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A New Upper Bound for Sampling Numbers
Foundations of Computational Mathematics ( IF 3 ) Pub Date : 2021-04-26 , DOI: 10.1007/s10208-021-09504-0
Nicolas Nagel , Martin Schäfer , Tino Ullrich

We provide a new upper bound for sampling numbers \((g_n)_{n\in \mathbb {N}}\) associated with the compact embedding of a separable reproducing kernel Hilbert space into the space of square integrable functions. There are universal constants \(C,c>0\) (which are specified in the paper) such that

$$\begin{aligned} g^2_n \le \frac{C\log (n)}{n}\sum \limits _{k\ge \lfloor cn \rfloor } \sigma _k^2,\quad n\ge 2, \end{aligned}$$

where \((\sigma _k)_{k\in \mathbb {N}}\) is the sequence of singular numbers (approximation numbers) of the Hilbert–Schmidt embedding \(\mathrm {Id}:H(K) \rightarrow L_2(D,\varrho _D)\). 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 of \(H^s_{\text {mix}}(\mathbb {T}^d)\) in \(L_2(\mathbb {T}^d)\) with \(s>1/2\). We obtain the asymptotic bound

$$\begin{aligned} g_n \le C_{s,d}n^{-s}\log (n)^{(d-1)s+1/2}, \end{aligned}$$

which improves on very recent results by shortening the gap between upper and lower bound to \(\sqrt{\log (n)}\). The result implies that for dimensions \(d>2\) any sparse grid sampling recovery method does not perform asymptotically optimal.



中文翻译:

采样数字的新上限

我们为采样数\((g_n)_ {n \ in \ mathbb {N}} \)提供了新的上限,该采样数与可分离的再生内核希尔伯特空间的紧凑嵌入到平方可积函数的空间中有关。有通用常数\(C,c> 0 \)(在纸张中指定),使得

$$ \ begin {aligned} g ^ 2_n \ le \ frac {C \ log(n)} {n} \ sum \ limits _ {k \ ge \ lfloor cn \ rfloor} \ sigma _k ^ 2,\ quad n \ ge 2,\ end {aligned} $$

其中\((\ sigma _k)_ {k \ in \ mathbb {N}} \)是Hilbert–Schmidt嵌入\(\ mathrm {Id}:H(K)\ rightarrow L_2(D,\ varrho _D)\)。实现边界的算法是基于一组特定采样节点的最小二乘算法。这些是由随机抽取构成的,并结合了著名的Weaver猜想证明所产生的下采样过程,该过程被证明等同于Kadison-Singer问题。我们的结果是非建设性的,因为我们仅显示了实现上述限制的线性采样算子的存在。例如一般的结果可以被应用到的公知的情况\(H 2 -S _ {\文本{混合}}(\ mathbb横置^ d)\)\(L_2(\ mathbb {T} ^ d)\)\(s> 1/2 \)。我们得到渐近界

$$ \ begin {aligned} g_n \ le C_ {s,d} n ^ {-s} \ log(n)^ {(d-1)s + 1/2},\ end {aligned} $$

通过缩短\(\ sqrt {\ log(n)} \)的上限和下限之间的距离,可以改善最近的结果。结果表明,对于维度\(d> 2 \),任何稀疏网格采样恢复方法都不会渐近最优。

更新日期:2021-04-26
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