Estimation of Stochastic Optimal Transport Maps

2025-12-10

AI Summary

Existing optimal transport (OT) mapping estimation theory heavily relies on Brenier’s theorem—which requires quadratic cost and absolutely continuous source distributions—rendering it inadequate for stochastic OT mappings with mass splitting, commonly encountered in real-world settings involving singular, discrete, or corrupted source/target distributions. Method: We propose a novel metric to quantify the quality of stochastic OT mappings and develop the first universal, robust, finite-sample optimal risk bound framework. Our approach integrates generalization error analysis, adversarially robust statistical learning, parameterized stochastic mapping modeling, and regularized empirical risk minimization. Contribution/Results: We derive near-optimal finite-sample risk bounds under minimal distributional assumptions. Experiments demonstrate substantial improvements in transport accuracy over conventional OT methods in challenging non-absolutely-continuous and corrupted-data regimes where standard approaches fail.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Calibration & Uncertainty Quantification
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• Develops a metric for evaluating stochastic optimal transport maps
• Provides efficient estimators with robust finite-sample risk bounds
• Extends theory to real-world applications with stochastic transport

Innovation

• Introduces novel metric for stochastic transport maps
• Develops computationally efficient estimators with risk bounds
• Accommodates adversarial contamination for robust guarantees