Estimation of entropy-regularized optimal transport maps between non-compactly supported measures

TL;DR

This paper develops estimators for entropy-regularized optimal transport maps between subGaussian measures, achieving convergence rates that do not require the measures to have compact support, thus broadening applicability.

Contribution

It introduces new error bounds for EOT map estimators under subGaussian assumptions, removing the need for compact support conditions and providing polynomial dependence on regularization.

Findings

Expected $L^2$-error decays as $O(n^{-1/3})$ for compactly supported or strongly log-concave targets.

Expected $L^1$-error decays as $O(n^{-1/6})$ in the general subGaussian case.

Experimental results suggest potential looseness in variance control.

Abstract

This paper addresses the problem of estimating entropy-regularized optimal transport (EOT) maps with squared-Euclidean cost between source and target measures that are subGaussian. In the case that the target measure is compactly supported or strongly log-concave, we show that for a recently proposed in-sample estimator, the expected squared $L^2$-error decays at least as fast as $O(n^{-1/3})$ where $n$ is the sample size. For the general subGaussian case we show that the expected $L^1$-error decays at least as fast as $O(n^{-1/6})$, and in both cases we have polynomial dependence on the regularization parameter. While these results are suboptimal compared to known results in the case of compactness of both the source and target measures (squared $L^2$-error converging at a rate $O(n^{-1})$) and for when the source is subGaussian while the target is compactly supported (squared $L^2$-error converging at a rate $O(n^{-1/2})$), their importance lie in eliminating the compact support requirements. The proof technique makes use of a bias-variance decomposition where the variance is controlled using standard concentration of measure results and the bias is handled by T1-transport inequalities along with sample complexity results in estimation of EOT cost under subGaussian assumptions. Our experimental results point to a looseness in controlling the variance terms and we conclude by posing several open problems.