Recent research has shown that alignment between the structure of graph data
and the geometry of an embedding space is crucial for learning high-quality
representations of the data. The uniform geometry of Euclidean and hyperbolic
spaces allows for representing graphs with uniform geometric and topological
features, such as grids and hierarchies, with minimal distortion. However,
real-world graph data is characterized by multiple types of geometric and
topological features, necessitating more sophisticated geometric embedding
spaces. In this work, we utilize the Riemannian symmetric space of symmetric
positive definite matrices (SPD) to construct graph neural networks that can
robustly handle complex graphs. To do this, we develop an innovative library
that leverages the SPD gyrocalculus tools \cite{lopez2021gyroSPD} to implement
the building blocks of five popular graph neural networks in SPD. Experimental
results demonstrate that our graph neural networks in SPD substantially
outperform their counterparts in Euclidean and hyperbolic spaces, as well as
the Cartesian product thereof, on complex graphs for node and graph
classification tasks. We release the library and datasets at
https://github.com/andyweizhao/SPD4GNNs.

使用对称正定矩阵（SPD）的 Riemannian 对称空间构建图神经网络以处理复杂图，并证明在结点和图分类任务上，其性能明显优于欧几里得空间和双曲空间及两者的笛卡尔积的图神经网络。

超越双曲线建模：对称正定矩阵中的图神经网络

Modeling Graphs Beyond Hyperbolic: Graph Neural Networks in Symmetric  Positive Definite Matrices

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。