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 dependen-
cies 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 interac-
tions from data remains challenging. Here, we introduce a hierarchy of d-order
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 inter-
action 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.