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Learning Continuous Hierarchies in the Lorentz Model of Hyperbolic Geometry

arXiv 1806.03417 ↗First posted 9 Jun 2018cs.AI cs.LG stat.MLcited by 261

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Abstract (arXiv)

We are concerned with the discovery of hierarchical relationships from large-scale unstructured similarity scores. For this purpose, we study different models of hyperbolic space and find that learning embeddings in the Lorentz model is substantially more efficient than in the Poincar'e-ball model. We show that the proposed approach allows us to learn high-quality embeddings of large taxonomies which yield improvements over Poincar'e embeddings, especially in low dimensions. Lastly, we apply our model to discover hierarchies in two real-world datasets: we show that an embedding in hyperbolic space can reveal important aspects of a company's organizational structure as well as reveal historical relationships between language families.

Key facts. Cited by 261 papers in the map (data through 7 Oct 2026). First posted on arXiv 9 Jun 2018.

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Citing papers per year: 2021: 37; 2022: 35; 2023: 34; 2024: 43; 2025: 60; 2026: 52.

261 papers.

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  1. 2021Hyperbolic Deep Neural Networks: A Surveycited by 138
  2. 2023Hyperbolic Image-Text Representationscited by 132
  3. 2022Hyperbolic Vision Transformers: Combining Improvements in Metric Learningcited by 109
  4. 2021Fully Hyperbolic Neural Networkscited by 107
  5. 2022Hyperbolic Image Segmentationcited by 101
  6. 2021Beltrami Flow and Neural Diffusion on Graphscited by 84
  7. 2023Hyperbolic Deep Learning in Computer Vision: A Surveycited by 72
  8. 2021Discrete-time Temporal Network Embedding via Implicit Hierarchical Learning in Hyperbolic Spacecited by 67
  9. 2021Lorentzian Graph Convolutional Networkscited by 66
  10. 2022Hyperbolic Contrastive Learning for Visual Representations beyond Objectscited by 58
  11. 2021Clipped Hyperbolic Classifiers Are Super-Hyperbolic Classifierscited by 56
  12. 2021A Hyperbolic-to-Hyperbolic Graph Convolutional Networkcited by 53
38 more
  1. 2021Hyperbolic Variational Graph Neural Network for Modeling Dynamic Graphscited by 49
  2. 2023Fully Hyperbolic Convolutional Neural Networks for Computer Visioncited by 48
  3. 2022The Numerical Stability of Hyperbolic Representation Learningcited by 46
  4. 2022Knowledge Graph Embedding: A Survey from the Perspective of Representation Spacescited by 41
  5. 2021Hyperbolic Busemann Learning with Ideal Prototypescited by 40
  6. 2022Hyperbolic Graph Neural Networks: A Review of Methods and Applicationscited by 40
  7. 2024Hypformer: Exploring Efficient Transformer Fully in Hyperbolic Spacecited by 39
  8. 2021Modeling Heterogeneous Hierarchies with Relation-specific Hyperbolic Conescited by 38
  9. 2022HRCF: Enhancing Collaborative Filtering via Hyperbolic Geometric Regularizationcited by 34
  10. 2025RiemannGFM: Learning a Graph Foundation Model from Riemannian Geometrycited by 32
  11. 2021ACE-HGNN: Adaptive Curvature Exploration Hyperbolic Graph Neural Networkcited by 24
  12. 2021Geometry-aware Bayesian Optimization in Robotics using Riemannian Mat'ern Kernelscited by 24
  13. 2022HICF: Hyperbolic Informative Collaborative Filteringcited by 24
  14. 2023HypLiLoc: Towards Effective LiDAR Pose Regression with Hyperbolic Fusioncited by 24
  15. 2024Hyperbolic Fine-Tuning for Large Language Modelscited by 24
  16. 2021Symmetric Spaces for Graph Embeddings: A Finsler-Riemannian Approachcited by 23
  17. 2021Entity Linking and Discovery via Arborescence-based Supervised Clusteringcited by 22
  18. 2022The Euclidean Space is Evil: Hyperbolic Attribute Editing for Few-shot Image Generationcited by 22
  19. 2025HELM: Hyperbolic Large Language Models via Mixture-of-Curvature Expertscited by 22
  20. 2023Hyperbolic Representation Learning: Revisiting and Advancingcited by 21
  21. 2022Enhancing Hyperbolic Graph Embeddings via Contrastive Learningcited by 20
  22. 2025Hyperbolic Deep Learning for Foundation Models: A Surveycited by 20
  23. 2021Multi-modal Entity Alignment in Hyperbolic Spacecited by 19
  24. 2024Emergent Visual-Semantic Hierarchies in Image-Text Representationscited by 19
  25. 2023Hyperbolic Diffusion Embedding and Distance for Hierarchical Representation Learningcited by 18
  26. 2024Lorentzian Residual Neural Networkscited by 17
  27. 2025Hyperbolic Safety-Aware Vision-Language Modelscited by 17
  28. 2025HLFormer: Enhancing Partially Relevant Video Retrieval with Hyperbolic Learningcited by 17
  29. 2022Hyperbolic Sliced-Wasserstein via Geodesic and Horospherical Projectionscited by 16
  30. 2024Learning Structured Representations with Hyperbolic Embeddingscited by 16
  31. 2021Unsupervised Hyperbolic Representation Learning via Message Passing Auto-Encoderscited by 15
  32. 2022HyperMiner: Topic Taxonomy Mining with Hyperbolic Embeddingcited by 15
  33. 2023Hyperbolic Contrastive Learningcited by 15
  34. 2023Curve Your Attention: Mixed-Curvature Transformers for Graph Representation Learningcited by 15
  35. 2025Position: Beyond Euclidean -- Foundation Models Should Embrace Non-Euclidean Geometriescited by 15
  36. 2021Nested Hyperbolic Spaces for Dimensionality Reduction and Hyperbolic NN Designcited by 14
  37. 2022Fully-Connected Network on Noncompact Symmetric Space and Ridgelet Transform based on Helgason-Fourier Analysiscited by 14
  38. 2022Hyperbolic VAE via Latent Gaussian Distributionscited by 14

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