# Symmetric Spaces for Graph Embeddings: A Finsler-Riemannian Approach

arXiv: https://arxiv.org/abs/2106.04941 · HTML: https://mlsanity.com/paper/2106.04941/

## Key facts

- Cited by 23 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 61 references that are in the map.
- First posted on arXiv 9 Jun 2021.

## Builds on (most cited first, 29 of 29)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Neural Collaborative Filtering](https://mlsanity.com/paper/1708.05031/) (2017)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [SphereFace: Deep Hypersphere Embedding for Face Recognition](https://mlsanity.com/paper/1704.08063/) (2017)
- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [Hyperbolic Graph Convolutional Neural Networks](https://mlsanity.com/paper/1910.12933/) (2019)
- [Learning Continuous Hierarchies in the Lorentz Model of Hyperbolic Geometry](https://mlsanity.com/paper/1806.03417/) (2018)
- [Representation Tradeoffs for Hyperbolic Embeddings](https://mlsanity.com/paper/1804.03329/) (2018)
- [Riemannian Adaptive Optimization Methods](https://mlsanity.com/paper/1810.00760/) (2018)
- [A Riemannian Network for SPD Matrix Learning](https://mlsanity.com/paper/1608.04233/) (2016)
- [Hyperbolic Entailment Cones for Learning Hierarchical Embeddings](https://mlsanity.com/paper/1804.01882/) (2018)
- [Poincar'e GloVe: Hyperbolic Word Embeddings](https://mlsanity.com/paper/1810.06546/) (2018)
- [Geoopt: Riemannian Optimization in PyTorch](https://mlsanity.com/paper/2005.02819/) (2020)
- [Constant Curvature Graph Convolutional Networks](https://mlsanity.com/paper/1911.05076/) (2019)
- [Mixed-curvature Variational Autoencoders](https://mlsanity.com/paper/1911.08411/) (2019)
- [Spherical Latent Spaces for Stable Variational Autoencoders](https://mlsanity.com/paper/1808.10805/) (2018)
- [DeepSphere: a graph-based spherical CNN](https://mlsanity.com/paper/2012.15000/) (2020)
- [From Trees to Continuous Embeddings and Back: Hyperbolic Hierarchical Clustering](https://mlsanity.com/paper/2010.00402/) (2020)
- [Building Deep Networks on Grassmann Manifolds](https://mlsanity.com/paper/1611.05742/) (2016)
- [Neural Embeddings of Graphs in Hyperbolic Space](https://mlsanity.com/paper/1705.10359/) (2017)
- [Spherical Text Embedding](https://mlsanity.com/paper/1911.01196/) (2019)
- [HyperML: A Boosting Metric Learning Approach in Hyperbolic Space for Recommender Systems](https://mlsanity.com/paper/1809.01703/) (2018)
- [Computationally Tractable Riemannian Manifolds for Graph Embeddings](https://mlsanity.com/paper/2002.08665/) (2020)
- [A Fully Hyperbolic Neural Model for Hierarchical Multi-Class Classification](https://mlsanity.com/paper/2010.02053/) (2020)
- [Manifold structure in graph embeddings](https://mlsanity.com/paper/2006.05168/) (2020)
- [Ultrahyperbolic Representation Learning](https://mlsanity.com/paper/2007.00211/) (2020)
- [Generalized Nonlinear and Finsler Geometry for Robotics](https://mlsanity.com/paper/2010.14745/) (2020)
- [Fine-Grained Entity Typing in Hyperbolic Space](https://mlsanity.com/paper/1906.02505/) (2019)
- [Change Detection in Graph Streams by Learning Graph Embeddings on Constant-Curvature Manifolds](https://mlsanity.com/paper/1805.06299/) (2018)

## Built on by (most cited first, 23 of 23)

- [Heterogeneous manifolds for curvature-aware graph embedding](https://mlsanity.com/paper/2202.01185/) (2022)
- [Motif-aware Riemannian Graph Neural Network with Generative-Contrastive Learning](https://mlsanity.com/paper/2401.01232/) (2024)
- [Vector-valued Distance and Gyrocalculus on the Space of Symmetric Positive Definite Matrices](https://mlsanity.com/paper/2110.13475/) (2021)
- [Transfer Learning in Robotics: An Upcoming Breakthrough? A Review of Promises and Challenges](https://mlsanity.com/paper/2311.18044/) (2023)
- [DeepRicci: Self-supervised Graph Structure-Feature Co-Refinement for Alleviating Over-squashing](https://mlsanity.com/paper/2401.12780/) (2024)
- [Small Transformers Compute Universal Metric Embeddings](https://mlsanity.com/paper/2209.06788/) (2022)
- [Modeling Graphs Beyond Hyperbolic: Graph Neural Networks in Symmetric Positive Definite Matrices](https://mlsanity.com/paper/2306.14064/) (2023)
- [Pulling back symmetric Riemannian geometry for data analysis](https://mlsanity.com/paper/2403.06612/) (2024)
- [Finsler Multi-Dimensional Scaling: Manifold Learning for Asymmetric Dimensionality Reduction and Embedding](https://mlsanity.com/paper/2503.18010/) (2025)
- [Non-linear Embeddings in Hilbert Simplex Geometry](https://mlsanity.com/paper/2203.11434/) (2022)
- [Invariant kernels on Riemannian symmetric spaces: a harmonic-analytic approach](https://mlsanity.com/paper/2310.19270/) (2023)
- [Augmenting the User-Item Graph with Textual Similarity Models](https://mlsanity.com/paper/2109.09358/) (2021)
- [Identifying latent distances with Finslerian geometry](https://mlsanity.com/paper/2212.10010/) (2022)
- [Normed Spaces for Graph Embedding](https://mlsanity.com/paper/2312.01502/) (2023)
- [Curvature Corrected Nonnegative Manifold Data Factorization](https://mlsanity.com/paper/2502.15124/) (2025)
- [Over-Squashing in Riemannian Graph Neural Networks](https://mlsanity.com/paper/2311.15945/) (2023)
- [Improving embedding of graphs with missing data by soft manifolds](https://mlsanity.com/paper/2311.17598/) (2023)
- [Siegel Neural Networks](https://mlsanity.com/paper/2511.09577/) (2025)
- [Pseudo-Riemannian Embedding Models for Multi-Relational Graph Representations](https://mlsanity.com/paper/2212.03720/) (2022)
- [Hyperbolic Graph Embeddings: a Survey and an Evaluation on Anomaly Detection](https://mlsanity.com/paper/2512.18826/) (2025)
- [Geometry-Preserving Neural Architectures on Manifolds with Boundary](https://mlsanity.com/paper/2602.03082/) (2026)
- [Batch Normalization for Neural Networks on Complex Domains](https://mlsanity.com/paper/2605.00467/) (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)

Learning faithful graph representations as sets of vertex embeddings has become a fundamental intermediary step in a wide range of machine learning applications. We propose the systematic use of symmetric spaces in representation learning, a class encompassing many of the previously used embedding targets. This enables us to introduce a new method, the use of Finsler metrics integrated in a Riemannian optimization scheme, that better adapts to dissimilar structures in the graph. We develop a tool to analyze the embeddings and infer structural properties of the data sets. For implementation, we choose Siegel spaces, a versatile family of symmetric spaces. Our approach outperforms competitive baselines for graph reconstruction tasks on various synthetic and real-world datasets. We further demonstrate its applicability on two downstream tasks, recommender systems and node classification.