s(t) for which the auxiliary input s can be freely selected. Let us now formalise the notion of a driving variable representation. We associate a driving variable representation with any given DS (1) by defining a tuple (Ad , Bd , Cu , Du , C) and setting     Ad Bd = M + A, = N, (14) Cu Du where N ∈ R(n+p)×p has orthonormal columns, that is N T N = I. For any given DS, this tuple defines the driving variable system ΣDV = (N, W, BΣDV ), which maintains the same set of initial states X0 and has dynamics x(t + 1) = Ad x(t) + Bd s(t) u(t) = Cu x(t) + Du s(t) (15) y(t) = Cx(t), thereby yielding the initialised behaviour BΣDV := {w ∈ WN |w =(u, x, y), ∃s ∈ SN s.t. (15) and x0 ∈ X0 }. Next, we propose the following assumption for DS, which will be used in the sequel to develop our main results. Assumption 1. The given DS Σ is a dynamical system with dynamics (E, A, B, C) such that M = [E −B] has full row rank. The relationship between a DS and its related DV system is characterised as follows. Theorem 9. Let the DS Σ be given as in (1) satisfying Assumption 1 and let ΣDV = (N, W, BΣDV ) be defined as in (15). Then (1) Σ and ΣDV are bisimilar, that is, Σ ∼ = ΣDV , (2) Σ and ΣDV have equal behaviour, i.e., BΣDV = BΣ , (3) Σ and ΣDV have equal output behaviour, that is, ΠY (BΣ ) = ΠY (BΣDV ). Proof. For the first statement (1), we define the diagonal relation as I := {(x, x) | x ∈ X}. Then I is a bisimulation relation between Σ and ΣDV , because by construction their state evolutions can be matched, hence stay in I; and they share the same output map. In addition, since they have the same set of initial states it follows that Σ ∼ = ΣDV . The second part (2) follows immediately from the derivation of ΣDV , because by construction all the transitions in Σ can be matched by those of ΣDV and vice versa, in addition, they have the same output map. Hence, they share the same signal space (U × X × Y) and we can conclude that Σ and ΣDV have equal behaviour. Additionally, we have that (2) implies (3); via Proposition 6 also (1) implies (3). ✷ 3.4 Main result: exact control refinement for DS Based on the results developed in the previous subsections, we now derive the solution to the exact control refinement problem in Problem 1. More precisely, subject to the assumption that there exists a simulation relation R from Σa to Σ, for which in addition holds that ∀x0 ∈ X0 , ∃xa0 ∈ Xa0 s.t. (xa0 , x0 ) ∈ R, we show that for any Σca ∈ C(Σa ), there exists a controller Σc for Σ that refines Σca such that Σc ∈ C(Σ) and ΠY (BΣ×Σc ) ⊆ ΠY (BΣa ×Σca ). In the case of Assumption 1, we construct DV systems ΣDV and ΣDVa for the respective DS systems Σ and Σa as a first step. For these systems, we develop the following results on exact control refinement: i) The exact control refinement for the DV systems: ∀ΣcDVa ∈ C(ΣDVa ), ∃ΣcDV ∈ C(ΣDV ), s.t.

ΠY BΣDV ×Σc ; ⊆ ΠY BΣDVa ×Σc DV DVa ii) The exact control refinement from Σa to ΣDVa : ∀Σca ∈ C(Σa ), ∃ΣcDVa ∈ C(ΣDVa ), s.t.

; ΠY BΣa ×Σca = ΠY BΣDVa ×Σc DVa iii) The exact control refinement from ΣDV to Σ: ∀ΣcDV ∈ C(ΣDV ), ∃Σc ∈ C(Σ), s.t.   = ΠY (BΣ×Σc ) . ΠY BΣDV ×Σc DV It will be shown that the combination of the elements i)–iii) also implies the construction of the exact control refinement for the concrete and abstract DS. i) Exact control refinement for the DV systems. From Theorem 9, we know that Σ ∼ = ΣDVa = ΣDV and Σa ∼ with respective diagonal relations I := {(x, x)|x ∈ X} and Ia := {(xa , xa )|xa ∈ Xa }. Hence as depicted in Fig. 2 and based on the transitivity of simulation relations, we also derive that R is a simulation relation from ΣDVa to ΣDV . Σ∼ = ΣDV , w.r.t. I Σ R = Ia ◦ R ◦ I (∃xa0 , ∀x0 ) ∈ R R (∃xa0 , ∀x0 ) ∈ R Σa ΣDV DVa ΣDVa Σa ∼ = ΣDVa , w.r.t. Ia Fig. 2. Connection between DS and DV systems for the exact control refinement. Since the DV systems ΣDV and ΣDVa share the same initial states as the respective DS Σ and Σa , it also holds that ∀x0 ∈ X0 , ∃xa0 ∈ Xa0 s.t. (xa0 , x0 ) ∈ R. According to Theorem 8, we know that we can do exact control refinement, that is, we have shown ∀ΣcDVa ∈ C(ΣDVa ), ∃ΣcDV ∈ C(ΣDV ), s.t.

⊆ ΠY BΣDVa ×Σc . ΠY BΣDV ×Σc DVa DV ii) Exact control refinement from Σa to ΣDVa . Denote with ΣDVa the abstract DV system related to Σa , with dynamics (Ada , Bda , Cua , Dua , Ca ) and initialised with Xa0 . We first derive the static function Sa mapping transitions of Σa to the auxiliary input sa of ΣDVa . From the definition of DV systems, we can also derive the transitions of ΣDVa indexed with a, which is similar to the derivation of (12).  + xa = Ma+ Aa xa + Na sa . (16) ua Multiplying NaT on both sides of (16), Sa is derived as  + T xa −NaT Ma+ Aa xa . (17) Sa : sa = Sa (x+ , u , x ) = N a a a a ua Sa maps the state evolutions of Σa ×wa Σca to the auxiliary input sa for ΣDVa , where wa = (ua , xa ). Now, we consider the exact control refinement from the abstract DS to the abstract DV system. Theorem 10. Let Σa be the abstract DS with dynamics (Ea , Aa , Ba , Ca ) satisfying the condition of Assumption 1 and let ΣDVa be its related DV system with dynamics (Ada , Bda , Cua , Dua , Ca ) such that both systems are initialised with Xa0 . Then, for any Σca ∈ C(Σa ), there exists a controller ΣcDVa ∈ C(ΣDVa ) that is an exact control refinement for Σca as defined in Definition 4 with

. ΠY BΣa ×Σca = ΠY BΣDVa ×Σc DVa Proof. Denote with xa and xda the state variables of Σa and ΣDVa , respectively. Next, we construct the controller ΣcDVa that achieves exact control refinement for Σca as ΣcDVa := (Σa ×wa Σca ) ×wa ΣSa , where wa = (ua , xa ) and where ΣSa := (N, W, BSa ) is a dynamical system with BSa := {(xa , ua , xda , sa ) ∈ W|xa0 = xda0 and sa = Sa (x+ a , ua , xa )}. c The dynamical system ΣDVa is a well-posed controller for ΣDVa with ΣDVa ×wad ΣcDVa sharing wad = (sa , xda ). Denote the behaviour of the controlled system. with BΣDVa ×Σc DVa By construction, we know that the set of the behaviour is non-empty and there is a unique continuation for any xda0 ∈ Xa0 . Further based on the construction of ΣSa , the behaviour is such that xda (t) = xa (t), ∀t ∈ N. Additionally, since Σa and ΣDVa share the same set of initial states  Xa0 ,
it holds that ΠY BΣa ×Σca = ΠY BΣDVa ×Σc . ✷ The proof is actually constructive in the design of the controller ΣcDVa that achieves exact control refinement for Σc a . iii) Exact control refinement from ΣDV to Σ. Now, we consider the exact control refinement from ΣDV to Σ. Suppose we are given a well-posed controller ΣcDV for ΣDV , which shares the free variable s and the state variable x with ΣDV . We want to design a well-posed controller for Σ over w = (u, x), for which we consider the dynamical system ΣC = (N, W, B) over the signal space W = U × X × S, the behaviour of which can be defined by BdT x(t + 1) = BdT Ad x(t) + BdT Bd s(t) (18) u(t) = Cu x(t) + Du s(t). Then the dynamics of the interconnected system Σ ×w ΣC as a function of x and as      s is derived BDu A + BCu E s(t). (19) x(t) + x(t + 1) = BdT Bd BdT BdT Ad Note that A + BCu = EAd and BDu = EBd by multiplying M = [E −B] on the left-hand side of the two equations in (14). Therefore, (19) is simplified to       E E E Bd s(t). (20) Ad x(t) + x(t + 1) = BdT BdT BdT T  Furthermore E T Bd has full column rank because the  T T is square and has full rank. Hence matrix M N  T T E Bd has a left inverse and the dynamics of Σ ×w ΣC in (20) can be simplified as x(t + 1) = Ad x(t) + Bd s(t), which is exactly the same as the state evolutions of ΣDV as shown in (15). Next we construct Σc := ΣC ×wd ΣcDV with wd = (s, xd ) and it is a well-posed controller for Σ. This allows us to state the following theorem regarding the control refinement from ΣDV to Σ. Theorem 11. Let Σ be the concrete DS with dynamics (E, A, B, C) satisfying Assumption 1 and let ΣDV be its related DV system with dynamics (Ad , Bd , Cu , Du , C) such that both systems are initialised with X0 . Then, for any ΣcDV ∈ C(ΣDV ), there exists a controller Σc ∈ C(Σ) that is an exact control refinement for ΣcDV as defined in Definition 4 with   ΠY BΣDV ×Σc DV = ΠY (BΣ×Σc ) . Proof. Denote with x and xd the state variables of the Σ and ΣDV , respectively. Next, we construct the controller Σc that achieves exact control refinement for ΣcDV as Σc := ΣC ×wd ΣcDV , where wd = (s, xd ) and the dynamics of ΣC is defined as (18). Then, we can show that the dynamical system Σc is a well-posed controller for Σ. Based on the analysis of (20), it is shown that Σ ×w ΣC = ΣDV with w = (u, x), then we can derive Σ ×w Σc = ΣDV ×wd ΣcDV . Therefore, we can

= ΠY BΣ×Σc conclude Σc ∈ C(Σ) with ΠY BΣDV ×Σc DV immediately follows from ΣcDV ∈ C(ΣDV ). ✷ Exact control refinement for descriptor systems. We can now argue that there exists exact control refinement from Σa to Σ, as stated in the following result. Theorem 12. Consider two DS Σa (abstract, initialised with Xa0 ) and Σ (concrete, initialised with X0 ) satisfying Assumption 1 and let R be a simulation relation from Σa to Σ, for which in addition holds that ∀x0 ∈ X0 , ∃xa0 ∈ Xa0 s.t. (xa0 , x0 ) ∈ R. Then, for any Σca ∈ C(Σa ), there exists a controller Σc ∈ C(Σ) such that
ΠY (BΣ×Σc ) ⊆ ΠY BΣa ×Σca . Proof. Based on Assumption 1, we first construct ΣDV and ΣDVa . Then to prove this we need to construct the exact control refinement. This can be done based on the subsequent control refinements given in Theorem 10, Theorem 8 and Theorem 11. ✷ Theorem 12 claims the existence of such controller Σc that achieves exact control refinement for Σca . More precisely, we have shown in the proof that the refined controller Σc is constructive, which provides the solution to Problem 1. To elucidate how such an exact control refinement is constructed, we consider the following example. Example 4. [Example 2,3: cont’d] Consider the DS of Example 2 and its related DV system (cf. Example 3) such that both systems are initialised with X0 = {x0 | x0 ∈ [−1, 1]3 ⊂ R3 }. According to Silverman-Ho algorithm (Dai, 1989), we can select an abstract DS Σa = (Ea , Aa , Ba , Ca ) that is the minimal realisation of Σ and is initialised with Xa0 = R2 , in addition [ 01 00 ], Aa [ 10 01 ], Ba [ 10 ], Ca T [ 0.7 0.2 ] . Ea = = = = Similarly, the related DV system ΣDVa of Σa is given as  0  xa (t + 1) = [ 00 10 ]xa (t) + −1 sa (t) (21) ua (t) = [ −1 0 ]xa (t) ya (t) = [ 0.7 0.2 ]xa (t). Subsequently, R := {(xa , x) | xa = Hx, xa ∈ Xa , x ∈ X} is a simulation relation from Σa to Σ with   1 H = 00 01 −1 . This can be proved through verifying the two properties of Definition 5. In addition, the condition ∀x0 ∈ X0 , ∃xa0 ∈ Xa0 s.t. (xa0 , x0 ) ∈ R holds. According to Theorem 12, we can refine any Σca ∈ C(Σa ) to attain a well-posed controller Σc for Σ that solves Problem 1 as follows: Define Σca ∈ C(Σa ) with dynamics as [ 1 1 ]xa (t + 1) = [ 0.5 0.5 ]xa (t) + ua (t). The controlled system Σa ×wa Σca is derived as  0  1 xa (t + 1) = −0.5 −0.5 xa (t) ya (t) = [ 0.7 0.2 ]xa (t), with wa = (ua , xa ) and ua (t) = [ −1 0 ]xa (t). Then Σa ×wa Σca is stable. According to Theorem 10, we derive the map Sa for ΣDVa as sa (t) = [ 0 −1 ]xa (t + 1) = [ 0.5 0.5 ]xa (t). Next, the related interface from ΣDVa to ΣDV is developed as s(t) = sa (t) − [ 0 1 −1 ]x(t). According to Theorem 11, we derive the well-posed controller Σc as [ 0 −1 0 ]x(t + 1) = [ 0 −1 1 ]x(t) + [ 0.5 0.5 ]xa (t) u(t) = [ 0 0 −1 ]x(t), and the interconnected system Σ ×w Σc with w = (u, x), is derived as i h1 0 1 i h 0 0 x(t + 1) = 0 1 −1 x(t) + −0.5 −0.5 xa (t) 0 1 −1 0 0 y(t) = [ 0 0.2 0.5 ]x(t). Since (xa , x) ∈ R, that is xa = Hx, Σ ×w Σc can be simplified by replacing xa (t): h1 0 1 i x(t + 1) = 0 0.5 −1 x(t) 0 1 −1 y(t) = [ 0 0.2 0.5 ]x(t).
Finally, Σc ∈ C(Σ) and ΠY (BΣ×Σc ) ⊆ ΠY BΣa ×Σca are achieved. 4. CONCLUSION In this paper, we have developed a control refinement procedure for discrete-time descriptor systems that is largely based on the behavioural theory of dynamical systems and the theory of simulation relations among dynamical systems. Our main results provide complete solutions of the control refinement problem for this class of discrete-time systems. The exact control refinement that has been developed in this work also opens the possibilities for approximate control refinement notions, to be coupled with approximate similarity relations: these promise to leverage general model reduction techniques and to provide more freedom for the analysis and control of descriptor systems. The future research includes a comparison of the control refinement approach for descriptor systems to results in perturbation theory, as well as control refinement for nonlinear descriptor systems. REFERENCES Abadir, K.M. and Magnus, J.R. (2005). Matrix algebra. Cambridge University Press. Antoulas, A.C. (2005). Approximation of large-scale dynamical systems. SIAM. Cao, X., Saltik, M., and Weiland, S. (2015). Hankel model reduction for descriptor systems. In 2015 54th IEEE CDC, 4668–4673. Dai, L. (1989). Singular control systems. Springer-Verlag New York, Inc. Fainekos, G.E., Girard, A., and Pappas, G.J. (2007). Hierarchical synthesis of hybrid controllers from temporal logic specifications. In International Workshop on HSCC, 203–216. Girard, A. and Pappas, G.J. (2009). Hierarchical control system design using approximate simulation. Automatica, 45(2), 566–571. Girard, A. and Pappas, G.J. (2011). Approximate bisimulation: A bridge between computer science and control theory. European Journal of Control, 17(5), 568–578. Kloetzer, M. and Belta, C. (2008). A fully automated framework for control of linear systems from temporal logic specifications. IEEE Transactions on Automatic Control, 53(1), 287–297. Kunkel, P. and Mehrmann, V.L. (2006). Differentialalgebraic equations: analysis and numerical solution. European Mathematical Society. Megawati, N.Y. and Van der Schaft, A. (2015). Bisimulation equivalence of DAE systems. arXiv:1512.04689. Tabuada, P. (2009). Verification and control of hybrid systems: a symbolic approach. Springer Science & Business Media. Van der Schaft, A. (2004). Equivalence of dynamical systems by bisimulation. IEEE transactions on automatic control, 49(12), 2160–2172. Weiland, S. (1991). Theory of approximation and disturbnace attenuation for linear systems. University of Groningen. Willems, J.C. and Polderman, J.W. (2013). Introduction to mathematical systems theory: a behavioral approach, volume 26. Springer Science & Business Media. 

Characterizing traits of coordination Raphael ‘kena’ Poss University of Amsterdam, The Netherlands arXiv:1307.4827v1 [] 18 Jul 2013 March 19, 2018 Abstract How can one recognize coordination languages and technologies? As this report shows, the common approach that contrasts coordination with computation is intellectually unsound: depending on the selected understanding of the word “computation”, it either captures too many or too few programming languages. Instead, we argue for objective criteria that can be used to evaluate how well programming technologies offer coordination services. Of the various criteria commonly used in this community, we are able to isolate three that are strongly characterizing: black-box componentization, which we had identified previously, but also interface extensibility and customizability of run-time optimization goals. These criteria are well matched by Intel’s Concurrent Collections and AstraKahn, and also by OpenCL, POSIX and VMWare ESX. Contents 1 Introduction 2 2 Qualifying criteria 3 3 Criteria evaluation 4 4 Problems of “coordination vs. computation” 6 5 Conclusion 9 Acknowledgements 9 References 9 1 1 Introduction The author of this report studies a research community whose specialization is the management of software components in multi-component applications. The members of this community have agreed on a common linguistic referent for their activities in this field: the word “coordination”. The main output from this community is a combination of programming languages and operating software aimed at optimizing the run-time execution of applications built by hierarchical composition of components. Example technologies whose authors self-identify as “working on coordination” include SNET [16, 4], AstraKahn [17] and Intel’s Concurrent Collections (CnC) [10, 2]. A recurring theme in the discussions within this community and with external observers is whether and how much coordination differs from other forms of programming. This topic is usually introduced with either of two questions: “what is coordination exactly?” and “what distinguishes research on coordination from other research on programming language design and implementation?” As it happens, different answers are used in these conversations depending on who is asking, who is answering and the topic at hand. This author has observed a consensus in the community that these answers are all accepted by the researchers as valid descriptions of their line of work. Of these explanations, we can recognize four groups: • self-referential explanations: a research activity is considered related to “coordination” if it self-identifies as such. For example, “this language is a coordination language because its designers call it a coordination language”; • negative space explanations: an existing field of study is selected ad hoc, then a research activity is considered related to “coordination” if it selfidentifies as “not related to” the selected research. For example, “this language is a coordination language because its designers do not focus on software modeling” (or functional programming, or model checking, etc.); • void explanations: a word is selected ad hoc with no well-defined meaning, then a research activity is considered related to “coordination” if it self-identifies as “not related to” the selected word. For example, “this language is a coordination language because its designers do not intend it to be a computation language” without a clear definition for the word “computation” (cf. section 4); • explanations by qualification: some well-defined, objective, observer-independent criteria on programming languages and operating software are identified, then “coordination” is defined based on the criteria. For example, “this language is a coordination language because it offers facilities to assemble applications from black-box components”, together with a careful definition of “black-box component”, constitutes a qualified explanation. The self-referential, negative space and void explanations are, by construction, factually vacuous: a newcomer audience exposed to them will not learn anything about what the researcher using the explanation actually does in their work. At best, the audience may understand that the researcher needs a keyword to motivate specialized attention and funding, but not more. These forms of explanations are thus not further considered here. Instead, this report reviews the criteria where consensus exists in the community that they can be used to recognize coordination objectively. 2 The discussion is presented in two parts. In section 2, we identify and detail the objective criteria that have been previously named and used by members of the research community. We then examine in section 3 how well this community’s technology matches their self-selected criteria, and also how well other technologies match the same criteria. We then use this analysis to isolate which criteria most strongly characterize the work of these researchers. Separately, in section 4 we examine the arguments that oppose “coordination” to “computation”, and we analyze how much objective understanding can be extracted from them. We then conclude in section 5. 2 Qualifying criteria We reuse below the definition of “component” and component-based design from [1, 15]: components are defined by their interface, which specifies how they can be used in applications, and one or more implementations which define their actual behavior. The two general principles of component-based design are then phrased as follows. The first is interface-based integration: when a designer uses a component for an application, he agrees to only assume what is guaranteed from the interface, so that another implementation can be substituted if needed without changing the rest of the application. The second is reusability: once a component is implemented, a designer can reuse the component in multiple applications without changing the component itself. Based on this definition, the word “coordination” is only used in the context of languages and infrastructures that enable component-based design1 . Separable provisioning (Sp): the language and its infrastructure enable the reuse of components provided by physically separate programmers, and where the considered technology is the only communication means between these providers. For example, a technology that offers the ability to build a specification from different files matches this criterion. Interface extensibility (Ie): the infrastructure enables an application designer to redefine and extend component interfaces independently from component providers, and extended interfaces can influence execution. For example, a technology that offers the ability to annotate a component to indicate post hoc that it is functionally pure (without state and deterministic), e.g. via a pragma or metadata, and which can exploit this annotation to increase execution parallelism, matches this criterion. Separable scheduling (Ss): the programmer can delegate to the technology the responsibility of choosing when (time) and where (space/resources) to execute concurrent component activities. A different but equivalent phrasing for the same criterion is the ability given to a programmer to define a partial scheduling order between component activities, and ability given to the technology to decide arbitrary actual schedules as long as the partial order is respected. 1 arguably, most contemporary programming technologies already enable component-based design; however we explicitly state the requirement to clearly scope the discussion. 3 For example, a language where programmers can declare a data-parallel operation and where the infrastructure decides how to schedule the operation matches this criterion. Adaptable optimization (Ao): the technology provides run-time optimization mechanisms that can adapt to different execution environments without changing the application specification. For example, a technology which can decide different placements when faced with different amounts of parallelism in hardware matches this criterion, and so does a technology able to decide different schedules over time when faced with different constraints on data locality (e.g. cache sizes). Customizable optimization goals (Cog): the application designer can specify different optimization goals at run-time (or no earlier than when the specification work has completed) and the technology chooses different execution strategies based on them. For example, a technology which enables to select between “run fast” and “use less memory” during execution matches this criterion. Black-box componentization (Bb): the application designer can specify an application using components only known to the technology by name and interface, and the technology provides a run-time interfacing mechanism previously agreed upon with component providers to integrate the components. For example, a technology which can link component codes compiled from different programming languages without requiring link-time cross-optimization matches this criterion. This is the main criterion proposed in [15]. Exploitable Turing-incompleteness (Eti): the specification language is not Turing complete but can still be used to define interesting / useful applications. For example, a technology whose advertised specification language can only define static acyclic data flow graphs of components matches this criterion. 3 Criteria evaluation We evaluate in table 1 how much different technologies match the criteria defined above: the criterion are listed in columns, the technologies in rows, each intersection states whether the technology matches the criterion, and a score column at the right side sums the number of criterion matched. Arrows in the score columns indicate the rows with highest scores. We review both technologies that self-identify as “coordination”, including SNET and CnC named previously, and other technologies that do not identify as such: various C and C++ implementations, Glasgow Haskell, Single-Assignment C (SAC), the standard Unix shell in a POSIX environment and VMWare ESX. While constructing table 1, we highlighted the following: • granularity: each technology may offer multiple levels of component granularities, and may not match the same criteria depending on the granularity considered. For example, the C language offers black-box componentization for entire functions but not for individual statements. To reflect this, 4 Technology Variant S-NET [16, 4] no filters / star S+NET [14, 13] no trans. / star AstraKahn [17] CnC [10, 2] C [6, 8, 12, 9, 5] C++ [7] Haskell [11] freestanding freestanding freestanding OpenMP OpenCL ISO11, hosted ISO99, POSIX ISO99, POSIX ISO11, POSIX GHC GHC SAC [3] Unix shell [5] VMWare ESX POSIX Criteria Sp Ie Granularity boxes networks boxes boxes networks boxes boxes networks steps I functions statements expressions statements kernels threads threads processes classes functions packages functions modules processes virtual machines I I I I I I I I I I I I I I I I I I I I I I I Ss Ao I I I I I I I I I I I I I I I I I I Cog Bb I I I I I I I I I E I I E I I I I I E I I I I I I I E E I I I I I E I I I I I I E E I E E E I E I I I I I I Eti E E I E I I I I I I I I I I E E E E Scores Total I only 4 2 5 5 4 6← 6← 3 6← 4 2 4 5 4 5 6← 3 6← 3 4 5 5 6 4 6 6 5 5 4 5 3 6 6 2 1 2 4 4 3 3 6← 2 5 4 5 3 6← 6← Table 1: How various technologies match the proposed criteria. multiple rows with different granularities are used for each technology in the table. • intent: a technology may happen to match a criterion although this match was not primarily intended by its designers. For example, a freestanding implementation of the C language (without library) happens to be Turing incomplete and still quite useful, although this was arguably not intended by its designer (nor commonly known of its users). To reflect this, we use the letters “I” (by intent) and “E” (emergent) at each intersection and provide two score columns in the right side. From this first evaluation table we observe the following. First, separable provisioning (Sp) is generally prevalent. Although it is a prerequisite to component-based design and thus coordination, its availability in a particular technology does not predict its score in our table. Therefore, it is a poor criterion to characterize coordination. Similarly, separable scheduling (Ss) and adaptable optimization (Ao) are also relatively prevalent. Although the benefits of separate scheduling and adaptable optimization wrt. performance speedups on parallel hardware is often used to highlight the benefits of coordination, other technologies which do not selfidentify as “coordination” (e.g. Haskell, OpenCL, SAC) also exhibit these features and can reap their associated benefits. These criteria may thus be phrased as “prerequisites” to recognize coordination but they are not characterizing. Also, the “exploitable Turing-incompleteness” (Eti) criterion is, perhaps surprisingly, difficult to match. The main reason, which we outline in section 4, is that it is actually quite difficult to design a programming language which is not Turing-equivalent. Finally, the table reveals that none of the proposed criteria clearly separates technology that self-identify as “coordinating” from those that don’t. The evaluation of whether a technology can be considered as coordination cannot yield a 5 ← ← ← ← ← Technology Variant boxes networks boxes networks boxes networks steps S-NET S+NET AstraKahn CnC C C++ Haskell freestanding freestanding freestanding OpenMP OpenCL ISO11, hosted ISO99, POSIX ISO99, POSIX ISO11, POSIX GHC GHC SAC Unix shell VMWare ESX Granularity POSIX functions statements expressions statements kernels threads threads processes classes functions packages functions modules processes virtual machines Criteria Ie Cog Bb I I I I I I I I I I I I I I I E E I I I I I E I E I I I I I I I I I I I I Scores Total I only 1 0 2 1 3← 1 3← 1 0 2 1 3← 1 3← 1 0 1 1 3 1 3 3 2 2 1 2 0 3 3 1 0 1 1 2 1 1 3← 1 2 1 2 0 3← 3← ← ← ← ← ← Table 2: How various technologies match the proposed criteria (simplified). boolean value and instead lies on a spectrum of “more or less able to coordinate”. From these observations, we can select the criteria most strongly matched by these technologies that the researchers would like to objectively describe as “strongly coordinating.” This suggests the criteria Ie, Cog and Bb and the summary in table 2. As the table shows, AstraKahn, CnC, OpenCL, POSIX and VMWare ESX can be considered strongly coordinating, each at their preferred component granularity: boxes, steps, kernels, threads/processes and virtual machines, respectively. 4 Problems of “coordination vs. computation” During the discussions around coordination, this author has observed a prevalent use of the following arguments by the members of the community: 1. “coordination technologies can be distinguished from computation technologies”; 2. “what differentiates coordination and computation technologies is the intent of the designer: the designers of coordination languages do not focus on computation”; 3. “there exist ‘pure’ coordination languages that cannot be used to specify computations.” All three arguments are motivated by a subjective, human desire of the involved individuals, that to create a “us-versus-them” vision of the research. The ulterior motive is to generate specialized attention and attract dedicated funding. In fairness to this community, we highlight here that this ulterior motive is shared by most academic researchers regardless of their field of study. However, despite and regardless of the motive and its subjectivity, the individuals involved claim (both implicitly and explicitly) that these three arguments can be recognized as objective by an external observer, i.e. they can stand and be defended at face value. 6 What interests us here is that all three arguments require some shared understanding of what is meant by “computation.” If no shared understanding can be found, then all three arguments are void and thus intellectually irrelevant. Moreover, if a shared understanding can be found, then only argument #3 is objectively qualified. Even with a shared understanding of computation, arguments #1 and #2 remain at best “negative space” arguments (cf. section 1) and still do not inform about what coordination actually entails. To see how much of argument #3 can be “saved” for the purpose of objective discussions, we need to investigate two points. The first is how much shared understanding can be gathered around the term “computation”. The second is whether, assuming some shared understanding of what “computation” entails, argument #3 actually holds: that languages that cannot express computation actually exist, and can be called coordination languages. 4.1 About the notion of computation As of this writing, there exists no formal definition of what constitutes a computation in general. What is known empirically is that for any function of mathematics it is often possible to build a machine which can calculate the value of this function for some input. What is known formally, is that for any given number function of mathematics it is always possible (in theory) to build a machine that can calculate the value of this function. What is not known however, is the set of all mathematical functions a given concrete (real-world) machine can reproduce; and whether it is possible to build a machine for all possible mathematical functions, not only number functions. Meanwhile, people can be observed to also build machines to perform work that is not described formally but is still considered useful. In this context, two approaches can be taken to define “computation”. One can seek formalism at all costs, and restrict the shared understanding to Church and Turing’s thesis: that the set of computations is exactly the set of all possible input-output transformations by any theoretical Turing machine. However, this Manichean approach excludes a range of machine activities that are commonly considered to be “computations” in practice, too: transformation and communication of real (physical) variables, non-deterministic operations over parallel hardware with loosely synchronized time, ongoing processes without a start event, etc. The other way to define “computation” is to identify some useful real-world artefacts and behaviors, call them “computation” axiomatically, then reverseengineer which languages and formal systems can be used to specify them. There are multiple ways to do so; here are the two such definitions that seem to gather most consensus: • “terminating value computations”: any operation which consumes a finite supply of static data as input, runs for a finite amount of time and produces a finite supply of static data as output. This includes but is not limited to the observable behavior of halting Turing machines; • “process computations”: any operation which is running within a wellformed space boundary (e.g. a specific component of a machine), running at a measurable cost and that is controllable: where an external agent (e.g. a person or another system) can start, stop, observe, accelerate, slow (etc.) the operation. 7 Chosen definition for “computation” Behavior of Turing machines Terminating value computations Process computations Computation-less languages (none known) (none known) some declarative languages, including Prolog, pure λ-calculus, HTML Incomplete computation languages few languages, but includes C most languages most languages Table 3: Languages that cannot specify computations. The choice of approach also defines the objective substance of any discussion that capitalizes on the notion of computation. Different choices result in different, possibly conflicting understandings. Therefore, any situation where the word “computation” is used casually to support negative space arguments should be reviewed with critical care; in particular, one should feel challenged to isolate and clarify explicitly what assumptions are being made. 4.2 Languages that “cannot specify computations” There are two interpretations for the phrase “cannot specify computations”: either “cannot specify any computation” or “cannot specify all computations”. The argument “There exists pure coordination languages that cannot specify computation” thus defines two classes of languages: computation-less languages which cannot be used to define any computation whatsoever; and incomplete languages which can only be used to specify a limited subset of computations. Both can only be discussed in the context of a specific, a priori chosen understanding of the word “computation” as described in the previous section. We collate in table 3 a condensed inventory of existing programming languages that are either computation-less or incomplete for the various definitions of “computation” isolated previously. Table 3 enables three observations. The first is that it is difficult to find concrete computation-less languages, for any definition of “computation”. In general, it is actually difficult to design a computation-less language: any language that is able to define a dynamic evaluation that can react to state, regardless of how dynamic its input is, can be tricked at a higher-level to define some computations. For example, with S-NET one can define operations using Peano arithmetic on the depth of the run-time expansion of a “star” combinator over a synchrocell, using only record types to perform choices. A computation-less language should either prevent its user from defining a dynamic evaluation, or restrict the evaluation to be stateinsensitive (or both). It is debatable whether languages with such restrictions can be called “programming” languages at all. The second is that if we consider process computations in general and we understand that “pure coordination languages are those languages that are computation-less”, we would need to accept languages like Prolog, λ-calculus or HTML as coordination languages. This does not appear compatible with the vision of the coordination community being studied. The third is that there are “too many” languages that are incomplete with regards to each definition of “computation” to hold a strong us-versus-them argument. For the two informal definitions, i.e. terminating value computations and process computations, if we understand that “pure coordination languages are those languages that are incomplete with regard to specifying computa- 8 tion”, then virtually any programming language in use today is a coordination language. If we take the formal definition instead (Turing-incompleteness), then C would also qualify as a coordination language because C is also Turingincomplete2 . Again, this does not appear compatible with the vision of this community. To summarize, it may not be possible to use argument #3 successfully to motivate specialized attention to the work of this community. 