Article

# Entropy functions and determinant inequalities

01/2012;
Source: arXiv

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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• ##### Article:Nonnegative entropy measures of multivariate symmetric correlations
[show abstract] [hide abstract]
ABSTRACT: A study of nonnegativity “in general” in the symmetric (correlative) entropy space as well as discussions of some related problems is presented. The main result is summarized as Theorems 4.1 and 5.3, which give the necessary and sufficient condition for an element of the symmetric (correlative) entropy space to be nonnegative. In particular, Theorem 4.1 may be regarded as establishing a mathematical foundation for information-theoretic analysis of multivariate symmetric correlation. On the basis of these results, we propose a “hierarchical structure” of probabilistic dependence relations where it is shown that any symmetric correlation associated with a nonnegative entropy is decomposed into pairwise conditional and/or nonconditional correlations. A systematic duality existing in the set of nonnegative entropies is also considerably clarified.
Information and Control.
• ##### Article:Balanced information inequalities.
IEEE Transactions on Information Theory. 01/2003; 49:3261-3267.

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### Keywords

$n \times n$ positive definite matrices

$n$ scalar Gaussian random variables

cone

convex cone

definite matrices

determinant inequalities

entropic functions

inner

representable functions

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