On the number of defects in optimal quantizers on closed surfaces: the hexagonal torus
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910901923217408 |
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| author | Tisdell, Jack Edward Choksi, Rustum Lu, Xin Yang |
| author_facet | Tisdell, Jack Edward Choksi, Rustum Lu, Xin Yang |
| contents | We present a strategy for proving an asymptotic upper bound on the number of defects (non-hexagonal Voronoi cells) in the $n$ generator optimal quantizer on a closed surface (i.e., compact 2-manifold without boundary). The program is based upon a general lower bound on the optimal quantization error and related upper bounds for the Löschian numbers $n$ (the norms of the Eisenstein integers) arising from the Goldberg-Coxeter construction. A gap lemma is used to reduce the asymptotics of the number of defects to precisely the asymptotics for the gaps between Löschian numbers. We apply this strategy on the hexagonal torus and prove that the number of defects is at most $O(n^{1/4})$ -- strictly fewer than surfaces with boundary -- and conjecture (based upon the number-theoretic Löschian gap conjecture) that it is in fact $O(\log n)$. Incidentally, the method also yields a related upper bound on the variance of the areas of the Voronoi cells. We show further that the bound on the number of defects holds in a neighborhood of the optimizers. Finally, we remark on the remaining issues for implementation on the 2-sphere. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_22680 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the number of defects in optimal quantizers on closed surfaces: the hexagonal torus Tisdell, Jack Edward Choksi, Rustum Lu, Xin Yang Metric Geometry Optimization and Control We present a strategy for proving an asymptotic upper bound on the number of defects (non-hexagonal Voronoi cells) in the $n$ generator optimal quantizer on a closed surface (i.e., compact 2-manifold without boundary). The program is based upon a general lower bound on the optimal quantization error and related upper bounds for the Löschian numbers $n$ (the norms of the Eisenstein integers) arising from the Goldberg-Coxeter construction. A gap lemma is used to reduce the asymptotics of the number of defects to precisely the asymptotics for the gaps between Löschian numbers. We apply this strategy on the hexagonal torus and prove that the number of defects is at most $O(n^{1/4})$ -- strictly fewer than surfaces with boundary -- and conjecture (based upon the number-theoretic Löschian gap conjecture) that it is in fact $O(\log n)$. Incidentally, the method also yields a related upper bound on the variance of the areas of the Voronoi cells. We show further that the bound on the number of defects holds in a neighborhood of the optimizers. Finally, we remark on the remaining issues for implementation on the 2-sphere. |
| title | On the number of defects in optimal quantizers on closed surfaces: the hexagonal torus |
| topic | Metric Geometry Optimization and Control |
| url | https://arxiv.org/abs/2503.22680 |