Multiradial (multi)filtrations and persistent homology
- UNCG Author/Contributor (non-UNCG co-authors, if there are any, appear on document)
- Joshua M. Martin (Creator)
- Institution
- The University of North Carolina at Greensboro (UNCG )
- Web Site: http://library.uncg.edu/
- Advisor
- Gregory Bell
Abstract: Motivated by the problem of optimizing sensor network covers, we generalize the persistent homology of simplicial complexes over a single radial parameter to the context of multiple radial parameters. The persistent homology of so-called multiradial (multi)filtrations is identified as a special case of multidimensional persistence. Specifically, we exhibit that the persistent homology of (multi)filtrations corresponds to both generalized persistence modules of the form [see full paper for form; could not be reproduced here] and (multi)graded modules over a polynomial ring. The stability of persistence barcodes/diagrams of multiradial filtrations is derived, along with explicit bounds associated to perturbations in both radii and vertex position. A strengthening of the Vietoris-Rips lemma of [DSG07, p. 346] to the setting of multiple radial parameters is obtained. We also use the categorical framework of [BdSS15] to show the persistent homology modules of multiradial (multi)filtrations are stable.
Multiradial (multi)filtrations and persistent homology
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Created on 8/1/2016
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Additional Information
- Publication
- Thesis
- Language: English
- Date: 2016
- Keywords
- Cech complex, Multidimensional persistence, Persistent homology, Sensor networks, Stability, Vietoris-Rips complex
- Subjects
- Homology theory
- Algebraic topology