Constrained quantization for a uniform distribution

TL;DR

This paper investigates the optimal constrained quantization of a uniform distribution on a triangle's side, deriving explicit solutions for optimal points, quantization errors, and related dimensions.

Contribution

It provides explicit solutions for constrained quantization problems on a triangle, including optimal point sets and error metrics, for all positive integers n.

Findings

Explicit optimal sets of points for constrained quantization

Formulas for nth constrained quantization errors

Calculation of constrained quantization dimension and coefficient

Abstract

Constrained quantization for a Borel probability measure refers to the idea of estimating a given probability by a discrete probability with a finite number of supporting points lying on a specific set. The specific set is known as the constraint of the constrained quantization. A quantization without a constraint is known as an unconstrained quantization, which traditionally in the literature is known as quantization. Constrained quantization has recently been introduced by Pandey and Roychowdhury. In this paper, for a uniform distribution with support lying on a side of an equilateral triangle, and the constraint as the union of the other two sides, we obtain the optimal sets of $n$-points and the $n$th constrained quantization errors for all positive integers $n$. We also calculate the constrained quantization dimension and the constrained quantization coefficient.