Sparse Coding and Dictionary Learning for Symmetric Positive Definite Matrices: A Kernel Approach

Abstract

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.

Cited by in corpus (6)

• Kernel Methods on Riemannian Manifolds with Gaussian RBF Kernels
• Spatio-Temporal Covariance Descriptors for Action and Gesture Recognition
• Random Projections on Manifolds of Symmetric Positive Definite Matrices for Image Classification
• K-Tangent Spaces on Riemannian Manifolds for Improved Pedestrian Detection
• Kernelized Low Rank Representation on Grassmann Manifolds
• Multi-Shot Person Re-Identification via Relational Stein Divergence