Permutation-Free High-Order Interaction Tests

2025-06-06

AI Summary

Traditional kernel-based higher-order interaction tests rely on computationally expensive permutation procedures, rendering them infeasible for high-dimensional settings. This paper proposes a novel permutation-free testing framework that constructs a V-statistic and incorporates cross-centered kernel estimation, ensuring the test statistic converges asymptotically to a standard normal distribution under the null hypothesis. The method reduces time complexity to linear in sample size, substantially improving scalability. Theoretical analysis establishes statistical consistency and power guarantees. Empirical evaluations on synthetic data—and downstream tasks including causal discovery and feature selection—demonstrate both high accuracy and computational efficiency. The key contribution is the first analytical derivation of the null distribution for kernel-based higher-order interaction testing, eliminating dependence on resampling strategies while preserving rigorous statistical validity.

Technology Category

• Machine Learning: Kernel Methods
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Causality

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Detect high-order interactions in multivariate data
• Eliminate computationally demanding permutation schemes
• Provide efficient tests for joint independence and partial factorizations

Innovation

• Kernel-based tests for high-order interactions
• Permutation-free using V-statistics and cross-centring
• Scalable with standard normal null distribution