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where • denote vector dot products. To describe the separating hyperplane let us
use the following form:
w
•
x
+
b
≥
0,
if
y
i
=+1
i
w
•
x
+
b
<
0
if
y
i
=-1
i
w
where
w
denote hyperplane's normal direction,
denote unit normal vector,
w
w
and
denote Euclid modular function.
We say that this set of vectors is separated by the optimal hyperplane (or the
maximal margin hyperplane) if the training data is separated without error and
the distance between the closest vector to the hyperplane is maximal (See Figure
8.4).
Optimal hyperplane
Optimal hyperplane
Figure 8.4. The optimal separating hyperplane
For linearly separable case, to find the optimal separating hyperplane is to solve
the following quadratic programming problem. Given training samples, find a
pair consisting of a vector
w
and a constant (threshold)
b
such that they minimize
the function
1
2
min
Φ
(
w
)
=
w
(8.20)
2
under the constraints of inequality type
y
(
w
•
x
+
b
)
−
1
≥
0
i
=
1
?
2
,
l
(8.21)
i
i
