# HopfE: Knowledge Graph Representation Learning using Inverse Hopf Fibrations

arXiv: https://arxiv.org/abs/2108.05774 · HTML: https://mlsanity.com/paper/2108.05774/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 58 references that are in the map.
- First posted on arXiv 12 Aug 2021.

## Builds on (most cited first, 23 of 23)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Efficient Estimation of Word Representations in Vector Space](https://mlsanity.com/paper/1301.3781/) (2013)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [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)
- [A Survey on Knowledge Graphs: Representation, Acquisition and Applications](https://mlsanity.com/paper/2002.00388/) (2020)
- [Low-Dimensional Hyperbolic Knowledge Graph Embeddings](https://mlsanity.com/paper/2005.00545/) (2020)
- [Quaternion Knowledge Graph Embeddings](https://mlsanity.com/paper/1904.10281/) (2019)
- [Learning Hierarchy-Aware Knowledge Graph Embeddings for Link Prediction](https://mlsanity.com/paper/1911.09419/) (2019)
- [A Re-evaluation of Knowledge Graph Completion Methods](https://mlsanity.com/paper/1911.03903/) (2019)
- [Beyond Fully-Connected Layers with Quaternions: Parameterization of Hypercomplex Multiplications with 1/n Parameters](https://mlsanity.com/paper/2102.08597/) (2021)
- [Quaternion Convolutional Neural Networks](https://mlsanity.com/paper/1903.00658/) (2019)
- [A Survey on Knowledge Graph Embeddings with Literals: Which model links better Literal-ly?](https://mlsanity.com/paper/1910.12507/) (2019)
- [Incorporating Literals into Knowledge Graph Embeddings](https://mlsanity.com/paper/1802.00934/) (2018)
- [RECON: Relation Extraction using Knowledge Graph Context in a Graph Neural Network](https://mlsanity.com/paper/2009.08694/) (2020)
- [KBLRN : End-to-End Learning of Knowledge Base Representations with Latent, Relational, and Numerical Features](https://mlsanity.com/paper/1709.04676/) (2017)
- [Answering Complex Questions by Joining Multi-Document Evidence with Quasi Knowledge Graphs](https://mlsanity.com/paper/1908.00469/) (2019)
- [Falcon 2.0: An Entity and Relation Linking Tool over Wikidata](https://mlsanity.com/paper/1912.11270/) (2019)
- [On Multi-Relational Link Prediction with Bilinear Models](https://mlsanity.com/paper/1709.04808/) (2017)
- [Connecting Embeddings for Knowledge Graph Entity Typing](https://mlsanity.com/paper/2007.10873/) (2020)
- [Knowledge Graph Embeddings and Explainable AI](https://mlsanity.com/paper/2004.14843/) (2020)

## Built on by (most cited first, 9 of 9)

- [A Survey of Knowledge Graph Reasoning on Graph Types: Static, Dynamic, and Multimodal](https://mlsanity.com/paper/2212.05767/) (2022)
- [Knowledge Graph Embedding: A Survey from the Perspective of Representation Spaces](https://mlsanity.com/paper/2211.03536/) (2022)
- [Knowledge Graph Embedding: An Overview](https://mlsanity.com/paper/2309.12501/) (2023)
- [MQuinE: a cure for "Z-paradox" in knowledge graph embedding models](https://mlsanity.com/paper/2402.03583/) (2024)
- [ModulE: Module Embedding for Knowledge Graphs](https://mlsanity.com/paper/2203.04702/) (2022)
- [Can Persistent Homology provide an efficient alternative for Evaluation of Knowledge Graph Completion Methods?](https://mlsanity.com/paper/2301.12929/) (2023)
- [Block-Diagonal Orthogonal Relation and Matrix Entity for Knowledge Graph Embedding](https://mlsanity.com/paper/2401.05967/) (2024)
- [Neural Probabilistic Logic Learning for Knowledge Graph Reasoning](https://mlsanity.com/paper/2407.03704/) (2024)
- [Pattern Over-Generalization of Knowledge Graph Embedding](https://mlsanity.com/paper/2609.03487/) (2026)

## Abstract (arXiv)

Recently, several Knowledge Graph Embedding (KGE) approaches have been devised to represent entities and relations in dense vector space and employed in downstream tasks such as link prediction. A few KGE techniques address interpretability, i.e., mapping the connectivity patterns of the relations (i.e., symmetric/asymmetric, inverse, and composition) to a geometric interpretation such as rotations. Other approaches model the representations in higher dimensional space such as four-dimensional space (4D) to enhance the ability to infer the connectivity patterns (i.e., expressiveness). However, modeling relation and entity in a 4D space often comes at the cost of interpretability. This paper proposes HopfE, a novel KGE approach aiming to achieve the interpretability of inferred relations in the four-dimensional space. We first model the structural embeddings in 3D Euclidean space and view the relation operator as an SO(3) rotation. Next, we map the entity embedding vector from a 3D space to a 4D hypersphere using the inverse Hopf Fibration, in which we embed the semantic information from the KG ontology. Thus, HopfE considers the structural and semantic properties of the entities without losing expressivity and interpretability. Our empirical results on four well-known benchmarks achieve state-of-the-art performance for the KG completion task.