Relaxed Greedy Randomized Kaczmarz with Signal Averaging for Solving Doubly-Noisy Linear Systems
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866915901159243776 |
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| author | Zhang, Lu Zeng, Jinchuan Zhang, Hui |
| author_facet | Zhang, Lu Zeng, Jinchuan Zhang, Hui |
| contents | Large-scale linear systems of the form $Ax=b$ are often doubly-noisy, in the sense that both its measurement matrix $A$ and measurement vector $b$ are noisy. In this paper, we extend the relaxed greedy randomized Kaczmarz (RGRK) method to the doubly-noisy systems to accelerate convergence. However, RGRK fails to converge to the least-squares solution for doubly-noisy systems. To address this limitation, we propose a simple modification: averaging multiple measurements instead of using a single measurement. The proposed RGRK with signal averaging (RGRK-SA) converges to the solution of doubly-noisy systems at a polynomial rate. Numerical experiments demonstrate that both RGRK and RGRK-SA outperform the classical randomized Kaczmarz method, and RGRK-SA has a higher accuracy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_29372 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Relaxed Greedy Randomized Kaczmarz with Signal Averaging for Solving Doubly-Noisy Linear Systems Zhang, Lu Zeng, Jinchuan Zhang, Hui Numerical Analysis 65F10, 65F20, 68W20 Large-scale linear systems of the form $Ax=b$ are often doubly-noisy, in the sense that both its measurement matrix $A$ and measurement vector $b$ are noisy. In this paper, we extend the relaxed greedy randomized Kaczmarz (RGRK) method to the doubly-noisy systems to accelerate convergence. However, RGRK fails to converge to the least-squares solution for doubly-noisy systems. To address this limitation, we propose a simple modification: averaging multiple measurements instead of using a single measurement. The proposed RGRK with signal averaging (RGRK-SA) converges to the solution of doubly-noisy systems at a polynomial rate. Numerical experiments demonstrate that both RGRK and RGRK-SA outperform the classical randomized Kaczmarz method, and RGRK-SA has a higher accuracy. |
| title | Relaxed Greedy Randomized Kaczmarz with Signal Averaging for Solving Doubly-Noisy Linear Systems |
| topic | Numerical Analysis 65F10, 65F20, 68W20 |
| url | https://arxiv.org/abs/2603.29372 |