Conditional Dependence via U-Statistics Pruning

TL;DR

This paper introduces a new measure of conditional dependence that avoids matrix inversion by using incomplete unbiased U-statistics, extending the Hilbert-Schmidt independence criterion with kernel operations on data 4-tuples.

Contribution

It proposes a novel conditional dependence measure based on incomplete U-statistics, enabling data pruning and avoiding ill-conditioned matrix inversions.

Findings

Avoids matrix inversion in conditional dependence testing

Extends Hilbert-Schmidt independence criterion with 4-tuple kernels

Enables data pruning based on confounder observations

Abstract

The problem of measuring conditional dependence between two random phenomena arises when a third one (a confounder) has a potential influence on the amount of information between them. A typical issue in this challenging problem is the inversion of ill-conditioned autocorrelation matrices. This paper presents a novel measure of conditional dependence based on the use of incomplete unbiased statistics of degree two, which allows to re-interpret independence as uncorrelatedness on a finite-dimensional feature space. This formulation enables to prune data according to observations of the confounder itself, thus avoiding matrix inversions altogether. The proposed approach is articulated as an extension of the Hilbert-Schmidt independence criterion, which becomes expressible through kernels that operate on 4-tuples of data.