A Kernel Two-Sample Test Invariant under Group Action with Applications to Functional Data

TL;DR

This paper develops a kernel-based two-sample test invariant under group actions, extending classical methods to non-compact groups and functional data, ensuring consistency and applicability in practical scenarios.

Contribution

It introduces invariant kernels for locally compact groups, extending Haar-based approaches beyond compact groups, and applies this to functional data analysis.

Findings

Invariant MMD test is consistent under natural conditions.

Method effectively handles invariances like temporal shifts in functional data.

Simulation studies demonstrate the test's practical effectiveness.

Abstract

We introduce a kernel-based two-sample test for comparing probability distributions up to group actions. Our construction yields invariant kernels for locally compact $\sigma$-compact groups and extends classical Haar-based approaches beyond the compact setting. The resulting invariant Maximum Mean Discrepancy (MMD) test is developed in a general framework where the sample space is assumed to be Polish. Under natural conditions, the invariant kernel induces a characteristic kernel on the quotient space, ensuring consistency of the associated MMD test. The method is well suited to functional data, where invariances such as temporal shifts arise naturally, and its effectiveness is illustrated through simulation studies.