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

Abstract

The optimal transport map between the standard Gaussian measure and an α-strongly logconcave 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.

Minimax estimation of discontinuous optimal transport maps: The semi-discrete case

“…The recent preprint by [RS22] proved parametetric rates of estimation between the empirical entropic Brenier map and its population counterpart, though with an exponentially poor dependence on the regularization parameter (see Remark 3.6). Using covariance inequalities, the entropic Brenier potentials were used give a new proof of Caffarelli's contraction theorem; see [CP22]; this approach was recently generalized in [Con22]. Entropic optimal transport has also come into contact with the area of deep generative modelling through the following works [FGOP20, DBTHD21], among others.…”