Distance covariance in metric spaces

TL;DR

This paper generalizes the concept of distance covariance from Euclidean spaces to general metric spaces, establishing conditions for independence testing and providing elementary proofs.

Contribution

It extends the theory of distance covariance to metric spaces and characterizes when it can be used for independence testing, answering a key open question.

Findings

Distance covariance applies to metric spaces of strong negative type.

Separable Hilbert spaces satisfy the strong negative type condition.

Elementary inequalities replace Fourier transform techniques.

Abstract

We extend the theory of distance (Brownian) covariance from Euclidean spaces, where it was introduced by Sz\'{e}kely, Rizzo and Bakirov, to general metric spaces. We show that for testing independence, it is necessary and sufficient that the metric space be of strong negative type. In particular, we show that this holds for separable Hilbert spaces, which answers a question of Kosorok. Instead of the manipulations of Fourier transforms used in the original work, we use elementary inequalities for metric spaces and embeddings in Hilbert spaces.