[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83160-en":3,"doc-seo-83160-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},83160,2336464648746,"Skyler","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","RPLSS: A Randomized Projected Linear Systems Solver","RPLSS (a unified randomized projected linear systems solver) addresses the projected linear system solver’s key bottleneck: repeated access to the full coefficient matrix. For consistent or inconsistent, underdetermined or overdetermined systems, RPLSS uses tailored randomized row/column selection schemes that rely only on partial matrix information per iteration. It develops randomized Gaussian Kaczmarz and an extended row-action variant, plus randomized coordinate descent column-action solvers. Theoretical results establish finite termination and exponential convergence, supported by experiments showing strong gains, especially with large missing data.","arXiv :2607 .06917v1 [math .NA] 8 Jul 2026  \nRPLSS: A RANDOMIZED PROJECTED LINEAR SYSTEMS SOLVER ∗  \nMENG-LONG XIAO†, TAO LI†, AND DEANNA NEEDELL‡  \nAbstract. The projected linear system solver (PLSS), by incrementally appending columns toa 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.  \nKey words. linear system; randomized PLSS; expected linear convergence; Kaczmarz method; coordinate descent  \nMSC codes. 65F10; 65F20; 94A08  \n1. Introduction. Consider a large, possibly consistent or not, general linear system  \n(1.1) Ax = b,  \nwhere A ∈ Rm ×n and b ∈ Rm are given, and x ∈ Rn is unknown. Such a linear system underpins widespread applications across image reconstruction [13, 19], signal processing [8, 6], and partial differential equations [9], etc. While the regular iterative solvers, such as Krylov subspace methods and Hermitian and skew-Hermitian methods [32, 33, 2, 30, 14], for (1.1) have been extensively matured, randomized solvers [36, 27, 12, 28, 11, 25, 5, 39, 29, 23, 20] are growing in popularity due to a revolutionary advantage. Specifically, the traditional iterative solvers, demanding full-matrix access per iteration, are costly to generate highly accurate solutions, but they are not needed or even desired since Eq.(1.1) may constitute a random subset of a larger dataset, or it may be contaminated by noise. Conversely, randomized solvers address only a sequence of small projected systems, rather than (1.1), while maintaining competitive convergence to yield the desired solutions. This makes them powerful for tackling large-scale, memory-constrained, or data-streaming applications. In this paper, we propose a family of randomized projection methods that solve a sequence of smaller systems and offer extraordinary advantages over state-of-the-art randomized solversin terms of accuracy and convergence.  \n1.1. Notations. Let Rm ×n be the collection of all real matrices with size m × n. Denote by A (i) and A (j), the i-th row and j-th column of a matrix A ∈ Rm ×n ,  \n∗ Corresponding author: T. Li ([tli@hainanu.edu.cn](tli@hainanu.edu.cn)).  \nFunding: This work was funded by the National Natural Science Foundation of China [grant number 12401493] .  \n†School of Mathematics and Statistics, Hainan University, Haikou 570228, P.R. China ([tli@hainanu.edu.cn](tli@hainanu.edu.cn), [xml@hainanu.edu.cn](xml@hainanu.edu.cn)).  \n‡Department of Mathematics, University of California, Los Angeles 90095, USA ([deanna@math.ucla.edu](deanna@math.ucla.edu)).  \n2 M.L. XIAO, T. LI, AND D. NEEDELL  \nrespectively. For index sets τk and τ˜k , Aτk and A: ,τ˜k represent the row and column submatrices of A, respectively. The symbol ej represents the j-th column of the identity matrix I, with dimension depending on the context. The Frobenius norm  \nn  \nof ","cbCaikeHyOoOLoOR","https://ap.wps.com/l/cbCaikeHyOoOLoOR","pdf",1784237,3,1,31,"English","en",105,"# Introduction\n## Notations\n# Sketch-and-Project methods","[{\"question\":\"What problem does RPLSS target in the projected linear system solver (PLSS)?\",\"answer\":\"RPLSS targets PLSS’s computational bottleneck of accessing the entire coefficient matrix at every iteration, which becomes prohibitive for very large or partially observed problems.\"},{\"question\":\"How does RPLSS reduce per-iteration computational requirements?\",\"answer\":\"RPLSS relies on randomized row or column selection strategies that require only partial matrix information per iteration, instead of using full matrix access.\"},{\"question\":\"What theoretical and practical performance properties are shown for the proposed methods?\",\"answer\":\"The methods inherit PLSS’s finite termination property and achieve an exponential convergence rate, and numerical experiments demonstrate superiority over state-of-the-art randomized methods, particularly with large missing 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problem does RPLSS target in the projected linear system solver (PLSS)?","Question",{"text":75,"@type":76},"RPLSS targets PLSS’s computational bottleneck of accessing the entire coefficient matrix at every iteration, which becomes prohibitive for very large or partially observed problems.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does RPLSS reduce per-iteration computational requirements?",{"text":80,"@type":76},"RPLSS relies on randomized row or column selection strategies that require only partial matrix information per iteration, instead of using full matrix access.",{"name":82,"@type":73,"acceptedAnswer":83},"What theoretical and practical performance properties are shown for the proposed methods?",{"text":84,"@type":76},"The methods inherit PLSS’s finite termination property and achieve an exponential convergence rate, and numerical experiments demonstrate superiority over state-of-the-art randomized methods, particularly with large missing 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