[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-119437-en":3,"doc-seo-119437-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},119437,1374391974468,"Eden","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","Localisation of Regularised and Multiview Support Vector Machine Learning - Representer Theorems for Operator Valued Kernels","The work establishes representer theorems for a localised semisupervised, manifold-regularised, multiview support vector machine learning framework built on operator-valued positive semidefinite kernels and vector-valued reproducing kernel Hilbert spaces. Results cover convex and nonconvex loss functions and both finite- and infinite-dimensional underlying Hilbert spaces, including cases where the losses are Gâteaux differentiable. Exponential least-squares calculations yield partially nonlinear systems solved via Newton-type approximations from interior-point methods, with experiments on a toy model demonstrating tractability.","Localisation of Regularised and Multiview Support Vector  \nMachine Learning  \nAurelian Gheondea [a.gheondea@imar.ro](a.gheondea@imar.ro)  \nInstitute of Mathematics of the Romanian Academy  \n21 Calea Grivit􀀘ei  \n010702 Bucharest, Romania and  \nDepartment of Mathematics Bilkent University  \n06800 Bilkent, Ankara, Turkey [aurelian@fen.bilkent.edu.tr](aurelian@fen.bilkent.edu.tr)  \nCankat Tilki [cankat@vt.edu](cankat@vt.edu)  \nDepartment of Mathematics and  \nDivision of Computational Modeling and Data Analytics Virginia Polytechnic Institute and State University Blacksburg Virginia, 24061 U.S.A.  \nEditor: Chris Oates  \nAbstract  \nWe prove some representer theorems for a localised version of a semisupervised, manifold regularised and multiview support vector machine learning problem introduced by  \nH.Q. Minh, L. Bazzani, and V. Murino, Journal of Machine Learning Research, 17(2016) 1{72, that involves operator valued positive semide􀀌nite kernels and their reproducing kernel Hilbert spaces. The results concern general cases when convex or nonconvex loss functions and 􀀌nite or in􀀌nite dimensional underlying Hilbert spaces are considered. We show that the general framework allows in􀀌nite dimensional Hilbert spaces and nonconvex loss functions for some special cases, in particular in case the loss functions are G^ateaux di􀀋erentiable. Detailed calculations are provided for the exponential least squares loss functions that lead to systems of partially nonlinear equations for which some Newton's approximation methods based on the interior point method can be used. Some numerical experiments are performed on a toy model that illustrate the tractability of the methods that we propose.  \nKeywords: operator valued reproducing kernel Hilbert spaces, manifold co-regularised and multiview learning, support vector machine learning, loss functions, representer theorem  \n1. Introduction  \nRepresenter theorems are of a special interest in machine learning due to the fact that they reduce the problem of 􀀌nding a minimiser for the learning map to the vector space spanned by the kernel functions, or operators, at the labeled and unlabeled input data. For classical  \n􀀍c2024 Aurelian Gheondea and Cankat Tilki.  \nLicense: CC-BY 4.0, see [https://creativecommons.org/licenses/by/4.0/](https://creativecommons.org/licenses/by/4.0/. Attribution)[. Attribution](https://creativecommons.org/licenses/by/4.0/. Attribution) requirements are provided  \nat [http://jmlr.org/papers/v25/23-0522.html](http://jmlr.org/papers/v25/23-0522.html).  \nA. Gheondea and C. Tilki  \nversions of representer theorem, we recommend the monographs of Sch􀁿olkopf and Smola (2002) and Steinwart and Christmann (2008) . There is a large literature on generalised representer theorems but in this article we refer to the unifying framework in vector valued reproducing kernel Hilbert spaces for semisupervised, manifold regularised and multiview machine learning, as investigated by Minh et al. (2016) and the vast literature cited there.  \nThe article Minh et al. (2016) has remarkable contributions to the domain of representer theorems in support vector machine learning, 􀀌rstly by unifying many variants of these theorems referring to semisupervised, regularised, manifold regularised, multiview machine learning and then by considering underlying Hilbert spaces that are in􀀌nite dimensional. Recently, in􀀌nite dimensional Hilbert spaces in learning with kernels have been of interest, e.g. see Lambert (2021) . However, although the general representer theorem, Theorem 2 in Minh et al. (2016), is stated for in􀀌nite dimensional spaces, this turns out to be problematic, as we will see in Remark 10 . Also, there is an interest for applications to learning problems in which loss functions may not be convex, cf. Zhao et al. (2010), or even inde􀀌nite, cf. Kwon and Zou (2023) .  \nIn this article we are concerned with questions triggered by the investigations in Minh et al. (2016) and Zhao et al. (2010), such as","cbCaiqPOHwGUoIeH","https://ap.wps.com/l/cbCaiqPOHwGUoIeH","pdf",729128,1,47,"English","en",105,"# Abstract\n# Introduction\n## Representer theorems and kernel operator reductions\n## Motivation for localisation and flexibility\n## Scope: infinite-dimensional spaces and nonconvex loss functions","[{\"question\":\"What problem does the paper study in multiview support vector machine learning?\",\"answer\":\"It studies a localised semisupervised, manifold-regularised, multiview SVM problem where operator-valued kernels define vector-valued reproducing kernel Hilbert spaces and labelled/unlabelled losses can vary locally.\"},{\"question\":\"How do the results handle infinite-dimensional Hilbert spaces and nonconvex loss functions?\",\"answer\":\"The paper proves representer theorems for general settings including infinite-dimensional Hilbert spaces and both convex and nonconvex losses, with special attention to cases where losses are Gâteaux differentiable.\"},{\"question\":\"What method is used for the exponential least squares loss, and what do experiments show?\",\"answer\":\"Exponential least squares leads to partially nonlinear equations; Newton approximation methods based on interior-point techniques are proposed, and numerical experiments on a toy model illustrate the tractability of the approach.\"}]","Localisation of Regularised and Multiview Support Vector Machine Learning - Representer Theorems for Operator Valued Kernels | PDF",1785724287,118,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"localisation-of-regularised-and-multiview-support-vector-machine-learning-representer-theorems-for-operator-valued-kernels","",{"@graph":36,"@context":85},[37,54,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/localisation-of-regularised-and-multiview-support-vector-machine-learning-representer-theorems-for-operator-valued-kernels/119437/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":62,"encodingFormat":61,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-03",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What problem does the paper study in multiview support vector machine learning?","Question",{"text":75,"@type":76},"It studies a localised semisupervised, manifold-regularised, multiview SVM problem where operator-valued kernels define vector-valued reproducing kernel Hilbert spaces and labelled/unlabelled losses can vary locally.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How do the results handle infinite-dimensional Hilbert spaces and nonconvex loss functions?",{"text":80,"@type":76},"The paper proves representer theorems for general settings including infinite-dimensional Hilbert spaces and both convex and nonconvex losses, with special attention to cases where losses are Gâteaux differentiable.",{"name":82,"@type":73,"acceptedAnswer":83},"What method is used for the exponential least squares loss, and what do experiments show?",{"text":84,"@type":76},"Exponential least squares leads to partially nonlinear equations; Newton approximation methods based on interior-point techniques are proposed, and numerical experiments on a toy model illustrate the tractability of the approach.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]