Hypothesis Test and Confidence Analysis with Wasserstein Distance on General Dimension

Abstract: We develop a general framework for statistical inference with the 1-Wasserstein distance. Recently, the Wasserstein distance has attracted considerable attention and has been widely applied to various machine learning tasks because of its excellent properties. However, hypothesis tests and a confidence analysis for the Wasserstein distance have not been established in a general multivariate setting. This is because the limit distribution of the empirical distribution with the Wasserstein distance is unavailable without strong restriction. To address this problem, in this study, we develop a novel non-asymptotic Gaussian approximation for the empirical 1-Wasserstein distance. Using the approximation method, we develop a hypothesis test and confidence analysis for the empirical 1-Wasserstein distance. Additionally, we provide a theoretical guarantee and an efficient algorithm for the proposed approximation. Our experiments validate its performance numerically.