The Generalized Matrix Separation Problem: Algorithms

The Generalized Matrix Separation Problem: Algorithms | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article The Generalized Matrix Separation Problem: Algorithms Xuemei Chen, Owen Deen This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7621464/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract When given a generalized matrix separation problem, which aims to recover a low rank matrix $L_0$ and a sparse matrix $S_0$ from $M_0=L_0+HS_0$, the work \cite{CW25} proposes a novel convex optimization problem whose objective function is the sum of the $\ell_1$-norm and nuclear norm. In this paper we detail the iterative algorithms and its associated computations for solving this convex optimization problem. We present various efficient implementation strategies, with attention to practical cases where $H$ is circulant, separable, or block structured. Notably, we propose a preconditioning technique that drastically improved the performance of our algorithms in terms of efficiency, accuracy, and robustness. While this paper serves as an illustrative algorithm implementation manual, we also provide theoretical guarantee for our preconditioning strategy. Numerical results demonstrate the effectiveness of the proposed approach. matrix separation robust PCA low rank sparse background removal Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-7621464","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":530212426,"identity":"927e140f-9ae1-4910-b18d-534069c04af7","order_by":0,"name":"Xuemei Chen","email":"","orcid":"","institution":"University of North Carolina Wilmington","correspondingAuthor":false,"prefix":"","firstName":"Xuemei","middleName":"","lastName":"Chen","suffix":""},{"id":530212427,"identity":"6c29a94f-8307-47c7-a965-912604a2b352","order_by":1,"name":"Owen Deen","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAnUlEQVRIiWNgGAWjYFCCBCCuAOIDDAwSJGg5Q7IWxjZStPCzJx98XDjvcDTfAeaDt3mI0SLZ8yzZeOa2w7kzD7AlWxOlxeBGjpk0L1DLhgM8ZtJEasn//pt3DkgL/zditeSwMfM2gG1hI04L0C/G0jzH0nNnHmYztpxDjBZgiD38zFNjndt3vPnhjTfEaEEAZtKUj4JRMApGwSjABwD2ZzMYWQGwnwAAAABJRU5ErkJggg==","orcid":"","institution":"University of Maryland, College Park","correspondingAuthor":true,"prefix":"","firstName":"Owen","middleName":"","lastName":"Deen","suffix":""}],"badges":[],"createdAt":"2025-09-15 14:08:28","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7621464/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7621464/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":93648381,"identity":"ed86ff02-0e6d-4a9f-8087-0c99ba89204a","added_by":"auto","created_at":"2025-10-16 05:01:54","extension":"json","order_by":0,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":3598,"visible":true,"origin":"","legend":"","description":"","filename":"8cf09a4b3af2478e99431d306aecd01a.json","url":"https://assets-eu.researchsquare.com/files/rs-7621464/v1/bb34ad0a005b583f5ccbb6f6.json"},{"id":93648938,"identity":"78664321-a21a-47fc-91bb-eff784e94f79","added_by":"auto","created_at":"2025-10-16 05:10:03","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2128811,"visible":true,"origin":"","legend":"","description":"","filename":"journalversion.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7621464/v1_covered_7cc22182-e822-4728-a4d9-5f43c78a6e6b.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"The Generalized Matrix Separation Problem: Algorithms","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"matrix separation, robust PCA, low rank, sparse, background removal","lastPublishedDoi":"10.21203/rs.3.rs-7621464/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7621464/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"When given a generalized matrix separation problem, which aims to recover a low rank matrix $L_0$ and a sparse matrix $S_0$ from $M_0=L_0+HS_0$, the work \\cite{CW25} proposes a novel convex optimization problem whose objective function is the sum of the $\\ell_1$-norm and nuclear norm. 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