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.

翻译：学习知识图谱（KG）的双曲嵌入因在捕获层次结构方面的优越性而受到越来越多的关注。然而，双曲空间中的一些重要操作仍缺乏良好的定义，使得现有方法无法充分利用双曲空间的优势。具体而言，它们存在两个主要局限：1）双曲空间中的现有图卷积网络方法依赖于切空间近似，这会在表示学习中引入近似误差；2）由于双曲空间中缺乏内积操作定义，现有方法只能通过双曲距离度量事实（链接）的可信度，难以捕获复杂的数据模式。在这项工作中，我们贡献了：1）一种全庞加莱多关系图卷积网络，可在无需任何近似的情况下实现双曲空间中的图信息传播；2）一种欧几里得内积的双曲泛化，有助于捕获层次和复杂模式。在此基础上，我们进一步开发了**全**和**灵活**的**双曲****表示**框架（**FFHR**），能够将近期基于欧几里得空间的进展迁移到双曲空间。我们通过将FFHR实例化为四种代表性的知识图谱补全方法来展示其能力。在基准数据集上的大量实验验证了我们的FFHR相比其欧几里得对应方法以及最先进的双曲嵌入方法的优越性。