# Characterization of translation invariant MMD on Rd and connections with Wasserstein distances

Thibault Modeste, Clément Dombry.

Year: 2024, Volume: 25, Issue: 237, Pages: 1−39

#### Abstract

Kernel mean embeddings and maximum mean discrepancies (MMD) associated with positive definite kernels are important tools in machine learning that allow to compare probability measures and sample distributions. We provide a full characterization of translation invariant MMDs on $\mathbb{R}^d$ that are parametrized by a spectral measure and a semi-definite positive symmetric matrix. Furthermore, we investigate the connections between translation invariant MMDs and Wasserstein distances on $\mathbb{R}^d$. We show in particular that convergence with respect to the MMD associated with the Energy Kernel of order $\alpha\in(0,1)$ implies convergence with respect to the Wasserstein distance of order $\beta<\alpha$. We also provide examples of kernels metrizing the Wasserstein space of order $\alpha\geq 1$. A short numerical experiment illustrates our findings in the framework of the one-sample-test.