Non-Euclidean Dissimilarities: Causes, Embedding and Informativeness

In many pattern recognition applications object structure is essential for the discrimination purpose. In such cases researchers often use recognition schemes based on template matching which lead to the design of non-Euclidean dissimilarity measures. A vector space derived from the embedding of the dissimilarities is desirable in order to use general classifiers. An isometric embedding of the symmetric non-Euclidean dissimilarities results in a pseudo-Euclidean space. More and better tools are available for the Euclidean spaces but they are not fully consistent with the given dissimilarities. In this chapter first a review is given of the various embedding procedures for the pairwise dissimilarity data. Next the causes are analyzed for the existence of nonEuclidean dissimilarity measures. Various ways are discussed in which the measures are converted into Euclidean ones. The purpose is to investigate whether the original non-Euclidean measures are informative or not. A positive conclusion is derived as examples can be constructed and found in real data for which the non-Euclidean characteristics of the data are essential for building good classifiers.1 Robert P.W. Duin Faculty of Electrical Engineering, Mathematics and Computer Sciences, Delft University of Technology, The Netherlands, e-mail: [email protected] Elżbieta P ↪ ekalska School of Computer Science, University of Manchester, United Kingdom, e-mail: [email protected] Marco Loog Faculty of Electrical Engineering, Mathematics and Computer Sciences, Delft University of Technology, The Netherlands, e-mail: [email protected] 1 This chapter is based on previous publications by the authors, [16, 17, 19, 21, 23, 43] and contains text, figures, equations and experimental results taken from these papers.