A Practical Introduction to Kernel Discrepancies: MMD, HSIC & KSD

This article provides a practical introduction to kernel discrepancies, focusing on the Maximum Mean Discrepancy (MMD), the Hilbert-Schmidt Independence Criterion (HSIC), and the Kernel Stein Discrepancy (KSD). Various estimators for these discrepancies are presented, including the commonly-used V-statistics and U-statistics, as well as several forms of the more computationallyefficient incomplete U-statistics. The importance of the choice of kernel bandwidth is stressed, showing how it affects the behaviour of the discrepancy estimation. Adaptive estimators are introduced, which combine multiple estimators with various kernels, addressing the problem of kernel selection. This paper corresponds to the introduction of my PhD thesis (Schrab, 2025a, Chapter 2) and is presented as a standalone article to introduce the reader to kernel discrepancies estimators. First, in Section 1, we define kernels, Reproducing Kernel Hilbert Spaces, mean embeddings and cross-covariance operators, and present kernel properties such as characteristicity, universality and translation invariance. Then, in Section 2, we introduce the Maximum Mean Discprecancy, the Hilbert-Schmidt Independence Criterion, and the Kernel Stein Discrepancy, as well as their estimators, and we discuss the importance of the choice of kernel for such measures. We then introduce a collection of statistics in Section 3, including the commonly-used complete statistics, as well as their incomplete counterparts which trade accuracy for computational efficiency. Finally, in Section 4, we construct adaptive estimators combining multiple statistics with various kernels, which is one method to address the problem of kernel selection.