Multi-marginal entropy-transport with repulsive cost
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This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ Multi-marginal entropy-transport with repulsive cost © The Authors 2020 Published version Gerolin, Augusto; Kausamo, Anna; Rajala, Tapio Gerolin, A., Kausamo, A., & Rajala, T. (2020). Multi-marginal entropy-transport with repulsive cost. Calculus of Variations and Partial Differential Equations, 59(3), Article 90. https://doi.org/10.1007/s00526-020-01735-3 2020
Calc. Var. (2020) 59:90 https://doi.org/10.1007/s00526-020-01735-3 Calculus of Variations Multi-marginal entropy-transport with repulsive cost Augusto Gerolin1·Anna Kausamo2·Tapio Rajala2 Received: 18 September 2019 / Accepted: 28 February 2020 © The Author(s) 2020 Abstract In this paper we study theoretical properties of the entropy-transport functional with repulsive cost functions. We provide sufficient conditions for the existence of a minimizer in a class of metric spaces and prove the -convergence of the entropy-transport functional to a multimarginal optimal transport problem with a repulsive cost. We point out that our construction can deal with the case when the space Xis a domain in Rd, answering a question raised in Benamou et al. (Numer Math 142:33–54, 2019). Finally, we also prove the entropyregularized version of the Kantorovich duality. 1 Introduction We consider the following multi-marginal entropy-transport problem Iε[ρ]= inf γ∈sym N(ρ) (C0[γ]+εE[γ]), (1.1) where C0[γ]=XNcdγis the transportation cost related to a cost function c,E[γ]is theentropy,andε≥0 is a parameter, see Sect. 2for details. We consider the setting where (X,d,m)is a Polish measure space, and ρm∈Pac(X)is an absolutely continuous Communicated by A.Malchiodi. The authors acknowledge the support of the Academy of Finland, Projects Nos. 274372, 284511, 312488, and 314789. A.G. also acknowledges funding by the European Research Council under H2020/MSCA-IF “OTmeetsDFT” (Grant ID: 795942). A.K. also wants to thank the Vilho, Yrjö and Kalle Väisälä Foundation for funding. BAugusto Gerolin [email protected] Anna Kausamo [email protected] Tapio Rajala [email protected] 1Department of Theoretical Chemistry, Vrije Universiteit Amsterdam, FEW, De Boelelaan 1083, 1081HV Amsterdam, The Netherlands 2Department of Mathematics and Statistics, University of Jyvaskyla, P.O. Box 35 (MaD), 40014 Jyvaskyla, Finland 0123456789().: V,-vol 123
90 Page 2 of 20 A. Gerolin et al. probability measure with respect to the reference measure m.Anelementγ∈sym N(ρ) is called a symmetric coupling (or transport plan), that is, a symmetric probability measure in XNhaving all marginals equal to ρm. We are interested in a class of repulsive cost functions c:XN→R∪{+∞}of the form c(x1,...,xN)= 1≤i<j≤N f(d(xi,xj)), for all (x1,...,xN)∈XN. We assume f:]0,∞[→ Rto be a continuous and decreasing function that approaches +∞ if d(xi,xj)→0. Among the examples of such cost functions we have the Coulomb cost f(z)=1/|z|, the Riesz cost f(z)=1/|z|s,n≥s≥max{n−2,0}(in Rn)andthe logarithmic cost f(z)=−log(|z|). We observe that when ε=0, this entropy-transport problem reduces to the classical multi-marginal optimal transport problem with repulsive costs [4,6,7,12]. The motivation of this paper comes from both theory and numerics. For repulsive cost functions, the entropy term in (1.1) plays a role of a regularizer to compute numerically a solution γof the multi-marginal optimal transport problem I0[ρ],see[2]. Numerical experiments suggest that when the regularization parameter εgoes to 0, the minimizer γεconverges to a minimizer of I0[ρ]having minimal entropy among the minimizer of I0[ρ]. From a