  Recent advances suggest that a wide range of computer vision problems can be
addressed more appropriately by considering non-Euclidean geometry. This paper
tackles the problem of sparse coding and dictionary learning in the space of
symmetric positive definite matrices, which form a Riemannian manifold. With
the aid of the recently introduced Stein kernel (related to a symmetric version
of Bregman matrix divergence), we propose to perform sparse coding by embedding
Riemannian manifolds into reproducing kernel Hilbert spaces. This leads to a
convex and kernel version of the Lasso problem, which can be solved
efficiently. We furthermore propose an algorithm for learning a Riemannian
dictionary (used for sparse coding), closely tied to the Stein kernel.
Experiments on several classification tasks (face recognition, texture
classification, person re-identification) show that the proposed sparse coding
approach achieves notable improvements in discrimination accuracy, in
comparison to state-of-the-art methods such as tensor sparse coding, Riemannian
locality preserving projection, and symmetry-driven accumulation of local
features.
