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Topological Hierarchies and Decomposition: From Clustering to Persistence

Brown, Kyle A.
http://rave.ohiolink.edu/etdc/view?acc_num=wright1650388451804736

Abstract Details

2022, Doctor of Philosophy (PhD), Wright State University, Computer Science and Engineering PhD.
Hierarchical clustering is a class of algorithms commonly used in exploratory data analysis (EDA) and supervised learning. However, they suffer from some drawbacks, including the difficulty of interpreting the resulting dendrogram, arbitrariness in the choice of cut to obtain a flat clustering, and the lack of an obvious way of comparing individual clusters. In this dissertation, we develop the notion of a topological hierarchy on recursively-defined subsets of a metric space. We look to the field of topological data analysis (TDA) for the mathematical background to associate topological structures such as simplicial complexes and maps of covers to clusters in a hierarchy. Our main results include the definition of a novel hierarchical algorithm for constructing a topological hierarchy, and an implementation of the MAPPER algorithm and our topological hierarchies in pure Python code as well as a web app dashboard for exploratory data analysis. We show that the algorithm scales well to high-dimensional data due to the use of dimensionality reduction in most TDA methods, and analyze the worst-case time complexity of MAPPER and our hierarchical decomposition algorithm. Finally, we give a use case for exploratory data analysis with our techniques.
Derek Doran, Ph.D. (Advisor)
Michael Raymer, Ph.D. (Committee Member)
Vincent Schmidt, Ph.D. (Committee Member)
Nikolaos Bourbakis, Ph.D. (Committee Member)
Thomas Wischgoll, Ph.D. (Committee Member)
145 p.
Computer Science
topological data analysis; hierarchical clustering; exploratory data analysis; topology; clustering; data science

Recommended Citations

Citations

  • Brown, K. A. (2022). Topological Hierarchies and Decomposition: From Clustering to Persistence [Doctoral dissertation, Wright State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=wright1650388451804736

    APA Style (7th edition)

  • Brown, Kyle. Topological Hierarchies and Decomposition: From Clustering to Persistence. 2022. Wright State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=wright1650388451804736.

    MLA Style (8th edition)

  • Brown, Kyle. "Topological Hierarchies and Decomposition: From Clustering to Persistence." Doctoral dissertation, Wright State University, 2022. http://rave.ohiolink.edu/etdc/view?acc_num=wright1650388451804736

    Chicago Manual of Style (17th edition)

wright1650388451804736
345
© 2022, some rights reserved.Creative Commons License Topological Hierarchies and Decomposition: From Clustering to Persistence by Kyle A. Brown is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Based on a work at etd.ohiolink.edu.
This open access ETD is published by Wright State University and OhioLINK.