AN INDEX OF TOPOLOGICAL PRESERVATION FOR FEATURE-EXTRACTION

This paper is about the ability of principal components analysis, the Sammon algorithm, and an extension of the Kohonen self-organizing feat ure map to preserve spatial order during feature extraction on unlabel ed data. Transformations to q-space that preserve the order of all pai rwise distances in any set of vectors in p-space are defined as metric topology preserving (MTP) transformations. We give a necessary and su fficient condition for this new property in terms of the Spearman rank correlation coefficient. Unlike many other measures of extracted feat ure 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 ex tent to which principal components, Sammon's algorithm and our extensi on 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.