  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.
