# Dual Quaternion Embeddings for Link Prediction
## 1 Introduction  
Knowledge graphs, which represent knowledge from real world applications, contain abundant facts. In knowledge graphs, each fact is represented by a triple \((h,r,t)\) which indicates that the relation \(r\) between the head entity \(h\) and tail entity \(t\). Knowledge graphs have been applied to various tasks such as explainable recommendation system [1], question answering [2] and prediction of future research collaborations [3]. Predicting missing facts (i.e., link prediction) is a fundamental task in knowledge graph research. Various models aiming at embedding entities and relations into low-dimension spaces have been proposed. For example, TransE [4] learned the embeddings of entities and relations by transforming head entity to tail entity according to the relation; RotatE [5] and QuatE [6] learned the embeddings of entities and relations by considering relations as rotations from head entities to tail entities. However, existing transformation based models fail to capture multiple relations between head and tail entities. For example, as shown in Figure 1, _[PERSON]_ is the director, the creator and an actor in the film _Mulohland Drive_, i.e., there are three relations: directed, created and actedin between _[PERSON]_ and _Mulohland Drive_. These relations between head entity _[PERSON]_ and tail entity _[PERSON]_ have no semantic connections with each other, which should be represented by spatially dispersed embeddings. Most existing transformation based models, however, assume that there is only one relation between each pair of head and tail entities. For instance, for each triple \((h,r,t)\), their corresponding embeddings are assumed to be satisfied with \(h+r\approx t\) in TransE, which indicates, for \((h,r_{1},t)\), \((h,r_{2},t)\), and \((h,r_{3},t)\), the embeddings of \(r_{1}\), \(r_{2}\), \(r_{3}\) are similar, as shown in Figure 1(c) (i.e., \(\mathbf{r_{1}}\approx\mathbf{r_{2}}\approx\mathbf{r_{3}}\)). To overcome this challenge, we propose a novel approach that considers multiple relations between head and tail entities in knowledge graph. In this paper, we propose a model called DualQuatE which utilizes the various combinations of distinct rotations and translations to represent multiple relations between head and tail entities. Based on this, easily to think of RotatE combined with TransE in complex space and real space. However, it is hard to find a uniform mathematical expression to convey their combination. Therefore, we propose DualQuatE which embeds entities and relations into dual quaternion space to combine rotation and translation. The dual quaternion consists of real part and dual part. More concretely, we embed entities with pure quaternions vectors in three-dimensional space to represent entity embeddings. To distinguish various relations between head entity \(h\) and tail entity \(t\), we design a score function to utilize dual quaternion Hamilton product to model relations as interaction of rotation and translation. We utilize distinct interactions of rotations and translations to represent various relations between head and tail entities. Compared with RotatE and TransE in two-dimensional space, the dual quaternions space is eight-dimensional with six real degrees of freedom, three for translation and three for rotation; we can explore the interaction of rotation and translation with more free degrees in higher dimensions. Summarized in Table 1, our model has rich expression abilities of relations (i.e., relation patterns and multiple relations). To conclude, the contributions of our proposed model are listed as follows:  
* We introduce dual quaternions to knowledge graph embeddings. * We propose a novel transformation based model DualQuatE to overcome the challenge of multiple relations between two entities. * Our experiments denote that DualQuatE is effective compared to the existing state-of-the-art models. Figure 1: Visualization of partial knowledge graph in YAGO3-10. The blue relations indicate that there can be multiple relations between two entities.