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74 Characteristic Properties by Examples

three compatible solutions s fulfilling Eq. 4.4, which is a bifurcation
with respect to the shift operation and a discontinuity with respect
to the mapping x x .

In view of the pure x projection, the final stage could be interpreted
as “topological defect” (see Sec. 5.4). Obviously, this consideration is
relative and depends very much on further circumstances, e.g. infor-
mation embedded in further X components.

5.7 Summary

The construction of the parameterized associative map using approxima-
tion polynomials shows interesting and unusual mapping properties. The
high-dimensional multi-directional mapping can be visualized by the help
of test-grids, shown in several construction examples.
The structure of the prototypical training examples is encoded in the
topological order, i.e. the correspondence to the location (a) in the map-
ping manifold S. This is the source of curvature information utilized by
the PSOM to embed a smooth continuous manifold in X. However, in
certain cases input-output mappings are non-continuous. The particular
manifold shape in conjunction with the associative completion and its op-
tional partial distance metric allows to select sub-spaces, which exhibit
multiple solutions. As described, the approximation polynomials (Sec. 4.5)
as choice of the PSOM basis function class bears the particular advan-
tage of multi-dimensional generalization. However, it limits the PSOM
approach in its extrapolation capabilities. In the case of a low-dimensional
input sub-space, further solutions may occur, which are compatible to the
given input. Fortunately, they can be easily discriminated by their their

remote s location.