Accelerated matrix completion algorithm using continuation strategy and randomized SVD
The matrix completion (MC) problem aims to recover the complete matrix from the observed matrix with missing entries. The unknown matrix is generally assumed to have a low-rank structure. It is well known that the low-rank minimization problem is NP-hard, so the rank function of the matrix is approx...
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| Vydané v: | Journal of computational and applied mathematics Ročník 429; s. 115215 |
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| Hlavní autori: | , , |
| Médium: | Journal Article |
| Jazyk: | English |
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Elsevier B.V
01.09.2023
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| ISSN: | 0377-0427 |
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| Abstract | The matrix completion (MC) problem aims to recover the complete matrix from the observed matrix with missing entries. The unknown matrix is generally assumed to have a low-rank structure. It is well known that the low-rank minimization problem is NP-hard, so the rank function of the matrix is approximated by a surrogate function. In this paper, we firs t apply the optimization transfer technique to formulate the minimization problem into a bi-variate minimization problem by adding an auxiliary variable. We apply the alternating minimization method to find the minimizer of the bi-variate minimization problem. For the auxiliary variable, the minimum can be formulated as a linear combination of the observation data and the original variable. For the original variable, the minimum is a generalized singular thresholding (GSVT) operator of the auxiliary variable. However, most existing iteration methods for matrix completion problem use fixed thresholding value and also suffer from full singular value decomposition (SVD), which need more computation time for iteration. Thus, they become inefficient for large-scale problems. In order to handle these limitations, we apply the continuation strategy and the randomized singular value decomposition (RSVD) to reduce the computational costs. The threshold value of the continuous strategy is reduced in each iteration until a given minimum thresholding value is reached, while the RSVD only needs to decompose small-scale matrices. Hence, using both strategies simultaneously can effectively reduce the computation time for iteration. Numerical experiments demonstrate that the proposed algorithm can effectively recover the missing entries both in synthetic and real data. |
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| AbstractList | The matrix completion (MC) problem aims to recover the complete matrix from the observed matrix with missing entries. The unknown matrix is generally assumed to have a low-rank structure. It is well known that the low-rank minimization problem is NP-hard, so the rank function of the matrix is approximated by a surrogate function. In this paper, we firs t apply the optimization transfer technique to formulate the minimization problem into a bi-variate minimization problem by adding an auxiliary variable. We apply the alternating minimization method to find the minimizer of the bi-variate minimization problem. For the auxiliary variable, the minimum can be formulated as a linear combination of the observation data and the original variable. For the original variable, the minimum is a generalized singular thresholding (GSVT) operator of the auxiliary variable. However, most existing iteration methods for matrix completion problem use fixed thresholding value and also suffer from full singular value decomposition (SVD), which need more computation time for iteration. Thus, they become inefficient for large-scale problems. In order to handle these limitations, we apply the continuation strategy and the randomized singular value decomposition (RSVD) to reduce the computational costs. The threshold value of the continuous strategy is reduced in each iteration until a given minimum thresholding value is reached, while the RSVD only needs to decompose small-scale matrices. Hence, using both strategies simultaneously can effectively reduce the computation time for iteration. Numerical experiments demonstrate that the proposed algorithm can effectively recover the missing entries both in synthetic and real data. |
| ArticleNumber | 115215 |
| Author | Li, Kexin Chen, Hefeng Wen, You-Wei |
| Author_xml | – sequence: 1 givenname: You-Wei orcidid: 0000-0002-9892-3668 surname: Wen fullname: Wen, You-Wei organization: School of Mathematics, Yunnan Normal University, Kunming, Yunnan, China – sequence: 2 givenname: Kexin surname: Li fullname: Li, Kexin email: likexin@hunnu.edu.cn organization: Key Laboratory of Computing and Stochastic Mathematics (LCSM), School of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan, China – sequence: 3 givenname: Hefeng surname: Chen fullname: Chen, Hefeng organization: School of Mathematics and Statistics, Guangdong University of Technology, Guangzhou, Guangdong, China |
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| Keywords | Alternating minimization Randomized singular value decomposition Non-convex surrogate Continuation strategy Matrix completion |
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| SubjectTerms | Alternating minimization Continuation strategy Matrix completion Non-convex surrogate Randomized singular value decomposition |
| Title | Accelerated matrix completion algorithm using continuation strategy and randomized SVD |
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