Related papers: Faster Randomized Block Sparse Kaczmarz by Averagi…
The Kaczmarz method is an iterative method for solving overcomplete linear systems of equations Ax=b. The randomized version of the Kaczmarz method put forth by Strohmer and Vershynin iteratively projects onto a randomly chosen solution…
Due to the ever growing amounts of data leveraged for machine learning and scientific computing, it is increasingly important to develop algorithms that sample only a small portion of the data at a time. In the case of linear least-squares,…
With the growth of large data as well as large-scale learning tasks, the need for efficient and robust linear system solvers is greater than ever. The randomized Kaczmarz method (RK) and similar stochastic iterative methods have received…
A greedy randomized augmented Kaczmarz (GRAK) method was proposed in [Z.-Z. Bai and W.-T. WU, SIAM J. Sci. Comput., 43 (2021), pp. A3892-A3911] for large and sparse inconsistent linear systems. However, one has to construct two new index…
The Kaczmarz algorithm is an iterative method that solves linear systems of equations. It stands out among iterative algorithms when dealing with large systems for two reasons. First, at each iteration, the Kaczmarz algorithm uses a single…
The sketch-and-project, as a general archetypal algorithm for solving linear systems, unifies a variety of randomized iterative methods such as the randomized Kaczmarz and randomized coordinate descent. However, since it aims to find a…
The Kaczmarz method is an iterative projection scheme for solving con-sistent system $Ax = b$. It is later extended to the inconsistent and ill-posed linear problems. But the classical Kaczmarz method is sensitive to the correlation of the…
The Extended Randomized Kaczmarz method is a well known iterative scheme which can find the Moore-Penrose inverse solution of a possibly inconsistent linear system and requires only one additional column of the system matrix in each…
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 $A$.…
Among recent developments centered around Randomized Kaczmarz (RK), a row-sampling iterative projection method for large-scale linear systems, several adaptions to the method have inspired faster convergence. Focusing solely on…
Existing block Kaczmarz methods face challenges in balancing computational efficiency and convergence for large sparse linear systems with scattered nonzero patterns, due to costly partitioning strategies and non-orthogonal projections. In…
The randomized Kaczmarz (RK) algorithm is one of the most computationally and memory-efficient iterative algorithms for solving large-scale linear systems. However, practical applications often involve noisy and potentially inconsistent…
The Kaczmarz algorithm is one of the most popular methods for solving large-scale over-determined linear systems due to its simplicity and computational efficiency. This method can be viewed as a special instance of a more general class of…
The Kaczmarz method is an iterative algorithm for solving systems of linear equations Ax=b. Theoretical convergence rates for this algorithm were largely unknown until recently when work was done on a randomized version of the algorithm. It…
By introducing a subsampling strategy, we propose a randomized block Kaczmarz-Motzkin method for solving linear systems. Such strategy not only determines the block size, but also combines and extends two famous strategies, i.e., randomness…
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 $\mathbf b\in{\mathbb{R}}^m$,…
Randomized block Kaczmraz method plays an important role in solving large-scale linear system. One of the key points of this type of methods is how to effectively select working rows. However, in most of the state-of-the-art randomized…
Randomized iterative algorithms have recently been proposed to solve large-scale linear systems. In this paper, we present a simple randomized extended block Kaczmarz algorithm that exponentially converges in the mean square to the unique…
The block Kaczmarz method and its variants are designed for solving the over-determined linear system. They involve iteratively projecting the current point onto the solution space of a subset of constraints. In this work, by alternately…
Recovery of a sparse signal from a nonlinear system arises in many practical applications including compressive sensing, image reconstruction and machine learning. In this paper, a fast block nonlinear Bregman-Kaczmarz method with averaging…