Growth estimates on optimal transport maps via concentration inequalities

TL;DR

This paper provides an alternative proof and extensions for polynomial growth bounds on optimal transport maps using concentration inequalities, building on previous work on Brenier maps and their growth behavior.

Contribution

It introduces new proof techniques based on concentration inequalities and extends existing results on growth bounds of optimal transport maps.

Findings

Polynomial upper bounds on Brenier maps established under various density conditions.

Monotonicity and concentration inequalities are effective tools for analyzing transport map growth.

Extensions to previous results on Gaussian and log-concave measures are demonstrated.

Abstract

We give an alternative proof and some extensions of results of Carlier, Figalli and Santambrogio on polynomial upper bounds on the Brenier map between probability measures under various conditions on the densities. The proofs are based on the monotonicity of the map and various concentration inequalities, as already used by Colombo and Fathi to prove quadratic growth for transport maps from the standard Gaussian onto log-concave measures.