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

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.