Optimal dual quantizers of$1D$$\log$ -concave distributions: uniqueness and Lloyd like algorithm - Jourdain, Benjamin; Pagès, Gilles (2020)
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,$L^r$ -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. Finally semi-closed forms of$L^r$ -optimal dual quantizers are established for power distributions on compacts intervals and truncated exponential distributions.