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68 Characteristic Properties by Examples

PSOM training samples are taken from the rectangular grid of the asterisk
markers depicted in Fig. 5.5ab.
The succeeding plots in the lower row present the situation, that the
PSOM only learned the randomly shifted sampling positions Fig. 5.5ef.
The mapping result is shown in the rightmost two plots: The 3 3 3 PSOM
can reconstruct the goal mapping fairly well, illustrating that there is no
necessity of sampling PSOM training points on any precise grid structure.
Here, the correspondence between X and S is weakened by the sampling,
but the topological order is still preserved.

5.3 Topological Order Introduces Model Bias

In the previous sections we showed the mapping manifolds for various
topologies which were already given. This stage of obtaining the topolog-
ical correspondence includes some important aspects:

1. Choosing a topology is the first step in interpreting a given data set.

2. It introduces a strong model bias and reduces therefore the variance.
This leads – in case of the correct choice – to an improved general-
ization.

3. The topological information is mediated by the basis functions. All
examples shown here, build on the high-dimensional extension to
approximation polynomials. Therefore, the examples are special in
the sense, that the basis functions are varying only within their class
as described in Sec. 4.5. Other topologies can require other types of
basis functions.

To illustrate this, let us consider a 2 D example with six training points.
If only such a small data set is available, one may find several topolog-
ical assignments. In Fig. 5.6 the six data points w a are drawn, and two
plausible, but different assignments to a 3 2 node PSOM are displayed.
In the vicinity of the training data points the mapping is equivalent,
but the regions interpolating and extrapolating differ, as seen for the cross-
marked example query point. Obviously, it needs further information to
resolve this ambiguity in topological ordering. Active sampling could re-
solve this ambiguity.