Riemannian Metric Learning for Symmetric Positive Definite Matrices
This work addresses a domain-specific problem in computer vision for improving distance measures on SPD matrices, with incremental contributions.
The paper tackled the problem of comparing symmetric positive definite matrices by proposing a data-driven approach to learn Riemannian metrics, specifically focusing on log-Euclidean geometry, and showed that the learned geodesic distance outperforms existing measures in face matching and clustering tasks.
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.