A coupling approach to Lipschitz transport maps

TL;DR

This paper introduces a probabilistic coupling method to bound the Lipschitz constant of Langevin flow maps, enabling construction of Lipschitz maps between certain probability measures with relaxed convexity conditions.

Contribution

It presents a novel coupling-based approach to estimate Lipschitz constants for Langevin flows, relaxing previous convexity and smoothness assumptions.

Findings

Bounded Lipschitz constants for Langevin flow maps in a dimension-free manner.

Constructed Lipschitz maps from log-concave measures to perturbations with weaker convexity.

Method applicable under weak asymptotic convexity, removing third derivative bounds.

Abstract

In this note, we propose a probabilistic approach to bound the (dimension-free) Lipschitz constant of the Langevin flow map on $\mathbb{R}^d$ introduced by Kim and Milman (2012). As example of application, we construct Lipschitz maps from a uniformly $\log$-concave probability measure to $\log$-Lipschitz perturbations as in Fathi, Mikulincer, Shenfeld (2024). Our proof is based on coupling techniques applied to the stochastic representation of the family of vector fields inducing the transport map. This method is robust enough to relax the uniform convexity to a weak asymptotic convexity condition and to remove the bound on the third derivative of the potential of the source measure.