Modeling Graphs Beyond Hyperbolic: Graph Neural Networks in Symmetric
  Positive Definite Matrices
Wei Zhao Federico Lopez J. Maxwell Riestenberg Michael Strube Diaaeldin Taha Steve Trettel
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
\url{https://github.com/andyweizhao/SPD4GNNs}.