# Dual Quaternion Embeddings for Link Prediction
## 4 Our DualQuatE Model
### Properties of DualQuatE  
In this part we describe the relation patterns and introduce how DualQuatE expresses those patterns. Recently, learning relation patterns including **symmetry/antisymmetry**, **inversion** and **composition** have been realized to the key of link prediction task. Our model DualQuatE can easily explain the relation patterns of the learned relation embeddings and proof of relation patterns can be found in the Appendix A.  
**Inversion**: If a relation \(r^{\prime}\in\mathcal{R}\) is the inverse to a relation \(r\in\mathcal{R}\), then we can infer \((h,r,t)\in\Omega^{+}\Leftrightarrow(t,r^{\prime},h)\in\Omega^{+}\). For example, the relation _has_part_ is inverse to the relation \(part\_of\). To \(r\) and \(r^{\prime}\), we infer that \((\mathbf{m^{\prime}}\mathbf{m})\mathbf{h}(\mathbf{m^{\prime}}\mathbf{m})^{*}+ \mathbf{m^{\prime}}\mathbf{nnn^{\prime}}+\mathbf{n^{\prime}}=\mathbf{h}\), which denotes the composition of component \(\mathbf{m}\) and \(\mathbf{m^{\prime}}\) have no rotation (i.e., \((\mathbf{m^{\prime}}\mathbf{m})\mathbf{h}(\mathbf{m^{\prime}}\mathbf{m})^{*}= \mathbf{h}\)) and the translation \(\mathbf{n}\) which is rotated by \(\mathbf{m^{\prime}}\) is the opposite number of the translation \(\mathbf{n^{\prime}}\) (i.e., \(\mathbf{m^{\prime}}\mathbf{nnn^{\prime}}+\mathbf{n^{\prime}}=\mathbf{0}\)).  
**Symmetry**: A relation \(r\in\mathcal{R}\) is symmetric, if \((h,r,t)\in\Omega^{+}\Leftrightarrow(t,r,h)\in\Omega^{+}\) holds. For instance, relations _similar_to_ and _verb_group_ from the dataset WN18 are symmetric. If a relation is symmetric, we reason that \((\mathbf{mm})\mathbf{h}(\mathbf{mm})^{*}+\mathbf{mm}^{*}+\mathbf{n}=\mathbf{h}\), which means no rotation of the self-composition of component \(\mathbf{m}\) (i.e., \((\mathbf{mm})\mathbf{h}(\mathbf{mm})^{*}=\mathbf{h}\)) and no translation of component \(\mathbf{n}\) (i.e., \(\mathbf{mm}\mathbf{n^{\prime}}+\mathbf{n}=\mathbf{0}\)).  
**Antisymmetry**: A relation \(r\in\mathcal{R}\) is antisymmetric, if \((h,r,t)\in\Omega^{+}\Rightarrow(h,r,t)\in\Omega^{-}\), which satisfies \((\mathbf{mm})\mathbf{h}(\mathbf{mm})^{*}+\mathbf{mm}^{*}+\mathbf{n}\
eq\mathbf{ h}\). For example, the relation \(part\_of\).  
**Composition**: A relation \(r_{3}\) is composed by the relation \(r_{1}\) and \(r_{2}\), which can be denoted by \(r_{3}=r_{1}\oplus r_{2}\) if \((h,r,s)\in\Omega^{+}\wedge(s,r,t)\Rightarrow(t,r,h)\in\Omega^{+}\). For example, relation uncle_of can be composied by brother_of and father_of such as if \(([PERSON], brother\_of, [PERSON])\), \(([PERSON]\_of,Abel)\) are true triples, we can reason \(([PERSON]\_of,Abel)\) is a true fact in the real world. Relation \(r_{3}\) can be composied by relation \(r_{1}\) and \(r_{2}\); they can be represented by \((\mathbf{m}_{2}\mathbf{m}_{1})\mathbf{h}(\mathbf{m}_{2}\mathbf{m}_{1})^{*}+ \mathbf{m}_{2}\mathbf{n}_{1}\mathbf{m}_{2}^{*}+\mathbf{n}_{2}=\mathbf{m}_{3} \mathbf{h}\mathbf{m}_{3}^{*}+\mathbf{n}_{3}\), which deduces that \(\mathbf{n}_{3}\) is equal to the sum of translation \(\mathbf{n}_{2}\) and translation \(\mathbf{n}_{1}\) which is rotated by the rotation \(\mathbf{m}_{2}\) (i.e., \(\mathbf{m}_{2}\mathbf{n}_{1}\mathbf{m}_{2}^{*}+\mathbf{n}_{2}=\mathbf{n}_{3}\)).