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 ($d \geq 2$) 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.

本研究提出了一种 $d$ 阶交互作用度量的层次结构，定义了非参数的基于核的检验方法来系统地确定 $d$ 阶交互作用的统计显著性，并与格理论建立数学联系，以提高计算效率。研究结果在合成数据验证和神经影像数据应用中得到了论证。

高阶交互作用的交互测量，划分格和核检验

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

Testing equality of two multivariate distributions is a classical problem for
which many non-parametric tests have been proposed over the years. Most of the
popular two-sample tests, which are asymptotically distribution-free, are based
either on geometric graphs constructed using inter-point distances between the
observations (multivariate generalizations of the Wald-Wolfowitz's runs test)
or on multivariate data-depth (generalizations of the Mann-Whitney rank test).
This paper introduces a general notion of distribution-free graph-based
two-sample tests, and provides a unified framework for analyzing and comparing
their asymptotic properties. The asymptotic (Pitman) efficiency of a general
graph-based test is derived, which include tests based on geometric graphs,
such as the Friedman-Rafsky test (1979), the test based on the $K$-nearest
neighbor graph, the cross-match test (2005), the generalized edge-count test
(2017), as well as tests based on multivariate depth functions (the Liu-Singh
rank sum statistic (1993)). The results show how the combinatorial properties
of the underlying graph effect the performance of the associated two-sample
test, and can be used to validate and decide which tests to use in practice.
Applications of the results are illustrated both on synthetic and real
datasets.

本文介绍了一种分布无关的基于图形的双样本检验方法，并提供了分析和比较其渐进性质的统一框架。该文章研究表明底层图形的组合特性对关联的双样本检验的性能产生影响，并且可以用于验证和决定在实践中使用哪种检验。