Minimax-Optimal Two-Sample Test with Sliced Wasserstein

Binh Thuan Tran, Nicolas Schreuder
Proceedings of The 29th International Conference on Artificial Intelligence and Statistics, PMLR 300:1927-1935, 2026.

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 $n^{-1/2}$ with respect to the sliced Wasserstein distance over multinomial and bounded-support alternatives. This matches the optimal minimax rate $n^{-1/2}$ achieved by kernel-based tests with respect to the MMD, while leveraging the geometric structure 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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v300-tran26c, title = { Minimax-Optimal Two-Sample Test with Sliced Wasserstein }, author = {Tran, Binh Thuan and Schreuder, Nicolas}, booktitle = {Proceedings of The 29th International Conference on Artificial Intelligence and Statistics}, pages = {1927--1935}, year = {2026}, editor = {Khan, Emtiyaz and Li, Yingzhen and Solin, Arno and Ramdas, Aaditya}, volume = {300}, series = {Proceedings of Machine Learning Research}, month = {02--05 May}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v300/main/assets/tran26c/tran26c.pdf}, url = {https://proceedings.mlr.press/v300/tran26c.html}, 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 $n^{-1/2}$ with respect to the sliced Wasserstein distance over multinomial and bounded-support alternatives. This matches the optimal minimax rate $n^{-1/2}$ achieved by kernel-based tests with respect to the MMD, while leveraging the geometric structure 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. } }
Endnote
%0 Conference Paper %T Minimax-Optimal Two-Sample Test with Sliced Wasserstein %A Binh Thuan Tran %A Nicolas Schreuder %B Proceedings of The 29th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2026 %E Emtiyaz Khan %E Yingzhen Li %E Arno Solin %E Aaditya Ramdas %F pmlr-v300-tran26c %I PMLR %P 1927--1935 %U https://proceedings.mlr.press/v300/tran26c.html %V 300 %X 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 $n^{-1/2}$ with respect to the sliced Wasserstein distance over multinomial and bounded-support alternatives. This matches the optimal minimax rate $n^{-1/2}$ achieved by kernel-based tests with respect to the MMD, while leveraging the geometric structure 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.
APA
Tran, B.T. & Schreuder, N.. (2026). Minimax-Optimal Two-Sample Test with Sliced Wasserstein . Proceedings of The 29th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 300:1927-1935 Available from https://proceedings.mlr.press/v300/tran26c.html.

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