Riemannian neural networks, which extend deep learning techniques to Riemannian spaces, have gained significant attention in machine learning.

Riemannian neural networks, which extend deep learning techniques to Riemannian spaces, have gained significant attention in machine learning. To better classify the manifold-valued features, researchers have started extending Euclidean multinomial logistic regression (MLR) into Riemannian manifolds. However, existing approaches suffer from limited applicability due to their strong reliance on specific geometric properties. This paper proposes a framework for designing Riemannian MLR over general geometries, referred to as RMLR. Our framework only requires minimal geometric properties, thus exhibiting broad applicability and enabling its use with a wide range of geometries. Specifically, we showcase our framework on the Symmetric Positive Definite (SPD) manifold and special orthogonal group, i.e., the set of rotation matrices. On the SPD manifold, we develop five families of SPD MLRs under five types of power-deformed metrics. On rotation matrices we propose Lie MLR based on the popular bi-invariant metric. Extensive experiments on different Riemannian backbone networks validate the effectiveness of our framework.

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.