Weighted Nuclear Norm Minimization and Its Applications to Low Level Vision
As a convex relaxation of the rank minimization model, the nuclear norm minimization (NNM) problem has been attracting significant research interest in recent years. The standard NNM regularizes each singular value equally, composing an easily calculated convex norm. However, this restricts its capa...
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| Published in: | International journal of computer vision Vol. 121; no. 2; pp. 183 - 208 |
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| Main Authors: | , , , , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
New York
Springer US
01.01.2017
Springer Springer Nature B.V |
| Subjects: | |
| ISSN: | 0920-5691, 1573-1405 |
| Online Access: | Get full text |
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| Abstract | As a convex relaxation of the rank minimization model, the nuclear norm minimization (NNM) problem has been attracting significant research interest in recent years. The standard NNM regularizes each singular value equally, composing an easily calculated convex norm. However, this restricts its capability and flexibility in dealing with many practical problems, where the singular values have clear physical meanings and should be treated differently. In this paper we study the weighted nuclear norm minimization (WNNM) problem, which adaptively assigns weights on different singular values. As the key step of solving general WNNM models, the theoretical properties of the weighted nuclear norm proximal (WNNP) operator are investigated. Albeit nonconvex, we prove that WNNP is equivalent to a standard quadratic programming problem with linear constrains, which facilitates solving the original problem with off-the-shelf convex optimization solvers. In particular, when the weights are sorted in a non-descending order, its optimal solution can be easily obtained in closed-form. With WNNP, the solving strategies for multiple extensions of WNNM, including robust PCA and matrix completion, can be readily constructed under the alternating direction method of multipliers paradigm. Furthermore, inspired by the reweighted sparse coding scheme, we present an automatic weight setting method, which greatly facilitates the practical implementation of WNNM. The proposed WNNM methods achieve state-of-the-art performance in typical low level vision tasks, including image denoising, background subtraction and image inpainting. |
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| AbstractList | As a convex relaxation of the rank minimization model, the nuclear norm minimization (NNM) problem has been attracting significant research interest in recent years. The standard NNM regularizes each singular value equally, composing an easily calculated convex norm. However, this restricts its capability and flexibility in dealing with many practical problems, where the singular values have clear physical meanings and should be treated differently. In this paper we study the weighted nuclear norm minimization (WNNM) problem, which adaptively assigns weights on different singular values. As the key step of solving general WNNM models, the theoretical properties of the weighted nuclear norm proximal (WNNP) operator are investigated. Albeit nonconvex, we prove that WNNP is equivalent to a standard quadratic programming problem with linear constrains, which facilitates solving the original problem with off-the-shelf convex optimization solvers. In particular, when the weights are sorted in a non-descending order, its optimal solution can be easily obtained in closed-form. With WNNP, the solving strategies for multiple extensions of WNNM, including robust PCA and matrix completion, can be readily constructed under the alternating direction method of multipliers paradigm. Furthermore, inspired by the reweighted sparse coding scheme, we present an automatic weight setting method, which greatly facilitates the practical implementation of WNNM. The proposed WNNM methods achieve state-of-the-art performance in typical low level vision tasks, including image denoising, background subtraction and image inpainting. |
| Audience | Academic |
| Author | Meng, Deyu Zuo, Wangmeng Feng, Xiangchu Gu, Shuhang Xie, Qi Zhang, Lei |
| Author_xml | – sequence: 1 givenname: Shuhang surname: Gu fullname: Gu, Shuhang organization: Department of Computing, The Hong Kong Polytechnic University – sequence: 2 givenname: Qi surname: Xie fullname: Xie, Qi organization: School of Mathematics and Statistics, Xi’an Jiaotong University – sequence: 3 givenname: Deyu surname: Meng fullname: Meng, Deyu organization: School of Mathematics and Statistics, Xi’an Jiaotong University – sequence: 4 givenname: Wangmeng surname: Zuo fullname: Zuo, Wangmeng organization: School of Computer Science and Technology, Harbin Institute of Technology – sequence: 5 givenname: Xiangchu surname: Feng fullname: Feng, Xiangchu organization: Department of Applied Mathematics, Xidian University – sequence: 6 givenname: Lei surname: Zhang fullname: Zhang, Lei email: cslzhang@comp.polyu.edu.hk organization: Department of Computing, The Hong Kong Polytechnic University |
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| SubjectTerms | Artificial Intelligence Computer Imaging Computer Science Convexity Image Processing and Computer Vision Low level Mathematical analysis Mathematical models Minimization Noise reduction Norms Optimization Pattern Recognition Pattern Recognition and Graphics Quadratic programming Solvers State of the art Subtraction Vision Weight |
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| Title | Weighted Nuclear Norm Minimization and Its Applications to Low Level Vision |
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