Entropy functions and determinant inequalities

Terence Chan; Dongning Guo; Raymond Yeung

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arXiv · 1201.5241

Entropy functions and determinant inequalities

Terence Chan·Dongning Guo·Raymond Yeung

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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.

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Terence Chan, Dongning Guo, Raymond Yeung. 2012-01-25. Entropy functions and determinant inequalities. https://arxiv.org/abs/1201.5241

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