Clustering symmetric positive definite matrices on the riemannian manifolds

Abstract

Using structured features such as symmetric positive definite (SPD) matrices to encode visual information has been found to be effective in computer vision. Traditional pattern recognition methods developed in the Euclidean space are not suitable for directly processing SPD matrices because they lie in Riemannian manifolds of negative curvature. The main contribution of this paper is the development of a novel framework, termed Riemannian Competitive Learning (RCL), for SPD matrices clustering. In this framework, we introduce a conscious competition mechanism and develop a robust algorithm termed Riemannian Frequency Sensitive Competitive Learning (rFSCL). Compared with existing methods, rFSCL has three distinctive advantages. Firstly, rFSCL inherits the online nature of competitive learning making it capable of handling very large data sets. Secondly, rFSCL inherits the advantage of conscious competitive learning which means that it is less sensitive to the initial values of the cluster centers and that all clusters are fully utilized without the “dead unit” problem associated with many clustering algorithms. Thirdly, as an intrinsic Riemannian clustering method, rFSCL operates along the geodesic on the manifold and the algorithms is completely independent of the choice of local coordinate systems. Extensive experiments show its superior performance compared with other state of the art SPD matrices clustering methods.