Nearly Minimax Optimal Wasserstein Conditional Independence Testing

TL;DR

This paper introduces a nearly minimax optimal Wasserstein-based method for conditional independence testing, expanding the class of distributions that can be tested under weaker smoothness conditions than previous total variation approaches.

Contribution

It proposes a novel Wasserstein-based test statistic that nearly achieves the optimal critical radius, broadening the scope of distributions for conditional independence testing.

Findings

Achieves nearly optimal minimax rate for Wasserstein conditional independence testing.

Introduces a weighted multi-resolution U-statistic as a new test statistic.

Expands the class of null and alternative distributions to include point masses.

Abstract

This paper is concerned with minimax conditional independence testing. In contrast to some previous works on the topic, which use the total variation distance to separate the null from the alternative, here we use the Wasserstein distance. In addition, we impose Wasserstein smoothness conditions which on bounded domains are weaker than the corresponding total variation smoothness imposed, for instance, by Neykov et al. [2021]. This added flexibility expands the distributions which are allowed under the null and the alternative to include distributions which may contain point masses for instance. We characterize the optimal rate of the critical radius of testing up to logarithmic factors. Our test statistic which nearly achieves the optimal critical radius is novel, and can be thought of as a weighted multi-resolution version of the U-statistic studied by Neykov et al. [2021].