  Recent advances in computer vision and machine learning suggest that a wide
range of problems can be addressed more appropriately by considering
non-Euclidean geometry. In this paper we explore sparse dictionary learning
over the space of linear subspaces, which form Riemannian structures known as
Grassmann manifolds. To this end, we propose to embed Grassmann manifolds into
the space of symmetric matrices by an isometric mapping, which enables us to
devise a closed-form solution for updating a Grassmann dictionary, atom by
atom. Furthermore, to handle non-linearity in data, we propose a kernelised
version of the dictionary learning algorithm. Experiments on several
classification tasks (face recognition, action recognition, dynamic texture
classification) show that the proposed approach achieves considerable
improvements in discrimination accuracy, in comparison to state-of-the-art
methods such as kernelised Affine Hull Method and graph-embedding Grassmann
discriminant analysis.
