Optimal Quantization of Finite Uniform Data on the Sphere
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908750639529984 |
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| author | Roychowdhury, Mrinal Kanti |
| author_facet | Roychowdhury, Mrinal Kanti |
| contents | 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 $\mathbb S^2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_03333 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Optimal Quantization of Finite Uniform Data on the Sphere Roychowdhury, Mrinal Kanti Optimization and Control Probability Primary 62H30, 41A25, Secondary 60Exx, 94A34, 68U05 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 $\mathbb S^2$. |
| title | Optimal Quantization of Finite Uniform Data on the Sphere |
| topic | Optimization and Control Probability Primary 62H30, 41A25, Secondary 60Exx, 94A34, 68U05 |
| url | https://arxiv.org/abs/2601.03333 |