# Translation-Based Embeddings with Octonion for Knowledge Graph Completion
## 3 Methodology
### Preliminaries
#### 3.1.2 Hyperbolic Geometry of Poincare Model
Figure 6 visualizes the mapping operations, and the detailed definitions of all operations are as follows:  
Mobius addition: There is no good definition of the native addition operation in the Poincare model. If two points in the Poincare model are added directly, the result will easily be out of the scope of space. Mobius addition provides an addition operation close  
Figure 6: An illustration of exponential mapping and logarithmic mapping on hyperbolic Riemannian manifold. \(M\) is a hyperbolic manifold and \(\mathcal{T}_{\mathbf{x}}M\) is the tangent space of \(M\) at point \(\mathbf{x}\). Point \(\mathbf{y}\) can be transformed into vector \(\mathbf{v}\) on tangent space \(\mathcal{T}_{\mathbf{x}}M\) by logarithmic mapping \(\log_{\mathbf{x}}(\mathbf{y})\), and vector \(\mathbf{v}\) can be transformed into point \(\mathbf{y}\) on manifold \(M\) by exponential mapping \(\exp_{\mathbf{x}}(\mathbf{v})\). Exponential mapping and logarithmic mapping are a pair of operations. to the native representation in the Poincare model, and ensures that the operation result is still within the scope:  
\[\mathbf{x}\oplus_{c}\mathbf{y}=\frac{\Big{(}1+2c\langle\mathbf{x},\mathbf{y} \rangle+c\|\mathbf{y}\|^{2}\Big{)}\mathbf{x}+\Big{(}1-c\|\mathbf{x}\|^{2}\Big{)} \mathbf{y}}{1+2c\langle\mathbf{x},\mathbf{y}\rangle+c^{2}\|\mathbf{x}\|^{2}\| \mathbf{y}\|^{2}} \tag{3}\]  
where \(\mathbf{x},\mathbf{y}\in\mathbb{B}^{d,c}\), \(c>0\) denotes the absolute value of negative curvature. Exponential mapping: \(\exp_{\mathbf{x}}^{c}(\mathbf{v})\) constructs a tangent space \(\mathcal{T}_{\mathbf{x}}^{c}\) of the Poincare ball \(\mathbb{B}^{d,c}\) on the point \(\mathbf{x}\), and then maps the vector \(\mathbf{v}\) in the tangent space back to the Poincare ball:  
\[\exp_{\mathbf{x}}^{c}(\mathbf{v})=\mathbf{x}\oplus_{c}\Big{(}\mathrm{tanh} \bigg{(}\sqrt{c}\frac{\lambda_{\mathbf{x}}^{c}\|\mathbf{v}\|}{2}\bigg{)}\frac {\mathbf{v}}{\sqrt{c}\|\mathbf{v}\|}\bigg{)} \tag{4}\]  
where \(\mathbf{x}\in\mathbb{B}^{d,c}\), \(\mathbf{v}\in\mathcal{T}_{\mathbf{x}}^{c}\), \(c>0\) denotes the absolute value of negative curvature, and \(\lambda_{\mathbf{x}}^{c}=2/\left(1-c\|\mathbf{x}\|^{2}\right)\) denotes the conformal factor. Logarithmic mapping: The processes of exponential mapping and logarithmic mapping are opposite. Logarithmic mapping \(\log_{\mathbf{x}}^{c}(\mathbf{y})\) maps a point \(\mathbf{y}\) from \(\mathbb{B}^{d,c}\) to \(\mathcal{T}_{\mathbf{x}}^{c}\):  
\[\log_{\mathbf{x}}^{c}(\mathbf{y})=\frac{2}{\sqrt{c}\lambda_{\mathbf{x}}^{c}} \mathrm{arctanh}\big{(}\sqrt{c}\|-\mathbf{x}\oplus_{c}\mathbf{y}\|\big{)} \frac{-\mathbf{x}\oplus_{c}\mathbf{y}}{\|-\mathbf{x}\oplus_{c}\mathbf{y}\|} \tag{5}\]  
where \(\mathbf{x},\mathbf{y}\in\mathbb{B}^{d,c}\), \(c>0\) denotes the absolute value of negative curvature, and \(\lambda_{\mathbf{x}}^{c}=2/\left(1-c\|\mathbf{x}\|^{2}\right)\) denotes the conformal factor. Simplified representation of mappings: Since exponential mapping and logarithmic mapping usually construct tangent space \(\mathcal{T}_{\mathbf{0}}^{c}\) at the origin \(\mathbf{0}\), these two mappings can be simplified in the form of \(\exp_{\mathbf{0}}^{c}(\mathbf{v})\) and \(\log_{\mathbf{0}}^{c}(\mathbf{y})\):  
\[\exp_{\mathbf{0}}^{c}(\mathbf{v})=\mathrm{tanh}\big{(}\sqrt{c}\|\mathbf{v}\| \big{)}\frac{\mathbf{v}}{\sqrt{c}\|\mathbf{v}\|} \tag{6}\]  
\[\log_{\mathbf{0}}^{c}(\mathbf{y})=\mathrm{arctanh}\big{(}\sqrt{c}\|\mathbf{y} \|\big{)}\frac{\mathbf{y}}{\sqrt{c}\|\mathbf{y}\|} \tag{7}\]  
Hyperbolic distance calculation: By using Mobius addition, the hyperbolic distance of two points \(\mathbf{x}\) and \(\mathbf{y}\) in Poincare ball \(\mathbb{B}^{d,c}\) can be defined:  
\[d_{\mathbb{B}}^{c}(\mathbf{x},\mathbf{y})=\frac{2}{\sqrt{c}}\mathrm{arctanh} \big{(}\sqrt{c}\|-\mathbf{x}\oplus_{c}\mathbf{y}\|\big{)} \tag{8}\]