Quaternion tensor low rank approximation

Quaternion tensor low rank approximation | 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 Quaternion tensor low rank approximation alaeddine Zahir, Khalide Jbilou, Ahmed Ratnani This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-5936577/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 31 Oct, 2025 Read the published version in Numerical Algorithms → Version 1 posted 9 You are reading this latest preprint version Abstract In this paper, we propose novel approaches for low-rank approximation of quaternion tensors. The first method employs quasi-norms to approximate a low-rank tensor using the QT-product, which generalizes the L-product to N-mode quaternions. The second method leverages Non-Convex norms to approximate both the Tucker and TT-rank for tensor completion. We demonstrate that the proposed methods provide more accurate tensor approximations compared to traditional convex relaxations of rank, such as the nuclear norm. Furthermore, we establish theoretical guarantees supporting the effectiveness of our models. To validate their performance, we conduct extensive numerical experiments, illustrating the superiority of our methods in inpainting and denoising applications. The results confirm that incorporating Non-Convex surrogate functions and quaternion tensor representations leads to enhanced reconstruction accuracy and robustness, making them valuable tools for high-dimensional data processing. Quaternion Low Rank Non-Convex Norm Tensor. Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 31 Oct, 2025 Read the published version in Numerical Algorithms → Version 1 posted Editorial decision: Revision requested 06 May, 2025 Reviews received at journal 06 May, 2025 Reviews received at journal 09 Apr, 2025 Reviewers agreed at journal 20 Mar, 2025 Reviewers agreed at journal 08 Feb, 2025 Reviewers invited by journal 07 Feb, 2025 Editor assigned by journal 07 Feb, 2025 Submission checks completed at journal 07 Feb, 2025 First submitted to journal 31 Jan, 2025 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. 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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-5936577","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":412897626,"identity":"1707a214-dd13-4410-b1d0-5305820721ce","order_by":0,"name":"alaeddine Zahir","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABBklEQVRIiWNgGAWjYDACdiDmYZMAs5kZDBjkgNQB/FqY0bQYMzCwJRCjBcFObGDgUcCrg7+Z+eGDN2UWefwM7A8/FxTcS99w7QzbA8YcO5xaJA6zGRvOOSdRLNnAkCw9w6A4d8Pt3OMGjNuScVtzmMFMmrdNInHDAYYD0jwGCUAteWkSjNuYceqQP8z+/TdIy/4DjM2/gVrSDW7nmAG11OPUYnCYx4wZbAsDMxvIlgSolsM4tRge5imWBPolccZhNjZroBbDmbfT0g0Stx3HqUXuePvGD2/K6hL729sf3+b5kyDPdzv52IOP26pxex8OkPzLxpBAhAYUwEaqhlEwCkbBKBjeAABhw05xEskobwAAAABJRU5ErkJggg==","orcid":"","institution":"Mohammed VI Polytechnic University","correspondingAuthor":true,"prefix":"","firstName":"alaeddine","middleName":"","lastName":"Zahir","suffix":""},{"id":412897628,"identity":"918686ec-146e-4ca5-887c-19faf0036d0c","order_by":1,"name":"Khalide Jbilou","email":"","orcid":"","institution":"University of the Littoral Opal Coast","correspondingAuthor":false,"prefix":"","firstName":"Khalide","middleName":"","lastName":"Jbilou","suffix":""},{"id":412897630,"identity":"7ab3adda-1477-4043-adc0-26ce0ab6dbc0","order_by":2,"name":"Ahmed Ratnani","email":"","orcid":"","institution":"Mohammed VI Polytechnic University","correspondingAuthor":false,"prefix":"","firstName":"Ahmed","middleName":"","lastName":"Ratnani","suffix":""}],"badges":[],"createdAt":"2025-01-31 13:23:05","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-5936577/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-5936577/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s11075-025-02226-2","type":"published","date":"2025-10-31T15:57:52+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":95040633,"identity":"417c24a4-c2f0-43ea-8179-01f6933409fd","added_by":"auto","created_at":"2025-11-03 16:10:02","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":512194,"visible":true,"origin":"","legend":"","description":"","filename":"QuaternionSpringerstyleNumericalAlgorithm.pdf","url":"https://assets-eu.researchsquare.com/files/rs-5936577/v1_covered_3fd31fea-f498-42ab-ad6e-f162289f7c9f.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Quaternion tensor low rank approximation","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"numerical-algorithms","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"numa","sideBox":"Learn more about [Numerical Algorithms](http://link.springer.com/journal/11075)","snPcode":"11075","submissionUrl":"https://submission.nature.com/new-submission/11075/3","title":"Numerical Algorithms","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Quaternion, Low Rank, Non-Convex Norm, Tensor.","lastPublishedDoi":"10.21203/rs.3.rs-5936577/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-5936577/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eIn this paper, we propose novel approaches for low-rank approximation of quaternion tensors. 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