A uniform kernel trick for high-dimensional two-sample problems

TL;DR

This paper introduces a simplified, uniform kernel trick for high-dimensional two-sample tests, providing theoretical distribution results and demonstrating advantages over existing methods in experiments.

Contribution

It proposes a novel uniform kernel trick for two-sample testing that simplifies kernel selection and offers theoretical and practical improvements.

Findings

Asymptotic distribution derived under null and alternative hypotheses

Method shows advantages over standard kernel and energy distance tests

Experimental results demonstrate improved performance

Abstract

We use a suitable version of the so-called "kernel trick" to devise two-sample (homogeneity) tests, especially focussed on high-dimensional and functional data. Our proposal entails a simplification related to the important practical problem of selecting an appropriate kernel function. Specifically, we apply a uniform variant of the kernel trick which involves the supremum within a class of kernel-based distances. We obtain the asymptotic distribution (under the null and alternative hypotheses) of the test statistic. The proofs rely on empirical processes theory, combined with the delta method and Hadamard (directional) differentiability techniques, and functional Karhunen-Lo\`eve-type expansions of the underlying processes. This methodology has some advantages over other standard approaches in the literature. We also give some experimental insight into the performance of our proposal compared to the original kernel-based approach \cite{Gretton2007} and the test based on energy distances \cite{Szekely-Rizzo-2017}.