# Cosine-Based Embedding for Completing Lightweight Schematic Knowledge in DL-Lite\({}_{core}\)+
###### Abstract  
Schematic knowledge, an important component of knowledge graphs (KGs), defines a rich set of logical axioms based on concepts and relations to support knowledge integration, reasoning, and heterogeneity elimination over KGs. Although several KGs consist of lots of factual knowledge, their schematic knowledge (e.g., \(subclassOf\) axioms, \(disjointWith\) axioms) is far from complete. Currently, existing KG embedding methods for completing schematic knowledge still suffer from two limitations. Firstly, existing embedding methods designed to encode factual knowledge pay little attention to the completion of schematic knowledge (e.g., axioms). Secondly, several methods try to preserve logical properties of relations for completing schematic knowledge, but they cannot simultaneously preserve the transitivity (e.g., \(subclassOf\)) and symmetry (e.g., \(disjointWith\)) of axioms well. To solve these issues, we propose a cosine-based embedding method named CosE tailored for completing lightweight schematic knowledge in DL-Lite\({}_{core}\). Precisely, the concepts in axioms will be encoded into two semantic spaces defined in CosE. One is called angle-based semantic space, which is employed to preserve the transitivity or symmetry of relations in axioms. The other one is defined as translation-based semantic space that is used to measure the confidence of each axiom. We design two types of score functions for these two semantic spaces, so as to sufficiently learn the vector representations of concepts. Moreover, we propose a novel negative sampling strategy based on the mutual exclusion between \(subclassOf\) and \(disjointWith\). In this way, concepts can obtain better vector representations for schematic knowledge completion. We implement our method and verify it on four standard datasets generated by real ontologies. Experiments show that CosE can obtain better results than existing models and keep the logical properties of relations for transitivity and symmetry simultaneously. s 20221202202212022120221202212212  
and YAGO [4]) mostly consists of lots of factual knowledge and little schematic knowledge. For the famous knowledge graph DBpedia ([[https://www.dbpedia.org/](https://www.dbpedia.org/)]([https://www.dbpedia.org/](https://www.dbpedia.org/)) accessed on 5 September 2022), it contains more than 1.89 billion triples and over 3.5 million entities among them. However, it contains only 768 concepts and 20 _disjointWith_ axioms asserted among them. The sparsity of schematic knowledge will limit the applications and services of KGs such as query-answering [5], recommendation system [6], and knowledge integration [7]. Hence, it is essential to improve the completeness of schematic knowledge. Nevertheless, it is hard for traditional reasoning-based methods to automatically infer all the remaining axioms. Take, for an example shown in Figure 1, there are three axioms (Farm_Boy, \(subclassOf\), Boy), (Boy, \(subclassOf\), Male_Person), and (Male_Person, \(subclassOf\), Person), defined in one schematic knowledge. If the relation \(subclassOf\) from Boy to Male_Person marked in red is missing, then it is hard to obtain the \(subclassOf\) relation from Boy to Person marked in blue using traditional reasoning-based methods. Knowledge graph embedding, which aims to encode the entities and relations of a KG into the low-dimensional and continuous vector space, has been widely studied and has been proven to be of great help in KG completion via link prediction [8] and other downstream tasks; see [9; 10; 11; 12]. The mainstream models are designed for factual knowledge embedding, including TransE [13], TransH [14], TransR [15], and so on, which regard the relation as a \"translation\" from the head entity to the tail entity. Another kind of model for factual knowledge embedding, such as RESCAL [16], DistMult [17], HolE [18], and ComplEx [19], design various operators to encode rich interactions among embedding vectors.