On inverse problems with unknown operators

Consider a problem of recovery of a smooth function (signal, image) f∈ℱ∈L₂([0, 1]^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_f^ε(t)=(Hf)(t)dt+εdW(t), t∈[0, 1]^d, ε>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 ε→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_l, Y(e_l )^σ), l=1, 2,…, n}, where {e_l} is an orthonormal system in L₂ and {Y(e_l)^σ} denote the solutions of stochastic differential equations of the above type with f=e_l and ε=σ is available. An optimal (in a minimax sense over considered operators and signals) data-driven recovery of the signal is suggested. The influence of ε, σ, 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