Wahba, G., 1999. Support Vector Machines, reproducing kernel Hilbert spaces and the randomized GACV. In: Scholkopf, B., Burgess, C., Smola, A.J. (Eds.), Advances in Kernel Methods---Support Vector Learning. MIT Press, pp. 69 -- 88.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....In practice, people usually choose hj (x) s to be the basis functions of a reproducing kernel Hilbert space. Then a kernel trick allows the dimension of the transformed feature space to be very large, even infinite in some cases (i.e. q = without causing extra computational burden ( 2] and [12]) In this paper, however, we will concentrate on the basis representation (3) rather than a kernel representation. Notice that (4) has the form loss penalty, and is the tuning parameter that controls the tradeoff between loss and penalty. The loss (1 y f) is called the hinge loss, and the ....

.... effect of controlling the variances of , hence possibly improves the fitted model s prediction accuracy, especially when there are many highly correlated features [6] So from a statistical function estimation point of view, the ridge penalty could possibly explain the success of the SVM ( 6] and [12]) On the other hand, computational learning theory has associated the good performance of the SVM to its margin maximizing property [11] a property of the hinge loss. 8] makes some effort to build a connection between these two different views. In this paper, we replace the ridge penalty in ....

Wahba, G. (1999) Support vector machine, reproducing kernel Hilbert spaces and the randomized GACV. Advances in Kernel Methods - Support Vector Learning, 69-88, MIT Press.

....it is used to derive bounds on the expected error not estimators. Jaakkola 12 and Haussler [Jaakkola and Haussler, 1999] present a generalized bound for inseparable data that is similar to that of lemma 1. Similarly, an approximation to the leave one out error of SVMs was recently proposed in [Wahba, 1999]. Nevertheless, like Vapnik s bound both are restricted to unbiased hyperplanes and do not apply to regular SVMs. While lemma 1 is valid for all kernel functions that return positive values, it is tightest when the minimum value is zero. The following lemma shows that this can always be achieved. ....

Wahba, G. (1999). Support vector machines, reproducing kernel hilbert spaces, and randomized gacv. In Scholkopf, B., Burges, C., and Smola, A., editors, Advances in Kernel Methods - Support Vector Learning, chapter 6, pages 69-88. MITPress.

....and it is used to derive bounds on the expected error not estimators. Jaakkola and Haussler [Jaakkola and Haussler, 1999] present a generalized bound for inseparable data that is similar to that of lemma 1. Similarly, an approximation to the leave one out error of SVMs was recently proposed in [Wahba, 1999]. Nevertheless, like Vapnik s bound both are restricted to unbiased hyperplanes and do not apply to regular SVMs. While lemma 1 is valid for all kernel functions that return positive values, it is tightest when the minimum value is zero. The following lemma shows that this can always be achieved. ....

Wahba, G. (1999). Support vector machines, reproducing kernel hilbert spaces, and randomized gacv. In Scholkopf, B., Burges, C., and Smola, A., editors, Advances in Kernel Methods - Support Vector Learning, chapter 6, pages 69--88. MITPress.

.... et al. 1996; Cristianini et al. 1998) The resulting algorithm is similar to Support Vector Machines (SVM) Cortes and Vapnik 1995; Vapnik 1995) which became popular in recent years due to their good generalization properties (Cortes and Vapnik 1995; Smola 1996; Scholkopf 1997; Girosi 1997; Wahba 1997; Weston et al. 1997; Pontil and Verri 1998; Weston and Watkins 1998) Since during learning and application of SVM s only inner products of object representations x i and x j have to be computed, the method of potential functions (the so called kernel trick ) can be applied (Aizerman et al. ....

Wahba, G. (1997). Support Vector Machines, Reproducing Kernel Hilbert Spaces and the randomized GACV. Technical report, Department of Statistics, University of Wisconsin, Madison. TR--NO--984.

....examples into (high or infinite dimensional) feature vectors and allows the classification to be carried out in the feature space without ever explicitly representing it. While support vector machines are non probabilistic classifiers they can be extended and formalized for probabilistic settings[12] (recently also [8] which is the topic of this paper. We can also identify the new formulations with other statistical methods such as Gaussian processes[11, 13, 4] We begin by defining the class of kernel regression techniques for binary classification, establish the connection to other ....

