A Lower Bound on the Entropy Rate for a Large Class of Stationary Processes and its Relation to the Hyperplane Conjecture

TL;DR

This paper establishes a new lower bound on the entropy rate of certain stationary processes, linking it to the hyperplane conjecture in convex geometry through information-theoretic measures.

Contribution

It introduces a novel lower bound on differential entropy rate for processes with regular density functions, connecting information theory and convex geometry.

Findings

Bound between entropy rate and Gaussian process with same autocovariance

Relation of entropy bounds to the hyperplane conjecture

Use of Wasserstein distance to relate divergence and entropy

Abstract

We present a new lower bound on the differential entropy rate of stationary processes whose sequences of probability density functions fulfill certain regularity conditions. This bound is obtained by showing that the gap between the differential entropy rate of such a process and the differential entropy rate of a Gaussian process with the same autocovariance function is bounded. This result is based on a recent result on bounding the Kullback-Leibler divergence by the Wasserstein distance given by Polyanskiy and Wu. Moreover, it is related to the famous hyperplane conjecture, also known as slicing problem, in convex geometry originally stated by J. Bourgain. Based on an entropic formulation of the hyperplane conjecture given by Bobkov and Madiman we discuss the relation of our result to the hyperplane conjecture.