Asymptotics of the quantization problem on metric measure spaces
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914328610865152 |
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| author | Aydin, Ata Deniz |
| author_facet | Aydin, Ata Deniz |
| contents | The problem of quantization of measures looks for best approximations of probability measures on a metric space by discrete measures supported on $N$ points, where the error of approximation is measured with respect to the Wasserstein distance. Zador's theorem states that, for measures on $\mathbb{R}^d$ or $d$-dimensional Riemannian manifolds satisfying appropriate integrability conditions, the quantization error decays to zero as $N \to \infty$ at the rate $N^{-1/d}$.
In this paper, we provide a general treatment of the asymptotics of quantization on metric measure spaces $(X, ν)$. We show that a weaker version of Zador's theorem involving the Hausdorff densities of $ν$ holds also in this general setting. We also prove Zador's theorem in full for appropriate $m$-rectifiable measures on Euclidean space, answering a conjecture by Graf and Luschgy in the affirmative. For both results, the higher integrability conditions of Zador's theorem are replaced with a general notion of $(p,s)$-quantizability, which follows from Pierce-type (non-asymptotic) upper bounds on the quantization error, and we also prove multiple such bounds at the level of metric measure spaces. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_18779 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotics of the quantization problem on metric measure spaces Aydin, Ata Deniz Metric Geometry Analysis of PDEs Optimization and Control Primary: 49Q22, 94A34, 53C23, Secondary: 28A75, 49Q20, 51F30 The problem of quantization of measures looks for best approximations of probability measures on a metric space by discrete measures supported on $N$ points, where the error of approximation is measured with respect to the Wasserstein distance. Zador's theorem states that, for measures on $\mathbb{R}^d$ or $d$-dimensional Riemannian manifolds satisfying appropriate integrability conditions, the quantization error decays to zero as $N \to \infty$ at the rate $N^{-1/d}$. In this paper, we provide a general treatment of the asymptotics of quantization on metric measure spaces $(X, ν)$. We show that a weaker version of Zador's theorem involving the Hausdorff densities of $ν$ holds also in this general setting. We also prove Zador's theorem in full for appropriate $m$-rectifiable measures on Euclidean space, answering a conjecture by Graf and Luschgy in the affirmative. For both results, the higher integrability conditions of Zador's theorem are replaced with a general notion of $(p,s)$-quantizability, which follows from Pierce-type (non-asymptotic) upper bounds on the quantization error, and we also prove multiple such bounds at the level of metric measure spaces. |
| title | Asymptotics of the quantization problem on metric measure spaces |
| topic | Metric Geometry Analysis of PDEs Optimization and Control Primary: 49Q22, 94A34, 53C23, Secondary: 28A75, 49Q20, 51F30 |
| url | https://arxiv.org/abs/2503.18779 |