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.

在Riemann流形上的深度神经网络已经在各个应用领域受到越来越多的关注，其中包括球面和双曲面流形上的DNN在计算机视觉和自然语言处理任务中的广泛应用。而球面和双曲面流形能够应用双翼运算和双翼向量空间的丰富代数结构，为成功的深度神经网络在这些流形上的推广提供了基础。最近的一些研究表明，双翼运算和双翼向量空间理论中的许多概念也能够推广到矩阵流形，比如对称正定和Grassmann流形。基于这些工作，我们设计了用于对称正定流形上的全连接和卷积层，并在Grassmann投影视角上提出了一种使用Grassmann对数映射进行反向传播的方法。我们在人类动作识别和节点分类任务中验证了这种方法的有效性。

矩阵流形神经网络++