The perturbation analysis of nonconvex low-rank matrix robust recovery

TL;DR

This paper introduces a perturbed nonconvex Schatten p-minimization method for low-rank matrix recovery under complete perturbations, providing theoretical guarantees and demonstrating superior performance over convex methods.

Contribution

It develops a new nonconvex Schatten p-minimization approach with theoretical recovery guarantees under complete perturbations, extending previous models.

Findings

The method guarantees recovery under certain RIP conditions.

Numerical results show better performance than convex nuclear norm minimization.

The analysis reveals optimal conditions for complete perturbation scenarios.

Abstract

In this paper, we bring forward a completely perturbed nonconvex Schatten $p$-minimization to address a model of completely perturbed low-rank matrix recovery. The paper that based on the restricted isometry property generalizes the investigation to a complete perturbation model thinking over not only noise but also perturbation, gives the restricted isometry property condition that guarantees the recovery of low-rank matrix and the corresponding reconstruction error bound. 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 $\delta_{2r}<1$ \cite{Recht et al 2010}. The numerical experiments are conducted to show better performance, and provides outperformance of the nonconvex Schatten $p$-minimization method comparing with the convex nuclear norm minimization approach in the completely perturbed scenario.