Database Reference
In-Depth Information
Distance between Feature Vectors.
The length of the line joining two
images
φ
(
x
)and
φ
(
z
) can be computed as
2
=
φ
(
x
)
−
φ
(
z
)
φ
(
x
)
−
φ
(
z
)
,φ
(
x
)
−
φ
(
z
)
=
φ
(
x
)
,φ
(
x
)
−
2
φ
(
x
)
,φ
(
z
)
+
φ
(
z
)
,φ
(
z
)
=
κ
(
x
,
x
)
−
2
κ
(
x
,
z
)+
κ
(
z
,
z
)
.
(1.1)
It is easy to find out that this is a special case of the norm. The algorithms
demonstrated at the end of this chapter are based on distance.
Norm and Distance from the Center of Mass.
Consider now the center
of mass of the set
φ
(
S
). This is the vector
1
φ
S
=
φ
(
x
i
)
.
i
=1
As with all points in the feature space we have not an explicit vector rep-
resentation of this point, but in this case there may not exist a point in
X
whose image under
φ
is
φ
S
. However we can compute the norm of the points
of
φ
S
using only evaluations of the kernel on the inputs:
=
1
φ
(
x
j
)
φ
(
x
i
)
,
1
2
2
=
φ
S
φ
S
,φ
S
i
=1
j
=1
i,j
=1
1
2
1
2
=
φ
(
x
i
)
,φ
(
x
j
)
=
κ
(
x
i
,
x
j
)
.
i,j
=1
Hence, the square of the norm of the center of mass is equal to the average
of the entries in the kernel matrix. This implies that this sum is equal to
zero if the center of mass is at the origin of the coordinate system and greater
than zero otherwise. The distance of the image of a point
x
from the center
of mass
φ
S
is:
2
=
φ
(
x
)
−
φ
S
φ
(
x
)
,φ
(
x
)
+
φ
S
,φ
S
−
2
φ
(
x
)
,φ
S
=
κ
(
x
,
x
)+
1
2
2
κ
(
x
i
,
x
j
)
−
κ
(
x
,
x
i
)
.
(1.2)
i,j
=1
i
=1
Linear Classification.
Classification, also called categorization in text
analysis, is one of the possible tasks that can be solved using kernel approach.
The aim is to assign any input of our training set to one of a finite set of
categories; the classification is binary if there are two categories, otherwise we
are considering a multi-class problem.
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