Gluing methods for quantitative stability of optimal transport maps
Abstract
Stability results have important applications in statistics (including on the estimation of OT maps from random data, and the matching problem), in numerical analysis, and in data science. In particular, it justifies the “linearized optimal transport” framework, which embeds the quadratic Wasserstein space in a Hilbert space through the optimal transport map from a reference measure.
In this talk, based on joint works with Alex Delalande and Cyril Letrouit, we present new quantitative stability results for quadratic optimal transport maps $T_\mu$ between a fixed source density $\rho$ and a target probability measure $\mu$ on $\mathbb{R}^d$. We establish that the map $\mu \mapsto T_\mu$ is bi-Hölder continuous with respect to the 2-Wasserstein distance under broad conditions on $\rho$. This include log-concave densities, densities on John domains, and some non-log-concave densities over the whole space. Our approach combines a local variance inequalities with innovative gluing techniques, leveraging tools such as Whitney decompositions and spectral graph theory to derive global stability bounds.
