Minimax-Optimal Two-Sample Test with Sliced Wasserstein

2025/10/31 by Tran, Binh Thuan, Schreuder, Nicolas
#FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Methodology (stat.ME) #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.2510.27498

Abstract

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.

Citations

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Related

• Nearly Minimax Optimal Wasserstein Conditional Independence Testing
• On Wasserstein Two Sample Testing and Related Families of Nonparametric\n Tests
• Permutation Tests at Nonparametric Rates
• Statistical and Computational Guarantees of Kernel Max-Sliced Wasserstein Distances
• A Scalable Nystrom-Based Kernel Two-Sample Test with Permutations