Quantitative Stability of Optimal Transport Maps under Variations of the Target Measure

TL;DR

This paper investigates how the optimal transport map between a fixed density and varying target measures remains stable under measure perturbations, establishing bi-Hölder continuity and equivalence with Wasserstein metrics.

Contribution

It proves bi-Hölder continuity of the optimal transport map with respect to the target measure under certain conditions, extending stability understanding in optimal transport theory.

Findings

The transport map Tμ is bi-Hölder continuous with respect to μ.

The linearized transport metric is equivalent to the Wasserstein distance on certain measure sets.

Stability estimates justify using the linearized metric in applications.

Abstract

This work studies the quantitative stability of the quadratic optimal transport map between a fixed probability density $\rho$ and a probability measure $\mu$ on R^d , which we denote T$\mu$. Assuming that the source density $\rho$ is bounded from above and below on a compact convex set, we prove that the map $\mu$ $\rightarrow$ T$\mu$ is bi-H{\"o}lder continuous on large families of probability measures, such as the set of probability measures whose moment of order p > d is bounded by some constant. These stability estimates show that the linearized optimal transport metric W2,$\rho$($\mu$, $\nu$) = T$\mu$ -- T$\nu$ L 2 ($\rho$,R d) is bi-H{\"o}lder equivalent to the 2-Wasserstein distance on such sets, justifiying its use in applications.