arXiv subjects

Meng-Long Xiao

Publications and source records attributed to Meng-Long Xiao.

3 recordsLinked to original sources

arXiv2026

RPLSS: A randomized projected linear systems solver

The projected linear system solver (PLSS), by incrementally appending columns to a random or deterministic sketching matrix, provides an attractive finite termination property for consistent linear systems. Nevertheless, a critical computational bottleneck of PLSS is accessing the whole coefficient matrix per iteration, making it prohibitive for extremely large-scale problems or applications with missing data. To alleviate this limitation, we propose a unified randomized PLSS (RPLSS) framework, built upon the tailored randomized row or column selection strategies that require only partial matrix information per iteration, for solving a general linear system, whether it is under- or overdetermined, and whether it is consistent or not. Within this framework, we develop a randomized Gaussian Kaczmarz method and its extended variant as row-action solvers, and randomized coordinate descent variants as column-action solvers. Theoretically, we prove that our methods inherit the finite termination property of PLSS, while achieving an exponential convergence rate, overcoming the sluggish convergence inherent in conventional randomized Kaczmarz and coordinate descent methods. Numerical experiments demonstrate the superiority of our method against state-of-the-art randomized methods, particularly in scenarios with large missing data.

math.NA↗

arXiv2025

Two-dimensional greedy randomized Kaczmarz methods for solving large-scale linear systems

In this paper, we consider a novel two-dimensional randomized Kaczmarz method and its improved version with simple random sampling, which chooses two active rows with probability proportional to the square of their cross-product-like constant, for solving large-scale linear systems. From the greedy selection strategy with grasping two larger entries of the residual vector at each iteration, we then devise a two-dimensional greedy randomized Kaczmarz method. To improve the above methods further, motivated by the semi-randomized Kaczmarz method and Chebyshev's law of large numbers, we propose a two-dimensional semi-randomized Kaczmarz method and its modified version with simple random sampling, which is particularly advantageous for big data problems. Theoretically, we prove that the proposed methods converge to the unique least-norm solution of the consistent linear systems. Numerical results on some practical applications illustrate the superiority of the proposed methods compared with some existing ones in terms of computing time.

math.NA↗

arXiv2025

Two-dimensional greedy randomized extended Kaczmarz methods

The randomized extended Kaczmarz method, proposed by Zouzias and Freris (SIAM J. Matrix Anal. Appl. 34: 773-793, 2013), is appealing for solving least-squares problems. However, its randomly selecting rows and columns of A with probability proportional to their squared norm is unattractive compared to the greedy strategy. In this paper, we first consider a novel two-dimensional greedy randomized extended Kaczmarz method for solving large linear least-squares problems. The proposed method randomly selects two rows and two columns of A by grasping two larger entries in the magnitude of the corresponding residual vector per iteration. To improve its convergence, we then propose a two-dimensional semi-randomized extended Kaczmarz method and its modified version with simple random sampling, which is particularly favorable for big data problems. The convergence analysis of which is also established. Numerical results on some practical applications illustrate the superiority of the proposed methods compared with state-of-the-art randomized extended Kaczmarz methods, especially in terms of computing time.

math.NA↗