Riemannian competitive learning for symmetric positive definite matrices clustering

Abstract

Abstract Symmetric positive definite (SPD) matrices have achieved considerable success in numerous computer vision applications including activity recognition, texture classification, and diffusion tensor imaging. Traditional pattern recognition methods developed in Euclidean space are not suitable for direct processing of SPD matrices because they lie in a Riemannian manifold of negative curvature. In this paper, we draw inspiration from neural network based simple competitive learning, which updates the weight of a neural network in Euclidean space, and propose Riemannian competitive learning which updates the weight in the Riemannian weight space, for SPD manifold clustering. This framework is based on the Fisher-Rao metric which is theoretically very good. We further introduce a conscious competition mechanism to enhance the performance of the RCL algorithm and have developed a robust algorithm termed Riemannian Frequency Sensitive Competitive Learning (rFSCL). Compared with existing methods, the advantages are threefold: (1) rFSCL inherits the online nature of competitive learning, which makes it capable of handling very large data sets; (2) rFSCL inherits the advantage of conscious competitive learning, which indicates 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; and (3) As an intrinsic Riemannian clustering method, rFSCL operates along the geodesic on the manifold in an closed-form analytic manner and is completely independent of the choice of local coordinate systems. Extensive experiments on both synthetic data sets and real data sets show its superior performance to that of other state-of-the-art SPD clustering methods.