# Dual Quaternion Embeddings for Link Prediction
## 3 Preliminaries  
In this part, we introduce several concepts used in this paper. * **Quaternion**: Quaternion [20], is a number system that extends complex numbers to four-dimensional numbers. Generally, a quaternion is a number of the form \(\text{q}=a+b\textbf{i}+c\textbf{j}+d\textbf{k}\), where \(a,b,c,d\) are real numbers and \(\textbf{i},\textbf{j},\textbf{k}\) satisfy that \(\textbf{i}^{2}=\textbf{j}^{2}=\textbf{k}^{2}=-1\). * **Quaternion conjugate**: The definition of conjugate to a quaternion \(\text{q}\) is \(\text{q}^{*}=\text{a}-\text{bi}-\text{cj}-\text{dk}\). * **Quaternion Multiplication:** Multiplication of two quaternions \(\text{p}=p_{0}+p_{1}\textbf{i}+p_{2}\textbf{j}+p_{3}\textbf{k}\) and \(\text{q}=q_{0}+q_{1}\textbf{i}+q_{2}\textbf{j}+q_{3}\textbf{k}\) is defined by: \[ \begin{split}\text{p}\text{q}&=(p_{0}q_{0}-p_{1}q_{1 }-p_{2}q_{2}-p_{3}q_{3})\\ &+(p_{0}q_{1}+p_{1}q_{0}+p_{2}q_{3}-p_{3}q_{2})\textbf{i}\\ &+(p_{0}q_{2}-p_{1}q_{3}+p_{2}q_{0}+p_{3}q_{1})\textbf{j}\\ &+(p_{0}q_{3}+p_{1}q_{2}-p_{2}q_{1}+p_{3}q_{0})\textbf{k}\end{split} \] (1)
* **Rotation with quaternions in three-dimensional space**: The point \(\text{v}^{\prime}\) is rotated by the point \(\text{v}\) along the unit vector \(\text{u}\) (i.e., rotation axis), which can use quaternion multiplication to represent. We define \(\text{v}\) and \(\text{v}^{\prime}\) as pure quaternion, i.e., quaternions with real part being zero, \(\text{m}=\cos\frac{\theta}{2}+\text{u}\sin\frac{\theta}{2}\) is a unit quaternion, then \[\text{v}^{\prime}=\text{mvm}^{*}\] (2)
* **Dual quaternion**: Dual quaternion [21] is an eight-dimensional real algebra to combine with quaternions. Formally, a dual quaternion \(\delta\) can be represented by \(\delta=\text{p}+\epsilon\text{q}\), where \(\epsilon\) is a dual unit with \(\epsilon^{2}=0\), both the real part \(\text{p}\) and the dual part \(\text{q}\) are quaternions. Therefore, a dual quaternion \(\delta\) is of the form \(\delta=p_{0}+p_{1}\textbf{i}+p_{2}\textbf{j}+p_{3}\textbf{k}+\epsilon(q_{0}+q_ {1}\textbf{i}+q_{2}\textbf{j}+q_{3}\textbf{k})\). * **Dual quaternion conjugate**: The conjugate of the dual quaternion \(\delta=\text{p}+\epsilon\text{q}\) is defined as: \(\delta^{\circ}=\text{p}^{*}-\epsilon\text{q}^{*}\), which can be represented by an 8-tuple: \(\delta^{\circ}=(p_{0},-p_{1},-p_{2},-p_{3},-q_{0},q_{1},q_{2},q_{3})\). * **Dual Quaternion Multiplication**: Dual Quaternion Hamilton product between \(\delta_{1}=\text{p}_{1}+\epsilon\text{q}_{1}\) and \(\delta_{2}=\text{p}_{2}+\epsilon\text{q}_{2}\) is defined as follows: \[\delta_{1}\otimes\delta_{2}=\text{p}_{1}\text{p}_{2}+\epsilon(\text{p}_{1} \text{q}_{2}+\text{q}_{1}\text{p}_{2})\] (3)* **Unit Dual Quaternion**: A dual quaternion \(\delta=\text{p}+\epsilon\text{q}\) is a unit dual quaternion, if \(\delta\otimes\delta^{*}=1\), namely, \(\delta\) satisfies the following conditions: \[ \begin{split} p_{0}^{2}+p_{1}^{2}+p_{2}^{2}+p_{3}^{2}& =1,\\ p_{0}q_{0}+p_{1}q_{1}+p_{2}q_{2}+p_{3}q_{3}&=0\end{split} \] (4) where \(\delta^{*}=\text{p}^{*}+\epsilon\text{q}^{*}\). In order to simplify the calculation process, we use another effective form to represent unit dual quaternion which defines as follows: \[\delta=\text{m}+\frac{\epsilon}{2}\text{mn}\] (5) where \(\text{m}=\cos\frac{\theta}{2}+\text{u}\sin\frac{\theta}{2}\) and n is a pure quaternion. We prove that \(\delta\) is a unit dual quaternion: \[ \begin{split}\delta\otimes\delta^{*}&=\left( \text{m}+\frac{\epsilon}{2}\text{nm}\right)\otimes\left(\text{m}^{*}+\frac{ \epsilon}{2}(\text{nm})^{*}\right)\\ &=\text{mm}^{*}+\frac{\epsilon}{2}(\text{mm}^{*}\text{n}^{*}+ \text{nmm}^{*})\\ &=1+\frac{\epsilon}{2}(\text{n}^{*}+\text{n})\\ &=1+\frac{\epsilon}{2}(-\text{n}+\text{n})\\ &=1\end{split} \] (6) We can easily verify \(\text{mm}^{*}=1\), as shown below: \[