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 $n^{1/2} ( \log n)^{-1/2}$.

翻译：我们证明了基于使用（随机）测量矩阵 $N \in \mathbb{R}^{n \times m}$ 提供的任意线性（因而非自适应）信息的算法，对嵌入 $\ell_1^m \rightarrow \ell_\infty^m$ 进行随机逼近的下界。这些下界反映了问题随 $m \to \infty$ 而增加的难度，即复杂度 $n$ 中包含 $\sqrt{\log m}$ 项。这一结果表明，任意巴拿赫空间之间的非紧算子无法使用非自适应蒙特卡洛方法进行逼近。我们还将这些非自适应方法的下界与基于自适应随机方法的重构上界进行了比较，其中后者的复杂度 $n$ 仅呈现 $(\log\log m)$ 依赖性。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蒙特卡洛方法的误差存在数量级为 $n^{1/2} ( \log n)^{-1/2}$ 的差距。