An entropic generalization of Caffarelli’s contraction theorem via covariance inequalities
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 Cramé...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Académie des sciences
2023-11-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.486/ |
| _version_ | 1797517900958924800 |
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| author | Chewi, Sinho Pooladian, Aram-Alexandre |
| author_facet | Chewi, Sinho Pooladian, Aram-Alexandre |
| author_sort | Chewi, Sinho |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-10T07:22:40Z |
| format | Article |
| id | doaj.art-e6acc656fc89479d8cfd7a6c24e048e5 |
| institution | Directory Open Access Journal |
| issn | 1778-3569 |
| language | English |
| last_indexed | 2024-03-10T07:22:40Z |
| publishDate | 2023-11-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj.art-e6acc656fc89479d8cfd7a6c24e048e52023-11-22T14:31:29ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692023-11-01361G91471148210.5802/crmath.48610.5802/crmath.486An entropic generalization of Caffarelli’s contraction theorem via covariance inequalitiesChewi, Sinho0Pooladian, Aram-Alexandre1School of Mathematics, Institute for Advanced Study, Princeton, USACenter for Data Science, New York University, New York, USAThe 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.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.486/ |
| spellingShingle | Chewi, Sinho Pooladian, Aram-Alexandre An entropic generalization of Caffarelli’s contraction theorem via covariance inequalities Comptes Rendus. Mathématique |
| title | An entropic generalization of Caffarelli’s contraction theorem via covariance inequalities |
| title_full | An entropic generalization of Caffarelli’s contraction theorem via covariance inequalities |
| title_fullStr | An entropic generalization of Caffarelli’s contraction theorem via covariance inequalities |
| title_full_unstemmed | An entropic generalization of Caffarelli’s contraction theorem via covariance inequalities |
| title_short | An entropic generalization of Caffarelli’s contraction theorem via covariance inequalities |
| title_sort | entropic generalization of caffarelli s contraction theorem via covariance inequalities |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.486/ |
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