Minimax-Optimal Two-Sample Test with Sliced Wasserstein
This provides a robust and efficient testing method for statisticians and machine learning practitioners, though it is incremental as it builds on existing sliced Wasserstein foundations.
The paper tackled the problem of nonparametric two-sample testing by proposing a permutation-based test using the sliced Wasserstein distance, achieving the minimax separation rate of n^{-1/2} and demonstrating competitive power and scalability in experiments.
We study the problem of nonparametric two-sample testing using the sliced Wasserstein (SW) distance. While prior theoretical and empirical work indicates that the SW distance offers a promising balance between strong statistical guarantees and computational efficiency, its theoretical foundations for hypothesis testing remain limited. We address this gap by proposing a permutation-based SW test and analyzing its performance. The test inherits finite-sample Type I error control from the permutation principle. Moreover, we establish non-asymptotic power bounds and show that the procedure achieves the minimax separation rate $n^{-1/2}$ over multinomial and bounded-support alternatives, matching the optimal guarantees of kernel-based tests while building on the geometric foundations of Wasserstein distances. Our analysis further quantifies the trade-off between the number of projections and statistical power. Finally, numerical experiments demonstrate that the test combines finite-sample validity with competitive power and scalability, and -- unlike kernel-based tests, which require careful kernel tuning -- it performs consistently well across all scenarios we consider.