arXiv 2308.017052023-08math.NAcs.NAmath.FAmath.PR

可和序列的随机逼近——自适应与非自适应

Randomized approximation of summable sequences -- adaptive and non-adaptive

Robert Kunsch, Erich Novak, Marcin Wnuk

AI总结：

该研究证明了基于非自适应线性信息的ℓ₁ᵐ到ℓ_∞ᵐ嵌入随机逼近的下界，表明非紧算子无法用非自适应Monte Carlo方法逼近，并对比了自适应方法的上界，揭示了两类方法的误差差距。

摘要：

我们证明了基于（随机化）测量矩阵$N \in \mathbb{R}^{n \times m}$提供的任意线性（即非自适应）信息的嵌入$\ell_1^m \rightarrow \ell_\infty^m$随机逼近的下界。这些下界反映了当$m \to \infty$时问题难度的增加，即复杂度$n$中存在$\sqrt{\log m}$项。该结果表明，任意巴拿赫空间之间的非紧算子无法用非自适应Monte Carlo方法逼近。我们还将这些非自适应方法的下界与基于自适应随机化恢复方法的上界进行了比较，后者的复杂度$n$仅呈现$(\log\log m)$依赖关系。在此过程中，我们给出了一个线性问题的例子，其中自适应与非自适应Monte Carlo方法的误差存在$n^{1/2} ( \log n)^{-1/2}$阶的差距。

英文摘要：

We prove lower bounds for the randomized approximation of the embedding $\ell_1^m \rightarrow \ell_\infty^m$ based on algorithms that use arbitrary linear (hence non-adaptive) information provided by a (randomized) measurement matrix $N \in \mathbb{R}^{n \times m}$. These lower bounds reflect the increasing difficulty of the problem for $m \to \infty$, namely, a term $\sqrt{\log m}$ in the complexity $n$. This result implies that non-compact operators between arbitrary Banach spaces are not approximable using non-adaptive Monte Carlo methods. We also compare these lower bounds for non-adaptive methods with upper bounds based on adaptive, randomized methods for recovery for which the complexity $n$ only exhibits a $(\log\log m)$-dependence. In doing so we give an example of linear problems where the error for adaptive vs. non-adaptive Monte Carlo methods shows a gap of order $n^{1/2} ( \log n)^{-1/2}$.

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