  Knowledge graphs are useful for many artificial intelligence (AI) tasks.
However, knowledge graphs often have missing facts. To populate the graphs,
knowledge graph embedding models have been developed. Knowledge graph embedding
models map entities and relations in a knowledge graph to a vector space and
predict unknown triples by scoring candidate triples. TransE is the first
translation-based method and it is well known because of its simplicity and
efficiency for knowledge graph completion. It employs the principle that the
differences between entity embeddings represent their relations. The principle
seems very simple, but it can effectively capture the rules of a knowledge
graph. However, TransE has a problem with its regularization. TransE forces
entity embeddings to be on a sphere in the embedding vector space. This
regularization warps the embeddings and makes it difficult for them to fulfill
the abovementioned principle. The regularization also affects adversely the
accuracies of the link predictions. On the other hand, regularization is
important because entity embeddings diverge by negative sampling without it.
This paper proposes a novel embedding model, TorusE, to solve the
regularization problem. The principle of TransE can be defined on any Lie
group. A torus, which is one of the compact Lie groups, can be chosen for the
embedding space to avoid regularization. To the best of our knowledge, TorusE
is the first model that embeds objects on other than a real or complex vector
space, and this paper is the first to formally discuss the problem of
regularization of TransE. Our approach outperforms other state-of-the-art
approaches such as TransE, DistMult and ComplEx on a standard link prediction
task. We show that TorusE is scalable to large-size knowledge graphs and is
faster than the original TransE.
