# Enhanced Knowledge Graph Embedding by Jointly Learning Soft Rules and Facts
## 5 Experiment
### Results
#### 5.3.3 Comparison of Forward Chaining and One Time Inference  
We further investigated the effectiveness of forward chaining by comparing it with one time inference. Since a small amount of rules was extracted from the dataset DB100K and the groundings generated by forward chaining and one time inference were the same, we only conducted the comparison experiment on FB15K. Table 4 shows the results achieved by SoLE with one time inference and forward chaining. From the results, we can observe that forward chaining slightly, but not statistically significantly outperformed one time inference.  
In order to figure out why the improvement was not obvious, we analyzed the different groundings generated by forward chaining and one time inference. We mainly explored the conclusions of the groundings that indicated the new derived triples. Figure 4 depicts the distributions of entities and relations in the conclusions. The X-axis denotes the entity/relation ID used in our code representing entities/relations; the Y-axis denotes the number of the entities/relations involved in the conclusions of the groundings generated by one time inference; and the Z-axis denotes the difference between the number of the entities/relations from forward chaining and the ones from one time inference. In the left picture of Figure 4, four entities, each one had a large number in the groundings generated by one time inference, accounting for 32% of the total difference, and 92% of 13,537 entities in the conclusions had the difference value less than one. Meanwhile, in the right picture, 14 out of 299 relations in the conclusions accounted for 99.99% of the total differences when they already had a large number in the conclusions w.r.t. one time inference.  
These data indicated that most of the entities and relations in the more derived triples from forward chaining had already shown up quite a few times in the conclusions of the groundings from one time inference where they may have contributed to learning good embeddings, while the embeddings of other entities and relations may not be improved too much because of their same low frequencies.  
Figure 4: The distributions of entities (**left**) and relations (**right**) in the conclusions of the groundings generated by forward chaining and one time inference.  
Figure 3: Results of MRR (**left**) and HITS@1 (**right**) achieved by SoLE with different confidence thresholds on the test set of FB15K.  
Though the improvement was not significant, forward chaining indeed helped learn better embeddings. We took out 14 relations from the groundings, which accounted for the mostly differences, and obtained the link prediction results on the test sets of FB15K only containing these relations. Table 5 gives these 14 relations, and Table 6 shows the results of link prediction. The results showed that forward chaining helped SoLE learn better embeddings for these relations even though many of them had shown up more than 3000 times in the groundings of SoLE_OTI. Furthermore, we chose the other three relations and constructed a query for each relation where its missing entity appeared more times in the groundings of SoLE than SoLE_OTI. Table 7 shows these queries and the results after performing the queries. From the top five entities and their scores in the results, we can observe that SoLE not only obtained the right entity as the first rank, but it also had a higher score than SoLE_OTI. The above analysis demonstrated the superiority of forward chaining to one time inference.