Spectral operators of matrices
The 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...
| Main Authors: | , , , |
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| Format: | Journal Article |
| Published: |
Springer
2017
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| Online Access: | http://hdl.handle.net/20.500.11937/55609 |
| _version_ | 1848759663866150912 |
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| author | Ding, C. Sun, D. Sun, Jie Toh, K. |
| author_facet | Ding, C. Sun, D. Sun, Jie Toh, K. |
| author_sort | Ding, C. |
| building | Curtin Institutional Repository |
| collection | Online Access |
| description | The 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 Löwner 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 Löwner 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 Fréchet-differentiability are systematically studied. © 2017 Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society. |
| first_indexed | 2025-11-14T10:03:28Z |
| format | Journal Article |
| id | curtin-20.500.11937-55609 |
| institution | Curtin University Malaysia |
| institution_category | Local University |
| last_indexed | 2025-11-14T10:03:28Z |
| publishDate | 2017 |
| publisher | Springer |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | curtin-20.500.11937-556092019-02-19T05:36:25Z Spectral operators of matrices Ding, C. Sun, D. Sun, Jie Toh, K. The 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 Löwner 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 Löwner 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 Fréchet-differentiability are systematically studied. © 2017 Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society. 2017 Journal Article http://hdl.handle.net/20.500.11937/55609 10.1007/s10107-017-1162-3 Springer fulltext |
| spellingShingle | Ding, C. Sun, D. Sun, Jie Toh, K. Spectral operators of matrices |
| title | Spectral operators of matrices |
| title_full | Spectral operators of matrices |
| title_fullStr | Spectral operators of matrices |
| title_full_unstemmed | Spectral operators of matrices |
| title_short | Spectral operators of matrices |
| title_sort | spectral operators of matrices |
| url | http://hdl.handle.net/20.500.11937/55609 |