  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.
