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Curvature Sets and Persistent Homology

Gomez Flores, Mario Roberto
http://orcid.org/0000-0001-6143-8995
http://rave.ohiolink.edu/etdc/view?acc_num=osu1689852191600607

Abstract Details

2023, Doctor of Philosophy, Ohio State University, Mathematics.
Given a metric space $(X,d_X)$, the $n$-th curvature set is the set of $n$-by-$n$ distance matrices generated by a sample from $X$ with $n$ or less points. Similarly, the $(n,k)$ persistence set of $X$ is the set of $k$-dimensional persistence diagrams of all $n$-point samples from $X$. This dissertation has two major parts, each dedicated to persistence sets or curvature sets. A major obstacle that hampers the widespread use of topological techniques in data science is the sometimes prohibitive computational cost of persistent homology. Persistence sets aim to circumvent this limitation while retaining useful geometric and topological information from the input space. We study the experimental and theoretical properties of persistence sets, and compare them with the standard VR-persistent homology from the perspectives of computational efficiency and discriminative power, including in a practical shape classification task. We characterize several persistence sets of the circle, higher-dimensional spheres, surfaces with constant curvature, and a specific family of metric graphs, and show spaces that have different persistence sets but are indistinguishable by persistent homology. All in all, we believe that persistence sets can aid in data science tasks where the shape is important but the standard persistent homology algorithms are impractical. In the second part, we study the curvature sets $\Kn_n(\Sp^1)$ of the circle $\Sp^1$ as a topological space. We give several characterizations of $\Kn_n(\Sp^1)$ as quotients of tori under group actions, and use them to compute the homology of $\Kn_n(\Sp^1)$ with Mayer-Vietoris arguments. We construct an abstract simplicial complex $\St_n(\Sp^1)$ whose geometric realization is $\Kn_n(\Sp^1)$. Moreover, we give an embedding of $\St_n(\Sp^1)$ in $\R^{n \times n}$ and show that $\Kn_n(\Sp^1)$ is the union of the convex hulls of the faces this embedding. Lastly, inspired by the characterization of a persistence set of surfaces with constant curvature, we show that the 4-point condition is as the tropicalization of Ptolemy's inquality. More precisely, there are generalizations of Ptolemy's inequality to $\cat(\kappa)$ spaces, and we show that the limit of the $\kappa$-Ptolemy inequality is the 4-point condition.
Facundo Mémoli (Advisor)
Matthew Kahle (Committee Member)
David Anderson (Committee Member)
284 p.
Mathematics
Curvature, Curvature Sets, Persistence Sets, VIetoris-Rips, Persistent Homology, subsampling, Topological Data Analysis, Generalized Ptolemy's inequality, Four-point condition, ultrametric spaces

Recommended Citations

Citations

  • Gomez Flores, M. R. (2023). Curvature Sets and Persistent Homology [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1689852191600607

    APA Style (7th edition)

  • Gomez Flores, Mario. Curvature Sets and Persistent Homology. 2023. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1689852191600607.

    MLA Style (8th edition)

  • Gomez Flores, Mario. "Curvature Sets and Persistent Homology." Doctoral dissertation, Ohio State University, 2023. http://rave.ohiolink.edu/etdc/view?acc_num=osu1689852191600607

    Chicago Manual of Style (17th edition)

osu1689852191600607
182
© 2023, some rights reserved.Creative Commons License Curvature Sets and Persistent Homology by Mario Roberto Gomez Flores 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.