# Knowledge Graph Completion Based on Entity Descriptions in Hyperbolic Space
## 3 Knowledge Representation Learning Based on Hyperbolic Space
### DMuRP Method  
**Step one:** Representation learning of entity descriptions (Text Representation). In hyperbolic space, the work of [PERSON] et al. [42] shows that the skip-gram algorithm for hyperbolic word embeddings has no obvious advantage compared with Euclidean counterpart. In addition, the research of [PERSON] et al. [43] finds that the effectiveness of text embeddings method in hyperbolic space depends on the specific downstream tasks, and this method did not achieve ideal results yet in word embeddings and sentence embeddings. Therefore, we choose to encode the description of entities in Euclidean space. For the text descriptions of entities, the description-based representations of entities is trained in Euclidean space through Siamese BERT networks. Different from others, we directly use sentence embedding models instead of word embedding models. For example, [PERSON] et al. [22] extract keywords in entity descriptions and use convolutional neural networks to embed these keywords, which will lose the semantic information of entities to some extent. Finally, we map the description-based representations of entities into hyperbolic space according to exponential function, and obtain the description-based representation of entities in hyperbolic space. Specifically, for processing the description of entities, we directly use the sentence embedding model SBERT [41]. The SBERT model uses the pre-trained BERT network, and adds a pooling operation to the output of BERT to generate a fixed-dimensional sentence embeddings. This paper directly takes the sentence representation after the pooling of the SBERT model. Our experimental results show that this is feasible and effective. **Step two:** Hyperbolic representation learning of triples (Structured Representation). For the structured representation of triples in hyperbolic space, we use the hyperbolic model to obtain the structured embeddings of triples (\(\mathbf{h}_{\mathbf{ss}}^{H}\), \(\mathbf{r}^{H}\), \(\mathbf{t}_{\mathbf{s}}^{H}\)). \(\mathbf{h}_{\mathbf{ss}}^{H}\) is introduced into DMuRP model after using Mobius multiplication. \(\mathbf{r}^{H}\) and \(\mathbf{t}_{\mathbf{s}}^{H}\) are translated directly into DMuRP model. The MuRP defines the calculation method of the triple distance function as \(d^{c}(\mathbf{R}\otimes_{c}\mathbf{h}_{\mathbf{ss}}^{H},\mathbf{t}_{\mathbf{s }}^{H}\oplus_{c}\mathbf{r}^{H})\). The research work of [PERSON] et al. [18] shows that the MuRP using this  
Figure 2: DMuRP overall frame. distance function is better than \(d^{c}(\mathbf{h}_{\mathbf{ss}}^{H},\mathbf{t}_{\mathbf{s}}^{H}\oplus_{c}\mathbf{r} ^{H})\), \(d^{c}(\mathbf{R}\otimes_{c}\mathbf{h}_{\mathbf{ss}}^{H},\mathbf{R}\otimes_{c} \mathbf{t}_{\mathbf{s}}^{H}\oplus_{c}\mathbf{r}^{H})\), This shows that the relation matrix is important for the head entity to obtain a better representation. Therefore, we use the calculation method of the triple distance function \(d^{c}(\mathbf{R}\otimes_{c}\mathbf{h}_{\mathbf{ss}}^{H},\mathbf{t}_{\mathbf{s} }^{H}\oplus_{c}\mathbf{r}^{H})\) to train triples and obtain the entity structure-based representations. **Step three:** Joint the entity text representation and structured representation. To make the learning process of description-based representations of entities to be compatible with structured representation of entities, we make use of the cross-fusion method in this paper. The head entity and the tail entity of the triple have two representations, respectively, as shown in Figure 2. Based on (\(\mathbf{h}_{\mathbf{s}}^{H}\), \(\mathbf{r}^{H}\), \(\mathbf{t}_{\mathbf{s}}^{H}\)), the head entity \(\mathbf{h}_{\mathbf{s}}^{H}\) is replaced by \(\mathbf{h}_{\mathbf{d}}^{H}\), and the triple embedding (\(\mathbf{h}_{\mathbf{d}}^{H}\), \(\mathbf{r}^{H}\), \(\mathbf{t}_{\mathbf{d}}^{H}\)) is obtained. Similarly, the tail entity \(\mathbf{t}_{\mathbf{s}}^{H}\) is replaced, we obtain the triple embedding (\(\mathbf{h}_{\mathbf{s}}^{H}\), \(\mathbf{r}^{H}\), \(\mathbf{t}_{\mathbf{d}}^{H}\)).