# Knowledge Graph Completion Based on Entity Descriptions in Hyperbolic Space
## 3 Knowledge Representation Learning Based on Hyperbolic Space
### DMuRP Method
 \tag{8}\]  
Among them, \(\mathbf{h}_{\mathbf{ss}}^{H}\) and \(\mathbf{t}_{\mathbf{s}}^{H}\) are the hyperbolic representation of the head and tail entities; \(\mathbf{h}_{\mathbf{d}}^{E}\), \(\mathbf{t}_{\mathbf{d}}^{E}\) is description-based representations of the entity in Euclidean space; \(\mathbf{R}\) is a diagonal relation matrix, which is combined with Mobius multiplication to adjust the head entity embeddings, whose value is determined by the specific triple relation; \(\mathbf{r}^{H}\) is the hyperbolic representation of the relation. In addition, \(\alpha_{1}\), \(\alpha_{2}\), \(\alpha_{3}\), and \(\alpha_{4}\) are constants, which are balance factors that control each energy function, and their values are determined by the performance of the energy functions \(E_{1}\), \(E_{2}\), \(E_{3}\), and \(E_{4}\). As shown in Figure 4, the balance of the triple embedding (\(\mathbf{h}_{\mathbf{s}}^{H}\), \(\mathbf{r}^{H}\), \(\mathbf{t}_{\mathbf{s}}^{H}\)) itself is adjusted by the balance factor \(\alpha_{1}\). For unbalanced energy function, for example, when we use the head entity description-based representations \(\mathbf{h}_{\mathbf{d}}^{H}\) to replace the head entity structure-based representations \(\mathbf{h}_{\mathbf{s}}^{H}\), \(\mathbf{h}_{\mathbf{d}}^{H}+\mathbf{r}^{H}\), and \(\mathbf{t}_{\mathbf{s}}^{H}\) are obtained. Because the structured representations of the entity and the described representations of the entity are different, \(\mathbf{h}_{\mathbf{d}}^{H}+\mathbf{r}^{H}\
eq\mathbf{t}_{\mathbf{s}}^{H}\) will be caused, resulting in an unbalanced energy function \(E_{2}\). It is adjusted by the balance factor \(\alpha_{2}\). Similarly, when the tail entity description-based representations replace the tail entity structure-based representations, the unbalanced energy function \(E_{3}\) is adjusted by the balance factor \(\alpha_{3}\). The head entity based on the described representations and the tail entity based on the described representations replace the head entity based on the structured representations and the tail entity based on the structured representations at the same time, and the unbalanced energy function \(E_{4}\) is generated, and it is adjusted by the balance factor \(\alpha_{4}\). The influence of each energy function is balanced by introducing balance factors. According to the above energy functions, the total score function of the DMuRP method is defined as \(\varphi\), Equation (11). Figure 4: Diagram of the energy function. \[\varphi=E(h,r,t)+\mathbf{b_{h}}+\mathbf{b_{t}} \tag{11}\]  
where \(\mathbf{b_{h}}\) and \(\mathbf{b_{t}}\) are head entity bias and tail entity bias [18]. In order to explore the influence of \(E_{1}\), \(E_{2}\), \(E_{3}\), and \(E_{4}\) on the energy function E, the score functions \(\Phi_{1}\), \(\Phi_{2}\), \(\Phi_{3}\), and \(\Phi_{4}\) are defined. The DMuRPss without containing entity descriptions is used as the baseline, and the DMuRPss score function is defined as \(\Phi_{1}\), Equation (12). It is worth noting that \(\Phi_{1}\) is also the definition of the MuRP [18] score function. \[\Phi_{1}=E_{1}(\mathbf{h_{ss}^{H}},\mathbf{r^{H}},\mathbf{t_{s}^{H}})+\mathbf{ b_{h}}+\mathbf{b_{t}} \tag{12}\]  
In the same way, in hyperbolic space, we define the score functions containing entity descriptions (DMuRPs, DMuRPsd, DMuRPdd) as \(\Phi_{2}\), \(\Phi_{3}\), and \(\Phi_{4}\), they are shown as Equations (13)-(15). \[\Phi_{2} =E_{2}(\mathbf{h_{d}^{E}},\mathbf{r^{H}},\mathbf{t_{s}^{H}})+ \mathbf{b_{h}}+\mathbf{b_{t}} \tag{13}\] \[\Phi_{3} =E_{3}(\mathbf{h_{ss}^{H}},\mathbf{r^{H}},\mathbf{t_{s}^{E}})+ \mathbf{b_{h}}+\mathbf{b_{t}}\] (14) \[\Phi_{4} =E_{4}(\mathbf{h_{d}^{E}},\mathbf{r^{H}},\mathbf{t_{d}^{E}})+ \mathbf{b_{h}}+\mathbf{b_{t}} \tag{15}\]  
The loss function is a negative log-likelihood loss, and it is defined as \(\mathcal{L}\), Equation (16).