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5.1 Illustrated Mappings – Constructed From a Small Number of Points 65

a) X 12 X 34 b) c) X 12 X 34 d)

Figure 5.2: A m dimensional PSOM with

training vectors in a d
dimensional embedding space X with the same corner training points as in the
previous figure.

parallel lines are not kept parallel. Any bending of M has disappeared. To
capture the proper “curvature” of the transformation, one should have at
least one further point, between the two endpoints of a mapping interval.

1
0.5
0 Figure 5.3: Isometric projection
-0.5 of the d , m dimen-
-1 sional manifolds M. The

PSOM manifold spans like a
0.5 1 soap film over the four corner-
-1 -0.5 -0.5 0
0 0.5 1 -1
ing reference vectors w a .

Fig. 5.3 visualizes the mapping capabilities of a m dimensional
mapping manifold spanned by

training vectors in the 3 D embed-
ding space X.. The resulting mapping belongs to the “harmonic func-
tions” with a zero second derivative (r s r s w s source free) and has
the characteristics of a “soap film membrane” spanned over the grid.
Fig. 5.4 illustrates the same mapping capabilities for the m dimen-
sional mapping manifold spanned by the 2 2 2 training vectors. The
nodes (corner vertices) span the m-dimensional hypercube configuration
m
with
nodes.
Summarizing, even with very small numbers of nodes, the PSOM man-
ifold has very rich mapping properties. Two and three-dimensional PSOM
were shown, with two and three nodes per axes. Using multi-dimensional