Spectral cut-off regularizations for ill-posed linear models

Abstract

This paper deals with recovering an unknown vector β from the noisy data Y = Xβ + σξ, where X is a known n × p matrix with n ≥ p and ξ is a standard white Gaussian noise. In order to estimate β, a spectral cut-off estimate βm̄(Y) with a data-driven cut-off frequency m̄(Y) is used. The cut-off frequency is selected as a minimizer of the unbiased risk estimate of the mean square prediction error, i.e., (Formula presented.). Assuming that β belongs to an ellipsoid W, we derive upper bounds for the maximal risk (Formula presented.) and show that βm̄}(Y) is a rate optimal minimax estimator overW. © 2014 Allerton Press, Inc.