An index of topological preservation and its application to self-organizing feature maps

We discuss topological preservation under feature extraction transformations. Transformations that preserve the order of all distances in any neighborhood of vectors in p-space are defined as metric topology preserving (MTP) transformations. We give a necessary and sufficient condition for this property in terms of Spearman´s rank correlation coefficient. A modification of Kohonen´s self-organizing feature map algorithm that extracts vectors in q-space from data in p-space is given. Three methods are empirically compared: principal components analysis; Sammon´s algorithm; and our extension of the self-organizing feature map algorithm. Our MTP index shows that the first two methods preserve distance ranks on six data sets much more effectively than extended SOFM.