Approximation rates of entropic maps in semidiscrete optimal transport

TL;DR

This paper improves the understanding of how quickly entropic optimal transport maps converge to the Brenier map in the semidiscrete setting, showing faster rates under a weaker norm and establishing a related central limit theorem.

Contribution

It establishes faster approximation rates of $O( ext{epsilon}^2)$ for entropic maps in the dual Lipschitz norm, improving upon previous $O( ext{epsilon}^{1/2})$ results, and derives a CLT for the estimator.

Findings

Faster $O( ext{epsilon}^2)$ approximation rate under the dual Lipschitz norm.

The $O( ext{epsilon}^2)$ rate is sharp in the dual Lipschitz space.

A central limit theorem for the entropic estimator of the Brenier map.

Abstract

Entropic optimal transport offers a computationally tractable approximation to the classical problem. In this note, we study the approximation rate of the entropic optimal transport map (in approaching the Brenier map) when the regularization parameter $\varepsilon$ tends to zero in the semidiscrete setting, where the input measure is absolutely continuous while the output is finitely discrete. Previous work shows that the approximation rate is $O(\sqrt{\varepsilon})$ under the $L^2$-norm with respect to the input measure. In this work, we establish faster, $O(\varepsilon^2)$ rates up to polylogarithmic factors, under the dual Lipschitz norm, which is weaker than the $L^2$-norm. For the said dual norm, the $O(\varepsilon^2)$ rate is sharp. As a corollary, we derive a central limit theorem for the entropic estimator for the Brenier map in the dual Lipschitz space when the regularization parameter tends to zero as the sample size increases.