# Translation-Based Embeddings with Octonion for Knowledge Graph Completion
## 1 Introduction
Figure 1: Examples of (**a**) inter-dependencies and (**b**) hierarchical information in KGs. Therefore, octonion has the potential to capture the neglected internal dependencies in KGs and improve the effectiveness of modeling. We propose a translation-based KGC model with octonion (TransO), which effectively takes advantage of the lightweight and easy-to-train characteristics of the translation-based thoughts, and improves the capturing capability of the model for latent internal dependencies. Since the form and operation rules of octonion do not match the matrix operations commonly used in machine learning, it is difficult to directly introduce octonion into the KGC method, so we refer to mathematical research [15; 16; 17], and use coordinate vectors and component matrices to perform octonion operations in equivalent form. Specifically, TransO regards the entities in KGs as octonion coordinate vectors, and the relations as the combination of octonion component matrices and coordinate vectors, both of which are based on a real number field. TransO divides the interaction between entities and relations into two steps: multiplication interaction and translation interaction. The former captures internal dependencies by the grouping product of octonion vector and matrix, forming a compact interaction pattern; the latter captures external dependencies using vector addition and l2-norm consistent with traditional Trans series methods. In addition, hierarchical information widely exists in KGs such as social networks and geographic information, which are usually represented as discrete tree-like structures. For example, in Figure 1b, the entity \(Silicon\)\(Valley\) is taken as the root node of the hierarchy, and extends downward layer by layer; the second layer includes two corporate entities \(Apple\)\(Inc.\) and \(Google\)\(Inc.\); the third layer includes four personal name entities such as \(Mary\), and finally forms a complete tree. Since there are multiple hierarchical structures in the KGs, some entities are at the top of one structure and may be at the bottom of the other. How to reasonably model hierarchical information to improve the embedding effect of KGs is also an important research direction. However, most of the current methods are based on Euclidean geometric space and are not suitable for modeling hierarchical information. On the one hand, due to the discrete tree-like data structure of hierarchical information, Euclidean geometric space cannot preserve the latent hierarchical features with high information distortion in the case of low-dimensional embedding. On the other hand, since the number of nodes at each layer of the hierarchical tree-like structure reflects an exponential growth trend, modeling hierarchical information using Euclidean geometry requires high memory and computing resources. Hyperbolic geometry in non-Euclidean mathematics has inherent advantages for modeling hierarchical information. Firstly, compared with Euclidean geometry, hyperbolic geometry is easier to capture and restore the features of hierarchical relational data, forming a natural tree-like hierarchical structure. Moreover, hyperbolic geometry has higher information-carrying capacity than Euclidean geometry, and can model with relatively lower dimensions and fewer parameters, i.e., hyperbolic geometry has less information distortion in low-dimensional embedding [18]. In recent years, KGC tasks in hyperbolic space have been studied. MuRP [19] introduces the theories of hyperbolic geometry, which embeds KGs into a hyperbolic Poincare ball. AttH [20] uses reflection and rotation operations of hyperbolic isometrics to replace the linear transformation in MuRP, and combines reflection and rotation through the attention mechanism. Although these methods are still inadequate in feature interaction, and the interactive level is relatively shallow, they show the great advantages of hyperbolic geometry in low-dimensional embedded space or hierarchical information, which further inspire our ideas of introducing hyperbolic geometry.