New RIC Bounds for Low Rank Matrix Recovery via Schatten-p Minimization with 0 < p < 1

Abstract

Low rank matrix recovery is widely applicable in many fields and has been the focus of much research recently. Schatten-p (0 < p ≤ 1) norm minimization are two main recovery methods. For case p = 1, i.e. nuclear norm minimization, there have been abundant results on exact and stable recovery conditions. The restricted isometry constants (RIC) play an important role in exact recovery theory via nuclear norm minimization. However, there are few results on Schatten-p (0 < p < 1) norm minimization. In this paper, we establish new RIC bounds to guarantee exact recovering low rank matrices via Schatten-p (0 < p < 1) norm minimization. First, we provide the RIC bound of δ[g(p,2r)](t-1) + 2r to guarantee exact recovery. Especially, the exact recovery can be succeed via Schatten-1/2 minimization if for any t > 1. Second, we obtain the RIC bound of δ2r to guarantee exact recovery. For Schatten-1/2 minimization, the bound is δ2r < 1/3. Moreover, we get the stable recovery results with bounded noise.