## 5 Experiments
### Model Validation
Additionally, ablation studies isolating the attribute projection vector and the dual-limit loss modules were carried out to quantify each component's independent impact on link prediction and various evaluation metrics, thereby ensuring the reliability and modular robustness of the conclusions. Model performance variations were recorded for each experimental condition. (3) Evaluation Metrics  
To quantitatively assess model performance, this study employs Mean Rank (MR), Mean Reciprocal Rank (MRR), Hits@K, triple classification accuracy (Accuracy), and F1-score as evaluation metrics. The Mean Rank (MR) is calculated as:  
\[MR=\frac{1}{N}\sum_{i=1}^{N}rank_{i} \tag{13}\]  
where \(N\) is the total number of test samples and \(rank_{i}\) denotes the rank position of the correct entity for the \(i\)-th sample within the list of candidate entities. A lower MR indicates that, on average, the model ranks correct entities closer to the top. The Mean Reciprocal Rank (MRR) is defined as:  
\[MRR=\frac{1}{N}\sum_{i=1}^{N}\frac{1}{rank_{i}} \tag{14}\]  
This metric emphasizes the cases where the correct entity appears at high positions, as higher reciprocal ranks (i.e., lower \(rank_{i}\)) yield larger contributions to MRR, reflecting the model's ability to assign higher scores to true answers. The Hits@K metric measures the proportion of test cases in which the correct entity appears among the top \(K\) candidates. It is defined as:  
\[Hits@K=\frac{1}{N}\sum_{i=1}^{N}\mathbb{I}(rank_{i}\leq K),(K=1,3,10) \tag{15}\]  
where \(\mathbb{I}\) (\(\cdot\)) is the indicator function. A higher _Hits@K_ value indicates that, in most cases, the model ranks the correct answer within the top \(K\), thereby improving retrieval effectiveness. Triple classification accuracy (Accuracy) reflects the overall correctness of distinguishing positive and negative triples, computed as:  
\[\text{Accuracy}=\frac{\text{Number of Currectly Classified Triples}}{\text{Total Number of Triples}} \tag{16}\]  
A higher accuracy implies that the model correctly classifies a larger proportion of triples, directly indicating enhanced overall discriminative capability. The F1-score harmonizes precision and recall and is defined as:  
\[\text{F1}=2\times\frac{\text{Precision}\ \times\ \text{Recall}}{\text{ Precision}+\text{Recall}} \tag{17}\]  
where  
\[\text{Precision}=\frac{\text{TP}}{\text{TP}+\text{FP}},\text{Recall}=\frac{ \text{TP}}{\text{TP}+\text{FN}}\]  
and TP, FP, and FN denote the counts of true positives, false positives, and false negatives, respectively. The F1-score balances precision and recall; a higher F1 indicates that the model maintains high precision while also achieving strong recall, yielding a more balanced and robust performance in classifying both positive and negative triples. (4) Results Analysis  
The experimental results demonstrate that, across varying embedding dimensions, the TransD model achieves significant performance gains on both FB15K-237 and WN18 RR datasets, as shown in Figure 9 for FB15K-237. Experimental results on the FB15K-237 dataset demonstrate that the enhanced TransD model (hereafter Enhanced TransD), which integrates attribute projection vectors and a dual-limit loss mechanism, yields substantial improvements in representational capacity and retrieval performance over classical translation-based embedding methods. The FB15K-237 dataset contains approximately 15,000 entities and 237 relation types, whose diversity and sparsity pose significant challenges for embedding models. Unlike TransE, TransH, and TransR--which typically exhibit a \"dimension-performance\" trade-off--Enhanced TransD maintains the original dynamic mapping mechanism while explicitly encoding entity attributes into projection vectors and employing a dual-threshold scoring loss to reinforce positive-negative sample separation. This design leads to superior performance across most evaluation metrics. For the Mean Rank (MR) metric, Enhanced TransD achieves a continuous decline from \(\sim\)320 at 50 dimensions to \(\sim\)230 at 300 dimensions.