Structure Preserving Embedding of Dissimilarity Data

Abstract

Partitioning methods for observations represented by pairwise dissimilarities are studied. Particular emphasis is put on their properties when applied to dissimilarity matrices that do not admit a loss-free embedding into a vector space. Specifically, the Pairwise Clustering cost function is shown to exhibit a shift invariance property which basically means that any symmetric dissimilarity matrix can be modified to allow a vector-space representation without distorting the optimal group structure. In an approximate sense, the same holds true for a probabilistic generalization of Pairwise Clustering, the so-called Wishart–Dirichlet Cluster Process. This shift-invariance property essentially means that these clustering methods are “blind” against Euclidean or metric violations. From the application side, such blindness against metric violations might be seen as a highly desired feature, since it broadens the applicability of certain algorithms. From the viewpoint of theory building, however, the same property might be viewed as a “negative” result, since studying these algorithms will not lead to any new insights on the role of metricity in clustering problems.