Optimal Quantization of Finite Uniform Data on the Sphere

TL;DR

This paper develops a comprehensive geometric theory for optimal quantization on the sphere, including existence, structure, stability, and algorithms for finite uniform distributions supported on the surface.

Contribution

It introduces new structural theorems, explicit distribution rules, and a spherical Lloyd's algorithm for optimal quantization on the sphere.

Findings

Optimal sets of n-means exist and are characterized by Voronoi tessellations.

A ring-allocation rule describes how representatives are distributed across latitudinal rings.

The stability theorem quantifies robustness of configurations under perturbations.

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 $\mathbb S^2$.