Extrinsic Methods for Coding and Dictionary Learning on Grassmann Manifolds
This work addresses the challenge of handling non-Euclidean data in visual recognition for researchers in computer vision and machine learning, though it is incremental as it bridges existing sparsity and manifold techniques.
The authors tackled the problem of sparse coding and dictionary learning on Grassmann manifolds by embedding them into symmetric matrices, extending sparse coding schemes and proposing closed-form dictionary learning solutions. Experiments on classification tasks like gender recognition and action recognition showed considerable accuracy improvements over state-of-the-art methods.
Sparsity-based representations have recently led to notable results in various visual recognition tasks. In a separate line of research, Riemannian manifolds have been shown useful for dealing with features and models that do not lie in Euclidean spaces. With the aim of building a bridge between the two realms, we address the problem of sparse coding and 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. This in turn enables us to extend two sparse coding schemes to Grassmann manifolds. Furthermore, we propose closed-form solutions for learning a Grassmann dictionary, atom by atom. Lastly, to handle non-linearity in data, we extend the proposed Grassmann sparse coding and dictionary learning algorithms through embedding into Hilbert spaces. Experiments on several classification tasks (gender recognition, gesture classification, scene analysis, face recognition, action recognition and dynamic texture classification) show that the proposed approaches achieve considerable improvements in discrimination accuracy, in comparison to state-of-the-art methods such as kernelized Affine Hull Method and graph-embedding Grassmann discriminant analysis.