## 3 Learned Semantic Index Structure
### Architecture of Learned Semantic Index Structure  
We propose a hybrid LOD storage strategy combining the index and distributed approaches. The index approach offers better performance, but the queried data might not be current because LOD has changed considerably. The distributed approach provides recent data, but the query execution is slower because data must be transmitted via the  
Figure 1: Entity and relation space of TransE, TransH, and TransR. network. Our proposed approach may solve these problems. By using a learned semantic index structure, our method can retrieve distributed LOD datasets efficiently. State-of-the-art RDF storage techniques may be adapted to develop an index structure for LOD queries. For instance, [PERSON] et al. [24] proposed a hybrid index structure, which was a combination of R*-tree and \(k\)-d trees. The goal of this method is to support efficient join query processing without significant storage demand. The core task in building the index is to transform RDF triples into points in a three-dimensional space by applying hash functions (e.g., the hashCode() function in Java) to the individual components of the RDF triples. However, this method considers nothing about the data distribution and does not exploit common patterns prevalent in real-world data. Typically, existing LOD index structures only consider the found relevant resources. Thus, there is no guarantee that the resources actually provide the RDF triples that were originally searched for (i.e., false positives). This is because the hash function does not describe exact coordinates in the multidimensional space. The input of LOD index structures is generally a triple pattern, and they output a list of data resources that potentially contribute to the result. Thus, a large number of resources can contribute to each triple pattern. Because accessing too many resources over the LOD cloud is expensive, we need to investigate a novel index structure that can efficiently process queries over the distributed LOD. It is not the intention of this paper to discuss which index structure is appropriate for RDF storage. We describe instead an extension of the existing index structures which integrates two index structures for clusters and the centroid of each cluster. In the existing index structures, the most popular R*-tree and \(k\)-d tree are selected for our index structure. As the cost of developing a new index structure can be more expensive than the cost of simply extending an existing one, adding new features to the existing index structure is an excellent alternative. Figure 2 depicts a workflow for constructing our learned semantic index structure. Our main idea is to turn heterogeneous hash codes into homogeneous sets. The heterogeneity of indexing keys is the main reason for long query processing time. If keys become homogeneous, query processing time can significantly reduce. Turning heterogeneous indexing keys into homogeneous sets entails semantically embedding numeric vectors in the adjacent embedding space, which we demonstrate to be achievable using KGE techniques. Figure 2: Workflow for constructing the learned semantic index structure. The proposed framework comprises three steps. The first step is to perform embedded processing for LOD datasets to obtain vectors of triples. The projected head and tail nodes are mapped by a project matrix \(M(r)\)[9]. With the mapping matrix, the method builds the projected entities:  
\[h(r)=hM(r),\;\;t(r)=tM(r) \tag{1}\]  
Its score function is as follows:  
\[f_{r}(h,\;t)=||h(r)+r-t(r)||_{2}^{2} \tag{2}\]  
The second step is to cluster vectorized triples and obtain each cluster centroid from a density-based spatial clustering algorithm. Given a set of points in some space, it groups all neighbors within a given radius.