Smoothing Brascamp-Lieb Inequalities and Strong Converses of Coding Theorems

The Brascamp-Lieb inequality in functional analysis can be viewed as a measure of the "uncorrelatedness" of a joint probability distribution. We define the smooth Brascamp-Lieb (BL) divergence as the infimum of the best constant in the Brascamp-Lieb inequality under a perturbation of the j...

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Bibliographic Details
Published in:IEEE transactions on information theory Vol. 66; no. 2; pp. 704 - 721
Main Authors: Liu, Jingbo, Courtade, Thomas A., Cuff, Paul, Verdu, Sergio
Format: Journal Article
Language:English
Published: New York IEEE 01.02.2020
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:
Brascamp-Lieb inequality
Channel coding
Codes
Coding
coding theorems
common randomness
Divergence
Entropy
finite blocklength
Functional analysis
Gray-Wyner network
hypercontractivity
Inequality
Infimum
Perturbation
Probability distribution
Random variables
Shannon theory
Smoothing methods
strong converse
Upper bounds
ISSN:0018-9448, 1557-9654
Online Access:Get full text
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Summary:The Brascamp-Lieb inequality in functional analysis can be viewed as a measure of the "uncorrelatedness" of a joint probability distribution. We define the smooth Brascamp-Lieb (BL) divergence as the infimum of the best constant in the Brascamp-Lieb inequality under a perturbation of the joint probability distribution. An information spectrum upper bound on the smooth BL divergence is proved, using properties of the subgradient of a certain convex functional. In particular, in the i.i.d. setting, such an infimum converges to the best constant in a certain mutual information inequality. We then derive new single-shot converse bounds for the omniscient helper common randomness generation problem and the Gray-Wyner source coding problem in terms of the smooth BL divergence, where the proof relies on the functional formulation of the Brascamp-Lieb inequality. Exact second-order rates are thus obtained in the stationary memoryless and nonvanishing error setting. These offer rare instances of strong converses/second-order converses for continuous sources when the rate region involves auxiliary random variables.
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ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2019.2953151