• Corpus ID: 235253864

A Note On The Randomized Kaczmarz Method With A Partially Weighted Selection Step

  title={A Note On The Randomized Kaczmarz Method With A Partially Weighted Selection Step},
  author={J. Gro{\ss}},
  • J. Groß
  • Published 30 May 2021
  • Computer Science, Mathematics
  • ArXiv
In this note we reconsider two known algorithms which both usually converge faster than the randomized Kaczmarz method introduced by Strohmer and Vershynin(2009), but require the additional computation of all residuals of an iteration at each step. As already indicated in the literature, e.g. arXiv:2007.02910 and arXiv:2011.14693, it is shown that the non-randomized version of the two algorithms converges at least as fast as the randomized version, while still requiring computation of all… 

Figures and Tables from this paper


A Weighted Randomized Kaczmarz Method for Solving Linear Systems
It is proved that if the $i-$th row is selected with likelihood proportional to $\left|\langle a_i, x_k \right\rangle - b_i\right|^{p}$, then the method de-randomizes and explains, among other things, why the maximal correction method works well.
A Randomized Kaczmarz Algorithm with Exponential Convergence
The Kaczmarz method for solving linear systems of equations is an iterative algorithm that has found many applications ranging from computer tomography to digital signal processing. Despite the
Paved with Good Intentions: Analysis of a Randomized Block Kaczmarz Method
This algorithm is the first block Kaczmarz method with an (expected) linear rate of convergence that can be expressed in terms of the geometric properties of the matrix and its submatrices.
A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria, 2021
  • URL https: //www.R-project.org/
  • 2021
Convergence of Adaptive, Randomized, Iterative Linear Solvers
This work provides a general set of assumptions under which highly adapted, randomized or deterministic, row-action or column-action linear solvers are guaranteed to converge with probability one, and provides worst case rates of convergence.
Greed Works: An Improved Analysis of Sampling Kaczmarz-Motkzin
This analysis improves upon previous known convergence bounds for SKM, capturing the benefit of partially greedy selection schemes and further generalizes previous known results, removing the theoretical assumptions that $\beta$ must be fixed at every iteration and that $A$ must have normalized rows.
A Kaczmarz Method with Simple Random Sampling for Solving Large Linear Systems
Numerical experiments demonstrate the superiority of the new algorithms over many state-of-the-art randomized Kaczmarz methods for large linear systems problems and ridge regression problems.
A Note On Convergence Rate of Randomized Kaczmarz Method
In this paper, we propose an alternative version of the randomized Kaczmarz method, which chooses each row of the coefficient matrix A with probability proportional to the square of the Euclidean
A new greedy Kaczmarz algorithm for the solution of very large linear systems
Numerical results show that the proposed algorithm is feasible and has faster convergence rate than the greedy randomized Kaczmarz algorithm.
Adaptive Sketch-and-Project Methods for Solving Linear Systems
New adaptive sampling rules for the sketch-and-project method for solving linear systems are presented and it is concluded that the max-distance sampling rule is superior both experimentally and theoretically to the capped sampling rule.