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222 5 Neural Networks

Figure 5.48. SVM solution for the classification of two classes of cork stoppers,
using C=10. The number of support vectors is 33 (a third of the total number of
patterns).

In the transformed feature space we now have the optimal weight vector:

i? = '&3;t; f (x,).
S vs
Using (5-105), the linear decision function in the transformed feature space is:

d(x) = w' y* = C&ti f (xi)' f (x),
S Vs
or
d(x) = w' y * = ZEiiti~(xi, x),
svs

where K(xi,x) is known as an inner-product kernel, which allows us to express the
decision functions in the original feature space.
The kernel K(x,x) will exist, provided some non-stringent conditions are
fulfilled (see e.g. Haykin, 1999). In particular, the following kernels can be used:

Polynomial
Gaussian radial basis function
Exponential radial basis function
Tanh sigmoid