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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Bibliographic Details
Published in:Linear algebra and its applications Vol. 698; pp. 220 - 260
Main Authors: Lok, Jackie, Rebrova, Elizaveta
Format: Journal Article
Language:English
Published: Elsevier Inc 01.10.2024
Subjects:
Corrupted linear systems
Kaczmarz algorithm
Quantile methods
Stochastic iterative methods
68W20
Stochastic iterative methods
65F10
65F20
Kaczmarz algorithm
Quantile methods
60B20
Corrupted linear systems
ISSN:0024-3795, 1873-1856
Online Access:Get full text
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Summary: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.
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2024.06.010