Minimax Rates of Estimation for Optimal Transport Map between Infinite-Dimensional Spaces

TL;DR

This paper establishes the minimax optimal rate for estimating smooth optimal transport maps in infinite-dimensional spaces, providing a practical estimator and validating its effectiveness through experiments.

Contribution

It introduces the first minimax rate analysis for smooth optimal transport maps in infinite-dimensional spaces and develops an estimator achieving this rate.

Findings

Minimax risk rate is polynomial in sample size for smooth maps.

Proposed estimator attains the minimax optimal rate.

Experiments confirm theoretical and practical effectiveness.

Abstract

We investigate the estimation of an optimal transport map between probability measures on an infinite-dimensional space and reveal its minimax optimal rate. Optimal transport theory defines distances within a space of probability measures, utilizing an optimal transport map as its key component. Estimating the optimal transport map from samples finds several applications, such as simulating dynamics between probability measures and functional data analysis. However, some transport maps on infinite-dimensional spaces require exponential-order data for estimation, which undermines their applicability. In this paper, we investigate the estimation of an optimal transport map between infinite-dimensional spaces, focusing on optimal transport maps characterized by the notion of $\gamma$-smoothness. Consequently, we show that the order of the minimax risk is polynomial rate in the sample size even in the infinite-dimensional setup. We also develop an estimator whose estimation error matches the minimax optimal rate. With these results, we obtain a class of reasonably estimable optimal transport maps on infinite-dimensional spaces and a method for their estimation. Our experiments validate the theory and practical utility of our approach with application to functional data analysis.