A multivariate generalization of Costa's entropy power inequality

TL;DR

This paper proves a multivariate extension of Costa's entropy power inequality, showing the entropy power's concavity with respect to component variances and providing a new expression for the Hessian matrix of entropy functions.

Contribution

It introduces a multivariate version of Costa's entropy power inequality and derives the Hessian matrix of entropy and entropy power functions with respect to variances.

Findings

Entropy power is a multidimensional concave function of variances.

Derived the Hessian matrix of entropy and entropy power functions.

Established a multivariate generalization of Costa's inequality.

Abstract

A simple multivariate version of Costa's entropy power inequality is proved. In particular, it is shown that if independent white Gaussian noise is added to an arbitrary multivariate signal, the entropy power of the resulting random variable is a multidimensional concave function of the individual variances of the components of the signal. As a side result, we also give an expression for the Hessian matrix of the entropy and entropy power functions with respect to the variances of the signal components, which is an interesting result in its own right.