Adaptive and non-adaptive randomized approximation of high-dimensional vectors

Abstract: 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.