Randomized Average Kaczmarz Algorithm for Tensor Linear Systems
For solving tensor linear systems under the tensor–tensor t-product, we propose the randomized average Kaczmarz (TRAK) algorithm, the randomized average Kaczmarz algorithm with random sampling (TRAKS), and their Fourier version, which can be effectively implemented in a distributed environment. We a...
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| Veröffentlicht in: | Mathematics (Basel) Jg. 10; H. 23; S. 4594 |
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01.12.2022
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| Abstract | For solving tensor linear systems under the tensor–tensor t-product, we propose the randomized average Kaczmarz (TRAK) algorithm, the randomized average Kaczmarz algorithm with random sampling (TRAKS), and their Fourier version, which can be effectively implemented in a distributed environment. We analyzed the relationships (of the updated formulas) between the original algorithms and their Fourier versions in detail and prove that these new algorithms can converge to the unique least F-norm solution of the consistent tensor linear systems. Extensive numerical experiments show that they significantly outperform the tensor-randomized Kaczmarz (TRK) algorithm in terms of both iteration counts and computing times and have potential in real-world data, such as video data, CT data, etc. |
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| AbstractList | For solving tensor linear systems under the tensor–tensor t-product, we propose the randomized average Kaczmarz (TRAK) algorithm, the randomized average Kaczmarz algorithm with random sampling (TRAKS), and their Fourier version, which can be effectively implemented in a distributed environment. We analyzed the relationships (of the updated formulas) between the original algorithms and their Fourier versions in detail and prove that these new algorithms can converge to the unique least F-norm solution of the consistent tensor linear systems. Extensive numerical experiments show that they significantly outperform the tensor-randomized Kaczmarz (TRK) algorithm in terms of both iteration counts and computing times and have potential in real-world data, such as video data, CT data, etc. |
| Audience | Academic |
| Author | Gao, Ying Zhang, Feiyu Bao, Wendi Li, Weiguo Wang, Qin |
| Author_xml | – sequence: 1 givenname: Wendi surname: Bao fullname: Bao, Wendi – sequence: 2 givenname: Feiyu surname: Zhang fullname: Zhang, Feiyu – sequence: 3 givenname: Weiguo orcidid: 0000-0002-7057-972X surname: Li fullname: Li, Weiguo – sequence: 4 givenname: Qin surname: Wang fullname: Wang, Qin – sequence: 5 givenname: Ying surname: Gao fullname: Gao, Ying |
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| Cites_doi | 10.1007/s10543-021-00877-w 10.1007/BF01396365 10.1016/0024-3795(81)90139-7 10.1016/j.cam.2021.113720 10.1002/nla.2470 10.1016/j.laa.2012.12.022 10.1016/j.cam.2022.114856 10.1080/01621459.2013.776499 10.1016/j.laa.2010.09.020 10.1137/110837711 10.1198/tas.2008.s262 10.1016/j.cam.2022.114372 10.1137/21M1398562 10.1007/s10543-016-0607-z 10.1137/120889897 10.1190/1.1441847 10.1007/s10543-010-0265-5 10.1007/s10543-020-00824-1 10.1177/016173468400600107 10.1137/19M1251643 10.1016/j.camwa.2017.04.017 |
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| SubjectTerms | Algorithms Analysis Fourier domain Fourier transforms Iterative methods least-norm problem Linear systems Mathematical analysis Methods Random sampling randomized average Kaczmarz method T-product tensor linear system Tensors Tensors (Mathematics) Video data |
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