A New Upper Bound for Sampling Numbers
A New Upper Bound for Sampling Numbers
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抽样数量的新上限
DOI：
10.1007/s10208-021-09504-0
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发表时间：
2021
影响因子：
3
通讯作者：
T. Ullrich
中科院分区：
文献类型：
--
作者：
N. Nagel;M. Schäfer;T. Ullrich
We provide a new upper bound for sampling numbersassociated with the compact embedding of a separable reproducing kernel Hilbert space into the space of square integrable functions. There are universal constants(which are specified in the paper) such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} g^2_n \le \frac{C\log (n)}{n}\sum \limits _{k\ge \lfloor cn \rfloor } \sigma _k^2,\quad n\ge 2, \end{aligned}$$\end{document}whereis the sequence of singular numbers (approximation numbers) of the Hilbert–Schmidt embedding. The algorithm which realizes the bound is a least squares algorithm based on a specific set of sampling nodes. These are constructed out of a random draw in combination with a down-sampling procedure coming from the celebrated proof of Weaver’s conjecture, which was shown to be equivalent to the Kadison–Singer problem. Our result is non-constructive since we only show the existence of a linear sampling operator realizing the above bound. The general result can for instance be applied to the well-known situation ofinwith. We obtain the asymptotic bound \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} g_n \le C_{s,d}n^{-s}\log (n)^{(d-1)s+1/2}, \end{aligned}$$\end{document}which improves on very recent results by shortening the gap between upper and lower bound to. The result implies that for dimensionsany sparse grid sampling recovery method does not perform asymptotically optimal.
We provide a new upper bound for sampling numbersassociated with the compact embedding of a separable reproducing kernel Hilbert space into the space of square integrable functions. There are universal constants(which are specified in the paper) such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} g^2_n \le \frac{C\log (n)}{n}\sum \limits _{k\ge \lfloor cn \rfloor } \sigma _k^2,\quad n\ge 2, \end{aligned}$$\end{document}whereis the sequence of singular numbers (approximation numbers) of the Hilbert–Schmidt embedding. The algorithm which realizes the bound is a least squares algorithm based on a specific set of sampling nodes. These are constructed out of a random draw in combination with a down-sampling procedure coming from the celebrated proof of Weaver’s conjecture, which was shown to be equivalent to the Kadison–Singer problem. Our result is non-constructive since we only show the existence of a linear sampling operator realizing the above bound. The general result can for instance be applied to the well-known situation ofinwith. We obtain the asymptotic bound \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} g_n \le C_{s,d}n^{-s}\log (n)^{(d-1)s+1/2}, \end{aligned}$$\end{document}which improves on very recent results by shortening the gap between upper and lower bound to. The result implies that for dimensionsany sparse grid sampling recovery method does not perform asymptotically optimal.
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DOI：
10.1016/j.jco.2020.101523
发表时间：
2020-01
期刊：
ArXiv
影响因子：
--
作者：
T. Kuehn;W. Sickel;T. Ullrich
通讯作者：
T. Kuehn;W. Sickel;T. Ullrich
关于L2中的采样离散化
影响因子：
1.3
作者：
I. Limonova;V. Temlyakov
通讯作者：
V. Temlyakov
关于 L2 的最佳恢复
影响因子：
1.7
作者：
V. Temlyakov
通讯作者：
V. Temlyakov
纯粹状态的延伸。
影响因子：
1.7
作者：
R. Kadison;I. Singer
通讯作者：
I. Singer
影响因子：
2.7
作者：
T. Kühn;W. Sickel;T. Ullrich
通讯作者：
T. Kühn;W. Sickel;T. Ullrich