Low-Rank Optimization With Convex Constraints.
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| Title: | Low-Rank Optimization With Convex Constraints. |
|---|---|
| Authors: | Grussler, Christian, Rantzer, Anders, Giselsson, Pontus |
| Source: | IEEE Transactions on Automatic Control. Nov2018, Vol. 63 Issue 11, p4000-4007. 8p. |
| Subjects: | Robust convex optimization, Machine learning, System identification, Singular value decomposition, Nonconvex programming |
| Abstract: | The problem of low-rank approximation with convex constraints, which appears in data analysis, system identification, model order reduction, low-order controller design, and low-complexity modeling is considered. Given a matrix, the objective is to find a low-rank approximation that meets rank and convex constraints while minimizing the distance to the matrix in the squared Frobenius norm. In many situations, this nonconvex problem is convexified by nuclear-norm regularization. However, we will see that the approximations obtained by this method may be far from optimal. In this paper, we propose an alternative convex relaxation that uses the convex envelope of the squared Frobenius norm and the rank constraint. With this approach, easily verifiable conditions are obtained under which the solutions to the convex relaxation and the original nonconvex problem coincide. A semidefinite programming representation of the convex envelope is derived, which allows us to apply this approach to several known problems. Our example on optimal low-rank Hankel approximation/model reduction illustrates that the proposed convex relaxation performs consistently better than nuclear-norm regularization and may outperform balanced truncation. [ABSTRACT FROM AUTHOR] |
| Copyright of IEEE Transactions on Automatic Control is the property of IEEE and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
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| Items | – Name: Title Label: Title Group: Ti Data: Low-Rank Optimization With Convex Constraints. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Grussler%2C+Christian%22">Grussler, Christian</searchLink><br /><searchLink fieldCode="AR" term="%22Rantzer%2C+Anders%22">Rantzer, Anders</searchLink><br /><searchLink fieldCode="AR" term="%22Giselsson%2C+Pontus%22">Giselsson, Pontus</searchLink> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22IEEE+Transactions+on+Automatic+Control%22">IEEE Transactions on Automatic Control</searchLink>. Nov2018, Vol. 63 Issue 11, p4000-4007. 8p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Robust+convex+optimization%22">Robust convex optimization</searchLink><br /><searchLink fieldCode="DE" term="%22Machine+learning%22">Machine learning</searchLink><br /><searchLink fieldCode="DE" term="%22System+identification%22">System identification</searchLink><br /><searchLink fieldCode="DE" term="%22Singular+value+decomposition%22">Singular value decomposition</searchLink><br /><searchLink fieldCode="DE" term="%22Nonconvex+programming%22">Nonconvex programming</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: The problem of low-rank approximation with convex constraints, which appears in data analysis, system identification, model order reduction, low-order controller design, and low-complexity modeling is considered. Given a matrix, the objective is to find a low-rank approximation that meets rank and convex constraints while minimizing the distance to the matrix in the squared Frobenius norm. In many situations, this nonconvex problem is convexified by nuclear-norm regularization. However, we will see that the approximations obtained by this method may be far from optimal. In this paper, we propose an alternative convex relaxation that uses the convex envelope of the squared Frobenius norm and the rank constraint. With this approach, easily verifiable conditions are obtained under which the solutions to the convex relaxation and the original nonconvex problem coincide. A semidefinite programming representation of the convex envelope is derived, which allows us to apply this approach to several known problems. Our example on optimal low-rank Hankel approximation/model reduction illustrates that the proposed convex relaxation performs consistently better than nuclear-norm regularization and may outperform balanced truncation. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of IEEE Transactions on Automatic Control is the property of IEEE and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1109/TAC.2018.2813009 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 8 StartPage: 4000 Subjects: – SubjectFull: Robust convex optimization Type: general – SubjectFull: Machine learning Type: general – SubjectFull: System identification Type: general – SubjectFull: Singular value decomposition Type: general – SubjectFull: Nonconvex programming Type: general Titles: – TitleFull: Low-Rank Optimization With Convex Constraints. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Grussler, Christian – PersonEntity: Name: NameFull: Rantzer, Anders – PersonEntity: Name: NameFull: Giselsson, Pontus IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 11 Text: Nov2018 Type: published Y: 2018 Identifiers: – Type: issn-print Value: 00189286 Numbering: – Type: volume Value: 63 – Type: issue Value: 11 Titles: – TitleFull: IEEE Transactions on Automatic Control Type: main |
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