Estimating the persistent homology of Rn-valued functions using function-geometric multifiltrations

Estimating the persistent homology of Rn-valued functions using function-geometric multifiltrations | 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 Estimating the persistent homology of R n -valued functions using function-geometric multifiltrations Ethan André, Jingyi Li, David Loiseaux, Steve Oudot This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8584070/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 7 You are reading this latest preprint version Abstract Given an unknown Rn-valued function f on a metric space X, can we approximate the persistent homology of f from a finite sampling of X with known pairwise distances and function values? This question has been answered in the case n = 1, assuming f is Lipschitz continuous and X is a sufficiently regular geodesic metric space, and using filtered geometric complexes with fixed scale parameter for the approximation. In this paper we answer the question for arbitrary n, under similar assumptions and using function-geometric multifiltrations. Our analysis offers a different view on these multifiltrations by focusing on their approximation properties rather than on their stability properties. We also leverage the multiparameter setting to provide insight into the influence of the scale parameter, whose choice is central to this type of approach. From a practical standpoint, we show that our approximation results are robust to input noise, and that functiongeometric multifiltrations have good statistical convergence properties. We also provide an algorithm to compute our estimators, and we use its implementation to conduct extensive experiments, on both synthetic and real biological data, in order to validate our theoretical results. Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Reviews received at journal 10 May, 2026 Reviewers agreed at journal 13 Feb, 2026 Reviewers agreed at journal 28 Jan, 2026 Reviewers invited by journal 28 Jan, 2026 Editor assigned by journal 20 Jan, 2026 Submission checks completed at journal 15 Jan, 2026 First submitted to journal 12 Jan, 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-8584070","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":581658991,"identity":"483fb5e8-9f13-4541-bae0-35f437253255","order_by":0,"name":"Ethan André","email":"","orcid":"","institution":"École Normale Supérieure - PSL","correspondingAuthor":false,"prefix":"","firstName":"Ethan","middleName":"","lastName":"André","suffix":""},{"id":581658998,"identity":"dc874e92-5bad-4826-87d8-a9f1f1699b63","order_by":1,"name":"Jingyi Li","email":"data:image/png;base64,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","orcid":"","institution":"École Polytechnique","correspondingAuthor":true,"prefix":"","firstName":"Jingyi","middleName":"","lastName":"Li","suffix":""},{"id":581659001,"identity":"2741f720-6b24-4ac7-82e5-2cc4d62efa38","order_by":2,"name":"David Loiseaux","email":"","orcid":"","institution":"Inria Saclay - Île-de-France Research Centre","correspondingAuthor":false,"prefix":"","firstName":"David","middleName":"","lastName":"Loiseaux","suffix":""},{"id":581659002,"identity":"d004af77-67cf-4c8c-a8d4-d7bcc86bcd32","order_by":3,"name":"Steve Oudot","email":"","orcid":"","institution":"Inria Saclay - Île-de-France Research Centre","correspondingAuthor":false,"prefix":"","firstName":"Steve","middleName":"","lastName":"Oudot","suffix":""}],"badges":[],"createdAt":"2026-01-12 16:53:31","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8584070/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8584070/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":101752510,"identity":"74d05506-e5d0-4d08-94b6-14f91f30bf13","added_by":"auto","created_at":"2026-02-03 10:27:52","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":3759585,"visible":true,"origin":"","legend":"","description":"","filename":"mphestimationJACT.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8584070/v1_covered_79a1a8cd-69f3-4e8f-adfe-d9a6a1c6e67d.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"\u003cp\u003eEstimating the persistent homology of R\u003csup\u003en\u003c/sup\u003e-valued functions using function-geometric multifiltrations\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":"journal-of-applied-and-computational-topology","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"apct","sideBox":"Learn more about [Journal of Applied and Computational Topology](https://www.springer.com/journal/41468)","snPcode":"41468","submissionUrl":"https://submission.springernature.com/new-submission/41468/3","title":"Journal of Applied and Computational Topology","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"","lastPublishedDoi":"10.21203/rs.3.rs-8584070/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8584070/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eGiven an unknown Rn-valued function f on a metric space X, can we approximate the persistent homology of f from a finite sampling of X with known pairwise distances and function values? This question has been answered in the case n = 1, assuming f is Lipschitz continuous and X is a sufficiently regular geodesic metric space, and using filtered geometric complexes with fixed scale parameter for the approximation. In this paper we answer the question for arbitrary n, under similar assumptions and using function-geometric multifiltrations. Our analysis offers a different view on these multifiltrations by focusing on their approximation properties rather than on their stability properties. We also leverage the multiparameter setting to provide insight into the influence of the scale parameter, whose choice is central to this type of approach. From a practical standpoint, we show that our approximation results are robust to input noise, and that functiongeometric multifiltrations have good statistical convergence properties. We also provide an algorithm to compute our estimators, and we use its implementation to conduct extensive experiments, on both synthetic and real biological data, in order to validate our theoretical results.\u003c/p\u003e","manuscriptTitle":"Estimating the persistent homology of Rn-valued functions using function-geometric multifiltrations","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-01-30 04:36:14","doi":"10.21203/rs.3.rs-8584070/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"editorInvitedReview","content":"","date":"2026-05-10T11:34:51+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"51167766659616226401447385013164062927","date":"2026-02-13T08:00:10+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"265721177277540387997385810466366898873","date":"2026-01-28T09:02:54+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2026-01-28T05:05:49+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2026-01-20T20:10:42+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2026-01-16T04:16:31+00:00","index":"","fulltext":""},{"type":"submitted","content":"Journal of Applied and Computational Topology","date":"2026-01-12T16:42:49+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"journal-of-applied-and-computational-topology","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"apct","sideBox":"Learn more about [Journal of Applied and Computational Topology](https://www.springer.com/journal/41468)","snPcode":"41468","submissionUrl":"https://submission.springernature.com/new-submission/41468/3","title":"Journal of Applied and Computational Topology","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"60f58a56-e6c4-4232-b437-283e333bb8ef","owner":[],"postedDate":"January 30th, 2026","published":true,"recentEditorialEvents":[{"type":"editorInvitedReview","content":"","date":"2026-05-10T11:34:51+00:00","index":15,"fulltext":""}],"rejectedJournal":[],"revision":"","amendment":"","status":"under-review","subjectAreas":[],"tags":[],"updatedAt":"2026-01-30T04:36:14+00:00","versionOfRecord":[],"versionCreatedAt":"2026-01-30 04:36:14","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8584070","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8584070","identity":"rs-8584070","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}