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retrieval where there is a small set of samples with a high level of correlation
between the samples. This new strategy is based on the following points:
The learning method formulates and solves the local approximator
K
(
x
,
z
)
from
available positive samples.
In order to obtain a dynamic weighting scheme, the Euclidean norm in
x
−
z
is replaced with the weighted Euclidean,
x
−
z
M
.
In order to take advantage of negative samples to improve the decision boundary,
a method of shifting centers is obtained, instead of employing linear weights.
The learning strategy for the ARBFN consists of two parts. First, the local
approximators
K
are constructed using
positive
samples. Second, in order to
improve the decision boundary,
negative
samples are used for shifting the centers,
based on anti-reinforced learning [
331
].
(
x
,
z
)
2.4.3.1
Construction of Local Approximators
N
p
i
X
+
=
{
x
i
}
Given the set of positive samples,
1
, each positive sample is assigned
=
to the local approximator
K
(
x
,
z
i
)
, so that the shape of each relevant cluster can be
described by:
exp
2
−
x
−
z
i
K
(
x
,
z
i
)=
,
(2.52)
i
2
˃
x
i
, ∀
z
i
=
i
∈ {
1
,...,
N
m
},
N
m
=
N
p
(2.53)
min
z
i
−
z
j
, ∀
j
∈{
˃
i
=
ʴ
·
1
,
2
,...,
N
p
},
i
=
j
(2.54)
where
5 is an overlapping factor.
Here, only the positive samples are assigned as the centers of the RBF functions.
Hence, the estimated model function
f
ʴ
=
0
.
(
x
)
is given by:
N
m
i
=
1
ʻ
i
K
(
x
,
z
i
)
f
(
x
)=
(2.55)
ʻ
i
=
1
, ∀
i
∈{
1
,...,
N
m
}
(2.56)
The linear weights are set to constant, indicating that all the centers (or the
positive samples) are taken into consideration. However, the degree of importance of
K
is indicated by the natural responses of the Gaussian-shaped RBF functions
and their superposition. For instance, if centers
z
a
and
z
b
are highly correlated (i.e.,
z
a
≈
(
x
,
z
i
)
z
b
), the magnitude of
f
(
x
)
will be biased for any input vector
x
located near
z
a
or
z
b
, i.e.,
f
(
x
)
≈
2
K
(
x
,
z
a
)
≈
2
K
(
x
,
z
b
)
.
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