The Theory of the Interleaving Distance on Multidimensional Persistence Modules

Michael Lesnick

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In 2009, Chazal et al. introduced ϵ-interleavings of persistence modules. ϵ-interleavings induce a pseudometric dI on (isomorphism classes of) persistence modules, the interleaving distance. The definitions of ϵ-interleavings and dI generalize readily to multidimensional persistence modules. In this paper, we develop the theory of multidimensional interleavings, with a view toward applications to topological data analysis. We present four main results. First, we show that on 1-D persistence modules, dI is equal to the bottleneck distance dB. This result, which first appeared in an earlier preprint of this paper, has since appeared in several other places, and is now known as the isometry theorem. Second, we present a characterization of the ϵ-interleaving relation on multidimensional persistence modules. This expresses transparently the sense in which two ϵ-interleaved modules are algebraically similar. Third, using this characterization, we show that when we define our persistence modules over a prime field, dI satisfies a universality property. This universality result is the central result of the paper. It says that dI satisfies a stability property generalizing one which dB is known to satisfy, and that in addition, if d is any other pseudometric on multidimensional persistence modules satisfying the same stability property, then d≤dI. We also show that a variant of this universality result holds for dB, over arbitrary fields. Finally, we show that dI restricts to a metric on isomorphism classes of finitely presented multidimensional persistence modules.

Original languageEnglish (US)
Pages (from-to)613-650
Number of pages38
JournalFoundations of Computational Mathematics
Issue number3
StatePublished - Jun 27 2015

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© 2015, SFoCM.


  • Algebraic stability
  • Interleavings
  • Isometry theorem
  • Multidimensional persistence
  • Persistence modules
  • Stability of persistent homology


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