Determining the Cause of Negative Dissimilarity Eigenvalues

Abstract

Pairwise dissimilarity representations are frequently used as an alternative to feature vectors in pattern recognition. One of the problems encountered in the analysis of such data, is that the dissimilarities are rarely Euclidean, and are sometimes non-metric too. As a result the objects associated with the dissimilarities can not be embedded into a Euclidean space without distortion. One way of gauging the extent of this problem is to compute the total mass associated with the negative eigenvalues of the dissimilarity matrix. However,this test does not reveal the origins of non-Euclidean or non-metric artefacts in the data. The aim in this paper is to provide simple empirical tests that can be used to determine the origins of the negative dissimilarity eigenvalues. We consider three sources of the negative dissimilarity eigenvalues, namely a) that the data resides on a manifold (here for simplicity we consider a sphere), b) that the objects may be extended and c) that there is Gaussian error. We develop three measures based on the non-metricity and the negative spectrum to characterize the possible causes of non-Euclidean data. We then experimentally test our measures on various real-world dissimilarity datasets.