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.

Lay Summary:

How can we understand complex relationships between multiple variables—beyond simple pairwise ones like correlation? These higher-order interactions are common in fields ranging from neuroscience to economics. Traditionally, detecting them has required computationally intensive permutation tests to check whether observed patterns are statistically meaningful.In our work, we introduce a much faster alternative using two mathematical tools: V-statistics and a novel cross-centring technique. This makes the tests permutation-free, resulting in speed-ups of over 100× without sacrificing accuracy.We demonstrate how this method can be used to uncover causal relationships and identify important features in machine learning quickly and accurately. This far more efficient testing method opens up new possibilities for exploring complex interactions in areas like financial markets, gene interactions, and brain activity.