The Perturbation Analysis of Nonconvex Low-Rank Matrix Robust Recovery

In this article, we bring forward a completely perturbed nonconvex Schatten
$p$
-minimization to address a model of completely perturbed low-rank matrix recovery (LRMR). This article based on the restricted isometry property (RIP) and the Schatten-
$p$
null space property (NSP) generalizes the investigation to a complete perturbation model thinking over not only noise but also perturbation, and it gives the RIP condition and the Schatten-
$p$
NSP assumption that guarantee the recovery of low-rank matrix and the corresponding reconstruction error bounds. In particular, the analysis of the result reveals that in the case that
$p$
decreases
$0$
and
$a>1$
for the complete perturbation and low-rank matrix, the condition is the optimal sufficient condition
$\delta_{2r}<1$
(Recht et al., 2010). In addition, we study the connection between RIP and Schatten-
$p$
NSP and discern that Schatten-
$p$
NSP can be inferred from the RIP. The numerical experiments are conducted to show better performance and provide outperformance of the nonconvex Schatten
$p$
-minimization method comparing with the convex nuclear norm minimization approach in the completely perturbed scenario.