Clarify correspondence between topological and information-theoretic higher-order interactions

Determine to what extent higher-order interactions identified via topological data analysis—such as cycles and cavities revealed by Vietoris–Rips persistent homology—correspond to redundancy- or synergy-dominated dependencies quantified by multivariate information-theoretic measures, including total correlation, dual total correlation, O-information, and S-information, in multivariate datasets.

Background

Topological data analysis (TDA) and multivariate information theory have both been developed to characterize higher-order interactions in complex systems, but largely in parallel and with limited cross-talk. TDA emphasizes topological features such as voids and cavities in point clouds, while information theory distinguishes redundancy and synergy using measures like total correlation, dual total correlation, and O-information.

The paper motivates a direct comparison of these frameworks, noting that despite similar aims, their different mathematical foundations have left it unclear whether the “higher-order” structures each detects are of the same kind. This problem asks for a precise understanding of the extent and nature of the correspondence between these two perspectives.