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：我们针对基于（随机）测量矩阵 $N \in \mathbb{R}^{n \times m}$ 提供任意线性（因此非自适应）信息的算法，证明了嵌入 $\ell_1^m \rightarrow \ell_\infty^m$ 的随机逼近下界。这些下界反映了当 $m \to \infty$ 时问题难度的增加，即复杂度 $n$ 中的 $\sqrt{\log m}$ 项。该结果意味着任意巴拿赫空间之间的非紧算子无法通过非自适应蒙特卡洛方法逼近。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 $(\log\log m)$ 依赖性。通过这种做法，我们给出了线性问题的一个示例，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。