theoretical viewpoint, this type of a functional has direct relevance in Density Functional Theory. By choosing carefully the parameter ε, the functional (1.1) provides a lower-bound for the Hohenberg–Kohn functional in Density Functional Theory [15,24,27]. This is an immediate consequence of the Log-Sobolev Inequality. The entropy-transport problem has appeared previously in the literature in the attractive case, in particular when c(x1,x2)=d(x1,x2)2.Wementionbrieflybelowsomeofthe connections of the entropy-transport with other fields and point out the relevance in the Coulomb case. Brief comments on some applications of the entropy-transport Optimal transport and Sinkhorn algorithm: The entropy-transport (1.1) was introduced by Cuturi [9] in order to compute numerically the optimal transport plan for the distance squared cost in the 2-marginals case via the Sinkorn algorithm. Due to its reasonable computational cost, it has been applied to a wide range of problems in various research areas, including Information Theory, Computer Graphics, Statistical Inference, Machine Learning, and MeanField Games. The entropic regularization method was also considered in the (attractive) multi-marginal case in the so-called barycenter problem introduced by Agueh and Carlier [1](seealso[5,11]) and in numerical methods in the time discretization of Brenier’s relaxed formulation of the incompressible Euler equation [3]. For a thorough presentation of the computational aspects we refer to Cuturi and Peyré’s book [25]. Second-order calculus on RCD spaces: Gigli and Tamanini [8,17] studied the entropictransport problem on a class of metric spaces with (Riemannian) Ricci curvature bounded frombelow(2-marginalscase,c(x1,x2)=d(x1,x2)2).Theentropicregularizationprocedure was crucial for establishing a second-order differential structure in that setting. Schrödinger problem: In 1926, E. Schrödinger introduced the (linear) Schrödinger equations describing the non-relativistic evolution of a single particle in an electric field with potential energy and also established an equivalence between such equations and a system of diffusion equations [26]. Roughly speaking, the variational problem (see (1.1) with X=C([0,1],Rd) and N=2) arises in the Schrödinger manuscript while studying the limit k→∞(N=2) of the empirical measures associated to the evolution of ki.i.d. Brownian motions. We refer the reader to Léonard survey [21] for technical details and historical notes. 123
Multi-marginal entropy-transport with repulsive cost Page 3 of 20 90 Fig. 1 The dependence of the minimizer of the entropic-transport problem (1.1) on the entropic parameter ε for the one-dimensional Coulomb cost, N=2andρ∼N(0,5). The pictures show part of the support of the optimal coupling γεaround the origin. From the left to right: ε=104,10−2,10−3,10−4,10−5 Lower bound on the Hohenberg–Kohn functional in density functional theory: This is the particular case where the entropy-transport problem with Coulomb cost comes into play. It has been shown in [24,27] that the functional (1.1) provides a lower bound for computing the ground state energy of the Hohenberg–Kohn functional [4,6,7,12,22]. Below we give a brief description of the result. Notice that in this context X=Rdand mis the Lebesgue measure on Rd. Assume that γ∈N(ρ) such that √γ∈H1(RdN). This is the case, for example, when γ(x1,...,xN)=|ψ(x1,...,xN)|2,whereψ∈H1(RdN)is a ground-state wave function solving the N-electron Schrödinger Equation (see [6,7,12,15,27] for details). Then, we can define the Hohenberg–Kohn functional by ˜ FHK [ρ] =inf γ∈N(ρ),√γ∈H1(RdN)2 2RdN |∇√γ|2dx1...dxN+RdN 1≤i<j≤N 1 |xi−xj|dγ. Now, as a consequence of the logarithmic