Randomized approximation of summable sequences – adaptive and non-adaptive
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We prove lower bounds for the randomized approximation of the embedding ℓ_1^m →ℓ_∞^m based on algorithms that use arbitrary linear (hence non-adaptive) information provided by a (randomized) measurement matrix N ∈ℝ^n × m. These lower bounds reflect the increasing difficulty of the problem for m →∞, namely, a term √(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 (loglog 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.
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