Hilbert space embeddings of independence tests of several variables

TL;DR

This paper develops a unified Hilbert space framework for independence tests involving multiple variables, extending existing measures like HSIC and distance covariance through new kernel constructions.

Contribution

It introduces a general theory for embedding independence tests with novel kernels on product spaces, broadening the scope of multivariate independence testing.

Findings

Unified framework for independence testing in Hilbert spaces

New classes of kernels based on Bernstein and monotone functions

Explicit methods for constructing these kernels

Abstract

In this paper, we present the general theory of embedding independence tests on Hilbert spaces that generalizes the concepts of distance covariance, distance multivariance and HSIC. This is done by defining new types of kernel on an $n$ Cartesian product called positive definite independent of order $k$. An emphasis is given on the continuous case in order to obtain a version of the Kernel Mean Embedding for this new classes of kernels. We also provide $2$ explicit methods to construct examples for this new type of kernel on a general space by using Bernstein functions of several variables and completely monotone functions of higher order.