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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2505.10801 |
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| _version_ | 1866910947502718976 |
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| author | Qian, Chenxing |
| author_facet | Qian, Chenxing |
| contents | We investigated the asymptotics of high-rate constrained quantization errors for a compactly supported probability measure P on Euclidean spaces whose quantizers are confined to a closed set S. The key tool is the metric projection of K onto S that assigns each source point to its nearest neighbor in S, allowing the errors to be transferred to the projection, where K = supp P. For the upper estimate, we establish a projection pull-back inequality that bounds the errors by the classical covering radius of the projection. For the lower estimate, a weighted distance function enables us to perturb any quantizer element lying on the projection slightly into the complement in S without enlarging the error, provided the projection is nowhere dense (automatically true when S and K are disjoint). Under mild conditions on the pushforward measure of P by T, obtained via a measurable selector T, we derive a uniform lower bound. If this set is Ahlfors regular of dimension d, the error decays like the reciprocal of the d-th root of n and every constrained quantization dimension equals d. The two estimates coincide, giving the first complete dimension comparison formula for constrained quantization and closing the gap left by earlier self-similar examples by Pandey-Roychowdhury while extending classical unconstrained theory to closed constraints under mild geometric assumptions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_10801 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotics of Constrained Quantization for Compactly Supported Measures Qian, Chenxing Metric Geometry Classical Analysis and ODEs Functional Analysis Optimization and Control Probability Primary 41A29, Secondary 37A50, 60D05, 28C20, 31E05 We investigated the asymptotics of high-rate constrained quantization errors for a compactly supported probability measure P on Euclidean spaces whose quantizers are confined to a closed set S. The key tool is the metric projection of K onto S that assigns each source point to its nearest neighbor in S, allowing the errors to be transferred to the projection, where K = supp P. For the upper estimate, we establish a projection pull-back inequality that bounds the errors by the classical covering radius of the projection. For the lower estimate, a weighted distance function enables us to perturb any quantizer element lying on the projection slightly into the complement in S without enlarging the error, provided the projection is nowhere dense (automatically true when S and K are disjoint). Under mild conditions on the pushforward measure of P by T, obtained via a measurable selector T, we derive a uniform lower bound. If this set is Ahlfors regular of dimension d, the error decays like the reciprocal of the d-th root of n and every constrained quantization dimension equals d. The two estimates coincide, giving the first complete dimension comparison formula for constrained quantization and closing the gap left by earlier self-similar examples by Pandey-Roychowdhury while extending classical unconstrained theory to closed constraints under mild geometric assumptions. |
| title | Asymptotics of Constrained Quantization for Compactly Supported Measures |
| topic | Metric Geometry Classical Analysis and ODEs Functional Analysis Optimization and Control Probability Primary 41A29, Secondary 37A50, 60D05, 28C20, 31E05 |
| url | https://arxiv.org/abs/2505.10801 |