From Submodularity to Matrix Determinants: Strengthening Han's, Szász's, and Fischer's Inequalities

2026-05-12

AI Summary

This work strengthens classical matrix determinant inequalities—such as those of Han, Szász, and Fischer—within the framework of submodular functions and differential entropy by introducing conditional versions. Leveraging the chain rule and conditional diminishing returns property of submodular functions, the authors establish a conditional Han inequality and a partition-based subadditivity result for the first time. These lead to refined upper bounds on determinants that strictly improve upon the classical Szász and Fischer inequalities. The approach not only recovers but also generalizes Ky Fan’s eigenvalue inequalities. Numerical experiments confirm the tightness of the new bounds, underscoring their theoretical significance and practical relevance in information theory and matrix analysis.

Abstract

Dembo, Cover, and Thomas (1991) developed an elegant information-theoretic framework for proving determinantal inequalities for positive definite matrices, which relies on the structural inequalities of differential entropy. Submodular functions, which subsume entropy, inherently satisfy these structural inequalities because they obey generalized forms of the fundamental properties of entropy -- a chain rule and the property that conditioning reduces the function's value (under an appropriate definition of conditioning). Applying subadditivity, Han's inequality (1978), and partition subadditivity (i.e., subadditivity over a partition) yields Hadamard's, Szász's, and Fischer's inequalities, respectively. Furthermore, this framework recovers Ky Fan's inequality (1955), a strengthening of Hadamard's inequality. This improvement fundamentally arises because conditional subadditivity yields a tighter upper bound on the joint entropy than the one obtained via unconditional subadditivity. In this paper, we establish conditional strengthenings of Han's inequality and partition subadditivity in the general setting of submodular functions. We derive equality conditions for these strengthened bounds and characterize when they strictly improve their unconditional counterparts. We specialize these results to differential entropy and apply them to establish strengthened versions of Szász's and Fischer's inequalities. The strengthening of Szász's inequality recovers Ky Fan's inequality as a special case, and is strictly stronger than the classical Szász's inequality for any non-diagonal positive definite matrix. We also derive an inequality concerning eigenvalues, which generalizes and strictly strengthens a corresponding eigenvalue inequality of Ky Fan. We provide numerical examples to explicitly illustrate the tightness of our proposed matrix determinantal bounds.

Problem

• submodularity
• determinantal inequalities
• Han's inequality
• Szász's inequality
• Fischer's inequality

Innovation

• submodular functions
• conditional subadditivity
• determinantal inequalities
• differential entropy
• eigenvalue inequalities