Global Optimization on Matrix Lie Groups via Intermittent Diffusion: A Geometric Stochastic Framework with Provable Convergence

Global Optimization on Matrix Lie Groups via Intermittent Diffusion: A Geometric Stochastic Framework with Provable Convergence | 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 Global Optimization on Matrix Lie Groups via Intermittent Diffusion: A Geometric Stochastic Framework with Provable Convergence Yinpu Ma, Cunlin Li, Zhichao Wang, Qian Li This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9250878/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 8 You are reading this latest preprint version Abstract This paper proposes a stochastic differential equation (SDE)-based global optimization method on matrix Lie groups, specifically addressing the challenge of local optima trapping in optimization over special orthogonal groups SO(n) and related structures. By combining Riemannian conjugate gradient methods with an intermittent diffusion mechanism, we develop an optimization framework that preserves geometric structures while maintaining global exploration capabilities. Theoretical analysis demonstrates first-order convergence between the continuous-time SDE and its discrete numerical implementation, with almost sure convergence to the global optimum under appropriate conditions. Experimental validation through two applications - multimodal function optimization and robotic path planning - confirms the effectiveness of the proposed ID-RCG algorithm, showing superior performance in both convergence rate and global search capability compared to conventional methods. matrix Lie groups stochastic optimization global convergence intermittent diffusion Riemannian conjugate gradient special orthogonal group geometric numerical integration robotic path planning Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Reviews received at journal 05 May, 2026 Reviewers agreed at journal 04 May, 2026 Reviewers agreed at journal 04 May, 2026 Reviewers agreed at journal 04 May, 2026 Reviewers invited by journal 16 Apr, 2026 Editor assigned by journal 30 Mar, 2026 Submission checks completed at journal 30 Mar, 2026 First submitted to journal 28 Mar, 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. 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. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. 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-9250878","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":625457460,"identity":"93c7dea6-ebe4-457e-961d-9688a6837405","order_by":0,"name":"Yinpu Ma","email":"","orcid":"","institution":"North Minzu University","correspondingAuthor":false,"prefix":"","firstName":"Yinpu","middleName":"","lastName":"Ma","suffix":""},{"id":625457461,"identity":"f6186474-c5d0-4cc8-96ff-bc3c672b7dfd","order_by":1,"name":"Cunlin Li","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAzUlEQVRIiWNgGAWjYBACeYbDBx///PdPjp+9gUgtho3Hko0Z2A4YS/YcINaaw2fMpIFaEg1uJBCpg7HtjIF0Ac+dBMmZjzfeYKixiSaohZ3nWIHxDIlnefzSacUWDMfSchsI2jLj8IYEHgPmYsnZOWYSjA2HCWthuP/A4ABPAnPihptniNVy4IhhM8+Bw4kbbvAQqcWw4Vgy48yGNGAgA/2SQIxfgFF5/MfHBhtgVB7eeONDjQ0RDkMCBhIJpCiHaCFVxygYBaNgFIwMAADr/UbAlSftzwAAAABJRU5ErkJggg==","orcid":"","institution":"North Minzu University","correspondingAuthor":true,"prefix":"","firstName":"Cunlin","middleName":"","lastName":"Li","suffix":""},{"id":625457462,"identity":"805b7b6d-7dcf-419b-8789-60a1a0443e6d","order_by":2,"name":"Zhichao Wang","email":"","orcid":"","institution":"Curtin University","correspondingAuthor":false,"prefix":"","firstName":"Zhichao","middleName":"","lastName":"Wang","suffix":""},{"id":625457467,"identity":"db28f266-3147-4e0d-a6e2-afd73f5cb5b0","order_by":3,"name":"Qian Li","email":"","orcid":"","institution":"North Minzu University","correspondingAuthor":false,"prefix":"","firstName":"Qian","middleName":"","lastName":"Li","suffix":""}],"badges":[],"createdAt":"2026-03-28 08:08:35","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-9250878/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9250878/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":107460606,"identity":"36a6fa55-6cee-4ac8-a3fe-a00b146da77d","added_by":"auto","created_at":"2026-04-21 16:53:22","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":958995,"visible":true,"origin":"","legend":"","description":"","filename":"snarticletemplate3.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9250878/v1_covered_c99487ac-59e7-4bd1-bb4c-15854ee219a8.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"\u003cp\u003eGlobal Optimization on Matrix Lie Groups via Intermittent Diffusion: A Geometric Stochastic Framework with Provable Convergence\u003c/p\u003e","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":"discover-computing","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"","sideBox":"Learn more about [Discover Computing](https://link.springer.com/journal/10791)","snPcode":"10791","submissionUrl":"https://submission.springernature.com/new-submission/10791/3","title":"Discover Computing","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Discover Series","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"matrix Lie groups, stochastic optimization, global convergence, intermittent diffusion, Riemannian conjugate gradient, special orthogonal group, geometric numerical integration, robotic path planning","lastPublishedDoi":"10.21203/rs.3.rs-9250878/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9250878/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"This paper proposes a stochastic differential equation (SDE)-based global optimization method on matrix Lie groups, specifically addressing the challenge of local optima trapping in optimization over special orthogonal groups SO(n) and related structures. 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