RMLR: Extending Multinomial Logistic Regression into General Geometries
This work addresses the limited applicability of existing Riemannian MLR methods for researchers in machine learning dealing with manifold-valued data, though it is incremental as it builds on prior extensions.
The paper tackles the problem of extending multinomial logistic regression to general Riemannian geometries, proposing the RMLR framework that requires minimal geometric properties and demonstrates broad applicability, achieving effective classification on SPD manifolds and rotation matrices.
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.