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b
b
Ð
Ð
K
(
x
,
y
)
g
(
x
)
g
(
y
)
dxdy
0
(8.37)
a
a
is valid.
This decision condition is not feasible. It is well known that polynomial
function satisfy Mercer's condition. Therefore,
K
(
x,y
) satisfy Mercer's condition
if it approaches some polynomial function.
The learning machines that construct decision functions of the type (8.36) are
called support vector machines (SVM). In SVM the complexity of the
construction depends on the number of support vectors rather than on the
dimensionality of the feature space. The scheme of SVM is shown in Figure 8.6.
In nonlinear case, SVM maps the input space vectors into a high-dimensional
feature space vectors through some nonlinear mapping defined inner product
functions. Thus, SVM can find the generalized optimal separating hyperplane in
feature space.
d(x)
Decision rule
α 1 y 1
α n y n
α 2 y 2
K(x 1 ,x)
K(x 2 ,x)
K(x n ,x)
Feature space
vector
Input vector
X 1
X 2
X 3
X d
Figure 8.6. Support Vectors Machine (SVM)
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