Randomized approximation of summable sequences — adaptive and non-adaptive

Abstract

We prove lower bounds for the randomized approximation of the embedding ℓ1m↪ℓ∞m based on algorithms that use arbitrary linear (hence non-adaptive) information provided by a (randomized) measurement matrix N∈Rn×m. These lower bounds reflect the increasing difficulty of the problem for m→∞, namely, a term logm 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 (loglogm)-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 n1/2(logn)−1/2.