Wahba, G. (1997). Support Vector Machines, Reproducing Kernel Hilbert Spaces and the Randomized GACV. University of Wisconsin - Madison technical report TR984rr.

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Wahba, G. (1998). Support vector machines, reproducing kernel Hilbert spaces, and randomized gacv, In B. Scholkopf, C.J.C. Burges, and A.J. Smola, editors, Advances in Kernel Methods - Support Vector Learning, MIT Press, chapter 6, pp. 69--87.

....can be cast as a variational regularization problem in a reproducing kernel Hilbert space (RKHS) see Kimeldorf Wahba (1971) Wahba (1990) Girosi (1997) Poggio Girosi (1998) the papers and references in Schoelkopf, Burges Smola (1999) and elsewhere. In this note, which is a sequel to Wahba (1998), we look at the SVM paradigm from the point of view of a regularization problem, which allows a comparison with penalized likelihood methods, as well as the application of model selection and tuning approaches which have been used with those and other regularization type algorithms to choose ....

....in SVM classification problems. We then review the Generalized Comparative Kullback Leibler Distance (GCKL) for the usual SVM paradigm, and observe that it is trivially a simple upper bound on the expected misclassification rate. Next we revisit the GACV as a proxy for the GCKL proposed in Wahba (1998) and the argument that it is a reasonable estimate of the GCKL. We found that it is not necessary to do the randomization of the GACV in Wahba (1998) because it can be replaced by an equally justifiable approximation which is readily computed exactly, along with the SVM solution to the dual ....

[Article contains additional citation context not shown here]

Wahba, G. (1998), Support vector machines, reproducing kernel Hilbert spaces and the randomized GACV, Technical Report 984rr, Department of Statistics, University of Wisconsin, Madison WI.

No context found.

Wahba, G., 1999. Support Vector Machines, reproducing kernel Hilbert spaces and the randomized GACV. In: Scholkopf, B., Burgess, C., Smola, A.J. (Eds.), Advances in Kernel Methods---Support Vector Learning. MIT Press, pp. 69 -- 88.

No context found.

Wahba, G. (1999) Support vector machines, reproducing kernel Hilbert spaces and the randomized GACV. In B. Scholkopf, C. Burges & A. Smola, eds, Advances in Kernel Methods - Support Vector Learning. MIT Press.

No context found.

Wahba, G. (1999). Support vector machines, reproducing kernel hilbert spaces, and randomized gacv. In Scholkopf, B., Burges, C., and Smola, A., editors, Advances in Kernel Methods - Support Vector Learning, chapter 6, pages 69-88. MITPress.

No context found.

Wahba, G. (1998). Support vector machines, reproducing kernel Hilbert spaces, and randomized GACV. In B. Scholkopf, C. J. C. Burges, and A. J. Smola (Eds.), Advances in Kernel Methods: Support Vector Learning, pp. 69--88. MIT Press.

No context found.

Wahba, G. (1999) Support vector machine, reproducing kernel Hilbert spaces and the randomized GACV. Advances in Kernel Methods-Support Vector Learning, 69-88, MIT press.

No context found.

Wahba, G. (1999) Support vector machine, reproducing kernel Hilbert spaces and the randomized GACV. Advances in Kernel Methods - Support Vector Learning, 69-88, MIT Press.

No context found.

Wahba, G. (1999). Support Vector Machines, Reproducing Kernel Hilbert Spaces and the Randomized GACV. In `Advances in Kernel Methods - Support Vector Learning', Scholkopf, Burges and Smola (eds.), MIT Press, 69-88.

No context found.

Wahba, G. (1998) Support Vector Machine, Reproducing Kernel Hilbert Spaces and the Randomized GACV. Technical Report 984rr, Department of Statistics, University of Wisconsin, Madison WI.

No context found.

Wahba, G. (1998). Support vector machines, reproducing kernel Hilbert spaces, and randomized gacv, In B. Scholkopf, C.J.C. Burges, and A.J. Smola, editors, Advances in Kernel Methods - Support Vector Learning, MIT Press, chapter 6, pp. 69--87.

No context found.

Wahba, G. (1998) Support Vector Machine, Reproducing Kernel Hilbert Spaces and the Randomized GACV. Technical Report 984rr, Department of Statistics, University of Wisconsin, Madison WI.

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