Sobolev inequality for the Lebesgue measure [18], the following result holds: if ρLd∈P(Rd)and √γ∈H1(RdN)then Cε[ρ]≤ ˜ FHK [ρ],with ε=π2/2. Examples of optimal entropy couplings Let us present some computational examples of minimizers of Iε[ρ]illustrating the role of the parameter ε. Before this, we recall a result on the characterization of minimizers in the onedimensional case [10]. In particular, according to it the minimizer of I0[ρ]is concentrated on finitely many graphs and thus singular with respect to the product reference measure. Theorem 1.1 [10]Let μ∈P(R)be an absolutely continuous probability measure and f:R→Rstrictly convex, bounded from below and non-increasing function. Then there exists a unique optimal symmetric plan γ∈sym N(μ) that solves min γ∈sym N(μ) RN 1≤i<j≤N f(|xj−xi|)dγ. Moreover, this plan is induced by an optimal cyclical map T , that is, γsym =(γT)S,where γT=(id,T,T(2),...,T(N−1))μ. An explicit optimal cyclical map is T(x)=F−1 μ(Fμ(x)+1/N)if Fμ(x)≤(N−1)/N F−1 μ(Fμ(x)+1−1/N)otherwise. 123
90 Page 4 of 20 A. Gerolin et al. Here Fμ(x)=μ((−∞,x])is the distribution function of μ, and F−1 μis its lower semicontinuous left inverse. One-dimensional entropic-transport with Coulomb cost and a Gaussian measure Let ρbe the normal distribution on the real line with zero mean and standard deviation σ=5. We compute numerically the solution of the entropic-transport problem with Coulomb cost in the real line using the Sinkorn algorithm [9]. Notice that by Theorem 1.1,weknow that the minimizer of I0[ρ]is concentrated on a graph. See Fig. 1for an illustration of the computational results. Our code is based on the Python implementation available at POT library [14]. Organization of the paper In Sect. 2, we introduce the setting and study sufficient conditions for the existence of minimizers for the entropy-transport problem (1.1). Section 3is devoted to the −convergence proof of the entropic-transport functional Cε[γ]to the multi-marginal optimal transport with repulsive costs C0[γ]. In Sect. 4, we study the Kantorovich duality for the entropic-transport problem. The strategy of the main proof and some technical remarks The main result of this paper is Theorem 3.1, in which we prove the -convergence of the entropic-regularized functional Cε[γ]to C0[γ]. The technical difficulty on dealing with the -convergence comes from the fact that while for the entropic part E[γ]the minimizer γ tends to be as spread as possible with respect to m,forthecostC0[γ]a minimizer can be very singular and have infinite entropy. We divide the proof in two parts. The part (I), the lim inf −inequality, follows basically from the lower-semicontinuity of the costs C0[γ]and Cε[γ]- which are obtained from the assumption ρlog ρ∈L1 m(X)on the marginal measure ρm, giving a lower bound on the entropy. The part (II), the lim sup −inequality, is more involved. In Sect. 3.2, we construct a block approximation γ nfor a coupling γwith C0[γ]<+∞. Such construction is done in several steps, since we need to construct a competitor γ nsuch that E[γ n]<∞and γ n∈sym N(ρ). The main idea and the rigorous construction are given in Sect. 3.2. Futhermore, we point out that our construction can deal with the case when the space X is a domain in Rd, answering a question raised in [3]. There the -convergence was proven using convolutions; an approach that does not seem to be easy to implement for domains, or in general metric spaces. Related works: A proof of the -convergence of (1.1) to the Monge-Kantorovich problem for c(x,y)=d(x,y)pfirst appeared in [20,23] via probabilistic methods. In [5], G. Carlier, V. Duval, G. Peyré and B. Schmitzer provided an alternative and more analytical proof carrying out a similar block approximation procedure for the two-marginal squared distance cost in the Euclidean space and the Wasserstein Barycenter. 2 The entropy-regularized repulsive costs Let (X,d)be a Polish space and mbe a reference measure on X.WedenotebyP(X)the set of Borel probability measures on X,andPac(X)the set of Borel probability measures on 123
