Minimax-Optimal Two-Sample Test with Sliced Wasserstein

2025-10-31

AI Summary

This paper addresses the lack of rigorous hypothesis testing theory for the sliced Wasserstein (SW) distance in nonparametric two-sample testing. We propose a permutation-based SW test that strictly controls Type-I error under finite samples. Crucially, we establish the first non-asymptotic power bound for SW testing, achieving the minimax-optimal separation rate of $n^{-1/2}$. Unlike kernel-based methods, our approach is parameter-free and inherently robust—adapting automatically to underlying data structures. We further prove its statistical optimality under multi-class settings and bounded-support alternatives. Empirical results demonstrate that the method maintains statistical validity while delivering high power and favorable scalability.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

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.

Problem

• Develops minimax-optimal two-sample test using sliced Wasserstein distance
• Establishes theoretical foundations for hypothesis testing with statistical guarantees
• Analyzes projection-power tradeoff while maintaining computational efficiency

Innovation

• Permutation-based sliced Wasserstein test for two-sample testing
• Achieves minimax separation rate with finite-sample error control
• Balances projection count and power without kernel tuning