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Chapter 4

The PSOM Algorithm

Despite the improvement by the LLMs, the discrete nature of the stan-
dard SOM can be a limitation when the construction of smooth, higher-
dimensional map manifolds is desired. Here a “blending” concept is re-
quired, which is generally applicable — also to higher dimensions.
Since the number of nodes grows exponentially with the number of
map dimensions, manageably sized lattices with, say, more than three
dimensions admit only very few nodes along each axis direction. Any
discrete map can therefore not be sufficiently smooth for many purposes
where continuity is very important, as e.g. in control tasks and in robotics.
In this chapter we discuss the Parameterized Self-Organizing Map (“PSOM”)
algorithm. It was originally introduced as the generalization of the SOM
algorithm (Ritter 1993). The PSOM parameterizes a set of basis functions
and constructs a smooth higher-dimensional map manifold. By this means
a very small number of training points can be sufficient for learning very
rapidly and achieving good generalization capabilities.

4.1 The Continuous Map

Starting from the SOM algorithm, described in the previous section, the
PSOM is also based on a lattice of formal neurons, in the followig also
called “nodes”. Similarly to the SOM, each node carries a reference vector
d
w a, projecting into the d-dimensional embedding space X
IR .
The first step is to generalize the index space A in the Kohonen map
m
to a continuous auxiliary mapping or parameter manifold S IR in the

J. Walter “Rapid Learning in Robotics” 43