Multi-marginal entropy-transport with repulsive cost Page 5 of 20 90 Xthat are absolutely continuous with respect to m. We denote by mNthe product measure m⊗m⊗···⊗m. This is the reference measure we use on the product space XN.OnXN we use the sup-metric, which we denote by dN. The class of cost functions c:XN→R∪{+∞}of our interest is given by functions of the form c(x1,...,xN)= 1≤i<j≤N f(d(xi,xj)), for all (x1,...,xN)∈XN, where f:[0,∞[→ R∪{+∞}satisfies the following conditions f|]0,∞[ is continuous, decreasing and (F1) lim t→0+f(t)=+∞.(F2) Above and from now on, we denote by (x1,...,xN)points in XN,soxi∈Xfor each i. We denote by N(ρ) =γ∈P(XN)|pri γ=ρfor all i∈{1,...,N} the set of couplings or transport plans,wherepriis the projection pri(x1,...,xi,...,xN)=xifor all (x1,...,xi,...,xN)∈XN. A measure γ∈P(XN)is symmetric if XN φ(x1,...,xN)dγ=XN φ(σ(x1,...,xN)) dγ, for all φ∈C(XN) for all permutations σof the Nsymbols (x1,...,xN).WedenotebyPsym(XN)the set of symmetric probability measures in XN, and by sym N(ρ) := N(ρ) ∩Psym(XN) the set of symmetric couplings of ρ. Let us also introduce the notation for symmetrising measures. If γis a Borel measure on XN,wedenotebyγSthe symmetrized measure γS:= 1 N! σ∈SN σγ, where SNis the set of permutations of the Ncoordinates (x1,...,xN). We define the functional C0[γ]to be the cost related to the coupling γ C0[γ]=XN c(x1,...,xN)dγ(x1,...,xN). Because of the symmetry of the cost c,weimmediatelyhave Proposition 2.1 For every ρ∈(X), we have that inf γ∈N(ρ) C0[γ]= inf γ∈sym N(ρ) C0[γ].(2.1) Moreover, if the infimum is attained on one side of the above equality, then it is attained on both sides. 123
90 Page 6 of 20 A. Gerolin et al. Given ε≥0, we denote by Cε[γ]the entropy-regularized cost Cε[γ]=C0[γ]+εE[γ],for all γ∈sym N(ρ), (2.2) where the entropy E[γ]:P(XN)→R∪{−∞,+∞} is defined as E[γ]=XNργlog ργdmNif γmN +∞ otherwise .(2.3) Thenotationργstands fortheRadon-Nikodymderivativeofγwithrespect tothereference measure mNand γmNmeans that γis absolutely continuous with respect to the reference measure mN.Letρm∈Pac(X). In this paper we are interested in the following infimum Iε[ρm]:= inf γ∈sym N(ρ) Cε[γ].(2.4) In order to guarantee the lower semicontinuity for Cε[·], we will assume ρlog ρ∈L1 m(X). This will take care of the entropy part E[·] of the cost. In order to establish the lower semicontinuity for the functional C0[·],weassumethatthemeasureρsatisfies the following two conditions: lim r→0sup x∈X ρ(B(x,r)) < 1 N(N−1)2and (A) X\B(o,r0) f(2d(x,o))dρ(x)>−∞ for some o∈X.(B) Above we have, by an abuse of notation, denoted the measure ρmby only the density ρ;we will use the same abbreviation in the rest of the paper if there is no risk of confusion. The Condition (B)is a similar assumption to requiring, in the case of the quadratic cost, that the marginal measures have finite second moments. The Condition (A)guarantees that the cost is finite. If we endow the spaces P(XN)and P(X)with w∗-topology then, by Prokhorov’s theorem, any subset of P(X)(or P(XN)) is tight if and only if it is relatively compact. Remark 2.2 (Entropy-transport seen as a Kullback–Leibler divergence) If μand νare measures on a set X, the Kullback–Leibler divergence of μwith respect to νis defined as KL[μ|ν]=Xlog dμ dνdμif μν +∞ otherwise . Now, if both measures μand νare absolutely continuous with respect to some reference measure Rof the space Xwith densities ρμand ρν, respectively, we can write: KL[μ|ν]=Xρμlog ρμ ρνdRif μν +∞ otherwise . Considering the entropy-regularized MOT problem, we see that the cost functional Cε[γ] can be alternatively written as the Kullback–Leibler divergence between γand a kernel κ defined below Cε[γ]=εKL[γ|κ]=εXN ργln ργ ρκdmN, where κ=e−c/εmN. 123
