Generalized vector quantization: jointly optimal quantization and estimation

Given a pair of random vectors X, Y, we study the problem of finding an efficient or optimal estimator of Y given X when the range of the estimator is constrained to be a finite set of values. A generalized vector quantizer (GVQ), with input dimension k, output dimension m, and size N maps input X∈ℛ^k, to output V(X)∈ℛ ^m. The output V(X) is constrained to be one of the estimation codevectors in the codebook, {y₁,y₂...y_N}. The performance of the GVQ is measured by the average distortion, D=E[d(Y,V(X))] for a suitable output-space distortion measure d(.,.). A GVQ reduces to a conventional vector quantizer in the special case where X=Y. The GVQ problem has been approached in the information theory literature from many different standpoints. In particular, it appears in the context of noisy source coding, which is the special case where we quantize X, the observable, noisy version of a source, Y