Sparse Coding on Symmetric Positive Definite Manifolds using Bregman Divergences

TL;DR

This paper develops a novel sparse coding framework for SPD matrices using Bregman divergences, enabling efficient dictionary learning and improving classification performance in computer vision tasks.

Contribution

It introduces a new sparse coding method for SPD matrices via Bregman divergences, with an online dictionary learning scheme, advancing the application of sparse coding on manifold data.

Findings

Outperforms state-of-the-art methods on classification tasks

Enables efficient sparse coding for SPD matrices

Provides an online, iterative dictionary learning algorithm

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.