# Knowledge Sheaves: A Sheaf-Theoretic Framework for Knowledge Graph Embedding

arXiv: https://arxiv.org/abs/2110.03789 · HTML: https://mlsanity.com/paper/2110.03789/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 59 references that are in the map.
- First posted on arXiv 7 Oct 2021.

## Builds on (most cited first, 24 of 24)

- [Modeling Relational Data with Graph Convolutional Networks](https://mlsanity.com/paper/1703.06103/) (2017)
- [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)
- [DeepPath: A Reinforcement Learning Method for Knowledge Graph Reasoning](https://mlsanity.com/paper/1707.06690/) (2017)
- [Holographic Embeddings of Knowledge Graphs](https://mlsanity.com/paper/1510.04935/) (2015)
- [Quaternion Knowledge Graph Embeddings](https://mlsanity.com/paper/1904.10281/) (2019)
- [Query2box: Reasoning over Knowledge Graphs in Vector Space using Box Embeddings](https://mlsanity.com/paper/2002.05969/) (2020)
- [Beta Embeddings for Multi-Hop Logical Reasoning in Knowledge Graphs](https://mlsanity.com/paper/2010.11465/) (2020)
- [Embedding Logical Queries on Knowledge Graphs](https://mlsanity.com/paper/1806.01445/) (2018)
- [Multi-Hop Knowledge Graph Reasoning with Reward Shaping](https://mlsanity.com/paper/1808.10568/) (2018)
- [PyKEEN 1.0: A Python Library for Training and Evaluating Knowledge Graph Embeddings](https://mlsanity.com/paper/2007.14175/) (2020)
- [Understanding Double Descent Requires a Fine-Grained Bias-Variance Decomposition](https://mlsanity.com/paper/2011.03321/) (2020)
- [Bringing Light Into the Dark: A Large-scale Evaluation of Knowledge Graph Embedding Models Under a Unified Framework](https://mlsanity.com/paper/2006.13365/) (2020)
- [A Semantic Matching Energy Function for Learning with Multi-relational Data](https://mlsanity.com/paper/1301.3485/) (2013)
- [Traversing Knowledge Graphs in Vector Space](https://mlsanity.com/paper/1506.01094/) (2015)
- [Orthogonal Relation Transforms with Graph Context Modeling for Knowledge Graph Embedding](https://mlsanity.com/paper/1911.04910/) (2019)
- [TorusE: Knowledge Graph Embedding on a Lie Group](https://mlsanity.com/paper/1711.05435/) (2017)
- [Research Commentary on Recommendations with Side Information: A Survey and Research Directions](https://mlsanity.com/paper/1909.12807/) (2019)
- [Measuring Social Bias in Knowledge Graph Embeddings](https://mlsanity.com/paper/1912.02761/) (2019)
- [On the Equivalence of Holographic and Complex Embeddings for Link Prediction](https://mlsanity.com/paper/1702.05563/) (2017)
- [Complex and Holographic Embeddings of Knowledge Graphs: A Comparison](https://mlsanity.com/paper/1707.01475/) (2017)
- [Knowledge Representation in Bicategories of Relations](https://mlsanity.com/paper/1706.00526/) (2017)
- [Group Representation Theory for Knowledge Graph Embedding](https://mlsanity.com/paper/1909.05100/) (2019)

## Built on by (most cited first, 11 of 11)

- [Neural Graph Reasoning: Complex Logical Query Answering Meets Graph Databases](https://mlsanity.com/paper/2303.14617/) (2023)
- [A Foundation Model for Zero-shot Logical Query Reasoning](https://mlsanity.com/paper/2404.07198/) (2024)
- [Copresheaf Topological Neural Networks: A Generalized Deep Learning Framework](https://mlsanity.com/paper/2505.21251/) (2025)
- [Sheaf-Theoretic Transport and Obstruction for Detecting Scientific Theory Shift in AI Agents](https://mlsanity.com/paper/2605.14033/) (2026)
- [On the Sheafification of Higher-Order Message Passing](https://mlsanity.com/paper/2509.23020/) (2025)
- [Feature Propagation on Knowledge Graphs using Cellular Sheaves](https://mlsanity.com/paper/2309.03773/) (2023)
- [Capability Sheaves for Compositional Agent-Harness Repair: Controlled Quotients and a Real-Repository Stress Test](https://mlsanity.com/paper/2608.13228/) (2026)
- [Toward Auto-Research: Mining Falsifiable Research Ideas from Paper Knowledge Graphs with Categorical Structure](https://mlsanity.com/paper/2608.20361/) (2026)
- [PAGR: Proof-Carrying Algebraic-Geometric Retrieval: A Quiver-, Provenance-, and Sheaf-Theoretic Framework for Grounded LLM Retrieval](https://mlsanity.com/paper/2609.06127/) (2026)
- [SheafPrune: Sheaf-Guided Graph Propagation for Visual Token Pruning](https://mlsanity.com/paper/2610.13263/) (2026)
- [Gluing Local Contexts into Global Meaning: A Sheaf-Theoretic Decomposition of Transformer Representations](https://mlsanity.com/paper/2610.17117/) (2026)

## Abstract (arXiv)

Knowledge graph embedding involves learning representations of entities -- the vertices of the graph -- and relations -- the edges of the graph -- such that the resulting representations encode the known factual information represented by the knowledge graph and can be used in the inference of new relations. We show that knowledge graph embedding is naturally expressed in the topological and categorical language of textitcellular sheaves: a knowledge graph embedding can be described as an approximate global section of an appropriate textitknowledge sheaf over the graph, with consistency constraints induced by the knowledge graph's schema. This approach provides a generalized framework for reasoning about knowledge graph embedding models and allows for the expression of a wide range of prior constraints on embeddings. Further, the resulting embeddings can be easily adapted for reasoning over composite relations without special training. We implement these ideas to highlight the benefits of the extensions inspired by this new perspective.