# CDSTAEP: Cross-Domain Spatial-Temporal Association Learning for Abnormal Events Prediction
## 4 Methodology
### Graph Representation Learning in Mixed-Curvature Spaces
#### 4.2.1 Metric and Map Representations in Curvature Spaces  
Theoretical foundations for graph representation learning in mixed-curvature spaces include:  
(1) Euclidean Space  
Euclidean space \(\mathbb{E}^{n}\) has curvature \(\kappa=0\), and the space is uniform and flat everywhere, which is the result of the extension of geometric space measurement in linear space. It has space vector isotropy and translation invariance and is suitable for modeling grid data such as mesh data. Euclidean distance means the real distance between two points in n-dimensional space. In Riemann space, it is the geodesic length. The induced distance is as follows:  
\[d_{\mathbb{E}}(x,y)=\left|\left|x-y\right|\right|=\sqrt{\sum\limits_{i}(x_{i}- y_{i})^{2}} \tag{7}\]  
The exponential map \(exp_{x}(v):Tx\mathcal{M}\rightarrow\mathcal{M}\) of the Euclidean manifold is as follows:  
\[exp_{x}(v)=x+v \tag{8}\]  
where \(v\) is the tangent space vector and \(x\) is the point in manifold space \(\mathcal{M}\). The logarithmic map \(log_{x}(y):\mathcal{M}\to Tx\mathcal{M}\) of the Euclidean manifold is as follows:  
\[\log_{x}(y)=y-x \tag{9}\]  
where \(y\) is the point in manifold space \(\mathcal{M}\). (2) Hyperbolic Space  
The curvature \(\kappa<0\) of the hyperbolic space \(\mathbb{H}_{k}^{n}\rightarrow\mathbb{R}^{n}\). Since hyperbolic space has exponential volume capacity, the distance metric of hyperbolic space is equivalent to power-law distribution, and the tree structure can be embedded losslessly, it is very suitable for modeling ultra-large-scale hierarchical data and scale-free networks [11; 15; 35]. The existing Euclidean space models can be transformed into representation models in hyperbolic space through basic operators. The essence of hyperbolic space is a Riemannian manifold, which has the characteristics of local Euclidean geometry and can be expressed by various models, the most common of which is the Poincare disc model. In the Riemannian space, the Poincare model \((\mathbb{P}^{n},g)\) can be defined by the Riemannian manifold \(\mathcal{M}=\left\{x\in\mathbb{R}^{n}:\left|\left|x\right|\right|=\sqrt{\left\{ x,x\right\}}<1\right\}\)[25], and the Riemannian metric is as follows:  
\[g_{x}=\lambda_{x}^{2}g^{E},\textit{where}\,\lambda_{x}^{2}:=\frac{2}{1-||x||^ {2}} \tag{10}\]  
where \(g^{E}=\mathbf{I}_{n}\) is the Euclidean metric tensor and \(\mathbf{I}_{n}\) is the identity matrix. The hyperbolic metric tensor and the Euclidean metric tensor are conformal when the Poincare sphere model represents the hyperbolic space embedding. The norm-induced distance in the hyperbolic space \(x,y\in\mathbb{P}^{n}\) is as follows:  
\[d_{\mathbb{H}}(x,y)=\cosh^{-1}\biggl{(}1+2\frac{||x-y||^{2}}{(1-||x||^{2})(1- ||y||^{2})}\biggr{)} \tag{11}\]For each tangent space \(Tx\mathcal{M}\), if \(x\in\mathcal{M}\), there is a specific one-to-one exponential mapping \(exp_{x}(v):Tx\mathcal{M}\rightarrow\mathcal{M}\) to map the space vector \(v\in Tx\mathcal{M}\) to the manifold \(\mathcal{M}\). The exponential mapping function is as follows:  
\[exp_{x}^{k}(v)=x\bigoplus_{k}\left(\tanh\left(\sqrt{|k|}\ \frac{\lambda_{x}^{k}|v| |}{2}\right)\frac{v}{\sqrt{|k|}||v||}\ \right) \tag{12}\]  
where the curvature \(k<0\), \(\bigoplus\) means the addition of space vectors, and any point \(x\in\mathbb{H}^{n}\), \(v\
eq 0\), \(||\cdot||\) is the Euclidean norm. Logarithmic mapping is the inverse operation of exponential mapping, which can map points in manifold space to tangent space, realize data dimensionality reduction, and simplify calculations. That is, at point \(x\), \(log_{x}(y):\mathcal{M}\to Tx\mathcal{M}\), map the vector \(y\) to tangent space \(Tx\mathcal{M}\). The logarithmic mapping function is as follows:  
\[log_{x}^{k}(y)=\frac{2}{\sqrt{|k|}\lambda_{x}^{k}}\tanh^{-1}\left(\sqrt{|k||} |-x\bigoplus_{k}y||\right)\frac{-x\bigoplus_{k}y}{||-x\bigoplus_{k}y||} \tag{13}\]  
where \(k\) is the curvature in the Riemann space, any point \(x\in\mathbb{H}^{n}\), \(y\
eq x\).