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Beyond Persistent Homology: More Discriminative Persistent Invariants

Zhou, Ling
http://orcid.org/0000-0001-6655-5162
http://rave.ohiolink.edu/etdc/view?acc_num=osu1689837764936381

Abstract Details

2023, Doctor of Philosophy, Ohio State University, Mathematics.
Persistent homology has been an important tool in topological and geometrical data analysis to study the shape of data. However, its ability to differentiate between various datasets is limited. To expand and enhance our toolkit, we study persistent invariants that arise from homotopy groups, rational homotopy groups, the cohomology ring, the Lyusternik-Schnirelmann (LS) category, and chain complexes, which can be more discriminative than persistent homology. Chapter 2 provides the necessary background from metric geometry, algebraic topology, and persistent theory. Chapter 3 discusses persistent homotopy groups of compact metric spaces, with a focus on the persistent fundamental groups, for which we obtain a more precise description via a persistent version of the notion of discrete fundamental groups due to Berestovskii-Plaut. Under fairly mild assumptions on the spaces, we prove that the persistent fundamental group admits a tree structure that encodes more information than its persistent homology counterpart. The rationalization of the persistent homotopy groups is also considered and completely characterized for the circle by invoking the results of Adamaszek-Adams and Serre. We establish that persistent homotopy groups enjoy stability in the Gromov-Hausdorff sense. Chapter 4 examines several persistent invariants that capture ring-theoretic information about the evolution of the cohomology structure across a filtration. The first one is the persistent cup-length invariant, which is a persistent version of the standard cup-length invariant and is computable from representative cocycles in polynomial time. The second one is the persistent LS-category invariant. Although not directly defined using the cup product operation, the persistent LS-category invariant is closely related to the persistent cup-length invariant by having the latter as a pointwise lower bound. The third one is the persistent cup module, which absorbs the cup product in a more refined way than the persistent cup-length invariant. In particular, it has a 2-dimensional persistence module structure associated with persistent cohomology, where one parameter is the cup-length and the other is the filtration parameter. We establish the stability of all the persistent invariants considered and present examples to compare their ability to distinguish between different metric spaces. Chapter 5 strengthens the usual stability theorem for Vietoris-Rips (VR) persistent homology of finite metric spaces, by incorporating the often-overlooked ephemeral features in data. Ephemeral features are features with zero persistence, which are accessible in the complete invariant of filtered chain complexes, called verbose barcodes by Usher and Zhang. To compare them, we introduce the pullback bottleneck distance ˆ dB between verbose barcodes, defined by first using tripods to pull back the underlying spaces to a common space and compare (via the matching distance) the verbose barcodes of the filtered chain complexes induced by the respective pullbacks. We show that ˆ dB between verbose barcodes is stable under the Gromov-Hausdorff distance between the underlying spaces, and it upper bounds the bottleneck distance between standard barcodes. This thesis is written as a comprehensive version of the author’s works [113, 48, 49, 114, 112, 115] during her Ph.D. study.
Facundo Mémoli (Advisor)
Matthew Kahle (Committee Member)
Jean-Francois Lafont (Committee Member)
401 p.
Mathematics
Persistent homotopy group, persistent cohomology ring, filtered chain complex, Gromov-Hausdorff distance, cup product, LS-category, discrete homotopy theory, Vietoris-Rips complexes, stability, barcode, bottleneck distance, matching distance

Recommended Citations

Citations

  • Zhou, L. (2023). Beyond Persistent Homology: More Discriminative Persistent Invariants [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1689837764936381

    APA Style (7th edition)

  • Zhou, Ling. Beyond Persistent Homology: More Discriminative Persistent Invariants. 2023. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1689837764936381.

    MLA Style (8th edition)

  • Zhou, Ling. "Beyond Persistent Homology: More Discriminative Persistent Invariants." Doctoral dissertation, Ohio State University, 2023. http://rave.ohiolink.edu/etdc/view?acc_num=osu1689837764936381

    Chicago Manual of Style (17th edition)

osu1689837764936381
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© 2023, some rights reserved.Creative Commons License Beyond Persistent Homology: More Discriminative Persistent Invariants by Ling Zhou 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 The Ohio State University and OhioLINK.