Information Technology Reference
In-Depth Information
The optimize function equality (8.26) and decision function equality (8.31) only
refer inner product
x
•
x
of training samples. Therefore, we calculate inner
product in high-dimensional space using inner product functions in input space.
On the basis of relational theory, a kernel function
i
j
K x x
•
is parallelism of
inner product in a certain space if it satisfies Mercer's condition (Vapnik, 1995).
(
)
i
j
Input space
Feature space
Φ
Figure 8.5. The SVM maps the input space into a feature space
K x x
•
, SVM maps the
input space vectors into a feature space vectors through some nonlinear mapping.
Moreover, the complexity of calculate does not increase. The object function
equality (8.26) is transformed to the following equality
(
)
Using some appropriate inner product function
i
j
l
1
l
l
Ã
Ã
Ã
max
W
(
α
)
=
α
−
α
α
y
y
K
(
x
,
x
)
(8.35)
i
i
j
i
j
i
j
2
α
i
=
1
i
=
1
j
=
1
Thus, the corresponding classifying function becomes the following equality.
l
=
Ã
=
*
*
d
(
x
)
y
α
K
(
x
,
x
)
+
b
(8.36)
i
i
i
i
1
It is called support vector machines (SVM).
For a given
), the corresponding function Φ(
x
) exists under certain
condition. It is necessary and sufficient that the condition as follow is satisfied.
K
(
x,y
b
Ð
2
g
(
x
)
dx
is finite,
Given an arbitrary function g(x), if
a
