An index of topological preservation for feature extraction

Abstract

This paper is about the ability of principal components analysis, the Sammon algorithm, and an extension of the Kohonen self-organizing feature map to preserve spatial order during feature extraction on unlabeled data. Transformations to q-space that preserve the order of all pairwise distances in any set of vectors in p-space are defined as metric topology preserving (MTP) transformations. We give a necessary and sufficient condition for this new property in terms of the Spearman rank correlation coefficient. Unlike many other measures of extracted feature quality, the MTP index is independent of the extraction method. A modification of the Kohonen self-organizing feature map algorithm that extracts vectors in q-space from data in p-space is developed. The extent to which principal components, Sammon's algorithm and our extension of the self-organizing feature map (SOFM) preserve the MTP property is discussed. Our MTP index shows that the first two methods preserve distance ranks on seven data sets much more effectively than extended SOFM.