  Deep neural networks for learning symmetric positive definite (SPD) matrices
are gaining increasing attention in machine learning. Despite the significant
progress, most existing SPD networks use traditional Euclidean classifiers on
approximated spaces rather than intrinsic classifiers that accurately capture
the geometry of SPD manifolds. Inspired by the success of hyperbolic neural
networks (HNNs), we propose Riemannian multiclass logistics regression (RMLR)
for SPD networks. We introduce a general unified framework for a family of
Riemannian metrics on SPD manifolds and showcase the specific
$\orth{n}$-invariant Log-Euclidean Metrics for SPD networks. Moreover, we
encompass the most popular classifier in existing SPD networks as a special
case of our framework. Extensive experiments on popular SPD learning benchmarks
demonstrate the superiority of our classifiers.

Machine Learning