Exact Low-rank Matrix Recovery via Nonconvex Mp-Minimization

The low-rank matrix recovery (LMR) arises in many fields such as signal and image processing, statistics, computer vision, system identification and control, and it is NP-hard. It is known that under some restricted isometry property (RIP) conditions we can obtain the exact low-rank matrix solution by solving its convex relaxation, the nuclear norm minimization. In this paper, we consider the nonconvex relaxations by introducing Mp-norm (0 < p < 1) of a matrix and establish RIP conditions for exact LMR via Mp-minimization. Specifically, letting A be a linear transformation from Rm×n into R and r be the rank of recovered matrix X ∈ Rm×n, and if A satisfies the RIP condition √ 2δmax{r+ 2k,2k} + ( k 2r ) 1 p− 12 δ2r+k < ( k 2r ) 1 p− 1 2 for a given positive integer k ≤ m − r, then r-rank matrix can be exactly recovered. In particular, we not only obtain a uniform bound on restricted isometry constant δ4r < √ 2 − 1 for any p ∈ (0, 1] for LMR via Mp-minimization, but also obtain the one δ2r < √ 2− 1 for any p ∈ (0, 1] for sparse signal recovery via lp-minimization. AMS Subject Classification: 62B10, 90C26, 90C59