# Entity Alignment Method Based on Joint Learning of Entity and Attribute Representations
## 3 Results  
### Evaluation Indicators and Experimental Setting  
Hits@k, mean rank (\(MR\)), and mean reciprocal rank (\(MRR\)) were used to evaluate the link prediction performance while comparing the methods.  
Hits@k is the ratio of correctly aligned entities ranked in the top \(k\). The larger the Hits@k value, the better is the performance of the model.  
\[Hits@k=\frac{1}{|S|}\sum\limits_{i=1}^{|S|}\prod(rank_{i}\leq k) \tag{25}\]  
where \(S\) is a set of triples, \(|S|\) is the number of triples, \(rank_{i}\) is the link prediction ranking of the \(i\)-th triple, and \(\prod(\cdot)\) is the indicator function, where \(1\) denotes that the condition is true and \(0\) denotes otherwise.  
\(MR\) is the mean rank of correctly aligned entities; smaller values are indicative of better performance.  
\[MR=\frac{1}{|S|}\sum\limits_{i=1}^{|S|}rank_{i}=\frac{1}{|S|}\Bigl{(}rank_{1} +rank_{2}+\cdots rank_{|S|}\Bigr{)} \tag{26}\]  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \multicolumn{2}{c}{**Dataset**} & \multicolumn{2}{c}{**Entities**} & **Relations** & **Attributes** & **Relation Triples** & **Attribute Triples** \\ \hline \multirow{8}{*}{DBP15K} & ZH-EN & Chinese & 66,469 & 2830 & 8113 & 153,929 & 379,684 \\  & English & 98,125 & 2317 & 7173 & 237,674 & 567,755 \\  & JA-EN & Japanese & 65,744 & 2043 & 5882 & 164,373 & 354,619 \\  & English & 95,680 & 2096 & 6066 & 233,319 & 497,230 \\  & French & 66,858 & 1379 & 4547 & 192,191 & 528,665 \\  & English & 105,889 & 2209 & 6422 & 278,590 & 576,543 \\  & DBP-WD & DBPedia & 100,000 & 330 & 351 & 463,294 & 381,166 \\  & Wikidata & 100,000 & 220 & 729 & 448,774 & 789,815 \\  & DBP-YG & DBpedia & 100,000 & 302 & 334 & 428,952 & 451,646 \\  & Yago3 & 100,000 & 31 & 23 & 502,563 & 118,376 \\ \hline \end{tabular}
\end{table}
Table 1: DBP15K and DWY100K datasets.  
\(MRR\) is the mean of the reciprocal rank of the correctly aligned entities; larger values indicate better performance.  
\[MRR=\frac{1}{|S|}\sum_{i=1}^{|S|}\frac{1}{rank_{i}}=\frac{1}{|S|}\Bigg{(}\frac{1} {rank_{1}}+\frac{1}{rank_{2}}+\cdots\frac{1}{rank_{|S|}}\Bigg{)} \tag{27}\]  
The metrics used to evaluate entity alignment were precision, recall, and F1 measure. Precision is the ratio of correctly aligned entities to all aligned entities, i.e., the accuracy of entity alignment, expressed as:  
\[Precision=\frac{T_{N}}{T_{N}+F_{N}} \tag{28}\]  
where \(T_{N}\) is the number of correctly aligned entities in the experimental results and \(F_{N}\) is the number of incorrectly aligned entities. Higher precision values indicate better performance. Recall is the ratio of correctly aligned entities to all aligned entities in the dataset, given as:  
\[Recall=\frac{T_{N}}{A_{N}} \tag{29}\]  
where \(A_{N}\) is the total number of aligned entity pairs in the dataset. Larger recall values indicate better performance. F1 measure is a combined measure of precision and recall, as shown in Equation (30):  
\[F1=\frac{2\times Recall\times Precision}{Recall+Precision} \tag{30}\]  
In the proposed entity alignment model, optimization was performed using stochastic gradient descent (SGD). The model parameters were configured as follows. Margin hyper-parameters: \(\gamma_{1}\) = 0.01 and \(\gamma_{2}\) = 2; balancing factor: \(\mu\) = 0.2; learning rate = 0.001; number of iterations: [500, 1500, 2500, 3500, 4500]; \(\theta\) = 1.0; search step length: \(k\) = 0.2; dimensionality of the entity, relation, attribute, and attribute value embeddings = 100.