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The Persistent Topology of Geometric Filtrations

Wang, Qingsong
http://rave.ohiolink.edu/etdc/view?acc_num=osu1657289975274281

Abstract Details

2022, Doctor of Philosophy, Ohio State University, Mathematics.
We study the theoretical foundation of the persistent topology of the geometric filtrations in Topological Data Analysis (TDA), such as Vietoris--Rips simplicial complexes, Vietoris--Rips metric thickenings. We introduce a $\ell_p$-relaxation to the Vietoris--Rips metric thickening where $p=\infty$ recovers the usual Vietoris--Rips metric thickening. We prove a stability theorem for the persistent homology of $\ell_p$ relaxed metric thickenings, which is novel even in the case $p=\infty$. The stability theorem then can be employed to show that the filtrations by Vietoris--Rips simplicial complexes and Vietoris--Rips metric thickenings have the same persistent diagram. Therefore, we can employ measure-theoretical methods to study the Vietoris--Rips complex. Some recent study also suggests that the persistent homology of Vietoris--Rips simplicial complex changes when the scale passes the diameter of some extremal configuration of the diameter functional. As an example, we study the extremal configurations on spheres. We implemented the diameter gradient flow and obtained nontrivial extremal configurations on $\Sp^2$ and $\Sp^3$. We find a natural condition for metric spaces that will guarantee the vanishing of the persistence diagram of Vietoris--Rips filtration over certain dimensions. We also demonstrate by a non-collapsing result that the persistent features can be utilized to obtain a quantitive lower bound for the Gromov--Hausdorff distance between Riemannian manifolds.
Facundo Mémoli (Advisor)
Jean-François Lafont (Committee Member)
Matthew Kahle (Committee Member)
159 p.
Mathematics
Persistent homology, Vietoris-Rips complex, metric thickening, Kuratowski embedding, Diameter extremal configurations, Generalized ultrametric.

Recommended Citations

Citations

  • Wang, Q. (2022). The Persistent Topology of Geometric Filtrations [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1657289975274281

    APA Style (7th edition)

  • Wang, Qingsong. The Persistent Topology of Geometric Filtrations . 2022. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1657289975274281.

    MLA Style (8th edition)

  • Wang, Qingsong. "The Persistent Topology of Geometric Filtrations ." Doctoral dissertation, Ohio State University, 2022. http://rave.ohiolink.edu/etdc/view?acc_num=osu1657289975274281

    Chicago Manual of Style (17th edition)

osu1657289975274281
205
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This open access ETD is published by The Ohio State University and OhioLINK.