Asymptotics for Optimal Empirical Quantization of Measures
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910574813642752 |
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| author | Quattrocchi, Filippo |
| author_facet | Quattrocchi, Filippo |
| contents | We investigate the minimal error in approximating a general probability measure $μ$ on $\mathbb{R}^d$ by the uniform measure on a finite set with prescribed cardinality $n$. The error is measured in the $p$-Wasserstein distance. In particular, when $1\le p<d$, we establish asymptotic upper and lower bounds as $n \to \infty$ on the rescaled minimal error that have the same, explicit dependency on $μ$.
In some instances, we prove that the rescaled minimal error has a limit. These include general measures in dimension $d = 2$ with $1 \le p < 2$, and uniform measures in arbitrary dimension with $1 \le p < d$. For some uniform measures, we prove the limit existence for $p \ge d$ as well.
For a class of compactly supported measures with Hölder densities, we determine the convergence speed of the minimal error for every $p \ge 1$.
Furthermore, we establish a new Pierce-type (i.e., nonasymptotic) upper estimate of the minimal error when $1 \le p < d$.
In the initial sections, we survey the state of the art and draw connections with similar problems, such as classical and random quantization. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_12924 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotics for Optimal Empirical Quantization of Measures Quattrocchi, Filippo Probability Optimization and Control 41A25 (Primary), 62E17, 49Q22 We investigate the minimal error in approximating a general probability measure $μ$ on $\mathbb{R}^d$ by the uniform measure on a finite set with prescribed cardinality $n$. The error is measured in the $p$-Wasserstein distance. In particular, when $1\le p<d$, we establish asymptotic upper and lower bounds as $n \to \infty$ on the rescaled minimal error that have the same, explicit dependency on $μ$. In some instances, we prove that the rescaled minimal error has a limit. These include general measures in dimension $d = 2$ with $1 \le p < 2$, and uniform measures in arbitrary dimension with $1 \le p < d$. For some uniform measures, we prove the limit existence for $p \ge d$ as well. For a class of compactly supported measures with Hölder densities, we determine the convergence speed of the minimal error for every $p \ge 1$. Furthermore, we establish a new Pierce-type (i.e., nonasymptotic) upper estimate of the minimal error when $1 \le p < d$. In the initial sections, we survey the state of the art and draw connections with similar problems, such as classical and random quantization. |
| title | Asymptotics for Optimal Empirical Quantization of Measures |
| topic | Probability Optimization and Control 41A25 (Primary), 62E17, 49Q22 |
| url | https://arxiv.org/abs/2408.12924 |