A proof of the Caffarelli contraction theorem via entropic regularization
Probability
2019-04-15 v1 Functional Analysis
Optimization and Control
Abstract
We give a new proof of the Caffarelli contraction theorem, which states that the Brenier optimal transport map sending the standard Gaussian measure onto a uniformly log-concave probability measure is Lipschitz. The proof combines a recent variational characterization of Lipschitz transport map by the second author and Juillet with a convexity property of optimizers in the dual formulation of the entropy-regularized optimal transport (or Schr{\"o}dinger) problem.
Keywords
Cite
@article{arxiv.1904.06053,
title = {A proof of the Caffarelli contraction theorem via entropic regularization},
author = {Max Fathi and Nathael Gozlan and Maxime Prodhomme},
journal= {arXiv preprint arXiv:1904.06053},
year = {2019}
}