An entropic generalization of Caffarelli’s contraction theorem via covariance inequalities

Sinho Chewi; Aram-Alexandre Pooladian

doi:10.5802/crmath.486

Analyse fonctionnelle

An entropic generalization of Caffarelli’s contraction theorem via covariance inequalities
[Une généralisation entropique du théorème de contraction de Caffarelli à l’aide d’inégalités de covariance]

Sinho Chewi ¹ ; Aram-Alexandre Pooladian ²

¹ School of Mathematics, Institute for Advanced Study, Princeton, USA
² Center for Data Science, New York University, New York, USA

Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1471-1482.

Résumé
Abstract

La fonction de transport optimale entre la mesure gaussienne standardisée et une mesure de probabilité $α$ -fortement log-concave est $α^{- 1 / 2}$ -Lipschitz, comme l’a noté Caffarelli dans le célèbre théorème qui porte désormais son nom. Dans ce travail, nous utilisons deux inégalités de covariance classiques (l’inégalité de Brascamp–Lieb ainsi de celle de Cramèr–Rao) pour établir une borne optimale sur la constante de Lipschitz de la fonction de transport associée au transport optimal avec régularisation entropique. En étudiant le cas limite où l’effet de la régularisation disparait, nous obtenons une démonstration courte et élegante du théorème de Caffarelli. De surcroît, cette approche nous permet d’étendre la validité du théoreme de Caffarelli au cas de log-densités dont les hessiens sont contrôlés par des matrices positives définies qui peuvent être choisies arbitrairement tant qu’elles commutent entre elles.

The optimal transport map between the standard Gaussian measure and an $α$ -strongly log-concave probability measure is $α^{- 1 / 2}$ -Lipschitz, as first observed in a celebrated theorem of Caffarelli. In this paper, we apply two classical covariance inequalities (the Brascamp–Lieb and Cramér–Rao inequalities) to prove a sharp bound on the Lipschitz constant of the map that arises from entropically regularized optimal transport. In the limit as the regularization tends to zero, we obtain an elegant and short proof of Caffarelli’s original result. We also extend Caffarelli’s theorem to the setting in which the Hessians of the log-densities of the measures are bounded by arbitrary positive definite commuting matrices.

Reçu le : 2022-11-14
Révisé le : 2023-02-17
Accepté le : 2023-03-02
Publié le : 2023-11-10

DOI : 10.5802/crmath.486

Affiliations des auteurs :

Sinho Chewi ¹ ; Aram-Alexandre Pooladian ²

¹ School of Mathematics, Institute for Advanced Study, Princeton, USA
² Center for Data Science, New York University, New York, USA

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Sinho Chewi and Aram-Alexandre Pooladian},
     title = {An entropic generalization of {Caffarelli{\textquoteright}s} contraction theorem via covariance inequalities},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1471--1482},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     year = {2023},
     doi = {10.5802/crmath.486},
     language = {en},
}

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Sinho Chewi; Aram-Alexandre Pooladian. An entropic generalization of Caffarelli’s contraction theorem via covariance inequalities. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1471-1482. doi : 10.5802/crmath.486. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.486/

Bibliographie
Cité par

[1] Adil Ahidar-Coutrix; Thibaut Le Gouic; Quentin Paris Convergence rates for empirical barycenters in metric spaces: curvature, convexity and extendable geodesics, Probab. Theory Relat. Fields, Volume 177 (2020) no. 1-2, pp. 323-368 | DOI | MR | Zbl

[2] Jason Altschuler; Sinho Chewi; Patrik Gerber; Austin Stromme Averaging on the Bures–Wasserstein manifold: dimension-free convergence of gradient descent (Advances in Neural Information Processing Systems), Volume 34, Curran Associates, Inc. (2021)

[3] Dominique Bakry; Ivan Gentil; Michel Ledoux Analysis and geometry of Markov diffusion operators, Grundlehren der Mathematischen Wissenschaften, 348, Springer, 2014 | DOI | Zbl

[4] Espen Bernton; Promit Ghosal; Marcel Nutz Entropic optimal transport: geometry and large deviations, Duke Math. J., Volume 171 (2022) no. 16, pp. 3363-3400 | MR | Zbl

[5] Sergey G. Bobkov; Michel Ledoux From Brunn–Minkowski to Brascamp–Lieb and to logarithmic Sobolev inequalities, Geom. Funct. Anal., Volume 10 (2000) no. 5, pp. 1028-1052 | DOI | MR | Zbl

[6] Luis A. Caffarelli Monotonicity properties of optimal transportation and the FKG and related inequalities, Commun. Math. Phys., Volume 214 (2000) no. 3, pp. 547-563 | DOI | MR | Zbl

[7] Sinho Chewi; Tyler Maunu; Philippe Rigollet; Austin J. Stromme Gradient descent algorithms for Bures–Wasserstein barycenters, Proceedings of Thirty Third Conference on Learning Theory (Jacob Abernethy; Shivani Agarwal, eds.) (Proceedings of Machine Learning Research), Volume 125, PMLR (2020), pp. 1276-1304

[8] Maria Colombo; Alessio Figalli; Yash Jhaveri Lipschitz changes of variables between perturbations of log-concave measures, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 17 (2017) no. 4, pp. 1491-1519 | MR | Zbl

[9] Dario Cordero-Erausquin Transport inequalities for log-concave measures, quantitative forms, and applications, Can. J. Math., Volume 69 (2017) no. 3, pp. 481-501 | DOI | MR | Zbl

