Strong Functional Representation Lemma and Applications to Coding Theorems

This paper shows that for any random variables <inline-formula> <tex-math notation="LaTeX">X </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">Y </tex-math></inline-formula>, it is possible to represent &l...

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Bibliographic Details
Published in:IEEE transactions on information theory Vol. 64; no. 11; pp. 6967 - 6978
Main Authors: Li, Cheuk Ting, Gamal, Abbas El
Format: Journal Article
Language:English
Published: New York IEEE 01.11.2018
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:
Channel coding
channel simulation
channel with state
Coding
Continuity (mathematics)
Electrical engineering
Functional representation lemma
Indexes
lossy source coding
one-shot achievability
Random variables
Representations
Source coding
Theorems
ISSN:0018-9448, 1557-9654
Online Access:Get full text
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Summary:This paper shows that for any random variables <inline-formula> <tex-math notation="LaTeX">X </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">Y </tex-math></inline-formula>, it is possible to represent <inline-formula> <tex-math notation="LaTeX">Y </tex-math></inline-formula> as a function of <inline-formula> <tex-math notation="LaTeX">(X,Z) </tex-math></inline-formula> such that <inline-formula> <tex-math notation="LaTeX">Z </tex-math></inline-formula> is independent of <inline-formula> <tex-math notation="LaTeX">X </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">I(X;Z|Y)\le \log (I(X;Y)+1)+4 </tex-math></inline-formula> bits. We use this strong functional representation lemma (SFRL) to establish a bound on the rate needed for one-shot exact channel simulation for general (discrete or continuous) random variables, strengthening the results by Harsha et al. and Braverman and Garg, and to establish new and simple achievability results for one-shot variable-length lossy source coding, multiple description coding, and Gray-Wyner system. We also show that the SFRL can be used to reduce the channel with state noncausally known at the encoder to a point-to-point channel, which provides a simple achievability proof of the Gelfand-Pinsker theorem.
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ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2018.2865570