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Hyperbolic Entailment Cones for Learning Hierarchical Embeddings

arXiv 1804.01882 ↗First posted 3 Apr 2018cs.LG stat.MLcited by 160

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Abstract (arXiv)

Learning graph representations via low-dimensional embeddings that preserve relevant network properties is an important class of problems in machine learning. We here present a novel method to embed directed acyclic graphs. Following prior work, we first advocate for using hyperbolic spaces which provably model tree-like structures better than Euclidean geometry. Second, we view hierarchical relations as partial orders defined using a family of nested geodesically convex cones. We prove that these entailment cones admit an optimal shape with a closed form expression both in the Euclidean and hyperbolic spaces, and they canonically define the embedding learning process. Experiments show significant improvements of our method over strong recent baselines both in terms of representational capacity and generalization.

Key facts. Cited by 160 papers in the map (data through 7 Oct 2026). First posted on arXiv 3 Apr 2018.

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 24; 2022: 17; 2023: 24; 2024: 22; 2025: 34; 2026: 39.

160 papers.

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  1. 2024The Geometry of Categorical and Hierarchical Concepts in Large Language Modelscited by 151
  2. 2021Hyperbolic Deep Neural Networks: A Surveycited by 138
  3. 2023Hyperbolic Image-Text Representationscited by 132
  4. 2022Hyperbolic Image Segmentationcited by 101
  5. 2022Deep Hierarchical Semantic Segmentationcited by 84
  6. 2021ConE: Cone Embeddings for Multi-Hop Reasoning over Knowledge Graphscited by 78
  7. 2023Hyperbolic Deep Learning in Computer Vision: A Surveycited by 72
  8. 2024Compositional Entailment Learning for Hyperbolic Vision-Language Modelscited by 72
  9. 2021Lorentzian Graph Convolutional Networkscited by 66
  10. 2022Hyperbolic Contrastive Learning for Visual Representations beyond Objectscited by 58
  11. 2021Learning the Predictability of the Futurecited by 48
  12. 2022Knowledge Graph Embedding: A Survey from the Perspective of Representation Spacescited by 41
38 more
  1. 2021Hyperbolic Busemann Learning with Ideal Prototypescited by 40
  2. 2022Hyperbolic Graph Neural Networks: A Review of Methods and Applicationscited by 40
  3. 2024Hypformer: Exploring Efficient Transformer Fully in Hyperbolic Spacecited by 39
  4. 2021Modeling Heterogeneous Hierarchies with Relation-specific Hyperbolic Conescited by 38
  5. 2021Modeling Fine-Grained Entity Types with Box Embeddingscited by 36
  6. 2022TaxoCom: Topic Taxonomy Completion with Hierarchical Discovery of Novel Topic Clusterscited by 30
  7. 2024Language Models as Hierarchy Encoderscited by 26
  8. 2023Poincar'e ResNetcited by 24
  9. 2024Hyperbolic Fine-Tuning for Large Language Modelscited by 24
  10. 2021Symmetric Spaces for Graph Embeddings: A Finsler-Riemannian Approachcited by 23
  11. 2023Rethinking Graph Lottery Tickets: Graph Sparsity Matterscited by 23
  12. 2021Probabilistic Box Embeddings for Uncertain Knowledge Graph Reasoningcited by 22
  13. 2022Improved Representation of Asymmetrical Distances with Interval Quasimetric Embeddingscited by 22
  14. 2025HELM: Hyperbolic Large Language Models via Mixture-of-Curvature Expertscited by 22
  15. 2022FALCON: Fast Visual Concept Learning by Integrating Images, Linguistic descriptions, and Conceptual Relationscited by 20
  16. 2025Hyperbolic Deep Learning for Foundation Models: A Surveycited by 20
  17. 2021Multi-modal Entity Alignment in Hyperbolic Spacecited by 19
  18. 2022Skin Lesion Recognition with Class-Hierarchy Regularized Hyperbolic Embeddingscited by 19
  19. 2023From Isolated Islands to Pangea: Unifying Semantic Space for Human Action Understandingcited by 19
  20. 2024Emergent Visual-Semantic Hierarchies in Image-Text Representationscited by 19
  21. 2024HYPE: Hyperbolic Entailment Filtering for Underspecified Images and Textscited by 17
  22. 2025Hyperbolic Safety-Aware Vision-Language Modelscited by 17
  23. 2022Hyperbolic Sliced-Wasserstein via Geodesic and Horospherical Projectionscited by 16
  24. 2022On the Learning and Learnability of Quasimetricscited by 15
  25. 2022HyperMiner: Topic Taxonomy Mining with Hyperbolic Embeddingcited by 15
  26. 2023Curve Your Attention: Mixed-Curvature Transformers for Graph Representation Learningcited by 15
  27. 2021Extracting Event Temporal Relations via Hyperbolic Geometrycited by 13
  28. 2021Generalization Error Bound for Hyperbolic Ordinal Embeddingcited by 12
  29. 2023Learning Weakly Supervised Audio-Visual Violence Detection in Hyperbolic Spacecited by 12
  30. 2024Ontology Embedding: A Survey of Methods, Applications and Resourcescited by 12
  31. 2021Word2Box: Capturing Set-Theoretic Semantics of Words using Box Embeddingscited by 11
  32. 2023Horospherical Decision Boundaries for Large Margin Classification in Hyperbolic Spacecited by 11
  33. 2024Hyperbolic Learning with Synthetic Captions for Open-World Detectioncited by 11
  34. 2021Supervised Tree-Wasserstein Distancecited by 10
  35. 2021Directed Graph Embeddings in Pseudo-Riemannian Manifoldscited by 9
  36. 2021Embedding Heterogeneous Networks into Hyperbolic Space Without Meta-pathcited by 9
  37. 2023Hyperbolic Random Forestscited by 9
  38. 2023Understanding and Mitigating Hyperbolic Dimensional Collapse in Graph Contrastive Learningcited by 9

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