L Regularization: A Thresholding Representation Theory and a Fast Solver
The special importance of L_{1/2} regularization has been recognized in recent studies on sparse modeling (particularly on compressed sensing). The L_{1/2} regularization, however, leads to a nonconvex, nonsmooth, and non-Lipschitz optimization problem that is difficult to solve fast and efficiently...
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| Vydané v: | IEEE transaction on neural networks and learning systems Ročník 23; číslo 7; s. 1013 - 1027 |
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| Hlavní autori: | , , , |
| Médium: | Journal Article |
| Jazyk: | English Japanese |
| Vydavateľské údaje: |
IEEE
01.07.2012
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| Predmet: | |
| ISSN: | 2162-237X, 2162-2388 |
| On-line prístup: | Získať plný text |
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| Abstract | The special importance of L_{1/2} regularization has been recognized in recent studies on sparse modeling (particularly on compressed sensing). The L_{1/2} regularization, however, leads to a nonconvex, nonsmooth, and non-Lipschitz optimization problem that is difficult to solve fast and efficiently. In this paper, through developing a threshoding representation theory for L_{1/2} regularization, we propose an iterative half thresholding algorithm for fast solution of L_{1/2} regularization, corresponding to the well-known iterative soft thresholding algorithm for L_{1} regularization, and the iterative hard thresholding algorithm for L_{0} regularization. We prove the existence of the resolvent of gradient of \Vert x\Vert^{1/2}_{1/2} , calculate its analytic expression, and establish an alternative feature theorem on solutions of L_{1/2} regularization, based on which a thresholding representation of solutions of L_{1/2} regularization is derived and an optimal regularization parameter setting rule is formulated. The developed theory provides a successful practice of extension of the well-known Moreau's proximity forward-backward splitting theory to the L_{1/2} regularization case. We verify the convergence of the iterative half thresholding algorithm and provide a series of experiments to assess performance of the algorithm. The experiments show that the {half} algorithm is effective, efficient, and can be accepted as a fast solver for L_{1/2} regularization. With the new algorithm, we conduct a phase diagram study to further demonstrate the superiority of L_{1/2} regularization over L_{1} regularization. |
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| AbstractList | The special importance of L_{1/2} regularization has been recognized in recent studies on sparse modeling (particularly on compressed sensing). The L_{1/2} regularization, however, leads to a nonconvex, nonsmooth, and non-Lipschitz optimization problem that is difficult to solve fast and efficiently. In this paper, through developing a threshoding representation theory for L_{1/2} regularization, we propose an iterative half thresholding algorithm for fast solution of L_{1/2} regularization, corresponding to the well-known iterative soft thresholding algorithm for L_{1} regularization, and the iterative hard thresholding algorithm for L_{0} regularization. We prove the existence of the resolvent of gradient of \Vert x\Vert^{1/2}_{1/2} , calculate its analytic expression, and establish an alternative feature theorem on solutions of L_{1/2} regularization, based on which a thresholding representation of solutions of L_{1/2} regularization is derived and an optimal regularization parameter setting rule is formulated. The developed theory provides a successful practice of extension of the well-known Moreau's proximity forward-backward splitting theory to the L_{1/2} regularization case. We verify the convergence of the iterative half thresholding algorithm and provide a series of experiments to assess performance of the algorithm. The experiments show that the {half} algorithm is effective, efficient, and can be accepted as a fast solver for L_{1/2} regularization. With the new algorithm, we conduct a phase diagram study to further demonstrate the superiority of L_{1/2} regularization over L_{1} regularization. |
| Author | Xu, Fengmin Xu, Zongben Zhang, Hai Chang, Xiangyu |
| Author_xml | – sequence: 1 givenname: Zongben surname: Xu fullname: Xu, Zongben email: zbxu@mail.xjtu.edu.cn organization: Institute for Information and System Science and the MOE Key Laboratory for Intelligent Networks and Network Security, Xi'an Jiaotong University, Xi'an, China – sequence: 2 givenname: Xiangyu surname: Chang fullname: Chang, Xiangyu email: xiangyuchang@gmail.com organization: Institute for Information and System Science and Department of Applied Mathematics, School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, China – sequence: 3 givenname: Fengmin surname: Xu fullname: Xu, Fengmin email: fengminxu@mail.xjtu.edu.cn organization: Institute for Information and System Science and Department of Applied Mathematics, School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, China – sequence: 4 givenname: Hai surname: Zhang fullname: Zhang, Hai email: zhanghai@nwu.edu.cn organization: Department of Mathematics, Northwest University, Xi'an, China |
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| CODEN | ITNNAL |
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| Snippet | The special importance of L_{1/2} regularization has been recognized in recent studies on sparse modeling (particularly on compressed sensing). The L_{1/2}... |
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| SubjectTerms | Compressed sensing Compressive sensing Convergence Convex functions half hard Iterative algorithms L_{q} regularization Learning systems Noise Optimization Signal processing algorithms soft sparsity thresholding algorithms thresholding representation theory |
| Title | L Regularization: A Thresholding Representation Theory and a Fast Solver |
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