QUANTIFYING THE NEIGHBORHOOD PRESERVATION OF SELF-ORGANIZING FEATURE MAPS

Neighborhood preservation from input space to output space is an essential element of such self-organizing feature maps as the Kohonen map. However, a measure for the preservation or violation of neighborhood relations, which is more systematic than just visual inspection of the map, has been lacking. We show that a topographic product P, first introduced in nonlinear dynamics, is an appropriate measure in this regard. It is sensitive to large-scale violations of the neighborhood ordering, but does not account for neighborhood ordering distortions caused by varying areal magnification factors. A vanishing value of the topographic product indicates a perfect neighborhood preservation; negative (positive) values indicate a too small (too large) output space dimensionality. In a simple example of maps from a 2-D input space onto 1-D, 2-D, and 3-D output spaces we demonstrate how the topographic product picks the correct output space dimensionality. In a second example we map 19-D speech data onto various output spaces and find that a 3-D output space (instead of 2-D) seems to be optimally suited to the data. This is in agreement with a recent speech recognition experiment on the same data set.