A Quantitative Entropy Power Inequality for Dependent Random Vectors

TL;DR

This paper develops a quantitative version of the entropy power inequality for dependent random vectors, especially those with log-supermodular joint densities, extending classical results to dependent settings.

Contribution

It introduces a new quantitative entropy power inequality for dependent vectors, broadening the scope of the classical inequality to include dependence structures.

Findings

Entropy power inequality extended to dependent vectors

Valid for vectors with log-supermodular joint densities

Provides bounds using conditional entropies

Abstract

The entropy power inequality for independent random vectors is a foundational result of information theory, with deep connections to probability and geometric functional analysis. Several extensions of the entropy power inequality have been developed for settings with dependence, including by Takano, Johnson, and Rioul. We extend these works by developing a quantitative version of the entropy power inequality for dependent random vectors. A notable consequence is that an entropy power inequality stated using conditional entropies holds for random vectors whose joint density is log-supermodular.