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

TL;DR

This paper investigates randomized algorithms for approximating high-dimensional vectors in different $\, ext{ell}_p$ to $ ext{ell}_q$ embeddings, achieving near-optimal complexity bounds with adaptive and non-adaptive methods.

Contribution

It introduces adaptive algorithms with complexity depending only logarithmically on the dimension and improves non-adaptive methods for $ ext{ell}_q$ approximation using denoising techniques.

Findings

Adaptive methods achieve $( ext{log} ext{log} ext{m})$-dependence in complexity.

Optimal polynomial order in $n$ for Monte Carlo error with $q< ext{infty}$.

Enhanced non-adaptive algorithms for $q< ext{infty}$ via denoising.

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.