Interaction Measures, Partition Lattices and Kernel Tests for High-Order Interactions (2306.00904v3)

Abstract: 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.

• The paper introduces a hierarchy of interaction measures that extend beyond pairwise analysis to capture complex multivariate dependencies.
• The authors develop kernel-based tests using RKHS embeddings and permutation strategies to efficiently detect high-order interactions.
• Experimental results on neuroimaging data validate the framework’s potential for uncovering non-trivial relationships in real-world systems.

High-Order Interaction Measures and Kernel Tests: A Systematic Approach

The paper "Interaction Measures, Partition Lattices and Kernel Tests for High-Order Interactions" presents a comprehensive paper on the limitations of pairwise statistical models in capturing the complexity inherent in multivariate data drawn from diverse real-world domains. The authors propose a rigorous framework for identifying high-order interactions by leveraging a hierarchy of interaction measures, partition lattices, and kernel-based statistical tests. This framework is critically intertwined with mathematical principles from lattice theory and non-parametric statistics, providing insights and methods applicable to socio-economic, ecological, and biomedical systems.

Interaction Measures and Partition Lattices

The paper develops upon the classical notion that interactions which extend beyond pairwise relationships can unveil important structures in complex systems. It introduces a hierarchy of $d$-order interaction measures, which systematically increase inclusivity of possible factorizations of joint probability distributions. At the core of this work are three key interaction measures: joint independence, Lancaster interaction, and Streitberg interaction. Each measures not just the absence of joint independence but explores intricate dependencies higher than two-order interactions.

• Joint Independence Measure: Captures the lack of complete factorization among variables, and is a traditional baseline for assessing independence.
• Lancaster Interaction: Extends beyond joint independence to tackle interactions that elude traditional pairwise relationships, albeit with some limitations at higher dimensions.
• Streitberg Interaction: Provides a comprehensive assessment by capturing all potential factorizations, leveraging a full partition lattice, making it superior for elucidating complex dependencies when $d \geq 4$.

The paper masterfully reinforces these measures through partition lattices—a mathematical structure that delineates the hierarchy of all possible interactions among variables—allowing for intuitive visualization and understanding of interaction complexities. These partition lattices are not trivially representable by traditional data structures like simplicial complexes, evidencing the richness of such interactions.

Kernel-Based Tests for High-Order Interactions

The formulation of kernel-based tests for these high-order interactions is a pivotal contribution of this research. By embedding interaction measures into Reproducing Kernel Hilbert Spaces (RKHS), the authors develop non-parametric tests that efficiently identify significant high-order relationships within data.

• Lancaster and Streitberg Kernel Tests: These tests innovate upon kernel embeddings, with the Streitberg test offering the most extensive coverage of interaction factorization, thus making it especially powerful in detecting non-trivial high-order interactions.

The application of these tests is underpinned by permutation-based strategies allowing for computationally feasible hypothesis testing in practice. The harnessing of the M\"obius inversion from lattice theory further enriches the test development process, establishing a firm statistical grounding.

Practical Implications and Theoretical Insights

Practically, the results from this paper are validated through experiments on both synthetic datasets and real-world neuroimaging data, exploring interactions among brain regions. Neuroimaging results imply that regions within certain resting state networks exhibit more significant interactions than when sampled randomly across the brain, suggesting potential applications in cognitive and functional neuroscience. The implications in theoretical modeling are substantial, providing a tangible method to incorporate high-order dependencies in the modeling of complex systems, challenging the traditional pairwise paradigm.

Future Developments

Looking forward, several avenues exist for developing these insights. A natural extension involves reducing computational complexity in practical applications, especially for large datasets. Moreover, addressing non-stationarity in real-world data remains an open question for enabling broader applicability. Finally, the paradigm can potentially integrate with causal discovery frameworks to further refine mechanistic understandings in complex systems.

In conclusion, this paper offers a sophisticated framework to tackle high-order dependencies in multivariate data, founded on robust mathematical bases and yielding practical tools for real-world data analysis. The potential for expanding its application across domains underscores the relevance of advanced statistical interactions in disentangling the complexity of multivariate systems.