# Interaction Measures, Partition Lattices and Kernel Tests for High-Order Interactions

arXiv: https://arxiv.org/abs/2306.00904 · HTML: https://mlsanity.com/paper/2306.00904/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 59 references that are in the map.
- First posted on arXiv 1 Jun 2023.

## Builds on (most cited first, 19 of 19)

- [Networks beyond pairwise interactions: structure and dynamics](https://mlsanity.com/paper/2006.01764/) (2020)
- [Random Walks on Simplicial Complexes and the normalized Hodge 1-Laplacian](https://mlsanity.com/paper/1807.05044/) (2018)
- [What are higher-order networks?](https://mlsanity.com/paper/2104.11329/) (2021)
- [The physics of higher-order interactions in complex systems](https://mlsanity.com/paper/2110.06023/) (2021)
- [Kernel-based Tests for Joint Independence](https://mlsanity.com/paper/1603.00285/) (2016)
- [Quantifying High-order Interdependencies via Multivariate Extensions of the Mutual Information](https://mlsanity.com/paper/1902.11239/) (2019)
- [Evolutionary dynamics of higher-order interactions in social networks](https://mlsanity.com/paper/2001.10313/) (2020)
- [A Kernel Independence Test for Random Processes](https://mlsanity.com/paper/1402.4501/) (2014)
- [Hypergraph reconstruction from network data](https://mlsanity.com/paper/2008.04948/) (2020)
- [Unveiling the higher-order organization of multivariate time series](https://mlsanity.com/paper/2203.10702/) (2022)
- [Random walks on hypergraphs](https://mlsanity.com/paper/1911.06523/) (2019)
- [A Kernel Test for Three-Variable Interactions](https://mlsanity.com/paper/1306.2281/) (2013)
- [Partial entropy decomposition reveals higher-order structures in human brain activity](https://mlsanity.com/paper/2301.05307/) (2023)
- [Tensor Balancing on Statistical Manifold](https://mlsanity.com/paper/1702.08142/) (2017)
- [The Partial Entropy Decomposition: Decomposing multivariate entropy and mutual information via pointwise common surprisal](https://mlsanity.com/paper/1702.01591/) (2017)
- [Kernel Two-Sample and Independence Tests for Non-Stationary Random Processes](https://mlsanity.com/paper/2010.00271/) (2020)
- [A Kernel Test for Three-Variable Interactions with Random Processes](https://mlsanity.com/paper/1603.00929/) (2016)
- [Algorithms for Tensor Network Contraction Ordering](https://mlsanity.com/paper/2001.08063/) (2020)
- [Kernel-based Joint Independence Tests for Multivariate Stationary and Non-stationary Time Series](https://mlsanity.com/paper/2305.08529/) (2023)

## Built on by (most cited first, 3 of 3)

- [Information-Theoretic Measures on Lattices for High-Order Interactions](https://mlsanity.com/paper/2408.07533/) (2024)
- [The Minimax Lower Bound of Kernel Stein Discrepancy Estimation](https://mlsanity.com/paper/2510.15058/) (2025)
- [Permutation-Free High-Order Interaction Tests](https://mlsanity.com/paper/2506.05963/) (2025)

## Abstract (arXiv)

Models that rely solely on pairwise relationships often fail to capture the complete statistical structure of the complex multivariate data found in diverse domains, such as socio-economic, ecological, or biomedical systems. Non-trivial dependencies between groups of more than two variables can play a significant role in the analysis and modelling of such systems, yet extracting such high-order interactions from data remains challenging. Here, we introduce a hierarchy of d-order (d ≥ 2) interaction measures, increasingly inclusive of possible factorisations of the joint probability distribution, and define non-parametric, kernel-based tests to establish systematically the statistical significance of d-order interactions. We also establish mathematical links with lattice theory, which elucidate the derivation of the interaction measures and their composite permutation tests; clarify the connection of simplicial complexes with kernel matrix centring; and provide a means to enhance computational efficiency. We illustrate our results numerically with validations on synthetic data, and through an application to neuroimaging data.