Statistical bounds for entropic optimal transport: sample complexity and the central limit theorem

TL;DR

This paper establishes new statistical bounds for entropic optimal transport, including sample complexity rates and a central limit theorem, extending previous results to unbounded measures and continuous spaces.

Contribution

It provides the first CLT for entropic OT in arbitrary dimensions and improves sample complexity bounds significantly over prior work.

Findings

Enhanced sample complexity bounds for entropic OT

First CLT established for continuous measures in arbitrary dimensions

Application to entropy estimation of noisy random variables

Abstract

We prove several fundamental statistical bounds for entropic OT with the squared Euclidean cost between subgaussian probability measures in arbitrary dimension. First, through a new sample complexity result we establish the rate of convergence of entropic OT for empirical measures. Our analysis improves exponentially on the bound of Genevay et al. (2019) and extends their work to unbounded measures. Second, we establish a central limit theorem for entropic OT, based on techniques developed by Del Barrio and Loubes (2019). Previously, such a result was only known for finite metric spaces. As an application of our results, we develop and analyze a new technique for estimating the entropy of a random variable corrupted by gaussian noise.