Knowledge graph embedding (KGE) is a increasingly popular technique that aims
to represent entities and relations of knowledge graphs into low-dimensional
semantic spaces for a wide spectrum of applications such as link prediction,
knowledge reasoning and knowledge completion. In this paper, we provide a
systematic review of existing KGE techniques based on representation spaces.
Particularly, we build a fine-grained classification to categorise the models
based on three mathematical perspectives of the representation spaces: (1)
Algebraic perspective, (2) Geometric perspective, and (3) Analytical
perspective. We introduce the rigorous definitions of fundamental mathematical
spaces before diving into KGE models and their mathematical properties. We
further discuss different KGE methods over the three categories, as well as
summarise how spatial advantages work over different embedding needs. By
collating the experimental results from downstream tasks, we also explore the
advantages of mathematical space in different scenarios and the reasons behind
them. We further state some promising research directions from a representation
space perspective, with which we hope to inspire researchers to design their
KGE models as well as their related applications with more consideration of
their mathematical space properties.

本文从数学空间的角度系统性地回顾了当前存在的 KGE 技术，并介绍了基本的数学空间定义。通过对不同的类别 KGE 方法的讨论，总结了它们在不同嵌入需求下如何发挥空间优势，并探讨了数学空间在不同场景下的优势和原因。最后，从表示空间的角度提出了一些有前途的研究方向，希望能激发研究人员在设计 KGE 模型及相关应用时更加考虑其数学空间特性。