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.

Long-memory errors dramatically slow down the convergence of minimax risks in a fixed design nonparametric regression. The problem becomes even more complicated for the case of adaptive estimation. This defines the curse of long-memory errors. I show that using a random design, instead of a fixed one, allows one to overcome this curse and make familiar data-driven estimators robust. Moreover, the result holds for a wide class of nonstationary errors with bounded moments (including bounded deterministic errors). Possible extensions are discussed.

Consider a channel where a continuous periodic input signal is passed through a linear filter and then is contaminated by an additive noise. The problem is to recover this signal when we observe n repeated realizations of the output signal. Adaptive efficient procedures, that are asymptotically minimax over all possible procedures, are known for channels with Gaussian noise and no filter (the case of direct observation). Efficient procedures, based on the smoothness of a recovered signal, are known for the case of Gaussian noise. Robust rate-optimal procedures are known as well. However, there are no results on robust and efficient data-driven procedures; moreover, the known results for the case of direct observation indicate that even a small deviation from Gaussian noise may lead to a drastic change. We show that for the considered case of indirect data and a particular class of so-called supersmooth filters there exists a procedure of recovery of an input signal that possesses the desired properties; namely, it is: adaptive to the smoothness of the input signal; robust to the distribution of the noise; globally and pointwise-efficient, that is, its minimax global and pointwise risks converge with the best constant and rate over all possible estimators as n/spl rarr//spl infin/; and universal in the sense that for a wide class of linear (not necessarily bounded) operators the efficient estimator is a plug-in one. Furthermore, we explain how to employ the obtained asymptotic results for the practically important case of small n (large noise).