A Unified Topological Approach to Data Science
Jelena Grbi\'c, Jie Wu, Kelin Xia, Guo-Wei Wei
TL;DR
This paper introduces a comprehensive topological framework for data science that unifies analysis of point cloud and graph data, extending traditional concepts to super-hypergraphs and developing super-persistent homology for practical applications.
Contribution
It generalizes simplicial complexes and hypergraphs to super-hypergraphs, establishing super-hypergraph homology and super-persistent homology for the first time.
Findings
Unified topological approach applicable to various data types
Extension of homology theories to super-hypergraphs
Introduction of super-persistent homology for data analysis
Abstract
We establish a new theory which gives a unified topological approach to 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 establish super-hypergraph homology as an extension of simplicial homology. Driven by applications, we also introduce super-persistent homology.
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Taxonomy
TopicsTopological and Geometric Data Analysis · Homotopy and Cohomology in Algebraic Topology · Alzheimer's disease research and treatments