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 $\hat \beta ^{\bar m} (Y)$ with a data-driven cut-off frequency $\bar m(Y)$ is used. The cut-off frequency is selected as a minimizer of the unbiased risk estimate of the mean square prediction error, i.e., $\bar m = \arg \min _m \left\{ {\left\| {Y - X\hat \beta ^m \left( Y \right)} \right\|^2 + 2\sigma ^2 m} \right\}$. Assuming that β belongs to an ellipsoid W, we derive upper bounds for the maximal risk $\sup _{\beta \in \mathbb{W}} E\left\| {\hat \beta ^{\bar m} \left( Y \right) - \beta } \right\|^2$ and show that $\hat \beta ^{\bar m} \left( Y \right)$ is a rate optimal minimax estimator overW.