RMLR: Extending Multinomial Logistic Regression into General Geometries

TL;DR

This paper introduces RMLR, a flexible framework for multinomial logistic regression on general Riemannian geometries, broadening applicability beyond specific geometric constraints.

Contribution

It develops a general RMLR framework applicable to various geometries, including SPD manifolds and rotation groups, with multiple metric-based models.

Findings

Effective classification on SPD manifolds with five new MLR families.

Successful application to rotation matrices using Lie MLR.

Validated framework's broad applicability through extensive experiments.

Abstract

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.