5 Conclusion We have reviewed in this report the commonly used, subjective argument that “coordination can be contrasted to computation”. We have revealed that this argument and all currently used related phrasings are largely intellectually unsound and we conclude they cannot be used to support specialized scientific attention towards “coordination” as a research activity. Instead, we have highlighted that research on “coordination” can be supported objectively using motivating arguments based on objective criteria. Of the various candidate criteria that have been proposed so far, we have shown that only three characterize the work of the researchers involved: • interface extensibility: the ability to extend or replace component interfaces arbitrarily after components are provided, and define valid composite behavior using the modified interfaces even if they conflict with the internal structure of the components; • customizable optimization goals: the ability to specify different optimization goals after the application has been specified, e.g. during execution, and the ability of the technology to use different execution strategies to match the custom goals; • black-box componentization: the ability to specify composite applications from components only known by name and interface, and the existence of run-time interfacing mechanisms that do not require the coordination technology to know anything about the internal structure of components. Of these three criteria, we had previously [15] identified the last as a clear objective criterion to recognize coordination, and we had recognized that programming technologies are “more or less coordinating” depending on how well they match the criterion. In the present report, we have extended this argument to the other two criteria, and recognized several concrete coordination technologies: AstraKahn and Intel’s CnC, but also OpenCL, POSIX and VMWare ESX. Acknowledgements This document reports on thoughts nurtured during successive discussions with Merijn Verstraaten, Alex Shafarenko, Sven-Bodo Scholz, Kath Knobe and Clemens Grelck. 2 we consider here the C language without its standard library as defined in [6, 8]. This author can provide a demonstration of C’s Turing-incompleteness upon request. 9 References [1] Don Batory and Sean O’Malley. The design and implementation of hierarchical software systems with reusable components. ACM Trans. Softw. Eng. 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Bounds on the Zero-Error List-Decoding Capacity of the q/(q − 1) Channel Siddharth Bhandari and Jaikumar Radhakrishnan arXiv:1802.08396v1 [] 23 Feb 2018 Tata Institute of Fundamental Research Homi Bhabha Road Mumbai 400005, INDIA Email: {siddharth.bhandari, jaikumar}@tifr.res.in Abstract—We consider the problem of determining the zero-error list-decoding capacity of the q/(q − 1) channel studied by Elias (1988). The q/(q − 1) channel has input and output alphabet consisting of q symbols, say, X = {x1 , x2 , . . . , xq }; when the channel receives an input x ∈ X , it outputs a symbol other than x itself. Let n(m, q, ℓ) be the smallest n for which there is a code C ⊆ X n of m elements such that for every list w1 , w2 , . . . , wℓ+1 of distinct code-words from C, there is a coordinate j ∈ [n] that satisfies {w1 [j], w2 [j], . . . , wℓ+1 [j]} = X . We show that for ǫ < 1/6, for all large q and large enough m, n(m, q, ǫq ln q) ≥ Ω(exp (q 1−6ǫ /8) log2 m). The lower bound obtained by Fredman and Komlós (1984) for perfect hashing implies that n(m, q, q − 1) = exp(Ω(q)) log 2 m; similarly, the lower bound obtained by Körner (1986) for nearlyperfect hashing implies that n(m, q, q) = exp(Ω(q)) log2 m. These results show that the zero-error list-decoding capacity of the q/(q − 1) channel with lists of size at most q is exponentially small. Extending these bounds, Chakraborty et al. (2006) showed that the capacity remains exponentially small even if the list size is allowed to be as large as 1.58q. Our result implies that the zero-error list-decoding capacity of the q/(q −1) channel with list size ǫq for ǫ < 1/6 is exp (Ω(q 1−6ǫ )). This resolves the conjecture raised by Chakraborty et al. (2006) about the zero-error listdecoding capcity of the q/(q − 1) channel at larger list sizes. I. I NTRODUCTION We study the zero-error-list-decoding capacity of the q/(q −1) channel. The input and output alphabet of this channel are a set of q symbols, namely X = {x1 , x2 , . . . , xq }; when the symbol x ∈ X is input, the output symbol can be anything other than x itself. We wish to design good error correcting codes for such a channel. For the q/(q − 1) channel it is impossible to recover the message without error if the code has at least two code-words: in fact, no matter how many letters are used for encoding, for every set of up to (q − 1) input code-words, one can construct an output word that is compatible with all of them. It is, however, possible to design codes where on receiving an output word from the channel, one can narrow down the input message to a set of size at most (q − 1)—that is, we can list-decode with lists of size (q − 1). Such codes have rate exponentially small in q. Definition I.1 (Code, Rate). A code C ⊆ {x1 , . . . , xq }n is an ℓ-list-decoding-code for the q/(q − 1) channel, if for every output word σ ′ ∈ X n , we have {σ ∈ X n : the input word σ is compatible with σ ′ } ≤ ℓ. Let n(m, q, ℓ) be the smallest n such that there exists an ℓ-list-decoding code for the q/(q − 1) channel with m code-words. The zero-errorlist-of-ℓ-rate of C, |C| = m, is given by n1 log2 (m/ℓ), and the list-of-ℓ-capacity of the q/(q − 1) channel, denoted by cap(q, ℓ), is the least upper bound on the attainable zero-errorlist-of-ℓ-rate across all ℓ-list-decoding-codes. The list-of-2-capacity of the 3/2 channel was studied by Elias [1], who showed that 0.08 ≈ log2 (3)−1.5 ≤ cap(3, 2) ≤ log2 (3) − 1 ≈ 0.58. For the 4/3 channel, Dalai, Guruswami and Radhakrishnan [2] showed that cap(4, 3) ≤ 6/19 ≈ 0.3158, improving slightly on an earlier upper bound of 0.3512 shown by Arikan [3]; it was shown by Körner and Marton [4] that cap(4, 3) ≥ (1/3) log2 (32/29) ≈ 0.0473. For general q, one can obtain the following upper bound using a routine probabilistic argument. Proposition I.1. n(m, q, q − 1) = exp(O(q)) lg m. This implies that the cap(q, q − 1) = exp(−O(q)). So for each fixed q we do have codes with positive rate, but the rate promised by this construction goes to zero exponentially with q. Fredman and Komlós [5] showed that this exponential deterioration is inevitable; Körner showed that cap(q, q) = exp(−Ω(q)). On the other hand, it can be shown that cap(q, ⌈q ln q⌉) = 1/q, and that for all functions ℓ : Z → Z we have cap(q, ℓ(q)) ≥ 1/q. Thus, the list-of-ℓcapacity of the q/(q − 1) channel cannot be better than 1/q unless ℓ is allowed to grow with m. We thus have the following situation. The list-of-ℓ-rate of any code reaches the optimal value of 1/q when the listsize is about q ln q; however, the list-of-(q − 1) (as well as list-of-q) rate is exponentially small in q. It is interesting, therefore, to study the trade-off between the list size and the rate, and determine how the rate changes from inverse polynomial in q to exponentially small in q. Chakraborty, Radhakrishnan, Raghunathan and Sasatte [6] addressed this question and showed the following. Theorem I.2. For every ǫ > 0, there is a δ > 0 such that for all large q and large enough m, we have n(m, q, (η − ǫ)q) ≥ exp(δq) log2 m, where η = e/(e − 1) ≈ 1.58. Thus, cap(q, (η − ǫ)q) = exp(−Ω(q)). We show the following. Theorem I.3 (Result). For every ǫ < 1/6, for all large q and large enough m, we have n(m, q, ǫq ln q) ≥ Ω(exp (q 1−6ǫ /8) log2 m). Thus, for all ǫ < 1/6, cap(q, ǫq ln q) = exp(−Ω(q 1−6ǫ )). This establishes both parts of the conjecture of Chakraborty et al. which states the following. Conjecture I.1. (a) For all constants c > 0, there is a constant α, such that for all large m, we have n(m, q, cq) ≥ exp (αq) log2 m. (b) For all functions ℓ(q) = o(q log2 q) and all large m, we have n(m, q, ℓ(q)) ≥ q ω(1) log2 m. A. Overview of our approach We extend the approach of Chakraborty et al., which in turn was based on the approach used by Fredman and Komlós [5] to obtain lower bounds on the size of families of perfect hash functions. To describe our adaptation of this approach, it will be convenient to reformulate the problem using matrix terminology. Consider C ⊆ X n with m code-words. We can build an m×n matrix C = (cij : i = 1, . . . , m and j = 1, . . . , n) (we use the name C both for the code and the associated matrix) by writing the code-words as rows of the matrix (the order does not matter): so cij = k iff the j-th component of the i-th codeword is xk ∈ X . Then, C is an ℓ-list-decoding code iff the matrix has the following property: for every choice R of ℓ + 1 rows, there is a column h such that {crh : r ∈ R} = X . In this reformulation, n(m, q, ℓ) is the minimum n so that there exists a matrix with this property. We refer to such a matrix as an ℓ-list-decoding matrix. Furthermore, instead of writing crh we write h(r); indeed, in the setting of hash families (originally considered by Fredman and Komlós), the columns correspond to hash functions that assign a symbol in X to each row-index in [m]. We can now describe the approach of Chakraborty et al. Fix a list-size ℓ = αq. Suppose there is an ℓ-list decoding matrix C with n = exp(βq) log2 m columns. We wish to show that if β is small then the matrix cannot have the required property; that is, we can find a set R of ℓ + 1 rows for which h(R) is a proper subset of [q] for every column h. To exhibit such a set R we will proceed in stages. In the first stage, we pick a subset R1 of q − 2 rows at random. Consider a column h. What can we expect? We expect to see a good number of collisions, where the same symbol appears in column h at two different rows in R1 . In fact, we expect h(R) to contain only about q(1 − 1/e) elements. By appealing to standard results (e.g., McDiarmid’s inequality), we may conclude that with probability exponentially close to 1 (that is, of the form 1 − exp(−γq)), h(R) is unlikely to have significantly more elements. So we might settle on a choice of R, so that h(R) deviates significantly (say by ǫq for some small ǫ) for at most exp(−γq) exp(βq) log2 m columns. If the original β is chosen to be much smaller than γ, this number is an exponentially small fraction of log2 m. The key idea now is to make these exceptional columns ineffective. We do this by focusing our attention on a reduced number of rows. For each exceptional column, we pick the symbol that appears most often in that column, and restrict attention to those rows that have this symbol in the exceptional column. This depletes the number of rows by a factor at 1/q for each exceptional column; after we do this sequentially for all the exp(−(γ − β)q) log2 m ≪ log2 m rows, we will be left with m′ rows, where log2 m′ = Ω(log2 m). We may now add more rows to our existing list R1 . If we choose these from the set of m′ rows, we are in no danger from the exceptional columns; in the other columns R1 spans about q(1 − 1/e) symbols, so we can add to R1 about q/e rows R2 (picked from the m′ -rows) and still ensure that in no column h, we are in danger of h(R1 ∪ R2 ) becoming X . It is clear that we can carry this approach further, e.g., by picking R2 randomly, expecting a significant number of internal collisions, making the exceptional columns ineffective, focusing attention on a smaller but still significant number of rows, etc., then picking R3 from the rows that survive, and so on. In fact, Chakraborty et al. derived Theorem I.2 using precisely this approach. In this paper, we follow the approach outlined above but implement the idea more precisely. Before we describe our contribution it will be useful to pin-point where the calculations in Chakraborty et al. were sub-optimal. We argued above that after R1 is picked, we expect to span only about q(1 − 1/e) symbols in a given column h. What about after R2 is picked? R1 ∪ R2 contains a total of q + q/e rows: if all symbols in column h appeared with the same frequency (and continued to do so in the m′ rows after the exceptional columns were eliminated), then we should expect h(R1 ∪ R2 ) to span about (q + q/e)(1 − exp(1 + 1/e)) symbols. Notice that this is roughly the expected number of distinct coupons collected in the classical coupon collector problem after q+q/e attempts. Unfortunately, there are technical difficulties that arise in claiming that this number will be reflected in our process because (i) R1 and R2 are not picked independently, and (ii) even if the symbols appeared with the same frequency initially, they may not do so after we focus on a depleted set of rows. Faced with these difficulties, Chakraborty et al. settled for less. Instead of matching the bound suggested by the coupon collector problem, when analysing the expected size of h(R1 ∪ R2 ), they estimated h(R2 ) separately and bounded |h(R1 ∪ R2 )| by |h(R1 )| + |h(R2 )|, thereby ignoring h(R1 ∩ R2 ). The loss in precision resulting from the use of this union bound increases as the number of phases increases. Indeed, when the coupon collector process is carried in phases by picking sets R1 , R2 , . . . , Rt for a large t, progress in collecting coupons is retarded more by collisions across sets (because for some i 6= j, h(Ri ) and h(Rj ) have elements in common) than by collisions within some h(Ri ). By neglecting collisions across phases, and by failing to track the coupon collector process closely, the argument in Chakraborty et al. were unable to push the list size in Theorem I.2 beyond e/(e − 1). What is new?: We attempt to track the progress of the coupon collector faithfully. Instead of the set R1 of size q − 2 that was picked earlier, we pick an ensemble (a collection of sets) R1 of sets of size q − 2. Similarly, in the later steps we will pick ensembles R2 , R3 , . . .. However, in the end we pick one set Ri from each of the ensembles Ri respectively, and assemble our list of rows: R1 ∪ R2 ∪ · · · ∪ Rt . That this process is more effective in bounding |h(R1 ∪ R2 ∪ . . . ∪ Rt )| will be formally verified in later sections. For now, let us qualitatively see how it helps in bounding |h(R1 ∪ R2 )|. We pick R1 at random: if the number of sets in the ensemble is large enough (we will set it to be exp(Θ(q))), then it should reflect a random set of rows that was obtained by picking rows independently (q − 2)-times from the set of all rows. Fix a choice for R2 , the set to be picked at the second stage. Consider X = |h(R1 ∪ R2 )| where R1 is picked uniformly from the ensemble R1 ; let Y = |h(R1 ∪ R2 )|, where R1 is picked uniformly from the set of all rows. Then, we expect X and Y to have similar distribution. So, we proceed as follows. We pick an ensemble R1 at random. If for a certain column h, the ensemble R1 fails to deliver a good sample, we will need to make that column ineffective as before. Further, if some set in R1 spans a significantly larger number of symbols in some column, we will again make that column ineffective. After this, we pick R2 from the remaining rows. We expect it to not only have a good number of internal collisions but also be such that |h(R1 ∪ R2 )| and |h(R1 ∪ R2 )| (where the set R2 is chosen uniformly from the available rows) are similar in expectation. Now, since we ensured that the ensemble R1 was good for column h, a random choice of R1 from the ensemble will deliver a value of |h(R1 ∪ R2 )| that, with high probability, can be bounded by the number of distinct coupons picked up at the same stage by the coupon collector; in particular, it accounts for symbols common to h(R1 ) and h(R2 ). The outline above illustrates the advantages of picking an ensemble instead of committing to just one randomly chosen set. However, a large ensemble comes with its drawbacks. We need to ensure that no set in the ensemble spans too many elements in any column, or rather, we need to eliminate any column where some set spans many elements. This forces a more drastic reduction in the number of rows than before (that is, now m′ when compared with m is much smaller than in the calculation in [6]). Thus, it is important to keep the sizes of the ensembles small. The tradeoff between these opposing concerns needs to be handled with some care. The argument is presented in detail below. II. P ROOF OF THE R ESULT In what follows we assume that q is a large natural number and m → ∞. We will need the following concentration result due to McDiarmid (1989). Lemma II.1 (McDiarmid). Let X1 , X2 , . . . , Xn be independent random variables where each Xk takes values in a finite set A. Let f : An → R be such that |f (x) − f (y)| ≤ c whenever x and y differ in only one coordinate. Let Y = f (X1 , X2 , . . . , Xn ); then, for all t > 0,   −2t2 Pr[E[Y ] − Y ≥ t], Pr[Y − E[Y ] ≥ t] ≤ exp . nc2 Let C be an ℓ-list-decoding-code for the q/(q−1) channel with ℓ < q ln q/6. As mentioned in the introduction, we will view C as an m×n matrix with entries from [q]. In other words, the rows are indexed by code-words and the columns are indexed by hash functions. Let wt P be a function from [q] to {0, 1}; for A ⊆ [q], let wt(A) := a∈A wt(a). Let R be a random variable taking values in P([m]). Sometimes we use R to also refer to the distribution of this random variable. Following the idea mentioned in the introduction, we intend to keep an ensemble R of sets of rows such that when we pick a new set of rows R2 from a depleted number of rows m′ , we not only observe the correct number of internal collisions within R2 but also observe the correct number of collision between members of R and R2 . This motivates the following definition. Definition II.1 (Sampler). We say that an ensemble R = (R1 , R2 , . . . , RL ), where each Ri ⊆ [m], is a (γ, δ)sampler for R wrt column h if (A1 , A2 , . . . , AL ) := (h(R1 ), h(R2 ), . . . , h(RL )) satisfies ∀wt : [q] → {0, 1}   h i Pr wt(Aj ) − E wt(h(R)) ≥ γq ≤ exp (−δq). j∈u [L] The definition makes provision for all functions wt, because it tries to anticipate the appropriate internal collisions (see Lemma II.2) with very little advance knowledge of what the distribution on [q] looks like in column h after a large number of rows have been discarded. Let π : S → [0, 1] be a probability mass function on a finite set S. Let k ≥ 1, and let X1 , X2 , . . . , Xk be independent random variables each distributed according to π. Then, let π {k} denote the probability mass function of the set {X1 , X2 , . . . , Xk }. For distributions A and B on P([m]), let A ∨ B be the distribution of S ∪ T where S ∼ A and T ∼ B, with S and T chosen independently. The following lemma will be the main workhorse for our argument. Lemma II.2 (Ensemble Composition Lemma). Let R be a distribution on P([m]) and let D be a distribution on [m]. Let R be a (γ, δ)-sampler for R wrt a column h; let (R1 , R2 , . . . , Rs ) be obtained by taking s independent samples from the ensemble R. Similarly, let R′ = (R1′ , R2′ , . . . , Rs′ ) be obtained by taking s independent samples according to R′ ∼ D{tq} where t < 1. Let γ ′ , δ ′ > 0 be such that δ ≤ 2(γ ′ )2 /t and δ > δ ′ Let s = exp(δ −δ ′ )q, γ e = γ +γ ′ , δe = δ −δ ′ . Then, ′ with probability 1 − 12 exp (−δ q) over the random choices, the composed ensemble e := R1 ∪ R1′ , R2 ∪ R2′ , . . . , Rs ∪ Rs′ ) R e of cardinality s, is a (e γ , δ)-sampler for R ∨R′ wrt the column h, and furthermore ∀i ∈ [s], h i h(Ri ∪ Ri′ ) − E h(R ∨ R′ ) ≤ e γ q. (1) Note that this ensemble is generated according to the product s distribution (R ∨ R′ ) .   Proof. Fix f : [q] → {0, 1} and let µf := E f (h(R ∪ R′ )) ; similarly, for R′ ⊆ [m] let µf (R′ ) := ER f (h(R ∪ R′ )) . First, we bound the probability that when R′ is chosen according to R′ , it fails to have µf (R′ ) close to µf . Using McDiarmid’s inequality over the tq primitive choices for R′ , we have     −2(γ ′ )2 q 2 ′ ′ Pr |µ (R ) − µ | ≥ γ q ≤ 2 exp f f R′ ∼R′ tq   −2(γ ′ )2 q = 2 exp . (2) t Now, let wt : [q] → 0, 1 be defined by wt(x) = f (x) if x 6∈ h(R′ ) and wt(x) = 0 otherwise. Then, for R ⊆ [m], we have, f (h(R ∪ R′ )) = f (h(R′ )) + wt(h(R)). Therefore (note here R′ is fixed and R varies randomly in R),   Pr f (h(R ∪ R′ )) − µf (R′ ) ≥ γq R∈u R     = Pr wt(h(R)) − E wt(h(R)) ≥ γq R∈u R and since R is a (γ, δ)-sampler wrt R, we have     Pr wt(h(R)) − E wt(h(R)) ≥ γq ≤ exp(−δq). R∈u R Thus, h i ′ ′ f (h(R ∪ R )) − µ ≥ (γ + γ )q ≤ f R∈u R,R ∼R′ i h |µf − µf (R′ )| ≥ γ ′ q + Pr R∈u R,R′ ∼R′ h i Pr ′ ′ µf (R′ ) − f (h(R ∪ R′ )) ≥ γq R∈u R,R ∼R   h
i −2(γ ′ )2 q + exp −δq ≤ 3 exp(−δq). (3) ≤ 2 exp t Pr ′ (We used δ ≤ 2(γ ′ )2 /t to justify the last inequality.) Let ∆ := 3 exp(−δq), the quantity on the right in (3). By taking f to be the all-1’s function, we conclude from (3) that for each i with   probability at least 1 − ∆, h(Ri ∪ Ri′ ) − E |h(R ∨ R′ )| ≤ (γ + γ ′ )q. Now, f (h(Ri ∪ Ri′ )) = h(Ri ) ∪ h(Ri′ ) , and µf =  E |h(R ∨ R′ )| . Now, by a union bound over the s choices for i, we obtain     Pr ∃i ∈ [s], h(Ri ∪ Ri′ ) − E h(R ∨ R′ ) ≥ (γ + γ ′ )q e R ≤ ∆s ≤ 3 exp(−δ ′ q). (4) This establishes (1). It remains to establish our first claim that whp the ensemble e picked according to (R ∨ R′ )s is a (e γ , δ)-sampler for R ∨ R′ . Fix f : [q] → {0, 1}. Now, (3) implies that for each i ∈ [s], the probability that |f (h(Ri ∪ Ri′ )) − µf | ≥ (γ + γ ′ )q is exponentially for  small in q.′ Then, the tail probabilities  Ps ′ Y := i=1 I |f (h(Ri ∪ Ri )) − µf | ≥ (γ + γ )q can be bounded by considering Bin(s, ∆). Therefore, e Pr [Y > exp (−δq)s] e R   s e ≤ (∆)exp (−δq)s e exp (−δq)s e exp (−δq)s e ≤ (e exp (δq)∆) ≤ 9 exp (−δ ′ q). (5) (We need to take a union bound against the 2q possible functions f : [q] → {0, 1}: by changing s to qs we may easily establsih this.) By (4) and (5), the probability that our e ensemble fails to be a (e γ , δ)-sampler, with γ e = γ + γ ′ and ′ δe = δ − δ , or fails to satisfy (1) is at most 12 exp (−δ ′ q). Let us recall the template of our argument. At any stage we will have an ensemble of sets of rows, say R, and a universe U ⊆ [m] to choose sets of rows from to add to R. We will add a specific number of randomly chosen sets of rows of a particular size from U and then declare those columns bad where the modified R deviates from its expected behaviour. Consider a set R ∈ R: we want to say that the couponcollector process at |R| probes into [q] is the gold standard for good behaviour, i.e., no set in R will have expansion more than the coupon-collector at the same stage. The expected number of elements that the coupon-collector process picks up after a i.i.d. uniform probes into [q] is approximately q (1 − exp(−a/q)): we will denote this as µcc q (a). So, we need the following lemma, which is proved in the appendix. Lemma II.3 (Phased Coupon Collector). Let a1 , a2 , . . . , ak be positive integers; let a = a1 + a2 + · · · + ak , and let π1 , π2 , . . . , πk be probability mass functions. Let A1 , A2 , . . . , Ak be independent random variables taking val{a } ues in P([q]), where Ai ∼ πi i . Suppose a ≤ ǫq ln q and ǫ k ≤ eq for some ǫ < 1/3, then, E[|A1 ∪ A2 ∪ · · · ∪ Ak |] ≤ q (1 − exp(−a/q)) + o(q 1−ǫ ) 1−ǫ = µcc ). q (a) + o(q Our next target is to understand the number of iterations we wish to perform, i.e., the number of times we need to enlarge the sizes of the sets surviving the ensemble R so that the list size hits the target of ǫq ln q, where ǫ < 1/6. At the first stage we will pick up sets of rows of size about ℓ1 = q, and expect the image size to be close to µcc q (ℓ1 ); we then prune out the exceptional columns. In the next stage, we pick sets of size about ℓ2 = q − µcc q (ℓ1 ) and expect the combined image size to be close to µcc q (ℓ1 + ℓ2 ). Hence, in the third iteration we pick sets of size close to ℓ3 = q − µcc q (ℓ1 + ℓ2 ), and so on for the subsequent iterations. P We are interested in the list size k after k iterations, i.e, ℓ≤k := i=1 ℓi . We have the following proposition, which is proved in the appendix. Proposition II.4. Let ℓ1 = q, and for i ≥ 1 let ℓi+1 = q − Pi ǫ µcc q ( j=1 ℓj ). Suppose k = eq for some ǫ < 1, then, ℓ≤k ≥ ǫq ln q. Proof. (The series {ℓ≤k } tends to q ln q.) Finally, we need a lemma where we glue all the steps mentioned in the introduction. At each iteration k, we maintain an ensemble Rk satisfying the requisite properties. We call a distribution D on P([m]) a (g1 , . . . , gk )-phased {g } {g } {g } coupon collector distribution if D = D1 1 ∨D2 2 . . .∨Dk k where each Di is a probability mass function on [m]. The following lemma tracks how the parameters change with each iteration. Lemma II.5 (Iteration Lemma). Let k ≤ q ǫ for some ǫ < 1/5. Let γ = γ ′ = q −2ǫ /2 and δ ′ = q −5ǫ /4. Assume n ≤ exp (δ ′ q) log2 m/(48 · q ǫ log2 q). Then, there exists a partition H1 (k) ⊔ H2 (k) of the columns of C, a universe of rows Uk ⊆ [m], an ensemble Rk = (R1 , R2 , . . . RLk ), integers (g1 , . . . , gk ) and a (g1 , . . . , gk )-phased coupon collector distribution Dk such that: a g1 = q − 2, and gi+1 = q − µcc q (gi ) − (i + 1)γq − 2 b ∀i ∈ [Lk ], |Ri | = g≤k ≥ ℓ≤k − 2k − k 2 γq/2 c ∀h ∈ H2 (k), ∀i ∈ [Lk ], |h(Ri ∪ Uk )| ≤ q − 1 d ∀h ∈ H1 (k), Rk is a ((k + 1)γ, γ 2 − kδ ′ )-sampler for Dk wrt h e ∀h ∈ H1 (k), ∀i ∈ [Lk ] |h(Ri )| − E [h(Dk )] ≤ (k + 1)γq f log2 |Uk | ≥ log2 m − k log2 q · 24 exp (−δ ′ q)n. Proof. We will use induction on k. For k = 1 we have g1 = q − 2. We use Lemma II.2 with R being the constant ∅, and R = {∅}. Clearly, R is a (γ, γ 2 )-sampler for R. Let D be the uniform distribution over [m] and let R′ = (R1′ , R2′ , . . . , Rs′ ) be obtained by taking s = exp ((γ 2 − δ ′ )q) independent samples according to R′ ∼ D{q−2} . So, D1 = D{q−2} . For a fixed column h we have the following: with probability 1 − 12 exp (−δ ′ q) over the random choices, the composed ensemble e = R1′ , R2′ , . . . , Rs′ ) R e is a (2γ, γ 2 − δ ′ )-sampler for R′ wrt is good wrt h, i.e., R the column h, and furthermore ∀i ∈ [s], h i h(Ri′ ) − E h(R′ ) ≤ 2γq. Hence, on expectation only 12 exp (−δ ′ q)n columns are bad. Therefore, with probability at least 1/2 at most 24 exp (−δ ′ q)n columns are bad. Also, the probability of an Ri′ ∈ R′ having size less than q − 2 (because some two of our q − 2 choices of rows picked the same row) is at most q 2 /m. Thus, by the union bound the probability of (b) not holding is at most s · q 2 /m e which which is less than 1/2. Therefore, there is choice of R, 1 ′ we call R , such that at most 24 exp (−δ q)n columns are bad and (b) holds. The set of bad columns is H2 (1) and the set of good columns is H1 (1). Then, clearly (d) and (e) are true. Let H2 (1) = {h1 , . . . , hb } where b ≤ 24 exp (−δ ′ q)n and WLOG assume that 1 is the most frequent symbol in h1 . Retain only those rows in U that correspond to the symbol 1 in h1 . Call this pruned universe U ′ : we have ensured that so long as we add rows to Ri ∈ R1 only from U ′ , the image size in h1 is at most h1 (R) + 1 ≤ q − 1. Thus, by taking a multiplicative hit of at most 1/q we have rendered h1 ineffective.  b Iterating this over H2 (1) we take a multiplicative hit of 1q . Hence, we obtain a universe U ′ , which will be U1 , such that log2 |U ′ | = log2 |U1 | ≥ log2 m − 24 exp (−δ ′ q)n log2 q. This establishes (c) and (f). This establishes the claims for k = 1; the induction step in general is similar. Now, as our IH let us assume that for (k − 1) we have the partition H1 (k − 1) ⊔ H2 (k − 1), Uk−1 ⊆ [m], Rk−1 , integers (g1 , . . . , gk−1 ) and Dk−1 such that (a) through (f) are satisfied. Then, we repeat the above argument. We have gk = q − µcc q (gk−1 ) − kγq − 2. We use Lemma II.2 for h ∈ H1 (k − 1) with R being Dk−1 , and R = Rk−1 which is a (kγ, γ 2 − (k − 1)δ ′ )-sampler for Dk−1 wrt h. Let (R1 , . . . , Rs ) be obtained by s = exp (γ 2 − kδ ′ ) independent samples from Rk−1 . Let D be the uniform distribution over Uk−1 and let R′ = (R1′ , R2′ , . . . , Rs′ ) be obtained by taking s independent samples according to R′ ∼ D{gk } . We let Dk = Dk−1 ∨ D{gk } . For a fixed column h we have the following: wp 1 − 12 exp (−δ ′ q) over the random choices, the composed ensemble e = R1 ∪ R1′ , R2 ∪ R2′ , . . . , Rs ∪ Rs′ ) R e is a ((k + 1)γ, γ 2 − kδ ′ )-sampler for is good wrt h, i.e., R Dk wrt h, and furthermore ∀i ∈ [s], h i h(Ri ∪ Ri′ ) − E h(Dk ) ≤ (k + 1)γq. Hence, on expectation only 12 exp (−δ ′ q)n columns of H1 (k − 1) are bad. Therefore, with probability at least 1/2 at most 24 exp (−δ ′ q)n columns of H1 (k − 1) are bad. Also, e having size less than g≤k the probability of an Ri ∪ Ri′ ∈ R cc (because some two of our q − µq (gk−1 ) − kγq − 2 choices of rows for Ri′ picked the same row of collided with some row in Ri ) is at most (q ln q)2 /|Uk−1 |. Thus, by the union bound the probability of (b) not holding is at most s · (q ln q)2 /|Uk−1 | e which which is less than 1/2. Therefore, there is choice of R, k ′ we call R , such that at most 24 exp (−δ q)n columns of H1 (k−1) are bad and (b) holds. Combining these bad columns with H2 (k−1) we obtain H2 (k) and the columns not in H2 (k) form the set H1 (k) = H2 (k). Then, clearly (d) and (e) are true. Let H2 (k) \ H2 (k − 1) = {h1 , . . . , hb } where b ≤ 24 exp (−δ ′ q)n and WLOG assume that 1 is the most frequent symbol in h1 . Retain only those rows in Uk−1 that correspond to the symbol 1 in h1 . Call this pruned universe U ′ : this pruning ensures that so long as we add rows to Ri ∈ Rk only from U ′ , the image size in h1 is at most q − 1. Thus, by taking a multiplicative hit of at most 1/q we have rendered h1 ineffective. Iterating this over H2 (k) we take a multiplicative  b hit of 1q . Hence, we obtain a universe U ′ , which will be Uk , such that log2 |U ′ | = log2 |Uk | ≥ log2 |Uk−1 | − 24 exp (−δ ′ q)n log2 q ≥ log2 m − k log2 q · 24 exp (−δ ′ q)n. Together with property (c) of Uk−1 this establishes (c) and (f). This completes the induction step. Proof of Theorem I.3 (main result of the paper). Fix an ǫ′ < 1/6 and let C be an ǫ′ q ln q-list-decoding-code for the q/(q−1) channel. Choose λ ≪ ǫ′ and let ǫ = ǫ′ +λ. Let q be sufficiently ′ large so that k = q ǫ ≥ eq ǫ +λ/2 . We will appeal to Lemma II.5 (with k and ǫ) and assume that n ≤ exp (δ ′ q) log2 m/(48 · q ǫ log2 q). Then, by choosing a set of rows R in the ensemble Rk and using (b) and Proposition II.4 we obtain that |R| ≥ ǫ′ q ln q. However, using (c) we have that for all columns h ∈ H2 (k), |h(R)| ≤ q − 1. Also, using (e) and Lemma II.3 we obtain that for all h ∈ H1 (k), |h(R)| < q. This is a contradiction and hence n > exp (δ ′ q) log2 m/(48 · q ǫ log2 q) or for ′ sufficiently large q we have n > Ω(exp (q 1−6ǫ /8) log2 m). We note that it is possible by a more careful analy′ sis to improve the bound of Ω(exp (q 1−6ǫ /8) log2 m) to 1−4ǫ′ Ω(exp (q /8) log2 m) in which case we may apply the bound till a list size of q ln q/4. This bound is obtained by modifying Lemma II.5 to accommodate γ ′ and δ ′ which vary across the induction steps and being more scrupulous about the argument in the preceding paragraph. ACKNOWLEDGEMENTS We are grateful to Prahladh Harsha for the numerous detailed discussions that led to the result reported in this paper, and also for proof-reading it. We also thank Ramprasad Saptharishi for his help with Lemma II.3. R EFERENCES [1] P. Elias, “Zero error capacity under list decoding,” IEEE Transactions on Information Theory, vol. 34, no. 5, pp. 1070–1074, Sep 1988. [2] M. Dalai, V. Guruswami, and J. Radhakrishnan, “An improved bound on the zero-error list-decoding capacity of the 4/3 channel,” in 2017 IEEE International Symposium on Information Theory, ISIT 2017, Aachen, Germany, June 25-30, 2017, 2017, pp. 1658–1662. [Online]. Available: https://doi.org/10.1109/ISIT.2017.8006811 [3] E. Arikan, “An upper bound on the zero-error list-coding capacity,” IEEE Transactions on Information Theory, vol. 40, no. 4, pp. 1237–1240, Jul 1994. [4] J. Korner and K. Marton, “New bounds for perfect hashing via information theory,” European Journal of Combinatorics, vol. 9, no. 6, pp. 523 – 530, 1988. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S0195669888800489 [5] M. L. Fredman and J. Komlós, “On the size of separating systems and families of perfect hash functions,” SIAM Journal on Algebraic Discrete Methods, vol. 5, no. 1, pp. 61–68, 1984. [Online]. Available: https://doi.org/10.1137/0605009 [6] S. Chakraborty, J. Radhakrishnan, N. Raghunathan, and P. Sasatte, “Zero error list-decoding capacity of the q/(q-1) channel,” in FSTTCS 2006: Foundations of Software Technology and Theoretical Computer Science, 26th International Conference, Kolkata, India, December 13-15, 2006, Proceedings, 2006, pp. 129–138. [Online]. Available: https://doi.org/10.1007/11944836_14 A PPENDIX Proof of Lemma II.3. Consider a constant λ ≪ ǫ. For i = 1, 2, . . . , k, let Bi be the set of q 1−2ǫ−λ elements of [q] taking the topmost values in πi . Let B = B1 ∪ B2 ∪ · · · ∪ Bk ; note that |B| ≤ kq 1−2ǫ−λ = o(q 1−ǫ ). Then, E[|A1 ∪ A2 ∪ . . . Ak |] is at most ! k X Y ai . (1 − πi (x)) |B| + 1− x6∈B i=1 Now, for x 6∈ B, we have πi (x) ≤ 1/q 1−2ǫ−λ , and

1 − πi (x) ≥ exp −πi (x)/ 1 − πi (x)

≥ exp −πi (x) 1 + 2/q 1−2ǫ−λ . Then, by the AM-GM inequality we have the upper-bound   X ai πi (x) . |B| + q − q exp −(1 + 2/q 1−2ǫ−λ )(1/q) i,x Our  claim follows from P √ exp −(1 + 2/ q)(1/q) i,x ai πi (x) o(1/q 1−2ǫ ) ≥ exp (−a/q) − o(q 1−ǫ )/q. ≥ this because exp(−a/q) − Proof of Proposition II.4. Suppose ℓ≤i ∈ [jq, (j + 1)q] for some j ≥ 0, then, ℓi+1 ≥ q/ej+1 . Therefore, the number of i’s for which ℓ≤i ∈ [jq, (j + 1)q] ≤ ej+1 . Suppose ℓ≤k < ǫq ln q, then as a contradiction we have k < e + e2 + · · · + eǫ ln q ≤ eq ǫ . 

