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In this article, 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.<\/jats:p>","DOI":"10.1145\/3643806","type":"journal-article","created":{"date-parts":[[2024,2,2]],"date-time":"2024-02-02T12:48:09Z","timestamp":1706878089000},"page":"1-42","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":125,"title":["Knowledge Graph Embedding: A Survey from the Perspective of Representation Spaces"],"prefix":"10.1145","volume":"56","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4338-4414","authenticated-orcid":false,"given":"Jiahang","family":"Cao","sequence":"first","affiliation":[{"name":"Sun Yat-sen University, Guangzhou, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6886-5882","authenticated-orcid":false,"given":"Jinyuan","family":"Fang","sequence":"additional","affiliation":[{"name":"Sun Yat-sen University, Guangzhou, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5374-0318","authenticated-orcid":false,"given":"Zaiqiao","family":"Meng","sequence":"additional","affiliation":[{"name":"University of Glasgow, Glasgow, United Kingdom"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1625-2168","authenticated-orcid":false,"given":"Shangsong","family":"Liang","sequence":"additional","affiliation":[{"name":"Sun Yat-sen University, Guangzhou, China and Mohamed bin Zayed University of Artificial Intelligence, Abu Dhabi, United Arab Emirates"}]}],"member":"320","published-online":{"date-parts":[[2024,3,13]]},"reference":[{"key":"e_1_3_2_2_2","first-page":"9649","article-title":"BoxE: A box embedding model for knowledge base completion","volume":"33","author":"Abboud Ralph","year":"2020","unstructured":"Ralph Abboud, Ismail Ceylan, Thomas Lukasiewicz, and Tommaso Salvatori. 2020. 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