Wasserstein Identity Testing

TL;DR

This paper introduces Wasserstein identity testing, a new approach for distribution testing in metric spaces that overcomes limitations of traditional methods by providing nearly optimal sample complexities, especially for distributions satisfying the Doubling Condition.

Contribution

It proposes the Wasserstein identity testing framework and establishes nearly optimal sample complexities, advancing distribution testing in large or continuous support spaces.

Findings

Nearly optimal worst-case sample complexity achieved.

Instance-optimal sample complexity for distributions satisfying the Doubling Condition.

Addresses limitations of $L_1$-distance testing in large or continuous supports.

Abstract

Uniformity testing and the more general identity testing are well studied problems in distributional property testing. Most previous work focuses on testing under $L_1$-distance. However, when the support is very large or even continuous, testing under $L_1$-distance may require a huge (even infinite) number of samples. Motivated by such issues, we consider the identity testing in Wasserstein distance (a.k.a. transportation distance and earthmover distance) on a metric space (discrete or continuous). In this paper, we propose the Wasserstein identity testing problem (Identity Testing in Wasserstein distance). We obtain nearly optimal worst-case sample complexity for the problem. Moreover, for a large class of probability distributions satisfying the so-called "Doubling Condition", we provide nearly instance-optimal sample complexity.