[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84145-en":3,"doc-seo-84145-105":29,"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":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},84145,2336464648746,"Skyler","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Adaptive Matrix-Free Low-Rank Approximation","Adaptive, matrix-free low-rank approximation studies how to build a QB factorization when a matrix A is available only through matrix-vector products. The work introduces adaptive randomized QB algorithms that estimate residual norms in Frobenius or spectral norm from random sketches with accuracy down to machine precision. A matrix-free rank-pruning step separates BLAS-3-friendly block sizes from the final rank, and an adjoint-free variant produces an orthonormal basis using only the forward operator. Experiments across matrices with varied singular-value decay achieve ranks near the truncated-SVD optimum while meeting prescribed tolerances with high probability.","arXiv :2607 .06758v1 [math .NA] 7 Jul 2026  \nADAPTIVE, MATRIX-FREE LOW-RANK APPROXIMATION  \nARNEL I. SMITH∗ , ELLY DO†, AND CHAO CHEN‡  \nAbstract. We study fixed-tolerance low-rank approximation in the matrix-free setting, where a matrix or linear operator A is accessible only through matrix-vector products and its rank must be determined adaptively to meet a prescribed error tolerance. We introduce a family of adaptive, matrix-free randomized QB algorithms. A randomized error indicator estimates the residual norm—in either the Frobenius or the spectral norm—directly from a random sketch, remaining accurate down to machine precision. A matrix-free rank-pruning step decouples the computational block size from the final rank, so that large, BLAS-3-friendly blocks can be used without over-estimating the rank, and an adjoint-free variant returns the orthonormal basis using only the forward operator. Across test matrices with diverse singular-value decays, the proposed methods attain ranks close to the truncated-SVD optimum while meeting the prescribed tolerance with high probability.  \nKey words. low-rank approximation, matrix-free algorithm, adaptive rank determination, adjoint-free range finder, fixed-precision problem, randomized norm estimation  \nAMS subject classifications. 65F55, 68W20, 15A23, 65F30, 15A18  \n1. Introduction. We consider the fixed-tolerance low-rank approximation problem. Given a matrix A ∈ Rm ×n and a target tolerance ε > 0, we seek to compute matrices Q ∈ Rm×k and B ∈ Rk ×n such that  \n(1.1) ∥A − QB∥ ≤ τ,  \nwhere the rank k ≪ min(m, n) is a priori unknown and must be determined dynamically. In (1.1), the tolerance τ is a user-prescribed parameter, specified either as an absolute threshold τ = ε or a relative threshold τ = ε∥A∥ . We focus on the Frobenius norm ∥ · ∥F and the spectral norm ∥ · ∥2 . As is typical of randomized methods, the algorithms we develop satisfy (1.1) with high probability and, in the spectral norm, up to a modest constant (Theorem 4.3) .  \nWe operate in the matrix-free setting: A is accessible only through matrixvector products (matvecs)—the forward and adjoint actions x →7 Ax and y →7 A⊤y—while its individual entries are not. This setting is ubiquitous in computational science. In Bayesian inverse problems and uncertainty quantification, for instance, the Hessian of the log-posterior is defined implicitly, yet a low-rank approximation of it accelerates downstream optimization and sampling [4, 12] . There, a single matvec may require a full forward and adjoint PDE solve, so the dominant cost is the number of operator applications, not the internal dense linear algebra. Our goal is therefore to solve (1.1) while minimizing the number of matvecs, and to do so accurately for ε down to machine precision.  \nWe focus on a specific structural form of (1.1), the QB approximation [33, 47], where Q has orthonormal columns; the choice B = Q⊤ A then minimizes the residual in the Frobenius norm. QB is a building block for many matrix factorizations, decoupling range identification (the range finder) from entry approximation (the projection): examples include the truncated SVD [15], the interpolative decomposition [8], the higher-order SVD for tensor compression [9], and hierarchical matrix approximations [21, 29] . Each relies on the fixed-tolerance QB primitive that the matrix-free methods of section 3 and section 4 provide.  \n∗ Department of Mathematics, North Carolina State University, Raleigh, NC ([aismith@ncsu.edu](aismith@ncsu.edu)).  \n†Department of Mathematics, North Carolina State University, Raleigh, NC ([ntdo2@ncsu.edu](ntdo2@ncsu.edu)) .  \n‡Department of Mathematics, North Carolina State University, Raleigh, NC ([cchen49@ncsu.edu](cchen49@ncsu.edu)) .  \n2 ARNEL I. SMITH, ELLY DO, AND CHAO CHEN  \n1.1. Existing Work and Limitations.  \nClassical deterministic methods. Direct factorization methods such as the singular value decomposition (SVD) and rank-revealing QR (RRQR) [16, 20] offer exc","cbCaieNJNGAOS7xE","https://ap.wps.com/l/cbCaieNJNGAOS7xE","pdf",919786,1,25,"English","en",105,"# Introduction\n## Existing Work and Limitations\n## Modern Randomized Methods","[{\"question\":\"What does “fixed-tolerance low-rank approximation” mean in this document?\",\"answer\":\"It means choosing an unknown rank k so that the QB approximation satisfies a user-prescribed error tolerance τ in either Frobenius or spectral norm.\"},{\"question\":\"What makes the approach “matrix-free” here?\",\"answer\":\"The algorithms assume A is not stored and only matrix-vector products are available for the forward action Ax and, when needed, the adjoint action A^T y.\"},{\"question\":\"How do the proposed methods determine the rank adaptively?\",\"answer\":\"They use a randomized error indicator that estimates the residual norm from a random sketch, followed by a matrix-free rank-pruning step to avoid over-estimating the final rank.\"}]",1784193403,63,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"adaptive-matrix-free-low-rank-approximation","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/adaptive-matrix-free-low-rank-approximation/84145/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What does “fixed-tolerance low-rank approximation” mean in this document?","Question",{"text":75,"@type":76},"It means choosing an unknown rank k so that the QB approximation satisfies a user-prescribed error tolerance τ in either Frobenius or spectral norm.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What makes the approach “matrix-free” here?",{"text":80,"@type":76},"The algorithms assume A is not stored and only matrix-vector products are available for the forward action Ax and, when needed, the adjoint action A^T y.",{"name":82,"@type":73,"acceptedAnswer":83},"How do the proposed methods determine the rank adaptively?",{"text":84,"@type":76},"They use a randomized error indicator that estimates the residual norm from a random sketch, followed by a matrix-free rank-pruning step to avoid over-estimating the final rank.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]