Entropy functions and determinant 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.

submitted to ISIT 2012

References in corpus (1)

• New inequalities for subspace arrangements