Optimal dual quantizers of$1D$$\log$ -concave distributions: uniqueness and Lloyd like algorithm

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$N$...

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Bibliographic Details
Main Authors Jourdain, Benjamin, Pagès, Gilles
Format Journal Article
LanguageEnglish
Published 21.10.2020
Subjects
Computer Science - Numerical Analysis
Mathematics - Numerical Analysis
Mathematics - Probability
Online AccessGet full text
DOI10.48550/arxiv.2010.10816

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Summary: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.
DOI:10.48550/arxiv.2010.10816