Entropy jumps for isotropic log-concave random vectors and spectral gap

TL;DR

This paper establishes a dimension-free entropy inequality for log-concave distributions, linking entropy jumps to spectral gaps, with implications for the isotropic constant and slicing problem.

Contribution

It provides a new quantitative bound on entropy for log-concave distributions using spectral gap analysis, advancing understanding of entropy production in high dimensions.

Findings

Dimension-free entropy bound for log-concave convolutions

Connection between spectral gap and entropy jumps

Implications for isotropic constant and slicing problem

Abstract

We prove a quantitative dimension-free bound in the Shannon-Stam Entropy inequality for the convolution of two log-concave distributions in dimension d interms of the spectral gap of the density. The method relies on the analysis of the Fisher Information production, which is the second derivative of the Entropy along the (normalized) Heat semi-group. We also discuss consequences of our result in the study of the isotropic constant of log-concave distributions (slicing problem).