Estimation of Stochastic Optimal Transport Maps

TL;DR

This paper develops a new framework for estimating stochastic optimal transport maps, providing efficient estimators with strong theoretical guarantees that work under minimal assumptions and in contaminated data scenarios.

Contribution

It introduces a novel metric for stochastic map quality, develops near-optimal estimators with risk bounds, and extends OT map estimation theory to stochastic and adversarial settings.

Findings

Proposed a new metric for stochastic transport quality.

Designed estimators with near-optimal finite-sample risk bounds.

Validated the approach through empirical experiments.

Abstract

The optimal transport (OT) map is a geometry-driven transformation between high-dimensional probability distributions which underpins a wide range of tasks in statistics, applied probability, and machine learning. However, existing statistical theory for OT map estimation is quite restricted, hinging on Brenier's theorem (quadratic cost, absolutely continuous source) to guarantee existence and uniqueness of a deterministic OT map, on which various additional regularity assumptions are imposed to obtain quantitative error bounds. In many real-world problems these conditions fail or cannot be certified, in which case optimal transportation is possible only via stochastic maps that can split mass. To broaden the scope of map estimation theory to such settings, this work introduces a novel metric for evaluating the transportation quality of stochastic maps. Under this metric, we develop computationally efficient map estimators with near-optimal finite-sample risk bounds, subject to easy-to-verify minimal assumptions. Our analysis further accommodates common forms of adversarial sample contamination, yielding estimators with robust estimation guarantees. Empirical experiments are provided which validate our theory and demonstrate the utility of the proposed framework in settings where existing theory fails. These contributions constitute the first general-purpose theory for map estimation, compatible with a wide spectrum of real-world applications where optimal transport may be intrinsically stochastic.