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

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4 Author(s)
Gyorgy, A. ; Dept. of Math. & Stat., Queen''s Univ., Kingston, Ont., Canada ; Linder, T. ; Chou, P.A. ; Betts, B.J.

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 Dh(R) = infQ{D(Q) : H(Q) ≤ 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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Information Theory, IEEE Transactions on  (Volume:49 ,  Issue: 11 )