Extending Kernel Testing To General Designs

TL;DR

This paper extends kernel-based non-parametric testing methods from simple two-sample scenarios to complex experimental designs by introducing a linear model in RKHS and a new test statistic, broadening applicability.

Contribution

It proposes a linear model in RKHS for general designs and introduces a truncated kernel Hotelling-Lawley statistic with proven asymptotic properties, enabling kernel testing beyond two-sample cases.

Findings

The new test statistic follows a chi-square distribution asymptotically.

The framework is demonstrated on single-cell RNA sequencing data.

Kernel-based diagnostic tools are generalized for complex designs.

Abstract

Kernel-based testing has revolutionized the field of non-parametric tests through the embedding of distributions in an RKHS. This strategy has proven to be powerful and flexible, yet its applicability has been limited to the standard two-sample case, while practical situations often involve more complex experimental designs. To extend kernel testing to any design, we propose a linear model in the RKHS that allows for the decomposition of mean embeddings into additive functional effects. We then introduce a truncated kernel Hotelling-Lawley statistic to test the effects of the model, demonstrating that its asymptotic distribution is chi-square, which remains valid with its Nystrom approximation. We discuss a homoscedasticity assumption that, although absent in the standard two-sample case, is necessary for general designs. Finally, we illustrate our framework using a single-cell RNA sequencing dataset and provide kernel-based generalizations of classical diagnostic and exploration tools to broaden the scope of kernel testing in any experimental design.