# Dual Quaternion Embeddings for Link Prediction
## 2 Related Work  
To gain high-quality knowledge graphs, approaches which utilize knowledge graph embedding to predict missing facts have been proposed recently. These methods fall into two broad categories in [10]: transformation based models and semantic matching models. Specifically, transformation based models transform head entity to tail entity by relations, while semantic matching models match entities and relations semantics in latent spaces. Compared to transformation based models, semantic matching models suffer from poor interpretability. Transformation based models usually embed entities and relations into vector space and model the relation as a transformation from head entity embeddings to tail entity embeddings. One of the most representative is TransE which mapped entities and relations to the same space \(\mathbb{R}^{k}\). For each triple \((h,r,t)\), entity embeddings \(\mathbf{h}\), \(\mathbf{t}\) and relation embedding \(\mathbf{r}\) hold \(\mathbf{h}+\mathbf{r}\approx\mathbf{t}\). Then a series of extensions based on TransE are presented to improve accuracy and interpretability. For instance, TransR [11] introduced relations-specific spaces. TransR modeled relations and entities into different spaces following the idea that TransE can only express 1-to-1 relations. RotatE mapped embeddings into complex space which focused on expressing relation patterns. HAKE [7] utilized the polar coordinate system to capture semantic hierarchies in the knowledge graphs. Semantic matching models that match latent semantics of entities and relations can be divided into two categories: bilinear models and neural network based models. Bi-linear models include DistMult [8], HolE [12], SimpIE [9], ComplEx [13] and QuatE and DihEdral [14]. DistMult represented each entity as a vector and each relation as a diagonal matrix. HolE matched latent semantics of entities by circular correlation operation and then the compositional vector interacted with relations latter. ComplEx, mapping knowledge graph embedding into complex space, leveraged Hermitian product to capture latent semantics of entities and relations which could express antisymmetry relation pattern. QuatE, extending knowledge graph embedding from complex space to quaternion space, modeled each relation as rotation in four-dimensional space with more degree of freedom. Compared with ComplEx, QuatE could express the main relation patterns except composition. For each entity, SimpLE proposed two embeddings and each of them learned latent semantics dependently. DihEdral mapped relations into dihedral group to capture composition relations. Neural network based models including ConvE [15], R-GCNs [16] and InteractE [17] are proposed recently. ConvE, R-GCNs introduced convolutional network and graph convolutional networks to knowledge graph embedding respectively. Compared with ConvE, InteractE introduced the feature permutation, \"checkered\" feature reshaping and circular convolution to increase interaction. Recently, some models introduced hyperbolic space to knowledge graph embeddings. MuRP [18] represented knowledge graph in Poincare ball of hyperbolic space. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Model** & **Symmetry** & **Antisymmetry** & **Inversion** & **Composition** & **Multiple Relations** \\ \hline TransE & ✓ & ✓ & ✓ & ✓ & \(\times\) \\ RotatE & ✓ & ✓ & ✓ & ✓ & \(\times\) \\ HAKE [7] & ✓ & ✓ & ✓ & ✓ & \(\times\) \\ \hline DistMult [8] & ✓ & \(\times\) & \(\times\) & \(\times\) & ✓ \\ ComplEx [9] & ✓ & ✓ & ✓ & \(\times\) & ✓ \\ QuatE & ✓ & ✓ & ✓ & \(\times\) & ✓ \\ \hline DualQuatE & ✓ & ✓ & ✓ & ✓ & ✓ \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The ability of expressing relation patterns and multiple relations between head and tail entities. [PERSON] et al. [19] attempted to capture hierarchical and logical patterns in hyperbolic space.