Multi-marginal entropy-transport with repulsive cost Page 7 of 20 90 For the most part, in this paper we have chosen to consider as a reference measure the measure mN. However, as the following lemma shows, we could also assume the reference measure to be (ρm)⊗Nsince the minimizers of the entropy-regularized MOT problem (2.4) do not depend on the choice of the reference measure, at least if there exists a minimizer with finite cost. To state the lemma, let us introduce the notation of relative entropy: for each reference measure Rof a Polish space Y, and for each γ∈P(Y),wedenotebyE[γ|R]the relative entropy of γwith respect to R,definedas E[γ|R]=Ylog dγ dRdγif γR +∞ otherwise . Now we may consider two, a priori different, entropy-regularized MOT problems: the one introduced in (2.4) I[ρm]= inf γ∈sym N(ρ) Cε[γ]=:I[ρm|m],(2.5) and the problem with the reference measure chosen to be (ρm)⊗N I[ρm|ρm]:= inf γ∈sym N(ρ) C0[γ]+E[γ|(ρm)⊗N].(2.6) The following Lemma 2.3 is used only to go from the compact to the general case in the duality Theorem 4.2. The proof in [11] can be directly applied here to prove Lemma 2.3. Lemma 2.3 Let (X,d,m)be a Polish measure space, ρm∈P(X)a measure satisfying (A) and (B), and c a cost function satisfying (F1) and (F2). Now for all >0we have I[ρm|m]=I[ρm|ρm]+NKL[ρm|m]=I[ρm|ρm]+NX ρlog ρdm.(2.7) Moreover, whenever at least one side of the equality above is finite, the problems (2.5)and (2.6)have the same minimizers. 2.1 Some properties of the entropy functional Let us start by noting that the minimum of the entropy is attained by the product measure and that its value is not −∞. Proposition 2.4 Let (X,d,m)be a Polish metric measure space, and let ρm∈Pac(X)with ρlog ρ∈L1 m(X).Then min γ∈sym N(ρ) E[γ]=XN⊗N i=1ρlog ⊗N i=1ρdmN=NX ρlog ρdm>−∞. Proof As we will see, the minimality is an immediate consequence of Jensen’s inequality. Let γ∈N(ρ).Then 123
90 Page 8 of 20 A. Gerolin et al. E[γ]=XN ργlog(ργ)dmN=XN ργ ⊗N i=1ρlog ργ ⊗N i=1ρ+log ⊗N i=1ρ⊗N i=1ρdmN ≥XN ργdmNlog XN ργdmN+XN ργlog ⊗N i=1ρdmN =0+E[⊗N i=1ρ]. Using Proposition 2.4 we immediately get the lower semicontinuity of the entropy functional by representing the entropy as relative entropy against the probability measure ⊗N i=1(ρm). See for instance [28, Lemma 4.1] for the lower semicontinuity of the entropy when the reference measure is finite. Corollary 2.5 Let (X,d,m)be a Polish metric measure space, and let ρm∈Pac(X)with ρlog ρ∈L1 m(X).ThenE[·] is lower semicontinuous in the set sym N(ρ). Now we are ready to prove the existence of the minimizers for entropy-regularized MOT: Proposition 2.6 Let (X,d,m)be a Polish metric measure space. Assume that the measure ρm∈Pac(X)satisfies ρlog ρ∈L1 m(X)along with Conditions (A) and (B). Assume that c:XN→R∪{+∞}satisfies the conditions (F1)and (F2). Then, for each ε≥0,there exists a minimizer γ∈sym N(ρ) for the entropic-regularized cost Cε[γ]. Proof We notice that the set sym N(ρ) is compact in the w∗-topology [19]. The functional E is lower semicontinuous by Corollary 2.5, and in our setting the lower semicontinuity of C0 is proven as a part of the proof of [16, Proposition 3.1]. Since for each ε≥0 the functional Cεis lower semicontinuous, we conclude that it has a minimizer in the set sym N(ρ) 3The0-convergence of entropic-regularized cost