Riemannian Dictionary Learning and Sparse Coding for Positive Definite Matrices

TL;DR

This paper introduces a Riemannian geometric approach for dictionary learning and sparse coding of symmetric positive definite matrices, improving classification and retrieval in computer vision tasks.

Contribution

It formulates a novel Riemannian optimization framework for sparse coding of SPD matrices and provides a simple algorithm for efficient computation.

Findings

Superior classification accuracy on multiple datasets.

Enhanced retrieval performance compared to non-Riemannian methods.

Effective sparse representations of SPD data.

Abstract

Data encoded as symmetric positive definite (SPD) matrices frequently arise in many areas of computer vision and machine learning. While these matrices form an open subset of the Euclidean space of symmetric matrices, viewing them through the lens of non-Euclidean Riemannian geometry often turns out to be better suited in capturing several desirable data properties. However, formulating classical machine learning algorithms within such a geometry is often non-trivial and computationally expensive. Inspired by the great success of dictionary learning and sparse coding for vector-valued data, our goal in this paper is to represent data in the form of SPD matrices as sparse conic combinations of SPD atoms from a learned dictionary via a Riemannian geometric approach. To that end, we formulate a novel Riemannian optimization objective for dictionary learning and sparse coding in which the representation loss is characterized via the affine invariant Riemannian metric. We also present a computationally simple algorithm for optimizing our model. Experiments on several computer vision datasets demonstrate superior classification and retrieval performance using our approach when compared to sparse coding via alternative non-Riemannian formulations.