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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.

Rate 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 mean-square error (MSE) with respect to m training vectors, then its MSE for the true source converges in expectation and 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 a real-valued source samples converges in expectation and almost surely to the distortion-rate function D(R) as O((√(loglog n/log n)). (4) Consider a training set of n real-valued source samples blocked into vectors of dimension k, and a k-dimension vector quantizer designed to minimize the empirical MSE with respect to the m=[n/k] training vectors. Then the per-letter MSE of this quantizer for the true source converges in expectation and 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] for any ε∈(0.1)

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Information Theory, IEEE Transactions on  (Volume:40 ,  Issue: 6 )