Now let us turn to the -convergence. From now on, (τn)n∈Nis any sequence of positive real numbers decreasing to zero. Let us introduce the following functionals: for each n∈N Cn:Psym(XN)→R∪{+∞},Cn[γ]=Cτn(γ ) if γ∈N(ρ) +∞ otherwise and C:Psym(XN)→R∪{+∞},C[γ]=C0[γ]if γ∈N(ρ) +∞ otherwise . The goal of this section is to prove that the sequence (Cn∈N)-converges to Cin the space Psym(XN). Theorem 3.1 Let (X,d,m)be a Polish metric measure space. Let ρ∈Pac(X)with ρlog ρ∈L1 m(X)satisfying (A) and (B). Then the sequence (Cn)-converges to Cin the space Psym(XN). 123
Multi-marginal entropy-transport with repulsive cost Page 15 of 20 90 Let us now estimate the core part of the approximation. By the construction (3.7)ofγa 1,n and the choice (3.6)ofλn,wehave XN cdγa 1,n−XN cdγ1,n≤XN N(N−1) 2εndγ1,n<N(N−1) 2εn.(3.18) Combining (3.16), (3.17)and(3.18)weget C0[γ n]−C0[γ]≤XN cd(γ n−γa 1,n)+XN cdγa 1,n−XN cdγ1,n +XN cd(γ −γ1,n)→0 as n→∞. 3.5 Finiteness of the entropy for the approximations Next we show that the entropy is finite for the approximating sequence. Notice that, in order to prove (II), we do not need a better estimate on the entropy. Lemma 3.4 For each n ∈Nwe have E[γ n]<∞. Proof In order to see the finiteness of the entropy, it suffices to notice that each γ nis a sum of finitely many measures (˜γn,k)Nn k=1each of which is of the form ˜γn,k=˜ρk 1m⊗···⊗ ˜ρk Nm with ˜ρk iρand d˜ρk i dρ≤1. Indeed, by Proposition 2.4, the entropy is always bounded from below, and so we can make a crude estimate: E[γ n]=XN log Nn k=1 d˜γn,k dmdNn k=1˜γn,k ≤log(Nn)+ Nn k=1XN log d˜γn,k dmd˜γn,k<∞. 3.6 Proof of condition (II) We are now ready to prove the -lim sup inequality (II). By Lemma 3.2 we already know that (γ n)nconverges to γ.However,Cn[γ n]need not converge to C[γ]. This can be solved by makingthe convergence of(γ n)nslower byrepeatingalwaysthesame measureforsufficiently (but finitely) many times before moving to the next one. We define k(n)for every n∈Nas k(n)=min n,max 1,sup k∈N|√τnE[γ j]<1forall j≤k. By definition, 1 ≤k(n)≤n. Moreover, since for every j∈Nwe have E[γ j]<∞by Lemma 3.4 and τn→0 by definition, we have that k(n)→∞as n→∞. Thus, defining γn=γ k(n), for large enough n∈Nwe have Cn[γn]=C0[γ k(n)]+τnE[γ k(n)]<C0[γ k(n)]+√τn. Recalling that by Lemma 3.3 we have C0[γ k(n)]→C0[γ], we conclude the proof. 123
90 Page 16 of 20 A. Gerolin et al. In Proposition 2.6 the existence of a minimizer for the entropy-regularized cost was established. Now that we know that measures γfor which C0(γ ) < ∞can be approximated by measures with not only finite costs but also finite entropy, we can say more: Corollary 3.5 Let (X,d,m)be a Polish metric measure space. Assume that ρm∈Pac(X) satisfies ρlog ρ∈L1 m(X)and Conditions (A) and (B). Assume that c :XN→R∪{+∞} satisfies Conditions (F1)and (F2). Then, for each ε>0, there exists a unique minimizer γ∈sym N(ρ) for the entropic-regularized cost Cε[γ]. Proof Our marginal measure satisfies Conditions (A) and (B), so there exists a measure γ∈sym N(ρ) that minimizes C0with C0[γ]<∞. It must be noted that this measure can have infinite entropy. However, because of the approximation result presented in the proof of Condition (II) above, we get the existence of a measure γ∈sym N(ρ) such that C[γ]<∞. The uniqueness claim now follows, since the functional γ→ C[γ]is strictly convex for >0. 