Central limit theorems for entropy-regularized optimal transport on finite spaces and statistical applications

TL;DR

This paper investigates the asymptotic behavior of entropy-regularized optimal transport (Sinkhorn divergence) on finite spaces, deriving distributional limits and proposing bootstrap tests for distribution comparison in statistical applications.

Contribution

It provides the first derivation of the distributional limits of empirical Sinkhorn divergence and introduces a bootstrap method for hypothesis testing of probability distributions.

Findings

Distributional limits of empirical Sinkhorn divergence derived.

Bootstrap procedure for distribution comparison developed.

Simulations and real data illustrate the methods.

Abstract

The notion of entropy-regularized optimal transport, also known as Sinkhorn divergence, has recently gained popularity in machine learning and statistics, as it makes feasible the use of smoothed optimal transportation distances for data analysis. The Sinkhorn divergence allows the fast computation of an entropically regularized Wasserstein distance between two probability distributions supported on a finite metric space of (possibly) high-dimension. For data sampled from one or two unknown probability distributions, we derive the distributional limits of the empirical Sinkhorn divergence and its centered version (Sinkhorn loss). We also propose a bootstrap procedure which allows to obtain new test statistics for measuring the discrepancies between multivariate probability distributions. Our work is inspired by the results of Sommerfeld and Munk (2016) on the asymptotic distribution of empirical Wasserstein distance on finite space using unregularized transportation costs. Incidentally we also analyze the asymptotic distribution of entropy-regularized Wasserstein distances when the regularization parameter tends to zero. Simulated and real datasets are used to illustrate our approach.