EXACT LOW-RANK MATRIX RECOVERY VIA NONCONVEX SCHATTEN p-MINIMIZATION

The low-rank matrix recovery (LMR) arises in many fields such as signal and image processing, quantum state tomography, magnetic resonance imaging, system identification and control, and it is generally NP-hard. Recently, Majumdar and Ward [Majumdar, A and RK Ward (2011). An algorithm for sparse MRI reconstruction by Schatten p-norm minimization. Magnetic Resonance Imaging, 29, 408–417]. had successfully applied nonconvex Schatten p-minimization relaxation of LMR in magnetic resonance imaging. In this paper, our main aim is to establish RIP theoretical result for exact LMR via nonconvex Schatten p-minimization. Carefully speaking, letting $\mathcal{A}$ be a linear transformation from ℝm×n into ℝs and r be the rank of recovered matrix X ∈ ℝm×n, and if $\mathcal{A}$ satisfies the RIP condition $\sqrt{2}\delta_{\max\{r+\lceil\frac{3}{2}k\rceil, 2k\}}+{(\frac{k}{2r})}^{\frac{1}{p}-\frac{1}{2}}\delta_{2r+k} <{ (\frac{k}{2r})}^{\frac{1}{p}-\frac{1}{2}}$ for a given positive integer k ∈ {1, 2, …, m – r}, then r-rank matrix can be exactly recovered. In particular, we obtain a uniform bound on restricted isometry constant $\delta_{4r}<\sqrt{2}-1$ for any p ∈ (0, 1] for LMR via Schatten p-minimization.