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Fig. 7.5.
The discrete topology of a 2D-topological map. The map features 10
×
10
neurons; each dot of the picture denotes a neuron
c
. The distance
δ
between two
neurons is defined on the grid. (
a
)shows
V
c
(1)
,V
c
(2)
,V
c
(3), which are neighborhoods
of order 1, 2 and 3 of neuron
c
;(
b
) shows some distances between neurons:
δ
(
c, c
1) =
4
,δ
(
c, c
2) = 1
,δ
(
c, c
3) = 2
,δ
(
c, c
4) = 3
an additional constraint is imposed to retain the topology of the map: two
neighboring neurons
r
and
c
are associated to reference vectors
w
c
and
w
r
that are close for the Euclidean distance in data space
D
.
That cursory description shows clearly that the self-organizing map algo-
rithm is an extension of
k
-means. We will further show that it minimizes an
appropriate cost function, which takes into account the inertia of the parti-
tion of the data set, and which guarantees that the topology of
C
is retained.
In order to design such a cost function, the inertia function of the
k
-means
algorithm will be generalized, by adding specific terms that take into account
the topology of the map, through the distance
δ
and the associated neighbor-
hoods.
The concept of neighborhood is taken into account through kernel func-
tions
K
which are positive and such that lim
|χ|→∞
K
(
x
) = 0. Those kernels
define influence regions around each neuron
c.
The distances
δ
(
c,r
) between
neurons
c
and
r
of the map allow the definition of the relative influence of the
neurons on elements of the data set. The quantity
K
(
δ
(
c,r
)) quantifies that
influence.
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