scholarly journals Restricted Isometry Property of Principal Component Pursuit with Reduced Linear Measurements

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Qingshan You ◽  
Qun Wan ◽  
Haiwen Xu
Keyword(s):  
Sparse Matrix ◽  
Principal Component ◽  
Restricted Isometry Property ◽  
Low Rank ◽  
Linear Measurements ◽  
Matrix Recovery ◽  
Strongly Convex ◽  
Rank Matrix ◽  
Principal Component Pursuit ◽  
Low Rank Matrix Recovery

The principal component prsuit with reduced linear measurements (PCP_RLM) has gained great attention in applications, such as machine learning, video, and aligning multiple images. The recent research shows that strongly convex optimization for compressive principal component pursuit can guarantee the exact low-rank matrix recovery and sparse matrix recovery as well. In this paper, we prove that the operator of PCP_RLM satisfies restricted isometry property (RIP) with high probability. In addition, we derive the bound of parameters depending only on observed quantities based on RIP property, which will guide us how to choose suitable parameters in strongly convex programming.

Principal Component Pursuit with Weighted Nuclear Norm

2014 ◽  
Vol 513-517 ◽  
pp. 1722-1726
Author(s):  
Qing Shan You ◽  
Qun Wan
Keyword(s):  
Sparse Matrix ◽  
Principal Component ◽  
Low Rank ◽  
Linear Measurements ◽  
Alternating Direction ◽  
Rank Matrix ◽  
Small Set ◽  
Principal Component Pursuit ◽  
Low Dimensional ◽  
Low Rank Matrix

Principal Component Pursuit (PCP) recovers low-dimensional structures from a small set of linear measurements, such as low rank matrix and sparse matrix. Pervious works mainly focus on exact recovery without additional noise. However, in many applications the observed measurements are corrupted by an additional white Gaussian noise (AWGN). In this paper, we model the recovered matrix the sum a low-rank matrix, a sparse matrix and an AWGN. We propose a weighted PCP for the recovery matrix, which is solved by alternating direction method. Numerical results show that the reconstructions performance of weighted PCP outperforms the classical PCP in term of accuracy.

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.

Research on Background Modeling Based on Low-Rank Matrix Recovery

2014 ◽  
Vol 635-637 ◽  
pp. 1056-1059 ◽  
Author(s):  
Bao Yan Wang ◽  
Xin Gang Wang
Keyword(s):  
Sparse Matrix ◽  
Background Modeling ◽  
Low Rank ◽  
Complex Scene ◽  
Matrix Recovery ◽  
Rank Matrix ◽  
Training Samples ◽  
The Individual ◽  
Low Rank Matrix Recovery ◽  
Low Rank Matrix

Key and difficult points of background subtraction method lie in looking for an ideal background modeling under complex scene. Stacking the individual frames as columns of a big matrix, background parts can be viewed as a low-rank background matrix because of large similarity among individual frames, yet foreground parts can be viewed as a sparse matrix as foreground parts play a small role in individual frames. Thus the process of video background modeling is in fact a process of low-rank matrix recovery. Background modeling based on low-rank matrix recovery can separate foreground images from background at the same time without pre-training samples, besides, the approach is robust to illumination changes. However, there exist some shortcomings in background modeling based on low-rank matrix recovery by analyzing numerical experiments, which is developed from three aspects.

Minimization of the difference of Nuclear and Frobenius norms for noisy low rank matrix recovery

2019 ◽  
Vol 18 (02) ◽  
pp. 1950056
Author(s):  
Yun Cai
Keyword(s):  
Linear Measurement ◽  
Low Rank ◽  
Bounded Noise ◽  
Linear Measurements ◽  
Minimization Method ◽  
Matrix Recovery ◽  
Rank Matrix ◽  
The Difference ◽  
Low Rank Matrix Recovery ◽  
Low Rank Matrix

This paper considers recovery of matrices that are low rank or approximately low rank from linear measurements corrupted with additive noise. We study minimization of the difference of Nuclear and Frobenius norms (abbreviated as [Formula: see text] norm) as a nonconvex and Lipschitz continuous metric for solving this noisy low rank matrix recovery problem. We mainly study two types of bounded observation noisy low rank matrix recovery problems, including the [Formula: see text]-norm bounded noise and the Dantizg Selector noise. Based on the matrix restricted isometry property (abbreviated as M-RIP), we prove that this [Formula: see text] norm-based minimization method can stably recover a (approximately) low rank matrix in the two types bounded noisy low rank matrix recovery problems. In addition, we use the truncated difference of Nuclear and Frobenius norms (denoted as the truncated [Formula: see text] norm) to recover a low rank matrix when the observation noise is the Dantizg Selector noise. We give the stable recovery result for this truncated [Formula: see text] norm minimization in Dantizg Selector noise case when the linear measurement map satisfies the M-RIP condition.

Low rank matrix recovery with adversarial sparse noise

2021 ◽  
Author(s):  
Hang Xu ◽  
Song Li ◽  
Junhong Lin
Keyword(s):  
Data Science ◽  
Low Rank ◽  
Bounded Noise ◽  
Linear Measurements ◽  
Absolute Deviation ◽  
Matrix Recovery ◽  
Rank Matrix ◽  
Iterative Hard Thresholding ◽  
Low Rank Matrix Recovery ◽  
Low Rank Matrix

Abstract Many problems in data science can be treated as recovering a low-rank matrix from a small number of random linear measurements, possibly corrupted with adversarial noise and dense noise. Recently, a bunch of theories on variants of models have been developed for different noises, but with fewer theories on the adversarial noise. In this paper, we study low-rank matrix recovery problem from linear measurements perturbed by $\ell_1$-bounded noise and sparse noise that can arbitrarily change an adversarially chosen $\omega$-fraction of the measurement vector. For Gaussian measurements with nearly optimal number of measurements, we show that the nuclear-norm constrained least absolute deviation (LAD) can successfully estimate the ground-truth matrix for any $\omega<0.239$. Similar robust recovery results are also established for an iterative hard thresholding algorithm applied to the rank-constrained LAD considering geometrically decaying step-sizes, and the unconstrained LAD based on matrix factorization as well as its subgradient descent solver.

scholarly journals Automated Tire Visual Inspection Based on Low Rank Matrix Recovery

2020 ◽  
Author(s):  
Guangxu Li ◽  
Zhouzhou Zheng ◽  
Yuyi Shao ◽  
Jinyue Shen ◽  
Yan Zhang
Keyword(s):  
Visual Inspection ◽  
Sparse Matrix ◽  
Industrial Applications ◽  
Convergence Speed ◽  
Low Rank ◽  
Inspection System ◽  
Matrix Recovery ◽  
Rank Matrix ◽  
Low Rank Matrix Recovery ◽  
Low Rank Matrix

Abstract Visual inspection is a challenging and widely employed process in industries. In this work, an automated tire visual inspection system is proposed based on low rank matrix recovery. Deep Network is employed to perform texture segmentation which benefits low rank decomposition in both quality and computational efficiency. We propose a dual optimization method to improve convergence speed and matrix sparsity by incorporating the improvement of the soft-threshold shrinkage operator by the weight matrix M. We investigated how incremental multiplier affects the decomposition accuracy and the convergence speed of the algorithm. On this basis, image blocks were decomposed into low-rank matrix and sparse matrix in which defects were separated. Comparative experiments have been performed on our dataset. Experimental results validate the theoretical analysis. The method is promising in false alarm, robustness and running time based on multi-core processor distributed computing. It can be extended to other real-time industrial applications.

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