1 Downlink Energy Efficiency of Power Allocation and Wireless Backhaul Bandwidth arXiv:1710.02942v1 [] 9 Oct 2017 Allocation in Heterogeneous Small Cell Networks Haijun Zhang, Senior Member, IEEE, Hao Liu, Julian Cheng, Senior Member, IEEE, and Victor C. M. Leung, Fellow, IEEE Abstract The widespread application of wireless services and dense devices access have triggered huge energy consumption. Because of the environmental and financial considerations, energy-efficient design in wireless networks becomes an inevitable trend. To the best of the authors’ knowledge, energy-efficient orthogonal frequency division multiple access heterogeneous small cell optimization comprehensively considering energy efficiency maximization, power allocation, wireless backhaul bandwidth allocation, and user Quality of Service is a novel approach and research direction, and it has not been investigated. In this paper, we study the energy-efficient power allocation and wireless backhaul bandwidth allocation in orthogonal frequency division multiple access heterogeneous small cell networks. Different from the existing resource allocation schemes that maximize the throughput, the studied scheme maximizes energy efficiency by allocating both transmit power of each small cell base station to users and bandwidth for backhauling, according to the channel state information and the circuit power consumption. The problem is first formulated as a non-convex nonlinear programming problem and then it is decomposed into two convex subproblems. A near optimal iterative resource allocation algorithm is designed to solve the Haijun Zhang is with the Beijing Engineering and Technology Research Center for Convergence Networks and Ubiquitous Services, University of Science and Technology Beijing, Beijing, 100083, China (e-mail: [email protected]). Hao Liu and Julian Cheng are with School of Engineering, The University of British Columbia, Kelowna, BC, Canada (e-mail: [email protected], [email protected]). Victor C. M. Leung is with the Department of Electrical and Computer Engineering, The University of British Columbia, Vancouver, BC V6T 1Z4 Canada (e-mail: [email protected]). October 10, 2017 DRAFT 2 resource allocation problem. A suboptimal low-complexity approach is also developed by exploring the inherent structure and property of the energy-efficient design. Simulation results demonstrate the effectiveness of the proposed algorithms by comparing with the existing schemes. Index Terms Bandwidth allocation, energy efficiency, heterogeneous network, power allocation, small cell, wireless backhaul. I. I NTRODUCTION Wireless communication networks have experienced tremendous growth in the past a few decades. It is shown that higher capacity wireless links are expected to meet the increasing quality of service (QoS) demands of multimedia applications, and these high data rate links also result in increasing device power consumption. The next generation communication systems need to provide higher data rate with limited power and bandwidth due to the rapidly increasing demands for multimedia services. Designing energy-efficient wireless communication system becomes an emerging trend, due to rapidly increasing system energy costs and rising requirements of communication capacity [1]–[3]. According to [4] and [5], the radio access part is a major energy consumer in conventional wireless cellular networks, and it accounts for up to more than 70 percent of the total energy consumption. Therefore, increasing the energy efficiency of typical wireless networks is important to overcome the challenges raised by the rising demands of energy consumption and communication throughput. To offload the overloaded traffics in macrocells and enhance the capacity and energy efficiency of the wireless networks, one proposed method is to shorten the distance between the base stations (BSs) and the user equipments. Small cells (e.g., picocells, femtocells and relay nodes) have been used to improve system capacity in hotspots for relieving the burden on overloaded macrocells, which is considered as a promising technique to provide an effective solution for the challenges in current macrocells [6], [7]. Therefore, there is no doubt that small cell has been paid much attention in recent years from academia and industry because it can help the system spatial reuse spectrum with low power consumption and improve the system coverage with low infrastructure cost deployment [8]. Heterogeneous small cell networks, where small cells are overlaid within October 10, 2017 DRAFT 3 a macrocell to improve coverage and increase system capacity beyond the initial deployment of macrocells, have been regarded as a promising approach to meet the increasing data traffic demand and reduce energy consumption. Although highly promising, many important problems related to heterogeneous small cell networks such as interference mitigation, resource allocation, and QoS provisioning [9]–[12] should be addressed to fully reap the potential gains. Resource allocation, such as power allocation and bandwidth allocation, has been widely used to maximize the energy efficiency under power limitation and QoS in heterogeneous small cell networks. Power allocation for energy efficiency has been widely studied in the literature. The distributed power control game was studied in [13] to maximize the energy efficiency of transmission for secondary users in cognitive radio networks and an optimal power control problem was formulated as a repeated game. In [14], based on the hardcore point process (HCPP), the authors investigated the maximum achievable energy efficiency of the considered multiuser multiantenna HCPP random cellular networks with the aforementioned minimum distance constraint for adjacent BSs. Different from the authors in [14], who took the minimum distance in adjacent BSs into consideration to maximize the energy efficient, we propose a suboptimal low-complexity approach of energy-efficient backhaul bandwidth allocation by optimizing the fraction of bandwidth allocated for wireless backhauling at all small cell BSs within a macrocell range. The authors in [15] studied energy-efficient power control and receiver design in cognitive radio networks, and a non-cooperative power control game for maximum energy efficiency of secondary users was considered with a fairness constraint and interference threshold. The authors of [16] formulated the energy-efficient spectrum sharing and power allocation in heterogeneous cognitive radio networks with femtocells as a Stackelberg game and they proposed a gradient based iteration algorithm to obtain the Stackelberg equilibrium solution to the energy-efficient resource allocation problem. Some works also have been done to consider bandwidth allocation for energy efficiency. In [17], the authors studied the joint service pricing and bandwidth allocation for energy and cost efficiency at the operator level in a multitier network where an operator deploys heterogeneous small cell networks, and they formulated the problem as a Stackelberg game. The problem of joint link selection, power and bandwidth allocation for energy efficiency maximization for Multi-Homing networks was investigated in October 10, 2017 DRAFT 4 [18]. A new energy-efficient scheme was presented in [19] to statistically meet the demands for QoS during the bandwidth allocation for wireless networks. In this paper, we define that wireless backhaul as the connection between macro BS and small cell BSs, and it is necessary to jointly consider the design of the radio access and backhaul network. Several related works considered the backhaul to improve energy efficiency in wireless networks. The authors of [20] studied energy efficiency of resource allocation in multi-cell orthogonal frequency division multiple access (OFDMA) downlink networks where the limited backhaul capacity, the circuit power consumption and the minimum required data rate are considered. The resource allocation problem for energy-efficient communication with limited backhaul capacity is formulated as an optimization problem. In [21], an energy efficiency model of small cell backhaul networks with Gauss–Markov mobile models has been proposed. In [22], the authors maximized system energy efficiency in OFDMA small cell networks by optimizing backhaul data rate and emission power, and they proposed a joint forward and backhaul link optimization scheme by taking both the power consumption of forward links and the backhaul links into consideration. To the best of the authors’ knowledge, energy efficiency of power allocation and wireless backhaul bandwidth allocation in heterogeneous small cell network has not been investigated. In this work, we study the power allocation and bandwidth allocation problem in a heterogeneous small cell network where the small cells use wireless backhauling to maximize energy efficiency of all small cell users. Similar to the paper in [23], we also use Gradient Assisted Binary Search (GABS) Algorithm to solve the energy-efficient power allocation problem. Reference [24] is a conference version of this paper. Different from the conference version, we provide the detailed proof for the theorem, complexity analysis for the proposed algorithms and more simulation results in this paper. The key contributions of our work can be summarized as follows. • Design of an energy-efficient OFDMA heterogeneous small cell optimization: This is a novel approach by considering energy efficiency maximization, power allocation, wireless backhaul bandwidth allocation, and user QoS into the design of OFDMA heterogeneous small cell optimization. We formulate the energy-efficient wireless backhaul bandwidth allocation and power allocation problem in a heterogeneous small cell as a nonlinear October 10, 2017 DRAFT 5 programming problem, where maximum transmit power constraints of each small cell BS to each small cell user, the downlink data rate constraint of small cell BSs and the minimum data rate between each small cell BS and its corresponding user are considered to provide reliable and low energy consumed downlink transmission for small cell users. The nonconvex optimization problem is then decomposed into two convex subproblems, and an algorithm is proposed for wireless backhual bandwidth allocation and power allocation. • Support of the small cell backhauling in the context of designing power allocation schemes for heterogeneous small cell networks: We study the wireless backhaul bandwidth allocation at the small cell BS, which means a fraction of bandwidth is scheduled for backhauling and the other is assigned for communication with corresponding users. We formulate the bandwidth allocation problem as a convex problem and obtain the optimum solution. • Design of suboptimal low-complexity algorithm by decomposing the power allocation and bandwidth allocation: The energy-efficient wireless backhaul bandwidth allocation and power allocation problem are decomposed and are optimized separately. Correspondingly, a suboptimal low-complexity algorithm is proposed. The effectiveness of the proposed suboptimal algorithm is demonstrated by simulations. The rest of this paper is organized as follows. Section II describes the system model. In Section III, the energy-efficient resource allocation and backhauling are presented, and in Section IV, optimization algorithms are proposed. Simulation results are discussed in Section V. Finally, Section VI concludes the paper. II. S YSTEM M ODEL In this section, we formulate the problem of downlink energy efficiency of power allocation and unified bandwidth allocation for wireless backhauling in heterogeneous small cell networks. We consider a heterogeneous small cell network as shown in Fig. 1 with a single macro BS, J small cells deployed within the macrocell range and K users randomly located in each small cell. The small cells share the same spectrum with macrocell. In this work, the unified wireless backhaul bandwidth allocation is investigated. The unified bandwidth allocation factor β ∈ [0, 1], October 10, 2017 DRAFT 6 which is the fraction of bandwidth allocated for wireless backhauling at all small cell BSs within a macrocell range. For simplicity, all small cells are assumed to have the same bandwidth allocation factor. We assume that the multiple antenna technology is used in the macro BS and each small cell corresponds to a beamforming group, so the interference for wireless backhaul between different small cells can be neglected. The antenna array size at macro BS is N, which is much greater than the beamforming group size B and the number of small cells, i.e., N ≫ B and N ≫ J. In this work, we also assume that B ≥ J. Each small cell BS is equipped with single antenna. OFDMA technology is used in each small cell to support the communication between BS and users. Let P0 be the equal transmit power of the macro BS transmit antenna targeted at corresponding small cell and σ 2 is the additive white Gaussian noise (AWGN) power. Then the received signalto-noise ratio (SNR) in the wireless backhaul downlink of small cell j is given by γj = P 0 Gj . σ2 (1) Let gj,k be the channel power gain between the jth small cell BS and its corresponding kth user, where j ∈ {1, 2, ..., J}, k ∈ {1, 2, ..., K}. Let pj,k denote the transmit power from the jth small cell BS to its corresponding kth user, and let P = [pj,k ]J×K denote the power allocation matrix. We assume that different users in each small cell use different subchannels and co-channel interference between small cells as part of the thermal noise because of the severe wall penetration loss and low power of small cell BSs [12]. The received signal-to-interference-plus-noise ratio (SINR) of small cell user k associated with small cell j is given by γj,k = pj,k gj,k σ 2 +Ij,k (2) where Ij,k is the interference introduced by macro BS, Ij,k = P0 Gj,k , where Gj,k is the channel power gain between macro BS and the kth user in the jth small cell. The achievable data transmission rate between the jth small cell BS and its corresponding kth user is determined by   1−β log2 (1 + γj,k ) . (3) rj,k = K Therefore, we have the relation between rj,k and pj,k October 10, 2017 DRAFT 7 Krj,k σ 2 +Ij,k pj,k = (2 1−β − 1) gj,k     pj,k gj,k 1−β log2 1 + 2 . rj,k = K σ +Ij,k (4) Besides the transmit power during the transmission, circuit energy consumption is also incurred by device electronics in small cell BSs [25], [26]. Circuit power represents the additional device power consumption of devices during transmissions [27], such as digital-to-analog converters, mixers and filters, and this portion of energy consumption is independent of the transmission state. If we denote the circuit power as PC , the overall power assumption of the jth small cell BS to the kth user is PC + pj,k . For energy-efficient communication, it is desirable to send the maximum amount of data with a given amount of energy for small cell BSs. Hence, given any amount of energy ∆e consumed in a duration ∆t in each small cell BS to each user, ∆e = ∆t(PC + pj,k ), the small cell BSs desire to send a maximum amount of data by choosing the power allocation vector and backhaul bandwidth to maximize J X K X rj,k (β, pj,k )∆t j=1 k=1 ∆e (5) which is equivalent to maximizing U(β, P) = J X K X Uj,k (β, pj,k ) (6) j=1 k=1 where Uj,k (β, pj,k ) = rj,k (β, pj,k ) . PC + pj,k (7) In (6), U(β, P) is called energy efficiency for all small cells; Uj,k (β, pj,k ) is the energy efficiency of the kth user of the jth small cell. The unit of the energy efficiency is bits per Hertz per Joule, which has been frequently used in the literature for energy-efficient communications [28]–[32]. When the downlink channel state information is estimated by the small cell BSs, the resource allocation is performed by a small cell BS under the following constraints. • Transmit power constraint of each small cell BS to each user: 0 ≤ pj,k ≤ Pmax , ∀j, k October 10, 2017 (8) DRAFT 8 where Pmax denotes the maximal transmit power of each small cell BS to each user. • The downlink data rate constraint of each small cell BS: The throughput of the small cell is given by Rj = K X rj,k . (9) k=1 Due to the inter-user interference within the overlapping areas of macrocell and small cell beamforming group, we use typical zero-forcing beamforming technique with equal-power allocation for each user transmission link to eliminate the interference significantly [33], [34], the capacity of the wireless backhaul downlink for small cell j is   N−B+1 Cj = βlog2 1 + γj . B (10) The downlink wireless backhaul constraint requires Rj ≤ Cj (11) such that the downlink traffic of the jth small cell can be accommodated by its wireless backhaul. • Heterogeneous QoS guarantee: The QoS requirement Rt should be guaranteed for each user in each small cell to maintain the performance of the communication system rj,k ≥ Rt . (12) Our target is to maximize the energy efficiency of power allocation and unified bandwidth allocation for wireless backhauling in heterogeneous small cell networks under power constraint and data rate requirements. Thus, the corresponding problem for the downlink can be formulated as the following nonlinear programming problem max U(β, P) = max β,P β,pj,k J X K X Uj,k (β, pj,k ) (13) j=1 k=1 s.t. C1 : 0 ≤ pj,k ≤ Pmax C2 : Rj ≤ Cj (14) C3 : rj,k ≥ Rt C4 : 0 ≤ β ≤ 1. October 10, 2017 DRAFT 9 III. E NERGY- EFFICIENT R ESOURCE A LLOCATION AND BACKHAULING Since the optimization problem formulated in (13) and (14) is non-convex and we notice that the continuous variables β and pj,k are separable in (13). Therefore, we consider a decomposition approach to solve the energy-efficient resource allocation problem. We decompose the nonconvex optimization problem into two convex subproblems: one for energy-efficient power allocation and one for energy-efficient wireless backhaul bandwidth allocation. Then, we solve the subproblems of energy-efficient power allocation and energy-efficient backhaul bandwidth allocation individually. A. Energy-Efficient Power Allocation The concept of quasiconcavity will be used in our discussion and is defined in [35]. Definition 1. A function f , which maps from a convex set of real n-dimensional vectors, D, to a real number, is called strictly quasiconcave if for any x1 , x2 ∈ D and x1 6= x2 , f (λx1 + (1 − λ)x2 ) > min{f (x1 ), f (x2 )} (15) for any 0 < λ < 1. Given a value β for unified wireless backhaul bandwidth allocation, the optimization algorithm begins with the power allocation subproblem P1.1 that is formulated as P1.1 : max U(P) = max pj,k P J X K X Uj,k (pj,k ) (16) j=1 k=1 s.t. C1 : 0 ≤ pj,k ≤ Pmax C2 : Rj ≤ Cj (17) C3 : rj,k (pj,k ) ≥ Rt where rj,k (pj,k ) =  1−β K    pj,k gj,k log2 1 + 2 σ +Ij,k (18) is strictly concave and monotonically increasing in pj,k with rj,k (0) = 0, when pj,k = 0. October 10, 2017 DRAFT 10 The optimal energy-efficient power allocation achieves the maximum energy efficiency, i.e. P∗ = arg max U(P). (19) P It is proved in Appendix A that U(P) has the following properties. Lemma 1. If rj,k (pj,k ) is strictly concave in pj,k , Uj,k (pj,k ) ∈ U(P) is strictly quasiconcave. Furthermore, Uj,k (pj,k ) is first strictly increasing and then strictly decreasing in any pj,k , i.e. the local maximum of U(P) for each pj,k exists at a positive finite value. For strictly quasiconcave functions, if a local maximum exists, it is also globally optimal [35]. Hence, a unique globally optimal transmission rate vector always exists and its characteristics are summarized in Theorem 1 according to the proofs in Appendix A. Theorem 1. If rj,k (pj,k ) is strictly concave, there exists a unique globally optimal transmission power vector P∗ = {p∗ j,k ; (j, k) ∈ J × K} for P∗ = arg max U(P), for each element in P∗ , P p∗j,k = arg max Uj,k (pj,k ) where p∗j,k is given by pj,k ∂Uj,k (pj,k ) ∂pj,k pj,k =p∗j,k i.e., Uj,k (p∗j,k ) = = 0, f (pj,k ) = 0, rj,k (p∗j,k ) PC +p∗j,k = ∂rj,k (pj,k ) ∂pj,k pj,k =p∗j,k . In order to solve the problem P1.1 for power allocation, we rewrite the objective function in (16) as max Uj,k (pj,k ) = max pj,k pj,k rj,k (pj,k ) . PC + pj,k (20) If each small cell user could reach the maximum energy efficiency, all small cell users could reach the maximum energy efficiency. The total data rate in each small cell could not exceed the capacity of the wireless backhaul downlink for small cell j, Rj ≤ Cj , we can approximate that the data rate for each user to be less than for user k in small cell j is PS . K Cj , K rj,k (pj,k ) ≤ and the maximum of power Thus, P1.1 is equivalent to P1.2 : max Uj,k (pj,k ) pj,k October 10, 2017 Cj , K (21) DRAFT 11 PS K Cj C2 : rj,k (pj,k ) ≤ K s.t. C1 : 0 ≤ pj,k ≤ (22) C3 : rj,k (pj,k ) ≥ Rt . We can rewrite C2 in (22) according to (4) as    2   P0 Gj β σ +Ij,k log2 1+ N−B+1 ( 2 1−β ) B σ 2 −1 . pj,k ≤ gj,k We can rewrite C3 in (22) according to (4) as  2   t σ +Ij,k  KR pj,k ≥ 2 1−β −1 . gj,k (23) (24) Therefore, we have Lj,k ≤ pj,k ≤ Hj,k (25) where   σ 2 +Ij,k  KR t 1−β −1 Lj,k = 2 gj,k  2      β N−B+1 P0 Gj σ +Ij,k log 1+ ) ( 2 2 B σ = min 2 1−β −1 , Pmax gj,k  Hj,k (26) (27) only if the following inequality is satisfied Lj,k ≤ Hj,k . (28) The energy-efficient power allocation is given by p̂∗j,k = arg max pj,k rj,k (pj,k ) PC + pj,k (29) subject to Lj,k ≤ pj,k ≤ Hj,k . (30) We can solve (20) by using Theorem 1 to find the optimal power allocation solution. We can also use the low-complexity iterative algorithms based on the GABS algorithm proposed in [23] to realize the energy-efficient power allocation for the kth user in the jth small cell BS as follows. If the output p̂∗j,k satisfies the power constraint, i.e. p̂∗j,k =p∗j,k ; otherwise, we can obtain the maximum Uj,k (pj,k ) by p∗j,k = Lj,k October 10, 2017 (31) DRAFT 12 Algorithm GABS Algorithm 1: Initialization: Each small cell BS allocates the same transmit power to each user, pj,k > 0. ∂Uj,k (pj,k ) (1) 2: Then do pj,k = pj,k , h1 ← (1) and c > 1 (let c = 2). ∂pj,k pj,k =pj,k 3: if h1 < 0 then 4: repeat 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19: 20: 21: (2) (1) (1) p (1) ∂U (p ) j,k , and h1 ← j,k pj,k ← pj,k , pj,k ← j,k (1) c ∂pj,k pj,k =pj,k until h1 ≥ 0 else ∂U (pj,k ) (2) (1) pj,k ← pj,k × c and h2 ← j,k (2) ∂pj,k pj,k =pj,k repeat ∂Uj (pj,k ) (1) (2) (2) (2) pj,k ← pj,k , pj,k ← pj,k × c and h2 ← ∂p (2) pj,k =pj,k j,k until h2 ≤ 0 end if while no convergence do p (1) +p (2) p̂∗j,k ← j,k 2 j,k , h′ ← if h′ > 0 then (1) pj,k = p̂∗j,k else (2) pj,k = p̂∗j,k end if end while Output p̂∗j,k . ∂Uj,k (pj,k ) pj,k =p̂∗j,k ∂pj,k if p̂∗j,k < Lj,k , or we can get the maximum Uj,k (pj,k ) by p∗j,k = Hj,k (32) if p̂∗j,k > Hj,k , since Uj,k (pj,k ) is first strictly increasing and then strictly decreasing in any positive finite pj,k . B. Energy-Efficient Wireless Backhaul Bandwidth Allocation Once the optimal solution P∗ = {p∗j,k ; (j, k) ∈ J × K} is obtained for the convex subproblem P1.2 parameterized by β, it can be used in the following subproblem P1.3 for the unified wireless backhaul bandwidth allocation P1.3 : max U(β, P∗ ) = max β October 10, 2017 β J X K X Uj,k (β, p∗j,k ) (33) j=1 k=1 DRAFT 13 s.t. C1 : 0 ≤ β ≤ 1 (34) C2 : Rj (β, P∗) ≤ Cj (β, P∗) C3 : rj,k (β, p∗j,k ) ≥ Rt where Rj (β, P∗) is the function value of Rj evaluated at P∗ , Cj (β, P∗) is the function value of Cj evaluated at P∗ . In order to obtain the solution to the original problem in (13) and (14), the two subproblems P1.2 and P1.3 are solved iteratively until convergence. Maximizing the objective function of P1.3 with respect to β is equivalent to maximizing (1 − β) only, since (33) is a monotonically decreasing function of β. Problem P1.3 reduces to a feasibility problem whose solution is the smallest feasible value of β given constraints (34). According to C2, Rj (β, P∗) ≤ Cj (β, P∗), we have   K P p∗ gj,k log2 1 + σ2j,k+Ij,k k=1 β≥    P K P 0 Gj Klog2 1 + N −B+1 + log 2 1+ B σ2 k=1 According to C3, rj,k (β, p∗j,k ) ≥ Rt , we have KRt β ≤ 1−  log2 1 + So we have p∗j,k gj,k σ2 +Ij,k p∗j,k gj,k σ2 +Ij,k . . (36) β = max {φj } where φj = K P k=1  Klog2 1 +  log2 1 + N −B+1 P0 Gj B σ2 only if Rt satisfies the following condition  + (37) p∗j,k gj,k σ2 +Ij,k K P k=1   log2 1 + (38) p∗j,k gj,k σ2 +Ij,k  Rt ≤ min {ϕj } where (35)     p∗j,k gj,k N −B+1 P0 Gj log2 1 + σ2 +Ij,k log2 1 + B σ2 ϕj =    P . K p∗j,k gj,k N −B+1 P0 Gj Klog2 1 + B + log2 1 + σ2 +Ij,k σ2 (39) (40) k=1 October 10, 2017 DRAFT 14 IV. A LGORITHM D ESIGN According to the analysis of power allocation and wireless backhaul bandwidth allocation discussed above, we propose an iterative optimization algorithm and a suboptimal low-complexity algorithm. A. Iterative Resource Allocation Algorithm The proposed iterative resource allocation algorithm is shown in Algorithm 1. Algorithm 1 Iterative Resource Allocation Algorithm 1: Initialization: Each small cell BS allocates the same transmit power to each user, pj,k > 0 and set l = 1. 2: repeat 3: Backhaul Bandwidth Allocation 4: Compute optimum β according to (37). 5: Macro BS broadcasts the updated wireles backhaul bandwidth allocation factor to all small cell BSs. 6: for each small cell BS do 7: for each small cell user do 8: Power Allocation 9: a) find p̂∗j,k = arg max Uj,k (pj,k ) according to GABS; 10: b) check power constraint; 11: if Lj,k ≤ p̂∗j,k ≤ Hj,k then 12: p∗j,k = p̂∗j,k 13: end if 14: if p̂∗j,k < Lj,k then 15: p∗j,k = Lj,k 16: end if 17: if p̂∗j,k > Hj,k then 18: p∗j,k = Hj,k 19: end if 20: end for 21: end for 22: l = l + 1. 23: until total energy efficiency convergence or l = Lmax In Algorithm 1, each small cell BS calculates φj according to (38) and then sends φj to macro BS. The macro BS chooses the maximum φj to be the optimal bandwidth allocation factor β and broadcasts β to all small cell BSs. October 10, 2017 DRAFT 15 B. Low-Complexity Optimization Algorithm To reduce the complexity of Algorithm 1, we propose a low-complexity optimization algorithm where bandwidth allocation factor is calculated from the equal power allocation and we fix β to calculate the power allocation according to the scheme proposed in Section IV. This lowcomplexity optimization algorithm is shown in Algorithm 2. Algorithm 2 Fixed β and Optimum Power Allocation Algorithm 1: Initialization: Each small cell BS allocates the same transmit power to each user, pj,k > 0. 2: Backhaul Bandwidth Allocation 3: Compute optimum β according to (37). 4: Macro BS broadcasts the wireles backhaul bandwidth allocation factor to all small cell BSs. 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19: 20: for each small cell BS do for each small cell user do Power Allocation a) find p̂∗j,k = arg max Uj,k (pj,k ) according to GABS; b) check power constraint; if Lj,k ≤ p̂∗j,k ≤ Hj,k then p∗j,k = p̂∗j,k end if if p̂∗j,k < Lj,k then p∗j,k = Lj,k end if if p̂∗j,k > Hj,k then p∗j,k = Hj,k end if end for end for C. Complexity Analysis Since the problem formulated in (13) and (14) is not convex, the only way to obtain the optimal solution is to use the method of exhaustion. If we assume that it costs P operations to calculate rj,k and it costs Q operations to calculate Cj , the complexity of checking C2 and C3 in (14) entails KP + K + Q operations and P + 1 operations, respectively. If we assume that it costs S operations to calculate Uj,k , the complexity of obtaining the total energy efficiency of all small cell users entails JKS + (J − 1) (K − 1) operations. The total complexity of getting the value of objective function in (13) under the constraints in (14) entails KP + K + P + 1 + October 10, 2017 DRAFT 16 JKS + (J − 1) (K − 1) operations for specific pj,k and β values. If we assume the value step
JK size for pj,k is a and the value step size for β is b, there are 1b Pmax choices for the values a 
 Pmax JK . of pj,k and β. Therefore, the complexity for the method of exhaustion is O JKS b a In Algorithm 1, the worst-case complexity of calculating bandwidth allocation factor β from (37) entails J operations in each iteration. If we assume that it costs Ω operations in each GABS to search the optimum power allocation without power constraint, then the worst-case complexity of finding the power allocation for every user in each small cell entails JK (Ω+4) operations in each iteration. Suppose the Algorithm 1 needs ∆ iterations to converge, so the total complexity of Algorithm 1 is O (JKΩ∆). Since iteration is not applied in Algorithm 2, the total complexity of Algorithm 2 is O (JKΩ), which is less than that of Algorithm 1. In the simulation, the typical value for ∆ is around 16, the typical value for Ω is less than 500, and the typical values for 1 b and Pmax b are both 100. So the complexities of Algorithm 1 and Algorithm 2 are always less than that of the method of exhaustion. When the number of small cells J and the number of users in each small cell K increase, the complexity of the method of exhaustion increases exponentially, so the complexity of the method of exhaustion is much larger than the complexities of two proposed algorithms. V. S IMULATION R ESULTS Simulation results are given in this section to evaluate the performance of the proposed power allocation and backhaul bandwidth allocation algorithms. In the simulations, it is assumed that small cells are uniformly distributed in the macrocell coverage area, and small cell users are uniformly distributed in the coverage area of their serving small cell. AWGN power σ 2 =3.9811× 10−14 W. The coverage radius of the macrocell is 500 m, and that of a small cell is 10 m. Small cell has a minimal distance of 50 m from the macro BS. The minimal distance between small cell BSs is 40 m. We assume that the channel fading is composed of path loss, shadowing fading, and Rayleigh fading. The pathloss model for small cell users is based on [36]. The lognormal shadowing between small cell BS and small cell users is 10 dB. At the macro BS, we assume that the transmit power is 33 dBm, the antenna array size N = 100 and beamforming group size is B = 20. We consider that all the small cell users have the same QoS requirement. October 10, 2017 DRAFT 17 Figure 2 shows the convergence in terms of the energy efficiency of all small cell users for the proposed Algorithm 1 versus the number of iterations, where J = 5, Rt = 0.01 bps/Hz, Pmax = 20 dBm. It can be observed that the proposed resource allocation algorithm takes nearly 16 iterations to converge to stable solutions. This result, together with the previous analysis, ensures that the proposed Algorithm 1 is applicable in heterogeneous small cell networks. Figure 3 shows the total energy efficiency of all small cell users when the number of users per small cell is increased from 2 to 10, for Algorithm 2 under Pmax = 7 dBm, Pmax = 10 dBm and Pmax = 20 dBm compared with Algorithm 1 under Pmax = 20 dBm. The simulation parameters are set as J = 5, Rt = 0.01 bps/Hz. Fig. 3 shows that the energy efficiency performance of Algorithm 1 is 20% more superior to that of Algorithm 2. It also can be seen from Fig. 3 that the more number of users in small cell is, the better performance is obtained because of the multi-user diversity. Figure 4 shows the total energy efficiency of all small cell users when the number of small cells is increased from 3 to 15, for Algorithm 2 under Pmax = 7 dBm, Pmax = 10 dBm and Pmax = 20 dBm when compared with Algorithm 1 under Pmax = 20 dBm. The simulation parameters are set as K = 5, Rt = 0.01 bps/Hz. Fig. 4 indicates that more number of small cell is, the better performance is obtained. It can also be seen from Fig. 4 that the energy efficiency performance of Algorithm 1 is always better than that of Algorithm 2 and the gap between them becomes larger when the number of small cells increases. The energy efficiency performance of Algorithm 1 is 30% superior to that of Algorithm 2 when the number of small cells is 10. Figure 5 shows the total downlink capacity of all small cell users when the number of users per small cell is increased from 2 to 10, for Algorithm 2 under Pmax = 7 dBm, Pmax = 10 dBm and Pmax = 20 dBm compared with Algorithm 1 under Pmax = 20 dBm. The simulation parameters are set as J = 5, Rt = 0.01 bps/Hz. Fig. 5 shows that the total downlink capacity of Algorithm 1 is more than 3 bps/Hz higher than that of Algorithm 2. It can also be seen from Fig. 5 that the more number of users in small cell is, the better performance is obtained due to the multi-user diversity. The total downlink capacity of Algorithm 1 is 21% higher than that of Algorithm 2 when the number of users in each small cell is over 10. Figure 6 shows the total downlink capacity of all small cell users when the number of small October 10, 2017 DRAFT 18 cells is increased from 3 to 15, for Algorithm 2 under Pmax = 7 dBm, Pmax = 10 dBm and Pmax = 20 dBm compared with Algorithm 1 under Pmax = 20 dBm. The simulation parameters are set as K = 5, Rt = 0.01 bps/Hz. Fig. 6 illustrates that Algorithm 1 is superior to Algorithm 2 in terms of the total downlink capacity and the gap between them becomes larger when the number of small cells increases. The total downlink capacity of Algorithm 1 is 29% larger than that of Algorithm 2 when there are 14 small cells in the heterogeneous network. Figure 7 shows the total energy efficiency of all the small cell users when using Algorithm 2 for power constraint Pmax ranging from 0 dBm to 12.79 dBm where the number of users in each small cell is 3, 4, 5. The simulation parameters are set as J = 5, Rt = 0.01 bps/Hz. Fig. 7 presents that the more users in each small cell are, the higher total energy efficiency can be obtained, which has already been shown in Fig. 3. It can also be seen from Fig. 7 that the larger power constraint is, the better performance is obtained. This is because the larger power constraint leads to the larger region of the optimizing variable. Figure 8 shows the total energy efficiency of all the small cell users when the number of users per small cell is increased from 2 to 10, for different algorithms. Algorithm 1 and Algorithm 2 are the iterative optimization algorithm and the low-complexity optimization algorithm, respectively, which we have proposed in Section IV. Algorithm 3 is an existing energy efficiency optimization algorithm with equal power allocation and Algorithm 4 is an algorithm that uses the optimum power allocation we proposed given a random β to optimize energy efficiency. All the algorithms are under the constraint Pmax = 20 dBm. Fig. 8 indicates that more users in each small cell are, the better performance can be obtained, which has already been shown in Fig. 3. It also can be seen from Fig. 8 that Algorithm 1 has the best performance, and then it follows by Algorithm 2, Algorithm 3 and Algorithm 4. The energy efficiency performance of Algorithm 1 is 30.5% and 56.6% higher than that of Algorithm 3 and Algorithm 4, respectively. Figure 9 shows the total energy efficiency of all the small cell users when the number of small cells is increased from 2 to 5, for the optimal solution and the two proposed algorithms. Since the complexity of the method of exhaustion is high, we only consider the situation with small dimension where there are two users located in each small cell, K = 2. All the algorithms are under the setting of Pmax = 20 dBm and Rt = 0.01 bps/Hz. From Fig. 9, we can observe that the October 10, 2017 DRAFT 19 difference between the optimal solution and Algorithm 1 in terms of energy efficiency is small, which ensures the effectiveness of the proposed algorithms. The energy efficiency performance of the optimal solution is only about 7% and 24% higher than that of Algorithm 1 and Algorithm 2 respectively when the number of small cells is 3. The difference between the optimal solution and the proposed Algorithm 1 is mainly caused by the approximation of C2 in (17). We can also observe that the performance of Algorithm 1 is slightly better than that of the existing algorithm, which is the backhaul bandwidth allocation in conjunction with the resource allocation algorithm in [14]. This phenomenon can be explained as follows. As the QoS requirement of small cell users increases, the more power is required to meet the higher QoS requirement. VI. C ONCLUSION In this paper, we investigated the energy-efficient wireless backhaul bandwidth allocation and power allocation in a heterogeneous small cell network. We demonstrated the existence of a unique globally optimal energy efficiency solution and provided an iterative algorithm to obtain this optimum. For the downlink scenario, we first found the near optimal energy-efficient resource allocation approach, and then developed a low-complexity suboptimal algorithm by exploring the inherent structure of the objective function and the feature of energy-efficient design. From the simulation results, we observed that energy efficiency is improved by increasing the number of small cells and the number of users per small cell, and the capacity is also improved by increasing the number of small cells and the number of users per small cell. Simulation results showed great energy efficiency improvement of the proposed iterative optimization algorithm than that of the low-complexity optimization algorithm and the existing schemes. The proposed low-complexity algorithms can achieve a promising tradeoff between performance and complexity. If the future, we will investigate the nonunified backhaul bandwidth allocation and inter-small-cell interference in heterogeneous small cell networks. A PPENDIX A P ROOF OF L EMMA 1 We first focus on Uj,k (pj,k ) and then we can obtain the properties of U(P). If every user in each small cell can reach the maximum energy efficiency, all small cell users can reach the October 10, 2017 DRAFT 20 maximum energy efficiency. Denote the α–superlevel sets of Uj,k (pj,k ) as Sα = { pj,k ≥ 0| Uj,k (pj,k ) ≥ α} (41) where pj,k is nonnegative. Based on the propositions in [35], Uj,k (pj,k ) is strictly quasiconcave if and only if Sα is strictly convex for any real number α. In this case, when α < 0, no points exist on the contour Uj,k (pj,k ) = α. When α = 0, only pj,k = 0 is on the contour Uj,k (0) = α. Hence, Sα is strictly convex when α ≤ 0. Now, we investigate the case when α > 0. We can rewrite the Sα as Sα = { pj,k ≥ 0| αPC + αpj,k − rj,k (pj,k ) ≤ 0}. Since rj,k (pj,k ) is strictly concave in pj,k , −rj,k (pj,k ) is strictly convex in pj,k ; therefore, Sα is strictly convex. Hence, we have the strict quasiconcavity of Uj,k (pj,k ). Next, we can obtain the partial derivative of Uj,k (pj,k ) with pj,k as f (pj,k ) ∂Uj,k (pj,k ) (PC + pj,k )r ′ j,k (pj,k ) − rj,k (pj,k ) = = 2 ∂pj,k (PC + pj,k ) (PC + pj,k )2 (42) where f (pj,k ) = (PC + pj,k )r ′ j,k (pj,k ) − rj,k (pj,k ), r ′ j,k (pj,k ) is the first partial derivative of rj,k (pj,k ) with respect to pj,k . If p∗j,k exists such that ∂Uj,k (pj,k ) ∂pj,k pj,k =p∗j,k = 0, it is unique, i.e. if there is a p∗j,k such that f (p∗j,k ) = 0. In the following, we investigate the conditions when p∗j,k exists. The derivative of f (pj,k ) is f ′ (pj,k ) = (PC + pj,k )r ′′ j,k (pj,k ) (43) where r ′′ j,k (pj,k ) is the second partial derivative of rj,k (pj,k ) with respect to pj,k . Since rj,k (pj,k ) is strictly concave in pj,k , so r ′′ j,k (pj,k ) < 0, f ′ (pj,k ) < 0. Hence, f (pj,k ) is strictly decreasing. lim f (pj,k ) = lim ((PC + pj,k )r ′ j,k (pj,k ) − rj,k (pj,k )) pj,k →∞ pj,k →∞ (44) ′ ′ = lim (PC r j,k (pj,k ) + pj,k r j,k (pj,k ) − rj,k (pj,k )) pj,k →∞ where r ′ j,k (pj,k ) =  1−β K  gj,k σ 2 + Ij,k  1 ln 2  1 1+ pj,k gj,k σ2 +Ij,k ! (45) and lim r ′ j,k (pj,k ) = 0. pj,k →∞ October 10, 2017 (46) DRAFT 21 So we have lim PC r ′ j,k (pj,k ) = 0. (47) pj,k →∞ According to the L’Hopital’s rule, it is easy to show that !     