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5.10 Support Vector Machines
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Notice that the preceding linear approaches can be viewed as particular cases of
the kernel approach, using a linear kernel, K(x,,x) = x'x,.
The connectionist structure of a generalized support vector machine, using a
kernel K(xi,x), is similar to the RBF network structure (see Figure 5.42) for a two
class problem, with the hidden layer neurons computing the kernels, which are next
linearly combined (5-107a) to provide the output .
Figure 5.49 exemplifies an SVM quadratic discrimination, K(xi,x) = (x'x~+~)~,
using two different values of C. The influence of C on the margin width and the
quadratic shape is evident. Using C=1000 for the same data a quadratic solution
similar to the C=m solution was obtained, with similar margin and misclassified
samples, in a clear demonstration that there may be many different "optimal"
values for C. The generalization properties of non-linear SVMs are still an open
issue.
The Support Vector Machine approach can also be applied to regression
problems. A description of this topic can be found in Gunn (1997) and Haykin
(1999).
Figure 5.49. SVM quadratic discrimination. (a) C=100, eleven support vectors, 4
misclassified patterns; (b) C=w, fifteen support vectors, 6 misclassified patterns.
5.1 1 Kohonen Networks
All the previous neural networks performed supervised classification or regression
tasks. Unlike these, Kohonen's selforganising feature map or Kohonen network
for short, constitutes a neural net approach to data clustering. As shown in Figure
5.50, these networks are constituted by just one layer of output neurons, arranged
as a two-dimensional grid. The main goal is to iteratively adjust the weights
connecting inputs to outputs, such that in the end these reflect the distance relations
among input patterns.