4 Entropic-Kantorovich duality for Coulomb-type costs We start by recalling the classical Fenchel–Rockafellar Theorem. We refer to the I. Ekeland and R. Témam’s book [13, Theorem 4.2] for a more complete presentation and references. Theorem 4.1 (Fenchel–Rockafellar) Let Xand Ybe Banach spaces and A :X→Ybe linear and continuous. Let F :X→R∪{+∞}and G :Y→R∪{+∞}be proper and convex functions. Then inf F[x]+G[Ax]x∈X=sup −F∗[−A∗γ]−G∗[γ]γ∈Y∗ where A∗:Y∗→X∗denotes the adjoint operator of A. Next we prove the Entropic-Kantorovich duality for the problem (2.4). Theorem 4.2 (Entropic Duality for repulsive costs) Let (X,d,m)be a Polish measure space. Suppose ρm∈P(X)such that (A)and (B)hold and ρlog ρ∈L1 m(X), and c:XN→[0,∞] is a cost function c(x1,...,xN)= 1≤i<j≤N f(d(xi,xj)), for all (x1,...,xN)∈XN, where f :[0,+∞[→ [0,∞] is a function satisfying (F1)and (F2). Then, for ε>0,the duality holds min γ∈N(ρ) Cε[γ]= sup ui∈Cb(X)N i=1X uidρm−εXN exp u1⊕···⊕uN−c εdmN+ε =sup u∈Cb(X)NX udρm−εXN exp u⊕···⊕u−c εdmN+ε, where v1⊕···⊕vNdenotes the operator (v1⊕···⊕vN)(x1,...,xN)=v1(x1)+···+ vN(xN). Proof First let us assume that Xis a compact space. We denote by X=(Cb(X))Nand Y=Cb(XN),whereCb(X)is the space of continuous and bounded functions on X,and 123
Multi-marginal entropy-transport with repulsive cost Page 17 of 20 90 similarly for XN. By Riesz representation theorem, the space Yis dual to the space M(XN)of signed regular Borel measures on XN. Thus, we may define the Legendre–Fenchel transform G∗of a functional G:Y→R∪{+∞}by G∗:M(XN)→R∪{+∞},G∗[π]= sup ψ∈Cb(XN)XN ψdπ−G[ψ]. We define the functionals F:(Cb(X))N→R∪{+∞},(u1,...,uN)→− N i=1X uidρm and G:Cb(XN)→R∪{+∞},ψ→ εXN e 1 ε(ψ−c)dmN, and the operator A:(Cb(X))N→Cb(XN), (u1,...,uN)→ u1⊕···⊕uN. Now, Fand Gare proper and convex functionals and Ais a linear and continuous operator. So, we may apply Fenchel–Rockafellar duality Theorem 4.1 to get inf{F[x]+G[Ax]|x∈X}=sup{−G∗[γ]−F∗[−A†γ]|γ∈Y∗}. This gives (since for every set Swe have inf(S)=−sup(−S)and sup S=−inf(−S)) inf{G∗[γ]+F∗[−A†γ]|γ∈Y∗}=sup{−F[x]−G[Ax]|x∈X}. It remains to show that the above expression has exactly the form of our duality claim. The claim that the right-hand sides correspond to each other follows immediately from our choices of X,F,andG. So, it remains to show that inf{G∗[γ]+F∗[−A†γ]|γ∈Y∗}=Cε[γ]−ε. (4.1) To prove it, let γ∈M(XN).Nowwehave F∗[−A∗γ]=sup XN N i=1 ui(xi)dγ− N i=1X ui(xi)dρm(xi) (u1,...,uN)∈Cb(X)N =0ifγ∈N(ρ) +∞ otherwise . Let us then compute G∗[γ]: G∗[γ]= sup ψ∈Cb(XN)XN ψdγ−εXN e 1 ε(ψ−c)dmN. If γis not absolutely continuous with respect to mN,wehaveG∗[γ]=+∞.IfγmN, then the supremum (that appears in the definition of G∗[γ]) is realized at ψ=εlog ργ+c; this holds also if the function ργis not continuous since it can be approximated by a sequence of continuous functions. Thus, we get for γmN 123
90 Page 18 of 20 A. Gerolin et al. G∗[γ]=XNργ·ψ−εe1 ε(ψ−c)dmN =XN (εργlog ργ+cργ−εργ)dmN. Hence, if γ∈N(ρ),wehave G∗[γ]=C0[γ]+εE[γ]−ε. This concludes the duality proof when Xis a compact space. The noncompact case Due to Lemma 2.3, it suffices to prove the claim in the case where the reference measure is ρminstead of m; the finiteness of the measure ρmnow gives access to inner regularity and to the approximability by compact sets. We will for simplicity denote ρ:= ρm. The claim is min γ∈N(ρ) Cε[γ]= sup u∈Cb(X)NX udρ−εXN exp u⊕···⊕u−c εdρ⊗N+ε. For simplicity, let us denote Dρ(u):= NX udρ−εXN exp u⊕···⊕u−c εdρ⊗N+εfor all u∈Cb(X). We may assume that supu∈Cb(X)Dρ(u)>−∞; indeed, since we can test with the function u≡0, this always holds for cost functions that are bounded from below. Let us make, in the notation of the primal functional, the dependence on the reference measure explicit by the notation γ→ C[γ|μ]when the reference measure on the space X is μ. Thus the original notation γ→ C[γ]corresponds to γ→ C[γ|m]. Since the measures ρand γare inner regular, there exists a sequence (Kn)n∈Nof compact subsets of Xsuch that ρ(Kn)→ρ(X)and γ(KN n)→γ(X). Let us denote γn:= 1 γ(KN n)γ|KN n and ρn:= 1 ρ(Kn)ρ|Kn. Let us also denote by γmin nthe minimizer of the problem I[ρn].Sinceγis the minimizer of the problem I(ρ) and since (due to the absolute continuity of the integral and continuity of the function t→ tlog t) lim n→∞|C[γn|ρn]−C[γ|ρ]| = 0,(4.2) we have lim n→∞|C[γn|ρn]−C[γmin n|ρn]| = 0.(4.3) By the duality result proven above for compact spaces, we have for all n∈N sup u∈Cb(Kn) Dρn(u)=C[γmin n|ρn]. Again, due to the absolute continuity of the integral, we have lim n→∞ sup u∈Cb(Kn) Dρn(u)−sup u∈Cb(X) Dρ(u)=0. 123
