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Low-rank matrix recovery via regularized nuclear norm minimization
Applied and Computational Harmonic Analysis  (IF3.055),  Pub Date : 2021-03-09, DOI: 10.1016/j.acha.2021.03.001
Wendong Wang, Feng Zhang, Jianjun Wang

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 $\mathbit{b}=\mathcal{A}\left(X\right)+\mathbit{n}$ with a bounded constraint ${‖\mathbit{n}‖}_{2}\le ϵ$, 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.