In this paper, we theoretically investigate the low-rank matrix recovery problem in the context of the unconstrained regularized nuclear norm minimization (RNNM) framework. Our theoretical findings show that, the RNNM method is able to provide a robust recovery of any matrix X (not necessary to be exactly low-rank) from its few noisy measurements $\mathit{b}=\mathcal{A}(X)+\mathit{n}$ with a bounded constraint ${\Vert \mathit{n}\Vert }_2\le ϵ$, provided that the tk-order restricted isometry constant (RIC) of $\mathcal{A}$ satisfies a certain constraint related to $t>0$. Specifically, the obtained recovery condition in the case of $t>4/3$ is found to be same with the sharp condition established previously by Cai and Zhang [10] to guarantee the exact recovery of any rank-k matrix via the constrained nuclear norm minimization method. More importantly, to the best of our knowledge, we are the first to establish the tk-order RIC based coefficient estimate of the robust null space property in the case of $0<t\le 1$.

在本文中，我们从理论上研究了无约束正则化核规范最小化（RNNM）框架下的低秩矩阵恢复问题。我们的理论发现表明，RNNM方法能够从其很少的噪声测量中提供任何矩阵X的稳健恢复（不一定是精确的低秩）。$\mathit{b}=\mathcal{一个}（X）+\mathit{ñ}$ 有限制的约束 ${\Vert \mathit{ñ}\Vert }_{\mathrm{2个}}\le ϵ$，前提是的tk阶受限等距常数（RIC）为$\mathcal{一个}$ 满足与 $Ť>0$。具体而言，在以下情况下获得的恢复条件$Ť>4/3$发现与蔡和张[10]先前建立的尖锐条件相同，以通过约束核范数最小化方法来保证任何等级k矩阵的精确恢复。更重要的是，据我们所知，在以下情况下，我们是第一个建立基于tk阶RIC的鲁棒零空间属性系数估计的方法$0<Ť\le \mathrm{1个}$。