The PSOM Algorithm
Despite the improvement by the LLMs, the
discrete nature of the standard SOM can be a limitation when the
construction of smooth, higherdimensional map manifolds
is desired. Here a “blending” concept is required, 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.
Figure 4.1:
The PSOM's starting
position is very much the same as for the SOM
depicted in Fig. 3.5. The gray shading indicates that the
index space A
, which is
discrete in the SOM, has been generalized to the
continuous space S
in the PSOM.
The space S
is referred to as
parameter space S.PSOM.
This is indicated by the grey shaded area on the right side of
Fig. 4.1.
Specifying for each training vector a node
location
a A
introduces a topological order
between the training points
wa:
training vectors assigned to nodes
a
and
a,
that are neighbors in the lattice
A,
are perceived to have this specific neighborhood relation. This
has an important effect: it allows the PSOM to draw extra
curvature information from the training set. Such
information is not available within other techniques, such as
the RBF approach (compare Fig. 3.3, and later examples, also in
Chap. 8). The topological organization of the given data points
is crucial for a good generalization behavior. For a general
data set the topological ordering of its points may be quite
irregular and a set of suitable basis functions
Ha s
difficult to construct. A suitable
set of basis functions can be constructed in several ways but
must meet two conditions:
