  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.
