球面上有限均匀数据的最优量化
Optimal Quantization of Finite Uniform Data on the Sphere
AI总结：
本文提出球面上有限均匀数据的最优量化理论，通过球面瓦片划分和Lipschitz稳定性定理，构建统一的几何和算法框架。
摘要：
本文发展了一种系统且几何化的球面上最优量化理论，重点研究支撑在球面表面的有限均匀概率分布，而非支撑在低维测地子集如圆或弧上。我们首先建立了n个均值的最优集的存在性，并通过重心球面瓦片划分来表征它们。获得了三个基本结构结果。第一，聚类-纯度定理表明，当支撑由良好分离的组件组成时，每个最优瓦片区域仍被限制在单一组件内。第二，环-分配（离散水填充）定理提供了一个显式规则，描述了最优代表如何分布在多个纬度环上，并伴有闭合形式的失真公式。第三，Lipschitz型稳定性定理量化了在支撑的小测地扰动下最优配置的鲁棒性。此外，还提出了一种球面的Lloyd算法变种，其中内禀（Karcher）均值取代欧几里得质心用于迭代优化。这些结果共同提供了一个统一且透明的框架，用于理解球面上最优量化的几何和算法结构。
英文摘要：
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 $\mathbb S^2$.
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