Optimal Concentration of Information Content For Log-Concave Densities

Abstract

An elementary proof is provided of sharp bounds for the varentropy of random vectors with log-concave densities, as well as for deviations of the information content from its mean. These bounds significantly improve on the bounds obtained by Bobkov and Madiman ({\it Ann. Probab.}, 39(4):1528--1543, 2011).

15 pages. Changes in v2: Remark 2.5 (due to C. Saroglou) added with more general sufficient conditions for equality in Theorem 2.3. Also some minor corrections and added references

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• Forward and Reverse Entropy Power Inequalities in Convex Geometry
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• Rényi Entropy Power Inequalities via Normal Transport and Rotation
• Analysis of Langevin Monte Carlo via convex optimization
• The Differential Entropy of Mixtures: New Bounds and Applications
• Sharp moment-entropy inequalities and capacity bounds for log-concave distributions
• A combinatorial approach to small ball inequalities for sums and differences
• Stochastic comparisons, differential entropy and varentropy for distributions induced by probability density functions
• Threshold for the expected measure of random polytopes
• Reversals of Rényi Entropy Inequalities under Log-Concavity
• Sumset estimates in convex geometry
• Dimension-free Information Concentration via Exp-Concavity
• Rényi Entropy Power and Normal Transport
• Support Recovery in the Phase Retrieval Model: Information-Theoretic Fundamental Limits