g 1 p 1 − β j,k j,k lim pj,k r ′j,k (pj,k ) = lim p g pj,k →∞ pj,k →∞ K σ 2 + Ij,k ln 2 1 + σ2j,k+Ij,k j,k !     1−β gj,k 1 1 = lim gj,k 2 pj,k →∞ K σ + Ij,k ln 2 σ2 +Ij,k    1 1−β = lim pj,k →∞ K ln 2      1−β pj,k gj,k lim (−rj,k (pj,k )) = lim − log2 1 + 2 = −∞. pj,k →∞ pj,k →∞ K σ +Ij,k (48) (49) So we have lim f (pj,k ) < 0. (50) pj,k →∞ Besides, lim f (pj,k ) = lim ((PC + pj,k )r ′ j,k (pj,k ) − rj,k (pj,k )) pj,k →0 pj,k →0 (51) (0) PC r ′ j,k (pj,k ) − 1−β K  gj,k 2 σ + Ij,k  1−β K  gj,k σ 2 + Ij,k  = (0) rj,k (pj,k ) (0) where pj,k denotes pj,k = 0 r ′ j,k (pj,k ) =  =  (0) (0) 1 ln 2  1 ln 2  1 1+ pj,k gj,k σ2 +Ij,k ! rj,k (pj,k ) = 0     1−β PC gj,k 1 lim f (pj,k ) = > 0. pj,k →0 K σ 2 + Ij,k ln 2 So, together with pj,k =0 (52) (53) (54) lim f (pj,k ) < 0, we see that p∗ j,k exists and Uj,k (pj,k ) is first strictly pj,k →∞ increasing and then strictly decreasing in pj,k . Lemma 1 is readily obtained. October 10, 2017  DRAFT 22 R EFERENCES [1] C. Jiang, H. Zhang, Y. Ren, and H.-H. Chen, “Energy-efficient non-cooperative cognitive radio networks: Micro, meso and macro views,” IEEE Commun. Mag., vol. 52, no. 7, pp. 14–20, July 2014. [2] F. R. Yu, X. Zhang, and V. C.M. Leung, Green Communications and Networking., CRC Press, 2012. [3] C. Xu, M. Sheng, C. Yang, X. Wang, and L. 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October 10, 2017 DRAFT 24 ... small cell base station small cell small cell user small cell macro cell user macrocell base station small cell small cell macrocell Fig. 1. Topology of a heterogeneous small cell network. October 10, 2017 DRAFT 25 Energy efficiency of all small cell users (bits/Hz/Joule) 200 180 160 140 120 100 Pmax=20 dBm, K=5 60 40 20 0 0 Fig. 2. Pmax=20 dBm, K=7 80 2 4 6 8 10 12 Iteration index 14 16 18 20 The convergence in terms of energy efficiency of all small cell users over the number of iterations. Energy efficiency of all small cell users (bits/Hz/Joule) 185 180 175 170 Algorithm 1, Pmax=20 dBm 165 Algorithm 2, Pmax=20 dBm Algorithm 2, Pmax=7 dBm 155 150 145 140 135 130 2 Fig. 3. Algorithm 2, Pmax=10 dBm 160 3 4 5 6 7 Number of users per small cell 8 9 10 Energy efficiency versus the number of users per small cell. October 10, 2017 DRAFT 26 500 Energy efficiency of all small cell users (bits/Hz/Joule) Algorithm 1, P max 450 400 Algorithm 2, Pmax=20 dBm Algorithm 2, Pmax=10 dBm Algorithm 2, Pmax=7 dBm 350 300 250 200 150 100 50 2 Fig. 4. =20 dBm 4 6 8 10 Number of small cells 12 14 16 Energy efficiency versus the number of small cells. Total downlink capacity of all small cell users (bps/Hz) 20 19 Algorithm 1, Pmax=20 dBm Algorithm 2, P max =10 dBm Algorithm 2, Pmax=7 dBm 17 16 15 14 2 Fig. 5. Algorithm 2, Pmax=20 dBm 18 3 4 5 6 7 Number of users per small cell 8 9 10 Capacity versus the number of users per small cell. October 10, 2017 DRAFT 27 Total downlink capacity of all small cell users (bps/Hz) 55 50 Algorithm 1, P max Algorithm 2, Pmax=20 dBm 45 Algorithm 2, Pmax=10 dBm 40 Algorithm 2, Pmax=7 dBm 35 30 25 20 15 10 5 0 2 Fig. 6. =20 dBm 4 6 8 10 Number of small cells 12 14 16 Capacity versus the number of small cells. Energy efficiency of all small cell users (bits/Hz/Joule) 150 145 140 135 125 120 115 0 Fig. 7. Algorithm 2, K=5 Algorithm 2, K=4 Algorithm 2, K=3 130 0.002 0.004 0.006 0.008 0.01 0.012 0.014 Power constraint Pmax (W) 0.016 0.018 0.02 Energy efficiency versus the power constraint. October 10, 2017 DRAFT 28 Energy efficiency of all small cell users (bits/Hz/Joule) 180 170 160 150 140 130 120 110 2 Fig. 8. Algorithm 1 Algorithm 2 Algorithm 3 Algorithm 4 3 4 5 6 7 Number of users per small cell 8 9 10 Energy efficiency comparison for different algorithms. October 10, 2017 DRAFT 29 Energy Efficiency of all small cell users (bits/Hz/Joule) 240 220 Algorithm 1, Pmax=20 dBm Algorithm 2, Pmax=20 dBm 200 Existing Algorithm, Pmax=20 dBm 180 160 140 120 100 80 60 Fig. 9. Optimal solution, Pmax=20 dBm 2 2.5 3 3.5 4 4.5 Number of small cells 5 5.5 6 Energy efficiency comparison for the optimal solution and proposed algorithms. October 10, 2017 DRAFT 

KAZHDAN CONSTANTS, CONTINUOUS PROBABILITY MEASURES WITH LARGE FOURIER COEFFICIENTS AND RIGIDITY SEQUENCES by arXiv:1804.01369v1 [math.DS] 4 Apr 2018 Catalin Badea & Sophie Grivaux To the memory of Jean-Pierre Kahane (1926-2017) Abstract. — Exploiting a construction of rigidity sequences for weakly mixing dynamical systems by Fayad and Thouvenot, we show that for every integers p1 , . . . , pr there exists a continuous probability measure µ on the unit circle T such that inf k1 ≥0,...,kr ≥0 |b µ(pk1 1 . . . pkr r )| > 0. ′ This results applies in particular to the Furstenberg set F = {2k 3k ; k ≥ 0, k′ ≥ 0}, and disproves a 1988 conjecture of Lyons inspired by Furstenberg’s famous ×2-×3 conjecture. We also estimate the modified Kazhdan constant of F and obtain general results on rigidity sequences which allow us to retrieve essentially all known examples of such sequences. 1. Introduction Denote by T the unit circle T = {λ ∈ C ; |λ| = 1}, by M(T) the set of (finite) complex Borel measures on T and by P(T) the set of Borel probability measures on T. The Fourier coefficients of µ ∈ M(T) are defined here as Z λn dµ(λ). µ̂(n) = T A measure µ ∈ P(T) is said to be continuous, or atomless, if µ({λ}) = 0 for every λ ∈ T. We denote the set of continuous probability measures on T by Pc (T). According to a theorem of Wiener and the Koopman-von Neumann lemma, µ is continuous if and only if µ̂(n) tends to zero as n tends to infinity along a sequence in N of density one. For 2000 Mathematics Subject Classification. — 43A25, 37A05, 37A25. Key words and phrases. — Fourier coefficients of continuous measures; non-lacunary semigroups of integers; Furstenberg Conjecture; rigidity sequences for weakly mixing dynamical systems; Kazhdan subsets of Z. This work was supported in part by the project FRONT of the French National Research Agency (grant ANR-17-CE40-0021) and by the Labex CEMPI (ANR-11-LABX-0007-01). We are grateful to Étienne Matheron for pointing out a simplification of our original proof of Theorem 2.4, and to Étienne Matheron, Martine Queffélec, Jean-Paul Thouvenot and Benjy Weiss for several interesting discussions. 2 CATALIN BADEA & SOPHIE GRIVAUX every µ ∈ P(T), we define µ e by setting µ e(A) = µ(Ac ) for every Borel set A ⊆ T, with e has the property that ν̂(n) = |µ̂(n)|2 ≥ 0 for every Ac = {λ ; λ ∈ A}. Then ν := µ ∗ µ n ∈ Z, and ν belongs to Pc (T) as soon as µ does. A conjecture of Russell Lyons. — Our aim in this paper is to study some nonlacunary sets of positive integers from a Fourier analysis point of view, and to construct some probability measures which have large Fourier coefficients on such sets. In particular, we disprove a 1988 conjecture of Lyons [31], called there Conjecture (C4), which reads as follows: Lyons’ Conjecture (C4): If S is a non-lacunary semigroup of integers, and if µ ∈ Pc (T), there exists an infinite sequence (nk )k≥1 of elements of S such that / +∞. / 0 as k µ b(nk ) This conjecture of Lyons is inspired by Furstenberg’s famous conjecture concerning common invariant probability measures for two commuting automorphisms Tp : λ ✤ / λp and Tq : λ ✤ / λq of the unit circle T when p and q are two multiplicatively independent integers (i.e. p and q are not both powers of the same integer). In this setting, Furstenberg’s conjecture states that the only continuous probability measure on T invariant by both Tp and Tq is the Lebesgue measure on T. Furstenberg himself proved [22] that if S is any non-lacunary semigroup of integers (i.e. if S is not contained in any semigroup of the form {an ; n ≥ 0}, a ≥ 2), the only infinite closed S-invariant subset of T is T itself. See [10] for an elementary proof of this result and the references mentioned in [15, Chapter 2] for ′ several extensions. Since S = {pk q k ; k, k ′ ≥ 0} is a non-lacunary semigroup whenever p and q are multiplicatively independent, the only infinite closed subset of T which is simultaneously Tp -invariant and Tq -invariant is T. Starting with the work of Lyons in [31], Furstenberg’s conjecture has given rise to an impressive amount of related questions and results, concerning in particular the dynamics of commuting group automorphisms. We refer the reader to the papers [35], [14] or [25] for example, as well as to the texts [29], [26] or [36] for surveys of results related to this conjecture, as well as for perspectives. As written in [31], conjecture (C4) is a natural version of Furstenberg’s conjecture about measures, but not involving invariance. If (C4) were true, it would obviously imply an affirmative answer to the Furstenberg conjecture. Kazhdan sets and modified Kazhdan constants. — It turns out that Lyons’ conjecture is related to an important property of subsets of Z, namely that of being or not a Kazhdan subset of Z. Kazhdan subsets Q of a second-countable topological group G are those for which there exists ε > 0 such that any strongly continuous representation π of G on a complex separable Hilbert space H admitting a vector x ∈ H with ||x|| = 1 which is ε-invariant on Q (i.e. supg∈Q ||π(g)x − x|| < ε) has a G-invariant vector. Such an ε is called a Kazhdan constant for Q, and the supremum of all ε’s with this property is the Kazhdan constant of Q. Groups with Property (T), also called Kazhdan groups, are those which admit a compact Kazhdan set. See the book [7] for more on Property (T) and its numerous important applications. As suggested in [7, Sec. 7.12], it is of interest to study Kazhdan sets in groups which do not have Property (T), such as locally compact abelian groups, Heisenberg groups, etc. See [4] and also [17] for a study of such problems. In the case of the group Z, the definition above is equivalent to the following one: CONTINUOUS PROBABILITY MEASURES WITH LARGE FOURIER COEFFICIENTS 3 Definition 1.1. — (Kazhdan sets and constants) A subset Q ⊂ Z is said to be a Kazhdan set if there exists ε > 0 such that any unitary operator U acting on a complex separable Hilbert space H satisfies the following property: if there exists a vector x ∈ H with ||x|| = 1 such that supn∈Q ||U n x − x|| < ε, then there exists a non-zero vector y ∈ H such that U y = y (i.e. 1 is an eigenvalue of U ). We will say in this case that (Q, ε) is a Kazhdan pair. We define the Kazhdan constant of Q as Kaz(Q) = inf inf sup kU q x − xk, U kxk=1 q∈Q where the first infimum is taken over all unitary operators U on H without fixed vectors. It follows from [7, p. 30] that 0 ≤ Kaz(Q) ≤ √ 2 for every Q ⊆ Z. Several characterizations of Kazhdan subsets of Z were obtained in [4] as consequences of results applying to a much wider class of groups; self-contained proofs of these characterizations of Kazhdan subsets of Z, involving only classical tools from harmonic analysis, were obtained in the paper [5]. One of the characterizations of generating Kazhdan sets obtained in [4, Th. 6.1] (see also [5, Th. 4.12]) runs as follows. Recall that Q is said to be generating in the group Z if the smallest subgroup containing Q is Z itself. Theorem 1.2 ([4]). — Let Q be a√generating subset of Z. Then Q is a Kazhdan subset of Z if and only if there exists ε′ ∈ (0, 2] such that (Q, ε′ ) is a modified Kazhdan pair, that is any unitary operator V acting on a complex separable Hilbert space H satisfies the following property: if there exists a vector x ∈ H with ||x|| = 1 such that supn∈Q ||V n x − x|| < ε′ , then V has at least one eigenvalue. We define now the modified Kazhdan constant of Q as g Kaz(Q) = inf inf sup kV q x − xk, V kxk=1 q∈Q where the first infimum is taken this time over unitary operators V on H without eigenvalues (that is, with continuous spectra). Therefore √ g 0 ≤ Kaz(Q) ≤ Kaz(Q) ≤ 2 g and for every Q ⊆ Z, Kaz(Q) = 0 if and only if Kaz(Q) = 0 if and only if Q is a nonKazhdan set. The property of being or not a Kazhdan set can also be expressed in terms of Fourier coefficients of probability measures; see Section 4 for a discussion. The characterization of Kazhdan subsets of Z obtained by the authors in [4] (see also [5]) implies that the generating subsets Q of Z which satisfy the property stated in (C4) (namely that there exists for every µ ∈ Pc (T) an infinite sequence (nk )k≥1 of elements / +∞) are exactly the Kazhdan subsets of Z with / 0 as k of Q such that µ b(nk ) √ √ g modified Kazhdan constant Kaz(Q) = 2. Since 2 is the modified Kazhdan constant √ of Z seen as a subset of itself, 2 is the maximal modified Kazhdan constant, and thus (C4) can be reformulated as: every generating non-lacunary semigroup S of integers is √ a Kazhdan subset of Z with maximal modified Kazhdan constant 2. The relationship between Furstenberg ×2-×3 conjecture and modified Kazhdan constants can be also seen directly from Proposition 4.4 below. CATALIN BADEA & SOPHIE GRIVAUX 4 2. Main results The first main result of this paper is the following: Theorem 2.1. — Let p1 , . . . , pr be positive distinct integers and set E = {pk11 . . . pkr r ; k1 ≥ 0, . . . , kr ≥ 0}. There exists a continuous probability measure µ on T such that inf k1 ≥0,...,kr ≥0 |b µ(pk11 . . . pkr r )| > 0. Equivalently, √ g Kaz(E) < 2. It should be noted that, as conjecture (C4) does not involve invariant measures, we do not assume in Theorem 2.1 that the integers pj are multiplicatively independent. Notice also that the statement of Theorem 2.1 is well-known in the lacunary case: if r = 1 it suffices to consider the classical Riesz product associated to the sequence (pk )k≥0 . In the non-lacunary case, Theorem 2.1 disproves Conjecture (C4), as well as the related conjectures (C5) and (C6) of [31] (which are both stronger than (C4)). It applies in ′ particular to the Furstenberg set F = {2k 3k ; k, k′ ≥ 0} and shows the existence of a measure µ ∈ Pc (T) such that ′ inf µ b(2k 3k ) > 0. ′ k,k ≥0 In view of this result, one may naturally wonder for which values of δ ∈ (0, 1) there exists a measure µ ∈ Pc (T) such that ′ inf µ b(2k 3k ) ≥ δ, ′ k,k ≥0 or, equivalently, whether the Furstenberg set F is a Kazhdan set in Z, and if yes, with which (modified) Kazhdan constant. In this direction, we prove the following result: ′ g ) ≤ 1. More precisely, there Theorem 2.2. — Let F = {2k 3k ; k, k′ ≥ 0}. Then Kaz(F exists for every δ ∈ (0, 1/2) a continuous probability measure µ on T with nonnegative Fourier coefficients such that ′ inf µ b(2k 3k ) > δ. ′ k,k ≥0 Rigidity sequences. — Our strategy for proving Theorem 2.1 is to construct measures µ ∈ Pc (T) whose Fourier coefficients tend to 1 along a substantial part of the set {pk11 . . . pkr r ; k1 ≥ 0, . . . , kr ≥ 0}. In other words, we show that certain large subsets of this set form, when taken in a strictly increasing order, rigidity sequences in the sense of [8] or [18]. Recall that a dynamical system (X, B, m; T ) on a Borel probability space is called rigid if there exists a strictly increasing sequence of integers (nk )k≥1 such / +∞ for every f ∈ L2 (X, B, m), where UT denotes as / 0 as k that ||UTnk f − f || ✤ / f ◦ T associated to T on L2 (X, B, m). Equivalently, usual the Koopman operator f −n / / m(T k A △ A) 0 as k +∞ for every A ∈ B. We say in this case that the system is rigid with respect to the sequence (nk )k≥1 , or that (nk )k≥1 is a rigidity sequence for (X, B, m; T ). The case where the system (X, B, m; T ) is weakly mixing is particularly interesting, and is the object of the works [8] and [18]. A strictly increasing sequence (nk )k≥1 of integers is called a rigidity sequence if there exists a weakly mixing system which is rigid with respect to (nk )k≥1 . CONTINUOUS PROBABILITY MEASURES WITH LARGE FOURIER COEFFICIENTS 5 Using Gaussian dynamical systems, one can show that (nk )k≥1 is a rigidity sequence / +∞. / 1 as k if and only if there exists a measure µ ∈ Pc (T) such that µ b(nk ) The study of rigidity sequences was initiated in [8] and [18]. Further works on this topic include the papers [1], [3], [2], [23], [21] and [20] among others. The paper [21] by Fayad and Thouvenot is especially relevant here: the authors re-obtain a result of Adams [3], showing that whenever (nk )k≥1 is a rigidity sequence for an ergodic rotation on the circle, it is a rigidity sequence for a weakly mixing system. The proof of this result in [3] relies on an involved construction of a suitable weakly mixing system by cutting and stacking, while the authors of [21] proceed by a direct construction of suitable continuous / 1 for some λ = e2iπθ ∈ T with θ ∈ R \ Q, probability measures: they show that if λnk / 1. We obtain the following theorem, which there exists µ ∈ Pc (T) such that µ b(nk ) generalizes the result of Fayad and Thouvenot: Theorem 2.3. — Let (nk )k≥0 be a strictly increasing sequence of integers. Suppose that the set / 1 as k / +∞ } C = {λ ∈ T ; λnk is dense in T. Then (nk )k≥1 is a rigidity sequence, and there exists a continuous probability / 1 as k / +∞. measure µ on T such that µ b(nk ) This result allows us to retrieve essentially all known examples of rigidity sequences (a notable exception being the examples of [20]). Notice that C, like every subgroup of the circle group, is dense in T as soon as it is infinite. We deduce from Theorem 2.3 the following two-dimensional statement, which is asymmetric and involves a uniformity assumption. Theorem 2.4. — Let (mk )k≥0 and (nk′ )k′ ≥0 be two strictly increasing sequences of inte/ +∞ as k / +∞, and set / N be such that ψ(k) gers. Let also ψ : N Dψ = {(k, k′ ) ∈ N2 ; 0 ≤ k′ ≤ ψ(k)}. Suppose that the set Cψ′ = {λ ∈ T ; λmk nk′ / 1 as k / +∞ , (k, k ′ ) ∈ Dψ } is dense in T. /1 Then there exists a continuous probability measure µ on T such that µ b(mk nk′ ) ′ / +∞ with (k, k ) ∈ Dψ . In other words, there exists for every ε > 0 an integer as k k0 such that |b µ(mk nk′ ) − 1| < ε for every (k, k′ ) ∈ N2 with k ≥ k0 and 0 ≤ k′ ≤ ψ(k). Remark that the assumption of Theorem 2.4 is in particular satisfied if the set C ′ = {λ ∈ T ; λmk nk′ / 1 as k / +∞ uniformly in k ′ } is dense in T. Theorem 2.1 is obtained by first observing that the set {pk11 . . . pkr r ; p1 ≥ 0, . . . , pr ≥ 0} can be split into r sets to which Theorem 2.4 applies, and then considering a convex combination of the continuous measures obtained in this way. In order to prove Theorem 2.2, one needs to refine the statement of Theorem 2.4, and to show that the sequences (mk nk′ )k,k′ ≥0 satisfying the assumption of Theorem 2.4 actually give rise to non-Kazhdan subsets of Z. More precisely, we prove the following strengthenings of Theorems 2.3 and 2.4 respectively: CATALIN BADEA & SOPHIE GRIVAUX 6 Theorem 2.5. — Under the assumption of Theorem 2.3, there exists for every ε > 0 a / +∞ and sup / 1 as k µ(nk ) − 1| < ε. In measure µ ∈ Pc (T) such that µ b(nk ) k≥0 |b particular, {nk ; k ≥ 0} is a non-Kazhdan subset of Z. Theorem 2.6. — Under the assumption of Theorem 2.4, there exists for every ε > 0 a / 1 as k / +∞ with (k, k ′ ) ∈ Dψ , and measure µ ∈ Pc (T) such that µ b(mk nk′ ) sup k≥0, 0≤k ′ ≤ψ(k) |b µ(mk nk′ ) − 1| < ε. In particular, {mk nk′ ; k ≥ 0, 0 ≤ k′ ≤ ψ(k)} is a non-Kazhdan subset of Z. Organization of the paper. — The paper is structured as follows. We give in Section 3 the proof of Theorems 2.3 and 2.4, which are very much inspired from the paper [21]. In Section 4, we recall a characterization of generating Kazhdan subsets of Z from [4], and detail the links between several natural constants involved in this characterization. We explain in particular why the generating subsets of Z which satisfy the property √ stated in (C4) are exactly the Kazhdan subsets of Z with modified Kazhdan constant 2. We then prove Theorems 2.5 and 2.6. Section 5 is devoted to applications: we prove Theorems 2.1 and 2.2, and show how to retrieve many examples of rigidity sequences, using Theorems 2.3 and 2.4. We also provide an application of Theorem 2.2 to the study of the size of the exceptional set of values θ ∈ R for which the sequence (nk θ)k≥0 is not almost uniformly distributed modulo 1 with respect to a (finite) complex Borel measure ν ∈ M(T), where (nk )k≥0 denotes the Furstenberg sequence: we show that this exceptional set is uncountable, thus providing a new example of a sublacunary sequence with uncountable exceptional set for (almost) uniform distribution. 3. Proof of Theorems 2.3 and 2.4 Given two integers a < b, we will when the context is clear denote by [a, b] the set of integers k such that a ≤ k ≤ b. Proof of Theorem 2.3. — The general strategy of the proof is the following: we construct a sequence (λi )i≥1 of pairwise distinct elements of C, as well as a strictly increasing sequence of integers (Np )p≥0 , such that the measures p −p µp = 2 2 X i=1 δ{λi } , p≥0 satisfy certain properties stated below. At step p, we determine the elements λi for 2p−1 < i ≤ 2p as well as the integer Np in such a way that λ1 = 1, N0 = 0, and (1) for every p ≥ 1, every j ∈ [0, p − 1] and every k ∈ [Nj , Nj+1 ], Z |λnk − 1| dµp (λ) < 2−(j−1) ; T (2) for every p ≥ 1, every q ∈ [0, p − 1], l ∈ [1, 2p−q ), r ∈ [1, 2q ], |λl2q +r − λr | < ηq where ηq = 1 inf 1≤i<j≤2q |λi − λj | for every q ≥ 1, and η0 = 1; 4 CONTINUOUS PROBABILITY MEASURES WITH LARGE FOURIER COEFFICIENTS 7 (3) for every p ≥ 1 and every k ≥ Np , Z |λnk − 1| dµp (λ) < 2−(p+1) . T Remark that property (2) implies that the sequence (λi )i≥1 satisfies (4) for every q ≥ 0, every l ≥ 0, and every r ∈ [1, 2q ], |λl2q +r − λr | < ηq . Suppose that the sequences (λi )i≥1 and (Np )p≥0 have been constructed so as to satisfy (1), (2), and (3) above, and let µ be a w∗ -limit point of the sequence (µp )p≥0 in P(T). Claim 3.1. — We have µ b(nk ) / 1 as k / +∞. Proof. — For every k ≥ 0, denote by jk ≥ 0 the unique integer j such that k ∈ [Nj , Nj+1 ). For every p > jk , we have by (1) Z Z nk −(jk −1) |λnk − 1| dµ(λ) ≤ 2−(jk −1) . |λ − 1| dµp (λ) < 2 so that Z T T / 0, i.e. µ / +∞, /1. / +∞ as k |λnk − 1| dµ(λ) b(nk ) Since jk T Claim 3.2. — The probability measure µ is continuous. Proof. — Fix q ≥ 1, and consider for every r ∈ [1, 2q ] the two arcs Γr and ∆r of T defined by 3 Γr = {λ ∈ T ; |λ − λr | ≤ ηq } and ∆r = {λ ∈ T ; |λ − λr | < ηq }. 2 The 2q arcs ∆r are pairwise disjoint. Indeed, for every r, r ′ ∈ [1, 2q ] with r 6= r ′ , every λ ∈ ∆r and every λ′ ∈ ∆r′ , we have by the definition of ηq that |λ − λ′ | ≥ |λr − λr′ | − 3ηq ≥ 4ηq − 3ηq = ηq > 0. So ∆r and ∆r′ do not intersect. Let us next estimate, for every r ∈ [1, 2q ] and every p ≥ q, the quantity µp (Γr ). We have µp (Γr ) = 2−p × #{i ∈ [1, 2p ] ; λi ∈ Γr }. Every i ∈ [1, 2p ] can be written as i = l2q + s for some l ≥ 0 and s ∈ [1, 2q ]. By (4), λi belongs to Γs . Since the arcs ∆r′ , r ′ ∈ [1, 2q ], are pairwise disjoint, it follows that Also, µp (∆r ) = µp (Γr ) = 2−p × #{i ∈ [1, 2p ] ; i ≡ r mod 2q } ≤ 2−q . µp 2q [ r=1  Γr = 1. Since the arcs Γr are closed while the arcs ∆r are open, going to the limit as p goes to infinity yields that µ(∆r ) ≤ 2−q for every r ∈ [1, 2q ] and µ 2q [ r=1  Γr = 1. If λ ∈ T is such that µ({λ}) > 0, there exists an r ∈ [1, 2q ] such that λ ∈ Γr ⊂ ∆r . So µ({λ}) ≤ µ(∆r ) ≤ 2−q , a contradiction if q is sufficiently large. It follows that the measure µ is continuous. CATALIN BADEA & SOPHIE GRIVAUX 8 By Claims 3.1 and 3.2, it suffices to construct (λi )i≥0 and (Np )p≥0 satisfying (1), (2), and (3) in order to prove Theorem 2.3. For p = 0, we set λ1 = 1 and N0 = 0, so that µ0 = δ{1} . For p = 1, we choose λ2 ∈ C distinct from λ1 with |λ2 − λ1 | < 1. Since µ1 = 12 (δ{1} + δ{λ2 } ), we have Z 1 |λnk − 1| dµ1 (λ) = |λn2 k − 1| ≤ 1 < 2 for every k ≥ 0. 2 T Hence property (1) is satisfied whatever the choice of N1 . Since η0 = 1 and |λ2 − λ1 | < 1, property (2) is satisfied. It remains to choose N1 in such a way that property (3) is satisfied. Since λ2 belongs to C, Z 1 / 0 as k / +∞, |λnk − 1| dµ1 (λ) = |λn2 k − 1| 2 T so we can choose N1 so large that Z |λnk − 1| dµ1 (λ) < 2−2 for every k ≥ N1 . T This terminates the construction for p = 1. Suppose now that the construction has been carried out until step p, i.e. that the quantities λi , i ∈ [1, 2p ], and Nl , l ∈ [0, p], have been constructed satisfying (1), (2), and (3). We construct by induction on s ∈ [1, 2p ] elements λ2p +s of C, measures µp,s ∈ P(T), and integers Np,s in such a way that the elements λi , i ∈ [1, 2p+1 ], are all distinct, Np < Np,1 < · · · < Np,2p , and the following three properties are satisfied: (a) for every j ∈ [0, p − 1] and every k ∈ [Nj , Nj+1 ], Z |λnk − 1| dµp,s (λ) < 2−(j−1) ; T (b) for every k ≥ Np , (c) for every k ≥ Np,s, Z Z T |λnk − 1| dµp,s (λ) < 2−(p−1) ; T |λnk − 1| dµp,s (λ) < 2−(p+2) . Let us start with the construction of λ2p +1 . By density of C, one can choose λ2p +1 distinct from all the elements λi , i ∈ [1, 2p ], with |λ2p +1 − λ1 | arbitrarily small. Consider the measure
µp,1 = µp + 2−(p+1) δ{λ2p +1 } − δ{λ1 } −(p+1) =2
p −p δ{λ1 } + δ{λ2p +1 } + 2 2 X δ{λi } i=2 obtained by splitting the point mass δ{λ1 } appearing in the expression of µp into 12 (δ{λ1 } + δ{λ2p +1 } ). We have for every k ≥ 0 Z Z nk |λnk − 1| dµp (λ) + 2−(p+1) |λn2pk+1 − λn1 k |. |λ − 1| dµp,1 (λ) ≤ (5) T T CONTINUOUS PROBABILITY MEASURES WITH LARGE FOURIER COEFFICIENTS 9 If |λ2p +1 − λ1 | is sufficiently small, we have by (1) that Z |λnk − 1| dµp,1 (λ) < 2−(j−1) T for every j ∈ [0, p−1] and every k ∈ [Nj , Nj+1 ], i.e. that (1) still holds true for the measure µp,1 . Also (5) and (3) imply that for every k ≥ Np , Z |λnk − 1| dµp,1 (λ) < 2−(p+1) + 2−p < 2−(p−1) , T which is (b). Since all the elements λi , i ∈ [1, 2p + 1], belong to C, there exists Np,1 > Np such that Z |λnk − 1| dµp,1 (λ) < 2−(p+2) for every k ≥ Np,1 . T Properties (a), (b), and (c) are thus satisfied for s = 1. Suppose now that λ2p +s′ , µ2p +s′ , and N2p +s′ have been constructed for s′ < s. Let λ2p +s ∈ C \ {λ1 , . . . , λ2p +s−1 } be very close to λs , and set
µp,s = µp,s−1 + 2−(p+1) δ{λ2p +s } − δ{λs } (this time, the mass point δ{λs } appearing in µp is split as 12 (δ{λs } + δ{λ2p +s } ). Since, for every k ≥ 0, (6) Z T |λ nk − 1| dµp,s (λ) ≤ Z T |λnk − 1| dµp,s−1 (λ) + 2−(p+1) |λn2pk+s − λns k |, the induction assumption implies that (a) holds true provided |λ2p +s − λs | is sufficiently small. As to (b), we have to consider separately the cases Np ≤ k < Np,s−1 and k ≥ Np,s−1 . If |λ2p +s − λs | is sufficiently small, Z |λnk − 1| dµp,s (λ) < 2−(p−1) for every Np ≤ k < Np,s−1 . T By property (c) at step s − 1 and (6), Z |λnk − 1| dµp,s (λ) < 2−(p+2) + 2−p < 2−(p−1) T for every k ≥ Np,s−1 . Hence (b) is satisfied at step s. Property (c) is satisfied if Np,s is chosen sufficiently large since all the elements λi , i ∈ [1, 2p + s], belong to C. Let us now set µp+1 = µp,2p and Np+1 = Np,2p . It remains to check that with these choices of λi , i ∈ [1, 2p+1 ], µp+1 and Np+1 , properties (1), (2), and (3) are satisfied. By (a), property (1) is satisfied for every j ∈ [0, p − 1]. The case where j = p follows from (b). So (1) is true. Property (3) follows immediately from (c), so it only remains to check (2). Fix q ∈ [0, p], l ∈ [1, 2p+1−q ) and r ∈ [1, 2q ]. Consider first the case where q = p. In this case l = 1, and the quantities under consideration have the form |λ2p +r − λr |, with r ∈ [1, 2p ]. One can ensure in the construction that |λ2p +r − λr | < ηp for every r ∈ [1, 2p ] and then (2) holds true for q = p. Suppose then that q ∈ [0, p − 1], and write l as l = l′ + ε2p−q with ε ∈ {0, 1} and l′ ∈ [1, 2p−q ). Then l2q +r = l′ 2q +r+ε2p . Set s = l′ 2q +r. Then 1 ≤ s ≤ (2p−q −1)2q +2q = 2p , i.e. s ∈ [1, 2p ]. We have |λl2q +r − λr | ≤ |λs+ε2p − λs | + |λl′ 2q +r − λr |. CATALIN BADEA & SOPHIE GRIVAUX 10 If ε = 0, the first term is zero; if ε = 1, it is equal to |λ2p +s − λs |, which can be assumed to be as small as we wish in the construction. As to the second term, it is less than ηq by property (2) at step p, since l′ ∈ [1, 2p−q ) and r ∈ [1, 2q ] with q ∈ [0, p − 1]. We can thus ensure that |λl2q +r − λr | < ηq for every q ∈ [0, p], l ∈ [1, 2p+1−q ), and r ∈ [1, 2q ]. This proves that property (2) is satisfied at step p + 1, and this concludes the proof of Theorem 2.3. Theorem 2.4 is now a formal consequence of Theorem 2.3. Proof of Theorem 2.4. — Recall that Dψ = {(k, k′ ) ∈ N2 ; 0 ≤ k′ ≤ ψ(k)} and Cψ′ = {λ ∈ T ; λmk nk′ / 1 as k / +∞ , (k, k ′ ) ∈ Dψ }. Order the set {mk nk′ ; (k, k′ ) ∈ Dψ } as a strictly increasing sequence (pl )l≥1 of integers. Since there exists for every integer k1 ≥ 0 an integer l1 ≥ 0 such that {pl ; l ≥ l1 } ⊆ {mk nk′ ; (k, k′ ) ∈ Dψ , k ≥ k1 }, / +∞. By Theorem 2.3 / 1 as l every element λ ∈ Cψ′ has the property that λpl / 1 as l / +∞. applied to the sequence (pl )l≥1 , there exists µ ∈ Pc (T) such that µ b(pl ) Using this time the fact that there exists for every integer l2 ≥ 0 an integer k2 ≥ 0 such that {mk nk′ ; (k, k′ ) ∈ Dψ , k ≥ k2 } ⊆ {pl ; l ≥ l2 }, we deduce that µ b(mk nk′ ) / 1 as k / +∞ with (k, k ′ ) ∈ Dψ . Remark 3.3. — Suppose that the set C ′ = {λ ∈ T ; λmk nk′ / 1 as k / +∞ uniformly in k ′ } is dense in T. It is natural to wonder whether there exists a measure µ ∈ Pc (T) such that / 1 as k / 1 uniformly in k ′ . The following example shows that it is not µ b(mk nk′ ) k the case: set mk = 2 and nk′ = k′ for every k, k′ ≥ 0. The set C ′ = {λ ∈ T ; λmk nk′ / 1 as k / +∞ uniformly in k ′ } contains all 2k -th roots of 1, and so is dense in T. Suppose that µ ∈ P(T) is such that / 1 as k / +∞ uniformly in k ′ . Then there exists an integer k0 ≥ 1 such that µ b(2k k′ ) k ′ |b µ(2 0 k )| ≥ 1/2 for every k′ ≥ 0. Consider the measure ν = T2k0 (µ). Since νb(n) = µ b(2k0 n) k for every n ∈ Z, ν cannot be continuous. Also, ν({λ0 }) = µ({λ ∈ T ; λ2 0 = λ0 }) for every λ0 ∈ T, and so the measure µ itself cannot be continuous. So the conclusion of Theorem 2.4 seems to be essentially optimal. 