Multiradial (multi)filtrations and persistent homology

• Main Author: Martin, Joshua M.
• Language: EN
• Published: The University of North Carolina at Greensboro 2016
• Subjects: Mathematics
• Online Access: http://pqdtopen.proquest.com/#viewpdf?dispub=10154660

<p> 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 <b>[special characters omitted]</b> 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.</p>