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

Abstruct-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 realvalued 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(\llogm/nl; (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(J/~); (3) There exists a fixed-rate universal lossy source coding scheme whose per-letter MSE on n real-valued source samples converges in expectation and almost surely to the distortion-rate function D(R) as O(\llog log n / log n ); (4) Consider a training set of II 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 ve.ctors. 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(Jloglog n /log n 1, if one chooses k = [(l / R)(l e)log n] for any E E (O,l).