  Knowledge graph embedding (KGE) is an 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.
