scholarly journals A multi-stage convex relaxation approach to noisy structured low-rank matrix recovery

2020 ◽  
Vol 12 (4) ◽  
pp. 569-602
Author(s):  
Shujun Bi ◽  
Shaohua Pan ◽  
Defeng Sun
Keyword(s):  
Convex Relaxation ◽  
Low Rank ◽  
Matrix Recovery ◽  
Multi Stage ◽  
Rank Matrix ◽  
Low Rank Matrix Recovery ◽  
Low Rank Matrix

scholarly journals Low-rank matrix recovery with Ky Fan 2-k-norm

2021 ◽  
Author(s):  
Xuan Vinh Doan ◽  
Stephen Vavasis
Keyword(s):  
Convex Relaxation ◽  
Low Rank ◽  
Optimization Approach ◽  
Proximal Point ◽  
Matrix Recovery ◽  
Rank Matrix ◽  
Low Rank Matrix Recovery ◽  
Low Rank Matrix ◽  
Ky Fan ◽  
Recovery Problem

AbstractLow-rank matrix recovery problem is difficult due to its non-convex properties and it is usually solved using convex relaxation approaches. In this paper, we formulate the non-convex low-rank matrix recovery problem exactly using novel Ky Fan 2-k-norm-based models. A general difference of convex functions algorithm (DCA) is developed to solve these models. A proximal point algorithm (PPA) framework is proposed to solve sub-problems within the DCA, which allows us to handle large instances. Numerical results show that the proposed models achieve high recoverability rates as compared to the truncated nuclear norm method and the alternating bilinear optimization approach. The results also demonstrate that the proposed DCA with the PPA framework is efficient in handling larger instances.

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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