Do optimal entropy-constrained quantizers have a finite or infinite number of codewords?

An entropy-constrained quantizer Q is optimal if it minimizes the expected distortion D(Q) subject to a constraint on the output entropy H(Q). We use the Lagrangian formulation to show the existence and study the structure of optimal entropy-constrained quantizers that achieve a point on the lower c...

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Vydané v:IEEE transactions on information theory Ročník 49; číslo 11; s. 3031 - 3037
Hlavní autori: Gyorgy, A., Linder, T., Chou, P.A., Betts, B.J.
Médium: Journal Article
Jazyk:English
Vydavateľské údaje: New York IEEE 01.11.2003
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
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Codes
Counters
Distortion
Distortion measurement
Entropy
Gaussian
Hulls
Hulls (structures)
Information theory
Lagrangian functions
Mathematical analysis
Mathematical models
Mathematics
Optimization
Probability distribution
Quantization
Source coding
Statistics
Theory
ISSN:0018-9448, 1557-9654
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Shrnutí:An entropy-constrained quantizer Q is optimal if it minimizes the expected distortion D(Q) subject to a constraint on the output entropy H(Q). We use the Lagrangian formulation to show the existence and study the structure of optimal entropy-constrained quantizers that achieve a point on the lower convex hull of the operational distortion-rate function D/sub h/(R) = inf/sub Q/{D(Q) : H(Q) /spl les/ R}. In general, an optimal entropy-constrained quantizer may have a countably infinite number of codewords. Our main results show that if the tail of the source distribution is sufficiently light (resp., heavy) with respect to the distortion measure, the Lagrangian-optimal entropy-constrained quantizer has a finite (resp., infinite) number of codewords. In particular, for the squared error distortion measure, if the tail of the source distribution is lighter than the tail of a Gaussian distribution, then the Lagrangian-optimal quantizer has only a finite number of codewords, while if the tail is heavier than that of the Gaussian, the Lagrangian-optimal quantizer has an infinite number of codewords.
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ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2003.819340