Related papers: Randomized block subsampling Kaczmarz-Motzkin meth…
The randomized Kaczmarz methods are a popular and effective family of iterative methods for solving large-scale linear systems of equations, which have also been applied to linear feasibility problems. In this work, we propose a new block…
The Sampling Kaczmarz Motzkin (SKM) algorithm is a generalized method for solving large scale linear systems of inequalities. Having its root in the relaxation method of Agmon, Schoenberg, and Motzkin and the randomized Kaczmarz method, SKM…
Quantile-based randomized Kaczmarz (QRK) was recently introduced to efficiently solve sparsely corrupted linear systems $\mathbf{A} \mathbf{x}^*+\mathbf{\epsilon} = \mathbf{b}$ [SIAM J. Matrix Anal. Appl., 43(2), 605-637], where…
The Kaczmarz algorithm is a popular solver for overdetermined linear systems due to its simplicity and speed. In this paper, we propose a modification that speeds up the convergence of the randomized Kaczmarz algorithm for systems of linear…
The Kaczmarz algorithm is popular for iteratively solving an overdetermined system of linear equations. The traditional Kaczmarz algorithm can approximate the solution in few sweeps through the equations but a randomized version of the…
Randomized linear system solvers have become popular as they have the potential to reduce floating point complexity while still achieving desirable convergence rates. One particularly promising class of methods, random sketching solvers,…
Iterative linear solvers have gained recent popularity due to their computational efficiency and low memory footprint for large-scale linear systems. The relaxation method, or Motzkin's method, can be viewed as an iterative method that…
In this paper, for solving large-scale nonlinear equations we propose a nonlinear sampling Kaczmarz-Motzkin (NSKM) method. Based on the local tangential cone condition and the Jensen's inequality, we prove convergence of our method with two…
A multi-step extended maximum residual Kaczmarz method is presented for the solution of the large inconsistent linear system of equations by using the multi-step iterations technique. Theoretical analysis proves the proposed method is…
To find the least squares solution of a very large and inconsistent system of equations, one can employ the extended Kaczmarz algorithm. This method simultaneously removes the error term, such that a consistent system is asymptotically…
Randomized iterative algorithms for solving a factorized linear system, $\mathbf A\mathbf B\mathbf x=\mathbf b$ with $\mathbf A\in{\mathbb{R}}^{m\times \ell}$, $\mathbf B\in{\mathbb{R}}^{\ell\times n}$, and $\mathbf b\in{\mathbb{R}}^m$,…
The randomized sparse Kaczmarz method, designed for seeking the sparse solutions of the linear systems $Ax=b$, selects the $i$-th projection hyperplane with likelihood proportional to $\|a_{i}\|_2^2$, where $a_{i}^T$ is $i$-th row of $A$.…
With a quite different way to determine the working rows, we propose a novel greedy Kaczmarz method for solving consistent linear systems. Convergence analysis of the new method is provided. Numerical experiments show that, for the same…
For solving a consistent system of linear equations, the classical row-action (also known as Kaczmarz) method is a simple while really effective iteration solver. Based on the greedy index selection strategy and Polyak's heavy-ball momentum…
Kaczmarz's alternating projection method has been widely used for solving a consistent (mostly over-determined) linear system of equations Ax=b. Because of its simple iterative nature with light computation, this method was successfully…
In this paper, an extension of Kaczmarz method, the Kaczmarz method with oblique projection (KO), is introduced and analyzed. Using this method, a number of iteration steps to solve the over-determined systems of linear equations are…
Matrix factorization techniques compute low-rank product approximations of high dimensional data matrices and as a result, are often employed in recommender systems and collaborative filtering applications. However, many algorithms for this…
To solve nonlinear problems, we construct two kinds of greedy capped nonlinear Kaczmarz methods by setting a capped threshold and introducing an effective probability criterion for selecting a row of the Jacobian matrix. The capped…
We propose a new method for preconditioning Kaczmarz method by sketching. Kaczmarz method is a stochastic method for solving overdetermined linear systems based on a sampling of matrix rows. The standard approach to speed up convergence of…
In this paper, we propose a federated algorithm for solving large linear systems that is inspired by the classic randomized Kaczmarz algorithm. We provide convergence guarantees of the proposed method, and as a corollary of our analysis, we…