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Conditional quantization for uniform distributions on line segments and regular polygons

Pigar Biteng, Mathieu Caguiat, Tsianna Dominguez, Mrinal Kanti Roychowdhury

TL;DR

The paper develops conditional quantization for uniform measures on one-dimensional line segments and on the boundaries of regular polygons, deriving explicit constructions of conditional optimal sets and exact distortion rates. By partitioning the support and exploiting symmetry, it provides closed-form placements of points and the corresponding conditional quantization errors $V_n$, and shows the conditional quantization dimension equals the ambient dimension ($D(P)=1$) with finite positive coefficients. For line segments with two interior conditioning points and for generalized $(k-1)$ interior points plus one boundary point, the authors obtain complete combinatorial rules for distributing points and computing $V_n$, while for polygon boundaries they derive edge-wise distortion formulas and a rotationally invariant construction leading to a coefficient $\frac{1}{3}m^2\sin^2(\pi/m)$. Overall, the work yields exact, constructive results and sharp asymptotics for conditional quantization in geometrically structured settings, linking geometric partitions to quantization performance.

Abstract

Quantization for a Borel probability measure refers to the idea of estimating a given probability by a discrete probability with support containing a finite number of elements. If in the quantization some of the elements in the support are preselected, then the quantization is called a conditional quantization. In this paper, we investigate the conditional quantization for the uniform distributions defined on the unit line segments and $m$-sided regular polygons, where $m\geq 3$, inscribed in a unit circle.

Conditional quantization for uniform distributions on line segments and regular polygons

TL;DR

The paper develops conditional quantization for uniform measures on one-dimensional line segments and on the boundaries of regular polygons, deriving explicit constructions of conditional optimal sets and exact distortion rates. By partitioning the support and exploiting symmetry, it provides closed-form placements of points and the corresponding conditional quantization errors , and shows the conditional quantization dimension equals the ambient dimension () with finite positive coefficients. For line segments with two interior conditioning points and for generalized interior points plus one boundary point, the authors obtain complete combinatorial rules for distributing points and computing , while for polygon boundaries they derive edge-wise distortion formulas and a rotationally invariant construction leading to a coefficient . Overall, the work yields exact, constructive results and sharp asymptotics for conditional quantization in geometrically structured settings, linking geometric partitions to quantization performance.

Abstract

Quantization for a Borel probability measure refers to the idea of estimating a given probability by a discrete probability with support containing a finite number of elements. If in the quantization some of the elements in the support are preselected, then the quantization is called a conditional quantization. In this paper, we investigate the conditional quantization for the uniform distributions defined on the unit line segments and -sided regular polygons, where , inscribed in a unit circle.
Paper Structure (10 sections, 20 theorems, 76 equations, 1 figure)

This paper contains 10 sections, 20 theorems, 76 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Delineation.
  3. Motivation and significance.
  4. Preliminaries
  5. Conditional optimal sets of $n$-points and the conditional quantization errors with two interior elements in the conditional set for all $n\geq 2$ on a unit line segment
  6. Conditional optimal sets of $n$-points and the $n$th conditional quantization errors
  7. Conditional optimal sets of $n$-points and the $n$th conditional quantization errors with $(k-1)$ interior elements and one boundary element in the conditional set for all $n\geq k$ on a unit line segment
  8. conditional optimal sets of $n$-points and the $n$th conditional quantization errors
  9. conditional quantization for uniform distributions on the boundaries of regular polygons inscribed in a unit circle
  10. Conditional optimal sets of $n$-points and the $n$th conditional quantization errors

Key Result

Proposition 2.2

Let $P$ be a uniform distribution on the closed interval $[a, b]$ and $c, d\in [a, b]$ be such that $a<c<d<b$. For $n\in \mathbb{N}$ with $n\geq 2$, let $\alpha_n$ be a conditional optimal set of $n$-points for $P$ with respect to the conditional set $\beta=\{c, d\}$ such that $\alpha_n$ contains $k with the conditional quantization error

Figures (1)

  • Figure 1: The regular $m$-sided polygon inscribed in a unit circle.

Theorems & Definitions (44)

  • Definition 1.1
  • Definition 1.2
  • Proposition 2.2
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • ...and 34 more