Embedding Knowledge Graphs in Degenerate Clifford Algebras
Louis Mozart Kamdem Caglar Demir Axel-Cyrille Ngonga
Clifford algebras are a natural generalization of the real numbers, the
complex numbers, and the quaternions. So far, solely Clifford algebras of the
form $Cl_{p,q}$ (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
$Cl_{p,q,r}$, allows generalizing over approaches based on dual numbers (which
cannot be modelled using $Cl_{p,q}$) 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.