# Leveraging Multi-Modal Information for Cross-Lingual Entity Matching across Knowledge Graphs
## 3 Method
### Multi-View Entity Matching
#### 3.4.1 Multi-View Intact Space Learning  
The information from different views can provide different aspects of one entity. For example, the attribute view can provide abstract attribute information of entities themselves, and the relation view can provide the relational structures among different entities. However, each view only captures the entity information from one aspect, and all views may share the common redundant information. We adopt the multi-view intact space learning (MISL) [14] to integrate multi-view entity representations to solve the insufficiency problem when using each individual view.  
We assume a multi-view training data set \(D=\{\mathbf{z}_{i}^{v}|1\leq i\leq n,1\leq v\leq m\}\), where \(\mathbf{z}_{i}^{v}\) denotes the \(v\)-th view of the \(i\)-th entity embedding, \(n\) is the number of the entity, and \(m\) is the number of the entity embedding views. Suppose that \(\mathbf{x}_{i}\) represents the embedding of an entity in the intact space, so the loss function of the \(m\)-view space being reconstructed by the intact space can be defined as follows:  
\[Loss_{MISL}=\frac{1}{mn}\sum_{i=1}^{n}\sum_{v=1}^{m}log(1+\frac{\left|\left| \mathbf{z}_{i}^{v}-\mathbf{W}_{v}\mathbf{x}_{i}\right|\right|^{2}}{c^{2}})+C _{1}\sum_{v=1}^{m}\left|\left|\mathbf{W}_{v}\right|\right|_{F}^{2}+C_{2}\sum _{i=1}^{n}\left|\left|\mathbf{x}_{i}\right|\right|_{2}^{2} \tag{14}\]  
where \(c\) is a constant scale parameter, \(C_{1}\) and \(C_{2}\) are non-negative constants, and \(\mathbf{W}_{v}\) is the \(v\)-th view generation matrix. The optimization problem can be decomposed into two sub-problems over the view generation function \(\mathbf{W}\) and the variable on entity embedding \(\mathbf{x}\) with the alternating optimization method. Given fixed view generation functions \(\{\mathbf{W}_{v}\}_{v=1}^{m}\), the loss function can be reduced as follows:  
\[min_{x}(Loss_{MISL})=\frac{1}{m}\sum_{v=1}^{m}log(1+\frac{\left|\left|\mathbf{ z}^{v}-\mathbf{W}_{v}\mathbf{x}\right|\right|^{2}}{c^{2}})+C_{2}||\mathbf{x}||_{2}^{2} \tag{15}\]  
Setting the gradient of \(Loss_{MISL}\) with respect to \(\mathbf{x}\) to \(0\), we can obtain the following equations:  
\[Q^{v}=\frac{1}{\left|\left|\mathbf{z}^{v}-\mathbf{W}_{v}\mathbf{x}\right| \right|^{2}+c^{2}} \tag{16}\]  
\[\mathbf{x}=(\sum_{v=1}^{m}\mathbf{W}_{v}^{T}Q^{v}\mathbf{W}_{v}+mC_{2})^{-1} \sum_{v=1}^{m}\mathbf{W}_{v}^{T}Q^{v}\mathbf{z}^{v} \tag{17}\]  
Given fixed entity embeddings \(\{\mathbf{x}_{i}\}_{i=1}^{n}\), the loss function can be reduced as follows:  
\[min_{W}(Loss_{MISL})=\frac{1}{n}\sum_{i=1}^{n}log(1+\frac{\left|\left|\mathbf{ z}_{i}-\mathbf{W}\mathbf{x}_{i}\right|\right|^{2}}{c^{2}})+C_{1}||\mathbf{W}||_{2}^{2} \tag{18}\]  
We can set the gradient of \(Loss_{MISL}\) with respect to \(\mathbf{W}\) to \(0\) and obtain the following equations:  
\[Q_{i}=\frac{1}{\left|\left|\mathbf{z}_{i}-\mathbf{W}\mathbf{x}_{i}\right| \right|^{2}+c^{2}} \tag{19}\]  
\[\mathbf{W}=\sum_{i=1}^{n}\mathbf{z}_{i}Q_{i}\mathbf{x}_{i}^{T}(\sum_{i=1}^{n} \mathbf{z}_{i}Q_{i}\mathbf{x}_{i}^{T}+nC_{1})^{-1} \tag{20}\]  
By constantly iterating the above formulas until the entity embedding \(\mathbf{x}\) converges, we can obtain the final integrated embedding for each entity (see Algorithm 1).