Geometric Mixture Classifier A Discriminative Per-Class Mixture of Hyperplanes for Fast, Transparent Classification

Geometric Mixture Classifier A Discriminative Per-Class Mixture of Hyperplanes for Fast, Transparent Classification | 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 Geometric Mixture Classifier A Discriminative Per-Class Mixture of Hyperplanes for Fast, Transparent Classification Prasanth K K This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7734425/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Many real-world categories are multimodal, with a single class occupying several disjoint regions of feature space. Classical linear models such as logistic regression or linear SVMs impose a single global hyperplane and therefore fail on such data, while kernel SVMs and deep networks can capture multimodality but often trade off interpretability, require extensive tuning, or incur high computational costs. We introduce the Geometric Mixture Classifier (GMC), a discriminative model that represents each class as a mixture of hyperplanes. Within a class, GMC combines plane scores using a temperature-controlled soft-OR (log-sum-exp), smoothly approximating the maximum; across classes, it applies a standard softmax to yield probabilistic posteriors. An optional Random Fourier Features (RFF) mapping equips GMC with nonlinear capacity while preserving linear inference in the number of planes and lifted dimensions. To make GMC practical, we develop a training recipe including geometry-aware ini tialization via k-means, automatic plane budgeting with silhouette score, alpha-annealing, usage-aware L2 regularization, label smoothing, and early stopping. Experiments on syn thetic multimodal benchmarks (moons, circles, anisotropic blobs, two-spirals) and real tabular/image datasets (iris, wine, WDBC breast cancer, digits) show that GMC consis tently outperforms linear baselines and k-NN, matches or exceeds RBF-SVM, Random Forests, and compact MLPs, and enables transparent geometric introspection via plane and class-level responsibility visualizations. Because inference scales linearly in the number of planes and features, GMC is CPU efficient, requiring only microseconds per example—comparable to or faster than RBF-SVM and compact MLPs. With post-hoc temperature scaling, calibration improves (ECE reduced from 0.06 to 0.02). GMC thus offers a favorable trade-off between accuracy, interpretability, and efficiency: more expressive than linear models, and lighter and more transparent than kernel or deep models. Artificial Intelligence and Machine Learning Machine Learning Artificial Intelligence Classification Clustering Algorithms Full Text Additional Declarations The authors declare no competing interests. Cite Share Download PDF Status: Posted 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-7734425","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":521753907,"identity":"3b353bfc-6763-4987-a56d-5442ab1e0269","order_by":0,"name":"Prasanth K K","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABCElEQVRIiWNgGAWjYDACCcYGKIuH4UCCgYQciHngAVFa2HgYHzyosDAGa0nAqwXGYONhNnxwpiIRbAI+Lfyzm9seV9TUyZvL9x6TSGyTSJ8fdvgh0BY7Od0G7Fok7hxsNzxz7LDhzja+NJCW3I230wyAWpKNzQ7gsOZGYptkA9sBxg3HeMwgWmYngLQcSNyGQ4s8WMu/OnuYlnTD2ekf8GoxAGlpbGNOBGoxNkg4I5EgL52D3xZDkF8a+w4nbziWY/ggoULCcIN0TgEwgnD7Re52+7OHDd/qbDccPmNw8IdBnbz87PTNHz5U2Mnh9D4wQtCcClZpgFM5Fi3yDXhVj4JRMApGwQgEALYvaMc8INAPAAAAAElFTkSuQmCC","orcid":"https://orcid.org/0009-0008-1839-6038","institution":"","correspondingAuthor":true,"prefix":"","firstName":"Prasanth","middleName":"K","lastName":"K","suffix":""}],"badges":[],"createdAt":"2025-09-28 12:45:24","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-7734425/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7734425/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":92554244,"identity":"5ede03b6-8be5-41eb-87b3-b5fb33b7b4fb","added_by":"auto","created_at":"2025-10-01 02:29:33","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":734987,"visible":true,"origin":"","legend":"","description":"","filename":"gmc19.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7734425/v1_covered_d14b8670-f39f-439f-9f17-63fb09563b5d.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003eGeometric Mixture Classifier A Discriminative Per-Class Mixture of Hyperplanes for Fast, Transparent Classification\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"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":"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":"Machine Learning, Artificial Intelligence, Classification, Clustering, Algorithms","lastPublishedDoi":"10.21203/rs.3.rs-7734425/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7734425/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eMany real-world categories are multimodal, with a single class occupying several disjoint \u0026nbsp;regions of feature space. Classical linear models such as logistic regression or linear SVMs \u0026nbsp;impose a single global hyperplane and therefore fail on such data, while kernel SVMs \u0026nbsp;and deep networks can capture multimodality but often trade off interpretability, require \u0026nbsp;extensive tuning, or incur high computational costs. \u0026nbsp;We introduce the Geometric Mixture Classifier (GMC), a discriminative model that \u0026nbsp;represents each class as a mixture of hyperplanes. Within a class, GMC combines plane \u0026nbsp;scores using a temperature-controlled soft-OR (log-sum-exp), smoothly approximating the \u0026nbsp;maximum; across classes, it applies a standard softmax to yield probabilistic posteriors. An \u0026nbsp;optional Random Fourier Features (RFF) mapping equips GMC with nonlinear capacity \u0026nbsp;while preserving linear inference in the number of planes and lifted dimensions. \u0026nbsp;To make GMC practical, we develop a training recipe including geometry-aware ini tialization via k-means, automatic plane budgeting with silhouette score, alpha-annealing, \u0026nbsp;usage-aware L2 regularization, label smoothing, and early stopping. Experiments on syn thetic multimodal benchmarks (moons, circles, anisotropic blobs, two-spirals) and real \u0026nbsp;tabular/image datasets (iris, wine, WDBC breast cancer, digits) show that GMC consis tently outperforms linear baselines and k-NN, matches or exceeds RBF-SVM, Random \u0026nbsp;Forests, and compact MLPs, and enables transparent geometric introspection via plane and class-level responsibility visualizations. \u0026nbsp;Because inference scales linearly in the number of planes and features, GMC is CPU efficient, requiring only microseconds per example—comparable to or faster than RBF-SVM \u0026nbsp;and compact MLPs. With post-hoc temperature scaling, calibration improves (ECE reduced \u0026nbsp;from 0.06 to 0.02). GMC thus offers a favorable trade-off between accuracy, interpretability, \u0026nbsp;and efficiency: more expressive than linear models, and lighter and more transparent than \u0026nbsp;kernel or deep models.\u003c/p\u003e","manuscriptTitle":"Geometric Mixture Classifier A Discriminative Per-Class Mixture of Hyperplanes for Fast, Transparent Classification","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-10-01 02:21:25","doi":"10.21203/rs.3.rs-7734425/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","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}}],"origin":"","ownerIdentity":"01a4479c-0602-43dd-ac37-a24e43fe5d38","owner":[],"postedDate":"October 1st, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":55455391,"name":"Artificial Intelligence and Machine Learning"}],"tags":[],"updatedAt":"2025-10-01T02:21:25+00:00","versionOfRecord":[],"versionCreatedAt":"2025-10-01 02:21:25","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-7734425","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7734425","identity":"rs-7734425","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}