# An Approach to Knowledge Base Completion
## 2 Related Work  
There have been a number of previous studies on knowledge base completion. One promising approach in the studies is to use knowledge graph embedding [13; 14; 15; 16]. Knowledge graph embedding represents all entities and relations of knowledge facts as low-dimensional vectors. These vectors of entities and relations are trained by preserving the inherent structure of a knowledge base. Thus, the plausibility of a knowledge fact can be measured using the vectors, and then knowledge base completion is done by filling up the facts with a high plausibility. Knowledge graph embeddings are clustered into two kinds of approach [17; 18]. One is a translation-based model and the other is a latent semantics-based model. In the translation-based model, the entities are translated according to their relation and the plausibility of facts is measured by a distance-based plausibility function. The best-known representative translation-based model is TransE [6]. TransE represents entities and relations as vectors in the same space assuming that the sum of a head entity vector and a relation vector is equal to a tail entity vector. Therefore, the plausibility function of TransE considers the distance between the sum vector and tail entity vector. It is simple and effective, especially for one-to-one relations. There have been many extensions of TransE since its first appearance [19; 20; 21; 22]. TransH [23], TransR [24], and TransD [25] are the extensions, but they all adopt relation-specific entity embedding. Since TransH projects entities onto relation-specific hyperplanes, it can represent an entity as a different vector according to its relation. Similarly, TransR adopts a relation-specific space rather than hyperplanes. It embeds entities into relation-specific spaces with a projection matrix. TransD simplifies TransR by decomposing the projection matrix into a product of two vectors, where the two vectors are mapping matrices of a head and a tail. Since entities are projected according to their role, it can represent one-to-\(n\), \(n\)-to-\(n\), and \(m\)-to-\(n\) relations. Latent semantics-based models capture the latent semantics of entities and relations using a similarity-based plausibility function [5; 26; 27]. In RESCAL [7], entities are represented as vectors, and relations are matrices derived from pairwise interactions between entities. It works well for all relation types from one-to-one to \(n\)-to-\(m\). However, it suffers from high complexity [28]. DistMult solves this problem by simplifying the relation matrices of RESCAL [29]. It represents a relation as diagonal matrices instead of matrices of relations. HoIE combines RESCAL and DistMult effectively [30]. The entities and relations are all represented as vectors, and the plausibility function adopts a circular correlation operation [31] to compress the pairwise interactions. By this operation, HoIE is able to model asymmetric relations that DistMult cannot. ComplEX is another method that extends DistMult to model asymmetric relations [32]. Since it embeds entities and relations into a complex vector space, its plausibility function is based on a Hermitian dot product. As a result, it is scalable to manage a large data set. Table 1 summarizes the knowledge graph embeddings explained above. For a given triple \(\langle h,r,t\rangle\), all methods but ComplEx embeds \(h\), \(r\), and \(t\) into a space \(\mathbb{R}^{d}\), where \(d\) is a space dimension. ComplEx embeds them into a complex space \(\mathbb{C}^{d}\). Then, \(h\) and \(t\) are represented as vectors \(\mathbf{h}\) and \(\mathbf{t}\), and \(r\) is as a vector \(\mathbf{r}\) or as projection matrices \(\mathbf{w}\) and \(\mathbf{M}\). In order to train these embedding vectors, each embedding has its own plausibility function \(f_{r}(\mathbf{h},\mathbf{t})\). In Table 1, \(diag(\cdot)\) implies a diagonal matrix, \(\otimes\) is the circular correlation operation [31], and \(Re(\cdot)\) is the real part of a complex value.