Multiradial (multi)filtrations and persistent homology

UNCG Author/Contributor (non-UNCG co-authors, if there are any, appear on document)
Joshua M. Martin (Creator)
The University of North Carolina at Greensboro (UNCG )
Web Site:
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.

Additional Information

Language: English
Date: 2016
Cech complex, Multidimensional persistence, Persistent homology, Sensor networks, Stability, Vietoris-Rips complex
Homology theory
Algebraic topology

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