Optimal dual quantizers of $1D$ $\log$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, $L^r$optimal dual quantizers are unique at each level $N$, the optimal grid being the unique critical point of the quantization error. An example of nonstrongly 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 semiclosed forms of $L^r$optimal dual quantizers are established for power distributions on compacts intervals and truncated exponential distributions.
 Publication:

arXiv eprints
 Pub Date:
 October 2020
 DOI:
 10.48550/arXiv.2010.10816
 arXiv:
 arXiv:2010.10816
 Bibcode:
 2020arXiv201010816J
 Keywords:

 Mathematics  Probability;
 Mathematics  Numerical Analysis