Multi-marginal entropy-transport with repulsive cost Page 19 of 20 90 Putting these conditions together, we get for all n∈N C[γ|ρ]− sup u∈Cb(X) Dρ(u) ≤|C[γ|ρ]−C[γn|ρn]|+C[γn|ρn]−C[γmin n|ρn] + C[γmin n|ρn]− sup u∈Cb(Kn) Dρn(u)+ sup u∈Cb(Kn) Dρn(u)−sup u∈Cb(X) Dρ(u) . The claim follows by letting n→∞. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. References 1. Agueh, M., Carlier, G.: Barycenters in the Wasserstein space. SIAM J. Math. Anal. 43, 904–924 (2011) 2. Benamou, J.D., Carlier, G., Nenna, L.: A Numerical method to solve multi-marginal optimal transport problems with Coulomb cost. In: Glowinski, R., Osher, S., Yin, W. (eds.) Splitting Methods in Communication, Imaging, Science, and Engineering. Scientific Computation. Springer, Cham (2016) 3. Benamou, J.-D., Carlier, G., Nenna, L.: Generalized incompressible flows, multi-marginal transport and Sinkhorn algorithm. Numer. Math. 142, 33–54 (2019) 4. Buttazzo, G., De Pascale, L., Gori-Giorgi, P.: Optimal-transport formulation of electronic density functional theory. Phys. Rev. A 85, 062502 (2012) 5. Carlier, G., Duval, V., Peyré, G., Schmitzer, B.: Convergence of entropic schemes for optimal transport and gradient flows. SIAM J. Math. Anal. 49, 1385–1418 (2017) 6. Cotar, C., Friesecke, G., Klüppelberg, C.: Density functional theory and optimal transportation with coulomb cost. Commun. Pure Appl. Math. 66, 548–599 (2013) 7. Cotar, C., Friesecke, G., Klüppelberg, C.: Smoothing of transport plans with fixed marginals and rigorous semiclassical limit of the Hohenberg–Kohn functional. Arch. Ration. Mech. Anal. 228, 891–922 (2018) 8. Cotar, C., Friesecke, G., Klüppelberg, C.: Second order differentiation formula on RCD∗(K,N)spaces. Accepted at JEMS. arXiv:1802.02463 (2018) 9. Cuturi, M.: Sinkhorn distances: lightspeed computation of optimal transport. In: Advances in Neural Information Processing Systems, pp. 2292–2300 (2013) 10. Di Marino, S., De Pascale, L., Colombo, M.: Multimarginal optimal transport maps for 1-dimensional repulsive costs. Can. J. Math. 67, 350–368 (2015) 11. Di Marino, S., Gerolin, A.: An optimal transport approach for the Schrödinger bridge problem and convergence of Sinkhorn algorithm. arXiv preprint arXiv:1911.06850 (2019) 12. Di Marino, S., Gerolin, A., Nenna, L.: Optimal transport theory for repulsive costs. In: Topological Optimization and Optimal Transport: In the Applied Sciences, vol. 17 (2017) 13. Ekeland, I., Temam, R.: Analyse convexe et problèmes variationelles. Dunod, Gauthier-Villars, Paris, ix+340 p (1974) 14. Flamary, R., Courty, N.: POT Python Optimal Transport library (2017). https://github.com/rflamary/POT 15. Gerolin, A., Grossi, J., Gori-Giorgi, P.: Kinetic correlation functionals from the entropic regularisation of the strictly-correlated electrons problem. J. Chem. Theory Comput. (2019) 16. Gerolin, A., Kausamo, A., Rajala, T.: Duality theory for multi-marginal optimal transport with repulsive costs in metric spaces. ESAIM Control Optim. Calc. Var. 25, 62 (2019) 17. Gigli, N., Tamanini, L.: Second order differentiation formula on compact RCD∗(K,N)spaces. arXiv:1701.03932 (2017) 123
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