[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-128854-105":59,"doc-detail-128854-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","more-efficency-in-multiple-kernel-learning","More Efficency in Multiple Kernel Learning","","An efficient and general multiple kernel learning (MKL) algorithm is studied to improve large-scale tractability and convergence behavior. Instead of relying on an iterative MKL procedure that may require many steps, the work formulates MKL with adaptive 2-norm regularization and incorporates kernel-specific weights into standard SVM empirical risk minimization under an ℓ1 constraint to promote sparsity. A dedicated solver is proposed, and equivalence to block ℓ1 regularization is proved. Experiments confirm rapid convergence and favorable efficiency versus related MKL methods.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/more-efficency-in-multiple-kernel-learning/128854/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/more-efficency-in-multiple-kernel-learning/128854.png","ImageObject",300,407,{"name":92,"@type":93},"Aria","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-09-19","2026-08-06",true,{"@type":102,"interactionType":103,"userInteractionCount":39},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"Why does the classical iterative MKL approach have convergence issues?","Question",{"text":112,"@type":113},"The paper explains that the iterative algorithm needs several iterations before converging toward a reasonable solution, motivating a different formulation.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How does the proposed method enforce sparsity across kernels?",{"text":117,"@type":113},"Kernel weights are included in SVM empirical risk minimization and constrained with an ℓ1 constraint, encouraging sparsity in the linear combination of kernels.",{"name":119,"@type":110,"acceptedAnswer":120},"What is the relationship between the new adaptive formulation and block 1-norm regularization?",{"text":121,"@type":113},"The work provides a variational argument showing the two formulations are equivalent, yielding new insight into MKL algorithms.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},128854,1786003916,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":39,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":39,"language":139,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":67,"update_tm":129,"read_time":46},2336474459895,"https://ap-avatar.wpscdn.com/avatar/22000baeef7a5ed0655?x-image-process=image/resize,m_fixed,w_180,h_180&k=1786071322749376916","More E􀀎ciency in Multiple Kernel Learning  \nAlain Rakotomamonjy [alain.rakotomamonjy@insa-rouen.fr](alain.rakotomamonjy@insa-rouen.fr)  \nLITIS EA 4051, UFR de Sciences, Universit􀀓e de Rouen, 76800 Saint Etienne du Rouvray, France  \nFrancis Bach [francis.bach@mines.org](francis.bach@mines.org)  \nCMM, Ecole des Mines de Paris, 35 rue Saint-Honor􀀓e, 77305 Fontainebleau, France  \nSt􀀓ephane Canu [stephane.canu@insa-rouen.fr](stephane.canu@insa-rouen.fr)  \nLITIS EA 4051, INSA de Rouen, 76801 Saint Etienne du Rouvray, France  \nYves Grandvalet [yves.grandvalet@idiap.ch](yves.grandvalet@idiap.ch)  \nIDIAP, Rue du Simplon 4, Case Postale 592, CH-1920 Martigny, Switzerland  \nAbstract  \nAn e􀀎cient and general multiple kernel learning (MKL) algorithm has been recently proposed by Sonnenburg et al. (2006) . This approach has opened new perspectives since it makes the MKL approach tractable for largescale problems, by iteratively using existing support vector machine code. However, it turns out that this iterative algorithm needs several iterations before converging towards a reasonable solution. In this paper, we address the MKL problem through an adaptive 2-norm regularization formulation. Weightson each kernel matrix are included in the standard SVM empirical risk minimization problem with a ℓ 1 constraint to encourage sparsity. We propose an algorithm for solving this problem and provide an new insight on MKL algorithms based on block 1-norm regularization by showing that the two approaches are equivalent. Experimental results show that the resulting algorithm converges rapidly and its e􀀎ciency compares favorably to other MKL algorithms.  \n1. Introduction  \nDuring the last few years, kernel methods, such as support vector machines (SVM) have proved to be e􀀎cient tools for solving learning problems like classi􀀌cation or  \nAppearing in Proceedings of the 24 th International Conference on Machine Learning, Corvallis, OR, 2007 . Copyright 2007 by the author(s)/owner(s) .  \nregression (Scholkopf & Smola, 2001) . For such tasks, the performance of the learning algorithm strongly depends on the data representation. In kernel methods, the data representation is implicitly chosen through the so-called kernel K (x, x′) . This kernel actually plays several roles: it de􀀌nes the similarity between two examples x and x′, while de􀀌ning an appropriate regularization term for the learning problem. For kernel algorithms, the solution of the learning problem is of the form:  \nℓ  \nf (x) =X α⋆iyi K (x, xi ) + b⋆ , (1)  \ni=1  \nwhere {xi , yi }ℓi=1 are the training examples, ℓ the number of learning examples, K (·, ·) is a given positive de􀀌nite kernel associated with a reproducing kernel Hilbert space (RKHS) H and {α⋆i}i , b⋆ some coe􀀎 -cients to be learned from examples.  \nRecent applications (Lanckriet et al., 2004a) and developments based on SVMs have shown that using multiple kernels instead of a single one can enhance interpretability of the decision function and improve classi􀀌er performance. In such cases, a common approach is to consider that the kernel K (x, x′) is actually a convex linear combination of other basis kernels:  \nM  \nK (x, x′) =XdkKk (x, x′) , with dk ≥ 0 , X dk = 1 ,  \nk=1 k  \nwhere M is the total number of kernels. Each basis kernel Kk may either use the full set of variables describing x or only a subset of these variables. Alternatively, kernels Kk can simply be classical kernels (such as Gaussian kernel) with di􀀋erent parameters,  \nor may rely on di􀀋erent data sources associated with the same learning problem (Lanckriet et al., 2004a) . Within this framework, the problem of data representation through the kernel is then transferred to the choice of weights dk . Learning both the coe􀀎cients αi and the weights dk in a single optimization problem is known as the multiple kernel learning (MKL) problem. This problem has been recently introduced by Lanckriet et al. (2004b) and the associated learning problem involves semi-de􀀌nite programm","cbCaifRJGERyIv8C","https://ap.wps.com/l/cbCaifRJGERyIv8C","pdf",179622,"English","# Introduction\n## Kernel methods and the role of kernels\n## Multiple kernel learning background and formulations\n# Multiple Kernel Learning framework\n## Problem formulation and adaptive regularization\n## Solving strategy and sparsity control","[{\"question\":\"Why does the classical iterative MKL approach have convergence issues?\",\"answer\":\"The paper explains that the iterative algorithm needs several iterations before converging toward a reasonable solution, motivating a different formulation.\"},{\"question\":\"How does the proposed method enforce sparsity across kernels?\",\"answer\":\"Kernel weights are included in SVM empirical risk minimization and constrained with an ℓ1 constraint, encouraging sparsity in the linear combination of kernels.\"},{\"question\":\"What is the relationship between the new adaptive formulation and block 1-norm regularization?\",\"answer\":\"The work provides a variational argument showing the two formulations are equivalent, yielding new insight into MKL algorithms.\"}]","More Efficency in Multiple Kernel Learning | PDF"]