[10] Imre Csiszár $I$ -divergence geometry of probability distributions and minimization problems, Ann. Probab., Volume 3 (1975), pp. 146-158 | MR | Zbl

[11] Marco Cuturi Sinkhorn distances: lightspeed computation of optimal transport (Advances in Neural Information Processing Systems), Volume 26, Curran Associates, Inc. (2013)

[12] Max Fathi; Nathael Gozlan; Maxime Prod’homme A proof of the Caffarelli contraction theorem via entropic regularization, Calc. Var. Partial Differ. Equ., Volume 59 (2020) no. 3, 96 | MR | Zbl

[13] Matthias Gelbrich On a formula for the $L^{2}$ Wasserstein metric between measures on Euclidean and Hilbert spaces, Math. Nachr., Volume 147 (1990) no. 1, pp. 185-203 | DOI | MR | Zbl

[14] Ivan Gentil; Christian Léonard; Luigia Ripani; Luca Tamanini An entropic interpolation proof of the HWI inequality, Stochastic Processes Appl., Volume 130 (2020) no. 2, pp. 907-923 | DOI | MR | Zbl

[15] Nathael Gozlan; Nicolas Juillet On a mixture of Brenier and Strassen theorems, Proc. Lond. Math. Soc., Volume 120 (2020) no. 3, pp. 434-463 | DOI | MR | Zbl

[16] Jan-Christian Hütter; Philippe Rigollet Minimax estimation of smooth optimal transport maps, Ann. Stat., Volume 49 (2021) no. 2, pp. 1166-1194 | MR | Zbl

[17] Hicham Janati; Boris Muzellec; Gabriel Peyré; Marco Cuturi Entropic optimal transport between unbalanced Gaussian measures has a closed form (Advances in Neural Information Processing Systems), Volume 33, Curran Associates, Inc. (2020)

[18] Young-Heon Kim; Emanuel Milman A generalization of Caffarelli’s contraction theorem via (reverse) heat flow, Math. Ann., Volume 354 (2012) no. 3, pp. 827-862 | MR | Zbl

[19] Bo’az Klartag Logarithmically-concave moment measures I, Geometric aspects of functional analysis (Lecture Notes in Mathematics), Volume 2116, Springer, 2014, pp. 231-260 | DOI | MR | Zbl

[20] Alexander V. Kolesnikov Mass transportation and contractions (2011) | arXiv

[21] Thibaut Le Gouic; Quentin Paris; Philippe Rigollet; Austin J. Stromme Fast convergence of empirical barycenters in Alexandrov spaces and the Wasserstein space, J. Eur. Math. Soc., Volume 25 (2023) no. 6, pp. 2229-2250 | DOI | MR | Zbl

[22] Michel Ledoux Remarks on some transportation cost inequalities, 2018

[23] Christian Léonard A survey of the Schrödinger problem and some of its connections with optimal transport, Discrete Contin. Dyn. Syst., Volume 34 (2014) no. 4, pp. 1533-1574 | DOI | Zbl

[24] Anton Mallasto; Augusto Gerolin; Hà Quang Minh Entropy-regularized 2-Wasserstein distance between Gaussian measures, Inf. Geom., Volume 5 (2022) no. 1, pp. 289-323 | DOI | MR | Zbl

[25] Gonzalo Mena; Jonathan Niles-Weed Statistical bounds for entropic optimal transport: sample complexity and the central limit theorem (Advances in Neural Information Processing Systems), Volume 32, Curran Associates, Inc. (2019)

[26] Dan Mikulincer; Yair Shenfeld The Brownian transport map (2021) | arXiv

[27] Dan Mikulincer; Yair Shenfeld On the Lipschitz properties of transportation along heat flows (2022) | arXiv

[28] Joe Neeman Lipschitz changes of variables via heat flow (2022) | arXiv

[29] Gabriel Peyré; Marco Cuturi Computational optimal transport, Found. Trends Mach. Learn., Volume 11 (2019) no. 5-6, pp. 355-607 | DOI

[30] Aram-Alexandre Pooladian; Marco Cuturi; Jonathan Niles-Weed Debiaser beware: pitfalls of centering regularized transport maps (Kamalika Chaudhuri; Stefanie Jegelka; Le Song; Csaba Szepesvari; Gang Niu; Sivan Sabato, eds.) (Proceedings of Machine Learning Research), Volume 162, Curran Associates, Inc. (2022), pp. 17830-17847

[31] Aram-Alexandre Pooladian; Jonathan Niles-Weed Entropic estimation of optimal transport maps (2021) | arXiv

[32] Maxime Prod’homme Contributions au problème du transport optimal et à sa régularité, Ph. D. Thesis (2021) (Thèse de doctorat dirigée par Max Fathi et Felix Otto, Mathématiques et Applications Toulouse 3 2021)

[33] R. Tyrrell Rockafellar Convex analysis, Princeton Landmarks in Mathematics, Princeton University Press, 1997 (reprint of the 1970 original, Princeton Paperbacks) | Zbl

[34] Vivien Seguy; Bharath B. Damodaran; Rémi Flamary; Nicolas Courty; Antoine Rolet; Mathieu Blondel Large-scale optimal transport and mapping estimation, International Conference on Learning Representations, Curran Associates, Inc. (2018)

[35] Stefán Ingi Valdimarsson On the Hessian of the optimal transport potential, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 6 (2007) no. 3, pp. 441-456 | Numdam | MR | Zbl

[36] Cédric Villani Topics in optimal transportation, Graduate Studies in Mathematics, 58, American Mathematical Society, 2003 | Zbl

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