Embedding Knowledge Graphs in Degenerate Clifford Algebras

<jats:p>Clifford algebras are a natural extension of division algebras, including real numbers, complex numbers, quaternions, and octonions. Previous research in knowledge graph embeddings has focused exclusively on Clifford algebras of a specific type, which do not include nilpotent base vectors—elements that square to zero. In this work, we introduce a novel approach by incorporating nilpotent base vectors with a nilpotency index of two, leading to a more general form of Clifford algebras named degenerate Clifford algebras. This generalization to degenerate Clifford algebras does allow for covering dual numbers and as such include translations and rotations models under the same generalization paradigm for the first time. We develop two models to determine the parameters that define the algebra: one using a greedy search and another predicting the parameters based on neural network embeddings of the input knowledge graph. Our evaluation on seven benchmark datasets demonstrates that this incorporation of nilpotent vectors enhances the quality of embeddings. Additionally, our method outperforms state-of-the-art approaches in terms of generalization, particularly regarding the mean reciprocal rank achieved on validation data. Finally, we show that even a simple greedy search can effectively discover optimal or near-optimal parameters for the algebra.</jats:p>