# Cosine-Based Embedding for Completing Lightweight Schematic Knowledge in DL-Lite\({}_{core}\)+
## 5 Experiments and Results
### The Results of Link Prediction  
Link prediction is a typical task for completing the axiom when one of the concepts or their relation is missing. We employ _MRR_ and _Hits@N_ as evaluation metrics proposed in TransE [13]. For each axiom \((C_{i},r,C_{j})\) in test datasets, we replace concept \(C_{i}\) or \(C_{j}\) with \(C_{n}\) in the set of concepts \(\mathcal{C}\) to generate _corrupted axioms_, and measure the confidences of these axioms using the score function. Then, the rank of correct concepts could be obtained by sorting the confidences of axioms in descending order. _MRR_ is a metric that calculates the mean reciprocal rank of all correct concepts. _Hits@N_ is a metric that counts the ratio of the correct concepts ranked in the _top N_. Notice that the corrupted axiom ranking above a test axiom is also valid, and should not be treated as a wrong axiom. Therefore, corrupted axioms asserted in schematic knowledge have been filtered before ranking. For convenience, we label the filtered result as \"Filter\", and the unfiltered one is denoted by \"Raw\". In our experiments of link prediction, all the models are required to infer the missing concept \(C_{i}\) or \(C_{j}\) for the axioms \((C_{i},r,C_{j})\) in the test datasets. For the metrics _MRR_ and _Hits@N_, a higher value of them indicates a better performance of the evaluated model. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{1}{c}{**Dataset**} & **YAGO-On [4]** & **FMA [43]** & **FoodOn [44]** & **HeLiS [45]** & **YAGO-On-t** & **YAGO-On-s** \\ \hline \multicolumn{1}{c}{\(\sharp\) Concept} & 46,109 & 78,988 & 28,182 & 17,550 & 46,109 & 46,109 \\ \hline \multirow{2}{*}{Train} & \(\sharp\)\(subclassOf\) & 29,181 & 29,181 & 20,844 & 14,222 & 11,898 & 0 \\  & \(\sharp\)\(disjointWith\) & 32,673 & 32,673 & 17,398 & 13,782 & 0 & 10,000 \\ \hline \multirow{2}{*}{Valid} & \(\sharp\)\(subclassOf\) & 1000 & 2000 & 1488 & 1015 & 1000 & 1000 \\  & \(\sharp\)\(disjointWith\) & 1000 & 2000 & 2714 & 1722 & 1000 & 1000 \\ \hline \multirow{2}{*}{Test} & \(\sharp\)\(subclassOf\) & 1000 & 2000 & 2978 & 2032 & 5949 & 0 \\  & \(\sharp\)\(disjointWith\) & 1000 & 1000 & 2174 & 1722 & 0 & 10,000 \\ \hline \hline \end{tabular} \(\sharp\): indicates number of concepts.  
\end{table} 
Table 1: The statistics of generated datasets for evaluation. Tables 2 and 3 list the results of link prediction on YAGO-On, FMA, FoodOn, and HeLaIS. Overall, CosE\({}^{-}\) and CosE have obviously surpassed other models in terms of _MRR_ and _Hits@N_. It indicates that both of them can preserve the logical properties (i.e., transitivity, symmetry) of relations via two designed semantic spaces, which can help our model to learn better vector representations of concepts for completing schematic knowledge. Compared with models employing the strategy of projection matrices (e.g., TransH, TransR and TransD), CosE is able to measure the confidences of related axioms more precisely. The possible reasons are that CosE attempts to project axioms with different relations into different translation-based semantic spaces, and the types of relations in schematic knowledge are relatively small. Hence, the projection strategy of CosE is more suitable than with other models. Furthermore, benefiting from our proposed strategy of negative sampling, the performances of CosE are slightly better than CosE\({}^{-}\) in terms of _MRR_ and _Hits@N_. We analyze that the mutual exclusion between \(subclassOf\) and \(disjointWith\) is useful in distinguishing the similar embedding of concepts in the semantic space. Tables 4 and 5 list the results of link prediction about the \(subclassOf\) axioms of four datasets. Overall, CosE\({}^{-}\) and CosE outperform most models in terms of _MRR_ and _Hits@N_.