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

arXiv: https://arxiv.org/abs/2306.14064 · HTML: https://mlsanity.com/paper/2306.14064/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 44 references that are in the map.
- First posted on arXiv 24 Jun 2023.

## Builds on (most cited first, 26 of 26)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [Simplifying Graph Convolutional Networks](https://mlsanity.com/paper/1902.07153/) (2019)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Link Prediction Based on Graph Neural Networks](https://mlsanity.com/paper/1802.09691/) (2018)
- [TUDataset: A collection of benchmark datasets for learning with graphs](https://mlsanity.com/paper/2007.08663/) (2020)
- [Hyperbolic Graph Convolutional Neural Networks](https://mlsanity.com/paper/1910.12933/) (2019)
- [Hyperbolic Graph Neural Networks](https://mlsanity.com/paper/1910.12892/) (2019)
- [A Riemannian Network for SPD Matrix Learning](https://mlsanity.com/paper/1608.04233/) (2016)
- [Graph-Coupled Oscillator Networks](https://mlsanity.com/paper/2202.02296/) (2022)
- [PDE-GCN: Novel Architectures for Graph Neural Networks Motivated by Partial Differential Equations](https://mlsanity.com/paper/2108.01938/) (2021)
- [Riemannian batch normalization for SPD neural networks](https://mlsanity.com/paper/1909.02414/) (2019)
- [DeepSphere: a graph-based spherical CNN](https://mlsanity.com/paper/2012.15000/) (2020)
- [A neural network based on SPD manifold learning for skeleton-based hand gesture recognition](https://mlsanity.com/paper/1904.12970/) (2019)
- [Tree! I am no Tree! I am a Low Dimensional Hyperbolic Embedding](https://mlsanity.com/paper/2005.03847/) (2020)
- [Weisfeiler and Leman Go Relational](https://mlsanity.com/paper/2211.17113/) (2022)
- [Symmetric Spaces for Graph Embeddings: A Finsler-Riemannian Approach](https://mlsanity.com/paper/2106.04941/) (2021)
- [Vector-valued Distance and Gyrocalculus on the Space of Symmetric Positive Definite Matrices](https://mlsanity.com/paper/2110.13475/) (2021)
- [A Statistical Recurrent Model on the Manifold of Symmetric Positive Definite Matrices](https://mlsanity.com/paper/1805.11204/) (2018)
- [Deep manifold-to-manifold transforming network for action recognition](https://mlsanity.com/paper/1705.10732/) (2017)
- [Learning a Robust Representation via a Deep Network on Symmetric Positive Definite Manifolds](https://mlsanity.com/paper/1711.06540/) (2017)

## Built on by (most cited first, 10 of 10)

- [Graph-level Representation Learning with Joint-Embedding Predictive Architectures](https://mlsanity.com/paper/2309.16014/) (2023)
- [Matrix Manifold Neural Networks++](https://mlsanity.com/paper/2405.19206/) (2024)
- [RMLR: Extending Multinomial Logistic Regression into General Geometries](https://mlsanity.com/paper/2409.19433/) (2024)
- [Sheaf Neural Networks on SPD Manifolds: Second-Order Geometric Representation Learning](https://mlsanity.com/paper/2604.20308/) (2026)
- [Normed Spaces for Graph Embedding](https://mlsanity.com/paper/2312.01502/) (2023)
- [Manifold GCN: Diffusion-based Convolutional Neural Network for Manifold-valued Graphs](https://mlsanity.com/paper/2401.14381/) (2024)
- [Geometry-Preserving Neural Architectures on Manifolds with Boundary](https://mlsanity.com/paper/2602.03082/) (2026)
- [Demystifying Oversmoothing in Sheaf Neural Networks: An Index-Theoretic Criterion](https://mlsanity.com/paper/2608.16180/) (2026)
- [Equivariant Covariance Tensors: Guaranteed SPD Uncertainty for Tensor-Valued Geometric Learning](https://mlsanity.com/paper/2608.24386/) (2026)
- [mathbbSL(n) Representation Learning: An Intrinsic Mixed-Curvature Space with Higher Curvature Capacities and Deeper Order-Aware Composition](https://mlsanity.com/paper/2609.15083/) (2026)

## Abstract (arXiv)

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 citelopez2021gyroSPD 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 urlhttps://github.com/andyweizhao/SPD4GNNs.