# Translation-Based Embeddings with Octonion for Knowledge Graph Completion
## 3 Methodology
### Preliminaries
#### 3.1.2 Hyperbolic Geometry of Poincare Model  
In addition to latent dependencies, we also attempt to improve the modeling of hierarchical information in KGs. Since hierarchical information is closely related to geometric space, we focus on the mathematical theories about geometric space. Euclidean space, as we know it, is a geometric space with constant curvature of 0, while the curvature of non-Euclidean spaces is not. Non-Euclidean spaces can be divided into two types, elliptic space and hyperbolic space, in which elliptic space has positive curvature greater than 0 and hyperbolic space has negative curvature less than 0. This paper mainly focuses on hyperbolic space, because, compared with elliptic space, hyperbolic space has the capability to model hierarchical information, and we will explain the reason later. Three typical equivalent models of hyperbolic space are the Lorentz model, Klein model, and Poincare model. Among them, the Poincare model is the most widely used equivalent model and is also the research object of this paper. It is is commonly known as the Poincare disk or Poincare ball; the former shows the construction results of the Poincare model in two-dimensional space, while the latter is in three-dimensional or higher-dimensional space. A \(d\) - dimensional Poincare ball with curvature \(-c(c>0)\) is defined as \(\mathbb{B}^{d,c}=\{\mathbf{x}\in\mathbb{R}^{d}:\left\|\mathbf{x}\right\|^{2}< 1/c\ \}\), where \(\left\|\cdot\right\|\) denotes \(l2\) - norm. \(\mathcal{T}_{\mathbf{x}}^{c}\) denotes the tangent space of \(\mathbb{B}^{d,c}\) at point \(\mathbf{x}\), which can be regarded as Euclidean space. The visualization of the Poincare model in the form of a two-dimensional disk is shown in Figure 5, in which all arcs connecting two points are regarded as equal-length lines by the Poincare model itself. Figure 5: Illustration of two-dimensional Poincaré model. The above figure reflects the two inherent advantages of hyperbolic space for modeling hierarchical information:  
1. Hyperbolic space has fine capabilities of information representation, modeling, and restoration for tree-like structures. A point in the Poincare disk shown in the figure forms a natural tree-like hierarchical structure in the process of connecting other points step by step to one side of the disk boundary. When the data being modeled also have a tree structure, hyperbolic space can more easily capture and restore the hierarchical features of the data than Euclidean space. In KGs, there are abundant tree-like hierarchical features. Thus, hyperbolic geometric space has natural consistency with the hierarchical information in KGs. 2. Hyperbolic space is an infinite metric space with higher information capacity. The Poincare disk shown in the figure has a quantity of hyperbolic spatial triangles with the same shape and different sizes, but for the Poincare disk, the sizes of these triangles are regarded as the same, only because the visualization gives the impression that the triangles are shrinking as they approach the boundary of the Poincare model. In fact, the closer to the boundary of the Poincare model, the capacity of information that the space can accommodate will increase exponentially. In KGs, since hierarchical information has a tree-like structure, the number of nodes in each layer also reflects an exponential growth trend relative to the previous layer. Traditional Euclidean geometry space is unable to model the KGs with a large amount of hierarchical information, and the spatial dimension of the embedded vectors has to be further increased to meet the exponential information growth. However, for hyperbolic space, since it has less information distortion than Euclidean space with lower dimensions and fewer parameters, it is theoretically more suitable for modeling KGs with hierarchical information. The operations involved in the Poincare model mainly include: Mobius addition, exponential and logarithmic mappings, and hyperbolic distance calculation.