Convergence of Vector Quantizers with Applications to Optimal Quantization

Abstract

Suppose that a sequence of probability distribution functions $\{ F_n \} $ converges weakly to a distribution function F. Does the sequence of optimal quantizers for the $F_n $’s converge to an optimal quantizer for F? If so, do the respective distortions converge to the optimal distortion for F? Sufficient conditions are given to guarantee the convergence for both scalar and vector quantizers with a general class of distortion measures. These results are used to prove the existence of minimum rth power distortion vector quantizers and the convergence of a proposed algorithm for constructing optimal quantizers.