Optimal dual quantizers of 1Dlog-concave distributions: Uniqueness and Lloyd like algorithm

• Published in: Journal of approximation theory Vol. 267
• Main Authors: Jourdain, Benjamin, Pagès, Gilles
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.07.2021
• ISSN: 0021-9045; 1096-0430
• DOI: 10.1016/j.jat.2021.105581

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.