Stability Bounds for Smooth Optimal Transport Maps and their Statistical Implications
Sivaraman Balakrishnan, Tudor Manole
TL;DR
This work addresses estimating the optimal transport map $T_0$ between two unknown distributions $P$ and $Q$ from samples by analyzing plugin and dual estimators through a new, unified stability bound. The authors show that, under Brenier regularity and smoothness, the discrepancy between an estimated coupling and the true OT map can be controlled by the Wasserstein distances between the estimated and true marginals, enabling unconditional guarantees that do not require smoothness or boundedness of the underlying measures. They prove sharp consequences for the smooth-density setting, achieving minimax-optimal rates (up to log factors in dimension 2) for plugin estimators and providing a tuning-parameter-free, practical estimator for strongly log-concave distributions. They also show that a nearest-neighbor OT estimator attains minimax rates under mild moment conditions, extending the results to log-smooth and log-strongly-concave cases. Overall, the paper unifies stability and growth analyses of semi-dual functionals with distribution estimation in Wasserstein distance, yielding practically applicable, sharp statistical guarantees for OT-map estimation from data.
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.