  Despite recent advances in representation learning in hypercomplex (HC)
space, this subject is still vastly unexplored in the context of graphs.
Motivated by the complex and quaternion algebras, which have been found in
several contexts to enable effective representation learning that inherently
incorporates a weight-sharing mechanism, we develop graph neural networks that
leverage the properties of hypercomplex feature transformation. In particular,
in our proposed class of models, the multiplication rule specifying the algebra
itself is inferred from the data during training. Given a fixed model
architecture, we present empirical evidence that our proposed model
incorporates a regularization effect, alleviating the risk of overfitting. We
also show that for fixed model capacity, our proposed method outperforms its
corresponding real-formulated GNN, providing additional confirmation for the
enhanced expressivity of HC embeddings. Finally, we test our proposed
hypercomplex GNN on several open graph benchmark datasets and show that our
models reach state-of-the-art performance while consuming a much lower memory
footprint with 70& fewer parameters. Our implementations are available at
https://github.com/bayer-science-for-a-better-life/phc-gnn.
