Enhanced Algorithmic Convergence Analysis for Federated Low-Rank Matrix Completion

Enhanced Algorithmic Convergence Analysis for Federated Low-Rank Matrix Completion | 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 Enhanced Algorithmic Convergence Analysis for Federated Low-Rank Matrix Completion Chris Junchi Li This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-5206289/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 In this work, we explore the low-rank matrix completion (LRMC) problem, which is crucial for data science and machine learning applications. We develop and analyze a Gradient Descent-based solution, termed Alternating GD and Minimization (AltGDmin), aimed at efficiently recovering an $n \times q$ rank-$r$ matrix $\Xstar$ from a subset of its entries, particularly when $r \ll \min (n, q)$. Our theoretical guarantees, including iteration and sample complexity bounds, demonstrate that AltGDmin is not only the most communication-efficient solution in a federated setting but also one of the fastest methods available, achieving the second-best sample complexity among all iterative solutions to LRMC. Furthermore, we establish convergence results for noisy LRMC and highlight how our lemmas improve the sample complexity guarantee for AltMin, the fastest centralized solution. Overall, our findings emphasize the importance of leveraging the low-dimensional structure of data to recover missing entries and enhance predictive capabilities in matrix completion. Low rank matrix completion Federated learning Alternating Gradient Minimization (AltGDMin) Alternating Minimization (AltMin) Full Text Additional Declarations The authors declare no competing interests. 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-5206289","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":362457118,"identity":"60b331e5-709e-463e-814d-abee9a595999","order_by":0,"name":"Chris Junchi Li","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABBElEQVRIiWNgGAWjYBACxgYGBmYGAwsefgYQg4HBgIEdLMFMSIsEj2QDTAszAS1QWQkGgwPEamGekXvwc0GBhIzx8eMPmAsq7hnzMzMfk2CosE5swOWwGXnJ0jOADjM7k2PAPONMsZlkM1uaBMOZdDxacsyYeUBabvAwMPO2JdgYHOYxk2BsO0xYi/EM9gfMvP8SbOwP83+TYPxHhBYDYAgw8zYkmBkw87BJMDbg0dLzxlgapEUC6JfDM44lGEscZjO2SDiWboxLi2F7juFnnj829vztxx8+LqhJMOxvb35440ONtSxOLcgSB6A0i0QCDuUgII9NkPkDHh2jYBSMglEw8gAAwlxKO7d2kLsAAAAASUVORK5CYII=","orcid":"","institution":"","correspondingAuthor":true,"prefix":"","firstName":"Chris","middleName":"Junchi","lastName":"Li","suffix":""}],"badges":[],"createdAt":"2024-10-04 22:48:22","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-5206289/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-5206289/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":66231406,"identity":"6881f128-4f3c-4638-85d2-cbdaa8dc6749","added_by":"auto","created_at":"2024-10-09 04:12:26","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":541295,"visible":true,"origin":"","legend":"","description":"","filename":"lrFedMatrixCompletion.pdf","url":"https://assets-eu.researchsquare.com/files/rs-5206289/v1_covered_a1a00ae2-79c1-406e-aef2-32a294161e99.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003eEnhanced Algorithmic Convergence Analysis for Federated Low-Rank Matrix Completion\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"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":"Low rank matrix completion, Federated learning, Alternating Gradient Minimization (AltGDMin), Alternating Minimization (AltMin)","lastPublishedDoi":"10.21203/rs.3.rs-5206289/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-5206289/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eIn this work, we explore the low-rank matrix completion (LRMC) problem, which is crucial for data science and machine learning applications. 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