Knowledge graph (KG) representation learning aims to encode entities and
relations into dense continuous vector spaces such that knowledge contained in
a dataset could be consistently represented. Dense embeddings trained from KG
datasets benefit a variety of downstream tasks such as KG completion and link
prediction. However, existing KG embedding methods fell short to provide a
systematic solution for the global consistency of knowledge representation. We
developed a mathematical language for KG based on an observation of their
inherent algebraic structure, which we termed as Knowledgebra. By analyzing
five distinct algebraic properties, we proved that the semigroup is the most
reasonable algebraic structure for the relation embedding of a general
knowledge graph. We implemented an instantiation model, SemE, using simple
matrix semigroups, which exhibits state-of-the-art performance on standard
datasets. Moreover, we proposed a regularization-based method to integrate
chain-like logic rules derived from human knowledge into embedding training,
which further demonstrates the power of the developed language. As far as we
know, by applying abstract algebra in statistical learning, this work develops
the first formal language for general knowledge graphs, and also sheds light on
the problem of neural-symbolic integration from an algebraic perspective.

本文提出了一种基于代数结构的 Knowledgebra（知识代数）数学语言，分析了五个不同的代数属性，证明了半群关系嵌入是一般知识图谱关系嵌入最合理的代数结构，提出了一个实例化模型 SemE，使用简单的矩阵半群具有最好的性能，同时提出了一种基于正则化的方法，在嵌入训练中融合人类知识派生的链式逻辑规则，为深度学习和代数推理的结合提供了一种可能性。