Neural network on Riemannian symmetric space such as hyperbolic space and the manifold of symmetric positive definite (SPD) matrices is an emerging subject of research in geometric deep learning. Based on the well-established framework of the Helgason-Fourier transform on the noncompact symmetric space, we present a fully-connected network and its associated ridgelet transform on the noncompact symmetric space, covering the hyperbolic neural network (HNN) and the SPDNet as special cases. The ridgelet transform is an analysis operator of a depth-2 continuous network spanned by neurons, namely, it maps an arbitrary given function to the weights of a network. Thanks to the coordinate-free reformulation, the role of nonlinear activation functions is revealed to be a wavelet function, and the reconstruction formula directly yields the universality of the proposed networks.

本文介绍在Riemann对称空间，例如双曲空间和对称正定矩阵流形上的神经网络，基于非紧致对称空间上的Helgason-Fourier变换框架，提出了一个全连接网络和其相关的ridgelet变换，覆盖了双曲神经网络(HNN)和SPDNet等特殊情况。

基于Helgason-Fourier分析的非紧对称空间全连接网络与Ridgelet变换