4. Some non-Kazhdan subsets of Z 4.1. Kazhdan constants and Fourier coefficients of probability measures. — We begin this section by recalling a characterization of generating Kazhdan subsets of Z, obtained in [4, Th. 6.1] (see also [5, Th. 4.12]) and presenting some facts concerning the (modified) Kazhdan constants of such sets. We state it here in a slightly modified way (condition (ii) is not exactly the same as in [5, Th. 4.12]), and include a discussion concerning the links between the various constants appearing in the equivalent conditions. CONTINUOUS PROBABILITY MEASURES WITH LARGE FOURIER COEFFICIENTS 11 Theorem 4.1. — Let Q be a generating subset of Z. Then Q is a Kazhdan subset of Z if and only if one of the following equivalent assertions holds true: √ (i) there exists ε ∈ (0, 2) such that (Q, ε) is a modified Kazhdan pair. Equivalently, g Kaz(Q) ≥ ε; (ii) there exists γ ∈ (0, 1) such that any measure µ ∈ P(T) with supn∈Q (1 − ℜe µ b(n)) < γ has a discrete part; (iii) there exists δ ∈ (0, 1) such that any measure µ ∈ P(T) with inf n∈Q |b µ(n)| > δ has a discrete part. Moreover: √ – (i) is satisfied for ε ∈ (0, 2) if and only if (ii) is satisfied for γ = ε2 /2; √ – if (ii) is satisfied for γ ∈ (0, 1), (iii) is satisfied for δ = 1 − γ, while if (iii) is satisfied for δ ∈ (0, 1), (ii) is satisfied for γ = 1 − δ; q √ – hence if (i) is satisfied for ε ∈ (0, 2), (iii) is satisfied for δ = 1 − ε2 /2 , while if p (iii) is satisfied for δ ∈ (0, 1), (i) holds true for ε = 2(1 − δ). We prove briefly here the statement concerning the relations between the constants ε, γ, and δ appearing in (i), (ii), and (iii) respectively, following [4] and [5]. √ Proof. — Suppose that (i) is satisfied for ε ∈ (0, 2), and let µ ∈ P(T). Consider the unitary operator U = Mλ of multiplication by λ on L2 (T, µ). Let f be the function constantly equal to 1. Then ||U n f − f ||2 = 2(1 − ℜe µ b(n)). If supn∈Q (1 − ℜe µ b(n)) < ε2 /2, g U has an eigenvalue since Kaz(Q) ≥ ε, and so µ has a discrete part. Conversely, suppose that (ii) is satisfied for γ ∈ (0, 1). Let U be a unitary operator on a separable Hilbert space H, and let x ∈ H with ||x|| = 1 be such that p sup ||U n x − x|| < 2γ. n∈Q The proof of [5, Th. 4.6] shows then that there exists µ ∈ P(T) such that 2 sup (1 − ℜe µ b(n)) = sup ||U n x − x||2 < 2γ. n∈Q n∈Q So supn∈Q (1 − ℜe µ b(n)) < γ. By (ii), µ has a discrete part, and so U has an eigenvalue. √ g Hence Kaz(Q) ≥ 2γ. Suppose next that √ property (ii) is satisfied for γ ∈ (0, 1). Let µ ∈ P(T) be such that inf n∈Q |b µ(n)| > 1 − γ. Set ν = µ ∗ µ e. Then inf n∈Q νb(n) > 1 − γ. It follows that supn∈Q (1 − νb(n)) < γ, and ν has a discrete part. So µ itself has a discrete part. Lastly, suppose that (iii) is satisfied for δ ∈ (0, 1). Let µ ∈ P(T) be a measure satisfying supn∈Q (1−ℜe µ b(n)) < 1 − δ. Then inf n∈Q |b µ(n)| ≥ inf n∈Q ℜe µ b(n) > δ, so µ has a discrete part. Remark 4.2. — Given a subset Q of Z, one can prove, using the spectral theorem for unitary operators, that the following assertions are equivalent (see [5, Th. 4.6]): √ (i’) Q is a Kazhdan subset of Z, i.e. there exists ε ∈ (0, 2) such that (Q, ε) is a Kazhdan pair; (ii’) there exists γ ∈ (0, 1) such that any measure µ ∈ P(T) with supn∈Q (1 − ℜe µ b(n)) < γ is such that µ({1}) > 0. CATALIN BADEA & SOPHIE GRIVAUX 12 √ Moreover (i’) holds true for a certain constant ε ∈ (0, 2) (i.e. Kaz(Q) ≥ ε) if and only if (ii’) holds true for γ = ε2 /2. It is interesting to note that these two conditions (i’) and (ii’) are not equivalent to the natural version (iii’) of (iii) (namely, that there exists δ ∈ (0, 1) such that any measure µ ∈ P(T) with inf n∈Q |b µ(n)| > δ satisfies µ({1}) > 0). Indeed, (iii’) is satisfied for any Dirac mass δ{λ} , λ ∈ T. The proof that (ii) implies (iii) in Theorem 4.1 above uses in a crucial way the fact that if µ ∈ P(T) is such that µ ∗ µ e has a discrete part, µ itself has a discrete part. But µ ∗ µ e may very well satisfy µ ∗ µ e({1}) > 0 while µ({1}) = 0, and so (ii’) does not imply (iii’). Theorem 4.1 is related to Conjecture (C4) in the following way: Corollary 4.3. — Let Q be a generating subset of Z. The following assertions are equivalent: √ g (α) Q is a Kazhdan subset of Z with Kaz(Q) = 2; (β) any measure µ ∈ Pc (T) satisfies inf n∈Q |b µ(n)| = 0; (γ) any measure µ ∈ Pc (T) satisfies lim inf |n|→+∞ |b µ(n)| = 0. n∈Q Proof. — The equivalence between (α) and (β) follows immediately from Theorem 4.1. So only the implication (β)=⇒(γ) requires a proof. Suppose that any µ ∈ Pc (T) satisfies inf n∈Q |b µ(n)| = 0. We want to show that the conclusion can be reinforced into lim inf |n|→+∞ |b µ(n)| = 0. Let ρ ∈ Pc (T) be a Rajchman measure with positive coeffin∈Q cients, that is such that lim|n|→+∞ ρb(n) = 0 and ρb(n) > 0 for every n ∈ Z. Consider the measure ν = (µ ∗ µ e + ρ)/2. It is continuous and satisfies ν(n) > 0 for every n ∈ Z. Since inf n∈Q νb(n) = 0 and ν(n) > 0 for every n ∈ Z, lim inf |n|→+∞ νb(n) = 0. Hence µ(n)|2 = 0, and the conclusion follows. lim inf |n|→+∞ |b n∈Q n∈Q So Conjecture (C4) is equivalent to the statement that any non-lacunary semigroup √ of integers has modified Kazhdan constant 2. We can also estimate the Fourier coefficients of a continuous probability measure on T which is T2 - and T3 -invariant in terms of the modified Kazhdan constant of the Furstenberg set. Notice that Proposition 4.4 is meaningful only if κ̃ > 0. ′ g ). Let µ be a Proposition 4.4. — Let F = {2k 3k ; k, k ′ ≥ 0} and set κ̃ = Kaz(F continuous probability measure on T which is T2 - and T3 -invariant. Then κ̃2 for every j ∈ Z \ {0}. 2 Proof of Proposition 4.4. — Set, for every j ∈ Z \ {0}, µj = Tj µ. Then µj is a continuous ′ measure which satisfies µ̂j (2k 3k ) = µ̂(j) for every k, k′ ≥ 0 It follows that if δ ∈ (0, 1) is such p that (iii) of Theorem 4.1 is satisfied, δ ≥ |µ̂(j)|. Hence, by Theorem 4.1 again, κ̃ ≤ 2(1 − |µ̂(j)|). |µ̂(j)| ≤ 1 − Remark 4.5. — Although a generating subset Q of Z is a Kazhdan set if and only if g Kaz(Q) > 0, there is no link between the Kazhdan constant and the modified Kazhdan constant of Q. Indeed, there exist Kazhdan subsets Q of Z with maximal modified con√ g stant Kaz(Q) = 2 and arbitrarily small Kazhdan constant Kaz(Q). This relies on the following observation, which can be extracted from the proof of [5, Th 7.1] and results from Proposition 5.9 below. CONTINUOUS PROBABILITY MEASURES WITH LARGE FOURIER COEFFICIENTS 13 Proposition 4.6. — Let (nk )k≥0 be a strictly increasing sequence of integers with n0 = 1 such that (nk θ)k≥0 is uniformly distributed modulo 1 for every θ ∈ R \ D, where D is countable subset of R. Then the set Q = {nk ; k ≥ 0} is a Kazhdan subset of Z which √ g satisfies Kaz(Q) = 2. Consider, for every integer p ≥ 2, the set Qp = p N + 1. By Proposition 4.6, Qp is a √ g p ) = 2. But the measure µ = δ 2iπ/p satisfies Kazhdan subset of Z with Kaz(Q {e } Hence Kaz(Qp ) ≤ p sup (1 − ℜe µ b(n)) = cos(2π/p). n∈Qp 2 cos(2π/p), which can be arbitrarily small if p is sufficiently large. 4.2. Proofs of Theorems 2.5 and 2.6. — We only sketch the proof of Theorem 2.5 (Theorem 2.6 is a formal consequence of it). Proof. — Fix ε ∈ (0, 1/2). Using the notation of the proof of Theorem 2.3, it suffices to construct the measures µp , p ≥ 0, in such a way that they satisfy the assertions (1ε ) for every p ≥ 1, every j ∈ [0, p − 1] and every k ∈ [Nj , Nj+1 ], Z |λnk − 1| dµp (λ) < 3ε(1 − ε)j ; T (2) for every p ≥ 1, every q ∈ [0, p − 1], l ∈ [1, 2p−q ), r ∈ [1, 2q ], |λl2q +r − λr | < ηq 1 inf 1≤i<j≤2q |λi − λj | for every q ≥ 1, and η0 = 1; 4 (3ε ) for every p ≥ 1 and every k ≥ Np , Z |λnk − 1| dµp (λ) < ε(1 − ε)p+1 ; where ηq = T (4ε ) µp ({λi }) ≤ (1 − ε)p for every i ∈ [1, 2p ]. Then any w∗ -limit point µ of (µp )p≥0 will be a continuous measure which simultaneously / 1 as k / +∞ and sup satisfies µ b(nk ) µ(nk )−1| ≤ 3ε. The main difference with k≥0 |b the proof of Theorem 2.3 is that the measures µp will be defined as p p µp = 2 X (p) ai δ{λi } for some suitable weights api > 0 with 2 X (p) ai = 1. i=1 i=1 For p = 0, we set λ1 = 1, N0 = 0, and µ0 = δ{1} . For p = 1, we choose λ2 ∈ C \ {λ1 } with |λ2 − λ1 | < 1 and set µ1 = (1 − ε)δ{1} + εδ{λ2 } . We have for every k ≥ 0 Z |λnk − 1| dµ1 (λ) = ε|λn2 k − 1| ≤ 2ε < 3ε. T Since |λ2 − λ1 | < 1, (2) is true. If N1 is chosen sufficiently large, µ1 satisfies properties (1ε ) and (3ε ). Moreover, µ1 ({1}) = 1 − ε and µ2 ({λ2 }) = ε < 1 − ε, so (4ε ) is true. Suppose now that the construction has been carried out until step p. We can then construct by induction on s ∈ [1, 2p ] measures µp,s which satisfy CATALIN BADEA & SOPHIE GRIVAUX 14 (aε ) for every j ∈ [0, p − 1] and every k ∈ [Nj , Nj+1 ], Z |λnk − 1| dµp,s (λ) < 3ε(1 − ε)j ; T (bε ) for every k ≥ Np , (cε ) for every k ≥ Np,s, Z Z T T |λnk − 1| dµp,s (λ) < 3ε(1 − ε)p ; |λnk − 1| dµp,s (λ) < 3ε(1 − ε)p+2 ; (dε ) µp,s ({λi }) = µp ({λi }) for every i ∈ (s, 2p ] and µp,s ({λi }) ≤ (1 − ε)p+1 for every i ∈ [1, s] ∪ [2p + 1, 2p + s]. We define µp,1 as µp,1 = µp + µp ({1}) ε (δ{λ2p +1 } − δ{λ1 } ) p = µp ({1}) (1 − ε) δ{λ1 } + 2 X i=2 µp ({λi }) δ{λi } + µp ({1}) ε δ{λ2p +1 } , where λ2p +1 ∈ C \ {λ1 , . . . , λ2p } is such that |λ2p +1 − λ1 | = |λ2p +1 − 1| is very small. Then for every k ≥ 0, Z Z nk |λnk − 1| dµp (λ) + (1 − ε)p ε |λn2pk+1 − λn1 k |. |λ − 1| dµp,1 (λ) ≤ (7) T T It follows that (aε ) holds true for µp,1 , provided that |λ2p +1 −λ1 | is sufficiently small. Also, we have by (7) and (3ε ) that for every k ≥ Np , Z |λnk − 1| dµp,1 (λ) < ε (1 − ε)p+1 + 2ε (1 − ε)p < 3ε (1 − ε)p T so that (bε ) holds true. If Np,1 is sufficiently large, (cε ) is true. Property (dε ) follows from the expression of µp,1 , since µp,1 ({1}) = µp ({1}) (1 − ε) ≤ (1 − ε)p+1 and µp,1 ({λ2p +1 }) = µp ({1}) ε ≤ ε (1 − ε)p ≤ (1 − ε)p+1 by (4ε ). Supposing now that s ≥ 2 and that the construction has been carried out for every s′ < s, we choose λ2p +s ∈ C \ {λ1 , . . . , λ2p +s−1 } very close to λs , and set µp,s = µp,s−1 + µp,s−1 ({λs }) ε (δ{λ2p +s } − δ{λs } ). Then, since µp,s−1 ({λs }) = µp ({λs }) ≤ (1 − ε)p by (dε ) and (4ε ), we have for every k ≥ 0 Z Z nk |λnk − 1| dµp,s−1 (λ) + (1 − ε)p ε |λn2pk+s − λns k |. |λ − 1| dµp,s (λ) ≤ (8) T T Thus (aε ) holds true provided that |λ2p +s − λs | is sufficiently small. In order to prove (bε ), suppose first that Np ≤ k < Np,s−1 . Then, if |λ2p +s − λs | is small enough, we have by (8) and (bε ) for s − 1 that Z |λnk − 1| dµp,s (λ) < 3ε (1 − ε)p for every Np ≤ k < Np,s−1 . T If k ≥ Np,s−1 we have by (cε ) at step s − 1 and (8) that Z |λnk − 1| dµp,s (λ) < ε (1 − ε)p+2 + 2ε (1 − ε)p < 3ε (1 − ε)p . T CONTINUOUS PROBABILITY MEASURES WITH LARGE FOURIER COEFFICIENTS 15 So (bε ) is satisfied at step s. Property (cε ) is true if Np,s is chosen sufficiently large. As to property (dε ), we have µp,s({λi }) = µp,s−1 ({λi }) for every i distinct from λs and λ2p +s . So µp,s ({λi }) = µp ({λi }) for every i ∈ (s, 2p ], and µp,s ({λi }) ≤ (1 − ε)p+1 for every i ∈ [1, s) ∪ [2p + 1, 2p + s). Also µp,s ({λs }) = µp,s−1 ({λs }) (1 − ε) = µp ({λs }) (1 − ε) ≤ (1 − ε)p+1 , while µp,s({λ2p +s }) = µp,s−1 ({λs }) ε ≤ (1 − ε)p+1 . So (dε ) holds true at step s. This terminates the construction of the measures µp,s . We then set µp+1 = µp,2p and Np+1 = Np,2p and check as in the proof of Theorem 2.3 that properties (1ε ), (2), and (3ε ) are satisfied. Since by (dε ) for s = 2p we have µp,2p ({λi }) ≤ (1 − ε)p+1 for every i ∈ [1, 2p+1 ], property (4ε ) is satisfied as well. This terminates the construction of the measures µp , and proves Theorem 2.5. We plan to come back to the study of the links between Kazhdan sets and rigidity sequences in a forthcoming preprint. As a direct corollary of Theorems 2.5 and 2.6, we obtain Corollary 4.7. — For any function ψ : N the sets ′ {2k 3k ; k ≥ 0, 0 ≤ k′ ≤ ψ(k)} and / N with ψ(k) / +∞ as k / +∞, ′ {2k 3k ; k′ ≥ 0, 0 ≤ k ≤ ψ(k′ )} are non-Kazhdan sets in Z. ′ It seems to be unknown whether the Furstenberg set {2k 3k ; k, k′ ≥ 0} is a Kazhdan set in Z (see Question 5.1 below). 5. Applications 5.1. Proof of Theorem 2.1. — Our first and main application of Theorem 2.4 is Theorem 2.1, which solves in particular Conjecture (C4) and shows that the invariance assumption on the measure is indeed essential in the statement of Furstenberg’s ×2 -×3 conjecture. Proof of Theorem 2.1. — If r = 1, Theorem 2.1 claims the existence, for every integer p ≥ 2, of a measure µ ∈ Pc (T) such that inf k≥0 |b µ(pk )| > 0. As mentioned in Section 2, this statement is well-known: it suffices to consider the classical Riesz product associated to the sequence (pk )k≥0 . One can also show, either as in [8] or [18], or as an application of Theorem 2.3, that (pk )k≥0 is a rigidity sequence, so that there exists µ ∈ Pc (T) with / +∞. / 1 as k µ b(pk ) Suppose now that r ≥ 2, and consider, for every fixed index 1 ≤ j ≤ r, the set −l Cj′ = {e2iπnpj ; n, l ≥ 0} of roots of all powers of pj . It is dense in T, and has the following property: there exists k1 k2 p2 ... pkr r for every λ ∈ Cj′ an integer lj such that λp1 1 ≤ i ≤ r with i 6= j. Hence sup ki ≥0 1≤i≤r, i6=j k1 λp 1 ... pkr r −1 /0 = 1 for every kj ≥ lj and ki ≥ 0, as kj / +∞. CATALIN BADEA & SOPHIE GRIVAUX 16 Consider the two sequences (mk )k≥0 and (nk′ )k′ ≥0 obtained by setting mk = pkj , k ≥ 0, and ordering the set  k1 kj−1 kj+1 p1 . . . pj−1 pj+1 . . . pkr r ; ki ≥ 0, 1 ≤ i ≤ r with i 6= j / N be a strictly increasing as a strictly increasing sequence (nk′ )k′ ≥0 , and let ψ : N function such that  k1 kj−1 kj+1 p1 . . . pj−1 pj+1 . . . pkr r ; 0 ≤ ki ≤ k, 1 ≤ i ≤ r with i 6= r is contained in the set {nk′ ; 0 ≤ k′ ≤ ψ(k)} for every k ≥ 0. By Theorem 2.4, there exists / 1 as kj / +∞ with 0 ≤ ki ≤ kj , bj (pk11 . . . pkr r ) a measure µj ∈ Pc (T) such that µ 1 ≤ i ≤ r with i 6= j. Replacing, for every 1 ≤ j ≤ r, µj by µj ∗ µ ej , we can suppose without loss of generality that µ bj (n) ≥ 0 for every n ∈ Z. Let now ρ ∈ Pc (T) be such that ρ(n) > 0 for every n ∈ Z, and set r  1 X µj + ρ . µ= r+1 j=1 Then µ is a continuous probability measure on T with µ b(n) > 0 for every n ∈ Z. Moreover, we have
1 / +∞. (9) lim inf µ b pk11 pk22 . . . pkr r ≥ as max(k1 , . . . , kr ) r+1 (l) (l) Indeed, if (k1 , . . . , kr )l≥1 is an infinite sequence of elements of Nr , one can extract from (l) (l) it a sequence (still denoted by (k1 , . . . , kr )l≥1 ) with the following property: there exists (l) (l) 1 ≤ j ≤ r such that ki ≤ kj for every 1 ≤ i ≤ r. Then (l) (l) (l)
(l)
1 1 k k lim inf µ b p11 . . . pkr r ≥ bj p11 . . . pkr r = lim inf µ · r + 1 l / +∞ r+1 l / +∞ This yields (9). Since µ b(n) > 0 for every n ≥ 0, it follows that
inf µ b pk11 . . . pkr r > 0, ki ≥0 1≤i≤r and Theorem 2.1 is proved. 5.2. The case of the Furstenberg set. — Theorem 2.1 applies to the Furstenberg ′ set F = {2k 3k ; k, k′ ≥ 0} and shows the existence of a measure µ ∈ Pc (T) such that ′ inf µ b(2k 3k ) > 0 ′ k,k ≥0 (the fact that the measure µ can be supposed to have nonnegative Fourier coefficients can be extracted from the proof of Theorem 2.1, or deduced formally from Theorem 2.1 by √ g considering the measure µ ∗ µ e). By Corollary 4.3, this means that Kaz(F ) < 2. As mentioned in the introduction, it is natural to look for the optimal constant δ ∈ (0, 1) for which there exists a measure µ ∈ Pc (T) such that (10) ′ inf µ b(2k 3k ) ≥ δ ′ k,k ≥0 This is equivalent to asking whether F is a Kazhdan set in Z, and if yes, with which (modified) Kazhdan constant. The best result which can be obtained via the methods CONTINUOUS PROBABILITY MEASURES WITH LARGE FOURIER COEFFICIENTS 17 presented here is that there exists a measure µ ∈ Pc (T) satisfying (10) for every δ ∈ (0, 1/2): this is the content of Theorem 2.2, which we now prove. Proof of Theorem 2.2. — The proof goes along the same lines as that of Theorem 2.1, but it involves Theorem 2.6 instead of Theorem 2.4. Fix δ ∈ (0, 1/2). There exist by Theorem 2.6 two measures µ1 , µ2 ∈ Pc (T) such that √ ′ |b µ1 (2k 3k )| ≥ 2δ for every k ≥ 0 and every 0 ≤ k′ ≤ k and ′ |b µ2 (2k 3k )| ≥ √ 2δ for every k′ ≥ 0 and every 0 ≤ k ≤ k′ . e1 + µ2 ∗ µ e2 ) has nonnegative Fourier coefficients and satisfies The measure µ = 12 (µ1 ∗ µ ′ µ b(2k 3k ) ≥ δ for every k, k′ ≥ 0. ′ It then follows from Theorem 4.1 that if {2k 3kp; k, k′ ≥ 0} is a Kazhdan subset of Z, its modified Kazhdan constant must be less than 2(1 − δ) for every δ ∈ (0, 1/2), so must be at most 1. That the bound 1/2 can be further improved does not seem clear at all, and we do not know whether there exists for every δ ∈ [1/2, 1) a measure µ ∈ Pc (T) such that ′ inf µ b(2k 3k ) ≥ δ. ′ k,k ≥0 ′ Question 5.1. — Is the Furstenberg set {2k 3k ; k, k′ ≥ 0} a Kazhdan set in Z? Note that a lacunary semigroup {an ; n ≥ 0}, a ≥ 2, cannot be a Kazhdan set (see [5, Ex. 5.2]). Along the same lines, one can also ask for which values of δ ∈ (0, 1] there exists a measure ′ / +∞. The proof of Theorem µ ∈ Pc (T) such that lim inf µ b(2k 3k ) ≥ δ as max(k, k′ ) 2.1 allows us to exhibit a measure µ ∈ Pc (T) with nonnegative Fourier coefficients (namely ′ / +∞. Again, we do µ = (µ1 + µ2 )/2) such that lim inf µ b(2k 3k ) ≥ 1/2 as max(k, k′ ) not know whether the constant 1/2 can be improved. The strongest statement which could be expected in this direction is the existence of a measure µ ∈ Pc (T) such that ′ / +∞. This would show that the Furstenberg sequence is / 1 as max(k, k ′ ) µ b(2k 3k ) a rigidity sequence for weakly mixing dynamical systems. This natural question is raised in Remark 3.12 (b) of [8] and we record it anew here: Question 5.2. — Is the Furstenberg sequence a rigidity sequence for weakly mixing dynamical systems? 5.3. Examples of rigidity sequences. — Theorems 2.3 and 2.4 allow us to retrieve directly all known examples of rigidity sequences from [8], [18], [2], [1] and [21]. The only examples of rigidity sequences not covered by our results are those of [20]. Indeed, Fayad and Kanigowski construct in [20] examples of rigidity sequences (nk )k≥0 such that {λnk ; k ≥ 0} is dense in T for every λ = e2iπθ ∈ T with θ ∈ R \ Q, and there exist for every integer p ≥ 2 infinitely many integers k such that p does not divide nk . So such sequences never satisfy the assumption of Theorem 2.3. We briefly list here some of the examples of rigidity sequences which can be obtained from Theorems 2.3 and 2.4. Our first example is that of Fayad and Thouvenot in [21]. 18 CATALIN BADEA & SOPHIE GRIVAUX Example 5.3. — [21] If the sequence (nk )k≥0 is such that there exists λ = e2iπθ ∈ T, / +∞, (nk )k≥0 is a rigidity sequence. / 1 as k with θ ∈ R \ Q, such that λnk / 1 with λ = This result of [21] follows directly from Theorem 2.3. Indeed, if λnk pn / 1 for every p ∈ Z. Since θ is irrational, the set {λp ; p ∈ Z} is θ ∈ R \ Q, λ k dense in T, and Theorem 2.3 applies. e2iπθ , Example 5.4. — [8], [18] If (nk )k≥0 is a strictly increasing sequence of integers such that nk |nk+1 for every k ≥ 0, (nk )k≥0 is a rigidity sequence. Indeed, under the assumption of Example 5.4, the set C = {λ ∈ T ; λnk all nk -th roots of 1, k ≥ 0, and is hence dense in T. Theorem 2.4 shows that Example 5.4 can be improved into / 1} contains Example 5.5. — Let (mk )k≥0 be a strictly increasing sequence of integers such that / N be a strictly increasing function. Order the mk |mk+1 for every k ≥ 0. Let ψ : N ′ ′ set {k mk ; k ≥ 0 and 1 ≤ k ≤ ψ(k)} as a strictly increasing sequence (nk )k≥0 . Then (nk )k≥0 is a rigidity sequence. ′ / +∞ uniformly in k ′ } contains all / 1 as k Indeed, the set C ′ = {λ ∈ T ; λk mk mk -th roots of 1, and is dense in T. So Theorem 2.4 applies. For instance, if (rk )k≥0 is any sequence of positive integers, the sequence (nk )k≥0 obtained by ordering the set {k′ 2k ; k ≥ 0, 1 ≤ k′ ≤ rk } in a strictly increasing sequence is a rigidity sequence. This provides new examples of rigidity sequences (nk )k≥0 such that nk+1 / +∞. / 1 as k nk / +∞ as Example 5.6. — Let (rk )k≥0 be any sequence of positive integers with rk / k +∞. The sequence (nl )l≥0 obtained by ordering in a strictly increasing fashion n / 1 as the set {j2k ; k ≥ 0, 1 ≤ j ≤ rk } is a rigidity sequence which satisfies nl+1 l / +∞. l Proof. — It suffices to show that for every ε > 0 and every l sufficiently large there exists n l′ > l such that nll′ < 1 + ε. – Suppose first that nl = j2k for some k ≥ 0 and some 1/ε < j < rk . Then taking n nl′ = (j + 1)2k , we have nll′ = j+1 j < 1 + ε. k ′ – Suppose next that nl = j2 for some k ≥ 0 and some 1 ≤ j ≤ 1/ε. Fix an integer p such that 2−p < ε. If l is sufficiently large, we have rk−p > 2p /ε. Set j ′ = j2p . Since j ′ ≤ 2p /ε < rk−p , the integer nl′ = (j ′ + 1)2k−p appears in the sequence (nl )l≥0 . Also, since nl′ = (j ′ + 1)2k−p > j2k = nl , we have l′ > l, and nl ′ (j ′ + 1)2k−p (j ′ + 1) −p j + 2−p = = 2 ≤ < 1 + 2−p < 1 + ε. k nl j2 j j – The last case we have to deal with is when nl = rk 2k for some k ≥ 0. Let j ′ ≥ 1 be such that j ′ ≤ rk /2 < j ′ + 1. Then j ′ < rk+1 , and if we set nl′ = (j ′ + 1)2k+1 , the integer nl′ appears in the sequence (nl )l≥0 . We have (j ′ + 1)2k+1 2(j ′ + 1) 2 nl ′ = = ≤1+ <1+ε k nl rk 2 rk rk if k is sufficiently large, and this terminates the proof. CONTINUOUS PROBABILITY MEASURES WITH LARGE FOURIER COEFFICIENTS 19 Example 5.7. — [1] (a) Let (dk )k≥0 be a strictly increasing sequence of positive integers of density zero. There exists a strictly increasing sequence of integers (nk )k≥0 which is a rigidity sequence and satisfies nk ≤ dk for every k ≥ 0. (b) Let (dk )k≥0 be a sequence of real numbers with dk ≥ k for every k ≥ 0 and limk→+∞ dk /k = +∞. There exists a strictly increasing sequence of integers (nk )k≥0 which is a rigidity sequence and satisfies nk ≤ dk for every k ≥ 0. This has been proved by Aaronson in [1, Th. 4]; a simpler construction with the weaker conclusion that nk ≤ dk for infinitely many k was given in [8, Prop. 3.18]. The proof given below uses Theorem 2.3 and a result of Bugeaud [16]. Proof. — As the statement (a) is a simple consequence of (b), we only give the proof of (b). Set g0 = 1 and gk = dk /k for every k ≥ 1. Then (gk )k≥0 is a sequence of reals with gk ≥ 1 for every k ≥ 0 which tends to infinity (notice that for (a) this holds since (dk )k≥0 is a sequence of density zero). Using (a particular case of) [16, Th. 1], we obtain that there exists for every fixed irrational number θ an increasing sequence (nk )k≥0 of positive / 1. It follows from integers such that nk ≤ kgk = dk for every k ≥ 1 and exp(2iπθ)nk Example 5.3 that (nk )k≥0 is a rigidity sequence. Example 5.8. — Let (mk )k≥0 be a strictly increasing sequence of positive integers with / +∞. There exists a strictly increasing sequence of integers (nk )k≥0 which mk+1 − mk is a rigidity sequence and satisfies mk ≤ nk < mk+1 for every k ≥ 0. Proof. — The proof is exactly the same as the preceding one, replacing the result from [16] by [9, Obs. 1.36]. 5.4. Exceptional sets for (almost) uniform distribution. — Let (nk )k≥0 be a strictly increasing sequence of integers, and let ν ∈ M(T) be a (finite) complex Borel measure on T. We stress that ν is not necessarily a probability measure. Given θ ∈ R, the sequence (nk θ)k≥0 is said ([30], [28, p. 53]) to be almost uniformly distributed with respect to ν if there exists a strictly increasing sequence (Nj )j≥1 of positive integers such that for every arc I ⊂ T whose endpoints are not atoms (mass-points) for ν one has 1 # {n ≤ Nj : exp(2iπnk θ) ∈ I} = ν(I). lim j→+∞ Nj The analog of Weyl’s criterion states that (nk θ)k≥0 is almost uniformly distributed with respect to ν if and only if there exists a strictly increasing sequence (Nj )j≥1 of positive integers such that Nj 1 X lim exp(m2iπnk θ) exists j→∞ Nj k=1 for every m ∈ Z. In this case, the limit is ν̂(m). It can also be proved that (nk θ)k≥0 is almost uniformly distributed with respect to ν if and only if there exists a strictly increasing sequence (Nj )j≥1 of positive integers such that Nj
1 X f e2iπnk θ Nj k=1 / Z f dµ T as j / +∞ for every f ∈ C(T). We now denote by W ((nk )k≥0 , ν), the exceptional set of almost uniform distribution of (nk ) with respect to ν. This is the set of all θ ∈ R such that (nk θ)k≥0 is not almost CATALIN BADEA & SOPHIE GRIVAUX 20 uniformly distributed with respect to ν. We will write U ((nk )k≥0 , ν) for the exceptional set of (classical) uniform distribution of (nk ) with respect to ν, which corresponds to the case where Nj = j for every j ≥ 1. The size of the exceptional set U ((nk )k≥0 , ν) has been studied in many works, in particular in the case where ν is the normalized Lebesgue measure on T. In this case, we write it as U ((nk )k≥0 ). If the sequence (nk )k≥0 is lacunary, U ((nk )k≥0 ) is uncountable, and even of Hausdorff dimension 1 ([19], see also [24]). See also [34] and [32] for a stronger result. On the other hand, it is known (see [11], [13]) that among various natural classes of random sequences of integers, almost all sequences (nk )k≥0 satisfy U ((nk )k≥0 ) = Q. / 1 as These typical random sequences (nk )k≥0 are sublacunary, i.e. satisfy nk+1 /nk / +∞. Nonetheless, examples of sublacunary sequences (nk )k≥0 with U ((nk )k≥0 ) unk countable were constructed in [19] (see also [6]). Concerning the size of W ((nk )k≥0 , ν) we refer for instance to [33], [24] and [27]. See also [15] for other references. Our results about the size of W ((nk )k≥0 , ν) rely on the following generalization of Proposition 4.6, which provides a link between the size of the exceptional set W ((nk )k≥0 , ν) and the modified Kazhdan constant of the set {nk ; k ≥ 0}. Proposition 5.9. — Let (nk )k≥0 be a strictly increasing sequence of positive integers with n0 = 1, and let ν ∈ M(T) with ν 6= δ{1} . If W ((nk )k≥0 , ν) is finite or countable infinite, Q = {nk ; k ≥ 0} is a Kazhdan subset of Z, and p g Kaz(Q) ≥ 2(1 − ℜe νb(1)). Proof. — Fix γ ∈ (0, 1 − ℜe νb(1)), and let µ be a probability measure on T such that supk≥0 (1 − ℜe µ b(nk )) < γ. Then 1 − ℜe Z  X N  1 λnk dµ(λ) < γ T N k=1 for every N ≥ 1. Suppose that the measure µ is continuous. Since there exists a strictly increasing sequence (Nj )j≥1 of integers such that Nj 1 X nk λ Nj k=1 /ν b(1) as j / +∞ for every λ ∈ T \ C, where C is a finite or countable infinite subset of T, we have 1 − ℜe νb(1) ≤ γ, which contradicts our initial assumption. So µ has a discrete part. p It then follows from Theorem 4.1 that the modified Kazhdan constant of Q is at least 2(1 − ℜe νb(1)). The following result provides an example of a nonlacunary semigroup (nk )k≥0 whose associated exceptional sets W ((nk )k≥0 , ν) with respect to ν are uncountable for a large class of measures ν ∈ M(T). Theorem 5.10. — Denote by (nk )k≥0 the sequence obtained by ordering the Furstenberg ′ set F = {2k 3k ; k, k ′ ≥ 0} in a strictly increasing fashion. For every measure ν ∈ M(T) such that ℜe νb(1) < 1/2, the set W ((nk )k≥0 , ν) is uncountable. Proof of Theorem 5.10. — Fix ν ∈ M(T), and suppose that U ((nk )k≥0 , ν) is at most g ) ≤ 1 by Theorem 2.2, it follows from Proposition 5.9 that countable. Since Kaz(F p 2(1 − ℜe νb(1)) ≤ 1, i.e. that ℜe νb(1) ≥ 1/2. This proves Theorem 5.10. CONTINUOUS PROBABILITY MEASURES WITH LARGE FOURIER COEFFICIENTS 21 References [1] J. Aaronson, Rational ergodicity, bounded rational ergodicity and some continuous measures on the circle, Israel J. Math. 33 (1979), 181–197. [2] J. Aaronson, M. Hosseini, M. Lemańczyk, IP-rigidity and eigenvalue groups, Erg. Th. Dynam. Syst. 34 (2014), 1057–1076. [3] T. Adams, Tower multiplexing and slow weak mixing, Colloq. Math. 138 (2015), 47–72. [4] C. Badea, S. Grivaux, Kazhdan sets in groups and equidistribution properties, J. Funct. Anal. 273 (2017), 1931–1969. [5] C. Badea, S. Grivaux, Sets of integers determined by operator-theoretical properties: Jamison and Kazhdan sets in the group Z (Actes du 1er congrès national de la SMF - Tours 2016), Séminaires et Congrès 31, Société Mathématique de France (2017). [6] R.C. Baker, On a theorem of Erdös and Taylor, Bull. London Math. Soc. 4 (1972), 373–374. [7] B. Bekka, P. de la Harpe, A. Valette, Kazhdan’s Property (T), New Mathematical Monographs 11, Cambridge University Press (2008). [8] V. Bergelson, A. del Junco, M. Lemańczyk, J. Rosenblatt, Rigidity and nonrecurrence along sequences, Erg. Th. Dynam. Syst. 34 (2014), 1464–1502. [9] V. Bergelson, D. Simmons, New examples of complete sets, with connections to a Diophantine theorem of Furstenberg, Acta Arith. 177 (2017), 101–131. [10] M. Boshernitzan, Elementary proof of Furstenberg’s Diophantine result, Proc. Amer. Math. Soc. 122 (1994), 67–70. [11] M. Boshernitzan, Density modulo 1 of dilations of sublacunary sequences, Adv. Math. 108, (1994) 104–117. [12] M. Boshernitzan, Homogeneously distributed sequences and Poincaré sequences of integers of sublacunary growth, Monatsh. Math. 96 (1983), 173–181. [13] J. Bourgain, On the maximal ergodic theorem for certain subsets of the integers, Israel J. Math. 61 (1988), 39–72. [14] J. Bourgain, E. Lindenstrauss, P. Michel, A. Venkatesh, Some effective results for ×a × b, Erg. Th. Dynam. Syst. 29 (2009), 1705–1722. [15] Y. Bugeaud, Distribution modulo one and Diophantine approximation, Cambridge Tracts in Mathematics 193, Cambridge University Press (2012). [16] Y. Bugeaud, On sequences (an ξ)n≥1 converging modulo 1, Proc. Amer. Math. Soc. 137 (2009), 2609–2612. [17] I. Chatterji, D. Witte Morris, R. Shah, Relative property (T) for nilpotent subgroups, preprint 2017, arXiv:1702.01801. [18] T. Eisner, S. Grivaux, Hilbertian Jamison sequences and rigid dynamical systems, J. Funct. 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Hochman, Geometric rigidity of ×m invariant measures, J. of the European Math. Soc. 14 (2012), 1539–1563. [26] M. Hochman, Lectures on dynamics, fractal geometry, and metric number theory, J. of Mod. Dyn. 8 (2014), 437–497. [27] J.-P. Kahane, Sur les mauvaises répartitions modulo 1, Ann. Inst. Fourier (Grenoble) 14 (1964), 519–526. [28] L. Kuipers, H. Niederreiter, Uniform distribution of sequences, Pure and Applied Mathematics, Wiley-Interscience (1974). [29] E. Lindenstrauss, Rigidity of multiparameter actions, Probability in mathematics, Israel J. Math. 149 (2005), 199–226. [30] R. Lyons, Fourier-Stieltjes coefficients and asymptotic distribution modulo 1, Ann. of Math. 122 (1985), 155–170. [31] R. Lyons, On measures simultaneously 2- and 3-invariant, Israel J. Math. 61 (1988), 219–224. [32] B. de Mathan, Sur un problème de densité modulo 1, C. R. Acad. Sci. Paris 287 (1978), 277–279. [33] I. Piatetski-Shapiro, On the laws of distribution of the fractional parts of an exponential function, Izv. Akad. Nauk SSSR Ser. Mat. 15 (1951), 47–52 (in Russian). [34] A. Pollington, On the density of sequence (nk ξ), Illinois J. Math. 23 (1979), 511–515. [35] D. Rudolph, ×2 and ×3 invariant measures and entropy, Erg. Th. Dynam. Syst. 10 (1990), 395–406. [36] A. Venkatesh, The work of Einsiedler, Katok and Lindenstrauss on the Littlewood conjecture, Bull. Amer. Math. Soc. 45 (2008), 117–134. Catalin Badea, Laboratoire Paul Painlevé, UMR 8524, Université de Lille, Cité Scientifique, Bâtiment M2, 59655 Villeneuve d’Ascq Cedex, France • E-mail : [email protected] Sophie Grivaux, CNRS, Laboratoire Paul Painlevé, UMR 8524, Université de Lille, Cité Scientifique, Bâtiment M2, 59655 Villeneuve d’Ascq Cedex, France E-mail : [email protected] 

Estimation of Risk Contributions with MCMC Takaaki Koike ∗and Mihoko Minami † February 13, 2017 arXiv:1702.03098v1 [q-fin.RM] 10 Feb 2017 Abstract Determining risk contributions by unit exposures to portfolio-wide economic capital is an important task in financial risk management. Despite its practical demands, computation of risk contributions is challenging for most risk models because it often requires rare-event simulation. In this paper, we address the problem of estimating risk contributions when the total risk is measured by Value-at-Risk (VaR). We propose a new estimator of VaR contributions, that utilizes Markov chain Monte Carlo (MCMC) method. Unlike the existing estimators, our MCMC-based estimator is computed by samples of conditional loss distribution given the rare event of our interest. MCMC method enables to generate such samples without evaluating the density of total risk. Thanks to these features, our estimator has improved sample-efficiency compared with the crude Monte Carlo method. Moreover, our method is widely applicable to various risk models specified by joint portfolio loss density. In this paper, we show that our MCMC-based estimator has several attractive properties, such as consistency and asymptotic normality. Our numerical experiment also demonstrates that, in various risk models used in practice, our MCMC estimator has smaller bias and MSE compared with these of existing estimators. Keywords: Quantitative risk management; Value-at-Risk; Economic capital; Risk allocation; risk contributions; VaR contributions; Markov chain Monte Carlo 1 Introduction In most financial institutions, the risk of their portfolios is measured by economic capital. For the purpose of further risk analysis, it is necessary to decompose the portfolio-wide economic capital into the sum of risk contributions by unit exposures. This risk-redistribution process is called capital allocation, and the allocated capitals are called risk contributions (see, e.g., Dev, 2004). Since it is non-trivial to allocate the economic capital in an economically meaningful way, various principles to determine the risk contributions have been proposed. Among these principles, the Euler principle is one of the most well-established rule proposed in Tasche (1999). Its financial justification is given from differing points of view (e.g., by Denault, 2001; Kalkbrener, 2005; Tasche, 1999). Detailed references for rationalizing the use of the Euler principle can also be found in Tasche (2008). Despite its good economic properties, actual computation of risk contributions poses theoretical and numerical difficulties, especially when the portfolio-wide risk is measured by Value-at-Risk ∗ Center for Mathematics, Graduate School of Science and Technology, Keio University, [email protected] † Department of Mathematical Engineering, Faculty of Science and Technology, Keio University, [email protected] 1 (VaR). Although the explicit formula of VaR contributions is derived by Tasche (2001), it can rarely be calculated analytically with a very few exceptions (such as CreditRisk+ studied in Tasche, 2004). Therefore, Monte Carlo (MC) method is often used to obtain numerical solution. However, the MC estimator of VaR contributions suffers from significant bias caused by sample-inefficiency and inevitable numerical modification (see, e.g., Yamai and Yoshiba, 2002 for the example). To overcome the difficulties, several methods have been proposed in the literature. Hallerbach (2003) and Tasche and Tibiletti (2004) proposed the approximation formulae by regarding the VaR contributions as the best prediction of losses given the total loss, under some model assumptions. In this paper, we call the estimator derived in such a way the generalized regression (GR) estimator. Evaluation of the approximation error caused by the model misspecification is quite difficult. Furthermore, it is also challenging to find a well-fitted model among losses, to accurize the approximation. Glasserman (2005) developed importance sampling (IS) estimators in the case of credit portfolios. Although the problem of sample-inefficiency is overcomed in the IS method, risk models where the IS technique are available are quite limited. Consequently, the IS estimator can not be applied to various risk models widely used in practice. Tasche (2009) proposed the Nadaraya-Watson (NW) estimator constructed by using kernel estimation method. Although the NW estimator is available to most risk models, incorporation of the importance sampling technique is still indispensable to achieve efficient estimation. Moreover, the asymptotic standard error of the NW estimator can not be computed easily. In this paper, we propose a new estimator of VaR contributions that utilizes Markov chain Monte Carlo (MCMC) method. Our MCMC-based estimator is available whenever the portfolio loss model is specified by the joint density. This case contains wide varieties of risk models since the joint density can be completely specified for all models having marginal densities and copula density (see, e.g., Yoshiba, 2013 for various examples of this type of models). In this case, the IS technique can hardly be available, and thus the problem of estimating VaR contributions is generally unsettled. We study theoretical properties of our MCMC estimator, and provide a guideline for the efficient application of MCMC to our problem. The proposed estimation method is then carried out within some risk models often used in practice. For these risk models, we compare the performance of the MCMC estimator with the existing estimators. The MCMC estimator has several attractive properties. First, the MCMC estimator is consistent. Since the MCMC method has an improved sample-efficieny compared with the MC method, we can expect the MCMC estimate to have much smaller bias than that of MC. Second, the MCMC estimator holds asymptotic normality, and its standard error can be consistently estimated by using