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4.4 Learning Phases 55

Assuming that a proper subset of input components I has been specified
(usually when the number of input components is equal to the mapping
dimension, jIj m), an input vector Px can be completed with a zero

distance (E s =0) best-match location Pw s Px in the
mapping man-
ifold M. Then, only the output components are effectively adapted by
Eq. 4.14 (equivalent to the selection matrix P). If the measurement of
the training vector components is differently reliable or noisy, the learning
parameter might be replaced by a suitable diagonal matrix controlling
the adaptation strength per component (we return to this issue in Sect. 8.2.)

Fig. 4.8 illustrates this procedure for an m dimensional three-node
PSOM. We consider the case of a d dimensional embedding space

x x and I f g, i.e., a mapping x x x . The embedded mani-

fold M is drawn before and after 300 adaptation steps (using Eq. 4.14 with
a learning steps parameter linearly decreased from 0.8 to 0.1) together
with the reference vectors and the noisy training data set.

Note, that Eq. 4.14 is taking the entire PSOM output w s into account,
in contrast to w a in the “locally acting” Kohonen step of Eq. 3.10. But the
alternatively proposed learning rule (Ritter 1993)

w a a s x H w a (4.15)

turns out to be unstable. A correctly trained PSOM is even corrupted by a
sequence of further training examples.
We demonstrate this fact by an example. For the same learning task,
Fig. 4.9 depicts the initially correct parabola shaped mapping manifold
M and the asterisk marked locations of the three reference vectors w a in
the front curve. The other curves, shifted to the back, show M after sev-
eral epochs of 25 adaptation steps, each using Eq. 4.15 (uniformly random
sampled training vectors, ). The adaptation leads to a displace-
ment of the reference vectors in such a way, that the symmetric parabola
becomes distorted. This shows that Eq. 4.15 is not able to maintain the
correct mapping. The reference vectors are shaken around too-much – in a
non-localized manner, and in both the output and input space. Note, that
the mapping manifold M in Fig. 4.9 does not display one simple parabola
but in fact the parameterized curve w s s w (the joint ordinate pro-

jection of two parabolas w s s with
w the invisible abscissa s in the
parameter manifold S; Initially w s is constructed here as a linear func-
tion (degenerated parabola) in s, similar as Fig. 4.8).