RMLR: Extending Multinomial Logistic Regression into General Geometries

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 SO(n), i.e.,the set of rotation matrices in Rn. On the SPD manifold, we develop five families of SPD MLRs under five types of power-deformed metrics. On SO(n), we propose Lie MLR based on the popula...