Sufficient conditions for uniqueness of a locally optimal quantizer for a class of convex error weighting functions

The simple condition of concavity of In p(x) is sufficient for uniqueness of a locally optimal quantizer being a stationary point of the quantization distortion measure E[f(x,\eta)].

Abstract

Sufficient conditions are presented for uniqueness of a locally optimal quantizer being a stationary point of the quantization distortion measure E[f(x,\eta)] , the expected value of an error weighting function f(x,\eta) , where x is a random variable to be quantized, where the probability density function p(x) describing x is continuous and positive on some finite or infinite interval and zero outside it, and where \eta is the quantization error. The function f(x,\eta) is assumed convex and symmetric in \eta and zero only for \eta = 0 . It is shown that in the cases of f(x,\eta)=\eta^{2} and f(x,\eta)=|\eta| , the simple condition of concavity of In p(x) is sufficient for uniqueness of a locally optimal quantizer.