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

Sinho Chewi; Aram-Alexandre Pooladian

doi:10.5802/crmath.486

Functional analysis

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

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.

Abstract
Résumé

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.

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.

Received: 2022-11-14
Revised: 2023-02-17
Accepted: 2023-03-02
Published online: 2023-11-10

DOI: 10.5802/crmath.486

Author's affiliations:

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

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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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/

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