Spherical embeddings for non-Euclidean dissimilarities

Many computer vision and pattern recognition problems may be posed by defining a way of measuring dissimilarities between patterns. For many types of data, these dissimilarities are not Euclidean, and may not be metric. In this paper, we provide a means of embedding such data. We aim to embed the data on a hypersphere whose radius of curvature is determined by the dissimilarity data. The hypersphere can be either of positive curvature (elliptic) or of negative curvature (hyperbolic). We give an efficient method for solving the elliptic and hyperbolic embedding problems on symmetric dissimilarity data. This method gives the radius of curvature and a method for approximating the objects as points on a hyperspherical manifold. We apply our method to a variety of data including shape-similarities, graph-similarity and gesture-similarity data. In each case the embedding maintains the local structure of the data while placing the points in a metric space.