# Cosine-Based Embedding for Completing Lightweight Schematic Knowledge in DL-Lite\({}_{core}\)+
## 5 Experiments and Results
### The Results of Link Prediction
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & \multicolumn{6}{c}{**YAGO-On**} & \multicolumn{6}{c}{**FMA**} \\ \hline \multirow{2}{*}{**Metric**} & \multicolumn{2}{c}{**MRR**} & \multicolumn{2}{c}{**Hits@N(\%)**} & \multicolumn{2}{c}{**MRR**} & \multicolumn{2}{c}{**Hits@N(\%)**} \\  & **Raw** & **Filter** & **10** & **3** & **1** & **Raw** & **Filter** & **10** & **3** & **1** \\ \hline TransE [13] & 0.241 \({}^{\dagger}\) & 0.501 \({}^{\dagger}\) & 0.784 \({}^{\dagger}\) & 0.582 \({}^{\dagger}\) & 0.343 \({}^{\dagger}\) & 0.066 \({}^{\dagger}\) & 0.325 \({}^{\dagger}\) & 0.474 \({}^{\dagger}\) & 0.371 \({}^{\dagger}\) & 0.247 \({}^{\dagger}\) \\ TransH [14] & 0.195 \({}^{\dagger}\) & 0.196 \({}^{\dagger}\) & 0.472 \({}^{\dagger}\) & 0.252 \({}^{\dagger}\) & 0.091 \({}^{\dagger}\) & 0.008 \({}^{\dagger}\) & 0.009 \({}^{\dagger}\) & 0.018 \({}^{\dagger}\) & 0.005 \({}^{\dagger}\) & 0.003 \({}^{\dagger}\) \\ TransR [15] & 0.090 \({}^{\dagger}\) & 0.428 \({}^{\dagger}\) & 0.588 \({}^{\dagger}\) & 0.433 \({}^{\dagger}\) & 0.355 \({}^{\dagger}\) & 0.060 \({}^{\dagger}\) & 0.411 \({}^{\dagger}\) & 0.490 \({}^{\dagger}\) & 0.440 \({}^{\dagger}\) & 0.370 \({}^{\dagger}\) \\ TransD [28] & 0.038 \({}^{\dagger}\) & 0.176 \({}^{\dagger}\) & 0.462 \({}^{\dagger}\) & 0.305 \({}^{\dagger}\) & 0.000 \({}^{\dagger}\) & 0.034 \({}^{\dagger}\) & 0.149 \({}^{\dagger}\) & 0.430 \({}^{\dagger}\) & 0.250 \({}^{\dagger}\) & 0.000 \({}^{\dagger}\) \\ RESCAL [16] & 0.080 \({}^{\dagger}\) & 0.339 \({}^{\dagger}\) & 0.525 \({}^{\dagger}\) & 0.392 \({}^{\dagger}\) & 0.244 \({}^{\dagger}\) & 0.047 \({}^{\dagger}\) & 0.317 \({}^{\dagger}\) & 0.469 \({}^{\dagger}\) & 0.377 \({}^{\dagger}\) & 0.236 \({}^{\dagger}\) \\ HolE [18] & 0.155 & 0.231 & 0.523 & 0.254 & 0.099 & 0.039 & 0.112 & 0.311 & 0.120 & 0.033 \\ ComplEx [19] & 0.034 \({}^{\dagger}\) & 0.237 \({}^{\dagger}\) & 0.491 \({}^{\dagger}\) & 0.403 \({}^{\dagger}\) & 0.058 \({}^{\dagger}\) & 0.033 \({}^{\dagger}\) & 0.201 \({}^{\dagger}\) & 0.484 \({}^{\dagger}\) & 0.372 \({}^{\dagger}\) & 0.011 \({}^{\dagger}\) \\ Analogy [47] & 0.037 \({}^{\dagger}\) & 0.301 \({}^{\dagger}\) & 0.496 \({}^{\dagger}\) & 0.429 \({}^{\dagger}\) & 0.160 \({}^{\dagger}\) & 0.037 \({}^{\dagger}\) & 0.277 \({}^{\dagger}\) & 0.487 \({}^{\dagger}\) & 0.415 \({}^{\dagger}\) & 0.130 \({}^{\dagger}\) \\ TransIt shows that the transitivity of \(subclassOf\) is preserved well in our defined semantic spaces. Relatively, the performances of some schematic knowledge embedding methods are not well. We analyze that most of their score functions focus on modeling \(subclassOf\) relations via spheres, but the external \(disjointWith\) axioms may influence the convergence of their score functions. Tables 6 and 7 list the results of the axioms with \(disjointWith\) relations. In terms of \(MRR\) and \(Hits@N\), \(\text{CoE}^{-}\) and \(\text{CoE}\) have outperformed all the models in most cases. For link prediction results on YAGO-On and FMA, the performances of \(\text{CoE}\) are a little worse than TransR and TransE in \(MRR\)_Raw_, but it obtains better results in terms of \(MRR\)_Filter_. Through further analysis, we discover that \(\text{CoE}\) tends to give a higher score to the correct corrupted axiom, so the value of \(MRR\)_Raw_ in \(\text{CoE}\) is much smaller than its \(MRR\)_Filter_ value. Overall, \(Hits@1\) of \(\text{CoE}\) has been improved from \(15\%\) to \(30\%\) in YAGO-On and FMA. It shows that \(\text{CoE}\) could preserve the symmetry of relations precisely in the angle-based semantic space. Nevertheless, the performances of all the KG embedding models on FoodOn and HeLiS are unsatisfactory. We observe that it may be related to the generation method [46] of \(disjointWith\) axioms. More importantly, it indicates that the current models still lack scalability for various large-scale ontologies to some extent, and we will leave these issues for future work.