publication . Preprint . 2017

Information-Theoretic Perspectives on Brascamp-Lieb Inequality and Its Reverse

Liu, Jingbo; Courtade, Thomas A.; Cuff, Paul; Verdu, Sergio;
Open Access English
  • Published: 20 Feb 2017
Abstract
We introduce an inequality which may be viewed as a generalization of both the Brascamp-Lieb inequality and its reverse (Barthe's inequality), and prove its information-theoretic (i.e.\ entropic) formulation. This result leads to a unified approach to functional inequalities such as the variational formula of R\'enyi entropy, hypercontractivity and its reverse, strong data processing inequalities, and transportation-cost inequalities, whose utility in the proofs of various coding theorems has gained growing popularity recently. We show that our information-theoretic setting is convenient for proving properties such as data processing, tensorization, convexity (R...
Subjects
free text keywords: Computer Science - Information Theory
Related Organizations
Funded by
NSF| CAREER: Digital Encoding of Information Signals for Security with Limited Resources
Project
  • Funder: National Science Foundation (NSF)
  • Project Code: 1350595
  • Funding stream: Directorate for Computer & Information Science & Engineering | Division of Computing and Communication Foundations
,
NSF| Emerging Frontiers of Science of Information
Project
  • Funder: National Science Foundation (NSF)
  • Project Code: 0939370
  • Funding stream: Directorate for Computer & Information Science & Engineering | Division of Computing and Communication Foundations
,
NSF| CIF: Small: Rate Distortion Paradigms for the Big Data Era
Project
  • Funder: National Science Foundation (NSF)
  • Project Code: 1528132
,
NSF| CIF: Small: Collaborative Research:Compressed databases for similarity queries: fundamental limits and algorithms
Project
  • Funder: National Science Foundation (NSF)
  • Project Code: 1319304
  • Funding stream: Directorate for Computer & Information Science & Engineering | Division of Computing and Communication Foundations
,
NSF| CIF: Small:Analog-to-Analog Compression: Fundamental Limits and Constructive Schemes
Project
  • Funder: National Science Foundation (NSF)
  • Project Code: 1319299
  • Funding stream: Directorate for Computer & Information Science & Engineering | Division of Computing and Communication Foundations
Download from
39 references, page 1 of 3

[1] R. Ahlswede and I. Csisza´r, “Common randomness in information theory and cryptography. II. CR capacity,” IEEE Transactions on Information Theory, vol. 44, no. 1, pp. 225-240, Jan. 1998.

[2] R. Ahlswede and P. Ga´cs, “Spreading of sets in product spaces and hypercontraction of the Markov operator,” The Annals of Probability, vol. 4, pp. 925-939, 1976. [OpenAIRE]

[3] R. Ahlswede, P. Ga´cs, and J. Ko¨rner, “Bounds on conditional probabilities with applications in multi-user communication,” Probability Theory and Related Fields, vol. 34, no. 2, pp. 157-177, 1976.

[4] V. Anantharam, A. Gohari, S. Kamath, and C. Nair, “On maximal correlation, hypercontractivity, and the data processing inequality studied by Erkip and Cover,” http://arxiv.org/pdf/1304.6133v1.pdf.

[5] R. Atar, K. Chowdhary, and P. Dupuis, “Robust bounds on risk-sensitive functionals via Re´nyi divergence,” SIAM J. Uncertainty Quant., vol. 3, pp. 18-33, 2015.

[6] R. Atar and N. Merhav, “Information-theoretic applications of the logarithmic probability comparison bound,” IEEE Transactions on Information Theory, vol. 61, no. 10, pp. 5366-5386, Oct. 2015. [OpenAIRE]

[7] K. Ball, “Volumes of sections of cubes and related problems,” in Geometric aspects of functional analysis. Springer, 1989, pp. 251-260.

[8] F. Barthe, “On a reverse form of the Brascamp-Lieb inequality,” Inventiones Mathematicae, vol. 134, no. 2, pp. 335-361, (see also arXiv:math/9 705 210 [math.FA]), 1998. [OpenAIRE]

[9] --, “Optimal Young's inequality and its converse: a simple proof,” Geometric and Functional Analysis, vol. 8, no. 2, pp. 234-242, 1998.

[37] E. Erkip and T. M. Cover, “The efficiency of investment information,” IEEE Transactions on Information Theory, vol. 44, no. 3, pp. 1026-1040, Mar. 1998.

[38] P. F. F. Chung, R. Graham and J. Shearer, “Some intersection theorems for ordered sets and graphs,” J. Combinatorial Theory Series A, vol. 43, no. 1, pp. 23-37, 1986. [OpenAIRE]

[39] E. Friedgut, G. Kalai, and A. Naor, “Boolean functions whose Fourier transform is concentrated on the first two levels,” Advances in Applied Mathematics, vol. 29, no. 3, pp. 427-437, 2002.

[40] A. Ganor, G. Kol, and R. Raz, “Exponential separation of information and communication,” in Foundations of Computer Science (FOCS), 2014 IEEE 55th Annual Symposium on, 2014, pp. 176-185. [OpenAIRE]

[41] C. Garban, G. Pete, and O. Schramm, “The Fourier spectrum of critical percolation,” Acta Mathematica, vol. 205, no. 1, pp. 19-104, 2010.

[42] R. Gardner, “The Brunn-Minkowski inequality,” Bulletin of the American Mathematical Society, vol. 39, no. 3, pp. 355-405, 2002.

39 references, page 1 of 3
Powered by OpenAIRE Open Research Graph
Any information missing or wrong?Report an Issue