Exact Statistical Inference for the Wasserstein Distance by Selective Inference

TL;DR

This paper introduces a novel exact statistical inference method for the Wasserstein distance that provides finite-sample valid confidence intervals in both one-dimensional and multi-dimensional settings, addressing limitations of asymptotic approaches.

Contribution

It presents the first non-asymptotic inference method for Wasserstein distance with finite-sample guarantees, applicable to multi-dimensional problems.

Findings

Valid finite-sample confidence intervals demonstrated on synthetic data.

Method performs well on real-world datasets.

Applicable to both one-dimensional and multi-dimensional cases.

Abstract

In this paper, we study statistical inference for the Wasserstein distance, which has attracted much attention and has been applied to various machine learning tasks. Several studies have been proposed in the literature, but almost all of them are based on asymptotic approximation and do not have finite-sample validity. In this study, we propose an exact (non-asymptotic) inference method for the Wasserstein distance inspired by the concept of conditional Selective Inference (SI). To our knowledge, this is the first method that can provide a valid confidence interval (CI) for the Wasserstein distance with finite-sample coverage guarantee, which can be applied not only to one-dimensional problems but also to multi-dimensional problems. We evaluate the performance of the proposed method on both synthetic and real-world datasets.