论文标题
基于Helgason-Fourier分析
Fully-Connected Network on Noncompact Symmetric Space and Ridgelet Transform based on Helgason-Fourier Analysis
论文作者
论文摘要
Riemannian对称空间的神经网络，例如双曲线空间和对称阳性（SPD）矩阵的流形，是几何深度学习研究的新兴主题。基于非政策对称空间上Helgason-Fourier变换的框架，我们在非策略对称空间上提出了一个完全连接的网络及其相关的Ridgelet变换，涵盖了双曲神经网络（HNN）和SPDNET作为特殊情况。 Ridgelet Transform是由神经元跨越的DEPTH-2连续网络的分析操作员，即，它将任意给定函数映射到网络的权重。得益于无坐标的重新制定，非线性激活函数的作用被揭示为小波函数，重建公式直接产生了提议的网络的普遍性。
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.