methods in the theory of Markov chain. Thanks to these features, we can construct approximated confidence interval of the true VaR contributions. Moreover, together with the sample-efficiency of the MCMC method, we can expect that the MCMC estimator has much smaller standard error compared with the existing methods. Finally, the MCMC estimator is free from modelmisspecification bias, and can flexibly incorporate the feature of the underlying loss distribution into the estimation. In contrast with the GR estimator, the MCMC estimator does not use any approximation. Therefore, the MCMC estimator is free from the approximation error caused by the model-misspecification. Moreover, by choosing an appropriate proposal distribution of the MCMC, we can directly capture the features of the underlying loss distributions, Thanks to these propertoes, the MCMC estimator can have stably good performance regardless of the underlying risk model. The key idea behind the MCMC estimator is to generate samples directly from the joint loss distribution given a rare event of interest. This approach is completely different from the crude MC method in which samples are generated from the loss distribution itself. In the MC method, large 2 portion of samples have to be discarded because VaR contributions depend only on losses given the rare event. Our method, in contrast, does not waste any samples for the estimation. Thanks to this difference, great sample-efficiency can be achieved in the MCMC method. How can we generate samples directly from the conditional density on the rare event? Usually it is quite difficult because our conditional density contains the density of the total loss, which is too cumbersome to compute. Nevertheless, MCMC method enables us to generate their samples directly by sequentially updating the samples, without evaluating the cumbersome term on the total loss. The paper is organized as follows. Section 2 introduces the mathematical setting of the capital allocation problem, and explains problems for estimating VaR contributions with the existing estimators. Section 3 is devoted to the brief introduction to Markov chain Monte Carlo method. Several families of the proposal distributions are also introduced for efficient use of MCMC. In Section 4, we propose the MCMC estimator, that combines the MCMC method with estimation of VaR contributions to achieve more sample efficiency. Based on the theory of MCMC, we also study some theoretical properties of our MCMC estimator. Numerical results and empirical studies are presented in Section 5. We demonstrate that, for widely used risk models in practice, the MCMC estimator has smaller bias and MSE than those of existing estimators. Through the empirical study, we provide a guideline of incorporating the MCMC method in our framework of estimating VaR contributions. Concluding remarks, future research and potential extensions of the MCMC estimator are summarized in Section 6. Throughout the paper, integration of vector-valued function is applied componentwise. Let r×c r×c (E) be the set of E-valued M (E) be the set of E-valued r × c matrices and Mr×c + (E) ⊂ M r×c (i,j) positive definite r×c matrices. For a matrix M ∈ M (E), we write M the (i, j)-th component of M for i = 1, 2, . . . , r and j = 1, 2, . . . , c. For any vector x ∈ Rd , write xj the j-th component of x. 2 Capital Allocation Problem Throughout the paper, we consider the following portfolio loss model: S= d X Xj , (2.1) j=1 where d ≥ 3 is the size of the portfolio, X1 , X2 , . . . , Xd are random variables that represent the losses incurred by the exposures j = 1, 2, . . . , d within a fixed time period. The random variable S defined in (2.1) stands for the portfolio-wide loss. In this paper, a positive value of loss random variable represents a financial loss, and a negative loss is interpreted as a profit. Let Fj be the cumulative distribution function (cdf) of Xj for j = 1, 2, . . . , d, FX be the joint cdf of the random vector X = (X1 , X2 , . . . , Xd ) and FS be the cdf of the total loss S. Let C be a copula of FX . By Sklar’s theorem (see, e.g., Nelsen, 2013), it holds that FX (x) = C(F1 (x1 ), . . . , Fd (xd )) (2.2) for any x ∈ Rd . This formula allows us to separate the modelling of the individual marginal losses from their dependence structure. See McNeil et al. (2015) for the benefits of this copula modelling approach, and its application in financial risk management. For the sake of argument, we impose two assumptions on the portfolio loss models (2.1) in this paper. One assumption is supp(FX ) = Rd+ , or Rd , where supp(F ) is the support of a function 3 F and Rd+ := {x ∈ Rd : x ≥ 0}. The former case focuses on modelling pure losses and the latter on modelling profits and losses (P&L). The other assumption is that, the distributions F1 , . . . , Fd , FX , FS and C are all absolutely continuous, i.e. they all have densities f1 , . . . , fd , fX , fS and c, respectively. In other words, we confine ourselves to continuous loss models, and discrete loss models are beyond the scope of this paper. Although typical credit loss models such as the factor models (see, e.g., Bluhm et al, 2016) are excluded, wide variety of loss models are still included in the sphere of our research. Under the assumption of continuity, we have the density-form of the Sklar’s formula: fX (x) = c(F1 (x1 ), . . . , Fd (xd )) · f1 (x1 ) · · · fd (xd ), (2.3) by differentiating the both sides of (2.2). As mentioned in Section 1, computing the risk contributions is important in the framework of integrated risk management. In practice, a two-step procedure is conducted for determining the risk contributions. First step is to compute the economic capital ρ(S), where ρ is a so-called risk measure. Risk measure is a map from a loss random variable to a capital buffer that is required to cover the loss with a certain high probability. One of the most popular risk measure is the Value-at-Risk (VaR), defined by VaRp (X) = inf{x ∈ R : P(X ≤ x) ≥ p}, (2.4) where p ∈ (0, 1) is called the confidence level. Second step is to allocate the capital ρ(S) to the individual d-exposures according to some principle. Mathematically, the capital allocation is the problem of determining the vector (AC1 , AC2 , . . . , ACd ) that satisfies the full allocation property: ρ(S) = d X ACj . (2.5) j=1 The Euler principle solves this problem by utilizing the well-known Euler rule for a function u 7→ ρ(uT X): d X ∂ρ(uT X) ρ(uT X) = uj , u∈Λ (2.6) ∂uj j=1 where Λ ⊂ Rd \{0} is an open set such that 1 ∈ Λ and ρ is a positive homogeneous risk measure. Define ∂ρ(uT X) j = 1, 2, . . . , d. (2.7) ACρj := ∂uj u=1 Then the full alocation property (2.5) holds for this vector (ACρ1 , . . . ,ACρd ) by taking u = 1 in the equation (2.6). Since VaRp is positive homogeneous, the Euler principle can be applied and its solution is given by VaRp ACj := ∂VaRp (uT X) ∂uj u=1 = E[Xj |X1 + · · · + Xd = VaRp (S)]. (2.8) VaR VaRp See Tasche (2001) for deriving the second equality. We call this vector ACVaRp := (AC1 p , . . . , ACd the VaR contributions. Since we focus only on this form of allocated capital in this paper, we drop the superscript ”VaRp ” and simply denote the VaR contributions by AC=(AC1 , . . . , ACd ). Although its good economical properties have been reported in the literature, VaR contributions can not be computed easily. Even when the joint density of the portfolio loss vector fX is analytically specified, theoretical computation of AC is difficult since deriving the joint distribution 4 ) of (Xj , S) is challenging in general. A possible numerical method to calculate the VaR contributions is the Monte Carlo (MC) method. In the crude MC method, we consider the pseudo VaR contributions, defined by ACδ = E[X| S ∈ [VaRp (S) − δ, VaRp (S) + δ] ] (2.9) instead of the true AC defined in (2.8), for a sufficiently small δ > 0. Since the probability P(S ∈ [VaRp (S) − δ, VaRp (S) + δ]) is positive, the right hand side of (2.9) can be written by ACδ = E[X · 1[S∈Aδ ] ] P(S ∈ Aδ ) where Aδ = [VaRp (S) − δ, VaRp (S) + δ]. (2.10) This expression allows us to construct the estimator of the pseudo VaR contributions given by MC c δ,N = AC 1 N PN (n) · 1[S (n) ∈Aδ ] n=1 X P N 1 n=1 1[S (n) ∈Aδ ] N = N 1 X (n) X · 1[S (n) ∈Aδ ] , Mδ,N n=1 (2.11) PN i.i.d. i.i.d. (n) (n) where Mδ,N = n=1 1[S (n) ∈Aδ ] , X (1) , . . . , X (N ) ∼ FX and {S (n) := X1 + · · · + Xd }N n=1 ∼ FS . We call (2.11) the MC estimator. The MC estimator is used to estimate the true VaR contributions by setting δ and N sufficiently small and large, respectively. Note that this method is available only when δ > 0 since P(S ∈ A0 ) = P(S = VaRp (S)) = 0 by absolute continuity of FS . Although we can compute the MC estimator whenever we can generate i.i.d. samples from the joint loss distribution FX , it has an inevitable bias to estimate the true VaR contributions. The bias of the MC estimator can be decomposed by MC c AC δ,N − AC = bδ (N ) + b(δ) (2.12) MC c δ,N − ACδ and b(δ) = ACδ − AC. Because we generally do not know the true where bδ (N ) = AC AC, the second term b(δ) can not be computed easily. Therefore, the δ should be as small as possible. However, when the δ is quite small, it is difficult to maintain a sufficient sample size Mδ,N that keeps the first term bδ (N ) small enough. This is because E[Mδ,N ] = N · P(S ∈ Aδ ) and P(S ∈ Aδ ) is usually much less than 1 − p. Due to this trade off relation, the two parts of the bias are difficult to eliminate simultaneously. To overcome this problem, several estimators have been proposed in the literature. We end up this section by introducing some existing estimators and their problems. Firstly, the NadarayaWatson (NW) kernel estimator proposed in Tasche (2009) is defined by NW c φ,h,N AC   (n) S −VaRp (S) X (n) · φ h  (n)  , PN S −VaRp (S) φ n=1 h PN = n=1 (2.13) where φ is the kernel density and h > 0 is the bandwidh. Since this estimator is a slight modification of the MC estimator (2.11), it has the same problem as the MC estimator explained above. Moreover, the bias and the standard error of the NW estimator (see, e.g. Hansen, 2009 for their expressions) can not be computed easily because it requires the evaluation of the total loss density fS (VaRp (S)). Secondly, Glasserman (2005) developed the importance sampling (IS) estimators to overcome the problem of sample-inefficiency. By using the IS technique to increase samples whose sums belong to Aδ , the bias of the IS estimators can be sufficiently small. However, risk models that can utilize this IS technique is limited to specific credit risk models (see, e.g., Huang 5 et al, 2007 ; Mausser and Rosen, 2008). Unfortunately, the IS estimators are hardly available in our case, where the loss model (2.1) has a joint loss density as specified in (2.3). Finallly, Hallerbach (2003) and Tasche and Tibiletti (2004) constructed estimators by supposing a generalized regression model among the losses: X = gβ (S) + ε, (2.14) where gβ (s) is a function parametrized by β ∈ Rq and ε is an error random vector such that E[ε|S = VaRp (S)] = 0. Let β̂N be an estimator of β. Then we call the following form of the estimator GR c (2.15) AC gβ ,N := gβ̂N (VaRp (S)) the generalized regression (GR) estimator. Although this estimator is intuitive and easy to apply, it also has an inevitable bias. Let g be the true function such that E[X|S = v] = g(v) and let β ∗ be the minimizer of the error term ε in some sense. If β̂N goes to β ∗ as N → +∞, the bias of the GR estimator can be decomposed by GR c AC gβ ,N − AC = b(N ) + b(gβ ), (2.16) where b(N ) = gβ̂N (VaRp (S)) − gβ∗ (VaRp (S)) and b(gβ ) = gβ∗ (VaRp (S)) − g(VaRp (S)). The first term b(N ) is caused by the estimation error and the second term b(gβ ) is by the misspecification of the model (2.14). Since we do not know the true function g in general, the second term b(gβ ) is difficult to evaluate. To avoid the model-misspecification, we have to choose the family of functions {gβ : β ∈ Rq } large enough to contain g. On the other hand, estimating the parameter β is not as easy as in the ordinary regression model. Since we usually do not have samples from FX|S=VaRp (S) , it is non-trivial to estimate β by minimizing the error term ε given {S = VaRp (S)}. Due to these obstacles, it is generally difficult to evaluate the bias of the GR estimator. A notable exception is the case when the risk model X follows an elliptical distribution. In this case, it holds that (e.g., Corollary 8.43 in McNeil et al., 2015) E[X|S = VaRp (S)] − E[X] = Cov(X, S) · (VaRp (S) − E[S]). Var(S) (2.17) Therefore, the true VaR contributions are provided when we choose gβ (x) = β0 + β1 x and β0∗ = E[X] − Cov(X, S) · E[S] Var(S) β1∗ = Cov(X, S) . Var(S) (2.18) Since this β ∗ is the minimizer of E[ε2 ] = E[(X − β0 − β1 S)2 ], the ordinary least squares (OLS) OLS estimator βN converges to β ∗ as N goes to infinity. As a consequence, the second term b(gβ ) is zero and the first term b(N ) can be eliminated by taking N sufficiently large. 3 General theory of MCMC Markov chain Monte Carlo (MCMC) is a method to generate samples from a probability distribution by constructing a Markov chain whose stationary distribution is the desired one. By allowing Markovian type dependence within the samples, MCMC enables to simulate wide variety of distributions even if they can not be simulated directly in practice. In this section, we briefly introduce the theory of MCMC method. We review consistency and asymptotic normality of estimators computed by Markov chain. To obtain an efficient estimator, it is important to choose an 6 appropriate proposal distribution on generating sample path of the Markov chain. In Section 3.2, we summarize the family of proposal distributions used in this paper. 3.1 Brief introduction to MCMC Let E ⊂ Rd be a set and E be a σ-algebra of subsets of E. Markov chain is a sequence of E-valued random variables (X (1) , X (2) , . . . ) satisfying the Markov property; P(X (n+1) ∈ A|X (k) = x(k) , k ≤ n) = P(X (n+1) ∈ A|X (n) = x(n) ), (3.1) for all n ≥ 1, A ∈ E, and x(1) , . . . , x(n) ∈ E. Markov chain is characterized by its stochastic kernel P : E × E → R+ , given by x × A 7→ P (x, A) := P(X (n+1) ∈ A|X (n) = x). If there exists a R probability distribution π such that π(A) = π(dx)P (x, A) for all x ∈ E and A ∈ E, then the π is called the stationary distribution. See, e.g., Nummelin (2004) for general theory of Markov chain. Markov chain Monte Carlo (MCMC) is a widely used statistical method for simulating probability distribution by generating Markov chain. MCMC is often used to estimate the quantity Z π(h) := h(x)π(dx), (3.2) for some distribution π and a π-measurable function h. Its MCMC estimator is given by π̂N (h) := 1 ( h(X (1) ) + · · · + h(X (N ) ) ), N (3.3) where (X (1) , . . . , X (N ) ) is an N -path of the Markov chain whose stationary distribution is π. In the framework of MCMC, we call this π the target distribution. Since the target distribution π is externally determined by the problem to solve, the problem of MCMC is to find the stochastic kernel P such that it has the stationary distribution π. Additionally, we require P that its sample path can be generated easily. One of the most popular stochastic kernel used in practice is the Metropolis-Hastings (MH) kernel (Metropolis et al., 1953; Hastings, 1970), defined by P (x, dy) = p(x, y)dy + r(x)δx (y), (3.4) where p(x, y) = q(x, y)α(x, y), α(x, y)  h i min π(y)q(y,x) , 1 π(x)q(x,y) = 0 r(x) = 1− if π(x)q(x, y) > 0 otherwise , Z p(x, y)dy, δx is the Dirac delta function, and q : E × E → R+ is a function such that x 7→ q(x, y) is a measurable function for all y ∈ E, and y 7→ q(x, y) is a probability density function (pdf) for all x ∈ E. This function q is called a proposal distribution. It can be shown that the MH kernel has a stationary distribution π (Tierney, 1994). The great popularity of the MH kernel comes from the fact that, we can easily generate sample path of the corresponding Markov chain by using the 7 so-called MH algorithm (see e.g., Chib and Greenberg, 1995 for the simple and intuitive exposition of the algorithm). Under the mild conditions that (i) a vector x(1) ∈ supp(π) is known, (ii) samples from q(x, ·) can be generated for all x ∈ E , and (iii) the ratio π(y)/π(x) can be calculated for all x, y ∈ E, we can generate N -sample path of the desired Markov chain by the following algorithm: Algorithm 1: MH Algorithm 1. Set N > 0, q: proposal distribution, and X (1) = x(1) ∈ supp(π) 2. For n = 1, 2, . . . , N , (n) 3. Generate X∗ 4. Set " αn := 5. ∼ q(X (n) , · ) and U ∼ U(0, 1) (n) α(X (n) , X∗ ) = min Set (n) (n) X (n+1) := 1[U ≤αn ] · X∗ 6. End for (n) π(X∗ ) q(X∗ , X (n) ) · , 1 π(X (n) ) q(X (n) , X∗(n) ) # + 1[U >αn ] · X (n) (3.5) (3.6) 7. Return (X (1) , . . . , X (N ) ) as an N -sample path of the Markov chain with the kernel (3.4) and the initial value X (1) = x(1) (n) We call αn := α(X (n) , X∗ ) in (3.5) the acceptance probability at n-th iteration. Based on the N -sample path (X (1) , . . . , X (N ) ) generated in Algorithm 1, we can compute the MCMC estimator π̂N (h) given by (3.3). The MCMC estimator π̂N (h) has several attractive properties such as consistency and asymptotic normality. Firstly, the MCMC estimator is consistent (see, e.g., Theorem 1 in Nummelin ;2002), i.e., lim π̂N (h) = π(h) with probability 1, (3.7) N →+∞ for all π-integrable functions h and all initial state X (1) = x(n) ∈ supp(π). Moreover, Central Limit Theorem (CLT) holds under some regularity conditions (see, e.g., Chapter 17 in Meyn and Tweedie, 2012), i.e., √ d N ( π̂N (h) − π(h) ) −→ Nd (0, Σh ) where Σh := Varπ [ h(X (1) )]+2 ∞ X as N → +∞, Covπ [ h(X (1) ), h(X (1+k) ) ] k=1 ∈ (0, ∞). (3.8) (3.9) Since we can rarely compute the asymptotic variance (3.9) in real situation, we have to estimate it from the same sample path (X (1) , . . . , X (N ) ) that we used to compute π̂N (h). One simple and popular estimator of Σh is the so-called batch means estimator (see, e.g., Geyer, 2012). For the N -path of the Markov chain (X (1) , . . . , X (N ) ), the batch means estimator Σ̂h,N is defined by B Σ̂h,N = N LN X ( π̂N,b (h) − π̂N (h) )T ( π̂N,b (h) − π̂N (h) ), BN − 1 (3.10) b=1 where LN , BN are positive integers satisfying N = LN · BN , and 1 π̂N,b (h) = LN bLX N −1 h(X (l) ) l=(b−1)LN 8 for b = 1, 2, . . . , BN . (3.11) LN is called the batch length and BN is the number of batches. Under some regularity conditions, the batch means estimator Σ̂h,N converges to Σh with probability 1 as N → +∞ (see, e.g., Jones et al., 2006; Vats et al., 2015). By using the asymptotic relation (3.8) and the consistency of the batch means estimator Σ̂h,N , we can construct an approximate 95% confidence interval of the true quantity π(h), given by σ̂h,N σ̂h,N π̂N (h) − 1.96 · √ < π(h) < π̂N (h) + 1.96 · √ , N N (3.12) (j,j) where σ̂h,N = (Σ̂h,N , j = 1, 2, . . . , d). 3.2 Choice of the proposal distribution When implementing the MCMC, an appropriate choice of the proposal function q is necessary. When choosing q, it is desirable that the Markov chain characterized by (3.4) theoretically holds the Central limit theorem (3.8), and also, its asymptotic variance (3.9) is as small as possible. Since the asymptotic variance Σh can rarely be calculated explicitly in real situation, we usually rely on post-implementation review, i.e. evaluating the goodness of the selected proposal distribution after performing the MCMC. In this section, we introduce two empirical methods for checking validity of the proposal selection. We also provide some families of the proposal distributions for an appropriate choice of q. Theoretical verification of CLT will be carried over into Section 4.3. In practice, there are two prevalent methods to check the validity of the proposal distribution (see, e.g., Geyer, 2012 in details). One method is to check that the autocorrelation plots of the marginal sample paths steadily decline . For an N -sample path (X (1) , . . . , X (N ) ) and the MCMC estimator π̂N (h) = (π̂N,1 (h), . . . , π̂N,d (h)), we draw plots of the sample autocorrelations r̂j (h) := R̂j (h)/R̂j (0), where R̂j (h) := N −k X 1 (n) (n+k) (hj (Xj ) − π̂N,j (h)) · (hj (Xj ) − π̂N,j (h)), N − k n=1 (3.13) versus the time lag k = 0, 1, 2, . . . , for j = 1, 2, . . . , d. We can expect that the asymptotic variance Σh should be small if the autocorrelation plots steadily decline to zero as the lag increases. The other method is to check the acceptance rate, i.e. the percentage of times a candidate is accepted. In general, extremely low acceptance rate implies that the chain tends to be stuck to one point. Conversely, extremely high acceptance rate occurs when the chain moves only around the mode of the target density and does not traverse the entire support (Chib and Greenberg, 1995). The first situation yields high asymptotic variance, and the second gives an illusionally low asymptotic variance. To avoid such situations, it is important to check that the acceptance rate takes a moderate value. Typically, proposal distribution q is selected from a certain class of distributions. To find a suitable q, several classes are in order. First, if q(x, y) = f (y − x) for some pdf f , then the candidate X∗ is drawn according to the process where Z ∼ f, X∗ = X + Z, (3.14) and X is the current state. We call this q the random walk proposal distribution. In the case when f is symmpetric around the origin, the acceptance probability (3.5) is given simply by π(y) α(x, y) = min[ π(x) , 1]. Second, when q(x, y) = f (y) for some pdf f , then the candidate X∗ is 9 updated by X∗ = Z, where Z ∼ f. (3.15) Since the candidate is drawn independently of the current position, this q is called the independent proposal distribution. The two proposal distributions, the random walk and the independent proposals, are widely used due to their simplicity. However, these proposal distributions often fail to perform well when the target distribution π is heavy-tailed. To overcome this problem, we finally introduce the mixed preconditioned Crank-Nicolson (MpCN) proposal distribution (proposed by Kamatani, 2014). This proposal distribution updates the candidate according to the following process: 1 1 1 X∗ = µ + ρ 2 · (X − µ) + (1 − ρ) 2 · Z − 2 · W , (3.16) where Z follows the gamma distribution with shape parameter d/2 and the scale parameter 1 ||Σ− 2 (X − µ)||2 /2 for µ := E[X] and Σ := Var[X], and W ∼ Nd (0, Σ). Note that, here we introduce the standardized version of the original MpCN in Kamatani (2014). The acceptance probability (3.5) can be written by  α(X, X∗ ) =  π(X∗ ) · π(X) ||Σ − 21 1 (X − µ)|| ||Σ− 2 (X∗ − µ)|| !−d  , 1 . (3.17) The major difference of this proposal distribution from the first two simple ones is that, not only the mean but also the variance of the candidate changes with the current state X. Since the MpCN proposal distribution admits larger jumps in the tail, we can expect better acceptance rate even when π is heavy-tailed (see, e.g., Livingstone, 2015 for such position-dependent proposal distributions). 4 Proposed method We have shown that estimators constructed by MCMC has several attractive properties, such as consistency and asymptotic normality. In this section, we propose a new estimator of VaR contributions, that utilizes MCMC method to achieve efficient estimation, We also show consistency and asymptotic normality of our MCMC-based estimator under a certain class of risk models. 4.1 The set-up We suppose the following situation: we have an explicit form of the joint loss density fX , and thus, we can compute the quantity fX (x) for any x ∈ Rd . We also can generate i.i.d. samples X (1) , . . . , X (N ) from FX up to sufficiently large N . Then, the i.i.d. samples S (1) , . . . , S (N ) from (n) (n) FS can be generated by taking S (n) = X1 + · · · + Xd for n = 1, 2, . . . , N . On the other hand, suppose that we have neither an explicit form of the total loss density fS nor the way to compute the quantity fS (s) for any s ∈ R. The situation described above typically occurs when we model the joint loss density fX by the copula approach, i.e. specifying the marginal loss densities f1 , f2 , . . . , fd and then specifying the copula density c. When we use this approach, the resulting joint loss density fX is speficied by the formula (2.3). Under the situation above, a two-step procedure (see, e.g., Glasserman, 2005) is conducted for computing the VaR contributions (2.8). The first step is to estimate VaRp (S) and the second step is to estimate the VaR contributions AC= E[X|S = VaRp (S)] with the VaRp (S) replaced by the estimated one in the first step. Estimation of the VaRp (S) in the first step is conducted 10 by Monte Carlo simulation. Based on the i.i.d. samples S (1) , . . . , S (N ) from FS , the estimator of d p (S) = S [N p] , where S [N p] is the N p-th largest sample among N samples VaRp (S) is given by VaR d p (S) as v in short, and regard it as a {S (1) , . . . , S (N ) }. Hereafter we refer to the estimate of VaR constant. In the second step, we aim at estimating AC= E[X|S = v]. According to the crude Monte Carlo method as explained in Section 2, the VaR contributions are estimated by MC c δ,N = AC i.i.d. N 1 X (n) X · 1[S (n) ∈Aδ ] , Mδ,N n=1 (n) (4.1) (n) where X (1) , . . . , X (N ) ∼ FX , S (n) := X1 + · · · + Xd for n = 1, . . . , N , Aδ = [v − δ, v + δ] and PN Mδ,N = n=1 1[S (n) ∈Aδ ] . As explained in Section 2, the problem of this two-step procedure is that the estimator of VaR contributions in the second step often has a significant bias. To address this issue, we develop a Markov chain Monte Carlo (MCMC)-based estimator, that achieves sample efficiency and consistency, by utilizing the MCMC method. The major difference of our MCMC-based estimator from the MC one is that, samples are generated not from the loss distribution FX itself, but from the conditional distribution of X given the sum constraint {S = v}. Although it is almost impossible in the MC method, the MCMC enables us to realize it. 4.2 The MCMC estimator We start to describe the MCMC-based estimator with reformulating the problem of computing VaR contributions. Let X 0 = (X1 , X2 , . . . , Xd0 ), where d0 = d − 1. By the full allocation property (2.5), we have that 0 T E[X|S = v] = ( E[X 0 |S = v], v − 1T d0 · E[X |S = v] ) , (4.2) where S = X1 + · · · + Xd . Therefore, computation of the VaR contributions AC=E[X|S = v] can be reduced to estimate the VaR contributions of the d0 -subportfolio, given by AC0 = E[X 0 |S = v]. In our method, we estimate this quantity AC0 by generating samples directly from FX 0 |S=v . The conditional joint density of X 0 given {S = v} can be written by fX 0 |S=v (x0 ) = 0 fX (x0 , v − 1T fX 0 ,S (x0 , v) d0 x ) = , fS (v) fS (v) 0 x0 ∈ R d (4.3) where the second equation follows from a linear transformation (X 0 , S) 7→ X. Since it is difficult to evaluate the total loss density fS (v), we can hardly generate samples directly from fX 0 |S=v . 0 By taking E = Rd , h(x) = x and π(x) = fX 0 |S=v (x) on the discussion in Section 3, our problem of estimating VaR contributions can be reduced to estimate π(h) = E[X 0 |S = v] in (3.2) by MCMC. Even though we can hardly compute fX 0 |S=v itself, we can compute the acceptance probability (3.5), given by  α(x, y) = min    fX 0 |S=v (y) · q(y, x) fX (y, v − 1T d0 y) · q(y, x) , 1 = min , 1 fX 0 |S=v (x) · q(x, y) fX (x, v − 1T d0 x) · q(x, y) (4.4) for all x, y. Note that the cumbersome term fS (v) disappears by taking the ratio of fX 0 |S=v (y) to fX 0 |S=v (x). Therefore, under some appropriate choice of the proposal function q, we can generate N -sample path of the Markov chain whose stationary distribution is π(x) = fX 0 |S=v (x). Based on the sample path of the Markov chain, we can construct the MCMC estimator π̂N (h) defined by 11 (3.3). Its consistency follows directly from the general theory of Markov chain presented in Section 3. The algorithm to compute the MCMC estimator of VaR contributions is summarized as follows: Algorithm 2: MCMC estimator of VaR contributions E[X|S = VaRp (S)] 1. Set v > 0, N > 0, q: proposal distribution, and X (1) = x(1) ∈ supp(π). 2. Perform Algorithm 1 for the given N , q, and x(1) to generate an N -sample path (X 0(1) , . . . , X 0(N ) ) of a Markov chain whose stationary distribution is fX 0 |S=v . 3. Define 0(n) T X (n) := (X 0(n) , v − 1T ) , d0 X and set MCMC c AC q,N = N 1 X (n) X N n=1 (4.5) (4.6) to estimate VaR contributions AC= E[X|S = v]. 4.3 Some theoretical results In this section, we provide some theoretical results to verify asymptotic normality of the MCMC estimator (4.6). We will see that CLT holds for various risk models when we model the pure losses. On the other hand, when we model the profits & losses, theoretical justification of CLT is challenging except for some special cases. We provide an example of justifying CLT only when we model P&L by some elliptical distribution. When we model pure losses, i.e. supp(FX ) = Rd+ , then the conditional distribution FX 0 |S=v is supported on the following bounded set, called the v-simplex: 0 Sv := {x ∈ Rd : x ≥ 0, 0 ≤ x1 + · · · + xd0 ≤ v}. (4.7) Thanks to the compactness of the support, CLT is verified under some mild conditions. The following theorem is a direct consequence of well-known results in the theory of MCMC. However, it clarifies the conditions on the marginal loss densities and the copula densities, that are easy to be checked in the framework of joint risk modeling with copulas. Theorem 4.1. Suppose that the joint distribution FX is supported on Rd+ , and has marginal √ densities f1 , f2 , . . . , fd and a copula density c. Then, N -CLT holds for the MCMC estimator (4.6) of VaR contributions if the following (C1) − (C3) hold: (C1)  := inf x,y∈[0,v]d0 q(x, y) > 0, (C2) fj (x) is positive and bounded above on [0, v] for all j = 1, 2, . . . , d, (C3) c(u) is positive and bounded above on F1 ([0, v]) × · · · × Fd ([0, v]). √ Proof. By Theorem 23 in Roberts and Rosenthal (2004), the N -CLT holds if the Markov chain is uniformly ergodic whenever E[||X 0 ||2 |S = v] < ∞. The moment condition is satisfied since E[Xi Xj |S = v] ≤ E[(X1 + · · · + Xd )2 |S = v] = v 2 < ∞ (4.8) for any i, j = 1, 2, . . . , d. Thus, it suffices to show that the Markov chain is uniformly ergodic. By Theorem 1.3 in Mengersen and Tweeide (1996), the Markov chain is uniformly ergodic if (and only 12 if) the minorization condition (see, e.g., Rosenthal, 1995) holds on the whole space Sv , i.e., there exist a positive integer n, δ > 0 and a probability measure ν such that P n (x, A) > δ · ν(A) for all 0 0 x ∈ Sv , A ∈ Bv , where Bv := B(Rd ) ∩ Sv . By the conditions (C2), (C3), and that Sv ⊂ [0, v]d , we have that l := inf π(x) > 0, u := sup π(x) < +∞. (4.9) x∈Sv x∈Sv Using (4.9) and the condition (C1), the minorization condition can be checked as follows. For any x ∈ Sv , define   π(y) q(y, x) Qx := y ∈ Sv : · <1 . (4.10) π(x) q(x, y) Then for all A ∈ Bv , we have P (x, A) ≥ = ≥ = Taking n = 1, δ =  u   π(y) q(y, x) · dy q(x, y) · min 1, π(x) q(x, y) Qx   Z π(y) q(y, x) · dy + q(x, y) · min 1, π(x) q(x, y) A\Qx Z Z π(y) · q(y, x)dy + q(x, y)dy A\Qx Qx π(x) Z Z  π(y) π(y)dy +  dy u Qx u A\Qx  π(A). u Z and ν = π completes the proof. We will see typical situations where the conditions (C1) − (C3) hold. Firstly, the condition 0 (C1) holds when, for example, q(x, y) = f (y − x), f is continuous, and positive on [−v, v]d . These conditions are satisfied for typical choices of f , such as Gaussian density or student’s t density. Secondly, the condition (C2) also holds when, for example f1 , f2 , . . . , fd follow Pareto distributions with the density given by κ fj (x) = κj γj j (x + γj )κj +1 κj > 0, γj > 0 for j = 1, 2, . . . , d. (4.11) Lastly, the condition (C3) can easily be checked for various copulas used in practice. For the use in Section 4, we especially focus on the following two examples: Example 4.1 (t copula). d-dimensional t copula density is of the form ct (u) = ν Γ( ν+d 2 )Γ( 2 ) 1 d |R| 2 Γ( ν+1 2 )  · 1+ xT R−1 x ν Πdj=1 (1 + − ν+d 2 x2j − ν+1 2 ν ) , ν>0 (4.12) −1 −1 where R ∈ Md×d + [0, 1] is a correlation matrix, x = (tν (u1 ), . . . , tν (ud )) and tν is a cdf of the student’s t distribution with degree of freedom ν. It is clear that (4.12) satisfies the condition (C3) if Fj (0) > 0 for j = 1, 2, 3. Example 4.2 (π-rotated Clayton copula). d-dimensional π-rotated Clayton copula has a density 13 of the form (Yoshiba, 2013)   − θ1 −d d d 1 Y X + d) Γ( cClayton (u) = θd · θ 1 ·  (1 − uj )−θ−1   (1 − uj )−θ − d + 1 , π Γ( θ ) j=1 j=1 0 < θ < ∞. (4.13) Since Clayton copula has a lower tail dependence, the π-rotated Clayton copula has an upper tail dependence. We will see that the π-rotated Clayton copula density (4.13) satisfies the condition (C3) under some parameter constraint. Let lj := 1 − Fj (v) ∈ (0, 1). Then, for any uj ∈ Fj ([0, v]), it holds that 0 < lj ≤ 1 − uj ≤ 1 − Fj (0) = 1 for all j = 1, 2, . . . , d. Thus, we have 1≤ and also d Y (1 − uj )−θ−1 ≤ j=1 d Y j=1 lj−θ−1 < ∞  − θ1 −d − θ1 −d  d d X X  ≤  (1 − uj )−θ − d + 1 ≤ 1. lj−θ − d + 1 j=1 (4.14) (4.15) j=1 Since v is the p-th quantile of FS , we have lj = 1 − Fj (v) ≤ 1 − p. Using this inequality, the lower bound of (4.15) is positive if θ < log(1 − p)/ log(1 − d1 ). Putting (4.14) and (4.15) together, the π-rotated Clayton copula density (4.13) satisfies the condition (C3) when 0 < θ < log(1 − p)/ log(1 − d1 ). In contrast to the pure-loss case, theoretical verification of CLT is challenging in the case when we model P&L, i.e. supp(FX ) = Rd . Since the conditional distribution FX 0 |S=v is supported on 0 the unbounded space Rd , we need a careful investigation on the tail-behavior of the conditional density fX 0 |S=v . When the proposal distribution is MpCN, we can justify the CLT of our MCMC estimator for a certain class of target distributions based on the result in Kamatani (2016). As its example, we consider the case where the underlying loss vector X follows a multivariate student’s t distribution. Example 4.3 (Multivariate student’s t distribution). Let X ∼ tν (µ, Σ), where ν > 2, µ ∈ Rd and Σ ∈ Md×d + . The density of the multivariate student’s t distribution is given by  − ν+d 2 Γ( ν+d (x − µ)T Σ−1 (x − µ) 2 ) p . fX (x) = · 1+ 1 ν ν |Σ| 2 Γ( 2 ) (π · d)d (4.16) Its conditional density fX 0 |S=v can be specified as follows. Firstly, let µ = 0 for simplicity. Write Σ−1 = 0 0 A1 AT 2 A2 A3 ! =: A, (4.17) 0 for A1 ∈ Md ×d (R), A2 ∈ Rd and A3 ∈ R. Then, it holds that x v − 1T d0 x !T A1 AT 2 A2 A3 ! ! x v − 1T d0 x 0 = (x − w)T V (x − w) + D, 0 (4.18) 0 d ×d T T where V := A1 − A2 1T , w := V −1 (vA3 1d0 − vA2 ) ∈ Rd , and d0 − 1d0 A2 + 1d0 1d0 ∈ M+ 14 D := v 2 A3 − wT V w ∈ R. Using (4.18), we have that fX 0 |S=v (x) ∝ fX (x, v − 1T d0 x)  − ν+d 2 (x − w)T V (x − w) + D ∝ 1+ ν  − ν+d 2 (x − w)T V (x − w) ∝ 1+ . ν+D (4.19) Provided ν + D > 0, X 0 |S = v follows a d0 -dimensional elliptical distribution with the location parameter w, scale parameter V −1 and the density generator g : R+ → R+ given by  g(x) = x 1+ ν+D − ν+d 2 . (4.20) This type of distribution is called the Pearson type VII distribution (see e.g., Schmidt, 2002). Consider the MCMC estimator (4.6) where π = fX 0 |S=v and the proposal distribution q is √ MpCN (3.16). By Theorem 25 in Roberts and Rosenthal (2004), a N -CLT holds if the Markov chain is geometrically ergodic and E[||X 0 ||2 |S = v] < ∞. By Proposition 3.4 in Kamatani (2016), the Markov chain with MpCN proposal distribution is geometrically ergodic if E[||X 0 ||δ |S = v] < ∞ for some δ > 0, π(x) is strictly positive and continuous, and moreover, it is symmetrically regularly varying, i.e., π(rx) lim = λ(x), (4.21) r→+∞ π(r1d ) 0 0 0 d −1 d −1 for some λ : R → (0, ∞) such that λ(x) = 1 for any x ∈ SW , where SW := {x ∈ Rd : 0 0 1 1 d ×d . We will firstly see that all the moment conditions ||W − 2 x|| = ||W − 2 1d0 ||} and W ∈ M+ hold, and then, the tail condition (4.21) is also satisfied for π = fX 0 |S=v . Write R := ||X 0 ||. It can be shown that g is regularly varying (see, e.g., Resnick, 2013) at ∞ with index α = − ν+d 2 , i.e. ν+d g(rx) = x− 2 lim x > 0. (4.22) r→∞ g(r) By Proposition 3.7 in Schmidt (2002), fR|S=v is regularly varying with index −(ν + 1). Then, by Karamata’s Theorem (see, e.g., Resnick, 2013, Theorem 0.6), FR|S=v is regularly varying with index −ν. Therefore, E[Rδ |S = v] < ∞ holds for all δ < ν (see, e.g., Mikosch, 1999). Thus all the moment conditions above are satisfied so long as ν > 2. In elliptical case, the tail condition (4.21) 0 is a direct consequence of (4.22). Since (x − w)T V (x − w) > 0 for all x ∈ Rd , it holds that fX 0 |S=v (rx) lim = r→+∞ fX 0 |S=v (r1d ) 1 ||V 2 x|| !