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

TL;DR

This paper proves the uniqueness of optimal dual quantizers for certain log-concave distributions, introduces an algorithm similar to Lloyd's for quadratic cases, and derives semi-closed forms for specific distributions.

Contribution

It establishes the uniqueness of $L^r$-optimal dual quantizers for log-concave distributions and proposes a Lloyd-like algorithm for quadratic cases.

Findings

Uniqueness of optimal dual quantizers for log-concave distributions.

Development of a Lloyd-like algorithm for quadratic dual quantization.

Semi-closed form solutions for specific distributions like power and truncated exponential distributions.

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 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.