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72 Characteristic Properties by Examples

guaranteed. However, for well-behaved mappings the corresponding s
values are “far away”, which leads to the advise: Be suspicious, if the best-

match s is found far outside the given node-set A.
Depending on the particular shape of the embedding manifold M,an
unfortunate gradient situation may occur in the vicinity of the border
training vectors. In some bad cases the local gradient may point to an-
other, far outside local minimum, producing a misleading completion re-
sult. Here the following heuristic proved useful:

In case the initial best-match node a (Sect. 4.3) has a marginal surface
position in A, the minimization procedure Eq. 4.4 should be started at a
shifted position

a a (5.2)
s t
The start-point correction a is chosen to move the start location inside

the node-set hyper-box, perpendicular to the surface. If a is an edge or
corner node, each surface normal contributes to a . The shift length is
uncritical: one third of the node-set interval, but maximal one inter-node
distance, is reasonable. This start-point correction is computationally neg-
ligible and helps to avoid critical border gradient situations, which could
otherwise lead to another, undesired remote minimum of Eq. 4.4.

5.6 Continuity Aspects

The PSOM establishes a smooth and continuous embedding manifold M
s w s . However, the procedure of associative completion bears several
cases of non-continuous responses of the PSOM.
They depend on the particular mapping and on the selection of the
in
input sub-space X , respectively P. The previous section already exhib-
ited the extrapolation case, where multiple solutions occured. What are
important cases, where discontinuous PSOM responses are possible?

Over-specified Input: Consider the case, where the specified input sub-
space X in over-determines the best-match point in the parameter
manifold S. This happens if the dimensionality of the input space
in
is higher than the parameter manifold S: dim X jIj . m
Fig. 5.9 illustrates this situation with a (m ) one-dimensional
PSOM and displays the two input space dimensions X in together