将逻辑回归拓展到任意黎曼几何，提升流形数据分类能力
RMLR: Extending Multinomial Logistic Regression into General Geometries
- 提出通用黎曼逻辑回归框架，仅需基础几何性质
- 在对称正定流形和旋转矩阵上构建五类新模型
- 适用于多种流形网络，适合流形学习研究者
黎曼神经网络近年来在机器学习中备受关注，其将深度学习方法扩展至黎曼空间。为更好分类流形值特征，研究者开始将欧氏多元逻辑回归（MLR）推广至黎曼流形。然而现有方法受限于特定几何特性，适用范围有限。本文提出一种可在一般几何结构上设计黎曼MLR的框架，称为RMLR。该框架仅需最低限度的几何假设，具备广泛适用性，可应用于多种几何结构。具体地，我们在对称正定（SPD）流形上基于五类幂变形度量构建了五种SPD MLR；在旋转矩阵（特殊正交群）上提出了基于双不变度量的李代数逻辑回归（Lie MLR）。在多种黎曼主干网络上的大量实验验证了本框架的有效性。
原文摘要 · Abstract (English)
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.
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