# Modeling Heterogeneous Hierarchies with Relation-specific Hyperbolic Cones

arXiv: https://arxiv.org/abs/2110.14923 · HTML: https://mlsanity.com/paper/2110.14923/

## Key facts

- Cited by 38 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 40 references that are in the map.
- First posted on arXiv 28 Oct 2021.

## Builds on (most cited first, 20 of 20)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [ConceptNet 5.5: An Open Multilingual Graph of General Knowledge](https://mlsanity.com/paper/1612.03975/) (2016)
- [Embedding Entities and Relations for Learning and Inference in Knowledge Bases](https://mlsanity.com/paper/1412.6575/) (2014)
- [Complex Embeddings for Simple Link Prediction](https://mlsanity.com/paper/1606.06357/) (2016)
- [RotatE: Knowledge Graph Embedding by Relational Rotation in Complex Space](https://mlsanity.com/paper/1902.10197/) (2019)
- [Convolutional 2D Knowledge Graph Embeddings](https://mlsanity.com/paper/1707.01476/) (2017)
- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [Knowledge Graphs](https://mlsanity.com/paper/2003.02320/) (2020)
- [Hyperbolic Neural Networks](https://mlsanity.com/paper/1805.09112/) (2018)
- [Hyperbolic Graph Convolutional Neural Networks](https://mlsanity.com/paper/1910.12933/) (2019)
- [TuckER: Tensor Factorization for Knowledge Graph Completion](https://mlsanity.com/paper/1901.09590/) (2019)
- [Learning Continuous Hierarchies in the Lorentz Model of Hyperbolic Geometry](https://mlsanity.com/paper/1806.03417/) (2018)
- [Modeling polypharmacy side effects with graph convolutional networks](https://mlsanity.com/paper/1802.00543/) (2018)
- [Low-Dimensional Hyperbolic Knowledge Graph Embeddings](https://mlsanity.com/paper/2005.00545/) (2020)
- [Hyperbolic Entailment Cones for Learning Hierarchical Embeddings](https://mlsanity.com/paper/1804.01882/) (2018)
- [Learning Hierarchy-Aware Knowledge Graph Embeddings for Link Prediction](https://mlsanity.com/paper/1911.09419/) (2019)
- [Multi-relational Poincar'e Graph Embeddings](https://mlsanity.com/paper/1905.09791/) (2019)
- [Order-Embeddings of Images and Language](https://mlsanity.com/paper/1511.06361/) (2015)
- [Probabilistic Embedding of Knowledge Graphs with Box Lattice Measures](https://mlsanity.com/paper/1805.06627/) (2018)
- [Improving Local Identifiability in Probabilistic Box Embeddings](https://mlsanity.com/paper/2010.04831/) (2020)

