On inverse problems with unknown operators

Consider a problem of recovery of a smooth function (signal, image) f is an element of F is an element of L-2 ([0, 1](d)) passed through an unknown fi lter and then contaminated by a noise. A typical model discussed in the pap er is described by a stochastic differential equation dY(f)(epsilon) (t) = (Hf)(t) dt + epsilon dW(t), t is an element of [0,1](d ), epsilon > 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 epsilon --> 0) minimax mean integrated squared error. Traditionally, the pr oblem is studied under the assumption that the operator H is known, then th e ill-posedness of the problem is the main concern. In this paper, a more c omplicated and more realistic case is considered where the operator is unkn own; instead, a training set of n pairs f {(e(l), Y-el(sigma)), l = 1, 2,.. ., n}, where {e(l)} is an orthonormal system in L-2 and {Y-el(sigma)} denot e the solutions of stochastic differential equations of the above type with f = epsilon (l) and epsilon = sigma is available. An optimal (in a minimax sense over considered operators and signals) data-driven recovery of the s ignal is suggested. The influence of epsilon, sigma, and n on the recovery is thoroughly studied; in particular, we discuss an interesting case of a l arger noise during the training and present formulas for threshold levels f or n beyond which no improvement in recovery of input signals occurs. We al so 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.