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

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-Loève-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}.

21 pages, 11 figures

References in corpus (8)

• Equivalence of distance-based and RKHS-based statistics in hypothesis testing
• Hilbert space embeddings and metrics on probability measures
• Universality, Characteristic Kernels and RKHS Embedding of Measures
• On the optimal estimation of probability measures in weak and strong topologies
• A Kernel Two-Sample Test for Functional Data
• On approximate validation of models: A Kolmogorov-Smirnov based approach
• MMD Aggregated Two-Sample Test
• MMD-FUSE: Learning and Combining Kernels for Two-Sample Testing Without Data Splitting