Deep neural networks (DNNs) on Riemannian manifolds have garnered increasing interest in various applied areas. For instance, DNNs on spherical and hyperbolic manifolds have been designed to solve a wide range of computer vision and nature language processing tasks. One of the key factors that contribute to the success of these networks is that spherical and hyperbolic manifolds have the rich algebraic structures of gyrogroups and gyrovector spaces. This enables principled and effective generalizations of the most successful DNNs to these manifolds. Recently, some works have shown that many concepts in the theory of gyrogroups and gyrovector spaces can also be generalized to matrix manifolds such as Symmetric Positive Definite (SPD) and Grassmann manifolds. As a result, some building blocks for SPD and Grassmann neural networks, e.g., isometric models and multinomial logistic regression (MLR) can be derived in a way that is fully analogous to their spherical and hyperbolic counterparts. Building upon these works, we design fully-connected (FC) and convolutional layers for SPD neural networks. We also develop MLR on Symmetric Positive Semi-definite (SPSD) manifolds, and propose a method for performing backpropagation with the Grassmann logarithmic map in the projector perspective. We demonstrate the effectiveness of the proposed approach in the human action recognition and node classification tasks.

翻译：黎曼流形上的深度神经网络（DNNs）在各种应用领域引起了日益增长的兴趣。例如，球面流形和双曲流形上的DNNs已被设计用于解决广泛的计算机视觉和自然语言处理任务。这些网络成功的关键因素之一是球面和双曲流形具有陀螺群和陀螺向量空间的丰富代数结构。这使得最成功的DNNs能够以有原则且有效的方式推广到这些流形上。最近，一些研究表明，陀螺群和陀螺向量空间理论中的许多概念也可以推广到矩阵流形，例如对称正定（SPD）流形和格拉斯曼流形。因此，SPD和格拉斯曼神经网络的一些构建模块，例如等距模型和多项逻辑回归（MLR），可以以一种完全类似于其球面和双曲对应物的方式推导出来。基于这些工作，我们为SPD神经网络设计了全连接（FC）层和卷积层。我们还在对称半正定（SPSD）流形上发展了MLR，并提出了一种在投影视角下使用格拉斯曼对数映射执行反向传播的方法。我们在人类动作识别和节点分类任务中证明了所提方法的有效性。