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A lower bound on the differential entropy for log-concave random variables with applications to rate-distortion theory

Marsiglietti, Arnaud and Kostina, Victoria (2017) A lower bound on the differential entropy for log-concave random variables with applications to rate-distortion theory. In: 2017 IEEE International Symposium on Information Theory (ISIT). IEEE , Piscataway, NJ, pp. 46-50. ISBN 978-1-5090-4096-4. http://resolver.caltech.edu/CaltechAUTHORS:20170816-162727008

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Abstract

We derive a lower bound on the differential entropy for symmetric log-concave random variable X in terms of the p-th absolute moment of X, which shows that entropy and p-th absolute moment of a symmetric log-concave random variable are comparable. We apply our bound to study the rate distortion function under distortion measure |x − x|^r for sources that follow a log-concave probability distribution. In particular, we establish that the difference between the rate distortion function and the Shannon lower bound is at most log(√2e) ≈ 1.9 bits, independently of r and the target distortion d. For mean-square error distortion, the difference is at most log √πe ≈ 1.55 bits, regardless of d. Our results generalize to the case of vector X. Our proof technique leverages tools from convex geometry.


Item Type:Book Section
Related URLs:
URLURL TypeDescription
https://doi.org/10.1109/ISIT.2017.8006487DOIArticle
http://ieeexplore.ieee.org/document/8006487/PublisherArticle
ORCID:
AuthorORCID
Kostina, Victoria0000-0002-2406-7440
Additional Information:© 2017 IEEE. [AM] Supported by the Walter S. Baer and Jeri Weiss CMI Postdoctoral Fellowship. [VK] Supported in part by the National Science Foundation (NSF) under Grant CCF-1566567.
Funders:
Funding AgencyGrant Number
Walter S. Baer and Jeri Weiss CMI Postdoctoral FellowshipUNSPECIFIED
NSFCCF-1566567
Subject Keywords:Differential entropy, rate-distortion function, Shannon lower bound, log-concave distribution
Record Number:CaltechAUTHORS:20170816-162727008
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20170816-162727008
Official Citation:A. Marsiglietti and V. Kostina, "A lower bound on the differential entropy for log-concave random variables with applications to rate-distortion theory," 2017 IEEE International Symposium on Information Theory (ISIT), Aachen, Germany, 2017, pp. 46-50. doi: 10.1109/ISIT.2017.8006487
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:80529
Collection:CaltechAUTHORS
Deposited By: Kristin Buxton
Deposited On:16 Aug 2017 23:32
Last Modified:16 Aug 2017 23:32

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