# Embedding Knowledge Graphs in Degenerate Clifford Algebras

arXiv: https://arxiv.org/abs/2402.04870 · HTML: https://mlsanity.com/paper/2402.04870/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 28 references that are in the map.
- First posted on arXiv 6 Feb 2024.

## Builds on (most cited first, 10 of 10)

- [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)
- [Knowledge Graphs](https://mlsanity.com/paper/2003.02320/) (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)
- [Multi-relational Poincar'e Graph Embeddings](https://mlsanity.com/paper/1905.09791/) (2019)
- [Convolutional Hypercomplex Embeddings for Link Prediction](https://mlsanity.com/paper/2106.15230/) (2021)
- [Knowledge Graph Embeddings in Geometric Algebras](https://mlsanity.com/paper/2010.00989/) (2020)

## Built on by (most cited first, 1 of 1)

- [Neural Reasoning for Robust Instance Retrieval in SHOIQ](https://mlsanity.com/paper/2510.20457/) (2025)

## Abstract (arXiv)

Clifford algebras are a natural generalization of the real numbers, the complex numbers, and the quaternions. So far, solely Clifford algebras of the form Clp,q (i.e., algebras without nilpotent base vectors) have been studied in the context of knowledge graph embeddings. We propose to consider nilpotent base vectors with a nilpotency index of two. In these spaces, denoted Clp,q,r, allows generalizing over approaches based on dual numbers (which cannot be modelled using Clp,q) and capturing patterns that emanate from the absence of higher-order interactions between real and complex parts of entity embeddings. We design two new models for the discovery of the parameters p, q, and r. The first model uses a greedy search to optimize p, q, and r. The second predicts (p, q,r) based on an embedding of the input knowledge graph computed using neural networks. The results of our evaluation on seven benchmark datasets suggest that nilpotent vectors can help capture embeddings better. Our comparison against the state of the art suggests that our approach generalizes better than other approaches on all datasets w.r.t. the MRR it achieves on validation data. We also show that a greedy search suffices to discover values of p, q and r that are close to optimal.