mlsanityData to 7 Oct 2026

Symmetric Spaces for Graph Embeddings: A Finsler-Riemannian Approach

arXiv 2106.04941 ↗First posted 9 Jun 2021cs.LG cs.CGcited by 23

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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.

Key facts. Cited by 23 papers in the map (data through 7 Oct 2026). Builds on 29 of its 61 references that are in the map. First posted on arXiv 9 Jun 2021.

Builds on

29 of 61 references.

References · 29

  1. 2019PyTorch: An Imperative Style, High-Performance Deep Learning Librarycited by 20,160
  2. 2017Neural Collaborative Filteringcited by 1,108
  3. 2016Geometric deep learning: going beyond Euclidean datacited by 1,042
  4. 2017SphereFace: Deep Hypersphere Embedding for Face Recognitioncited by 733
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  6. 2019Hyperbolic Graph Convolutional Neural Networkscited by 400
  7. 2018Learning Continuous Hierarchies in the Lorentz Model of Hyperbolic Geometrycited by 261
  8. 2018Representation Tradeoffs for Hyperbolic Embeddingscited by 204
  9. 2018Riemannian Adaptive Optimization Methodscited by 187
  10. 2016A Riemannian Network for SPD Matrix Learningcited by 162
  11. 2018Hyperbolic Entailment Cones for Learning Hierarchical Embeddingscited by 160
  12. 2018Poincar'e GloVe: Hyperbolic Word Embeddingscited by 155
17 more
  1. 2020Geoopt: Riemannian Optimization in PyTorchcited by 109
  2. 2019Constant Curvature Graph Convolutional Networkscited by 106
  3. 2019Mixed-curvature Variational Autoencoderscited by 85
  4. 2018Spherical Latent Spaces for Stable Variational Autoencoderscited by 66
  5. 2020From Trees to Continuous Embeddings and Back: Hyperbolic Hierarchical Clusteringcited by 65
  6. 2020DeepSphere: a graph-based spherical CNNcited by 65
  7. 2016Building Deep Networks on Grassmann Manifoldscited by 58
  8. 2017Neural Embeddings of Graphs in Hyperbolic Spacecited by 58
  9. 2019Spherical Text Embeddingcited by 49
  10. 2018HyperML: A Boosting Metric Learning Approach in Hyperbolic Space for Recommender Systemscited by 47
  11. 2020Computationally Tractable Riemannian Manifolds for Graph Embeddingscited by 29
  12. 2020A Fully Hyperbolic Neural Model for Hierarchical Multi-Class Classificationcited by 23
  13. 2020Manifold structure in graph embeddingscited by 17
  14. 2020Ultrahyperbolic Representation Learningcited by 17
  15. 2020Generalized Nonlinear and Finsler Geometry for Roboticscited by 10
  16. 2019Fine-Grained Entity Typing in Hyperbolic Spacecited by 9
  17. 2018Change Detection in Graph Streams by Learning Graph Embeddings on Constant-Curvature Manifoldscited by 8

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 2; 2022: 5; 2023: 6; 2024: 3; 2025: 4; 2026: 3.

23 papers.

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  1. 2022Heterogeneous manifolds for curvature-aware graph embeddingcited by 23
  2. 2024Motif-aware Riemannian Graph Neural Network with Generative-Contrastive Learningcited by 23
  3. 2021Vector-valued Distance and Gyrocalculus on the Space of Symmetric Positive Definite Matricescited by 22
  4. 2023Transfer Learning in Robotics: An Upcoming Breakthrough? A Review of Promises and Challengescited by 17
  5. 2024DeepRicci: Self-supervised Graph Structure-Feature Co-Refinement for Alleviating Over-squashingcited by 15
  6. 2022Small Transformers Compute Universal Metric Embeddingscited by 12
  7. 2023Modeling Graphs Beyond Hyperbolic: Graph Neural Networks in Symmetric Positive Definite Matricescited by 10
  8. 2024Pulling back symmetric Riemannian geometry for data analysiscited by 8
  9. 2025Finsler Multi-Dimensional Scaling: Manifold Learning for Asymmetric Dimensionality Reduction and Embeddingcited by 7
  10. 2023Invariant kernels on Riemannian symmetric spaces: a harmonic-analytic approachcited by 6
  11. 2022Non-linear Embeddings in Hilbert Simplex Geometrycited by 5
  12. 2021Augmenting the User-Item Graph with Textual Similarity Modelscited by 3
11 more
  1. 2022Identifying latent distances with Finslerian geometrycited by 2
  2. 2023Normed Spaces for Graph Embeddingcited by 2
  3. 2025Curvature Corrected Nonnegative Manifold Data Factorizationcited by 2
  4. 2023Over-Squashing in Riemannian Graph Neural Networkscited by 1
  5. 2023Improving embedding of graphs with missing data by soft manifoldscited by 1
  6. 2025Siegel Neural Networkscited by 1
  7. 2022Pseudo-Riemannian Embedding Models for Multi-Relational Graph Representationscited by 0
  8. 2025Hyperbolic Graph Embeddings: a Survey and an Evaluation on Anomaly Detectioncited by 0
  9. 2026Geometry-Preserving Neural Architectures on Manifolds with Boundarycited by 0
  10. 2026Batch Normalization for Neural Networks on Complex Domainscited by 0
  11. 2026mathbbSL(n) Representation Learning: An Intrinsic Mixed-Curvature Space with Higher Curvature Capacities and Deeper Order-Aware Compositioncited by 0