Adaptive spectral regularizations of high dimensional linear models

Abstract

This paper focuses on recovering an unknown vector β from the noisy data Y=Xβ+σξ, where X is a known n×p-matrix, ξ is a standard white Gaussian noise, and σ is an unknown noise level. In order to estimate β, a spectral regularization method is used, and our goal is to choose its regularization parameter with the help of the data Y. In this paper, we deal solely with regularization methods based on the so-called ordered smoothers (see [13]) and extend the oracle inequality from [11] to the case, where the noise level is unknown.