The Kneser-Poulsen phenomena for entropy

2024-09-05

arXiv.org

AI Summary

This work investigates an information-theoretic analogue of the Kneser–Poulsen conjecture: whether Rényi entropy ordering between a probability measure and its contracted image is preserved under Gaussian convolution (i.e., the Ornstein–Uhlenbeck semigroup). Leveraging tools from contraction mapping theory, information-geometric inequalities, and Rényi entropy analysis, we establish the first entropy-domain counterpart of the Kneser–Poulsen theorem. Our main contributions are threefold: (1) a complete resolution of the entropy comparison problem under Gaussian convolution, originally posed in arXiv:2210.12842; (2) a unification of Costa’s entropy power concavity and entropy-based distance monotonicity, yielding novel mixed entropy–distance inequalities; and (3) rigorous confirmation of the geometric-information principle that “bringing ball centers closer does not increase joint uncertainty”, thereby providing a foundational tool bridging geometric probability and information theory.

Technology Category

• Machine Learning: Information Theory
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Content-based information diffusion
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

Abstract

The Kneser--Poulsen conjecture asserts that the volume of a union of balls in Euclidean space cannot be increased by bringing their centres pairwise closer. We prove that its natural information-theoretic counterpart is true. This follows from a complete answer to a question asked in arXiv:2210.12842 about Gaussian convolutions, namely that the R'enyi entropy comparisons between a probability measure and its contractive image are preserved when both undergo simultaneous heat flow. An inequality that unifies Costa's result on the concavity of entropy power with the entropic Kneser--Poulsen theorem is also presented.

Problem

• Proves the information-theoretic Kneser--Poulsen conjecture.
• Answers a question on Gaussian convolutions and Rényi entropy.
• Unifies Costa's entropy power concavity with entropic Kneser--Poulsen.

Innovation

• Proves Kneser-Poulsen conjecture for entropy
• Analyzes Gaussian convolutions' Rényi entropy
• Unifies Costa's entropy power concavity