Matrix manifolds, such as manifolds of Symmetric Positive Definite (SPD) matrices and Grassmann manifolds, appear in many applications. Recently, by applying the theory of gyrogroups and gyrovector spaces that is a powerful framework for studying hyperbolic geometry, some works have attempted to build principled generalizations of Euclidean neural networks on matrix manifolds. However, due to the lack of many concepts in gyrovector spaces for the considered manifolds, e.g., the inner product and gyroangles, techniques and mathematical tools provided by these works are still limited compared to those developed for studying hyperbolic geometry. In this paper, we generalize some notions in gyrovector spaces for SPD and Grassmann manifolds, and propose new models and layers for building neural networks on these manifolds. We show the effectiveness of our approach in two applications, i.e., human action recognition and knowledge graph completion.

翻译：矩阵流形，例如对称正定矩阵流形和格拉斯曼流形，出现在众多应用中。近来，通过应用旋转群与旋转向量空间理论（一种用于研究双曲几何的强大框架），已有工作尝试在矩阵流形上构建欧几里得神经网络的原则性推广。然而，由于所考虑的流形在旋转向量空间中缺乏许多概念（例如内积和旋转角），这些工作所提供的技术与数学工具相较于双曲几何研究中发展的方法仍显局限。本文针对对称正定矩阵流形和格拉斯曼流形，推广了旋转向量空间中的若干概念，并提出了在这些流形上构建神经网络的新模型与层结构。我们在两个应用场景——即人体动作识别与知识图谱补全——中展示了本方法的有效性。