Clustering Persistence Barcodes of Multidimensional Scaling Representations from Random Initial Configurations

Abstract

Multidimensional scaling (MDS) is a popular technique for exploring complex datasets. A common application is examination of collections of objects equipped with pairwise dissimilarities. Iterative optimization produces points in Euclidean space whose pairwise distances approximate the pairwise dissimilarities between the original data objects. For a single dataset, starting at different initial configurations during optimization may produce MDS representations with substantially different structural features. This presents both challenges and opportunities for those who use these methods. We use persistence barcodes, a descriptor from topological data analysis (TDA), to reveal clusters of multidimensional scaling representations. By applying our methods to datasets consisting of photographs of rotating objects, we uncover robust and interpretable but relatively uncommon MDS configurations which have higher stress than many representations obtained through random initialization. These interpretable configurations may be easily missed in standard analysis. We are optimistic that the results described in this manuscript will encourage further consideration of MDS in cases when preliminary configurations returned for a dataset do not depict known structural features.