Quantization and centroidal Voronoi tessellations for probability measures on dyadic Cantor sets

TL;DR

This paper studies optimal quantization and centroidal Voronoi tessellations for probability measures supported on dyadic Cantor sets, providing insights into their structure and optimal configurations.

Contribution

It introduces methods to analyze and compute optimal quantization and CVT for measures on dyadic Cantor sets with self-similar structure.

Findings

Derived explicit formulas for optimal quantizers.

Characterized centroidal Voronoi tessellations on Cantor sets.

Established bounds on similarity ratios for optimality.

Abstract

Quantization of a probability distribution is the process of estimating a given probability by a discrete probability that assumes only a finite number of levels in its support. Centroidal Voronoi tessellations (CVT) are Voronoi tessellations of a region such that the generating points of the tessellations are also the centroids of the corresponding Voronoi regions. In this paper, we investigate the optimal quantization and the centroidal Voronoi tessellations with $n$ generators for a Borel probability measure $P$ on $\mathbb R$ supported by a dyadic Cantor set generated by two self-similar mappings with similarity ratios $r$, where $0<r\leq \frac{5-\sqrt{17}}2$.