Accelerated reweighted nuclear norm minimization algorithm for low rank matrix recovery

2015 ◽  
Vol 114 ◽  
pp. 24-33 ◽  
Author(s):  
Xiaofan Lin ◽  
Gang Wei
Keyword(s):  
Nuclear Norm ◽  
Low Rank ◽  
Minimization Algorithm ◽  
Nuclear Norm Minimization ◽  
Matrix Recovery ◽  
Norm Minimization ◽  
Rank Matrix ◽  
Low Rank Matrix Recovery ◽  
Low Rank Matrix

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.

scholarly journals Recovery analysis of damped spectrally sparse signals and its relation to MUSIC

2020 ◽  
Author(s):  
Shuang Li ◽  
Hassan Mansour ◽  
Michael B Wakin
Keyword(s):  
Classical Music ◽  
Frequency Estimation ◽  
Low Rank ◽  
Sparse Signals ◽  
Music Algorithm ◽  
Matrix Recovery ◽  
Norm Minimization ◽  
Rank Matrix ◽  
Low Rank Matrix Recovery ◽  
Low Rank Matrix

Abstract One of the classical approaches for estimating the frequencies and damping factors in a spectrally sparse signal is the MUltiple SIgnal Classification (MUSIC) algorithm, which exploits the low-rank structure of an autocorrelation matrix. Low-rank matrices have also received considerable attention recently in the context of optimization algorithms with partial observations, and nuclear norm minimization (NNM) has been widely used as a popular heuristic of rank minimization for low-rank matrix recovery problems. On the other hand, it has been shown that NNM can be viewed as a special case of atomic norm minimization (ANM), which has achieved great success in solving line spectrum estimation problems. However, as far as we know, the general ANM (not NNM) considered in many existing works can only handle frequency estimation in undamped sinusoids. In this work, we aim to fill this gap and deal with damped spectrally sparse signal recovery problems. In particular, inspired by the dual analysis used in ANM, we offer a novel optimization-based perspective on the classical MUSIC algorithm and propose an algorithm for spectral estimation that involves searching for the peaks of the dual polynomial corresponding to a certain NNM problem, and we show that this algorithm is in fact equivalent to MUSIC itself. Building on this connection, we also extend the classical MUSIC algorithm to the missing data case. We provide exact recovery guarantees for our proposed algorithms and quantify how the sample complexity depends on the true spectral parameters. In particular, we provide a parameter-specific recovery bound for low-rank matrix recovery of jointly sparse signals rather than use certain incoherence properties as in existing literature. Simulation results also indicate that the proposed algorithms significantly outperform some relevant existing methods (e.g., ANM) in frequency estimation of damped exponentials.

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.

Line survey joint denoising via low-rank minimization

Geophysics ◽  
2019 ◽  
Vol 84 (1) ◽  
pp. V21-V32 ◽  
Author(s):  
Zhao Liu ◽  
Jianwei Ma ◽  
Xueshan Yong
Keyword(s):  
Minimization Problem ◽  
Nuclear Norm ◽  
Low Rank ◽  
Single Shot ◽  
Rank Minimization ◽  
Nuclear Norm Minimization ◽  
Norm Minimization ◽  
Rank Matrix ◽  
Line Survey ◽  
Low Rank Matrix

Prestack seismic data denoising is an important step in seismic processing due to the development of prestack time migration. Reduced-rank filtering is a state-of-the-art method for prestack seismic denoising that uses predictability between neighbor traces for each single frequency. Different from the original way of embedding low-rank matrix based on the Hankel or Toeplitz transform, we have developed a new multishot gathers joint denoising method in a line survey, which used a new way of rearranging data to a matrix with low rank. Inspired by video denoising, each single-shot record in the line survey can be viewed as a frame in the video sequence. Due to high redundancy and similar event structure among the shot gathers, similar patches can be selected from different shot gathers in the line survey to rearrange a low-rank matrix. Then, seismic denoising is formulated into a low-rank minimization problem that can be further relaxed into a nuclear-norm minimization problem. A fast algorithm, called the orthogonal rank-one matrix pursuit, is used to solve the nuclear-norm minimization. Using this method avoids the computation of a full singular value decomposition. Our method is validated using synthetic and field data, in comparison with [Formula: see text] deconvolution and singular spectrum analysis methods.

scholarly journals Weighted Nuclear Norm Minimization on Multimodality Clustering

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Lei Du ◽  
Songsong Dai ◽  
Haifeng Song ◽  
Yuelong Chuang ◽  
Yingying Xu
Keyword(s):  
Spectral Clustering ◽  
Clustering Algorithm ◽  
Transition Probability ◽  
Nuclear Norm ◽  
Low Rank ◽  
Nuclear Norm Minimization ◽  
Norm Minimization ◽  
Rank Matrix ◽  
Low Rank Matrix ◽  
Weighted Nuclear Norm Minimization

Generally, multimodality data contain different potential information available and are capable of providing an enhanced analytical result compared to monosource data. The way to combine the data plays a crucial role in multimodality data analysis which is worth investigating. Multimodality clustering, which seeks a partition of the data in multiple views, has attracted considerable attention, for example, robust multiview spectral clustering (RMSC) explicitly handles the possible noise in the transition probability matrices associated with different views. Spectral clustering algorithm embeds the input data into a low-dimensional representation by dividing the clustering problem into k subproblems, and the corresponding eigenvalue reflects the loss of each subproblem. So, the eigenvalues of the Laplacian matrix should be treated differently, while RMSC regularizes each singular value equally when recovering the low-rank matrix. In this paper, we propose a multimodality clustering algorithm which recovers the low-rank matrix by weighted nuclear norm minimization. We also propose a method to evaluate the weight vector by learning a shared low-rank matrix. In our experiments, we use several real-world datasets to test our method, and experimental results show that the proposed method has a better performance than other baselines.

An Algorithm of Schatten ρ-Norm Regularized Least Squares Problems for Video Restoration

2013 ◽  
Vol 718-720 ◽  
pp. 2308-2313
Author(s):  
Lu Liu ◽  
Wei Huang ◽  
Di Rong Chen
Keyword(s):  
Convex Relaxation ◽  
Nuclear Norm ◽  
Low Rank ◽  
Rank Minimization ◽  
Nuclear Norm Minimization ◽  
Video Restoration ◽  
Norm Minimization ◽  
Rank Matrix ◽  
Low Rank Matrix ◽  
Nonconvex Relaxation

Minimizing the nuclear norm is recently considered as the convex relaxation of the rank minimization problem and arises in many applications as Netflix challenge. A closest nonconvex relaxation - Schatten norm minimization has been proposed to replace the NP hard rank minimization. In this paper, an algorithm based on Majorization Minimization has be proposed to solve Schatten norm minimization. The numerical experiments show that Schatten norm with recovers low rank matrix from fewer measurements than nuclear norm minimization. The numerical results also indicate that our algorithm give a more accurate reconstruction

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