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Chapter 5

Characteristic Properties by

Examples

As explained in the previous chapter, the PSOM builds a parameterized
associative map. Employing the described class of generalized multi-dimensional
Lagrange polynomials as basis functions facilitates the construction of very
versatile PSOM mapping manifolds. Resulting unusual characteristics
and properties are exposed in this chapter.
Several aspects find description: topological order introduces model
bias to the PSOM and strongly influences the interpretation of the training
data; the influence of non-regularities in the process of sampling the train-
ing data is explored; “topological defects” can occur, e.g. if the correspon-
dence of input and output data is mistaken; The use of basis polynomials
affects the PSOM extrapolation properties. Furthermore, in some cases the
given mapping task faces the best-match procedure with the problem of
non-continuities or multiple solutions.
Since the visualization of multi-dimensional mappings embedded in
even higher dimensional spaces is difficult, the next section illustrates
stepwise several PSOM examples. They start with small numbers of nodes.

5.1 Illustrated Mappings – Constructed From a

Small Number of Points

The first mapping example is a two-dimensional (m ) PSOM mapping
manifold with reference vectors in a four-dimensional (d ) dimen-

J. Walter “Rapid Learning in Robotics” 63