# Sparse Coding on Symmetric Positive Definite Manifolds using Bregman Divergences

arXiv: https://arxiv.org/abs/1409.0083 · HTML: https://mlsanity.com/paper/1409.0083/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Aug 2014.

## Built on by (most cited first, 6 of 6)

- [Learning with symmetric positive definite matrices via generalized Bures-Wasserstein geometry](https://mlsanity.com/paper/2110.10464/) (2021)
- [Geometric Sparse Coding in Wasserstein Space](https://mlsanity.com/paper/2210.12135/) (2022)
- [Collaborative Representation for SPD Matrices with Application to Image-Set Classification](https://mlsanity.com/paper/2201.08962/) (2022)
- [Approximate Nearest Neighbour Search on Dynamic Datasets: An Investigation](https://mlsanity.com/paper/2404.19284/) (2024)
- [Riemannian statistics meets random matrix theory: towards learning from high-dimensional covariance matrices](https://mlsanity.com/paper/2203.00204/) (2022)
- [Optimal Stability of KL Divergence under Gaussian Perturbations](https://mlsanity.com/paper/2604.11026/) (2026)

## Abstract (arXiv)

This paper introduces sparse coding and dictionary learning for Symmetric Positive Definite (SPD) matrices, which are often used in machine learning, computer vision and related areas. Unlike traditional sparse coding schemes that work in vector spaces, in this paper we discuss how SPD matrices can be described by sparse combination of dictionary atoms, where the atoms are also SPD matrices. We propose to seek sparse coding by embedding the space of SPD matrices into Hilbert spaces through two types of Bregman matrix divergences. This not only leads to an efficient way of performing sparse coding, but also an online and iterative scheme for dictionary learning. We apply the proposed methods to several computer vision tasks where images are represented by region covariance matrices. Our proposed algorithms outperform state-of-the-art methods on a wide range of classification tasks, including face recognition, action recognition, material classification and texture categorization.