An Efficient Iterative Method for Direct INDSCAL with Missing Values in Metric Multidimensional Scaling

An Efficient Iterative Method for Direct INDSCAL with Missing Values 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 An Efficient Iterative Method for Direct INDSCAL with Missing Values in Metric Multidimensional Scaling Jiao-fen Li, Meng-xue Chen, Xue-lin Zhou, Chao-qian Li This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6590776/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 23 Feb, 2026 Read the published version in Statistics and Computing → Version 1 posted 9 You are reading this latest preprint version Abstract The classical INdividual Differences SCALing (INDSCAL) model is widely used for simultaneous metric multidimensional scaling (MDS) of multiple doubly centered squared dissimilarity matrices. An alternative approach, called for short direct INDSCAL, is proposed for analyzing directly the input matrices of squared dissimilarities. An important consequence is that missing values can be easily handled. In this study, we reformulate the fitting of the direct INDSCAL model with missing values as a Riemannian optimization problem defined on a product manifold consisting of Stiefel sub-manifold of zero column-sums matrices and non-negative diagonal matrices. To address this problem, we propose a simple and efficient Riemannian gradient algorithm incorporating the Zhang-Hager nonmonotone line search strategy. The global convergence of the method is established. Extensive numerical experiments are provided to illustrate the computational effectiveness of the proposed approach and to benchmark its performance against several state-of-the-art methods. Mathematics subject classification:15A24, 15A57, 65C60, 65F30 Metric multidimensional scaling Direct individual differences scaling Missingdata Riemannian optimization Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 23 Feb, 2026 Read the published version in Statistics and Computing → Version 1 posted Editorial decision: Revision requested 16 Aug, 2025 Reviews received at journal 16 Aug, 2025 Reviews received at journal 15 Aug, 2025 Reviewers agreed at journal 27 May, 2025 Reviewers agreed at journal 16 May, 2025 Reviewers invited by journal 05 May, 2025 Editor assigned by journal 05 May, 2025 Submission checks completed at journal 05 May, 2025 First submitted to journal 04 May, 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. 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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-6590776","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":452401148,"identity":"66dd82b5-122d-4836-9073-4766e69e31e6","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":452401149,"identity":"d192ec66-6584-4d0a-9a91-b635c176fdb7","order_by":1,"name":"Meng-xue Chen","email":"","orcid":"","institution":"Guilin University of Electronic Technology","correspondingAuthor":false,"prefix":"","firstName":"Meng-xue","middleName":"","lastName":"Chen","suffix":""},{"id":452401150,"identity":"da7a2d4c-83e5-4eed-a9ee-151f3a26ea3a","order_by":2,"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":"Yunnan University","correspondingAuthor":true,"prefix":"","firstName":"Xue-lin","middleName":"","lastName":"Zhou","suffix":""},{"id":452401151,"identity":"188053ba-2e3c-4c01-aafd-9ed4bb9ad6a8","order_by":3,"name":"Chao-qian Li","email":"","orcid":"","institution":"Yunnan University","correspondingAuthor":false,"prefix":"","firstName":"Chao-qian","middleName":"","lastName":"Li","suffix":""}],"badges":[],"createdAt":"2025-05-05 02:38:17","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-6590776/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-6590776/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s11222-026-10838-4","type":"published","date":"2026-02-23T15:59:03+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":103765702,"identity":"457bf3cc-e161-474e-9fa1-090f534f3fe8","added_by":"auto","created_at":"2026-03-02 16:07:52","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":5686190,"visible":true,"origin":"","legend":"","description":"","filename":"StatisticsandComputing.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6590776/v1_covered_31ba5e77-5b8c-4ad8-b8fc-df726f4e5797.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"An Efficient Iterative Method for Direct INDSCAL with Missing Values in Metric Multidimensional Scaling","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":"statistics-and-computing","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"stco","sideBox":"Learn more about [Statistics and Computing](http://link.springer.com/journal/11222)","snPcode":"11222","submissionUrl":"https://submission.nature.com/new-submission/11222/3","title":"Statistics and Computing","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Metric multidimensional scaling, Direct individual differences scaling, Missingdata, Riemannian optimization","lastPublishedDoi":"10.21203/rs.3.rs-6590776/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-6590776/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThe classical INdividual Differences SCALing (INDSCAL) model is widely used for simultaneous metric multidimensional scaling (MDS) of multiple doubly centered squared dissimilarity matrices. An alternative approach, called for short direct INDSCAL, is proposed for analyzing directly the input matrices of squared dissimilarities. An important consequence is that missing values can be easily handled. In this study, we reformulate the fitting of the direct INDSCAL model with missing values as a Riemannian optimization problem defined on a product manifold consisting of Stiefel sub-manifold of zero column-sums matrices and non-negative diagonal matrices. To address this problem, we propose a simple and efficient Riemannian gradient algorithm incorporating the Zhang-Hager nonmonotone line search strategy. The global convergence of the method is established. 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