{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,4,7]],"date-time":"2024-04-07T14:36:59Z","timestamp":1712500619561},"reference-count":42,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2010,7,11]],"date-time":"2010-07-11T00:00:00Z","timestamp":1278806400000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J Glob Optim"],"published-print":{"date-parts":[[2012,10]]},"DOI":"10.1007\/s10898-010-9575-z","type":"journal-article","created":{"date-parts":[[2010,7,11]],"date-time":"2010-07-11T22:47:23Z","timestamp":1278888443000},"page":"235-250","source":"Crossref","is-referenced-by-count":25,"title":["Regularized learning in Banach spaces as an optimization problem: representer theorems"],"prefix":"10.1007","volume":"54","author":[{"given":"Haizhang","family":"Zhang","sequence":"first","affiliation":[]},{"given":"Jun","family":"Zhang","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2010,7,11]]},"reference":[{"key":"9575_CR1","unstructured":"Argyriou, A., Micchelli, C.A., Pontil, M.: When is there a representer theorem? Vector versus matrix regularizers. Preprint, arXiv:0809.1590v1 (2008)"},{"key":"9575_CR2","doi-asserted-by":"crossref","first-page":"337","DOI":"10.1090\/S0002-9947-1950-0051437-7","volume":"68","author":"N. Aronszajn","year":"1950","unstructured":"Aronszajn N.: Theory of reproducing kernels. Trans. Amer. Math. Soc. 68, 337\u2013404 (1950)","journal-title":"Trans. Amer. Math. Soc."},{"key":"9575_CR3","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-0913-3","volume-title":"Gr\u00f6bner Bases: A Computational Approach to Commutative Algebra","author":"T. Becker","year":"1993","unstructured":"Becker T., Weispfenning V.: Gr\u00f6bner Bases: A Computational Approach to Commutative Algebra. Springer-Verlag, New York (1993)"},{"key":"9575_CR4","first-page":"57","volume-title":"Proceeding of the Seventeenth International Conference on Machine Learning","author":"K. Bennett","year":"2000","unstructured":"Bennett, K., Bredensteiner, E.: Duality and geometry in SVM classifier. In: Langley, P. (ed.) Proceeding of the Seventeenth International Conference on Machine Learning, pp. 57\u201364. Morgan Kaufmann, San Francisco (2000)"},{"key":"9575_CR5","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4419-9096-9","volume-title":"Reproducing Kernel Hilbert Spaces in Probability and Statistics","author":"A. Berlinet","year":"2004","unstructured":"Berlinet A., Thomas-Agnan C.: Reproducing Kernel Hilbert Spaces in Probability and Statistics. Kluwer Academic Publishers, Norwell, MA (2004)"},{"key":"9575_CR6","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1016\/S0747-7171(86)80014-1","volume":"1","author":"W. Boege","year":"1986","unstructured":"Boege W., Gebauer R., Kredel H.: Some examples for solving systems of algebraic equations by calculating Gr\u00f6bner bases. J. Symb. Comput. 1, 83\u201398 (1986)","journal-title":"J. Symb. Comput."},{"key":"9575_CR7","unstructured":"Canu, S., Mary, X., Rakotomamonjy, A.: Functional learning through kernel. In: Suykens, J., Horvath, G., Basu, S., Micchelli, C.A., Vandewalle, J. (eds.) Advances in Learning Theory: Methods, Models and Applications. NATO Science Series III: Computer and Systems Sciences, vol. 190, pp. 89\u2013110. IOS Press, Amsterdam (2003)"},{"key":"9575_CR8","volume-title":"A Course in Functional Analysis","author":"J.B. Conway","year":"1990","unstructured":"Conway J.B.: A Course in Functional Analysis, 2nd edn. Springer-Verlag, New York (1990)","edition":"2"},{"key":"9575_CR9","doi-asserted-by":"crossref","first-page":"1676","DOI":"10.1214\/aos\/1176347872","volume":"18","author":"D. Cox","year":"1990","unstructured":"Cox D., O\u2019Sullivan F.: Asymptotic analysis of penalized likelihood and related estimators. Ann. Statist. 18, 1676\u20131695 (1990)","journal-title":"Ann. Statist."},{"key":"9575_CR10","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1090\/S0273-0979-01-00923-5","volume":"39","author":"F. Cucker","year":"2002","unstructured":"Cucker F., Smale S.: On the mathematical foundations of learning. Bull. Amer. Math. Soc. 39, 1\u201349 (2002)","journal-title":"Bull. Amer. Math. Soc."},{"key":"9575_CR11","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1090\/S0002-9904-1963-10948-9","volume":"69","author":"D.F. Cudia","year":"1963","unstructured":"Cudia D.F.: On the localization and directionalization of uniform convexity. Bull. Amer. Math. Soc. 69, 265\u2013267 (1963)","journal-title":"Bull. Amer. Math. Soc."