Generalization of the HSIC and distance covariance using positive definite independent kernels

TL;DR

This paper introduces a generalized independence criterion using positive definite independent kernels, expanding the theoretical framework of HSIC and distance covariance for analyzing variable independence.

Contribution

It generalizes HSIC and distance covariance by defining positive definite independent kernels and characterizes radial kernels that are positive definite independent on Euclidean spaces.

Findings

Characterization of radial kernels that are positive definite independent

Examples of positive definite independent kernels provided

Theoretical framework for independence testing expanded

Abstract

Hilbert-Schmidt independence criterion and distance covariance are methods to describe independence of random variables using either the Kronecker product of positive definite kernels or the Kronecker product of conditionally negative definite kernels. In this paper we generalize both methods by providing an independence criteria using a new concept, of positive definite independent kernels. We provide a characterization of the radial kernels that are positive definite independent on all Euclidean spaces and we present several examples.