Nystr\"om $M$-Hilbert-Schmidt Independence Criterion

TL;DR

This paper introduces a Nyström-based estimator for the Hilbert-Schmidt independence criterion (HSIC) that efficiently handles multiple variables, providing theoretical guarantees and demonstrating its effectiveness in various applications.

Contribution

It extends HSIC to the case of multiple variables ($M  2$), offers a consistent estimator, and applies it to real-world problems with theoretical backing.

Findings

The proposed estimator is consistent.

It effectively handles $M  2$ variables.

Demonstrated success in dependency testing and causal discovery.

Abstract

Kernel techniques are among the most popular and powerful approaches of data science. Among the key features that make kernels ubiquitous are (i) the number of domains they have been designed for, (ii) the Hilbert structure of the function class associated to kernels facilitating their statistical analysis, and (iii) their ability to represent probability distributions without loss of information. These properties give rise to the immense success of Hilbert-Schmidt independence criterion (HSIC) which is able to capture joint independence of random variables under mild conditions, and permits closed-form estimators with quadratic computational complexity (w.r.t. the sample size). In order to alleviate the quadratic computational bottleneck in large-scale applications, multiple HSIC approximations have been proposed, however these estimators are restricted to $M=2$ random variables, do not extend naturally to the $M\ge 2$ case, and lack theoretical guarantees. In this work, we propose an alternative Nystr\"om-based HSIC estimator which handles the $M\ge 2$ case, prove its consistency, and demonstrate its applicability in multiple contexts, including synthetic examples, dependency testing of media annotations, and causal discovery.