},{"key":"9575_CR12","unstructured":"Der, R., Lee, D.: Large-margin classification in Banach spaces. JMLR Workshop and Conference Proceedings 2: AISTATS, 91\u201398 (2007)"},{"key":"9575_CR13","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1023\/A:1018946025316","volume":"13","author":"T. Evgeniou","year":"2000","unstructured":"Evgeniou T., Pontil M., Poggio T.: Regularization networks and support vector machines. Adv. Comput. Math. 13, 1\u201350 (2000)","journal-title":"Adv. Comput. Math."},{"key":"9575_CR14","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4757-3480-5","volume-title":"Functional Analysis and Infinite-Dimensional Geometry","author":"M. Fabian","year":"2001","unstructured":"Fabian M. et\u00a0al.: Functional Analysis and Infinite-Dimensional Geometry. Springer, New York (2001)"},{"key":"9575_CR15","first-page":"213","volume":"2","author":"C. Gentile","year":"2001","unstructured":"Gentile C.: A new approximate maximal margin classification algorithm. J. Mach. Learn. Res. 2, 213\u2013242 (2001)","journal-title":"J. Mach. Learn. Res."},{"key":"9575_CR16","doi-asserted-by":"crossref","first-page":"436","DOI":"10.1090\/S0002-9947-1967-0217574-1","volume":"129","author":"J.R. Giles","year":"1967","unstructured":"Giles J.R.: Classes of semi-inner-product spaces. Trans. Amer. Math. Soc. 129, 436\u2013446 (1967)","journal-title":"Trans. Amer. Math. Soc."},{"key":"9575_CR17","doi-asserted-by":"crossref","first-page":"333","DOI":"10.1016\/j.jcss.2004.10.013","volume":"71","author":"M. Hein","year":"2005","unstructured":"Hein M., Bousquet O., Sch\u00f6lkopf B.: Maximal margin classification for metric spaces. J. Comput. System Sci. 71, 333\u2013359 (2005)","journal-title":"J. Comput. System Sci."},{"key":"9575_CR18","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1016\/0304-3975(95)00021-N","volume":"148","author":"D. Kimber","year":"1995","unstructured":"Kimber D., Long P.M.: On-line learning of smooth functions of a single variable. Theoret. Comput. Sci. 148, 141\u2013156 (1995)","journal-title":"Theoret. Comput. Sci."},{"key":"9575_CR19","doi-asserted-by":"crossref","first-page":"82","DOI":"10.1016\/0022-247X(71)90184-3","volume":"33","author":"G. Kimeldorf","year":"1971","unstructured":"Kimeldorf G., Wahba G.: Some results on Tchebycheffian spline functions. J. Math. Anal. Appl. 33, 82\u201395 (1971)","journal-title":"J. Math. Anal. Appl."},{"key":"9575_CR20","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1090\/S0002-9947-1961-0133024-2","volume":"100","author":"G. Lumer","year":"1961","unstructured":"Lumer G.: Semi-inner-product spaces. Trans. Amer. Math. Soc. 100, 29\u201343 (1961)","journal-title":"Trans. Amer. Math. Soc."},{"key":"9575_CR21","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-0603-3","volume-title":"An Introduction to Banach Space Theory","author":"R.E. Megginson","year":"1998","unstructured":"Megginson R.E.: An Introduction to Banach Space Theory. Springer, New York (1998)"},{"key":"9575_CR22","doi-asserted-by":"crossref","first-page":"415","DOI":"10.1098\/rsta.1909.0016","volume":"209","author":"J. Mercer","year":"1909","unstructured":"Mercer J.: Functions of positive and negative type and their connection with the theorey of integral equations. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 209, 415\u2013446 (1909)","journal-title":"Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci."},{"key":"9575_CR23","doi-asserted-by":"crossref","unstructured":"Micchelli, C.A., Pontil, M.: A function representation for learning in Banach spaces. In: Learning Theory, pp. 255\u2013269. Lecture Notes in Computer Science, 3120, Springer, Berlin (2004)","DOI":"10.1007\/978-3-540-27819-1_18"},{"key":"9575_CR24","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1007\/s10994-006-0679-0","volume":"66","author":"C.A. Micchelli","year":"2007","unstructured":"Micchelli C.A., Pontil M.: Feature space perspectives for learning the kernel. Machine Learning 66, 297\u2013319 (2007)","journal-title":"Machine Learning"},{"key":"9575_CR25","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1090\/S0002-9904-1916-02867-9","volume":"23","author":"E.H. Moore","year":"1916","unstructured":"Moore E.H.: On properly positive Hermitian matrices. Bull. Amer. Math. Soc. 23, 59 (1916)","journal-title":"Bull. Amer. Math. Soc."