Hypothesis Test and Confidence Analysis with Wasserstein Distance with General Dimension

TLDR: A novel nonasymptotic gaussian approximation for the empirical 1-Wasserstein distance is developed and a hypothesis test and confidence analysis are developed and an efficient algorithm is provided for the proposed approximation.

Abstract: We develop a general framework for statistical inference with the Wasserstein distance. Recently, the Wasserstein distance has attracted much attention and been applied to various machine learning tasks due to its celebrated properties. Despite the importance, hypothesis tests and confidence analysis with the Wasserstein distance have not been available in a general setting, since a limit distribution of empirical distribution with Wasserstein distance has been unavailable without strong restrictions. In this study, we develop a novel \textit{non-asymptotic Gaussian approximation} for the empirical Wasserstein distance, which can avoid the problem of unavailable limit distribution. By the approximation method, we develop a hypothesis test and confidence analysis for the empirical Wasserstein distance. We also provide a theoretical guarantee and an efficient algorithm for the proposed approximation. Our experiments validate its performance numerically.