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k
(
δ
(
i
,
j
))
k
(
δ
(
i
,
j
))
Interaction with the
cells of the map
Interaction with the
cells of the map
T
(2)
T
(1)
T
(0)
Temperature
T
i
i
δ
(
i
,
j
)
δ
(
i
,
j
)
Location of cell
i
Location of cell
i
Fig. 7.6.
Threshold neighborhood kernel (
left picture
) and Gaussian neighborhood
kernel (
right picture
). In the case of the threshold kernel, neurons either belong to the
neighborhood and share the same influence, or do not belong to the neighborhood,
hence have no influence at all. In the case of Gaussian kernels, the influence between
two neurons depends on their mutual distance
To take advantage of the size of the neighborhood, the family of kernels
K
T
that is parameterized by
T
(where
T
stands for temperature) will be used:
K
T
(
δ
)=
K
(
δ/T
)
Figure 7.6 shows kernel functions that are commonly used in applications:
K
(
δ
)=
1 f
δ<
1
0
hence
K
T
(
δ
)=
1 f
δ<T
0
•
otherwise
otherwise
) hence
K
T
(
δ
)=exp
−
|δ|
T
•
K
(
δ
)=exp(
−|
δ
|
δ
2
) hence
K
T
(
δ
)=exp
.
−
δ
2
T
2
•
K
(
δ
)=exp(
−
Figure 7.7 shows graphs of various kernels, for different values of parameter
T
.Ifwechoosealevel
α
such that the influence of a neuron that is below
α
is considered negligible (
K
T
(
δ
)
<α
), the radius of the effective neighborhood
of a neuron can be computed for each value of
T
. For neuron
c
, that influence
zone is exactly the ball
V
c
. Figure 7.7 shows
that the size of the neighborhood decreases with
T
: the smaller
T
, the fewer
the neurons that belong to the neighborhood
V
c
. The self-organizing map
training algorithms minimize a cost function. When the minimum is reached,
one gets a partition that is made of sets that are compact enough, and, in
addition, it is possible to define an order that stems from the topology of the
map. That cost function will be hereinafter noted as
J
som
. It plays the role
of the cost function
I
of the
k
-means algorithm that was described in the
previous section. We will now consider the most popular function
J
som
, which
is
=
{
r
∈
C/K
T
(
δ
(
c,r
))
>α
}
J
som
(
χ,W
)=
z
i
∈A
2
.
K
T
(
δ
(
c,χ
(
z
i
)))
z
i
−
w
c
c∈C
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