},{"key":"9575_CR26","unstructured":"Moore, E.H.: General Analysis. Memoirs of the American Philosophical Society, Part I (1935), Part II (1939)"},{"key":"9575_CR27","unstructured":"Pardalos, P.M., Hansen, P. (eds.): Data Mining and Mathematical Programming. Papers from the workshop held in Montreal, QC, October 10\u201313, 2006. CRM Proceedings & Lecture Notes 45. American Mathematical Society, Providence, RI (2008)"},{"key":"9575_CR28","doi-asserted-by":"crossref","unstructured":"Sch\u00f6lkopf, B., Herbrich, R., Smola, A.J.: A generalized representer theorem. Proceeding of the Fourteenth Annual Conference on Computational Learning Theory and the Fifth European Conference on Computational Learning Theory, pp. 416\u2013426. Springer-Verlag, London, UK (2001)","DOI":"10.1007\/3-540-44581-1_27"},{"key":"9575_CR29","volume-title":"Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond","author":"B. Sch\u00f6lkopf","year":"2002","unstructured":"Sch\u00f6lkopf B., Smola A.J.: Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond. MIT Press, Cambridge, Mass (2002)"},{"key":"9575_CR30","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511809682","volume-title":"Kernel Methods for Pattern Analysis","author":"J. Shawe-Taylor","year":"2004","unstructured":"Shawe-Taylor J., Cristianini N.: Kernel Methods for Pattern Analysis. Cambridge University Press, Cambridge (2004)"},{"key":"9575_CR31","unstructured":"Tikhonov, A.N., Arsenin, V.Y.: Solutions of Ill-posed Problems. V. H. Winston & Sons (distributed by Wiley), New York (1977)"},{"key":"9575_CR32","doi-asserted-by":"crossref","first-page":"1030","DOI":"10.1109\/TIT.2005.864420","volume":"52","author":"J.A. Tropp","year":"2006","unstructured":"Tropp J.A.: Just relax: convex programming methods for identifying sparse signals in noise. IEEE Trans. Inform. Theory 52, 1030\u20131051 (2006)","journal-title":"IEEE Trans. Inform. Theory"},{"key":"9575_CR33","volume-title":"Statistical Learning Theory","author":"V.N. Vapnik","year":"1998","unstructured":"Vapnik V.N.: Statistical Learning Theory. Wiley, New York (1998)"},{"key":"9575_CR34","first-page":"669","volume":"5","author":"U. Luxburg von","year":"2004","unstructured":"von Luxburg U., Bousquet O.: Distance-based classification with Lipschitz functions. J. Mach. Learn. Res. 5, 669\u2013695 (2004)","journal-title":"J. Mach. Learn. Res."},{"key":"9575_CR35","first-page":"69","volume-title":"Advances in Kernel Methods\u2013Support Vector Learning","author":"G. Wahba","year":"1999","unstructured":"Wahba G.: Support vector machines, reproducing kernel Hilbert spaces and the randomized GACV. In: Sch\u00f6lkopf, B., Burge, C., Smola, A.J. (eds) Advances in Kernel Methods\u2013Support Vector Learning., pp. 69\u201388. MIT Press, Cambridge, Mass (1999)"},{"key":"9575_CR36","first-page":"2083","volume":"8","author":"Y. Xu","year":"2007","unstructured":"Xu Y., Zhang H.: Refinable kernels. J. Mach. Learn. Res. 8, 2083\u20132120 (2007)","journal-title":"J. Mach. Learn. Res."},{"key":"9575_CR37","first-page":"107","volume":"10","author":"Y. Xu","year":"2009","unstructured":"Xu Y., Zhang H.: Refinement of reproducing kernels. J. Mach. Learn. Res. 10, 107\u2013140 (2009)","journal-title":"J. Mach. Learn. Res."},{"key":"9575_CR38","volume-title":"An Introduction to Nonharmonic Fourier Series","author":"R.M. Young","year":"1980","unstructured":"Young R.M.: An Introduction to Nonharmonic Fourier Series. Academic Press, New York (1980)"},{"key":"9575_CR39","first-page":"2741","volume":"10","author":"H. Zhang","year":"2009","unstructured":"Zhang H., Xu Y., Zhang J.: Reproducing kernel Banach spaces for machine learning. J. Mach. Learn. Res. 10, 2741\u20132775 (2009)","journal-title":"J. Mach. Learn. Res."},{"key":"9575_CR40","doi-asserted-by":"crossref","unstructured":"Zhang, H., Zhang, J.: Generalized semi-inner products with applications to regularized learning. J. Math. Anal. Appl., accepted","DOI":"10.1016\/j.jmaa.2010.04.075"},{"key":"9575_CR41","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1023\/A:1012498226479","volume":"46","author":"T. Zhang","year":"2002","unstructured":"Zhang T.: On the dual formulation of regularized linear systems with convex risks. Machine Learning 46, 91\u2013129 (2002)","journal-title":"Machine Learning"},{"key":"9575_CR42","unstructured":"Zhou, D., Xiao, B., Zhou, H., Dai, R.: Global geometry of SVM classifiers. Technical Report 30-5-02. Institute of Automation, Chinese Academy of Sciences (2002)"}],"container-title":["Journal of Global Optimization"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10898-010-9575-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10898-010-9575-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10898-010-9575-z","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,31]],"date-time":"2019-05-31T04:58:56Z","timestamp":1559278736000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10898-010-9575-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,7,11]]},"references-count":42,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2012,10]]}},"alternative-id":["9575"],"URL":"https:\/\/doi.org\/10.1007\/s10898-010-9575-z","relation":{},"ISSN":["0925-5001","1573-2916"],"issn-type":[{"value":"0925-5001","type":"print"},{"value":"1573-2916","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,7,11]]}}}