Information-Theoretic Matrix Inequalities and Diffusion Processes on Unimodular Lie Groups

Abstract

Unimodular Lie groups admit natural generalizations of many of the core concepts on which classical information-theoretic inequalities are built. Specifically, they have the properties of shift-invariant integration, an associative convolution operator, well-defined diffusion processes, and concepts of Entropy, Fisher information, Gaussian distribution, and Fourier transform. Equipped with these definitions, it is shown that many inequalities from classical information theory generalize to this setting. Moreover, viewing the Fourier transform for noncommutative unimodular Lie groups as a matrix-valued function, relationships between trace inequalities, diffusion processes, and convolution are examined.