  Learning hyperbolic embeddings for knowledge graph (KG) has gained increasing
attention due to its superiority in capturing hierarchies. However, some
important operations in hyperbolic space still lack good definitions, making
existing methods unable to fully leverage the merits of hyperbolic space.
Specifically, they suffer from two main limitations: 1) existing Graph
Convolutional Network (GCN) methods in hyperbolic space rely on tangent space
approximation, which would incur approximation error in representation
learning, and 2) due to the lack of inner product operation definition in
hyperbolic space, existing methods can only measure the plausibility of facts
(links) with hyperbolic distance, which is difficult to capture complex data
patterns. In this work, we contribute: 1) a Full Poincar\'{e} Multi-relational
GCN that achieves graph information propagation in hyperbolic space without
requiring any approximation, and 2) a hyperbolic generalization of Euclidean
inner product that is beneficial to capture both hierarchical and complex
patterns. On this basis, we further develop a \textbf{F}ully and
\textbf{F}lexible \textbf{H}yperbolic \textbf{R}epresentation framework
(\textbf{FFHR}) that is able to transfer recent Euclidean-based advances to
hyperbolic space. We demonstrate it by instantiating FFHR with four
representative KGC methods. Extensive experiments on benchmark datasets
validate the superiority of our FFHRs over their Euclidean counterparts as well
as state-of-the-art hyperbolic embedding methods.
