Sparse Coding on Symmetric Positive Definite Manifolds using Bregman Divergences

2014/08/30 by Mehrtash Harandi, Harandi, Mehrtash, Richard Hartley +5
Computer Science · Engineering · #Advanced SAR Imaging Techniques #Computer Vision and Pattern Recognition (cs.CV) #FOS: Computer and information sciences #Face and Expression Recognition #Sparse and Compressive Sensing Techniques #Video Surveillance and Tracking Methods

paper · pdf · doi:10.48550/arxiv.1409.0083

openalex publication_date 2014/08/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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.

Related

• Riemannian Dictionary Learning and Sparse Coding for Positive Definite\n Matrices
• Dictionary Learning and Sparse Coding on Grassmann Manifolds: An Extrinsic Solution
• Dictionary Learning and Sparse Coding on Statistical Manifolds
• An information theoretic formulation of the Dictionary Learning and Sparse Coding Problems on Statistical Manifolds
• Extrinsic Methods for Coding and Dictionary Learning on Grassmann Manifolds