Optimal quantizers for a nonuniform distribution on a Sierpinski carpet

TL;DR

This paper investigates the optimal quantization of a nonuniform probability measure supported on a Sierpinski carpet, determining optimal sets of means and quantization errors for all n ≥ 2.

Contribution

It provides a detailed analysis of optimal quantizers and quantization errors for a specific nonuniform measure on a Sierpinski carpet, extending quantization theory to fractal supports.

Findings

Explicit optimal n-means sets for all n ≥ 2.

Quantization errors characterized for the measure.

Insights into quantization on fractal structures.

Abstract

The purpose of quantization for a probability distribution is to estimate the probability by a discrete probability with finite support. In this paper, a nonuniform probability measure $P$ on $\mathbb R^2$ which has support the Sierpi\'nski carpet generated by a set of four contractive similarity mappings with equal similarity ratios has been considered. For this probability measure, the optimal sets of $n$-means and the $n$th quantization errors are investigated for all $n\geq 2$.