On two-subspace randomized extended Kaczmarz method for solving large linear least-squares problems

For solving the large-scale linear least-squares problem, we propose a block version of the randomized extended Kaczmarz method, called the two-subspace randomized extended Kaczmarz method, which does not require any row or column paving. Theoretical analysis and numerical results show that the two-...

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Bibliographic Details
Published in:Numerical algorithms 2022, Vol.89 (1), p.1-31
Main Author: Wu, Wen-Ting
Format: Article
Language:English
Subjects:
Algebra
Algorithms
Coefficients
Computer Science
Convergence
Iterative methods
Least squares method
Mathematical analysis
Matrices (mathematics)
Norms
Numeric Computing
Numerical Analysis
Original Paper
Sequences
Subspaces
Theory of Computation
Upper bounds
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Summary:For solving the large-scale linear least-squares problem, we propose a block version of the randomized extended Kaczmarz method, called the two-subspace randomized extended Kaczmarz method, which does not require any row or column paving. Theoretical analysis and numerical results show that the two-subspace randomized extended Kaczmarz method is much more efficient than the randomized extended Kaczmarz method. When the coefficient matrix is of full column rank, the two-subspace randomized extended Kaczmarz method can also outperform the randomized coordinate descent method. If the linear system is consistent, we remove one of the iteration sequences in the two-subspace randomized extended Kaczmarz method, which approximates the projection of the right-hand side vector onto the orthogonal complement space of the range space of the coefficient matrix, and obtain the generalized two-subspace randomized Kaczmarz method, which is actually a generalization of the two-subspace randomized Kaczmarz method without the assumptions of unit row norms and full column rank on the coefficient matrix. We give the upper bound for the convergence rate of the generalized two-subspace randomized Kaczmarz method which also leads to a better upper bound for the convergence rate of the two-subspace randomized Kaczmarz method.
ISSN:1017-1398
1572-9265
DOI:10.1007/s11075-021-01104-x