[...]... positive de nite kernels, such as reproducing kernel, Mercer kernel, or support vector kernel The de nitions for positive de nite kernels and positive matrices di er only in the fact that in the former case, we are free to choose the points on which the kernel is evaluated Positive de nitness implies positivity on the diagonal, kx1 ; x1   0 for all x1 2 X ; 74 use m = 1 in 72, and symmetry,... convergent in that norm, and thus gets a Hilbert space H , usually called a reproducing kernel Hilbert space.2 The case of real-valued kernels is included in the above; in that case, H can be chosen as a real Hilbert space 9 Examples of Kernels Besides 65, 8 and 28 suggest the usage of Gaussian radial basis function kernels 1 0  = exp , kx , x0 k2 kx; x 89 2 2 and sigmoid kernels kx; x0  = tanhx... nite kernel, its real part is a real-valued positive de nite kernel Kernels can be regarded as generalized dot products Indeed, any dot product can be shown to be a kernel; however, linearity does not carry over from dot products to general kernels Another property of dot products, the CauchySchwarz inequality, does have a natural generalization to kernels: Proposition 5 If k is a positive de nite kernel, ... nite kernels Second, their relationship to positive de nite kernels is a rather interesting one, and a number of connections between the two classes provide understanding of kernels in general Third, they are intimately related to a question which is a variation on the central aspect of positive de nite kernels: the latter can be thought of as dot products in feature spaces; the former, on the other hand,... recognition learning Automation and Remote Control, 25:821 837, 1964 2 N Alon, S Ben-David, N Cesa-Bianchi, and D Haussler Scale sensitive Dimensions, Uniform Convergence, and Learnability Journal of the ACM, 444:615 631, 1997 3 N Aronszajn Theory of reproducing kernels Trans Amer Math Soc., 68:337 404, 1950 4 P L Bartlett and J Shawe-Taylor Generalization performance of support vector machines and other... B Scholkopf Improving the accuracy and speed of support vector learning machines In M Mozer, M Jordan, and T Petsche, editors, Advances in Neural Information Processing Systems 9, pages 375 381, Cambridge, MA, 1997 MIT Press 10 C Cortes and V Vapnik Support vector networks Machine Learning, 20:273 297, 1995 11 F Girosi, M Jones, and T Poggio Regularization theory and neural networks architectures Neural... Support Vector Learning R Oldenbourg Verlag, Munchen, 1997 Doktorarbeit, TU Berlin B Scholkopf, C Burges, and V Vapnik Extracting support data for a given task In U M Fayyad and R Uthurusamy, editors, Proceedings, First International Conference on Knowledge Discovery & Data Mining, Menlo Park, 1995 AAAI Press B Scholkopf, C J C Burges, and A J Smola Advances in Kernel Methods | Support Vector Learning MIT... Neural Computation, 12:1083 1121, 2000 A Smola, B Scholkopf, and K.-R Muller The connection between regularization operators and support vector kernels Neural Networks, 11:637 649, 1998 A J Smola and B Scholkopf On a kernel based method for pattern recognition, regression, approximation and operator inversion Algorithmica, 22:211 231, 1998 A J Smola and B Scholkopf A tutorial on support vector regression... and optimizers are listed on the web page http: www .kernel- machines.org For more details on the optimization problem, see 19 On the theoretical side, the least understood part of the SV algorithm initially was the precise role of the kernel, and how a certain kernel choice would in uence the generalization ability In that respect, the connection to regularization theory provided some insight For kernel- based... Scholkopf, C J C Burges, and A J Smola, editors, Advances in Kernel Methods | Support Vector Learning, pages 43 54, Cambridge, MA, 1999 MIT Press 5 C Berg, J.P.R Christensen, and P Ressel Harmonic Analysis on Semigroups Springer-Verlag, New York, 1984 6 D P Bertsekas Nonlinear Programming Athena Scienti c, Belmont, MA, 1995 7 V Blanz, B Scholkopf, H Bultho , C Burges, V Vapnik, and T Vetter Comparison