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任意线性（即非自适应）信息的算法，对嵌入 $\ell_1^m \rightarrow \ell_\infty^m$ 进行随机逼近的下界，其中线性信息由（随机化）测量矩阵 $N \in \mathbb{R}^{n \times m}$ 提供。这些下界反映了该问题在 $m \to \infty$ 时难度增加的特性，具体表现为复杂度 $n$ 中出现 $\sqrt{\log m}$ 项。该结果意味着任意巴拿赫空间之间的非紧算子无法使用非自适应的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恢复方法的上界进行了比较，后者的复杂度 $n$ 仅表现出 $(\log\log m)$ 依赖关系。通过这一比较，我们给出了一个线性问题的示例，其中自适应与非自适应蒙特卡洛方法的误差呈现出 $n^{1/2} ( \log n)^{-1/2}$ 量级的差距。