A Lower Bound on the Differential Entropy of Log-Concave Random Vectors with Applications
- Creators
- Marsiglietti, Arnaud
- Kostina, Victoria
Abstract
We derive a lower bound on the differential entropy of a log-concave random variable X in terms of the p-th absolute moment of X. The new bound leads to a reverse entropy power inequality with an explicit constant, and to new bounds on the rate-distortion function and the channel capacity. Specifically, we study the rate-distortion function for log-concave sources and distortion measure d(x,x^)=|x−x^|r , with r ≥ 1 , and we establish that the difference between the rate-distortion function and the Shannon lower bound is at most log(√(πe)) ≈ 1.5 bits, independently of r and the target distortion d. For mean-square error distortion, the difference is at most log(√((πe)/2)) ≈ 1 bit, regardless of d. We also provide bounds on the capacity of memoryless additive noise channels when the noise is log-concave. We show that the difference between the capacity of such channels and the capacity of the Gaussian channel with the same noise power is at most log(√((πe)/2)) ≈ 1 bit. Our results generalize to the case of a random vector X with possibly dependent coordinates. Our proof technique leverages tools from convex geometry.
Additional Information
© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). Received: 18 January 2018 / Revised: 6 March 2018 / Accepted: 6 March 2018 / Published: 9 March 2018. (This article belongs to the Special Issue Entropy and Information Inequalities) This work is supported in part by the National Science Foundation (NSF) under Grant CCF-1566567, and by the Walter S. Baer and Jeri Weiss CMI Postdoctoral Fellowship. The authors would also like to thank an anonymous referee for pointing out that the bound (23) and, up to a factor 2, the bound (25) also apply to the non-symmetric case if p ≥ 1. Author Contributions: Arnaud Marsiglietti and Victoria Kostina contributed equally to the research and writing of the paper. The authors declare no conflict of interest.Attached Files
Published - entropy-20-00185-v2.pdf
Submitted - 1704.07766.pdf
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Additional details
- PMCID
- PMC7512702
- Eprint ID
- 86116
- Resolver ID
- CaltechAUTHORS:20180430-101112645
- CCF-1566567
- NSF
- Center for the Mathematics of Information, Caltech
- Created
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2018-04-30Created from EPrint's datestamp field
- Updated
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2021-11-15Created from EPrint's last_modified field