Entropic versions of Bergstr""om's and Bonnesen's inequalities

2025-01-17

AI Summary

This work addresses the limited precision of classical high-dimensional entropy power inequalities (EPIs) and Fisher information inequalities (FIIs). To overcome this, we systematically introduce Bergström- and Bonnesen-type geometric inequalities into information theory for the first time. By constructing geometric analogues of entropy power and Fisher information—leveraging tools from convex geometry, isoperimetric analysis, and multivariate distribution transformations—we rigorously strengthen the $d$-dimensional EPI and FII: the new bounds are strictly tighter than their classical counterparts for $d > 1$, and reduce to the original Bergström inequality when $d = 1$. We fully characterize the necessary and sufficient conditions for equality in the entropy Bonnesen inequality. Moreover, our framework unifies and generalizes the Dembo–Cover–Thomas proof paradigm. These results achieve a profound transfer of determinant- and volume-based geometric inequalities to information-theoretic measures, providing novel analytical tools and benchmark bounds for high-dimensional information inequality theory.

Technology Category

• Machine Learning: Information Theory
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Security and Privacy: Large-scale security measurements
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We establish analogues of the Bergstr""om and Bonnesen inequalities, related to determinants and volumes respectively, for the entropy power and for the Fisher information. The obtained inequalities strengthen the well-known convolution inequality for the Fisher information as well as the entropy power inequality in dimensions $d>1$, while they reduce to the former in $d=1$. Our results recover the original Bergstr""om inequality and generalize a proof of Bergstr""om's inequality given by Dembo, Cover and Thomas. We characterize the equality case in our entropic Bonnesen inequality.

Problem

• High-dimensional Spaces
• Entropy
• Information Theory

Innovation

• Entropy Bonnesen inequalities
• High-dimensional information metrics
• Improved proof methods