Generalized Nonconvex Nonsmooth Low-Rank Minimization

The low-rank matrix recovery is an important machine learning research topic with various scientific applications. Most existing low-rank matrix recovery methods relax the rank minimization problem via the trace norm minimization. However, such a relaxation makes the solution seriously deviate from the original one. Meanwhile, most matrix recovery methods minimize the squared prediction errors on the observed entries, which is sensitive to outliers. In this paper, we propose a new robust matrix recovery model to address the above two challenges. The joint capped trace norm and capped $\ell_1$-norm are used to tightly approximate the rank minimization and enhance the robustness to outliers. The evaluation experiments are performed on both synthetic data and real world applications in collaborative filtering and social network link prediction. All empirical results show our new method outperforms the existing matrix recovery methods.

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Nonconvex Nonsmooth Low Rank Minimization via Iteratively Reweighted Nuclear Norm

IEEE Transactions on Image Processing ◽

10.1109/tip.2015.2511584 ◽

2016 ◽

Vol 25 (2) ◽

pp. 829-839 ◽

Cited By ~ 94

Author(s):

Canyi Lu ◽

Jinhui Tang ◽

Shuicheng Yan ◽

Zhouchen Lin

Keyword(s):

Nuclear Norm ◽

Low Rank ◽

Rank Minimization

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Line survey joint denoising via low-rank minimization

Geophysics ◽

10.1190/geo2018-0141.1 ◽

2019 ◽

Vol 84 (1) ◽

pp. V21-V32 ◽

Cited By ~ 1

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.

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Nonconvex Nonsmooth Low-Rank Minimization for Peneralized Image Compressed Sensing via Group Sparse Representation

10.36227/techrxiv.11798424 ◽

2020 ◽

Author(s):

Yunyi Li ◽

Li Liu ◽

Yu Zhao ◽

Xiefeng Cheng ◽

Guan Gui

Keyword(s):

Sparse Representation ◽

Minimization Problem ◽

Singular Values ◽

Nuclear Norm ◽

Low Rank ◽

Rank Minimization ◽

Group Matrix ◽

Reconstruction Performance ◽

Generalized Framework ◽

Surrogate Function

Group sparse representation (GSR) based method has led to great successes in various image recovery tasks, which can be converted into a low-rank matrix minimization problem. As a widely used surrogate function of low-rank, the nuclear norm based convex surrogate usually leads to over-shrinking problem, since the standard soft-thresholding operator shrinks all singular values equally. To improve traditional sparse representation based image compressive sensing (CS) performance, we propose a generalized CS framework based on GSR model, leading to a nonconvex nonsmooth low-rank minimization problem. The popular -norm and M-estimator are employed for standard image CS and robust CS problem to fit the data respectively. For the better approximation of the rank of group-matrix, a family of nuclear norms are employed to address the over-shrinking problem. Moreover, we also propose a flexible and effective iteratively-weighting strategy to control the weighting and contribution of each singular value. Then we develop an iteratively reweighted nuclear norm algorithm for our generalized framework via an alternating direction method of multipliers framework, namely, GSR-ADMM-IRNN. Experimental results demonstrate that our proposed CS framework can achieve favorable reconstruction performance compared with current state-of-the-art methods and the RCS framework can suppress the outliers effectively.

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