Stability bounds for smooth optimal transport maps and their statistical implications

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-Ampère 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. [41], 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 provide a novel analysis of the tuning parameter-free nearest neighbor estimator of the OT map between two strongly log-concave and log-smooth distributions.