We study approximation of the embedding $\ell_p^m \hookrightarrow \ell_q^m$, $1 \leq p < q \leq \infty$, based on randomized algorithms that use up to $n$ arbitrary linear functionals as information on a problem instance where $n \ll m$. By analysing adaptive methods we show upper bounds for which the information-based complexity $n$ exhibits only a $(\log\log m)$-dependence. In the case $q < \infty$ we use a multi-sensitivity approach in order to reach optimal polynomial order in $n$ for the Monte Carlo error. We also improve on non-adaptive methods for $q < \infty$ by denoising known algorithms for uniform approximation.

翻译：我们研究基于随机算法的嵌入 $\ell_p^m \hookrightarrow \ell_q^m$（其中 $1 \leq p < q \leq \infty$）的逼近问题，这些算法使用最多 $n$ 个任意线性泛函作为问题实例的信息，且满足 $n \ll m$。通过分析自适应方法，我们给出了信息复杂度 $n$ 仅具有 $(\log\log m)$ 依赖性的上界。在 $q < \infty$ 的情形下，我们采用多灵敏度方法以获得蒙特卡洛误差关于 $n$ 的最优多项式阶。此外，通过对已知均匀逼近算法进行去噪处理，我们改进了 $q < \infty$ 时的非自适应方法。