Average Complexity for Linear Problems in a Model with Varying Information Noise

Abstract

We study the minimal cost of information (called the information complexity) for approximating values S( f) of a linear operator S: F → G, where F is a Banach and G is a Hilbert space, based on noisy observations of linear functionals. The noise of each observation is Gaussian and its (known) variance σ 2 influences the cost via a nonincreasing cost function c(σ 2). Adaptive choice of the functionals L i and the variances of the noise, σ 2 i , is allowed. The error of approximation is equal to the average squared distance between exact and approximate solutions, with respect to the noise and Gaussian measure μ on F. We show that adaptive methods are not (much) better than nonadaptive methods, if the minimal information cost of obtaining approximation with accuracy √ϵ using only nonadaptive methods is an (almost) convex function of ϵ. The main complexity results are obtained in the case where the functionals L i are in the ball, || L i || 2 μ = ∫ F L 2 i ( f) μ( df) ≤ 1. In particular, we give formulas for the information complexity of multivariate function approximation in the case where μ is the folded Wiener sheet measure and the cost function is c(σ 2) = (1 + σ −2) α for some α ≥ 0.