Dictionary Learning and Sparse Coding on Grassmann Manifolds: An Extrinsic Solution

Recent advances in computer vision and machine learning suggest that a wide range of problems can be addressed more appropriately by considering non-Euclidean geome-try. 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 en-ables us to devise a closed-form solution for updating a Grassmann dictionary, atom by atom. Furthermore, to han-dle non-linearity in data, we propose a kernelised version of the dictionary learning algorithm. Experiments on sev-eral classification tasks (face recognition, action recogni-tion, dynamic texture classification) show that the proposed approach achieves considerable improvements in discrim-ination accuracy, in comparison to state-of-the-art meth-ods such as kernelised Affine Hull Method and graph-embedding Grassmann discriminant analysis. 1