## 3 Proposed Method
### Triplets Mapping Properties Weighting  
For previous work that did not take into account the different mapping properties of each triplet, we associate each training triplet with the weight that represents the degree of mapping. The main issue we consider is that positive triplets and negative triplets both conform to the triplet constraint _h+r_\(\approx\)_t_ in the embedding space. As shown in Figure 7, \(h\) denotes the head entity vector, \(r\) denotes the relation vector, \(t\) denotes the tail entity vector, and \(i\) denotes a real representation of _h+r_ in the vector space. As with the family of translation models, we expect a small distance between the performance of \(i\) and \(t\) in each positive triplet. We consider that the model has a better learning ability when \(i\) \(\approx\)_t_, and the features of entities and relations are well represented by vectors. On the other hand, we expect a large distance between the performance of \(i\) and the performance of \(t\) in each negative triplet.  
For each triplet, we take _h+r_ as the input sample and \(t\) as the real label. We focus on triplets with better vector representation, which contribute more to the training goal. We regard that the model has a better performance when the correlation between both \(i\) and \(t\) in the positive triplet is strong, while the correlation between both \(i\) and \(t\) in the negative triplet is weak. We evaluate the correlation of two vectors, \(i\) and \(t\), in the embedding space, by calculating their cosine values, and the higher the cosine similarity, the stronger the correlation of the head and tail entities in the relation \(r\). We  
Figure 6: An example of negative sampling. There is a positive triplet in (**a**), and (**b**) shows the candidate negative samples in the cache, which are generated by negative sampling of the positive triplet. (**c**) represents the embedded space. \(a\) denotes the positive triplet, \(\approx\) denotes the negative triplet.  
Figure 7: \(h\) denotes the head entity, \(r\) denotes the relation, \(t\) denotes the tail entity, and \(i\) denotes a real representation of _h+r_ in the vector space.  
that reflects the degree of the triplet vector representation as the triplet-mapping property. Thus, we define the triplet-mapping-property calculation as shown in Equation (5).  
\[tmp=\frac{\hat{\mathbf{t}}\cdot\mathbf{t}}{\|\hat{\mathbf{t}}\|\|\mathbf{t}\|}+\alpha \tag{5}\]  
Since the range of cosine values is [\(-\)1,1], to avoid introducing other biases, we use the hyperparameter \(\alpha\) as an equilibrium coefficient, so that \(tmp>0\). When the \(tmp\) value is larger, we consider that the input sample being closer to the true label means there is a higher probability that the triplet is correct. We extend the method to the Trans series model, as shown in Equation (6).  
\[tmp=\frac{(\mathbf{h}_{\perp}+\mathbf{r}_{\perp})\cdot\mathbf{t}_{\perp}}{\|\mathbf{h}_{\perp }+\mathbf{r}_{\perp}\|\|\mathbf{t}_{\perp}\|}+\alpha \tag{6}\]  
where \(\mathbf{h}_{\perp}\), \(\mathbf{r}_{\perp}\), \(\mathbf{t}_{\perp}\) denote, respectively, the head entity, relation, and tail entity mapping vectors in the particular space. A simple way to measure the degree of mapping for the triplet is to count the \(tmp\) of each triplet. Ideally, the positive triplet would have a low score. We take \(1/tmp\) as the triplet-mapping weight. We need to pre-calculate the specific weight of each triplet according to its mapping properties; thus, we constructed a new score function for the triplet, as shown in Equation (7).  
\[f(h,r,t)=\frac{1}{tmp}\|\mathbf{h}_{\perp}+\mathbf{r}_{\perp}-\mathbf{t}_{\perp}\|_{1/2} \tag{7}\]  
In experiments, we enforce constraints as \(\|\mathbf{h}_{\perp}\|\leq 1,\|\mathbf{r}_{\perp}\|\leq 1,\|\mathbf{t}_{\perp}\|\leq 1\). For a positive triplet \((h,r,t)\) in the training set \(\Delta\), we expect that the score of \(f(h,r,t)\) will be much lower than any negative triplet \((h^{\prime},r,t^{\prime})\).