Optimal dual quantizers of [formula omitted][formula omitted]-concave distributions: Uniqueness and Lloyd like algorithm

Abstract

We establish for dual quantization the counterpart of Kieffer’s uniqueness result for compactly supported one dimensional probability distributions having a log-concave density (also called strongly unimodal): for such distributions, Lr-optimal dual quantizers are unique at each level N, the optimal grid being the unique critical point of the quantization error. An example of non-strongly unimodal distribution for which uniqueness of critical points fails is exhibited. In the quadratic r=2 case, we propose an algorithm to compute the unique optimal dual quantizer. It provides a counterpart of Lloyd’s method I algorithm in a Voronoi framework (see [14] and [15]). Finally semi-closed forms of Lr-optimal dual quantizers are established for power distributions on compacts intervals and truncated exponential distributions.