A Continuous Gaussian Mixture Approach to Sample Multivariate Gaussians constrained by linear inequalities

A Continuous Gaussian Mixture Approach to Sample Multivariate Gaussians constrained by linear inequalities | 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 A Continuous Gaussian Mixture Approach to Sample Multivariate Gaussians constrained by linear inequalities Mehdi Chahine AMROUCHE, Jérôme IDIER, Hervé CARFANTAN This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7264784/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 9 You are reading this latest preprint version Abstract We propose a new MCMC method to sample from truncated multivariate Gaussian (TMG) distributions under linear inequality constraints. Unlike existing approaches, our method is applicable even when the underlying, unconstrained Gaussian distribution is improper, and whatever the number of constraints. Our algorithm relies on continuous Gaussian mixture (CGM) decompositions of the target TMG distribution, derived from novel integral identities. Such decompositions are exact within the admissible domain, which allows us to get (asymptotically) exact samples from the TMG using blocked Gibbs updates followed by a rejection step to discard samples outside the admissible domain. Empirical results demonstrate that the proposed method outperforms state-of-the-art alternatives over a set of challenging settings. Markov-chain Monte Carlo Truncated Multivariate Gaussians Continuous Gaussian Mixtures Linear Inequality Constraints Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Revision requested 18 Dec, 2025 Reviews received at journal 17 Dec, 2025 Reviews received at journal 20 Nov, 2025 Reviewers agreed at journal 27 Sep, 2025 Reviewers agreed at journal 07 Aug, 2025 Reviewers invited by journal 07 Aug, 2025 Editor assigned by journal 02 Aug, 2025 Submission checks completed at journal 01 Aug, 2025 First submitted to journal 31 Jul, 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. 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-7264784","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":497383067,"identity":"94ed1a0d-3b65-43d0-8246-1a2e288d54a3","order_by":0,"name":"Mehdi Chahine AMROUCHE","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA2klEQVRIiWNgGAWjYBCDBAb2xgZStfAcJFmLRAKRSnXbT6d9+Nhml8cv+bjtw88dDHK6sxsYH/Pg0WJ2JnfzzJltycWSsxObZ/aeYTA2u3OA2RivlgO5m5l52w4kbrid2MzA28aQuO1GAps0Xi3n30K07L95sJnxL0QL+2+8Wm7AbJFgbGaG2cKMX8vbzYwzziUnzjiT2Mwse0YC6JeDzZJz8DosdzPDhzK7xP72448Z3+6wkTO73Xzwwxs8WlABY4MEA4MEYwPRGkBagIQECRpGwSgYBaNgRAAAaUNRx6BrFzIAAAAASUVORK5CYII=","orcid":"","institution":"Research Institute in Astrophysics and Planetology","correspondingAuthor":true,"prefix":"","firstName":"Mehdi","middleName":"Chahine","lastName":"AMROUCHE","suffix":""},{"id":497383068,"identity":"b99a9d3c-5ab2-43d9-9d97-1d26699e0579","order_by":1,"name":"Jérôme IDIER","email":"","orcid":"","institution":"Laboratoire des Sciences du Numérique de Nantes","correspondingAuthor":false,"prefix":"","firstName":"Jérôme","middleName":"","lastName":"IDIER","suffix":""},{"id":497383069,"identity":"fda7b7fa-b8a5-4ab3-b0ce-12da8062c8ec","order_by":2,"name":"Hervé CARFANTAN","email":"","orcid":"","institution":"Research Institute in Astrophysics and Planetology","correspondingAuthor":false,"prefix":"","firstName":"Hervé","middleName":"","lastName":"CARFANTAN","suffix":""}],"badges":[],"createdAt":"2025-07-31 17:53:15","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7264784/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7264784/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":88936081,"identity":"d3ce36a3-1114-47b1-bd5f-5d8a1b172251","added_by":"auto","created_at":"2025-08-13 01:40:37","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2840407,"visible":true,"origin":"","legend":"","description":"","filename":"TruncatedGaussianSamplingSubmitted.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7264784/v1_covered_dba41649-1059-42ed-9e03-6c438033a96d.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"A Continuous Gaussian Mixture Approach to Sample Multivariate Gaussians constrained by linear inequalities","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":"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":"Markov-chain Monte Carlo, Truncated Multivariate Gaussians, Continuous Gaussian Mixtures, Linear Inequality Constraints","lastPublishedDoi":"10.21203/rs.3.rs-7264784/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7264784/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"We propose a new MCMC method to sample from truncated multivariate Gaussian (TMG) distributions under linear inequality constraints. Unlike existing approaches, our method is applicable even when the underlying, unconstrained Gaussian distribution is improper, and whatever the number of constraints. Our algorithm relies on continuous Gaussian mixture (CGM) decompositions of the target TMG distribution, derived from novel integral identities. Such decompositions are exact within the admissible domain, which allows us to get (asymptotically) exact samples from the TMG using blocked Gibbs updates followed by a rejection step to discard samples outside the admissible domain. Empirical results demonstrate that the proposed method outperforms state-of-the-art alternatives over a set of challenging settings.","manuscriptTitle":"A Continuous Gaussian Mixture Approach to Sample Multivariate Gaussians constrained by linear inequalities","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-08-13 01:32:26","doi":"10.21203/rs.3.rs-7264784/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2025-12-19T02:39:38+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-12-17T20:19:47+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-11-21T04:04:16+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"14232284220380593546415476078853526407","date":"2025-09-27T17:49:57+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"213564401788161179344810828659079375671","date":"2025-08-07T23:06:45+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2025-08-07T14:33:01+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2025-08-03T03:37:10+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2025-08-01T05:12:32+00:00","index":"","fulltext":""},{"type":"submitted","content":"Statistics and Computing","date":"2025-07-31T17:41:12+00:00","index":"","fulltext":""}],"status":"published","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}}],"origin":"","ownerIdentity":"021a05c3-b313-40aa-b382-c2da16ff9f14","owner":[],"postedDate":"August 13th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"under-review","subjectAreas":[],"tags":[],"updatedAt":"2026-04-06T18:23:28+00:00","versionOfRecord":[],"versionCreatedAt":"2025-08-13 01:32:26","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-7264784","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7264784","identity":"rs-7264784","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}