An NPDo Approach for Principal Joint Block Diagonalization

An NPDo Approach for Principal Joint Block Diagonalization | 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 NPDo Approach for Principal Joint Block Diagonalization Ren-Cang Li, Ding Lu, Li Wang, Lei-Hong Zhang This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7463403/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 14 Apr, 2026 Read the published version in BIT Numerical Mathematics → Version 1 posted You are reading this latest preprint version Abstract Matrix joint block-diagonalization (jbd) frequently arises from diverse applications such as independent component analysis, blind source separation, and common principal component analysis (CPCA), among others. Particularly, CPCA aims at joint diagonalization, i.e., each block size being 1-by-1. This paper is concerned with principal joint block-diagonalization (pjbd), which aim to achieve two goals: 1) partial joint block-diagonalization, and 2) identification of dominant common block-diagonal parts for all involved matrices. This is in contrast to most existing methods, especially the popular ones based on Givens rotation, which focus on full joint diagonalization and quickly become impractical for matrices of even moderate size (300-by-300 or larger). An NPDo approach is proposed and it is built on a nonlinear polar decomposition with orthogonal polar factor dependency that characterizes the solutions of the optimization problem designed to achieve pjbd, and it is shown the associated SCF iteration is globally convergent to a stationary point while the objective function increases monotonically during the iterative process. Numerical experiments are presented to illustrate the effectiveness of the NPDo approach and its superiority to Givens rotation-based methods. MSC Classification: 62H25 , 65F30 , 65K05 , 90C26 principal joint block-diagonalization pjbd principal joint diagonalization pjd common principal component analysis NPDo SCF Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 14 Apr, 2026 Read the published version in BIT Numerical Mathematics → 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. 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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-7463403","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":525311848,"identity":"9d493385-9448-4085-b589-3992af5464c8","order_by":0,"name":"Ren-Cang Li","email":"","orcid":"","institution":"The University of Texas at Arlington","correspondingAuthor":false,"prefix":"","firstName":"Ren-Cang","middleName":"","lastName":"Li","suffix":""},{"id":525311849,"identity":"3a9d2009-39ba-4d29-8c04-1a0ace51bf9f","order_by":1,"name":"Ding Lu","email":"","orcid":"","institution":"University of Kentucky","correspondingAuthor":false,"prefix":"","firstName":"Ding","middleName":"","lastName":"Lu","suffix":""},{"id":525311852,"identity":"ab8a00e2-7a54-4e61-bf63-a58020687b81","order_by":2,"name":"Li Wang","email":"","orcid":"","institution":"The University of Texas at Arlington","correspondingAuthor":false,"prefix":"","firstName":"Li","middleName":"","lastName":"Wang","suffix":""},{"id":525311853,"identity":"8ef05adf-c4ac-41b5-a1bb-482c3442a7c4","order_by":3,"name":"Lei-Hong Zhang","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA1UlEQVRIiWNgGAWjYBACPjBZcEAOSBoAMTNhLWxg0uCAMelaEhuI18Lee/jlD4M76fNnJG/8wFBhndjAfvYAfi0859IsJAye5W64kVYswXAmPbGBJy8BvxaJHDMDA4PDuRskcgwkGNsOJzZI8BgQ1pJgcDhdfkaO8Q/Gf8RpMX5wwOBwAsONHDMJxgZitPCcMWNsMDhsuOHMszKLhGPpxm08Ofi18LP3GH/8UXFYXr49efONDzXWsv3sZ/BrAbsNzkxggMUUfsD8gQhFo2AUjIJRMJIBAKWFQt93T9DUAAAAAElFTkSuQmCC","orcid":"","institution":"Soochow University","correspondingAuthor":true,"prefix":"","firstName":"Lei-Hong","middleName":"","lastName":"Zhang","suffix":""}],"badges":[],"createdAt":"2025-08-26 13:23:32","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7463403/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7463403/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s10543-026-01124-w","type":"published","date":"2026-04-14T15:57:16+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":93067374,"identity":"5830da66-05bf-40a8-9aa3-f113d7caa897","added_by":"auto","created_at":"2025-10-08 17:01:17","extension":"json","order_by":0,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":5443,"visible":true,"origin":"","legend":"","description":"","filename":"b8b68da79cfa491d9fc49e12694e5158.json","url":"https://assets-eu.researchsquare.com/files/rs-7463403/v1/6434d7177beb4069613d6d65.json"},{"id":107351023,"identity":"328d5734-6de7-4c47-8c4b-58acfa6c6d0c","added_by":"auto","created_at":"2026-04-20 16:07:53","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":637036,"visible":true,"origin":"","legend":"","description":"","filename":"NPDo4JBDvBIT.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7463403/v1_covered_c9b73f73-cd37-4a8f-a73f-c049738550df.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"An NPDo Approach for Principal Joint Block Diagonalization","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"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":"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":"principal joint block-diagonalization, pjbd, principal joint diagonalization, pjd, common principal component analysis, NPDo, SCF","lastPublishedDoi":"10.21203/rs.3.rs-7463403/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7463403/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eMatrix joint block-diagonalization (jbd) frequently arises from diverse applications such as independent component analysis, blind source separation, and common principal component analysis (CPCA), among others. Particularly, CPCA aims at joint diagonalization, i.e., each block size being 1-by-1. This paper is concerned with principal joint block-diagonalization (pjbd), which aim to achieve two goals: 1) partial joint block-diagonalization, and 2) identification of dominant common block-diagonal parts for all involved matrices. This is in contrast to most existing methods, especially the popular ones based on Givens rotation, which focus on full joint diagonalization and quickly become impractical for matrices of even moderate size (300-by-300 or larger). An NPDo approach is proposed and it is built on a nonlinear polar decomposition with orthogonal polar factor dependency that characterizes the solutions of the optimization problem designed to achieve pjbd, and it is shown the associated SCF iteration is globally convergent to a stationary point while the objective function increases monotonically during the iterative process. 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