A Tutorial on Support Vector Machines for Pattern Recognition

214.3. Some Examples of Nonlinear SVMsThe rst kernels investigated for the pattern recognition problem were the following:K(x y) =(x y +1) p (74)K(x y) =e ;kx;yk2 =2 2 (75)K(x y) = tanh(x y ; ) (76)Eq. (74) results in a classier that is a polynomial of degree p in the data Eq. (75) givesa Gaussian radial basis function classier, and Eq. (76) gives a particular kind of two-layersigmoidal neural network. For the RBF case, the number of centers (N S in Eq. (61)),the centers themselves (the s i ), the weights ( i ), and the threshold (b) are all producedautomatically by the SVM training and give excellent results compared to classical RBFs,for the case of Gaussian RBFs (Scholkopf et al, 1997). For the neural network case, therst layer consists of N S sets of weights, each set consisting of d L (the dimension of thedata) weights, and the second layer consists of N S weights (the i ), so that an evaluationsimply requires taking a weighted sum of sigmoids, themselves evaluated on dot products ofthe test data with the support vectors. Thus for the neural network case, the architecture(number of weights) is determined by SVM training.Note, however, that the hyperbolic tangent kernel only satises Mercer's condition forcertain values of the parameters and (and of the data kxk 2 ). This was rst noticedexperimentally (Vapnik, 1995) however some necessary conditions on these parameters forpositivity are now known 14 .Figure 9 shows results for the same pattern recognition problem as that shown in Figure7, but where the kernel was chosen to be a cubic polynomial. Notice that, even thoughthe number of degrees of freedom is higher, for the linearly separable case (left panel), thesolution is roughly linear, indicating that the capacity is being controlled and that thelinearly non-separable case (right panel) has become separable.Figure 9.Degree 3 polynomial kernel. The background colour shows the shape of the decision surface.Finally, note that although the SVM classiers described above are binary classiers, theyare easily combined to handle the multiclass case. A simple, eective combination trains