Entropy functions and determinant inequalities

TL;DR

This paper establishes a connection between determinant inequalities for positive definite matrices and entropy functions of Gaussian variables, providing tight bounds and complete characterizations for small matrix sizes.

Contribution

It demonstrates the equivalence between determinant inequalities and entropy function cones, and derives explicit bounds for matrices up to size 3.

Findings

Tight bounds on the cone of entropy functions for n ≤ 3.

Complete characterization of determinant inequalities for 3×3 matrices.

Equivalence of determinant inequalities with Shannon-type information inequalities.

Abstract

In this paper, we show that the characterisation of all determinant inequalities for $n \times n$ positive definite matrices is equivalent to determining the smallest closed and convex cone containing all entropy functions induced by $n$ scalar Gaussian random variables. We have obtained inner and outer bounds on the cone by using representable functions and entropic functions. In particular, these bounds are tight and explicit for $n \le 3$, implying that determinant inequalities for $3 \times 3$ positive definite matrices are completely characterized by Shannon-type information inequalities.