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

TL;DR

This paper extends Caffarelli's contraction theorem using covariance inequalities, providing a new proof via entropic regularization and broadening its applicability to measures with bounded Hessians.

Contribution

It introduces a novel entropic regularization approach to prove and generalize Caffarelli's contraction theorem using covariance inequalities.

Findings

Sharp Lipschitz bounds for entropically regularized optimal transport maps

A new proof of Caffarelli's original theorem as regularization vanishes

Extension of the theorem to measures with Hessians bounded by commuting matrices

Abstract

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 Cram\'er-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.