# Minimax-Optimal Two-Sample Test with Sliced Wasserstein

arXiv: https://arxiv.org/abs/2510.27498 · HTML: https://mlsanity.com/paper/2510.27498/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 86 references that are in the map.
- First posted on arXiv 31 Oct 2025.

## Builds on (most cited first, 24 of 24)

- [MMD GAN: Towards Deeper Understanding of Moment Matching Network](https://mlsanity.com/paper/1705.08584/) (2017)
- [Generalized Sliced Wasserstein Distances](https://mlsanity.com/paper/1902.00434/) (2019)
- [On Wasserstein Two Sample Testing and Related Families of Nonparametric Tests](https://mlsanity.com/paper/1509.02237/) (2015)
- [Max-Sliced Wasserstein Distance and its use for GANs](https://mlsanity.com/paper/1904.05877/) (2019)
- [Subspace Robust Wasserstein Distances](https://mlsanity.com/paper/1901.08949/) (2019)
- [MMD Aggregated Two-Sample Test](https://mlsanity.com/paper/2110.15073/) (2021)
- [Statistical optimal transport](https://mlsanity.com/paper/2407.18163/) (2024)
- [Model Equality Testing: Which Model Is This API Serving?](https://mlsanity.com/paper/2410.20247/) (2024)
- [Statistical, Robustness, and Computational Guarantees for Sliced Wasserstein Distances](https://mlsanity.com/paper/2210.09160/) (2022)
- [MMD-FUSE: Learning and Combining Kernels for Two-Sample Testing Without Data Splitting](https://mlsanity.com/paper/2306.08777/) (2023)
- [Transductive Rademacher Complexity and its Applications](https://mlsanity.com/paper/1401.3441/) (2014)
- [A Permutation-free Kernel Two-Sample Test](https://mlsanity.com/paper/2211.14908/) (2022)
- [Two-sample Test using Projected Wasserstein Distance](https://mlsanity.com/paper/2010.11970/) (2020)
- [Compress Then Test: Powerful Kernel Testing in Near-linear Time](https://mlsanity.com/paper/2301.05974/) (2023)
- [A Permutation-Free Kernel Independence Test](https://mlsanity.com/paper/2212.09108/) (2022)
- [Differentially Private Permutation Tests](https://mlsanity.com/paper/2310.19043/) (2023)
- [Sharp bounds for max-sliced Wasserstein distances](https://mlsanity.com/paper/2403.00666/) (2024)
- [Refereeing the Referees: Evaluating Two-Sample Tests for Validating Generators in Precision Sciences](https://mlsanity.com/paper/2409.16336/) (2024)
- [Shedding a PAC-Bayesian Light on Adaptive Sliced-Wasserstein Distances](https://mlsanity.com/paper/2206.03230/) (2022)
- [A Scalable Nystrom-Based Kernel Two-Sample Test with Permutations](https://mlsanity.com/paper/2502.13570/) (2025)
- [Computational-Statistical Trade-off in Kernel Two-Sample Testing with Random Fourier Features](https://mlsanity.com/paper/2407.08976/) (2024)
- [Statistical and Computational Guarantees of Kernel Max-Sliced Wasserstein Distances](https://mlsanity.com/paper/2405.15441/) (2024)
- [Minimax Optimal Kernel Two-Sample Tests with Random Features](https://mlsanity.com/paper/2502.20755/) (2025)
- [Two-sample tests for repeated measurements of histogram objects with applications to wearable device data](https://mlsanity.com/paper/2106.12768/) (2021)

## Built on by (most cited first, 1 of 1)

- [Model-Agnostic Signal Discovery with Machine Learning: Bridging the Gap Between Theory and Practice](https://mlsanity.com/paper/2605.31103/) (2026)

## Abstract (arXiv)

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.