Recent knowledge graph (KG) embeddings have been advanced by hyperbolic geometry due to its superior capability for representing hierarchies. The topological structures of real-world KGs, however, are rather heterogeneous, i.e., a KG is composed of multiple distinct hierarchies and non-hierarchical graph structures. Therefore, a homogeneous (either Euclidean or hyperbolic) geometry is not sufficient for fairly representing such heterogeneous structures. To capture the topological heterogeneity of KGs, we present an ultrahyperbolic KG embedding (UltraE) in an ultrahyperbolic (or pseudo-Riemannian) manifold that seamlessly interleaves hyperbolic and spherical manifolds. In particular, we model each relation as a pseudo-orthogonal transformation that preserves the pseudo-Riemannian bilinear form. The pseudo-orthogonal transformation is decomposed into various operators (i.e., circular rotations, reflections and hyperbolic rotations), allowing for simultaneously modeling heterogeneous structures as well as complex relational patterns. Experimental results on three standard KGs show that UltraE outperforms previous Euclidean- and hyperbolic-based approaches.

翻译：近代知识图形( KG) 嵌入器由于超偏斜几何制而得到进步, 因为它具有代表等级的超强能力。 然而, 真实世界 KG 的地形结构是相当多样化的, 也就是说, KG 是由多个不同的等级和非等级的图形结构组成的。 因此, 光化( 不管是Euclidean 或 superblic) 几何制不足以公平代表这种多元结构。 为了捕捉 KGs 的表层异质性, 我们展示了超超超超偏偏( 伪Riemannian) 中超超超超超超超偏偏KG 嵌入( UltraE) 嵌入( Ulphyperblic( or- Riemannian) ) 的多元结构。 特别是, 我们将每种关系建模为假正统的( Ecolorphonal) 变形, 以保存伪Riemann 双线形式的形式。 为了向不同的操作者( 即循环旋转、 反射镜反射和双曲旋转), 允许同时建模的K- cloan- climal- bal- beal- beal- beal- 以及前的模型显示。