−(ν+d) , 1 2 ||V 1d0 || 0 x ∈ Rd . (4.23) Thus, taking 1 λ(x) := ||V 2 x|| !−(ν+d) , 1 ||V 2 1d0 || W = V −1 (4.24) in (4.21) yields that π = fX 0 |S=v is symmetrically regularly varying. Putting them together, we √ can find that the MCMC estimator with MpCN proposal distribution satisfies N -CLT when the underlying loss vector follows multivariate student’s t distribution with ν > 2 and D > −ν. 15 5 Simulation and numerical studies We have shown that the MCMC estimator has various attractive properties for the estimation of VaR contributions. In this section, we apply the proposed estimator to various risk models used in practice. Our numerical experiment shows that the MCMC estimator has smaller bias and Mean Squared Error (MSE) compared with the existing estimators that have been proposed in the previous studies. Then, based on some numerical results, we study how to choose an appropriate proposal distribution given risk model. 5.1 Numerical study of the MCMC estimator In the simulation study, we consider four risk models which are modeled with marginal densities and copula densities. As is often done in risk management, we adopt heavy-tailed marginal distributions and copulas with tail dependence. In all risk models, we set the size of the portfolio d = 3. The models are described as follows: (1) The loss random variable Xj follows Pareto distribution (4.11) with κj = 4 and γj = 3 for all j = 1, 2, 3. The loss vector (X1 , X2 , X3 ) has a π-rotated Clayton copula (4.13) with θ = 0.5. (2) Xj , (j = 1, 2, 3) have the same marginals as in the case (1). However, their copula is a student’s t copula (4.12) with ν = 4 and the correlation matrix given by  1  P = −0.5 0.3 −0.5 1 0.5  0.3  0.5 . 1 (5.1) (3) Xj follows student’s t distribution with νj = 4, µj = 0 and σj = 1 for all j = 1, 2, 3. (X1 , X2 , X3 ) has a π-rotated Clayton copula with θ = 0.5. (4) The loss random vector (X1 , X2 , X3 ) follows a multivariate student’s t distribution (4.16) with ν = 4, µ = 0, and Σ given by (5.1). The former two models (1) and (2) treat pure losses, and the latter two models (3) and (4) handle P&L. In all models, marginal distributions have variance 2.0. Moreover, they are heavy-tailed with tail index 5.0. The models (1) and (3) possess homogeneous upper tail dependence with tail coefficient λU i,j = 0.025 (see, e.g., Joe, 2014). The models (2) and (4) have symmetric upper, lower, L U L and upper-lower tail dependences with tail coefficients λU 1,2 = λ1,2 = 0.012, λ1,3 = λ1,3 = 0.162, L UL LU UL LU UL LU λU 2,3 = λ2,3 = 0.253, λ1,2 = λ1,2 = 0.253, λ1,3 = λ1,3 = 0.029, and λ2,3 = λ2,3 = 0.012. For each risk model, we compute several estimators of VaR contributions AC = E[X|S = VaRp (S)] for p = 0.999, with the VaRp (S) replaced by its Monte Carlo estimator v = S [N p] . In this setting, the condition (C3) in Theorem 4.1 holds in the risk models (1) and (2) since log(1 − p)/ log(1 − 1/d) = 17.037 > 0.5 = θ. Also, as shown in Example 4.3, the risk model (4) with MpCN proposal distribution satisfies CLT since D + ν = 137.935 > 0. The estimators that we study are the Monte Carlo (MC) estimator (2.11), Nadaraya-Watson (NW) estimator (2.13) (Tasche, 2009), Generalized regression (GR) estimator (2.15) (Tasche and Tibiletti, 2004; Hallerbach, 2003), and the MCMC estimator (4.6) proposed in this paper: MC c δ,N = AC 1 Mδ,N N X n=1 NW X (n) · 1[S (n) ∈Aδ ] , c φ,h,N = AC 16  (n)  S −v (n) X · φ n=1 h   , PN S (n) −v n=1 φ h PN MCMC GR c AC gβ ,N = gβ̂N (v), i.i.d. c AC q,N = (n) i.i.d. (n) N 1 X (n) X , N n=1 |S=v (n) where X (n) ∼ FX , S (n) := X1 + · · · + Xd ∼ FS , and {X|S=v }n=1,...,N is an N -sample path of a Markov chain with stationary distribution FX|S=v . For all estimators, we fix the sample size N = 106 . Other parameters of the estimators above are determined as follows. First, on the MC estimator, we set δ > 0 such that the MC sample size Mδ,N is around 103 . As in Glasserman (2003), asymptotic normality p MC c δ,N − ACδ ) → Nd (0, Σδ ), Mδ,N · (AC as N → +∞ (5.2) MC c holds for a fixed δ. Using this result, we report the estimate of AC δ,N and its approximated (j,j) p standard error sM C / Mδ,N for j = 1, 2, 3, where sM C is the sample standard error defined by sM C v u u =t N 1 X (n) c MC T (n) c MC (X − ACδ,N ) (X − ACδ,N ) · 1[S (n) ∈Aδ ] . Mδ,N n=1 (5.3) Second, on the NW estimator, we choose the kernel density φ the standard normal density. We decide the bandwidth h > 0 according to the Silverman’s rule of thumb h = 1.06 · σ̂S · N −1/5 (Silverman, 1986). Although asymptotic normality holds for the NW estimator, its asymptotic variance can hardly be computed because it requires the calculation of fS (v). Therefore, we report NW c φ,h,N . Third, on the GR estimator, we choose gβ (s) = β0 + β1 · s and its only the estimate of AC coefficients are estimated by PN (n) − X̄N )(S (n) − n=1 (X PN (n) − S̄ )2 N n=1 (S β̂N,1 = S̄N ) β̂N,0 = X̄N − β̂N,1 · S̄N , (5.4) PN PN where X̄N = N1 n=1 X (n) and S̄N = N1 n=1 S (n) . According to the general theory of OLS, it holds under some regularity conditions that √    (j) β0  → N2 (02 , σε2 Q−1 ), − N ·  j (j) (j) β̂N,1 β1 (j) β̂N,0  (j) as N → +∞ (5.5) (j) for j = 1, 2, 3, where β̂N,k and βk is the j-th component of β̂N,k and βk for k = 1 and 2, respectively, εj is the j-th component of the error term ε, σε2j is its conditional variance given S(1) , . . . , S(N ) , and T Q := lim Y Y /N N →∞ where 1 S (1) Y = ··· ··· 1 S (N ) !T . (5.6) GR c According to this result, we report the estimate of AC gβ ,N and its approximated standard error (j) √ sGR / N for j = 1, 2, 3, where (j) sGR q (1,1) (1,2) (2,2) := Σ̂GR,j + 2v · Σ̂GR,j + v 2 · Σ̂GR,j , (5.7) Σ̂GR,j = σ̂ε2j · (Y T Y /N )−1 and σ̂εj is the sample standard error of the j-th residuals. Finally, on the MCMC estimator, we choose different proposal distributions depending on the risk models 17 (1)-(4): (1) Random walk proposal q(x, y) = f (y − x) with f ∼ Nd (0, Σ̂v ), where Σ̂v := s2M C , for sM C defined in (5.3), (2) Independent proposal q(x, y) = f (y), where f is the density of the Dirichlet distribution with parameters (0.2, 0.28, 0.6), (3) and (4) MpCN proposal with ρ = 0.8, GR GR c g ,N 0 , v − 1T0 AC c g ,N 0 )T and Σ := s2 . In the Algorithm 2, we set the initial state µ = (AC d β MC β x(1) = (v/3, v/3, v/3)T . We suppose that asymptotic normality (3.8) holds for all risk models. Also, its asymptotic variance is estimated by the batch means estimator Σ̂N defined by (3.10). 1 Following the recommendation of Jones et al. (2006), we choose LN := bN 2 c = 103 and BN := MCMC 1 c bN/LN c = bN 2 c = 103 . We report the estimate of AC and its approximated standard error q,N (j,j) √ Σ̂N / N for j = 1, 2, 3. As mentioned in Section 3.2, the validity of the MCMC method can empirically be checked by autocorrelation plots and the acceptance rate. Fig. 1 show the autocorrelation plots of the Markov chains generated by Algorithm 2. The acceptance rate of the MH algorithm in each risk model was 1.0 (b) ○ : X1 |S = v 4 : X2 |S = v △ + : X3 |S = v ＋ ● ● ○ : X1 |S = v 4 : X2 |S = v △ ● 0.8 0.8 1.0 (a) ● + : X3 |S = v ＋ ● 0.6 0.6 ● ACF ● ● ● 0.4 ● 0.4 ACF ACF ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ●●● ●●●● ●●●●●●●● ●●●●●●●●●●●●●●●● ● 0 10 20 30 40 0.2 0.0 0.0 0.2 ● ● ● ● ● ● ●● ●● ●●● ●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●● 50 0 10 20 Lag 40 50 1.0 (d) ○ : X1 |S = v 4 : X2 |S = v △ ● ● ○ : X1 |S = v 4 : X2 |S = v △ ● + : X3 |S = v ＋ ● ● + : X3 |S = v ＋ 0.8 1.0 (c) ● 0.8 30 Lag ● 0.0 0 0.6 10 20 30 40 50 Lag Lag ● ● 0.4 ● ● ● ● ● ● ● ● ● ●● ●● ●● ●● ●●●● ●●●●●●● ●●●●●●●●●●●●●●●●●● 0.2 0.4 ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ●● ●●● ●●● ●●●●●● ●●●●●●● ●●●●● 0.0 ● ACF ● 0.2 ACF ACF 0.6 ● 0 10 20 30 40 50 Lag Lag Figure 1: Autocorrelation plots of the Markov chains generated by algorithm 2: (a) Pareto + π-rotated Clayton, (b) Pareto + t copula, (c) Student’s t + π-rotated Clayton, and (d) Student’s t + t copula. The dotted line represents y = 0.1. (1) 0.556, (2) 0.186, (3) 0.643, and (4) 0.757. In Fig. 1, we can observe that the autocorrelation plots steadily decline below 0.1 by the lag h around 30 for all risk models. Together with the moderate acceptance rates, we can confirm that the choice of the proposal distributions above are appropriate for all risk models. Before showing the estimation results, we check the shapes of the conditional distributions FX 0 |S=v by plotting the Markov chains generated by Algorithm 2. Fig. 2 shows the contour plots of the generated Markov chains. According to these plots, the features of the conditional 18 (b) 20 20 X2 |S X2 = v X2 15 04 10 15 0.0 055 10 00 5 0.0 03 0.0 01 0.00 5 45 5 0.00 3 0.0025 0.0015 5 10 0. 0.0 0 4 00 2 0.004 0.001 0.002 0.00 3 0 0 5e−04 0 0.006 0.0 0. 05 0.0 0.002 0.0035 0.01 25 25 30 35 (a) 15 20 25 30 35 0 5 10 15 X1 X1 25 20 (d) 0.001 15 15 20 (c) 20 0.001 10 0.003 0.003 X2 0.01 0.00 5 0.0 1 8 0.009 4 0.002 −10 0.004 0.002 0.001 −15 0 5 0.00 −5 00 0.00 0.006 0 0.01 5 5 07 1 10 X2 |S X2 = v 6 00 0.0 0.007 0. 0.005 0. 0.006 0 5 10 15 20 X1 |SX1= v −15 −10 −5 0 5 10 15 20 X1 |SX1= v Figure 2: Contour plots of the Markov chains generated by algorithm 2: (a) Pareto + π-rotated Clayton, (b) Pareto + t copula, (c) Student’s t + π-rotated Clayton, and (d) Student’s t + t copula. When drawing the contour plots, we used subsamples, which are picked up every 100-th points of the original Markov chains, to elliminate the dependence among samples. We plot only the first and second variables of the 3-dimensional Markov chains since the last variable can be determined by the sum-constraint. The gray diagonal lines in the plots represent y = −x + v, where v is the estimate of VaRp (S). The contour plots approximately express the conditional distributions FX 0 |S=v . distributions FX 0 |S=v are summarized as follows: (1) Pareto + π-rotated Clayton : the contour plot Fig. 2 (a) shows that FX 0 |S=v has a unique mode at the center of the Simplex. Most probability of FX 0 |S=v is concentrated around the mode, and the probability decreases as going far away from the center. Also, the contour plot is almost symmetric at the diagonal line y = x. (2) Pareto + t copula : Unlike the case (1), FX 0 |S=v possesses two distinct modes near the xand y-axes. Especially near the y-axis, there is a peak around which the probability changes significantly. Also, the contour plot Fig. 2 (b) is asymmetric at the diagonal line y = x. (3) Student’s t + π-rotated Clayton : Although X 0 |S = v can take negative values, it is supported almost on the bounded simplex as in the case (1). The contour plot Fig. 2 (c) is unimodal and almost symmetric around y = x. Also, the tail of FX 0 |S=v is apparently light. (4) Student’s t + t copula : As discussed in Section 4.3, the conditional distribution FX 0 |S=v in this case follows the Pearson type VII distribution given by (4.19). From the contour plot Fig. 2 (d), we can observe some properties of this type of distribution, such as elliptical symmetry 19 and tail-heaviness. Unlike the case (3), X 0 |S = v takes large negative values beyond the bounded simplex. The results of our estimation are summarized in Table 1. In the four different risk models (1)(4), we report the estimates, its approximated standard errors, biases, and Root MSE (RMSE)s of the four different estimators MC, NW, GR, and MCMC. In the first risk model where marginals follow Pareto distributions and they have a π-rotated Clayton copula, true VaR contributions can be computed explicitly. The true value is obtained by allocating the total VaR homogeneously because the marginal distributions are homogeneous and the copula is exchangeable. In this case, we observe that the MC and the NW estimators have relatively large biases. In contrast, the GR and the MCMC estimators have smaller biases compared with the first two. The GR estimator still has some bias caused by the model-misspecification, although its standard error is quite small. In the viewpoint of MSE, the MCMC estimator outperforms all the other estimators. The second risk model has the same marginal distributions as in the case (1), but their copula is t copula having upper-lower tail dependence. Even though the true VaR contributions are not available in this case, we can construct its 95% confidence interval based on the MCMC estimator by using (3.12). Based on this interval of the true value, we computed the range of the bias and RMSE for other estimators. The reported ranges of bias and RMSE in Table 1 (2) show that the MCMC estimator significantly improves the performance of other estimators. Under the risk model (2), all estimators suffer from bias caused by asymmetry and multi-modality of the conditional distribution FX 0 |S=v as seen in Fig. 2 (b). Especially, in contrast with the good performance of the GR in the case (1), the GR estimator has relatively large bias and RMSE in this case. The third risk model examines the case where the marginal distributions are student’s t distribution and their copula is π-rotated Clayton. The true VaR contributions can be computed similarly as in the case (1). Thanks to symmetry and uni-modality of the conditional distribution FX 0 |S=v , all estimators remain small bias and RMSE. Together with the results in the cases (1) and (2), it can be found that the GR estimator performs well so long as FX 0 |S=v does not have irregular shape, such as asymmetry and multi-modality. Also, the MCMC estimator reduces bias and RMSE compared with the MC and the NW estimators. The final risk model illustrates the case where the underlying portfolio loss vector follows the multivariate student’s t distribution. In this case, the true VaR contributions is available by the formula (2.17). In such an elliptical case, the GR estimator provides a quite accurate estimate. Although the conditional distribution FX 0 |S=v is heavy-tailed as seen in Fig. 2 (d), the MCMC estimator keeps high performance compared with the MC and the NW estimators. Its bias is significantly improved compared with the MC and the NW. Additionally, its standard error and RMSE are also lower than those of the MC estimator. Throughout the numerical study, the MCMC estimator provided significantly small bias and RMSE regardless of the shape of the conditional distribution FX 0 |S=v . Under the case when FX 0 |S=v is unimodal and symmetric, the GR estimator also had good performance. On the other hand, at least in our numerical study, the MC and the NW estimator had larger bias and RMSE compared with other estimators. At the end of this section, we summarize the advantages of the MCMC estimator compared with the other estimators. First, the MCMC estimator is consistent although the other estimators are not always. As explained in Section 2, the MC, the NW, and the GR estimators have biases which can not be evaluated easily. In Table 1, we observed that these estimators have some 20 Table 1: Performance of the four different estimators of VaR contributions under four different risk models† . √ Standard error ( M SE): Estimate of AC (Bias): Estimator MC NW GR M CM C MC GR M CM C 0.166 (0.169) 0.164 (0.497) 0.167 (0.655) 0.008 (0.033) 0.008 (0.061) 0.008 (0.029) 0.016 (0.017) 0.017 (0.017) 0.016 (0.018) 0.247 (0.250, 0.694) 0.226 (0.229, 0.687) 0.142 (0.147, 0.664) 0.010 (0.034, 0.532) 0.010 (0.010, 0.532) 0.006 (0.033, 0.532) 0.050 (0.050, 0.110) 0.031 (0.031, 0.103) 0.030 (0.030, 0.102) (1) Pareto + π-rotated Clayton: True AC = (11.265, 11.265, 11.265) AC1 AC2 AC3 11.298 (0.033) 10.796 (-0.469) 10.631 (-0.634) 11.550 (0.2859) 11.481 (0.216) 10.762 (-0.503) 11.297 (0.033) 11.204 (-0.061) 11.293 (0.028) 11.258 (-0.007) 11.263 (-0.001) 11.273 (0.008) (2) Pareto + t copula: True AC is not available ‡ AC1 AC2 AC3 7.475 (0.039, 0.234) 8.156 (-0.649, -0.526) 11.803 (-0.371, -0.255) 6.907 (-0.529, -0.333) 8.878 (0.072, 0.196) 12.415 (0.241, 0.357) 7.772 (0.336, 0.532) 8.715 (-0.090, 0.033) 11.710 (-0.463, -0.347) 7.338 (-0.098, 0.098) 8.744 (-0.062, 0.062) 12.115 (-0.058, 0.058) (3) Student’s t + π-rotated Clayton : True AC = (5.917, 5.917, 5.917) AC1 AC2 AC3 5.919 (0.002) 5.710 (-0.208) 5.638 (-0.279) 6.046 (0.129) 6.009 (0.092) 5.702 (-0.215) 5.921 (0.003) 5.918 (0.001) 5.913 (-0.004) 5.915 (-0.002) 5.940 (0.023) 5.897 (-0.021) 0.077 (0.077) 0.077 (0.221) 0.078 (0.290) 0.005 (0.006) 0.005 (0.006) 0.005 (0.007) 0.016 (0.016) 0.016 (0.028) 0.014 (0.025) 0.007 (0.019) 0.006 (0.015) 0.002 (0.005) 0.034 (0.046) 0.033 (0.044) 0.011 (0.011) (4) Student’s t + t copula : True AC = (2.972, 3.715, 6.686) AC1 AC2 AC3 † 3.028 (0.056) 3.482 (-0.233) 6.526 (-0.161) 3.559 (0.587) 3.053 (-0.662) 6.736 (0.050) 2.989 (0.017) 3.701 (-0.013) 6.682 (-0.004) 3.004 (0.032) 3.687 (-0.028) 6.683 (-0.004) 0.125 (0.136) 0.112 (0.259) 0.046 (0.167) The estimate is computed for the Monte Carlo M C, the Nadaraya Watson N W , the generalized re- gression GR, and the Markov chain Monte Carlo M CM C estimators. The standard error is computed except for the NW estimator. The bias and RMSE are also computed for the cases (1), (3), and (4), where true VaR contributions are available. The sample size is N = 106 . The proposal distributions of the MCMC estimators are (1) Random Walk, (2) Independent, (3) MpCN, and (4) MpCN. ‡ In the case (2), where true VaR contributions are not available, the range of bias and RMSE are computed based on the 95% confidence interval of the true VaR contributions derived by the MCMC estimator (3.12). c N , we compute the range of Based on the confidence interval ACL < AC < ACU and the estimate AC c N − ACU , AC c N − ACL ). The range of RMSE is computed in the same way. the bias (AC 21 biases even when their standard errors are sufficiently small. In contrast, the MCMC estimator provides the accurate estimate of the VaR contributions due to its consistency. Together with the CLT, we also can construct the confidence interval of the true VaR contributions. Second, the MCMC estimator has great sample efficiency compared with the MC estimator. Since the MCMC estimator generates sample not from FX but from FX 0 |S=v , no sample have to be discarded. In addition, unlike the MC, there is no trade-off between the bias and the sample efficiency. Thanks to these properties, the MCMC estimator can hold low standard error without increasing the bias. Finally, the MCMC estimator can maintain high performance even when the conditional distribution FX 0 |S=v has irregular shape, such as multi-modality and heavy-tailed. As compared with the results in the case (2) and others in Table 1, the performance of the GR estimator highly depends on the shape of FX 0 |S=v . For the existing estimators, it is generally difficult to reflect the shape of FX 0 |S=v to their hyper-parameters. On the other hand, for the MCMC estimator, we can directly capture the shape of FX 0 |S=v through the proposal distribution q. By choosing an appropriate proposal distribution q, the MCMC estimator has stably good performance regardess of the shape of FX 0 |S=v . 5.2 Empirical study of the proposal distribution In Section 5.1, we have shown that the MCMC estimator improves the performance of estimating VaR contributions. However, the performance of the MCMC estimator highly depends on the appropriate choice of the proposal distribution q. Additionally, the choice of proposal distribution is less apparent compared with that of hyper-parameters for the other estimators. In this section, we firstly investigate the symptoms caused by the bad choice of q. Then, we consider how to overcome this problem based on the numerical results. Bad choice of q is largely classified into two cases. One is that, the proposal distribution q often generates a candidate at which the probability measured by π is quite small. This case occurs for example when the variance of q is much larger than that of π, or when q does not capture the irregular shape of π as appeared in Fig. 2 (b). In such cases, the Markov chain moves quite slowly, yielding high asymptotic standard error of the MCMC estimator. This symptom appears as a quite low acceptance rate and high autocorrelation plots. The other case is that the proposal distribution q generates only some parts of the whole support of the target distribution π. This case happens for example when π has distinct local modes and the variance of q is so small that the chain can not pass between them. In this case, the estimate has a significant bias, although the acceptance rate and the autocorrelation plots are seemingly fine. This symptom appears as a distorted plot of the MCMC samples whose shape is clearly different from the target distribution π. How can we detect and avoid such fallacious estimates? First of all, as mentioned in Section 3.2, it is indispensable to check the acceptance rate and the autocorrelation plots. Additionally, it is also important to draw the plots of the generated Markov chain, and compare them with the plots of the MC samples whose sums belong to Aδ = [v − δ, v + δ]. Since such MC samples follow the distribution FX 0 |S∈Aδ , we can recognize the distortion of the generated Markov chain by comparing with the two plots of the samples. In the case of our numerical study, Fig. 3 shows the scatter plots of the MC samples whose sums belong to [v − δ, v + δ], overlaid on the scatter plots of the MCMC samples. We can check in Fig. 3 that the shape of the scatter plots of the MCMC samples bear striking resemblance to those of the MC samples for all risk models. Finally, we found that dependence information of the underlying risk model is quite helpful for the selection of q. When the copula C of the underlying risk model represents only upper 22 (b) 35 (a) 0 5 10 15 20 25 30 25 20 X2 5 10 15 ● ●● ● ● ● ● ●● ●● ●●● ●●● ●●● ●● ● ●●●●● ● ● ●● ●● ●● ● ●● ● ●● ● ● ● ● ● ●●● ●● ● ● ● ● ● ●●●● ● ● ● ● ● ● ● ● ● ●●● ● ●●●● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ●● ●●●● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ●● ● ● ● ● ● ● ●● ●● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ●● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ●● ●●● ● ● ● ● ●● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 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We plot the MC samples generated from FX such that their sums belong to Aδ = [v − δ, v + δ]. In the four risk models, the value of δ are (1) 4.8, (2) 3.9, (3) 2.2, and (4) 1.7. When drawing the scatter plots of the MCMC samples, we used subsamples, which are picked up every 100-th points among the original Markov chains. We plot only the first and second variables of 3-dimensional samples since the last variable is determined by the sum-constraint. The plots of the MC, and the MCMC samples express the distribution FX 0 |S∈Aδ and FX 0 |S=v , respectively. and/or lower tail dependence, then the target distribution π is likely to be tractable. For example in the cases of risk models (1) and (3), where the copula C has an upper tail dependence, the contour plots Fig. 2 (a) and (c) show that the target distribution π is unimodal and light-tailed. These features facilitate the estimation with MCMC since simple proposal distributions such as Random Walk proposal (3.14) perform well. Conversely, when the copula C has upper-lower tail dependence, then the target π tends to be complicated. In the cases (2) and (4), where the copula C has upper-lower tail dependence, the Fig. 2 (b) indicates that the π is bimodal, and also the contour plot Fig. 2 (d) shows that the π is heavy-tailed. In such cases, careful selection of the proposal function q is required for the good performance of the MCMC estimator. For instance, to capture the bi-modality appeared in the contour plot Fig. 2 (b), one of the good choice of q is an independent proposal distribution (3.15) with f the Dirichlet distribution on the v-simplex Sv . Since Dirichlet distribution can possess two distinct modes around the edges of the simplex, we can generate a candidate which often visits the two separated modes of the π. In the case (4), to overcome the problem of heavy-tailedness of π seen in the contour plot Fig. 2 (d), it is necessary to adopt some specialized MCMC for heavy-tailed target distribution, such as MpCN (3.16). 23 6 Concluding remarks Computing VaR contributions for a continuous risk model is a difficult task in general. There is no standard method since existing estimators have their own deficiencies. Especially, it can be seen that the crude Monte Carlo estimator suffers from unavoidable bias to the true VaR contributions. Our MCMC estimator proposed enables the consistent estimation of VaR contributions. Its sample efficiency is significantly improved because the MCMC estimator is computed by samples generated directly from the conditional density of a portfolio loss given the sum constraint. Moreover, since the MCMC estimator can capture the features of the risk model more directly than the existing estimators, it can maintain high performance even when the shape of the underlying loss distribution is far from elliptical. We showed some theoretical properties of the MCMC estimator, such as consistency and asymptotic normality. The consistency follows naturally from the construction of the estimator. We provided some conditions and example where CLT holds. By the general theory of MCMC, its asymptotic variance can be consistently estimated. We numerically compared the performance of various estimators of VaR contributions for typical risk models used in practice. The simulation results showed that, in most risk models we considered, the MCMC estimator had smaller bias and MSE compared with the existing estimators. Although the MCMC estimator has good theoretical properties and numerically shows great performance, some remaining issues merit future research. Firstly, although we provided guidelines to choose good proposal distribution, they are hardly applicable when the dimension of the target distribution is too high to visualize. For such high-dimensional cases, asymptotic analysis and adaptive method should be developed to chooce good proposal distribution. Secondly, theoretical investigation of the conditional joint distribution given the sum constraint is still unsatisfactory. Our concern mainly lies with the influence of the underlying copula of a risk model to the shape of the conditional joint distribution, such as tail behavior and multi-modality. We believe that revealing these relations helps to find a better proposal distribution in our context of estimating VaR contributions. Finally, our MCMC method can potentially be extended to the computation of Expected shortfall (ES) contributions. The ES contributions can be derived as a conditional expectation of losses given that the total loss is equal to or greater than VaR (see Tasche, 2001). Due to its similar structure to the VaR contributions, the ES contributions can also be efficiently estimated in an analogous method proposed in this paper. This, and other potential extensions are currently being studied by us. Acknowledgements We would like to thank Paul Embrechts for his valuable comments and Kengo Kamatani for a fruitful discussion on MCMC. Our research is supported by the Core-to-Core program (Japan Society for the Promotion of Science: JSPS) in Keio Univeristy. References [1] C. Bluhm, L. Overbeck, and C. Wagner. Introduction to credit risk modeling. CRC Press, 2016. [2] S. Chib and E. Greenberg. Understanding the metropolis-hastings algorithm. The american statistician, 49(4), 327-335, 1995. [3] M. Denault. Coherent allocation of risk capital. Journal of Risk, 4(1):1-34, 2001. 24 [4] A. Dev, editor. 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A Scheduling and Tiling Reductions on Realistic machines arXiv:1801.05909v1 [] 18 Jan 2018 Nirmal Prajapati October 25, 2016 Computations, where the number of results is much smaller than the input data and are produced through some sort of accumulation, are called Reductions. Reductions appear in many scientific applications. Usually, reductions admit an associative and commutative binary operator over accumulation. Reductions are therefore highly parallel. Given unbounded fan-in, one can execute a reduction in constant/linear time provided that the data is available. However, due to the fact that real machines have bounded fan-in, accumulations cannot be performed in one time step and have to be broken into parts. Thus, a (partial) serialization of reductions becomes necessary. This makes scheduling reductions a difficult and interesting problem. There have been a number of research works in the context of scheduling reductions. We focus on the scheduling techniques presented in [Gupta et al. 2002], identify a potential issue in their scheduling algorithm and provide a solution. In addition, we demonstrate how these scheduling techniques can be extended to “tile" reductions and briefly survey other studies that address the problem of scheduling reductions. 1. INTRODUCTION Reductions are those computations in which an associative and commutative operator accumulate a set of points into a single value. Following example illustrates such a computation. for i = 1 to N A[i] += B[i-1]; The reduction operator is associative and commutative, which implies that accumulations need not admit any order. Therefore, we should be able to exploit parallelism. With an unbounded number of processors and unbounded fan-in operator, accumulations can be done in a single time step. However, for a real machine, both the number of processors and the fan-in are bounded. This necessitates ordering of accumulations. Many scientific and engineering applications spend most of their execution time in nested loops. The task of optimizing such nested loops involve dataflow analysis of the program [Feautrier 1991]. The dataflow analysis of the above program reflects loop-carried data dependences that prevent parallelism. Therefore, scheduling reductions is a difficult problem. A multidimensional affine function that imposes a particular order of execution is called a Schedule. A schedule must satisfy the precedence constraints imposed by the dependences. Polyhedral model renders a powerful abstraction that enables precise reasoning for the legality of transformations. Iterations of the nested loops can be viewed as integer points in a polyhedron. The computations impose dependence constraints which are represented as affine functions of the indices. Reductions can also be described as Systems of Affine Recurrence Equations (SAREs) over polyhedral domains. Reduction dependences are implicit in the SAREs. Such a representation allows us to focus on non-reduction dependences that must be satisfied by the schedule. [Redon and Feautrier 1994] present a scheduling technique that optimally schedules reductions over CRCW PRAM model. They assumed that accumulations can happen in one time step. This approach was extended by [Gupta et al. 2002] to work on realistic machines. They invented a scheduling technique for machines with binary operators and exclusive writes. They claim that on such machines, their scheduling method gives efficient solutions with a constant factor slowdown compared to best possible schedules on a CRCW PRAM model. We discover a flaw in their technique which violates their claim of “exclusive writes". Using a counter example, we prove that their scheduling technique requires a machine with concurrent writes. We solve this problem by introducing additional causality constraints and show that the modified scheduling constraints guarantee a schedule with exclusive writes. Furthermore, we extend the scheduling approach to tile reductions. Tiling [Wolfe 1987; Irigoin and Triolet 1988] is a classic iteration space partitioning technique which combines a set of points into tiles, where each tile can be executed atomically. Tiling comes in handy for exploiting data locality [Wolf and Lam 1991], minimizing communication [Andonov et al. 2001; Xue 1997] and maximizing parallelism. Reductions are usually serialized before tiling. Most of the tiling techniques such as [Bondhugula et al. 2008a; Doerfert et al. 2015] take as input serialized reduction programs. Loop transformation techniques are used to find tiling hyperplanes. In the majority of the cases, serializing imposes uniform dependences which are tileable. Serializing reductions, however, negates the fact that accumulations can be carried out in any order and imposes unnecessary intra-tile and inter-tile dependences. The tiling techniques developed in this paper maximize parallelism as well as improve data locality in a work efficient manner. We also discuss other limitations of Gupta’s scheduling algorithm and suggest possible improvements. We highlight the unexplored areas and address other related work in the context of scheduling reductions. The rest of the paper is organized as follows. Section 2 describes necessary background. In section 3, we describe [Gupta et al. 2002] scheduling technique using examples. Section 4 exposes the flaw in their technique using a counter example which shows that “exclusive writes" condition is violated and proposes a solution that is guaranteed to give schedules for machines with “exclusive writes". Section 5 extends this scheduling technique to tile reductions. Section 6 suggests other possible improvements, section 7 discusses related work and section 8 concludes the paper. 2. BACKGROUND A reduction of variable X can be represented as X = reduce(⊕, f, R) (1) where ⊕ is an associative and commutative accumulation operator, the body of the reduction is some variable R, and f is a projection function that maps a subset of the points in the domain of R, represented as Dom(R), to zX ∈ Dom(X). For every zX ∈ Dom(X), the Parametrized Reduction Domain of zX , referred   as P (zX ), is the set of points in R that are z mapped to zX by the function f = z → AP + cP where AP is a constant matrix, cP is p a constant vector and p is a vector of the size parameters. 