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Rates of convergence in the source coding theorem, in empirical quantizer design, and in universal lossy source coding

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3 Author(s)
Linder, T. ; Dept. of Telecommun., Tech. Univ. Budapest, Hungary ; Lugosi, G. ; Zeger, K.

Rates of convergence results are established for vector quantization. Convergence rates are given for an increasing vector dimension and/or an increasing training set size. In particular, the following results are shown for memoryless real valued sources with bounded support at transmission rate R. (1) If a vector quantizer with fixed dimension k is designed to minimize the empirical MSE with respect to m training vectors, then its MSE for the true source converges almost surely to the minimum possible MSE as O(√(log m/m)); (2) The MSE of an optimal k-dimensional vector quantizer for the true source converges, as the dimension grows, to the distortion-rate function D(R) as O(√(log k/k)); (3) There exists a fixed rate universal lossy source coding scheme whose per letter MSE on n real valued source samples converges almost surely to the distortion-rate function D(R) as O(√(log log n/log n)); and (4) Consider a training set of n real valued source samples blocked into vectors of dimension k, and a k-dimensional vector quantizer designed to minimize the empirical MSE with respect to the m=[n/k] training vectors. Then the MSE of this quantizer for the true source converges almost surely to the distortion-rate function D(R) as O(√(log log n/log n)), if one chooses k=[1/R(1-ε)(log n)] ∀ε ε(0,1)

Published in:

Information Theory, 1994. Proceedings., 1994 IEEE International Symposium on

Date of Conference:

27 Jun-1 Jul 1994

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