Generalized Wedderburn Rank Reduction
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| Materias: | |
| Acceso en línea: | |
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| _version_ | 1866929376183975936 |
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| author | Kędzierski, Oskar |
| author_facet | Kędzierski, Oskar |
| contents | We generalize the Wedderburn rank reduction formula by replacing the inverse with the Moore--Penrose pseudoinverse. In particular, this allows one to remove the non--singularity of a certain matrix from assumptions. The results implies in a straightforward way Nystroem, CUR decompositions, meta-factorization, and a result of Ameli, Shadden. We investigate which properties of the matrix are inherited by the generalized Wedderburn reduction. Reductions leading to the best low-rank approximation are explicitly described in terms of singular vectors. We give a self--contained calculation of the range and the nullspace of the projection $A(BA)^+B$ and prove that any projection can be expressed in this way. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_03992 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generalized Wedderburn Rank Reduction Kędzierski, Oskar Numerical Analysis 15A09, 15A04, 15A10, 15-04 We generalize the Wedderburn rank reduction formula by replacing the inverse with the Moore--Penrose pseudoinverse. In particular, this allows one to remove the non--singularity of a certain matrix from assumptions. The results implies in a straightforward way Nystroem, CUR decompositions, meta-factorization, and a result of Ameli, Shadden. We investigate which properties of the matrix are inherited by the generalized Wedderburn reduction. Reductions leading to the best low-rank approximation are explicitly described in terms of singular vectors. We give a self--contained calculation of the range and the nullspace of the projection $A(BA)^+B$ and prove that any projection can be expressed in this way. |
| title | Generalized Wedderburn Rank Reduction |
| topic | Numerical Analysis 15A09, 15A04, 15A10, 15-04 |
| url | https://arxiv.org/abs/2406.03992 |