# TorusE: Knowledge Graph Embedding on a Lie Group

arXiv: https://arxiv.org/abs/1711.05435 · HTML: https://mlsanity.com/paper/1711.05435/

## Key facts

- Cited by 36 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Nov 2017.

## Built on by (most cited first, 30 of 36)

- [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)
- [Logical Message Passing Networks with One-hop Inference on Atomic Formulas](https://mlsanity.com/paper/2301.08859/) (2023)
- [Geometry Interaction Knowledge Graph Embeddings](https://mlsanity.com/paper/2206.12418/) (2022)
- [Link Prediction on N-ary Relational Data Based on Relatedness Evaluation](https://mlsanity.com/paper/2104.10424/) (2021)
- [Knowledge Sheaves: A Sheaf-Theoretic Framework for Knowledge Graph Embedding](https://mlsanity.com/paper/2110.03789/) (2021)
- [Knowledge Graph Embedding: An Overview](https://mlsanity.com/paper/2309.12501/) (2023)
- [Negative Sampling in Knowledge Graph Representation Learning: A Review](https://mlsanity.com/paper/2402.19195/) (2024)
- [SpherE: Expressive and Interpretable Knowledge Graph Embedding for Set Retrieval](https://mlsanity.com/paper/2404.19130/) (2024)
- [PHyCLIP: ℓ1-Product of Hyperbolic Factors Unifies Hierarchy and Compositionality in Vision-Language Representation Learning](https://mlsanity.com/paper/2510.08919/) (2025)
- [M"obiusE: Knowledge Graph Embedding on M"obius Ring](https://mlsanity.com/paper/2101.02352/) (2021)
- [PIE: a Parameter and Inference Efficient Solution for Large Scale Knowledge Graph Embedding Reasoning](https://mlsanity.com/paper/2204.13957/) (2022)
- [ModulE: Module Embedding for Knowledge Graphs](https://mlsanity.com/paper/2203.04702/) (2022)
- [Normed Spaces for Graph Embedding](https://mlsanity.com/paper/2312.01502/) (2023)
- [Arbitrary Time Information Modeling via Polynomial Approximation for Temporal Knowledge Graph Embedding](https://mlsanity.com/paper/2405.00358/) (2024)
- [Knowledge Graph Embeddings: A Comprehensive Survey on Capturing Relation Properties](https://mlsanity.com/paper/2410.14733/) (2024)
- [SpaceE: Knowledge Graph Embedding by Relational Linear Transformation in the Entity Space](https://mlsanity.com/paper/2204.10245/) (2022)
- [Tight and fast generalization error bound of graph embedding in metric space](https://mlsanity.com/paper/2305.07971/) (2023)
- [Estimating Text Similarity based on Semantic Concept Embeddings](https://mlsanity.com/paper/2401.04422/) (2024)
- [Harnessing the Power of Large Language Model for Uncertainty Aware Graph Processing](https://mlsanity.com/paper/2404.00589/) (2024)
- [Knowledge Graph Embedding by Normalizing Flows](https://mlsanity.com/paper/2409.19977/) (2024)
- [Two-dimensional Taxonomy for N-ary Knowledge Representation Learning Methods](https://mlsanity.com/paper/2506.05626/) (2025)
- [KG-EDAS: A Meta-Metric Framework for Evaluating Knowledge Graph Completion Models](https://mlsanity.com/paper/2508.15357/) (2025)
- [Normalized Maximum Likelihood Code-Length on Riemannian Data Spaces](https://mlsanity.com/paper/2508.21466/) (2025)
- [Multimodal Cultural Heritage Knowledge Graph Extension with Language and Vision Models](https://mlsanity.com/paper/2605.17669/) (2026)
- [Scalable knowledge base completion with superposition memories](https://mlsanity.com/paper/2110.12341/) (2021)
- [Semi-supervised Training for Knowledge Base Graph Self-attention Networks on Link Prediction](https://mlsanity.com/paper/2209.01350/) (2022)
- [ProjB: An Improved Bilinear Biased ProjE model for Knowledge Graph Completion](https://mlsanity.com/paper/2209.02390/) (2022)
- [Joint embedding in Hierarchical distance and semantic representation learning for link prediction](https://mlsanity.com/paper/2303.15655/) (2023)
- [Mitigating Heterogeneity among Factor Tensors via Lie Group Manifolds for Tensor Decomposition Based Temporal Knowledge Graph Embedding](https://mlsanity.com/paper/2404.09155/) (2024)

## Abstract (arXiv)

Knowledge graphs are useful for many artificial intelligence (AI) tasks. However, knowledge graphs often have missing facts. To populate the graphs, knowledge graph embedding models have been developed. Knowledge graph embedding models map entities and relations in a knowledge graph to a vector space and predict unknown triples by scoring candidate triples. TransE is the first translation-based method and it is well known because of its simplicity and efficiency for knowledge graph completion. It employs the principle that the differences between entity embeddings represent their relations. The principle seems very simple, but it can effectively capture the rules of a knowledge graph. However, TransE has a problem with its regularization. TransE forces entity embeddings to be on a sphere in the embedding vector space. This regularization warps the embeddings and makes it difficult for them to fulfill the abovementioned principle. The regularization also affects adversely the accuracies of the link predictions. On the other hand, regularization is important because entity embeddings diverge by negative sampling without it. This paper proposes a novel embedding model, TorusE, to solve the regularization problem. The principle of TransE can be defined on any Lie group. A torus, which is one of the compact Lie groups, can be chosen for the embedding space to avoid regularization. To the best of our knowledge, TorusE is the first model that embeds objects on other than a real or complex vector space, and this paper is the first to formally discuss the problem of regularization of TransE. Our approach outperforms other state-of-the-art approaches such as TransE, DistMult and ComplEx on a standard link prediction task. We show that TorusE is scalable to large-size knowledge graphs and is faster than the original TransE.