Constrained Quantization for Uniform Distributions with Two Constraint Families

TL;DR

This paper investigates optimal quantization strategies for uniform distributions under two types of geometric constraints—lines and circles—deriving explicit solutions and key quantization metrics for all numbers of points.

Contribution

It provides a comprehensive analysis of constrained quantization for uniform distributions with line and circle constraints, including explicit optimal sets and quantization errors for all n.

Findings

Explicit optimal sets for line constraints

Explicit optimal sets for circle constraints

Quantization errors and dimensions computed for all n

Abstract

In this paper, we first consider a family of constraints given by straight lines. For a uniform probability distribution, we determine the constrained optimal sets of $n$-points and the corresponding $n$th constrained quantization errors for all positive integers $n$. In addition, we calculate the constrained quantization dimension and the constrained quantization coefficient with respect to this family of constraints. Next, we turn to another family of constraints, consisting of concentric circles. For the same probability distribution, we present a methodology to compute the constrained optimal sets of $n$-points and the corresponding $n$th constrained quantization errors for all positive integers $n$. Finally, we conclude the paper with a summary of the results and a discussion of future research directions.