A Scalable Nystrom-Based Kernel Two-Sample Test with Permutations

TL;DR

This paper introduces a scalable kernel two-sample test using Nyström approximation of MMD, achieving computational efficiency while maintaining statistical guarantees, suitable for large-scale scientific data.

Contribution

It proposes a Nyström-based kernel two-sample test with finite-sample guarantees and optimal separation rates, addressing computational challenges in large datasets.

Findings

Finite-sample bound on test power for well-separated distributions

Separation rate matches minimax optimal rate

Numerical experiments demonstrate applicability to real scientific data

Abstract

Two-sample hypothesis testing-determining whether two sets of data are drawn from the same distribution-is a fundamental problem in statistics and machine learning with broad scientific applications. In the context of nonparametric testing, maximum mean discrepancy (MMD) has gained popularity as a test statistic due to its flexibility and strong theoretical foundations. However, its use in large-scale scenarios is plagued by high computational costs. In this work, we use a Nystr\"om approximation of the MMD to design a computationally efficient and practical testing algorithm while preserving statistical guarantees. Our main result is a finite-sample bound on the power of the proposed test for distributions that are sufficiently separated with respect to the MMD. The derived separation rate matches the known minimax optimal rate in this setting. We support our findings with a series of numerical experiments, emphasizing applicability to realistic scientific data.