Aspects of topological approaches for data science

Grbic, Jelena, Wu, Jie, Xia, Kelin and Wei, Guowei (2022) Aspects of topological approaches for data science. Foundations of Data Science, 4 (2), 165-216. (doi:10.3934/fods.2022002).

Record type: Article

Abstract

We establish a new theory which unifies various aspects of topo- logical approaches for data science, by being applicable both to point cloud data and to graph data, including networks beyond pairwise interactions. We generalize simplicial complexes and hypergraphs to super-hypergraphs and es- tablish super-hypergraph homology as an extension of simplicial homology. Driven by applications, we also introduce super-persistent homology.

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Hypergraph2022_01_23 - Accepted Manuscript

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Accepted/In Press date: 22 January 2022

e-pub ahead of print date: 1 February 2022

Published date: 1 June 2022

Additional Information: Funding Information: The work of JW was supported in part by Natural Science Foundation of China (NSFC grant no. 11971144) and High-level Scientific Research Foundation of Hebei Province. The third author was supported by Nanyang Technological University Startup Grant M4081842 and Singapore Ministry of Education Academic Research fund Tier 1 RG109/19, Tier 2 MOE-T2EP20220-0010, and Tier 2 MOE-T2EP20120-0013. The work of GWW was supported by NIH grant GM126189, NSF grants DMS-1761320, IIS-1900473, and DMS-2052983, and NASA grant 80NSSC21M0023. We wish to thank the referees most warmly for important suggestions that have improved the exposition of this paper. ∗Corresponding author: Jie Wu. † JG, JW, KX and GW should be considered joint first author. Publisher Copyright: © 2022 American Institute of Mathematical Sciences. All right reserved.

Keywords: Delta set, Topological data analysis, hypergraph, persistent homology, scoring scheme, simplicial complex, super persistent homology, super-hypergraph