Permutation-Free High-Order Interaction Tests

TL;DR

This paper introduces permutation-free high-order interaction tests using V-statistics and cross-centring, enabling scalable, non-parametric detection of complex multivariate dependencies without permutation-based null approximations.

Contribution

The authors develop a novel class of permutation-free tests for high-order interactions that are computationally efficient and have a standard normal null distribution, advancing multivariate independence testing.

Findings

Tests are scalable and effective on synthetic datasets.

Applications demonstrate importance of high-order interactions in causal discovery.

Method outperforms permutation-based approaches in efficiency.

Abstract

Kernel-based hypothesis tests offer a flexible, non-parametric tool to detect high-order interactions in multivariate data, beyond pairwise relationships. Yet the scalability of such tests is limited by the computationally demanding permutation schemes used to generate null approximations. Here we introduce a family of permutation-free high-order tests for joint independence and partial factorisations of $d$ variables. Our tests eliminate the need for permutation-based approximations by leveraging V-statistics and a novel cross-centring technique to yield test statistics with a standard normal limiting distribution under the null. We present implementations of the tests and showcase their efficacy and scalability through synthetic datasets. We also show applications inspired by causal discovery and feature selection, which highlight both the importance of high-order interactions in data and the need for efficient computational methods.