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

arXiv: https://arxiv.org/abs/1304.4344 · HTML: https://mlsanity.com/paper/1304.4344/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Apr 2013.

## Built on by (most cited first, 9 of 9)

- [On Riemannian Optimization over Positive Definite Matrices with the Bures-Wasserstein Geometry](https://mlsanity.com/paper/2106.00286/) (2021)
- [Riemannian Local Mechanism for SPD Neural Networks](https://mlsanity.com/paper/2201.10145/) (2022)
- [Riemannian Self-Attention Mechanism for SPD Networks](https://mlsanity.com/paper/2311.16738/) (2023)
- [Collaborative Representation for SPD Matrices with Application to Image-Set Classification](https://mlsanity.com/paper/2201.08962/) (2022)
- [DreamNet: A Deep Riemannian Network based on SPD Manifold Learning for Visual Classification](https://mlsanity.com/paper/2206.07967/) (2022)
- [A Geometric-Aware Perspective and Beyond: Hybrid Quantum-Classical Machine Learning Methods](https://mlsanity.com/paper/2504.06328/) (2025)
- [Reconciliation of Statistical and Spatial Sparsity For Robust Image and Image-Set Classification](https://mlsanity.com/paper/2106.00256/) (2021)
- [Kernel Recursive Least Squares Dictionary Learning Algorithm](https://mlsanity.com/paper/2507.01636/) (2025)
- [Geometric Dictionary Learning of Dynamical Systems with Optimal Transport](https://mlsanity.com/paper/2605.18276/) (2026)

## Abstract (arXiv)

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.