From Submodularity to Matrix Determinants: Strengthening Han's, Sz\'asz's, and Fischer's Inequalities

TL;DR

This paper extends determinantal inequalities for positive definite matrices by leveraging submodular functions and entropy-based frameworks, resulting in stronger bounds and new eigenvalue inequalities.

Contribution

It introduces conditional strengthenings of Han's inequality and partition subadditivity for submodular functions, leading to improved matrix determinantal bounds.

Findings

Strengthened Szász's and Fischer's inequalities for positive definite matrices.

Recovered Ky Fan's inequality as a special case of the new bounds.

Provided numerical examples demonstrating the tightness of the bounds.

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, Sz\'asz'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 Sz\'asz's and Fischer's inequalities. The strengthening of Sz\'asz's inequality recovers Ky Fan's inequality as a special case, and is strictly stronger than the classical Sz\'asz'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.