Sparse Coding and Dictionary Learning for Symmetric Positive Definite Matrices: A Kernel Approach
This addresses the problem of handling non-Euclidean data in computer vision for researchers and practitioners, offering an incremental improvement over existing methods.
The paper tackles sparse coding and dictionary learning for symmetric positive definite matrices by embedding Riemannian manifolds into reproducing kernel Hilbert spaces using the Stein kernel, resulting in a convex kernel Lasso problem. Experiments on classification tasks like face recognition and texture classification show notable improvements in discrimination accuracy compared to state-of-the-art methods.
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.