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Spectral operators of matrices
Ding, Chao1; Sun, Defeng2,3; Sun, Jie4; Toh, Kim-Chuan5
AbstractThe class of matrix optimization problems (MOPs) has been recognized in recent years to be a powerful tool to model many important applications involving structured low rank matrices within and beyond the optimization community. This trend can be credited to some extent to the exciting developments in emerging fields such as compressed sensing. The Lowner operator, which generates a matrix valued function via applying a single-variable function to each of the singular values of a matrix, has played an important role for a long time in solving matrix optimization problems. However, the classical theory developed for the Lowner operator has become inadequate in these recent applications. The main objective of this paper is to provide necessary theoretical foundations from the perspectives of designing efficient numerical methods for solving MOPs. We achieve this goal by introducing and conducting a thorough study on a new class of matrix valued functions, coined as spectral operators of matrices. Several fundamental properties of spectral operators, including the well-definedness, continuity, directional differentiability and Fr,chet-differentiability are systematically studied.
KeywordSpectral operators Directional differentiability Frechet differentiability Matrix valued functions Proximal mappings
Funding ProjectNational Natural Science Foundation of China[11301515] ; National Natural Science Foundation of China[11671387] ; National Natural Science Foundation of China[11531014]
WOS Research AreaComputer Science ; Operations Research & Management Science ; Mathematics
WOS SubjectComputer Science, Software Engineering ; Operations Research & Management Science ; Mathematics, Applied
WOS IDWOS:000426071000020
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Document Type期刊论文
Corresponding AuthorSun, Defeng
Affiliation1.Chinese Acad Sci, Acad Math & Syst Sci, Inst Appl Math, Beijing, Peoples R China
2.Natl Univ Singapore, Dept Math, Singapore, Singapore
3.Natl Univ Singapore, Risk Management Inst, Singapore, Singapore
4.Curtin Univ, Dept Math & Stat, Bentley, WA, Australia
5.Natl Univ Singapore, Dept Math, Singapore, Singapore
Recommended Citation
GB/T 7714
Ding, Chao,Sun, Defeng,Sun, Jie,et al. Spectral operators of matrices[J]. MATHEMATICAL PROGRAMMING,2018,168(1-2):509-531.
APA Ding, Chao,Sun, Defeng,Sun, Jie,&Toh, Kim-Chuan.(2018).Spectral operators of matrices.MATHEMATICAL PROGRAMMING,168(1-2),509-531.
MLA Ding, Chao,et al."Spectral operators of matrices".MATHEMATICAL PROGRAMMING 168.1-2(2018):509-531.
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