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

• Published in: IEEE transactions on information theory Vol. 49; no. 11; pp. 3031 - 3037
• Main Authors: Gyorgy, A., Linder, T., Chou, P.A., Betts, B.J.
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.11.2003; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Associate members; 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
• DOI: 10.1109/TIT.2003.819340

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.