## Built on by (most cited first, 30 of 38)

- [Hyperbolic Image-Text Representations](https://mlsanity.com/paper/2304.09172/) (2023)
- [A*Net: A Scalable Path-based Reasoning Approach for Knowledge Graphs](https://mlsanity.com/paper/2206.04798/) (2022)
- [Heterogeneous Graph Masked Autoencoders](https://mlsanity.com/paper/2208.09957/) (2022)
- [Hypformer: Exploring Efficient Transformer Fully in Hyperbolic Space](https://mlsanity.com/paper/2407.01290/) (2024)
- [Hyperbolic Graph Neural Networks: A Review of Methods and Applications](https://mlsanity.com/paper/2202.13852/) (2022)
- [Answering Complex Logical Queries on Knowledge Graphs via Query Computation Tree Optimization](https://mlsanity.com/paper/2212.09567/) (2022)
- [Hyperbolic Representation Learning: Revisiting and Advancing](https://mlsanity.com/paper/2306.09118/) (2023)
- [Ultrahyperbolic Knowledge Graph Embeddings](https://mlsanity.com/paper/2206.00449/) (2022)
- [UUKG: Unified Urban Knowledge Graph Dataset for Urban Spatiotemporal Prediction](https://mlsanity.com/paper/2306.11443/) (2023)
- [Hyperbolic Deep Learning for Foundation Models: A Survey](https://mlsanity.com/paper/2507.17787/) (2025)
- [Position: Beyond Euclidean -- Foundation Models Should Embrace Non-Euclidean Geometries](https://mlsanity.com/paper/2504.08896/) (2025)
- [HYPE: Hyperbolic Entailment Filtering for Underspecified Images and Texts](https://mlsanity.com/paper/2404.17507/) (2024)
- [KG-FIT: Knowledge Graph Fine-Tuning Upon Open-World Knowledge](https://mlsanity.com/paper/2405.16412/) (2024)
- [Multi-task Pre-training Language Model for Semantic Network Completion](https://mlsanity.com/paper/2201.04843/) (2022)
- [kHGCN: Tree-likeness Modeling via Continuous and Discrete Curvature Learning](https://mlsanity.com/paper/2212.01793/) (2022)
- [Dual-Geometric Space Embedding Model for Two-View Knowledge Graphs](https://mlsanity.com/paper/2209.08767/) (2022)
- [Geometric Relational Embeddings: A Survey](https://mlsanity.com/paper/2304.11949/) (2023)
- [Start from Zero: Triple Set Prediction for Automatic Knowledge Graph Completion](https://mlsanity.com/paper/2406.18166/) (2024)
- [Large Language Model Enhanced Knowledge Representation Learning: A Survey](https://mlsanity.com/paper/2407.00936/) (2024)
- [SQUIRE: A Sequence-to-sequence Framework for Multi-hop Knowledge Graph Reasoning](https://mlsanity.com/paper/2201.06206/) (2022)
- [Rethinking Regularization Methods for Knowledge Graph Completion](https://mlsanity.com/paper/2505.23442/) (2025)
- [Self-supervised Topic Taxonomy Discovery in the Box Embedding Space](https://mlsanity.com/paper/2408.15050/) (2024)
- [Shadow Cones: A Generalized Framework for Partial Order Embeddings](https://mlsanity.com/paper/2305.15215/) (2023)
- [Efficient Relation-aware Neighborhood Aggregation in Graph Neural Networks via Tensor Decomposition](https://mlsanity.com/paper/2212.05581/) (2022)
- [A Self-matching Training Method with Annotation Embedding Models for Ontology Subsumption Prediction](https://mlsanity.com/paper/2402.16278/) (2024)
- [FFHR: Fully and Flexible Hyperbolic Representation for Knowledge Graph Completion](https://mlsanity.com/paper/2302.04088/) (2023)
- [Lorentz Equivariant Model for Knowledge-Enhanced Hyperbolic Collaborative Filtering](https://mlsanity.com/paper/2302.04545/) (2023)
- [NumCoKE: Ordinal-Aware Numerical Reasoning over Knowledge Graphs with Mixture-of-Experts and Contrastive Learning](https://mlsanity.com/paper/2411.12950/) (2024)
- [KG-BiLM: Knowledge Graph Embedding via Bidirectional Language Models](https://mlsanity.com/paper/2506.03576/) (2025)
- [HyBIRD: Hyperbolic Bridge Retrieval and Diagnosis for Methodology Inspiration Retrieval](https://mlsanity.com/paper/2606.28336/) (2026)

## Abstract (arXiv)

Hierarchical relations are prevalent and indispensable for organizing human knowledge captured by a knowledge graph (KG). The key property of hierarchical relations is that they induce a partial ordering over the entities, which needs to be modeled in order to allow for hierarchical reasoning. However, current KG embeddings can model only a single global hierarchy (single global partial ordering) and fail to model multiple heterogeneous hierarchies that exist in a single KG. Here we present ConE (Cone Embedding), a KG embedding model that is able to simultaneously model multiple hierarchical as well as non-hierarchical relations in a knowledge graph. ConE embeds entities into hyperbolic cones and models relations as transformations between the cones. In particular, ConE uses cone containment constraints in different subspaces of the hyperbolic embedding space to capture multiple heterogeneous hierarchies. Experiments on standard knowledge graph benchmarks show that ConE obtains state-of-the-art performance on hierarchical reasoning tasks as well as knowledge graph completion task on hierarchical graphs. In particular, our approach yields new state-of-the-art Hits@1 of 45.3% on WN18RR and 16.1% on DDB14 (0.231 MRR). As for hierarchical reasoning task, our approach outperforms previous best results by an average of 20% across the three datasets.