  The choice of geometric space for knowledge graph (KG) embeddings can have
significant effects on the performance of KG completion tasks. The hyperbolic
geometry has been shown to capture the hierarchical patterns due to its
tree-like metrics, which addressed the limitations of the Euclidean embedding
models. Recent explorations of the complex hyperbolic geometry further improved
the hyperbolic embeddings for capturing a variety of hierarchical structures.
However, the performance of the hyperbolic KG embedding models for
non-transitive relations is still unpromising, while the complex hyperbolic
embeddings do not deal with multi-relations. This paper aims to utilize the
representation capacity of the complex hyperbolic geometry in multi-relational
KG embeddings. To apply the geometric transformations which account for
different relations and the attention mechanism in the complex hyperbolic
space, we propose to use the fast Fourier transform (FFT) as the conversion
between the real and complex hyperbolic space. Constructing the attention-based
transformations in the complex space is very challenging, while the proposed
Fourier transform-based complex hyperbolic approaches provide a simple and
effective solution. Experimental results show that our methods outperform the
baselines, including the Euclidean and the real hyperbolic embedding models.
