One motivation for using an index of continuity of the mapping from
the input space to the map grid as a measure of the goodness of the
mapping stems from the fact that the lower-dimensional SOM grid tends
to fold when trying to follow a distribution of a higher
dimensionality. Discontinuities in the mapping could then indicate the
presence of such folds. This motivation may be misleading for
high-dimensional data spaces which are necessarily sparse because of
the ``curse of dimensionality'' (cf. Bellman,
1961). There may be too few samples for
distinguishing between a non-linear curve and a sample from a
higher-dimensional distribution, for instance. The goodness measure
proposed in Publication 7 may, however, be a useful
index of the *regularity* of the mapping even in such cases, and
if there are local lower-dimensional regions in the input space the
measure will be useful in those regions.

Mon Mar 31 23:43:35 EET DST 1997