B-splines and discretization in an inverse problem for Poisson processes
Journal of Multivariate Analysis
Deconvolution from panel data with unknown error distribution
Journal of Multivariate Analysis
Blind Recover Conditions for Images with Side Information
IEICE - Transactions on Information and Systems
Thresholding projection estimators in functional linear models
Journal of Multivariate Analysis
Application of second generation wavelets to blind spherical deconvolution
Journal of Multivariate Analysis
Hi-index | 754.84 |
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 dYfε(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 {(el, Y(el )σ), l=1, 2,…, n}, where {el} is an orthonormal system in L2 and {Y(el)σ} denote the solutions of stochastic differential equations of the above type with f=el 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