Formal links between information theory and topology

Establish formal mathematical relationships between multivariate information-theoretic quantities that characterize redundancy and synergy (such as O-information, total correlation, and dual total correlation) and topological invariants derived from persistent homology of point clouds (such as the birth–death lifetimes and counts of three-dimensional cavities), moving beyond empirical correlations to rigorous theorems.

Background

The paper reports robust empirical associations between synergistic information and three-dimensional cavities in point clouds, across toy manifolds and fMRI data. However, the analysis is explicitly correlational and does not provide formal proofs linking the information-theoretic measures to topological features.

The authors conjecture that formal theoretical connections may exist, highlighting the need for rigorous results that unify these domains beyond observational evidence.