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

Chewi, Sinho; Pooladian, Aram-Alexandre

Comptes Rendus. Mathématique, Tome 361 (2023) no. G9, pp. 1471-1482

The optimal transport map between the standard Gaussian measure and an $\alpha$-strongly log-concave probability measure is ${\alpha }^{-1/2}$-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.

La fonction de transport optimale entre la mesure gaussienne standardisée et une mesure de probabilité $\alpha$-fortement log-concave est ${\alpha }^{-1/2}$-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.

[1] Ahidar-Coutrix, Adil; Le Gouic, Thibaut; Paris, Quentin 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 | Zbl | MR | DOI

[2] Altschuler, Jason; Chewi, Sinho; Gerber, Patrik; Stromme, Austin 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] Bakry, Dominique; Gentil, Ivan; Ledoux, Michel Analysis and geometry of Markov diffusion operators, Grundlehren der Mathematischen Wissenschaften, 348, Springer, 2014 | Zbl | DOI

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

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

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

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

[8] Colombo, Maria; Figalli, Alessio; Jhaveri, Yash 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 | Zbl | MR

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

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

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

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

[13] Gelbrich, Matthias 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 | Zbl | DOI | MR

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

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

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

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

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

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

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

[21] Le Gouic, Thibaut; Paris, Quentin; Rigollet, Philippe; Stromme, Austin J. 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] Ledoux, Michel Remarks on some transportation cost inequalities, 2018

[23] Léonard, Christian 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 | Zbl | DOI

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

[25] Mena, Gonzalo; Niles-Weed, Jonathan 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] Mikulincer, Dan; Shenfeld, Yair The Brownian transport map (2021) | arXiv

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

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

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

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

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

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

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

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

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