# Translation-Based Embeddings with Octonion for Knowledge Graph Completion
## 3 Methodology
### Preliminaries
#### 3.1.1 Background and Calculation Rules of Octonion
In form, the embeddings of KGs and signals are both multi-dimensional features, and the components of embedding have the meaning of projection in space, which is similar to the meaning of channels in the signal. Since signals can be represented by octonions, it is likely that the embeddings in KGs are also represented by octonions; in content, there are also a large number of complex  
Figure 2: Illustrations of (\(\mathbf{a}\)) inner product, (\(\mathbf{b}\)) tensor product, (\(\mathbf{c}\)) Hadamard product, and (\(\mathbf{d}\)) octonion product. correlations in KGs, such as external and internal dependencies. Since octonions are good at capturing the correlations of different channels in the signal, they have the potential to capture those between different components of embedded vectors in KGs. At present, most of the technologies and frameworks currently used in KGs are based on real-valued representation and operation, and the research of octonion has great potential and broad prospects. However, the calculation rules of octonion restrict its application in computer science to a certain extent, as shown below. The addition operation of octonion adds the corresponding coefficients, similar to complex number or quaternion. However, the multiplication operation of octonion is determined by the specific multiplication rules of the unit octonion. There are also specific multiplication rules in quaternion, but the rules of octonion are obviously more complex, as they are the extension of quaternion. From Figure 2(a), we can preliminarily understand the multiplication rules of quaternion. When two multipliers are selected along the arrow direction in the figure, the multiplication result is the next element adjacent to the two multipliers, but when two multipliers are selected against the arrow direction, the multiplication result is the opposite number of the next element adjacent to the two multipliers--for example, \(ij=k\), \(ik=-j\). Similar to quaternion, the multiplication rules of octonion can be understood from the Fano plane in Figure 2(b)--for example, \(klj=il\), \(kli=-jl\). Moreover, from Figure 4, the multiplication rules of octonion can also be understood by looking at the table. Select multiplier 1 from the first column and multiplier 2 from the first row, and then the multiplication result is the element at the intersection of the row and column; for example, if we select \(il\) from the first column as multiplier 1 and \(kl\) from the first row as multiplier 2, the operation result is \(j\), which is consistent with that obtained in Figure 2(b). Figure 3: Illustrations of (**a**) quaternion multiplication and (**b**) octonion multiplication (Fano plane). According to the above rules, the multiplication of octonion is neither commutative nor associative, e.g., \(ij=-ji\
eq ji\), or \((ij)l=-i(jl)\
eq i(jl)\). Furthermore, these rules of octonion are more complex, while most current machine learning methods use a matrix to operate the dataset; their forms do not match. If the machine learning model is directly introduced into the octonion space, it may not bring additional benefits to the model. Even the loss of some algebraic properties and the complex operation rules might bring side effects to the model. In recent years, there have been studies [15; 16; 17] on transforming the operations between octonions into those between vectors and matrices in the real number field. These innovative operations can be performed using the knowledge of linear algebra and are much more convenient than the direct octonion multiplication shown in Figures 3b and 4. With this, the calculation difficulty between octonions can be reduced to a great extent.