# Translation-Based Embeddings with Octonion for Knowledge Graph Completion
## 1 Introduction  
Knowledge graphs (KGs) are semantic networks composed of entities and relations, which provide an accurate description of objects in the real world [1]. Since their concept was first proposed by Google [2] in 2012, KGs have become an important part of search engines, personalized recommendation systems, and conversational robots [3; 4; 5]. KGs use a triple \((h,r,t)\) to describe a fact, where \(h\) denotes the head entity, \(t\) denotes the tail entity, and \(r\) denotes the relation between the head entity and the tail entity [6]. Knowledge graph embeddings embed the entities and relations of triples into continuous vector space [7]. By defining the scoring function \(f_{r}(h,t)\) and calculating the score of any embedded triple, the embedding method can judge whether this triple is true or not. Since the embedding process obtains the vector representation of the entities and relations in the knowledge graph through machine learning, it is also called knowledge representation learning [8]. The combination of knowledge representation learning and link predictioncan allow the missing facts in the KGs to be predicted, thus realizing knowledge graph completion (KGC). Recently, in the research field of KGC, the inter-dependencies and hierarchical information in KGs have gained attention. The dependency information in KGs can be divided into external dependencies and internal dependencies. On the one hand, external dependencies refer to the dependencies between multi-dimensional embedded vectors from the KGs. For example, in Figure 1a, there is a triple (_[PERSON]_, _painted_, _The stream_), and the entities _[PERSON]_ and _The stream_ are connected through the relation _painted_, i.e., there is an external dependency between the embedded vectors of _[PERSON]_ and _The stream_. To some extent, the embedded vector representation of _[PERSON]_ depends on _The stream_, and vice versa. On the other hand, there are internal dependencies between the components or elements within the multi-dimensional embedded vector. For example, the value of a one-dimensional component in the embedded vector of _[PERSON]_ depends on other dimensional components, and the dimensions are not independent of each other. Current KGC methods tend to focus more on the interaction between the individuals of entities or relations, i.e., external dependencies, while ignoring the latent internal dependencies between elements or components. For example, the translation-based TransE [9] method regards the relation in the triples of KGs as the translation vector from the head entity to the tail entity, and captures the distance feature between the head entity after the relational interaction and the tail entity; the semantics-based RESCAL [10] method represents the entity as a low-dimensional vector and relation as a matrix that captures the interaction between the head and the tail entity. By contrast, neural network-based methods can learn complex input-output mapping, and usually have better capabilities in capturing internal dependencies, such as ConvE [11], ConvKB [12]. However, they require a large number of parameters and complex calculations, and are not suitable for low-resource application scenarios. Octonion can be regarded as a hypercomplex with one real and seven imaginary components, which takes the correlation between components into account in the structure. Therefore, it is widely used in the feature extraction of color images and multi-dimensional speech signals. For example, reference [13] proposes an octonion neural network to replace the real-valued neural network, and reference [14] proposes an octonion BP neural network to extract fine palmprint lines in color palmprint images. These studies show that by taking advantage of the specific rules of octonion multiplication, octonion can effectively encode the internal dependencies in input features and the external dependencies between features with relatively fewer parameters, and are suitable for applications with lower resources.