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

TL;DR

This paper introduces a kernel-based method for sparse coding and dictionary learning on the space of symmetric positive definite matrices, leveraging Riemannian geometry and Stein kernels to improve classification accuracy in computer vision tasks.

Contribution

It proposes a novel kernel embedding approach for sparse coding on Riemannian manifolds of SPD matrices, enabling efficient convex optimization and improved classification performance.

Findings

Achieves notable accuracy improvements over state-of-the-art methods.

Demonstrates effectiveness on face recognition, texture classification, and re-identification.

Provides an efficient algorithm for Riemannian dictionary learning.

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.