ICML 2025

Permutation-Free High-Order Interaction Tests
Zhaolu Liu 1 Robert L. Peach 2 3 Mauricio Barahona 1
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 factorisa-
tions of d variables. Our tests eliminate the need
for permutation-based approximations by lever-
aging V-statistics and a novel cross-centring tech-
nique 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 dis-
covery and feature selection, which highlight both
the importance of high-order interactions in data
and the need for efficient computational methods.
1. Introduction
Many complex real-world systems involve interactions be-
yond pairwise relationships, necessitating an explicit under-
standing of interactions among three or more entities (Bat-
tiston et al., 2020; 2021; Rosas et al., 2022; Thibeault et al.,
2024). Such high-order interactions amongst d > 2 vari-
ables have been shown to affect processes in social (Benson
et al., 2016), ecological (Grilli et al., 2017) and biological
systems (Luff et al., 2024; Arnaudon et al., 2022), often
in ways that cannot be reduced to combinations of pair-
wise effects (Schaub et al., 2020; Bick et al., 2023). Given
the relative scarcity of relational datasets that record high-
order phenomena, it is important to develop unsupervised,
mathematically grounded methods for their detection in ob-
1Department of Mathematics, Imperial College London,
United Kingdom 2Department of Neurology, University Hospi-
tal W¨urzburg, Germany 3Depar...