Title: Fractional Subadditivity of Submodular Functions: Equality Conditions and Their Applications
Abstract: Submodular functions are known to satisfy various forms of fractional subadditivity. This work investigates the conditions for equality to hold exactly or approximately in the fractional subadditivity of submodular functions. We establish that a small gap in the inequality implies that the function is close to being modular, and that the gap is zero if and only if the function is modular. We then present 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 to hold 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), Han's dual total correlation (1978), and Csiszar and Narayan's shared information (2004), 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.
Bio: Gowtham R. Kurri is an Assistant Professor with the Signal Processing and Communications Research Centre at IIIT Hyderabad. His research focuses on information theory and its applications to computer science and machine learning. He received his M.Sc. and Ph.D. degrees from the Tata Institute of Fundamental Research, Mumbai in 2020. He was a Post-Doctoral Researcher at the School of Electrical, Computer and Energy Engineering at Arizona State University from 2020-2023. He is a recipient of the ANRF Prime Minister's Early Career Research Grant (2025). Earlier, he worked as an Associate Engineer at Qualcomm India Private Limited, Hyderabad (2011-2012) and as a Research Intern in the Blockchain Technology Group at IBM Research, Bangalore (2019).