Generalization of the energy distance by Bernstein functions

TL;DR

This paper provides a new perspective on the energy distance metric on Hilbert spaces, generalizing it through Bernstein functions and analyzing its behavior with various kernels.

Contribution

It introduces a novel approach to generalize the energy distance using Bernstein functions and conditionally negative definite kernels, expanding its applicability.

Findings

Reproves the energy distance as a metric on probability measures in Hilbert spaces.

Generalizes energy distance to kernels related to Bernstein functions.

Analyzes the energy distance behavior for kernels with powers greater than two.

Abstract

We reprove the well known fact that the energy distance defines a metric on the space of Borel probability measures on a Hilbert space with finite first moment by a new approach, by analyzing the behavior of the Gaussian kernel on Hilbert spaces and a Maximum Mean Discrepancy analysis. From this new point of view we are able to generalize the energy distance metric to a family of kernels related to Bernstein functions and conditionally negative definite kernels. We also explain what occurs on the energy distance on the kernel $\|x-y\|^{\alpha}$ for every $\alpha >2$, where we also generalize the idea to a family of kernels related to derivatives of completely monotone functions and conditionally negative definite kernels.