A Geometry Preserving Kernel over Riemannian Manifolds

Kernel trick and projection to tangent spaces are two choices for linearizing the data points lying on Riemannian manifolds. These approaches are used to provide the pre-requisites for applying the standard machine learning methods on Riemannian manifolds. Classical kernels implicitly project data to a high-dimensional feature space without considering the intrinsic geometry of the data points. Projection to tangent spaces truly preserves topology along radial geodesics. In this paper, we propose a method for extrinsic inference on Riemannian manifold based on the kernel approach. We show that computing the Gramian matrix using geodesic distances, on a complete Riemannian manifold with unique minimizing geodesic between each pair of points, provides a feature mapping that is proportional with the topology of data points in the input space. The proposed approach is evaluated on real datasets composed of EEG signals of patients with two different mental disorders, texture, and visual object classes. To assess the effectiveness of our scheme, the extracted features are examined by other state-of-the-art techniques for extrinsic inference over symmetric positive definite (SPD) Riemannian manifold. The experimental results obtained show the superior accuracy of the proposed approach over approaches that use the kernel trick to compute similarity on SPD manifolds without considering the topology of dataset or partially preserving the topology