A subspace constrained randomized Kaczmarz method for structure or external knowledge exploitation
We study a version of the randomized Kaczmarz algorithm for solving systems of linear equations where the iterates are confined to the solution space of a selected subsystem. We show that the subspace constraint leads to an accelerated convergence rate, especially when the system has approximately l...
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| Vydané v: | Linear algebra and its applications Ročník 698; s. 220 - 260 |
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| Hlavní autori: | , |
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| Jazyk: | English |
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01.10.2024
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| ISSN: | 0024-3795, 1873-1856 |
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| Abstract | We study a version of the randomized Kaczmarz algorithm for solving systems of linear equations where the iterates are confined to the solution space of a selected subsystem. We show that the subspace constraint leads to an accelerated convergence rate, especially when the system has approximately low-rank structure. On Gaussian-like random data, we show that it results in a form of dimension reduction that effectively increases the aspect ratio of the system. Furthermore, this method serves as a building block for a second, quantile-based algorithm for solving linear systems with arbitrary sparse corruptions, which is able to efficiently utilize external knowledge about corruption-free equations and achieve convergence in difficult settings. Numerical experiments on synthetic and realistic data support our theoretical results and demonstrate the validity of the proposed methods for even more general data models than guaranteed by the theory. |
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| AbstractList | We study a version of the randomized Kaczmarz algorithm for solving systems of linear equations where the iterates are confined to the solution space of a selected subsystem. We show that the subspace constraint leads to an accelerated convergence rate, especially when the system has approximately low-rank structure. On Gaussian-like random data, we show that it results in a form of dimension reduction that effectively increases the aspect ratio of the system. Furthermore, this method serves as a building block for a second, quantile-based algorithm for solving linear systems with arbitrary sparse corruptions, which is able to efficiently utilize external knowledge about corruption-free equations and achieve convergence in difficult settings. Numerical experiments on synthetic and realistic data support our theoretical results and demonstrate the validity of the proposed methods for even more general data models than guaranteed by the theory. |
| Author | Lok, Jackie Rebrova, Elizaveta |
| Author_xml | – sequence: 1 givenname: Jackie orcidid: 0009-0009-1022-2209 surname: Lok fullname: Lok, Jackie email: jackie.lok@princeton.edu – sequence: 2 givenname: Elizaveta surname: Rebrova fullname: Rebrova, Elizaveta email: elre@princeton.edu |
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| SubjectTerms | Corrupted linear systems Kaczmarz algorithm Quantile methods Stochastic iterative methods |
| Title | A subspace constrained randomized Kaczmarz method for structure or external knowledge exploitation |
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