A coupling approach to Lipschitz transport maps
Giovanni Conforti, Katharina Eichinger
TL;DR
This work develops a probabilistic, coupling-based approach to bound the Lipschitz constant of the Langevin transport map between a log-concave base measure and a log-Lipschitz perturbation, providing a dimension-free bound without requiring uniform convexity or a bound on the third derivative. The method leverages a stochastic control representation of the transport vector fields via the Hamilton-Jacobi-Bellman equation, and employs reflection coupling to obtain gradient and Hessian estimates for the associated value function. These estimates yield explicit Lipschitz constants for both the Langevin transport map and its inverse, with relaxations of convexity and regularity assumptions compared to previous results. The framework shows robustness to weakened convexity and perturbations, and offers a constructive path to Lipschitz transport maps in high-dimensional, perturbed log-concave settings with potential applications in transferring analytic inequalities between measures.
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.