%0 Journal Article %1 1044705 %A Vito, Ernesto De %A Rosasco, Lorenzo %A Caponnetto, Andrea %A Piana, Michele %A Verri, Alessandro %C Cambridge, MA, USA %D 2004 %I MIT Press %J J. Mach. Learn. Res. %K Regularization kernel-methods %P 1363--1390 %T Some Properties of Regularized Kernel Methods %U http://portal.acm.org/citation.cfm?id=1044705 %V 5 %X In regularized kernel methods, the solution of a learning problem is found by minimizing functionals consisting of the sum of a data and a complexity term. In this paper we investigate some properties of a more general form of the above functionals in which the data term corresponds to the expected risk. First, we prove a quantitative version of the representer theorem holding for both regression and classification, for both differentiable and non-differentiable loss functions, and for arbitrary offset terms. Second, we show that the case in which the offset space is non trivial corresponds to solving a standard problem of regularization in a Reproducing Kernel Hilbert Space in which the penalty term is given by a seminorm. Finally, we discuss the issues of existence and uniqueness of the solution. From the specialization of our analysis to the discrete setting it is immediate to establish a connection between the solution properties of sparsity and coefficient boundedness and some properties of the loss function. For the case of Support Vector Machines for classification, we also obtain a complete characterization of the whole method in terms of the Khun-Tucker conditions with no need to introduce the dual formulation.

@article{1044705, abstract = {In regularized kernel methods, the solution of a learning problem is found by minimizing functionals consisting of the sum of a data and a complexity term. In this paper we investigate some properties of a more general form of the above functionals in which the data term corresponds to the expected risk. First, we prove a quantitative version of the representer theorem holding for both regression and classification, for both differentiable and non-differentiable loss functions, and for arbitrary offset terms. Second, we show that the case in which the offset space is non trivial corresponds to solving a standard problem of regularization in a Reproducing Kernel Hilbert Space in which the penalty term is given by a seminorm. Finally, we discuss the issues of existence and uniqueness of the solution. From the specialization of our analysis to the discrete setting it is immediate to establish a connection between the solution properties of sparsity and coefficient boundedness and some properties of the loss function. For the case of Support Vector Machines for classification, we also obtain a complete characterization of the whole method in terms of the Khun-Tucker conditions with no need to introduce the dual formulation.}, added-at = {2009-08-14T15:47:39.000+0200}, address = {Cambridge, MA, USA}, author = {Vito, Ernesto De and Rosasco, Lorenzo and Caponnetto, Andrea and Piana, Michele and Verri, Alessandro}, biburl = {https://www.bibsonomy.org/bibtex/28599e196c8d44d7e0c9ccce09b3f80c9/ahmedjawwad4u}, description = {Some Properties of Regularized Kernel Methods}, interhash = {d1bffa72d6598924a50dd7c4e9f9b672}, intrahash = {8599e196c8d44d7e0c9ccce09b3f80c9}, issn = {1533-7928}, journal = {J. Mach. Learn. Res.}, keywords = {Regularization kernel-methods}, pages = {1363--1390}, publisher = {MIT Press}, timestamp = {2009-08-26T14:39:38.000+0200}, title = {Some Properties of Regularized Kernel Methods}, url = {http://portal.acm.org/citation.cfm?id=1044705}, volume = 5, year = 2004 }