Stability Bounds for Smooth Optimal Transport Maps and their Statistical Implications

2025-02-17

AI Summary

This paper investigates stability bounds and statistical implications of optimal transport (OT) map estimation, relaxing classical strong assumptions—such as density smoothness and boundedness. Methodologically, it reformulates OT map estimation as a density estimation problem under the Wasserstein metric, leveraging Brenier’s potential and the semi-dual functional to derive novel stability bounds while circumventing regularity analysis of the Monge–Ampère equation. Key contributions include: (i) the first sharp stability bound for OT maps without smoothness or boundedness assumptions on the densities; (ii) a precise characterization of stability in terms of the growth rate of the semi-dual functional; and (iii) the construction of the first parameter-free, adaptive plug-in OT map estimator for strongly log-concave distributions, achieving minimax-optimal statistical convergence rates—significantly extending both the scope and accuracy of prior work by Manole et al.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization
• Machine Learning: Learning with Manifolds

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Sustainability of Web economics

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.

Problem

• Estimating optimal transport maps
• Developing stability bounds
• Analyzing plugin estimators

Innovation

• Develops novel stability bounds
• Reduces transport map estimation
• Analyzes tuning parameter-free estimator