Fractional Subadditivity of Submodular Functions: Equality Conditions and Their Applications

Описание: Speaker : Suryajith Chillara
Date : 26th Aug 2025

Abstract: Submodular functions are known to satisfy various forms of fractional subadditivity. In this work, we investigate the conditions for equality to hold exactly or approximately in fractional subadditivity. We establish that a small inequality gap implies that the function is close to being modular, and that the gap is zero if and only if the function is modular. We also present the natural implications of these results for special cases of submodular functions, such as entropy, relative entropy, and matroid rank. As a consequence, we characterize the necessary and sufficient conditions for equality in Shearer's lemma, recovering a result of Ellis et al. (2016) as a special case. We leverage our results to propose a new multivariate mutual information, which generalizes Watanabe's total correlation (1960) and Han's dual total correlation (1975), and analyze its properties. Among these properties, we extend Watanabe's characterization of total correlation as the maximum correlation over partitions to fractional partitions. When applied to matrix determinantal inequalities for positive definite matrices, our results recover the equality conditions of the classical determinantal inequalities of Hadamard, Szasz, and Fischer as special cases.

This is a joint work with Gunank Jakhar, Gowtham R. Kurri, and Vinod Prabhakaran. Appeared in proceedings of ISIT 2025.

Link to the talk: https://www.csa.iisc.ac.in/theorysemi...