Page 74 - Rapid Learning in Robotics

60 The PSOM Algorithm

From here the coefficients of the linear system Eq. 4.13 can be assembled

d

X
k ws

E p k x k w k s (4.27)

s k
s

and

d

X
k ws
k ws
w k s

E p k x k w k s (4.28)

s
s
s
s
s
s

k
Care should be taken to efficiently skip any computing of non-input com-
ponents k with p k . Later, we present several examples where the

PSOM ability to augment the embedding space is extensively used. In
Chap. 9 the PSOM will find a hierarchical structure where it is important,
that a few hundred output components (d) can be completed without sig-
nificant operational load.
Other numerical algorithms exist, which optimize the evaluation of
special polynomials, for example the recursive Neville's algorithm (Press
et al. 1988). Those are advantageous for the situation m d . But
the requirement to deal with multiple dimensions m and even very large
embedding dimensions d makes the presented algorithm superior with
respect to computational efficiency.

4.6 Summary

The key points of the “Parameterized Self-Organizing Map” algorithm are:

The PSOM is the continuous analog of the standard discrete “Self-
Organizing Map” and inherits the SOM's unsupervised learning ca-
pabilities (Kohonen 1995). It shows excellent generalization capabil-
ities based on continuous attractor manifolds instead of just attractor
points.

Based on a strong bias, introduced by structuring the training data in
a topological order, the PSOM can generalize from very few examples
— if this assumed topological model is a good approximation to the
system.