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Next: Note 3: Computational complexity. Up: Choosing good maps Previous: Note 1: Sparse data.

Note 2: High dimensionality.

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.

Sami Kaski
Mon Mar 31 23:43:35 EET DST 1997