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Optimal Quantization of Finite Uniform Data on the Sphere

Mrinal Kanti Roychowdhury

TL;DR

This work develops a geometrically transparent theory of optimal quantization on the sphere for finite uniform data, addressing how to place a finite set of representative points on $\mathbb{S}^2$ to minimize geodesic distortion. By establishing existence and a centroidal spherical Voronoi framework, the paper reveals a three-part structural theory (cluster purity, ring-allocation via discrete water-filling, and Lipschitz stability) and provides a spherical Lloyd-type algorithm that uses intrinsic (Karcher) means instead of Euclidean centroids. It shows that optimal quantizers decompose cleanly across latitudinal rings, with no cross-ring mixing and a precise ring-level allocation rule, and proves stability under small perturbations of the data. The results yield a practical, curvature-aware toolkit for spherical data analysis, with confirmed behavior on symmetric and irregular datasets and clear guidance for implementation and future extensions to nonuniform densities and higher-dimensional spheres.

Abstract

This paper develops a systematic and geometric theory of optimal quantization on the unit sphere $\mathbb S^2$, focusing on finite uniform probability distributions supported on the spherical surface - rather than on lower-dimensional geodesic subsets such as circles or arcs. We first establish the existence of optimal sets of $n$-means and characterize them through centroidal spherical Voronoi tessellations. Three fundamental structural results are obtained. First, a cluster - purity theorem shows that when the support consists of well-separated components, each optimal Voronoi region remains confined to a single component. Second, a ring - allocation (discrete water - filling) theorem provides an explicit rule describing how optimal representatives are distributed across multiple latitudinal rings, together with closed-form distortion formulas. Third, a Lipschitz - type stability theorem quantifies the robustness of optimal configurations under small geodesic perturbations of the support. In addition, a spherical analogue of Lloyd's algorithm is presented, in which intrinsic (Karcher) means replace Euclidean centroids for iterative refinement. These results collectively provide a unified and transparent framework for understanding the geometric and algorithmic structure of optimal quantization on $\mathbb S^2$.

Optimal Quantization of Finite Uniform Data on the Sphere

TL;DR

This work develops a geometrically transparent theory of optimal quantization on the sphere for finite uniform data, addressing how to place a finite set of representative points on to minimize geodesic distortion. By establishing existence and a centroidal spherical Voronoi framework, the paper reveals a three-part structural theory (cluster purity, ring-allocation via discrete water-filling, and Lipschitz stability) and provides a spherical Lloyd-type algorithm that uses intrinsic (Karcher) means instead of Euclidean centroids. It shows that optimal quantizers decompose cleanly across latitudinal rings, with no cross-ring mixing and a precise ring-level allocation rule, and proves stability under small perturbations of the data. The results yield a practical, curvature-aware toolkit for spherical data analysis, with confirmed behavior on symmetric and irregular datasets and clear guidance for implementation and future extensions to nonuniform densities and higher-dimensional spheres.

Abstract

This paper develops a systematic and geometric theory of optimal quantization on the unit sphere , focusing on finite uniform probability distributions supported on the spherical surface - rather than on lower-dimensional geodesic subsets such as circles or arcs. We first establish the existence of optimal sets of -means and characterize them through centroidal spherical Voronoi tessellations. Three fundamental structural results are obtained. First, a cluster - purity theorem shows that when the support consists of well-separated components, each optimal Voronoi region remains confined to a single component. Second, a ring - allocation (discrete water - filling) theorem provides an explicit rule describing how optimal representatives are distributed across multiple latitudinal rings, together with closed-form distortion formulas. Third, a Lipschitz - type stability theorem quantifies the robustness of optimal configurations under small geodesic perturbations of the support. In addition, a spherical analogue of Lloyd's algorithm is presented, in which intrinsic (Karcher) means replace Euclidean centroids for iterative refinement. These results collectively provide a unified and transparent framework for understanding the geometric and algorithmic structure of optimal quantization on .
Paper Structure (27 sections, 13 theorems, 144 equations)

This paper contains 27 sections, 13 theorems, 144 equations.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Geodesic Distance
  4. Finite Discrete Uniform Distributions on $\mathbb{S}^2$
  5. Distortion and Optimal $n$--Means
  6. Spherical Voronoi Partitions
  7. Intrinsic (Karcher) Mean on the Sphere
  8. Notation Summary
  9. Existence and Characterization of Optimal $n$--Means
  10. Spherical Voronoi partition and centroidal property
  11. Quantization on Finite Latitudinal Rings
  12. Discrete ring configuration
  13. Stability of Optimal Sets under Perturbation
  14. Perturbed distributions
  15. Algorithmic Construction of Optimal $n$--Means
  16. ...and 12 more sections

Key Result

Proposition 3.1

Let $P$ be a finite discrete uniform distribution on $\mathbb{S}^2$ and $n \ge 1$. Then there exists at least one configuration ${Q}^* \subset \mathbb{S}^2$ with $| {Q}^*| \le n$ such that Each such ${Q}^*$ is called an optimal set of $n$--means for $P$.

Theorems & Definitions (48)

  • Remark 2.4
  • Remark 2.6
  • Remark 2.8: Standing assumption on intrinsic means
  • Definition 2.9: Centroidal Voronoi configuration
  • Remark 2.10
  • Proposition 3.1: Existence of optimal set
  • proof
  • Theorem 3.3: Necessary Centroidal Condition
  • Remark 3.4: Centroidal condition is not sufficient
  • Remark 3.5
  • ...and 38 more