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（SPD）矩阵流形和格拉斯曼流形，在许多应用中都有出现。近年来，通过应用陀螺群和陀螺向量空间理论（一种研究双曲几何的强大框架），一些研究尝试在矩阵流形上建立欧几里得神经网络的原理性推广。然而，由于所考虑流形在陀螺向量空间中缺乏许多概念（例如内积和陀螺角），这些研究提供的技术与数学工具相较于双曲几何领域的发展仍较为有限。本文针对SPD流形和格拉斯曼流形，推广了陀螺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新层。我们在人体动作识别和知识图谱补全两个应用中验证了该方法的有效性。