# Translation-Based Embeddings with Octonion for Knowledge Graph Completion
## 3 Methodology
### Preliminaries
#### 3.1.1 Background and Calculation Rules of Octonion  
Octonion is a non-associative generalization of quaternion, which is usually denoted as \(\mathbb{O}\). An octonion can be regarded as a tuple with eight elements. Each octonion is a linear combination of unit octonion \(\{1,i,j,k,l,il,jl,kl\}\), i.e., the octonion \(x=x_{0}+x_{1}i+x_{2}j+x_{3}k+x_{4}l+x_{5}il+x_{6}jl+x_{7}kl\), where \(x_{i}\in\mathbb{R},i=0,1,\cdots,7\). As a hypercomplex number in abstract algebra, octonion has the following good mathematical properties:  
* As the generalized form of complex number and quaternion, octonion is the only normed division algebra except real number, complex number, and quaternion [36], which can define norm and multiplicative inverse, and can also perform inner product operation, with rich algebraic definitions. * In geometric space, octonions can represent the combination of rotation and translation [37], which has good geometric significance, while the quaternion, which is also one of the hypercomplex numbers, can only represent rotation. Consequently, compared with the complex number and quaternion, octonion has higher degrees of freedom and more flexible expressive capability. * The representation of octonion has internal correlations and dependencies among its constituent elements [38]. On the premise of not increasing the amount of parameters, octonion can add additional information capacity and interactive features to the graph data. When performing mathematical operations, octonion can encourage the compact interaction between the objects being calculated. It can be seen that octonion has good interpretability, because it is well defined algebraically and has both algebraic and geometric interpretations. Moreover, octonion adds additional interactive features and encourages compact interactive patterns mainly through its specific product operation rules. As shown in Figure 2, we compare the common operations among the inner product, tensor product, Hadamard product, and octonion product, where \(\mathbf{a}\) and \(\mathbf{b}\) denote multiplier vectors, which are represented in 8-dimensional form for comparison with octonions. \(c\), \(\mathbf{C}\), or \(\mathbf{c}\) denotes the product number, matrix, or vector. Figure 2: _Cont._  
As can be seen from the comparison in the figure, the octonion product \(\mathbf{c}\) is still an octonion vector, and each one-dimensional component of \(\mathbf{c}\) is related to each one-dimensional component of the multiplier octonion vectors \(\mathbf{a}\) and \(\mathbf{b}\). Assuming that \(\mathbf{a}\) and \(\mathbf{b}\) denote two interactive input features, the octonion vector \(\mathbf{c}\) denotes the output feature after interaction. Each component in \(\mathbf{c}\) contains the information of each component in the input features, thus forming a compact interactive pattern, which can fully mine the deep features hidden in the input vectors. Moreover, after multiplicative interaction, the internal components of octonion product \(\mathbf{c}\) are no longer relatively independent individuals, but have internal correlation, i.e., the value of a one-dimensional component depends on other components, which has significant advantages in modeling internal dependencies. These characteristics are not found in the inner product, tensor product, and Hadamard product. Octonion is used in many scientific fields, such as string theory, special relativity, and quantum logic. In the field of signal processing, the representation of signals in the octonion domain can provide global stability; thus, octonion has a rich application background. From the perspective of signal processing, if one multi-dimensional input feature is regarded as a signal, in this way, some components of this feature can represent a channel of the signal, and there are tight correlations between different channels of the signal. These correlations are exactly what the representation of octonion is good at capturing. The embedded representations in KGs are similar to signals.