Minimization of the difference of Nuclear and Frobenius norms for noisy low rank matrix recovery

Abstract

This paper considers recovery of matrices that are low rank or approximately low rank from linear measurements corrupted with additive noise. We study minimization of the difference of Nuclear and Frobenius norms (abbreviated as [Formula: see text] norm) as a nonconvex and Lipschitz continuous metric for solving this noisy low rank matrix recovery problem. We mainly study two types of bounded observation noisy low rank matrix recovery problems, including the [Formula: see text]-norm bounded noise and the Dantizg Selector noise. Based on the matrix restricted isometry property (abbreviated as M-RIP), we prove that this [Formula: see text] norm-based minimization method can stably recover a (approximately) low rank matrix in the two types bounded noisy low rank matrix recovery problems. In addition, we use the truncated difference of Nuclear and Frobenius norms (denoted as the truncated [Formula: see text] norm) to recover a low rank matrix when the observation noise is the Dantizg Selector noise. We give the stable recovery result for this truncated [Formula: see text] norm minimization in Dantizg Selector noise case when the linear measurement map satisfies the M-RIP condition.