On inverse problems with unknown operators

Consider a problem of recovery of a smooth function (signal, image) f/spl isin//spl Fscr//spl isin/L/sub 2/([0, 1]/sup d/) passed through an unknown filter and then contaminated by a noise. A typical model discussed in the paper is described by a stochastic differential equation dY/sub f//sup /spl epsi//(t)=(Hf)(t)dt+/spl epsi/dW(t), t/spl isin/[0, 1]/sup d/, /spl epsi/>0 where H is a linear operator modeling the filter and W is a Brownian motion (sheet) modeling a noise. The aim is to recover f with asymptotically (as /spl epsi//spl rarr/0) minimax mean integrated squared error. Traditionally, the problem is studied under the assumption that the operator H is known, then the ill-posedness of the problem is the main concern. In this paper, a more complicated and more realistic case is considered where the operator is unknown; instead, a training set of n pairs {(e/sub l/, Y(e/sub l/)/sup /spl sigma//), l=1, 2,..., n}, where {e/sub l/} is an orthonormal system in L/sub 2/ and {Y(e/sub l/)/sup /spl sigma//} denote the solutions of stochastic differential equations of the above type with f=e/sub l/ and /spl epsi/=/spl sigma/ is available. An optimal (in a minimax sense over considered operators and signals) data-driven recovery of the signal is suggested. The influence of /spl epsi/, /spl sigma/, and n on the recovery is thoroughly studied; in particular, we discuss an interesting case of a larger noise during the training and present formulas for threshold levels for n beyond which no improvement in recovery of input signals occurs. We also discuss the case where H is an unknown perturbation of a known operator. We describe a class of perturbations for which the accuracy of recovery of the signal is asymptotically the same (up to a constant) as in the case of precisely known operator.