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70 Characteristic Properties by Examples

If we have insufficient information about the correct node assignment
and are forced to make a some specification, we may introduce “topologi-
cal defects”.

5.4 “Topological Defects”

What happens if the training vectors w a are not properly assigned to the
node locations a? What happens, if the topological order is mistaken
and the neighborhood relations do not correspond the closeness in input
space? Let us consider here the case of exchanging two node correspon-
dences.
Fig. 5.7a-b and Fig. 5.7c-d depict two previous examples, where two
reference vectors got swapped. One the left side, the

PSOM exhibits
a complete twist, pinching all vertical lines. The right pictures show, how
the embedding manifold of the PSOM in Fig. 5.1 becomes distorted
in the lower right part. The PSOM manifold follows nicely all “topolog-
ical defects” given and resembles an “elastic net” or cover, pinned at the
supporting training vectors.

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

Figure 5.7: “Topological defects” by swapping two training vectors: a–b the
PSOM of Fig. 5.2 and c–d the PSOM of Fig. 5.1

Note, that the node points are still correctly mapped, as one can expect
from Eq. 4.2, but in the inter-node areas the PSOM does not generalize
well. Furthermore, if the opposite mapping direction is chosen, the PSOM

has in certain areas more than one unique best-match solution s . The
result, found by Eq. 4.4, will depend on the initial start point s t .
Can we algorithmically test for topological defects? Yes, to a certain
extent. Bauer and Pawelzik (1991) introduced a method to compare the