Stability Bounds for Smooth Optimal Transport Maps and their Statistical Implications

TL;DR

This paper introduces new stability bounds for smooth optimal transport maps, linking the estimation of the transport map to density estimation in Wasserstein distance, and applies these bounds to develop a tuning-parameter-free estimator for strongly log-concave distributions.

Contribution

The paper develops novel stability bounds for OT maps that generalize previous results and do not require smoothness or boundedness assumptions on the measures.

Findings

Stability bounds connect OT map estimation to density estimation in Wasserstein distance.

New bounds apply under more general conditions, relaxing smoothness assumptions.

A novel tuning parameter-free estimator for strongly log-concave distributions is proposed.

Abstract

We study estimators of the optimal transport (OT) map between two probability distributions. We focus on plugin estimators derived from the OT map between estimates of the underlying distributions. We develop novel stability bounds for OT maps which generalize those in past work, and allow us to reduce the problem of optimally estimating the transport map to that of optimally estimating densities in the Wasserstein distance. In contrast, past work provided a partial connection between these problems and relied on regularity theory for the Monge-Ampere equation to bridge the gap, a step which required unnatural assumptions to obtain sharp guarantees. We also provide some new insights into the connections between stability bounds which arise in the analysis of plugin estimators and growth bounds for the semi-dual functional which arise in the analysis of Brenier potential-based estimators of the transport map. We illustrate the applicability of our new stability bounds by revisiting the smooth setting studied by Manole et al., analyzing two of their estimators under more general conditions. Critically, our bounds do not require smoothness or boundedness assumptions on the underlying measures. As an illustrative application, we develop and analyze a novel tuning parameter-free estimator for the OT map between two strongly log-concave distributions.