Induced matchings of barcodes and the algebraic stability of persistence

Ulrich Bauer, Michael Lesnick

Research output: Chapter in Book/Report/Conference proceedingConference contribution

14 Scopus citations

Abstract

We define a simple, explicit map sending a morphism f : M → N of pointwise finite dimensional persistence modules to a matching between the barcodes of M and N. Our main result is that, in a precise sense, the quality of this matching is tightly controlled by the lengths of the longest intervals in the barcodes of ker f and coker f . As an immediate corollary, we obtain a new proof of the algebraic stability theorem for persistence barcodes [5, 9], a fundamental result in the theory of persistent homology. In contrast to previous proofs, ours shows explicitly how a δ-interleaving morphism between two persistence modules induces a δ-matching between the barcodes of the two modules. Our main result also specializes to a structure theorem for submodules and quotients of persistence modules. Copyright is held by the owner/author(s).

Original languageEnglish (US)
Title of host publicationProceedings of the 30th Annual Symposium on Computational Geometry, SoCG 2014
PublisherAssociation for Computing Machinery
Pages355-364
Number of pages10
ISBN (Print)9781450325943
DOIs
StatePublished - 2014
Event30th Annual Symposium on Computational Geometry, SoCG 2014 - Kyoto, Japan
Duration: Jun 8 2014Jun 11 2014

Publication series

NameProceedings of the Annual Symposium on Computational Geometry

Other

Other30th Annual Symposium on Computational Geometry, SoCG 2014
Country/TerritoryJapan
CityKyoto
Period6/8/146/11/14

Keywords

  • Induced matching
  • Interleaving distance
  • Persistence module

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