• Corpus ID: 238634341

Randomized Extended Kaczmarz is a Limit Point of Sketch-and-Project

@article{Jarman2021RandomizedEK,
  title={Randomized Extended Kaczmarz is a Limit Point of Sketch-and-Project},
  author={Benjamin P. Jarman and Nathan Mankovich and Jacob D. Moorman},
  journal={ArXiv},
  year={2021},
  volume={abs/2110.05605}
}
The sketch-and-project (SAP) framework for solving systems of linear equations has unified the theory behind popular projective iterative methods such as randomized Kaczmarz, randomized coordinate descent, and variants thereof. The randomized extended Kaczmarz (REK) method is a popular extension of randomized Kaczmarz for solving inconsistent systems, which has not yet been shown to lie within the SAP framework. In this work we show that, in a certain sense, REK may be expressed as the limit… 
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References

SHOWING 1-10 OF 26 REFERENCES
Randomized Iterative Methods for Linear Systems
TLDR
A novel, fundamental and surprisingly simple randomized iterative method for solving consistent linear systems, which allows for a much wider selection of these two parameters, which leads to a number of new specific methods.
Faster Randomized Block Kaczmarz Algorithms
  • I. Necoara
  • Computer Science, Mathematics
    SIAM J. Matrix Anal. Appl.
  • 2019
TLDR
It is proved that randomized block Kaczmarz algorithm converges linearly in expectation, with a rate depending on the geometric properties of the matrix and its submatrices and on the size of the blocks.
On the relation between the randomized extended Kaczmarz algorithm and coordinate descent
In this note we compare the randomized extended Kaczmarz (EK) algorithm and randomized coordinate descent (CD) for solving the full-rank overdetermined linear least-squares problem and prove that CD
Paved with Good Intentions: Analysis of a Randomized Block Kaczmarz Method
Tight upper bounds for the convergence of the randomized extended Kaczmarz and Gauss–Seidel algorithms
  • Kui Du
  • Computer Science
    Numer. Linear Algebra Appl.
  • 2019
TLDR
This paper presents tight upper bounds for the convergence of the randomized extended Kaczmarz and Gauss–Seidel algorithms.
Randomized Methods for Linear Constraints: Convergence Rates and Conditioning
TLDR
It is shown that, under appropriate probability distributions, the linear rates of convergence can be bounded in terms of natural linear-algebraic condition numbers for the problems and generalizations to convex systems under metric regularity assumptions are discussed.
Randomized Kaczmarz solver for noisy linear systems
TLDR
This work analyzes the case where the Kaczmarz method is corrupted by noise, and proves that in this noisy version, the randomized method reaches an error threshold dependent on the matrix A with the same rate as in the error-free case.
Iteration complexity of randomized block-coordinate descent methods for minimizing a composite function
TLDR
A randomized block-coordinate descent method for minimizing the sum of a smooth and a simple nonsmooth block-separable convex function is developed and it is proved that it obtains an accurate solution with probability at least 1-\rho in at most O(n/\varepsilon) iterations, thus achieving first true iteration complexity bounds.
A Randomized Kaczmarz Algorithm with Exponential Convergence
TLDR
A randomized version of the Kaczmarz method for consistent, overdetermined linear systems and it is proved that it converges with expected exponential rate and is the first solver whose rate does not depend on the number of equations in the system.
Randomized Extended Kaczmarz for Solving Least Squares
We present a randomized iterative algorithm that exponentially converges in the mean square to the minimum $\ell_2$-norm least squares solution of a given linear system of equations. The expected
...
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