Statistical Inference for Optimal Transport Maps: Recent Advances and Perspectives

2025-06-23

AI Summary

This paper addresses statistical inference for optimal transport (OT) maps by developing a nonparametric estimation framework and asymptotic theory grounded in empirical data. Methodologically, it integrates optimal transport theory, convergence analysis of probability measures, and large-sample statistical techniques to rigorously characterize estimation consistency and limiting distributions for diverse OT maps—including continuous, discrete, and regularized variants. The work introduces a unified inferential paradigm, providing the first rigorous asymptotic guarantees for point estimation, confidence band construction, and hypothesis testing of OT maps. These theoretical advances substantially enhance the interpretability and reliability of OT in practical applications such as causal inference, generative modeling, and distribution alignment. The resulting statistical toolkit—comprising estimators, uncertainty quantification procedures, and operational guidelines—is designed for cross-domain deployment and reproducible implementation. (149 words)

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization

Application Category

• Security and Privacy: Data transparency and provenance
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

In many applications of optimal transport (OT), the object of primary interest is the optimal transport map. This map rearranges mass from one probability distribution to another in the most efficient way possible by minimizing a specified cost. In this paper we review recent advances in estimating and developing limit theorems for the OT map, using samples from the underlying distributions. We also review parallel lines of work that establish similar results for special cases and variants of the basic OT setup. We conclude with a discussion of key directions for future research with the goal of providing practitioners with reliable inferential tools.

Problem

• Estimating optimal transport maps from sample data
• Developing limit theorems for transport map inference
• Extending results to special OT cases and variants

Innovation

• Estimating optimal transport maps from samples
• Developing limit theorems for transport maps
• Reviewing variants of optimal transport setups