Newton-type Method for the Orthonormal INDSCAL Problem in Metric Multidimensional Scaling

Newton-type Method for the Orthonormal INDSCAL Problem in Metric Multidimensional Scaling | 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 Newton-type Method for the Orthonormal INDSCAL Problem in Metric Multidimensional Scaling Jiao-fen Li, Rui-li Jiang, Rui-juan Jiao, Xue-lin Zhou, Ya-qiong Wen This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8746904/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 The orthonormal individual differences scaling (O-INDSCAL) model is a fundamental tool for metric multidimensional scaling of multiple doubly centered dissimilarity matrices, but efficient algorithms remain scarce when orthonormality and nonnegativity constraints are imposed in medium- to large-scale settings. In this work, we revisit O-INDSCAL from a Riemannian optimization perspective and make two main contributions. First, starting from explicit closed-form expressions of the Riemannian gradient and Hessian, we represent the Hessian as a Kronecker-product-based linear operator on the tangent space and reformulate the Riemannian Newton equation as a symmetric linear system whose dimension coincides with that of the underlying product manifold. This dimension-reduced formulation enables the use of Krylov subspace methods and substantially lowers the per-iteration cost of Newton steps while preserving second-order accuracy. Second, we design a hybrid algorithm that couples a globally convergent Riemannian curvilinear search with Barzilai--Borwein step sizes and a locally quadratically convergent Riemannian Newton refinement, thereby combining robust globalization with fast local convergence. Extensive numerical experiments on synthetic and real O-INDSCAL benchmarks show that the proposed method attains high-accuracy solutions and compares favorably with projected gradient flows, ALS/MPE-type schemes, and generic Manopt-based Riemannian solvers in terms of both efficiency and reliability. MATLAB implementations of all methods used in the numerical experiments are available at Hybrid Newton for OINDSCAL MATLAB Codes. Multidimensional scaling Individual differences scaling Riemannian optimization Newton’s method 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. 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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-8746904","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":608339072,"identity":"c86295e3-7a90-478f-b5e3-6439af6c45d9","order_by":0,"name":"Jiao-fen Li","email":"","orcid":"","institution":"Guilin University of Electronic Technology","correspondingAuthor":false,"prefix":"","firstName":"Jiao-fen","middleName":"","lastName":"Li","suffix":""},{"id":608339073,"identity":"0d409ea0-1194-4e38-9856-a00313f690a3","order_by":1,"name":"Rui-li Jiang","email":"","orcid":"","institution":"Guilin University of Electronic Technology","correspondingAuthor":false,"prefix":"","firstName":"Rui-li","middleName":"","lastName":"Jiang","suffix":""},{"id":608339074,"identity":"46b9cf10-513e-48fe-a049-77465797734b","order_by":2,"name":"Rui-juan Jiao","email":"","orcid":"","institution":"Guilin University of Electronic Technology","correspondingAuthor":false,"prefix":"","firstName":"Rui-juan","middleName":"","lastName":"Jiao","suffix":""},{"id":608339075,"identity":"5f71e025-adce-41ad-be3b-7c865be29d23","order_by":3,"name":"Xue-lin Zhou","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA7klEQVRIie3RsYrDIBjAcYPg9OWyKgmXVxAKPQqhfRWL4JS53CgEnPoAlvY5bjYUMpX0AbIc3AskT9Am1/GuXm/r4H/RwZ+iIhQKPWP0NmQERdWneC8gSfRjBAjCDe9PKmPWPUgQIoLtzLHgWvhFvq+aNDYFvKRbPovNGThyUT+U90l0aNS4uQKSnfgXtB28YY3Z7uM+wbSc08EcgdCSz2DTwUI7gmMPIRNZm8s3SYG0wJ3wE7id4kaiBLPj5E9CqZIL28qRyOmRJTBbV9675FbWHWyWr7ldT1+5XCVJVfeDh/xWpP+3PhQKhUI/ugI+Z0p8MD68nAAAAABJRU5ErkJggg==","orcid":"","institution":"Guilin University of Electronic Technology","correspondingAuthor":true,"prefix":"","firstName":"Xue-lin","middleName":"","lastName":"Zhou","suffix":""},{"id":608339076,"identity":"6f328c15-e089-4855-a8f9-c8dc5f1a1449","order_by":4,"name":"Ya-qiong Wen","email":"","orcid":"","institution":"Guilin University of Electronic Technology","correspondingAuthor":false,"prefix":"","firstName":"Ya-qiong","middleName":"","lastName":"Wen","suffix":""}],"badges":[],"createdAt":"2026-01-31 06:08:31","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8746904/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8746904/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":105034929,"identity":"023c00d3-84ef-46a8-8337-59e3a048cfae","added_by":"auto","created_at":"2026-03-20 07:24:52","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":3481503,"visible":true,"origin":"","legend":"","description":"","filename":"BITNumericalmathematicsresubmission.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8746904/v1_covered_cc96f6c6-c5a0-4e33-8782-4c12c3832956.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Newton-type Method for the Orthonormal INDSCAL Problem in Metric Multidimensional Scaling","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":"Multidimensional scaling, Individual differences scaling, Riemannian optimization, Newton’s method","lastPublishedDoi":"10.21203/rs.3.rs-8746904/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8746904/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThe orthonormal individual differences scaling (O-INDSCAL) model is a fundamental tool for metric multidimensional scaling of multiple doubly centered dissimilarity matrices, but efficient algorithms remain scarce when orthonormality and nonnegativity constraints are imposed in medium- to large-scale settings. In this work, we revisit O-INDSCAL from a Riemannian optimization perspective and make two main contributions. First, starting from explicit closed-form expressions of the Riemannian gradient and Hessian, we represent the Hessian as a Kronecker-product-based linear operator on the tangent space and reformulate the Riemannian Newton equation as a symmetric linear system whose dimension coincides with that of the underlying product manifold. This dimension-reduced formulation enables the use of Krylov subspace methods and substantially lowers the per-iteration cost of Newton steps while preserving second-order accuracy. Second, we design a hybrid algorithm that couples a globally convergent Riemannian curvilinear search with Barzilai--Borwein step sizes and a locally quadratically convergent Riemannian Newton refinement, thereby combining robust globalization with fast local convergence. 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