An outlier detection and recovery method for manifold-based function approximation data

An outlier detection and recovery method for manifold-based function approximation data | 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 An outlier detection and recovery method for manifold-based function approximation data Sanpeng Zheng, Guifang Liang, Qingqing Wang This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9573269/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 5 You are reading this latest preprint version Abstract Manifold-based function approximation can utilize the intrinsic low-dimensional manifold structure of high-dimensional data to improve approximation accuracy. However, outliers may significantly affect the approximation results. Most existing outlier detection methods are constructed under Euclidean metrics, which limits their effectiveness for data with manifold structures. Moreover, many outlier detection methods for manifold-based function approximation lack rigorous theoretical analysis. To address these issues, this paper proposes an outlier detection and recovery method for manifold-based function approximation data. The method achieves dimensionality reduction by projecting the points on manifold onto a local affine space, and it utilizes the sensitivity of moving least squares to outliers and the sparsity of the $l_0$-minimization model to accurately identify outliers and restore their true values. Furthermore, under a mild assumption, we address the theoretical difficulties arising from varying local affine spaces by establishing a consistency relation among them, which provides theoretical support for the proposed method. Numerical experiments show that the proposed method can accurately detect outliers and achieve high-accuracy recovery in function approximation on non-orientable closed surfaces as well as high-dimensional data with small sample sizes. MSC Classification: 65D15 , 41A10 , 65K10 , 41A63 Outlier detection Manifold-based function approximation Moving least squares l0-minimization. Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Reviewers agreed at journal 07 May, 2026 Reviewers invited by journal 07 May, 2026 Editor assigned by journal 01 May, 2026 Submission checks completed at journal 01 May, 2026 First submitted to journal 30 Apr, 2026 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. 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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-9573269","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":636082416,"identity":"57a275b1-7765-436d-b65f-70c7ac721aa8","order_by":0,"name":"Sanpeng Zheng","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABCElEQVRIiWNgGAWjYDACCWTOB2yCeLUwziBZCzMPMVrkZzcfe/jlj12eOXvv4dc2v+ryDA4wH7zNw2CXh0sL45xj6caybcnFlj3n0qxz+9iKDQ6wJVvzMCQX49LCLJFjJi3ZwJy44UaOmXFuD0/ihgM8ZtI8DAcSG3BoYZPI/yYt8ac+ccP9N2bGlj0SQC383/Bq4ZHIYZP8wHYYaAuP8WOGHwYgW9jwapGQSDOTZmw7nrizJ8eMsbchIXHmYTZjyzkGyTi1yM9Ifib540914nb2M8YffvypS+w73vzwxpsKO5xawEEAig4DkL8Y20BcBggXH2D8AVHD/IHhD36Vo2AUjIJRMDIBAMQdVz7sQmYJAAAAAElFTkSuQmCC","orcid":"","institution":"School of Mathematical Sciences, Guizhou Normal University","correspondingAuthor":true,"prefix":"","firstName":"Sanpeng","middleName":"","lastName":"Zheng","suffix":""},{"id":636082417,"identity":"69bf8a70-ff64-4e66-ac4c-499e5d8dd2dd","order_by":1,"name":"Guifang Liang","email":"","orcid":"","institution":"School of Mathematical Sciences, Guizhou Normal University","correspondingAuthor":false,"prefix":"","firstName":"Guifang","middleName":"","lastName":"Liang","suffix":""},{"id":636082418,"identity":"888109f5-8e6f-4361-a71a-67d3e8215f9b","order_by":2,"name":"Qingqing Wang","email":"","orcid":"","institution":"School of Mathematical Sciences, Beihang University","correspondingAuthor":false,"prefix":"","firstName":"Qingqing","middleName":"","lastName":"Wang","suffix":""}],"badges":[],"createdAt":"2026-04-30 07:08:54","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-9573269/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9573269/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":109405645,"identity":"d89c15cd-6048-4de3-8542-6e83f6b2a7e1","added_by":"auto","created_at":"2026-05-17 13:19:31","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1057974,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9573269/v1_covered_5cc8f4b9-c9b4-40bd-8c88-c36325463d24.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"An outlier detection and recovery method for manifold-based function approximation data","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"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":"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":"Outlier detection, Manifold-based function approximation, Moving least squares, l0-minimization.","lastPublishedDoi":"10.21203/rs.3.rs-9573269/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9573269/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eManifold-based function approximation can utilize the intrinsic low-dimensional manifold structure of high-dimensional data to improve approximation accuracy. 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