Optimal Sets and Quantization Errors under Geometric Constraints for Discrete Distributions

TL;DR

This paper investigates constrained quantization for discrete distributions on real line subsets, computing optimal sets and errors under geometric constraints like semicircular arcs and triangle sides, with explicit constructions and a framework for linear constraints.

Contribution

It provides explicit solutions for constrained optimal quantizers for various discrete distributions and introduces a general framework for linear constrained quantization with a zero quantization dimension.

Findings

Explicit optimal quantizers for finite and infinite distributions.

Quantization errors computed under specific geometric constraints.

Constrained quantization dimension is zero for linear constraints.

Abstract

This paper presents a detailed study of constrained quantization for both finite and infinite discrete probability distributions supported on subsets of the real line. Under specific geometric constraints - namely, a semicircular arc and the union of two sides of an equilateral triangle - we compute constrained optimal sets of $n$-points and the corresponding $n$th constrained quantization errors. For finite discrete distributions, we consider both uniform and nonuniform cases with support on $\{-3, -2, \dots, 3\}$. For infinite discrete distributions, two cases are analyzed: one supported on $\left\{ \frac{1}{n} : n \in \mathbb{N} \right\}$ and the other on the set of natural numbers $\mathbb{N}$. Explicit constructions and numerical computations of optimal quantizers and errors are provided. Furthermore, for the infinite discrete distribution supported on $\mathbb{N}$, we develop a general framework for constrained quantization under the linear constraint $y = mx + c$ and prove that the constrained quantization dimension in this setting is zero. Our results highlight how geometric constraints influence the structure and existence of optimal quantizers and pave the way for further investigations into constrained quantization theory.