EXACT LOW-RANK MATRIX RECOVERY VIA NONCONVEX SCHATTEN p-MINIMIZATION

2013 ◽  
Vol 30 (03) ◽  
pp. 1340010 ◽  
Author(s):  
LINGCHEN KONG ◽  
NAIHUA XIU
Keyword(s):  
Magnetic Resonance Imaging ◽  
Magnetic Resonance ◽  
Low Rank ◽  
Magnetic Resonance Imaging System ◽  
Resonance Imaging ◽  
Matrix Recovery ◽  
Rank Matrix ◽  
Restricted Isometry Constant ◽  
Low Rank Matrix Recovery ◽  
Low Rank Matrix

The low-rank matrix recovery (LMR) arises in many fields such as signal and image processing, quantum state tomography, magnetic resonance imaging, system identification and control, and it is generally NP-hard. Recently, Majumdar and Ward [Majumdar, A and RK Ward (2011). An algorithm for sparse MRI reconstruction by Schatten p-norm minimization. Magnetic Resonance Imaging, 29, 408–417]. had successfully applied nonconvex Schatten p-minimization relaxation of LMR in magnetic resonance imaging. In this paper, our main aim is to establish RIP theoretical result for exact LMR via nonconvex Schatten p-minimization. Carefully speaking, letting [Formula: see text] be a linear transformation from ℝm×n into ℝs and r be the rank of recovered matrix X ∈ ℝm×n, and if [Formula: see text] satisfies the RIP condition [Formula: see text] for a given positive integer k ∈ {1, 2, …, m – r}, then r-rank matrix can be exactly recovered. In particular, we obtain a uniform bound on restricted isometry constant [Formula: see text] for any p ∈ (0, 1] for LMR via Schatten p-minimization.

scholarly journals Guarantees of Fast Band Restricted Thresholding Algorithm for Low-Rank Matrix Recovery Problem

2020 ◽  
Vol 2020 ◽  
pp. 1-14
Author(s):  
Fujun Zhao ◽  
Jigen Peng ◽  
Kai Sun ◽  
Angang Cui
Keyword(s):  
Combinatorial Problem ◽  
Low Rank ◽  
Linear Measurements ◽  
Matrix Recovery ◽  
Rank Matrix ◽  
Restricted Isometry Constant ◽  
Wide Range ◽  
Thresholding Algorithm ◽  
Low Rank Matrix Recovery ◽  
Low Rank Matrix

Affine matrix rank minimization problem is a famous problem with a wide range of application backgrounds. This problem is a combinatorial problem and deemed to be NP-hard. In this paper, we propose a family of fast band restricted thresholding (FBRT) algorithms for low rank matrix recovery from a small number of linear measurements. Characterized via restricted isometry constant, we elaborate the theoretical guarantees in both noise-free and noisy cases. Two thresholding operators are discussed and numerical demonstrations show that FBRT algorithms have better performances than some state-of-the-art methods. Particularly, the running time of FBRT algorithms is much faster than the commonly singular value thresholding algorithms.

scholarly journals Low-Rank Matrix Denoising Algorithm-Based Magnetic Resonance Imaging Combined with Computed Tomography Images in the Diagnosis of Cerebral Aneurysm

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Daigui Zhang ◽  
Lihua Zhou ◽  
Tingdi Zhang ◽  
Shuai Wang ◽  
Yue Li
Keyword(s):  
Magnetic Resonance Imaging ◽  
Computed Tomography ◽  
Magnetic Resonance ◽  
Low Rank ◽  
Resonance Imaging ◽  
Ct Diagnosis ◽  
Rank Matrix ◽  
Mri Diagnosis ◽  
Low Rank Matrix ◽  
Better Than

This study was to analyze the diagnostic effects of computed tomography (CT) and magnetic resonance imaging (MRI) in patients with cerebrovascular diseases (CVDs) based on low-rank matrix denoising (LRMD) algorithm. The LRMD algorithm was adopted for MRI diagnosis and CT diagnosis for comparative analysis. 129 CVD patients were selected as the research objects, 43 cases were diagnosed by CT, 43 cases were diagnosed by MRI under LRMD, and the other 43 cases were diagnosed by CT + MRI. The results showed that the diagnostic compliance rates (DCRs) of CT group in the cerebral hemorrhage (CH), cerebral infarction (CI), and cerebral aneurysm (CA) were 95.1%, 94.7%, and 70%, respectively, while those in the MRI group were 99.01%, 97.71%, and 100%, respectively. Thus, it was obtained that MRI diagnosis was much better than CT diagnosis, and CT + MRI showed the best diagnosis efficacy, showing statistical differences ( P < 0.05 ). The accuracy, sensitivity, and specificity of MRI diagnosis under the LRMD algorithm were 96.28%, 88.76%, and 90.62%, respectively, which were superior to those of CT diagnosis (92.71%, 84.94%, and 80.71%, respectively). The diagnosis cost per case (DC/C) (799.73 ± 100.02 yuan) and the total diagnosis cost (TDC) (58,521.67 ± 301.62 yuan) in the MRI group were higher than those in the CT group (601.42 ± 83.61 yuan and 39,819.2 ± 198.72, respectively) ( P < 0.05 ). In conclusion, CT + MRI under the LRMD algorithm showed good potential in diagnosis of CVD; MRI based on the LRMD algorithm showed a higher positive rate in the diagnosis of CA and was better than CT diagnosis, and CT + MRI showed the best diagnosis effect and could improve the clinical diagnosis rate.

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