Hard margin nonlinear SVM classification

The mathematical formulation of the hard margin nonlinear SVM classification is similar to that presented for the SVM classification for linearly separable datasets, only now input patterns x are replaced with feature functions, X — 4)(x), and the dot product for two feature functions () (x,) i xj) is replaced with a kernel function K xi, Xj), Eq. [64]. Analogously with Eq. [28], the dual problem is 

The vector w that determines the optimum separation hyperplane is 

Therefore, the threshold b can be obtained by averaging the b values obtained for all support vector patterns, i.e., the patterns with Ij 0  

The SVM classifier obtained with a kernel K is defined by the support vectors from the training set (7,- 0) and the corresponding values of the Lagrange multipliers Xf. 

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