Neural Networks: An Overview - Neural Networks: Methodology and Applications - page 79

Fig. 1.51. Gaussian isotropic RBF y =exp[ − ( x 1 + x 2 )] : w 0 = w 1 =0 ,w 3 =1 / √ 2

The term radial basis function arises from approximation theory; they can
be chosen so as to form a mathematical basis of functions. In regression, RBF's
are generally not chosen so as to satisfy that requirement; however, following
the current use, we will keep the term radial basis function.

1.6.2 The Ho and Kashyap Algorithm

The Ho and Kashyap algorithm finds, in a finite number of iterations, whether
two given sets of observations are linearly separable in feature space. If they
are, the algorithm provides a solution (among an infinity of possible solu-
tions), which is not optimized (as opposed to algorithms that are explained
in Chap. 6). Therefore, that algorithm is mainly used for finding out whether
sets are linearly separable. If such is the case, it is advisable to use one of the
optimized algorithms described in Chap. 6.
Consider two sets of examples, having n A and n B elements respectively,
belonging to two classes A and B ; if the examples are described by n features,
each of them is described by an n -vector. We denote by x k the vector that
represents the k -th example of class A ( k =1to n A ), and by w the vector of
parameters of a linear separator; if such a separator exists, i.e., if the examples
are linearly separable, then one has

x k w > 0 for all k,

x k w < 0 for all k.

We define matrix M whose rows are the vectors that represent the examples
of A and the opposites of the vectors representing the examples of B , i.e.,

M =[ x 1 , x 2 ,..., x n a , x 1 , x 2 ,..., x nb ] T ,