Gluing methods for quantitative stability of optimal transport maps

TL;DR

This paper establishes quantitative stability bounds for optimal transport maps, showing bi-Hölder continuity of the transport map with respect to the measure, under various conditions on the density, and introduces new techniques for the proof.

Contribution

It proves bi-Hölder stability of optimal transport maps under general conditions and develops novel gluing methods using Whitney decompositions and spectral graph theory.

Findings

Bi-Hölder continuity of transport maps for log-concave densities.

Stability results on John domains and certain polynomial decay densities.

Counterexamples on non-John domains showing limits of stability.

Abstract

We establish quantitative stability bounds for the quadratic optimal transport map $T_\mu$ between a fixed probability density $\rho$ and a probability measure $\mu$ on $\mathbb{R}^d$. Under general assumptions on $\rho$, we prove that the map $\mu\mapsto T_\mu$ is bi-H\"older continuous, with dimension-free H\"older exponents. The linearized optimal transport metric $W_{2,\rho}(\mu,\nu)=\|T_\mu-T_\nu\|_{L^2(\rho)}$ is therefore bi-H\"older equivalent to the $2$-Wasserstein distance, which justifies its use in applications. We show this property in the following cases: (i) for any log-concave density $\rho$ with full support in $\mathbb{R}^d$, and any log-bounded perturbation thereof; (ii) for $\rho$ bounded away from $0$ and $+\infty$ on a John domain (e.g., on a bounded Lipschitz domain), while the only previously known result of this type assumed convexity of the domain; (iii) for some important families of probability densities on bounded domains which decay or blow-up polynomially near the boundary. Concerning the sharpness of point (ii), we also provide examples of non-John domains for which the Brenier potentials do not satisfy any H\"older stability estimate. Our proofs rely on local variance inequalities for the Brenier potentials in small convex subsets of the support of $\rho$, which are glued together to deduce a global variance inequality. This gluing argument is based on two different strategies of independent interest: one of them leverages the properties of the Whitney decomposition in bounded domains, the other one relies on spectral graph theory.