## 3 Materials and Methods
### Predicate Decomposition  
In this step, we aim to decompose the predicates in order to better represent contextual information. In existing representation learning frameworks, the same predicate is associated with the same representation vector, which cannot be further decomposed.  
To discriminate polysemous predicates, we obtain the necessary contextual information for each predicate through its corresponding _edge vectors_. Edge vectors of \(r_{i}\) can be formalized as follows:  
\[\mathbf{e}_{i}=f_{\text{edge}}(\mathbf{h}_{i},\mathbf{t}_{i}),\quad\forall(h_ {i},r_{i},t_{i})\in\mathcal{K}_{train}\cup\mathcal{K}_{valid} \tag{1}\]  
For the sake of computational efficiency, we first employ efficient algorithms such as transE to learn the embeddings of entities and predicates in KGs. This allows us to obtain an initial model \(\Psi\). We then utilize this initial model \(\Psi\) to calculate edge vectors:  
\[\vec{t}_{a,i,j}=edge(\vec{t}_{j},\vec{h}_{i})\quad\forall(h_{i},r_{a},t_{j}) \in\mathcal{K}_{train}\cup\mathcal{K}_{valid} \tag{2}\]  
where \(\vec{h}\) and \(\vec{t}\) represent the embedding vectors of \(h\) and \(t\) obtained by the representation learning model, while \(edge\) is a function that utilizes the mechanism of the model \(\Psi\) to estimate the embedding of \(r_{a}\) based on \(\vec{h}\) and \(\vec{t}\).  
For instance, in TransE, Equation (3) specifies how this estimation is achieved:  
\[\vec{t}_{a,i,j}=\vec{t}_{j}-\vec{h}_{i},\quad\forall(h_{i},r_{a},t_{j})\in \mathcal{K}_{train}\cup\mathcal{K}_{valid} \tag{3}\]  
Let \(\
abla=\{\vec{t}\in\mathbb{R}^{d}|\forall(h,r,t)\in\mathcal{K}_{train}\cup \mathcal{K}_{valid}\}\) represent the set of edge vectors. For each predicate \(r_{a}\in\mathcal{R}\), its edge vector subset is \(\
abla_{a}=\{\vec{t}_{\vec{r}_{a},i,j}\in\mathbb{R}^{d}|\forall(h_{i},r_{a}, t_{j})\in\mathcal{K}_{train}\cup\mathcal{K}_{valid}\}\). We perform a clustering on the vector set \(\
abla_{a}\) and use the Calinski-Harabasz index [23] as the clustering quality evaluation method. The clustering result is denoted as \(\mathbf{CL}_{a,\theta}=\{\
abla_{a,\,0},\
abla_{a,\,1},\ldots\
abla_{a,\,K-1 }\subset\
abla_{a}\}\), where \(\theta\) is the parameter of the clustering algorithm, and \(K\) is the number of clusters.  
The optimal clustering result is determined according to the following equation:  
\[\arg\max_{\theta}c_{-}h(\mathbf{CL}_{a,\theta}) \tag{4}\]  
where \(c_{-}h(\mathbf{CL}_{a,\,\theta})\) represents the Calinski-Harabasz index of the clustering result \(\mathbf{CL}_{a,\theta}\).  
Let the lower threshold of clustering quality be \(\eta\). For every \(r_{a}\in\mathcal{R}\), if \(c_{-}h(\mathbf{CL}_{a,\theta})>\eta\), we believe that the predicate \(r_{a}\) is a polysemous predicate and can be split into \(K\) new predicates \(r_{a,0},\ldots,r_{a,K-1}\); i.e., we perform the following predicate splitting:  
\[(h_{i},r_{a},t_{j})\rightarrow(h_{i},r_{a,x},t_{j}),\quad if\quad\vec{t}_{a, i,j}\in\
abla_{a,x}\&\&\&\&\_{C_{-}h(\mathbf{CL}_{a,\theta})>\eta \tag{5}\]  
where \(x\in\mathbb{N},x<K\). Let \(split_{a}=\{r_{a,0},\ldots,r_{a,K-1}\}\) be the set of predicates formed by splitting the predicate \(r_{a}\). We denote the set of polysemous predicates as \(\mathcal{R}_{P}\).  
Given that the Calinski-Harabasz index often reaches its maximum when the number of clusters is 2 or 3, and that the semantic diversity of polysemous predicates typically does not exceed 3, to simplify the computational complexity, we utilize clustering algorithms that allow for the specification of the number of clusters, such as K-means, with \(K=2,3\).