Fractional Subadditivity of Submodular Functions: Equality Conditions and Their Applications

2025-01-21

AI Summary

This paper investigates fractional subadditivity of submodular functions and its equality conditions, aiming to characterize their approximation to modular functions. Methodologically, it establishes a quantitative relationship between the fractional subadditivity gap and the modular distance, thereby providing the first necessary and sufficient condition for equality: the gap vanishes if and only if the function is modular; a small gap implies proximity to modularity. The analysis integrates submodular function theory, information theory, determinant inequalities, and fractional partition theory. Key contributions include: (i) a complete characterization of equality conditions for Shearer’s lemma; (ii) a generalization of Watanabe’s total correlation to the fractional partition framework; (iii) recovery of classical determinantal equality criteria (Hadamard, Szász, Fischer); and (iv) introduction of generalized multivariate mutual information, which unifies several classical information measures and systematically analyzes their structural properties.

Technology Category

• Machine Learning: Information Theory
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Security and Privacy: Large-scale security measurements

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 emph{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 Csisz'ar 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, Sz'asz, and Fischer as special cases.

Problem

• Submodular Functions
• Fractional Subadditivity
• Information Theory

Innovation

• Multidimensional Mutual Information
• Fractional Subadditivity
• Matrix Inequalities