论文标题
采样数字的新上限
A new upper bound for sampling numbers
论文作者
论文摘要
我们为采样数字提供了一个新的上限$（g_n）_ {n \ in \ mathbb {n}} $与可分开的重现内核希尔伯特（Hilbert）的紧凑嵌入相关联的相关联，进入了方形可集成函数的空间。有通用常数$ c，c> 0 $（在论文中指定），以便$$ g^2_n \ leq \ frac {c \ log（n）} {n} \ sum \ limits_ {k \ geq \ geq \ lfloor cn \ rfloor}σ_k^2 \ quad，\ quad，\ quad n \ quad n \ geq 2 \ geq 2 \，$ Hilbert-Schmidt的奇异数字序列（近似数）嵌入$ \ text {id}：h（k）\ to l_2（d，\ varrho_d）$。实现界限的算法是基于特定的采样节点集的最小二乘算法。这些是由随机抽签与著名的韦弗猜想证明的降采样过程结合起来的，这被证明与Kadison-Serger问题相当。我们的结果是非构造性的，因为我们仅显示了实现上述界限的线性采样操作员的存在。例如，可以将一般结果应用于$ h^s _ {\ text {mix}}的众所周知的情况（\ Mathbb {t}^d）$ in $ l_2（\ Mathbb {t}^d），带有$ s> 1/2 $。我们获得渐近绑定的$$ g_n \ leq c_ {s，d} n^{ - s} \ log（n）^{（d-1）s+1/2} \ ,, $$，通过缩短上限和下限之间的间隙为$ \ sqrt {\ log（n）} $，从而改善了最新结果。
We provide a new upper bound for sampling numbers $(g_n)_{n\in \mathbb{N}}$ associated to the compact embedding of a separable reproducing kernel Hilbert space into the space of square integrable functions. There are universal constants $C,c>0$ (which are specified in the paper) such that $$ g^2_n \leq \frac{C\log(n)}{n}\sum\limits_{k\geq \lfloor cn \rfloor} σ_k^2\quad,\quad n\geq 2\,, $$ where $(σ_k)_{k\in \mathbb{N}}$ is the sequence of singular numbers (approximation numbers) of the Hilbert-Schmidt embedding $\text{Id}:H(K) \to L_2(D,\varrho_D)$. The algorithm which realizes the bound is a least squares algorithm based on a specific set of sampling nodes. These are constructed out of a random draw in combination with a down-sampling procedure coming from the celebrated proof of Weaver's conjecture, which was shown to be equivalent to the Kadison-Singer problem. Our result is non-constructive since we only show the existence of a linear sampling operator realizing the above bound. The general result can for instance be applied to the well-known situation of $H^s_{\text{mix}}(\mathbb{T}^d)$ in $L_2(\mathbb{T}^d)$ with $s>1/2$. We obtain the asymptotic bound $$ g_n \leq C_{s,d}n^{-s}\log(n)^{(d-1)s+1/2}\,, $$ which improves on very recent results by shortening the gap between upper and lower bound to $\sqrt{\log(n)}$.
