scholarly journals s-Goodness for Low-Rank Matrix Recovery

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Lingchen Kong ◽  
Levent Tunçel ◽  
Naihua Xiu
Keyword(s):  
Linear Transformation ◽  
Optimization Problems ◽  
Sufficient Conditions ◽  
Nuclear Norm ◽  
Low Rank ◽  
Matrix Recovery ◽  
Rank Matrix ◽  
Necessary And Sufficient ◽  
Low Rank Matrix Recovery ◽  
Low Rank Matrix

Low-rank matrix recovery (LMR) is a rank minimization problem subject to linear equality constraints, and it arises in many fields such as signal and image processing, statistics, computer vision, and system identification and control. This class of optimization problems is generally𝒩𝒫hard. A popular approach replaces the rank function with the nuclear norm of the matrix variable. In this paper, we extend and characterize the concept ofs-goodness for a sensing matrix in sparse signal recovery (proposed by Juditsky and Nemirovski (Math Program, 2011)) to linear transformations in LMR. Using the two characteristics-goodness constants,γsandγ^s, of a linear transformation, we derive necessary and sufficient conditions for a linear transformation to bes-good. Moreover, we establish the equivalence ofs-goodness and the null space properties. Therefore,s-goodness is a necessary and sufficient condition for exacts-rank matrix recovery via the nuclear norm minimization.

scholarly journals Truncated Nuclear Norm Minimization for Image Restoration Based on Iterative Support Detection

2014 ◽  
Vol 2014 ◽  
pp. 1-17 ◽  
Author(s):  
Yilun Wang ◽  
Xinhua Su
Keyword(s):  
Convex Relaxation ◽  
Nuclear Norm ◽  
Low Rank ◽  
Nuclear Norm Minimization ◽  
Matrix Recovery ◽  
Norm Minimization ◽  
Rank Matrix ◽  
Low Rank Matrix Recovery ◽  
Low Rank Matrix ◽  
Truncated Nuclear Norm

Recovering a large matrix from limited measurements is a challenging task arising in many real applications, such as image inpainting, compressive sensing, and medical imaging, and these kinds of problems are mostly formulated as low-rank matrix approximation problems. Due to the rank operator being nonconvex and discontinuous, most of the recent theoretical studies use the nuclear norm as a convex relaxation and the low-rank matrix recovery problem is solved through minimization of the nuclear norm regularized problem. However, a major limitation of nuclear norm minimization is that all the singular values are simultaneously minimized and the rank may not be well approximated (Hu et al., 2013). Correspondingly, in this paper, we propose a new multistage algorithm, which makes use of the concept of Truncated Nuclear Norm Regularization (TNNR) proposed by Hu et al., 2013, and iterative support detection (ISD) proposed by Wang and Yin, 2010, to overcome the above limitation. Besides matrix completion problems considered by Hu et al., 2013, the proposed method can be also extended to the general low-rank matrix recovery problems. Extensive experiments well validate the superiority of our new algorithms over other state-of-the-art methods.

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