s-Goodness for Low-Rank Matrix Recovery
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 generally 𝒩𝒫 hard. A pop...
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| Format: | Article |
| Language: | English |
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Wiley
2013-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2013/101974 |
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| _version_ | 1832549947617050624 |
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| author | Lingchen Kong Levent Tunçel Naihua Xiu |
| author_facet | Lingchen Kong Levent Tunçel Naihua Xiu |
| author_sort | Lingchen Kong |
| collection | DOAJ |
| description | 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 generally 𝒩𝒫 hard. A popular approach replaces the rank function with the nuclear norm of the matrix
variable. In this paper, we extend and characterize the concept of s-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 characteristic s-goodness constants, γs and γ^s, of a linear transformation, we derive necessary and sufficient conditions for a linear transformation to be s-good. Moreover, we establish the equivalence of s-goodness and the null space properties. Therefore, s-goodness is a necessary and sufficient condition for exact s-rank matrix recovery via the nuclear norm minimization. |
| format | Article |
| id | doaj-art-82abf92a5ca741f9b2f673a332714e6a |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2013-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-82abf92a5ca741f9b2f673a332714e6a2025-02-03T06:08:10ZengWileyAbstract and Applied Analysis1085-33751687-04092013-01-01201310.1155/2013/101974101974s-Goodness for Low-Rank Matrix RecoveryLingchen Kong0Levent Tunçel1Naihua Xiu2Department of Applied Mathematics, Beijing Jiaotong University, Beijing 100044, ChinaDepartment of Combinatorics and Optimization, Faculty of Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1, CanadaDepartment of Applied Mathematics, Beijing Jiaotong University, Beijing 100044, ChinaLow-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 generally 𝒩𝒫 hard. A popular approach replaces the rank function with the nuclear norm of the matrix variable. In this paper, we extend and characterize the concept of s-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 characteristic s-goodness constants, γs and γ^s, of a linear transformation, we derive necessary and sufficient conditions for a linear transformation to be s-good. Moreover, we establish the equivalence of s-goodness and the null space properties. Therefore, s-goodness is a necessary and sufficient condition for exact s-rank matrix recovery via the nuclear norm minimization.http://dx.doi.org/10.1155/2013/101974 |
| spellingShingle | Lingchen Kong Levent Tunçel Naihua Xiu s-Goodness for Low-Rank Matrix Recovery Abstract and Applied Analysis |
| title | s-Goodness for Low-Rank Matrix Recovery |
| title_full | s-Goodness for Low-Rank Matrix Recovery |
| title_fullStr | s-Goodness for Low-Rank Matrix Recovery |
| title_full_unstemmed | s-Goodness for Low-Rank Matrix Recovery |
| title_short | s-Goodness for Low-Rank Matrix Recovery |
| title_sort | s goodness for low rank matrix recovery |
| url | http://dx.doi.org/10.1155/2013/101974 |
| work_keys_str_mv | AT lingchenkong sgoodnessforlowrankmatrixrecovery AT leventtuncel sgoodnessforlowrankmatrixrecovery AT naihuaxiu sgoodnessforlowrankmatrixrecovery |