## 3 Materials and Methods
### Theoretical Proof of PDC  
We conducted a theoretical analysis to demonstrate the effectiveness of PDEC in the TransE model. Consider the toy KG \(K_{t}\subseteq\mathcal{E}_{t}\times\mathcal{R}_{t}\times\mathcal{E}_{t}\) represented in Figure 2. If we use TransE for learning entity embeddings, the optimization objective would be as follows:  
\[\arg\max_{\theta}\|x+r_{0}-a\|+\|x+r_{0}-b\|+\|x+r_{0}-c\|+\|x+r_{0}-d\| \tag{10}\]  
where \(x,a,b,c,d\in\mathcal{E}_{t}\) are entities in the KG, and \((x,r_{0},a),(x,r_{0},b),(x,r_{0},c),(x,r_{0},d)\in K_{t}\). The predicate \(r_{0}\) is represented by an orange line in Figure 2.  
It is easy to observe that after optimization, the representation of the predicate \(r_{0}\) approaches the vector from \(x\) to the Fermat point \(F\) of the quadrilateral formed by \(a,b,c\), and \(d\). The deviation \(dev_{0}\) is equivalent to the sum of distances from \(F\) to \(a,b,c\), and \(d\), that is, \(dev_{0}=|Fa|+|Fb|+|Fc|+|Fd|\).  
After PDEC processing, the edge vectors \(\vec{tr}_{0,x,a}\) and \(\vec{tr}_{0,x,b}\) form one cluster, while \(\vec{tr}_{0,x,d}\) and \(\vec{tr}_{0,x,d}\) form another cluster, thus decomposing the predicate \(r_{0}\) into \(r_{1}\) and \(r_{2}\). This means that the triples \((x,r_{1},a),(x,r_{1},b),(x,r_{2},c),(x,r_{2},d)\in K_{t}\). Consequently, the optimization objective transforms to the following:  
\[\arg\max_{b}\|x+r_{1}-a\|+\|x+r_{1}-b\|\|x+r_{2}-c\|+\|x+r_{2}-d\| \tag{11}\]  
where the final optimized predicate \(r_{1}\) is actually approaching the vector from \(x\) to the Fermat point \(F_{1}\) of the line formed by \(a,b\), and \(F_{1}\) must be on the line \(\vec{ab}\). Therefore, the loss is equivalent to the sum of the distances from \(F_{1}\) to \(a\) and \(b\), i.e., \(dev_{1}=|F_{1}a|+|F_{1}b|\). Similarly, the deviation of embedding learning for \(r_{2}\) is \(dev_{2}=|F_{2}c|+|F_{2}d|\), and \(F_{2}\) must be on the line \(\vec{cd}\). As shown in Figure 3, according to the geometric properties of triangles, it can be seen as follows:  
\[\begin{array}{l}dev_{1}+dev_{2}=|F_{1}a|+|F_{1}b|+|F_{2}c|+|F_{2}d|\\ <|Fa|+|Fb|+|Fc|+|Fd|=dev_{0}\end{array} \tag{12}\]  
The theoretical analysis of the toy KG reveals that in the case where the head entity of the triplet associated with the predicate \(r\) in KG is unique, TransE's optimization process aims to find the Fermat points \(F\) for all tail entities. The obtained embedding of \(r\) corresponds to the vector between the head entity and the Fermat point \(F\), with the error measured as the distance between \(F\) and all tail entities.  
Figure 3: Advantages of predicate decomposition. Based on the geometric characteristics of triangles, PDEC reduces the theoretical error of TransE algorithm optimization (represented by dashed lines).  
Figure 2: Schematic diagram of the predicate decomposition principle. The theoretical error of TransE algorithm optimization can be approximately estimated using the length of dashed lines.  
By clustering edge vectors, predicate decomposition transforms the task of finding Fermat points for all global tail entities into finding local Fermat points for several clusters. The error is now the sum of distances from the Fermat points of each cluster to their corresponding tail entities. This approach inevitably reduces the total error and improves TransE's performance.  
When there are multiple head entities for the predicate \(r\), the analysis can be applied to each individual head entity. In this case, the embedding of the original predicate obtained by TransE corresponds to the vector between Fermat points of clusters formed by head entities and Fermat points of clusters formed by tail entities. This observation further supports the claim that PDEC reduces the error in the TransE model even when dealing with multiple head entities.