2.1. Schedule Schedule of a variable X is a vector that represents the time instant at which zX ∈ Dom(X) is computed and is given by λX (zX ), a multidimensional affine function. For any two variables X and Y , if X(z) depends on Y (z 0 ) then Y (z 0 ) must be computed before X(z). The dependence imposes the following causality constraints on the schedule λX (z)  λY (z 0 ) + TeqX (z, z 0 ) (2) where TeqX (z, z 0 ) is the time to compute the RHS of the equation X(z) after Y (z 0 ) becomes available. [Redon and Feautrier 1994] scheduling algorithm assumes a CRCW PRAM with unbounded number of processors and unbounded fan-in operators. The reductions can, therefore, be performed in a single time step. The causality constraints for equation (1) on such a machine are given by 2 zX ∈ Dom(X), zR ∈ P (zX ), λX (zX )  λR (zR ) + 1 (3) However, this technique is not applicable for scheduling reductions on real machines. [Gupta et al. 2002] developed a technique that schedules reductions on realistic machines with bounded fan-in (binary) operators and exclusive writes. They claim that their scheduling algorithm generates efficient schedules with a constant fold slow down compared to optimal schedules obtained on a CRCW PRAM model. The following section explains in detail the scheduling algorithm as presented in [Gupta et al. 2002]. 3. GUPTA’S SCHEDULING ALGORITHM Let λR (zR ) be the schedule of zR ∈ Dom(R). λR (zR ) = t are the equitemporal hyperplanes, or “slices", defined as the set of points in zR ∈ P (zX ) that become available for accumulation at time t. A temporary variable T empX(zX , t) is defined as follows          AP z c T empX = reduce ⊕, z → + P ,R (4) ΛR p αR   z where λR (zR ) = ΛR R + αR is the schedule for R. p T empX(zX , t) are the partial accumulations of equitemporal hyperplanes in P (zX ). These intermediate results are accumulated to get final answer zX ∈ Dom(X). Equation (1) is modified as X = reduce(⊕, (zX , t → zX ), T empX) (5) Consider, X as the following reduction: X= i=N,j=i X R(i, j) (6) i=0,j=0 Dom(R) = {i, j|0 ≤ j ≤ i ≤ N } Assume, λR (zR ) = i are the equitemporal hyperplanes. With this information, we can deduce that there are N equitemporal hyperplanes. i.e. Dom(T empX) has N elements where the t-th element is a partial accumulation of the t-th equitemporal hyperplane in P (zX ). Figure 1(b) shows the new set of equations obtained after decomposing the reduction as shown in equations (4) and (5). Correspondingly, the causality constraints in equation (2) are updated to accommodate T empX. f (zX ) ≤ t ≤ l(zX ), λT empX (zX , t)  t + TeqT empX (zX , t) λX (zX )  λT empX (zX , t) + TeqX (zX , t) (7) (8) where f (zX ) and l(zX ) are the first and last time steps at which values in P(zX ) are available. TeqT empX (zX , t) is the time to compute the reduction of all the values in the equitemporal hyperplane (zX , t). For a machine with binary operators, size(zX , t) gives the time to accumulate the equitemporal hyperplane at (zX , t). The number of time steps 3 Fig. 1: Gupta’s scheduling technique for the equation in (a). required for linear accumulation of all values in any box B is defined as its size, which is also equal to the 1-norm of its principal diagonal. For the example in Figure 1(a), the size(zX , t) of a hyperplane is given by t = i. Figure 1(c) shows the iteration space and the table in Figure 1(d) give the sizes of the equitemporal hyperplanes at (zX , t). The scheduling constraints in equations (7) and (8) get reduced to size(zX , t)  λX (zX ) − t (9) The slack of an equitemporal hyperplane is defined as sl(zX , t) = λX (zX ) − t. The scheduling constraints in (9) are further modified to sl(zX , t)  size(zX , t) (10) Note that by definition size(zX , t) is always one-dimensional scalar. If slack is truly multidimensional then the constraint in (10) is trivially satisfied. However, if the slack is not multidimensional, then the value of the innermost dimension of slack should be greater than the size(zX , t). These causality constraints can be formulated as an integer linear program and solved using a PIP [Feautrier 1988] solver. Using the above constraints to formulate causality for the example in Figure 1, we obtain schedule λX (zX ) as shown in Figure 1(d). For the same example, assume that λR (zR ) = i − j are the equitemporal hyperplanes. Figure 2(a) shows the hyperplanes at which values in P (zX ) become available for accumu4 Fig. 2: Gupta’ scheduling technique for equation (6), given λR (zR ) = i − j. lation. For this example, P (zX ) = Dom(R). Again, there are N equitemporal hyperplanes. i.e. Dom(T empX) has N elements where the t − th element is a partial accumulation of the t − th equitemporal hyperplane in zX . The size(zX , t) of each hyperplane is shown in Figure 2(b). We obtain the schedule for zX such that it requires a machine with concurrent writes. As shown in Figure 2(b), the accumulations are scheduled at the same time step. In such cases, [Gupta et al. 2002] suggests that we slow down the schedule by a factor of 2 in the innermost dimension of time and obtain enough time for the accumulations in equation (5) on a machine with binary operators. Figures 1(c) and (d) show how this modification easily solves the problem. 5 In the next section, we provide a counter example to show that this modification does not solve the problem of concurrent writes in all cases. 4. COUNTER EXAMPLE Consider, variable X as the following reduction equation: X(i) = j=i X R(i, j) (11) j=0 Dom(R) = {i, j|0 ≤ j ≤ i ≤ N } Here, X is a one-dimensional variable. Dom(R) is again two-dimensional. Assume, λR (zR ) = j are the equitemporal hyperplanes. With this information, we can see that there are j equitemporal hyperplanes. i.e. the slice at (zX , t) has a single element. The t − th element is a partial accumulation of the t − th equitemporal hyperplane in zX which is a single point, therefore, there are no partial accumulations. The size of all the equitemporal hyperplanes is equal to 1. Figure 3(c) shows the iteration space of equation (11) and Figure 3(d) shows the desird schedule. Using these constraints, we will now solve for t. Assume, we get λR (zR ) = t = 0. The schedule λX (zX ) is calculated using the constraint in equation (10). This constraint is trivially satisfied with a size of 1 for all equitemporal hyperplanes. Figures 3(e) and (f) show the issue. Here the schedule is zero-dimensional and we can no longer slow it down by a constant factor to accumulate the intermediate results. This violates the “exclusive write" condition and requires a machine with Concurrent Writes!. In the previous example, note that the number of values to be accumulated into zX is more than then number of time steps between f (zX ) and l(zX ). This prevents linear accumulation. We define size0 (zX ) as the total number of time steps required for linear accumulation of all values in the reduction body T empX(zX , t) of X in equation (5). Let the total number of time steps between f (zX ) and l(zX ) be TzX . If the condition ∀zX ∈ Dom(X), TzX + 1 > size0 (zX ) (12) is satisfied, then concurrent writes in zX can be avoided. Suppose, f (zX ) = 1 and l(zX ) = N . Therefore, TzX = N . If size0 (zX ) < N , then the condition is trivially satisfied. However, this constraint cannot be satisfied for the example (11) where TzX = 1 and size0 (zX ) = N . Let the total number of equitemporal hyperplanes in P (zX ) be EzX . Again, if EzX < size0 (zX ) < TzX , then exclusive writes cannot be guaranteed by equation (12). Now let’s see what happens if the condition ∀zX ∈ Dom(X), EzX + 1 > size0 (zX ) (13) is satisfied. There would be enough time steps for the accumulation of the intermediate values into the final answer zX . However, if there is only one equitemporal hyperplane like our example (11), then the constraint in (13) will not be satisfied. Neither the number of equitemporal hyperplanes EzX nor the total number of time steps TzX can be guaranteed to be more than size0 (zX ). Therefore, we suggest the following scheduling constraints λX (zX )  size0 (zX ) (14) in addition to the constraints in (10). Linear accumulations are now guaranteed on a machine with bounded fan-in. The above scheduling technique can be further optimized to get better schedules. In the next section, we will see how this scheduling technique can be extended to tile reductions in order to maximize parallelism and improve data locality. 6 Fig. 3: Gupta’s scheduling technique violates Exclusive Write condition for equation (11). 7 Fig. 4: [Gupta et al. 2002] scheduling technique for equation in (a), given λR (zR ) = i. 5. TILING REDUCTIONS Tiling or blocking computations is a strategy of dividing the iteration space into tiles where each tile is a set of points [Wolfe 1987; Irigoin and Triolet 1988]. A tiling is considered to be legal if there are no dependence based cycles between tiles and if all tiles can be executed atomically. → − Let θX ( i ) define a set of tiling hyperplanes that tile the iteration space of a variable X. For any two variables X and Y , if X depends on Y then the following tiling legality constraint must be satisfied for all the dependencies between X and Y → − → − θX ( i ) − θY ( i ) ≥ 0 (15) The above condition ensures the legality of tiling as shown in [Bondhugula et al. 2008b]. With the knowledge that accumulations can be carried out in any order, we can eliminate the reduction dependences from the dependence set. However, (15) must hold for all other dependencies. 8 Consider the equation shown in Figure 4(a). Figure 4(b) shows the decomposition of X as per equation (4). Figures 4(c) and (d) show the iteration space and schedule obtained using the formulation discussed in Section 3. We provide an incremental approach to finding tiling hyperplanes for reductions. We first show how equitemporal hyperplanes can be tiled, succeeded by tiling P (zX ) and finally suggest possible tilings for the reduction body R. 5.1. Tiling Equitemporal Hyperplanes Tiling an equitemporal hyperplane (zX , t) using any tiling hyperplane is a legal tiling. This is due to the fact that all values in an equitemporal hyperplane are available for accumulation at the same time and that all of them contribute to a single value in T empX. Therefore, the tiling legality condition (15) holds for any tiling hyperplane. We are left with many possible choices. We choose orthogonal tiling hyperplanes with tiles of size s in every dimension. We introduce a new variable T ileT empX such that (zX , t, b) ∈ Dom(T ileT empX ) maps to the bth tile in T empX, where (zX , t) ∈ Dom(T empX). T ileT empX = reduce ⊕, z → AP ΛR γ ! z p s !  c + P αR ! ! ,R (16) where γ is a function that divides every dimension of the equitemporal hyperplane at (zX , t) into tiles of size s such that 1 ≤ b ≤ τ (zX , t), where τ (zX , t) is the total number of tiles in the slice at (zX , t). With this definition of variable T ileT empX , equation (4) is modified as T empX = reduce(⊕, (zX , t, b → zX , t), T ileT empX ) (17) T empX(zX , t) is the accumulation of τ (zX , t) tiles. Equation (5) remains unchanged. Consider the equation shown in Figure 5(a). Figure 5(b) shows the decomposition of X as per equations (16) and (17). We now obtain causality constraints for equations (16) and (17). The precedence constraints on T ileT empX state that 1 ≤ b ≤ τ (zX , t), f (zX )  t  l(zX ), zx ∈ Dom(X) λT ileT empX (zX , t, b)  t + TeqT ileT empX (zX , t, b) (18) λT empX (zX , t)  λT ileT empX (zX , t, b) + TeqT empX (zX , t) (19) In an equitemporal hyperplane, all the tiles can be executed simultaneously. TeqT ileT empX (zX , t, b) is given by the size of the tile. We assume that tile size is s in every dimension. Let the number of dimension of the equitemporal hyperplane be d(zX , t). Hence, size of a tile (zX , t, b) will be given by d(zX , t) × s. We get λT ileT empX (zX , t, b)  t + [d(zX , t) × s] (20) as the constraints for λT ileT empX (zX , t, b). TeqT empX (zX , t) of T empX, defined as the size(zX , t), is the time it takes to accumulate all the partial answers produced by each tile, which is also equal to τ (zX , t). Therefore, causality constraints on the schedule for equation (17) can be formulated as: λT empX (zX , t)  t + τ (zX , t) + [d(zX , t) × s] 9 (21) Fig. 5: Extending the scheduling techniques for tiling equitemporal hyperplanes for the equation in (a), given λR (zR ) = i. Similarly, we deduce the following as the causality constraints for equation (5). λX (zX )  t + τ (zX , t) + [d(zX , t) × s] 0 λX (zX )  size (zX ) (22) (23) Figures 5(c) and (d) show the iteration space and schedule obtained using the formulation discussed above. Using tile size s = 3 and N = 9, we get the number of tiles τ = 3 and hence λX (zX ) can be scheduled as early as 15. Note, when s does not equally divide every dimension of an equitemporal hyperplane, we get partial tiles whose size is smaller than full tiles. Therefore, above causality constraints are satisfied trivially for partial tiles. With the above formulation, there are many possible choices for tile size s. If we choose s = 1, then there will be only one point in each tile. We provide a cost function that leads to good tile size and maximizes parallelism. 10 5.1.1. Towards finding good solutions. To minimize the total execution time, reductions of an equitemporal hyperplane can be implemented using a binary-tree-like algorithm. However, this is not work-efficient. To accumulate N elements, binary-tree algorithms take logN steps. Brent’s theorem suggests N/logN parallel instances where each instance performs logN work [Brent 1999]. Later, all N/logN parallel instances contribute to the final accumulation of the partial results. The cost is now given by (N/logN )∗logN = N , which is work efficient. Our proposed cost function provides such work efficient solutions. We seek to minimize the following cost function Let, ω = [d(zX , t) × s] − τ (zX , t) minimize(ω) (24) The tile size s used in Figures 5(c) and (d) reflect the result of the cost function (24). Note that our tile size optimization function assumes equal tile size in every dimension. This restriction can be lifted to enable rectangular tiling. This technique of tiling equitemporal hyperplanes applies to those cases where the equitemporal hyperplane is at least one dimensional. If the hyperplanes are zero-dimensional then above formulation does not apply. This motivates tiling across equitemporal hyperplanes. 5.2. Tiling Parametrized Reduction Domain of zX As shown above, tiling equitemporal hyperplanes is straightforward. Let us consider tiling the Parameterized reduction domain of zX . If there are no dependencies between equitemporal hyperplanes, then orthogonal tiling hyperplane can be chosen and all tiles can be launched independently. The partial answers can be accumulated to get the final answer zX . However, if there exists dependence in t, then orthogonal tiling hyperplanes might not be legal. Notice, the only dependences that affect tiling legality are between equitemporal hyperplanes (as per the definition of slices, all values become available for accumulation in an equitemporal hyperplane at the same time). The problem of tiling P (zX ) is thus reduced to time tiling. A well-known approach is to enforce forward communication only constraint as presented in [Griebl et al. 2005]. If we want to maximize parallelism and minimize synchronization, then we can use the time-partitioning technique of [Lim and Lam 1998]. If minimizing communication is also desired optimization, then the cost optimization techniques of [Bondhugula et al. 2008a] can be used. If the dependences are uniform, then time tiling techniques for stencil computations such as [Tang et al. 2011; Grosser et al. 2014; Bondhugula et al. 2016] can be used for maximizing parallelism together with the concurrent start. The dependences in t impose additional constraints on tile sizes. The tile size along each tiling hyperplane must be greater than the length of the longest dependence in the hyperplane. Using these tiling hyperplanes and additional dependence based constraints, we can now optimize for tile size with our cost function. We can eliminate the variable T empX and use only one variable T ileX to tile P (zX ). − T ileX is defined using an affine function → γ which represents the tiling hyperplanes and the tile size s. X is now an accumulation of the partial answers produced by each tile. Additional analysis is needed to tile the reduction body. Let R be the reduction body of X. We want to tile all the d dimensions of the reduction body using the same approach as mentioned above. In order to do so, the schedule of tiles must admit the schedules of both variables R and X. Tiling hyperplanes such that the tiling legality constraint (15) is satisfied can be found. Reduction body can be partitioned using these tiling hyperplanes to decompose the reduction. 11 6. FURTHER IMPROVEMENTS We identify the following potential areas of improvements to the scheduling techniques presented in this paper. (1) Reiterating over the scheduling approach discussed in section 4, the causality constraints are formulated by making some assumptions regarding λR and then these constraints are simultaneously resolved to find schedules for all the variables. Hence, while formulating the causality constraints it is not possible to recognize the exact equitemporal hyperplanes without the knowledge of λR . The scheduling techniques do not show how to make an optimal choice of equitemporal hyperplanes while formulating the causality constraints. The analysis is, therefore, sub-optimal and can be improved. (2) The suggested scheduling and tiling techniques do not consider the program size parameters. Reconsider the example in Figure 4 (a) with modified size parameters such that 0 ≤ i ≤ M and 0 ≤ i ≤ N . Assume λR (zR ) = i as the equitemporal hyperplanes as shown in Figure 4 (c). The number of equitemporal hyperplanes will be given by M and the size of an equitemporal hyperplane will be given by N . The causality constraints in the equation (10) will impose the condition that the slack must be greater than or equal to the size. Either M ≥ N or M < N . Without the knowledge of the values of N and M , it is not possible to find a schedule that satisfies both the inequalities. Assuming that the values of N and M are known and that M > N , after solving if we get λR (zR ) = t = j which used N as the size of equitemporal hyperplanes, we get an incorrect schedule of X. Therefore, it becomes necessary to consider size parameters. (3) In the situations where equitemporal hyperplanes have different slacks, it is suggested that equitemporal hyperplanes be scheduled in the decreasing order of slack. However, the method does not discuss ordering of equitemporal hyperplanes when they have the same slack. (4) Furthermore, if the reduction operator is not commutative then accumulations must admit an order. The scheduling techniques presented in this paper can be extended to consider non-commutative operators. For example, a reduction computation with an associative but non-commutative operator can be tiled using the techniques presented in Section 5 with additional constraint such that accumulations within a tile are lexicographically ordered. The accumulations of partial answers into the final answer can also be ordered in the lexicographically increasing order of tiles. Such an ordering imposed by non-commutative operator might slow down the schedule by a constant factor. Parallelism can, however, be exploited irrespective of the commutativity of the operator. 7. RELATED WORK The scheduling technique of [Karp et al. 1967] solved the problem of scheduling Systems of Uniform Recurrence Equations (SUREs). Using polyhedral [Rajopadhye et al. 1986] presented a technique for synthesizing systolic architectures from recurrence rquations which enable scheduling Affine Recurrence Equations. [Feautrier 1992a; 1992b] give closed form schedules as affine functions of the indices of a nested loop program. The reader is referred to the book [Darte et al. 2000] which details scheduling algorithms for recurrence equations. The problem of finding schedules in the presence of reductions was initially tackled by [Redon and Feautrier 1994]. They assumed a CRCW PRAM model where accumulations can be carried out in a single time step. They also show that their technique can be extended to machines with a bounded number of processors by serializing reductions using a partialbinary-tree algorithm. However, they did not show how this can be done efficiently. Building over their scheduling technique, [Gupta et al. 2002] developed an algorithm to determine effective serialization of reductions to achieve the fastest possible linear schedules on an exclusive writes machines with bounded fan-in. Scheduling SAREs do not need to consider 12 memory based dependence like [Doerfert et al. 2015; Sato and Iwasaki 2011; Bondhugula et al. 2008a] and hence provide more flexibility. While scheduling SAREs with reductions, the reduction dependences are implicit which also allows for maximal parallelism. Other techniques such as [Pugh and Wonnacott 1994; Stock et al. 2014] make the dependences explicit to improve parallelization. Automatic Parallelization technique such as Polly’s polyhedral optimizer [Doerfert et al. 2015] tries to achieve parallelism by introducing privatization. The problem of finding optimal schedules are directed towards optimizing some cost function that miniminze latency or delay, or maximize fine-grained parallelism [Feautrier 1992a; 1992b; Redon and Feautrier 1994]. Tiling improves data locality and works such as [Bondhugula et al. 2008a] process loops where reductions are serialized. In certain cases, it becomes necessary to find piecewice linear schedules. It is, however, difficult to determine the pieces automatically. In the paper [Wonnacott et al. 2015], the authors show how to find the optimal piece-wise schedule for Optimal String Parenthesization problem and use the Mostly-Tileable technique for tiling. The schedule was, however, found by hand. 8. CONCLUSION We studied previous works that address the problem of scheduling SAREs in the presence of reductions. We show that method of scheduling reductions developed in [Gupta et al. 2002] has an error in the formulation of causality constraints which leads to concurrent writes. We exposed this error with an example and provided a solution that guarantees exclusive writes. The scheduling techniques presented in this paper gives optimal linear schedules. Above all, reductions remain memory-bound computations. Therefore, exploiting data locality using tiling techniques can improve the performance [Wolf and Lam 1991]. Using the knowledge of reduction operator being associative and commutative, we extended Gupta’s scheduling technique to scheduling as well as tiling reductions. Tiling is also useful for coarse-grained parallelism [Lim and Lam 1998; Xue 2000]. We demonstrated that tiling the equitemporal hyperplanes renders maximal parallelism. 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PROBABILITY THEORY AND MATHEMATICAL STATISTICS 2017 International Kazan conference Improved nonparametric estimation of the drift in diffusion processes 1 Evgeny Pchelintsev, arXiv:1710.03550v1 [] 10 Oct 2017 1 1 Svyatoslav Perelevskiy and 1 Irina Makarova Tomsk State University, 36 Lenina avenue, Tomsk, 634050, Russian Federation; [email protected], [email protected], [email protected] http://en.tsu.ru/ Abstract. In this paper, we consider the robust adaptive non parametric estimation problem for the drift coefficient in diffusion processes. An adaptive model selection procedure, based on the improved weighted least square estimates, is proposed. Sharp oracle inequalities for the robust risk have been obtained. Keywords: Improved estimation, stochastic diffusion process, mean-square accuracy, model selection, sharp oracle inequality. 1 Introduction Let (Ω, F , (F )t≥0, P) be a filtred probability space on which the following stochastic differential equation is defined: y. = S(yt ) t. + w.t , 0 ≤ t ≤ T , (1) t where (wt )t≥0 is ascalar standard Wiener process, the initial value y0 is some given constant, and S(·) is an unknown function.The problem is to estimate the function S(x), xin[a, b], from the observations (yt )0≤t≤T . The calibration problem for the model (1) is important in various applications. In particular, it appears, when constructing optimal strategies for investor behavior in diffusion financial markets. It is known that the optimal strategy depends on unknown market parameters, in particular, on unknown drift coefficient S. Therefore, in practical financial calculations it is necessary to use statistical estimates for the function S which are reliable on some fixed time interval [0, T ] [6]. Earlier, the problem of non-asymptotic estimation of the parameters of diffusion processes was studied in [9]. Here it was shown that many difficulties of asymptotic estimation of parameters for one-dimensional diffusion processes can be overcome by using a sequantial approach. It turns out that the theoretical analysis of successive estimates is simpler than the analysis of classical procedures. In particular, it is possible to calculate non-asymptotic bounds for quadratic risk. Owing to the use of a sequential approach, the problems of non-asymptotic estimation of parameters were studied in [1] for multidimensional diffusion processes and recently in [2] for multidimensional continuous and discrete semimartingales. In [7] a truncated sequential method for estimating the parameters of diffusion processes was developed. Now about nonparametric estimation. A consistent approach to nonparametric criteria for minimax estimation of the drift coefficient in (ergodic) diffusion processes was developed in [3]. In this article, sequential pointwise kernel estimates are considered. For such estimates, non-asymptotic upper bounds of the root-mean-square risk are obtained, and these estimates give the optimal convergence rate as T → ∞. This paper deals with the estimating the unknown function S(x), a ≤ x ≤ b, in the sense of the mean square risk Z b 2 2 b b R(ST , S) = ES kST − Sk , kSk = S 2 (x)x. , (2) a 2 E. Pchelintsev, S. Perelevskiy, I. Makarova where SbT is the estimate of S by observations (yt )0≤t≤T , a < b are some real numbers. Here ES is the expectation with respect to the distribution PS of the random process (yt )0≤t≤T given the drift function S. The goal of this paper is to construct an adaptive estimate S ∗ of the drift coefficient S in (1) and to show that the quadratic risk of this estimate is less then the one of the estimate proposed in [3], i.e. we construct the improved estimate in the mean square acuracy sence. For this we use the improved estimation approach proposed in [10] and [8] for parametric regression models and recently developted in [11] for a nonparametric estimation problem. Moreover in this paper we consider the estimation problem in adaptive setting, i.e. when the regulary of S is unknown. For this we use a model selection method proposed in [4]. Such approach provides adaptive solution for the nonparametric estimation through oracle inequalities which give the nonparametric upper bound for the quadratic risk of estimate. The rest of the paper is organized as follows. In section 2 we reduce the initial problem to an estimation problem in a discrete time nonparametric regression model. In section 3 we construct the improved weigted least square estimates. In section 4 the sharp nonasymptotic oracle inequality for quadratic risk of model selection procedure is given. 2 Passage to a discrete time regression model To obtain a reliable estimate of the function S, it is necessary to impose on it certain conditions that are analogous to the periodicity of the deterministic signal in the white noise model [5]. One of the conditions sufficient for this purpose is the assumption that the process (yt )t≥0 in (1) returns to any neighborhood of each points x ∈ [a, b]. As in [3] to get the ergodicity of the process (1) we define the following functional class: ΣL,N = {S ∈ LipL (R) : |S(N)| ≤ L ; ∀|x| ≥ N, ∃ Ṡ(x) ∈ C(R) such that − L ≤ inf Ṡ(x) ≤ sup Ṡ(x) ≤ −1/L} , |x|≥N (3) |x|≥N where L > 1, N > |a| + |b|, Ṡ(x)− derivative S(x), ( LipL (R) = ) |f (x) − f (y)| f ∈ C(R) : sup ≤ L . |x − y| x,y∈R We note that if S ∈ ΣL,N , then there exists an invariant density Rx exp{2 0 S(z)z.} . q(x) = qS (x) = R +∞ Ry exp{2 0 S(z)z.}y. −∞ (4) We note that the functions in ΣL,N are uniformly bounded on [a, b], i.e. s∗ = sup sup S 2 (x) < ∞ . a≤x≤b S∈ΣL,N We start with the partition of the interva [a, b] by the points (xk )1≤k≤n , defined as xk = a + k (b − a) , n (5) Improved nonparametric estimation of the drift in diffusion processes 3 where n = n(T ) is an integer-valued function of T such that n(T ) ≤ T n(T ) = 1. T →∞ T and lim (6) Now at any point xk we estimate the function S by a sequential kernel estimation. We fix some 0 < t0 < T and put  Rt y −x
τk = inf{t ≥ t0 : t Q s h k s. ≥ Hk } ;   0  (7) 1 R τk  ys −xk
  Sek = Q y. , h s Hk t0 where Q(z) = 1{|z|≤1}, 1A is an indicator of the set A, h = (b − a)/(2n) and Hk is a positive threshold, which will be indicated below. From (1) it is easy to obtain that Sek = S(xk ) + ζk . The error ζk is represented as a sum of the approximating and stochastic parts, i.e.  Z τk  ys − xk 1 1 Q ξk , Bk = ∆S(ys , xk )s. , ζk = Bk + p Hk t h Hk 0 where ∆S(y, x) = S(y) − S(x) and 1 ξk = p Hk Z τk Q t0  ys − xk h  w.s . Taking into account that S is Lipshitz function , we obtain an upper bound for the approximating part as |Bk | ≤ Lh . It is easy to see that random variables (ξk )1≤k≤n are independent identically distributed from N (0, 1). In [3] it is established that an effective kernel estimate of the form (7) has a stochastic part distributed as N (0, 2T hqS (xk )), where qS (xk ) is the ergodic density defined in (4). Therefore, for an effective estimate at each point xk by the kernel estimate (7), we need to estimate the density (4) from observations (yt )0≤t≤t0 .To this end, we establish that where ǫT is positive, 0 < ǫT < 1, qeT (xk ) = max{b q(xk ) , ǫT } , 1 qb(xk ) = 2t0 h Z 0 t0 Q  ys − xk h  s. . Now choose the threshold Hk in (7): Hk = (T − t0 )(2e qT (xk ) − ǫ2T )h . Suppose that the parameters t0 = t0 (T ) and ǫT satisfy the following conditions: 4 E. Pchelintsev, S. Perelevskiy, I. Makarova H1 ) For any T ≥ 32, 16 ≤ t0 ≤ T /2 and H2 ) lim t0 (T ) = ∞ , T →∞ √ 1/8 2/t0 lim ǫT = 0 , T →∞ ≤ ǫT ≤ 1 . lim T ǫT /t0 (T ) = ∞ . T →∞ H3 ) For any ν > 0 and m > 0, lim T ǫm = ∞ and T T →∞ lim T m e−ν √ t0 For example, for T ≥ 32, t0 = max{min{ln4 T , T /2} , 16} and ǫT = Let = 0. T →∞ √ −1/8 2 t0 . Γ = { max τl ≤ T } and Yk = Sek 1Γ . (8) 1≤l≤n Then on the set Γ there exists a temporary heteroscedastic regression model Yk = S(xk ) + ζk , with δk = Bk and σk2 = ζk = σk ξk + δk (9) n . (T − t0 )(e qT (xk ) − ǫ2T /2)(b − a) It should be noted that from (6) and H1 ), we get the following upper bound max σk2 ≤ 1≤k≤n 4 = σ∗ (b − a)ǫT (10) for which, by condition H3 ), σ∗ = 0 for any m > 0 . T →∞ T m lim To estimate the S function from the observations of (9) should study some properties of the set Γ in (8). Proposition 1. Suppose that the parameters t0 and ǫT satisfy the following conditions: H1 ) – H3 ). Then sup PS (Γ c ) ≤ ΠT , S∈ΣL,N where limT →∞ T m ΠT = 0 for any m > 0. 3 Improved estimates In this section we consider the estimation problem for the model (9). The function S(·) is unknown and has to be estimated from observations Y1 , . . . , Yn . The accuracy of any estimator Sb will be measured by the empirical squared error of the form n b−aX b kSb − Sk2n = (Sb − S, Sb − S)n = (S(xl ) − S(xl ))2 . n l=1 Improved nonparametric estimation of the drift in diffusion processes 5 Now we fix a basis (φj )1≤j≤n which is orthonormal for the empirical inner product: n b−aX φi (xl )φj (xl ) = Krij , (φi , φj )n = n l=1 where Krij is Kronecker’s symbol. By making use of this basis we apply the discrete Fourier transformation to (9) and we obtain the Fourier coefficients n b−aX θbj,n = Yl φj (xl ) , n n θj,n l=1 b−aX = S(xl ) φj (xl ) . n l=1 From (9) it follows directly that these Fourier coefficients satisfy the following equation r b−a θbj,n = θj,n + ζj,n with ζj,n = ξ + δj,n , n j,n where ξj,n = r n n l=1 l=1 b−aX b−aX σl ξl φj (xl ) and δj,n = δl φj (xl ) . n n Note that the upper bound (10) and the Bounyakovskii-Cauchy-Schwarz inequality imply that |δj,n | ≤ kδkn kφj kn = kδkn . We estimate the function S in (9) on the sieve (5) by the weighted least squares estimator Sbλ (xl ) = n X j=1 λ(j) θbj,n φj (xl ) 1Γ , 1 ≤ l ≤ n, where the weight vector λ = (λ(1), . . . , λ(n)) belongs to some finite set Λ ⊂ [0, 1]n . We set for any a≤x≤b n X b b Sλ (x) = Sλ (x1 )1{a≤x≤x1 } + Sbλ (xl )1{xl−1 <x≤xl } . (11) l=2 Further we suppose that the first d ≤ n components of the weight vector λ are equal to 1, i.e. λ(j) = 1 for any 1 ≤ j ≤ d. We consider a new estimate for the function S in (9) of the form Sλ∗ (xl ) = n X ∗ λ(j) θj,n φj (xl ) 1Γ , j=1 where ∗ θj,n = c(d) 1− 1{1≤j≤d} kθe k n where (d − 1)σ∗2 L(b − a)1/2 p , c(d) = n(s∗ + dσ∗ /n) ! 1 ≤ l ≤ n, θbj,n , kθen k2 = d X j=1 2 θbj,n . 6 E. Pchelintsev, S. Perelevskiy, I. Makarova Now we define the estimate for S in (1). We set for any a ≤ x ≤ b Sλ∗ (x) = Sλ∗ (x1 )1{a≤x≤x1 } + n X Sλ∗ (xl )1{xl−1 <x≤xl } . (12) l=2 We denote the difference of quadratic risks of the estimates (12) and (11) as ∆n (S) := ES kSλ∗ − Sk2n − ES kSbλ − Sk2n . The choice of estimate (12) is motivated by the desire to control the quadratic risk. Theorem 1. The estimate (12) outperforms in mean square accuracy the estimate (11), i.e. sup ∆n (S) < −c2 (d). S∈ΣL,N 4 Oracle inequalities In order to obtain a good estimator, we have to write a rule to choose a weight vector λ ∈ Λ in (12). It is obvious, that the best way is to minimize the empirical squared error with respect to λ: Errn (λ) = kSλ∗ − Sk2n → min . Making use of (12) and the Fourier transformation of S imply Errn (λ) = n X λ 2 ∗2 (j)θj,n j=1 −2 n X ∗ λ(j)θj,n θj,n + j=1 n X 2 θj,n . j=1 ∗ θj,n by some its estimator Since the coefficient θj,n is unknown, we need to replace the term θj,n which we choose as n b−a b−a X 2 2 ∗ θej,n = θbj,n θj,n − σl φj (xl ) . sj,n with sj,n = n n l=1 One has to pay a penalty for this substitution in the empirical squared error. Finally, we define the cost function of the form n n X X 2 ∗2 Jn (λ) = λ (j)θj,n − 2 λ(j) θej,n + ρPn (λ) , j=1 j=1 where the penalty term is defined as n Pn (λ) = b−aX 2 λ (j)sj,n n j=1 and 0 < ρ < 1 is some positive constant which will be chosen later. We set b = argmin λ J (λ) λ∈Λ n and define an estimator of S of the form (11): S ∗ (x) = Sλb∗ (x) for a ≤ x ≤ b . (13) Now we obtain the non asymptotic upper bound for the quadratical risk of the estimator (13). Improved nonparametric estimation of the drift in diffusion processes 7 Theorem 2. Let Λ ⊂ [0, 1]n be any finite set such that the first d ≤ n components of the weight vector λ are equal to 1. Then, for any n ≥ 3 and 0 < ρ < 1/6, the estimator (13) satisfies the following oracle inequality ES kS ∗ − Sk2n ≤ where limn→∞ Ψn (ρ)/n = 0. 1 + 6ρ Ψ (ρ) min ES kSbλ − Sk2n + n , 1 − 6ρ λ∈Λ n Now we consider the estimation problem (1) via model (9). We apply the estimating procedure (13) with special weight set introduced in [3] to the regression scheme (9). Denoting Sα∗ = Sλ∗ we α set S ∗ = Sαb∗ with α b = argminα∈A Jn (λα ) . ε We obtain through Theorem 2 the following oracle inequality. Theorem 3. Assume that S ∈ ΣL,N and the number of the points n = n(T ) in the model (9) satisfies (6). Then the procedure S ∗ satisfies, for any T ≥ 32, the following inequality R(S ∗ , S) ≤ B (ρ) (1 + ρ)2 (1 + 6ρ) min R(Sα∗ , S) + T , 1 − 6ρ n α∈Aε where limT →∞ BT (ρ)/n(T ) = 0. Acknowledgement The results of Section 3 of this work are supported by the RSF grant number 17-11-01049. 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