提出基于切片Wasserstein的两样本检验，兼具统计最优与计算高效。
Minimax-Optimal Two-Sample Test with Sliced Wasserstein
- 采用置换检验框架，保证小样本下第一类错误控制。
- 在多项分布和有界支撑下达到最小极大分离率n^{-1/2}。
- 无需调参，对不同数据场景表现稳定，适合实际应用。
研究使用切片Wasserstein（SW）距离进行非参数两样本检验。尽管已有理论和实证研究表明SW距离在统计保障与计算效率之间具有良好平衡，但其在假设检验中的理论基础仍不充分。本文提出一种基于置换的SW检验方法，并对其性能进行分析。该检验继承了置换原则的有限样本第一类错误控制能力。进一步建立了非渐近幂界，证明该方法在多项分布和有界支撑替代假设下可实现最小极大分离率n^{-1/2}，与核基测试的最优保证一致，同时保留了Wasserstein距离的几何基础。分析还量化了投影数量与统计功效之间的权衡。数值实验表明，该检验兼具有限样本有效性、竞争性功效和可扩展性；且相较于需精细调参的核基测试，其在所有测试场景中均表现稳健。
原文摘要 · Abstract (English)
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.
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