Spherical and hyperbolic embeddings of data

Many computer vision and pattern recognition problems may be posed as the analysis of a set of {\bf dissimilarities} between objects. For many types of data, these dissimilarities are not Euclidean (i.e. they do not represent the distances between points in a Euclidean space), and therefore cannot be isometrically embedded in a Euclidean space. Examples include shape-dissimilarities, graph distances and mesh geodesic distances. In this paper, we provide a means of embedding such non-Euclidean data onto surfaces of constant curvature. We aim to embed the data on a space whose radius of curvature is determined by the dissimilarity data. The space can be either of positive curvature (spherical) or of negative curvature (hyperbolic). We give an efficient method for solving the spherical and hyperbolic embedding problems on symmetric dissimilarity data. Our approach gives the radius of curvature and a method for approximating the objects as points on a hyperspherical manifold without optimisation. For objects which do not reside exactly on the manifold, we develop a optimisation-based procedure for approximate embedding on a hyperspherical manifold. We use the exponential map between the manifold and its local tangent space to solve the optimisation problem locally in the Euclidean tangent space. This process is efficient enough to allow us to embed datasets of several thousand objects. We apply our method to a variety of data including time warping functions, 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.