# Hyperbolic Entailment Cones for Learning Hierarchical Embeddings

arXiv: https://arxiv.org/abs/1804.01882 · HTML: https://mlsanity.com/paper/1804.01882/

## Key facts

- Cited by 162 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Apr 2018.

## Built on by (most cited first, 30 of 162)

- [The Geometry of Categorical and Hierarchical Concepts in Large Language Models](https://mlsanity.com/paper/2406.01506/) (2024)
- [Hyperbolic Deep Neural Networks: A Survey](https://mlsanity.com/paper/2101.04562/) (2021)
- [Hyperbolic Image-Text Representations](https://mlsanity.com/paper/2304.09172/) (2023)
- [Hyperbolic Image Segmentation](https://mlsanity.com/paper/2203.05898/) (2022)
- [Deep Hierarchical Semantic Segmentation](https://mlsanity.com/paper/2203.14335/) (2022)
- [ConE: Cone Embeddings for Multi-Hop Reasoning over Knowledge Graphs](https://mlsanity.com/paper/2110.13715/) (2021)
- [Compositional Entailment Learning for Hyperbolic Vision-Language Models](https://mlsanity.com/paper/2410.06912/) (2024)
- [Hyperbolic Deep Learning in Computer Vision: A Survey](https://mlsanity.com/paper/2305.06611/) (2023)
- [Lorentzian Graph Convolutional Networks](https://mlsanity.com/paper/2104.07477/) (2021)
- [Hyperbolic Contrastive Learning for Visual Representations beyond Objects](https://mlsanity.com/paper/2212.00653/) (2022)
- [Learning the Predictability of the Future](https://mlsanity.com/paper/2101.01600/) (2021)
- [Hypformer: Exploring Efficient Transformer Fully in Hyperbolic Space](https://mlsanity.com/paper/2407.01290/) (2024)
- [Hyperbolic Busemann Learning with Ideal Prototypes](https://mlsanity.com/paper/2106.14472/) (2021)
- [Knowledge Graph Embedding: A Survey from the Perspective of Representation Spaces](https://mlsanity.com/paper/2211.03536/) (2022)
- [Hyperbolic Graph Neural Networks: A Review of Methods and Applications](https://mlsanity.com/paper/2202.13852/) (2022)
- [Modeling Heterogeneous Hierarchies with Relation-specific Hyperbolic Cones](https://mlsanity.com/paper/2110.14923/) (2021)
- [Modeling Fine-Grained Entity Types with Box Embeddings](https://mlsanity.com/paper/2101.00345/) (2021)
- [TaxoCom: Topic Taxonomy Completion with Hierarchical Discovery of Novel Topic Clusters](https://mlsanity.com/paper/2201.06771/) (2022)
- [Language Models as Hierarchy Encoders](https://mlsanity.com/paper/2401.11374/) (2024)
- [Hyperbolic Fine-Tuning for Large Language Models](https://mlsanity.com/paper/2410.04010/) (2024)
- [HELM: Hyperbolic Large Language Models via Mixture-of-Curvature Experts](https://mlsanity.com/paper/2505.24722/) (2025)
- [Poincar'e ResNet](https://mlsanity.com/paper/2303.14027/) (2023)
- [Improved Representation of Asymmetrical Distances with Interval Quasimetric Embeddings](https://mlsanity.com/paper/2211.15120/) (2022)
- [Symmetric Spaces for Graph Embeddings: A Finsler-Riemannian Approach](https://mlsanity.com/paper/2106.04941/) (2021)
- [Rethinking Graph Lottery Tickets: Graph Sparsity Matters](https://mlsanity.com/paper/2305.02190/) (2023)
- [Probabilistic Box Embeddings for Uncertain Knowledge Graph Reasoning](https://mlsanity.com/paper/2104.04597/) (2021)
- [FALCON: Fast Visual Concept Learning by Integrating Images, Linguistic descriptions, and Conceptual Relations](https://mlsanity.com/paper/2203.16639/) (2022)
- [Hyperbolic Deep Learning for Foundation Models: A Survey](https://mlsanity.com/paper/2507.17787/) (2025)
- [Multi-modal Entity Alignment in Hyperbolic Space](https://mlsanity.com/paper/2106.03619/) (2021)
- [Skin Lesion Recognition with Class-Hierarchy Regularized Hyperbolic Embeddings](https://mlsanity.com/paper/2209.05842/) (2022)

## 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.