Spherical Embedding and Classification

Most problems in pattern recognition can be posed in terms of using the dissimilarities between the set of objects of interest. A vector-space representation of the objects can be obtained by embedding them as points in Euclidean space. However many dissimilarities are non-Euclidean and cannot be represented accurately in Euclidean space. This can lead to a loss of information and poor performance. In this paper, we approach this problem by embedding the points in a non-Euclidean curved space, the hypersphere. This is a metric but non-Euclidean space which allows us to define a geometry and therefore construct geometric classifiers. We develop a optimisation-based procedure for embedding objects on hyperspherical manifolds from a given set of dissimilarities. We use the Lie group representation of the hypersphere and its associated Lie algebra to define the exponential map between the manifold and its local tangent space. We can then solve the optimisation problem locally in Euclidean space. This process is efficient enough to allow us to embed large datasets. We also define the nearest mean classifier on the manifold and give results for the embedding accuracy, the nearest mean classifier and the nearest-neighbor classifier on a variety of indefinite datasets.