Riemannian Metric Learning for Symmetric Positive Definite Matrices

TL;DR

This paper introduces a data-driven method to learn Riemannian metrics for symmetric positive definite matrices, improving face matching and clustering performance over traditional measures.

Contribution

It proposes a novel approach to learn Riemannian metrics for SPD matrices based on log-Euclidean geometry, enhancing similarity measures in computer vision tasks.

Findings

Learned geodesic distances outperform existing measures in face matching.

The approach improves clustering accuracy for SPD matrix data.

Demonstrates the effectiveness of data-driven metric learning in Riemannian geometry.

Abstract

Over the past few years, symmetric positive definite (SPD) matrices have been receiving considerable attention from computer vision community. Though various distance measures have been proposed in the past for comparing SPD matrices, the two most widely-used measures are affine-invariant distance and log-Euclidean distance. This is because these two measures are true geodesic distances induced by Riemannian geometry. In this work, we focus on the log-Euclidean Riemannian geometry and propose a data-driven approach for learning Riemannian metrics/geodesic distances for SPD matrices. We show that the geodesic distance learned using the proposed approach performs better than various existing distance measures when evaluated on face matching and clustering tasks.