Foundations of Computational Mathematics

, Volume 15, Issue 3, pp 613–650 | Cite as

The Theory of the Interleaving Distance on Multidimensional Persistence Modules



In 2009, Chazal et al. introduced \(\epsilon \)-interleavings of persistence modules. \(\epsilon \)-interleavings induce a pseudometric \(d_\mathrm{I}\) on (isomorphism classes of) persistence modules, the interleaving distance. The definitions of \(\epsilon \)-interleavings and \(d_\mathrm{I}\) 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, \(d_\mathrm{I}\) is equal to the bottleneck distance \(d_\mathrm{B}\). 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 \(\epsilon \)-interleaving relation on multidimensional persistence modules. This expresses transparently the sense in which two \(\epsilon \)-interleaved modules are algebraically similar. Third, using this characterization, we show that when we define our persistence modules over a prime field, \(d_\mathrm{I}\) satisfies a universality property. This universality result is the central result of the paper. It says that \(d_\mathrm{I}\) satisfies a stability property generalizing one which \(d_\mathrm{B}\) is known to satisfy, and that in addition, if \(d\) is any other pseudometric on multidimensional persistence modules satisfying the same stability property, then \(d\le d_\mathrm{I}\). We also show that a variant of this universality result holds for \(d_\mathrm{B}\), over arbitrary fields. Finally, we show that \(d_\mathrm{I}\) restricts to a metric on isomorphism classes of finitely presented multidimensional persistence modules.


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

Mathematics Subject Classification

55 68 


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Copyright information

© SFoCM 2015

Authors and Affiliations

  1. 1.Institute for Mathematics and its ApplicationsMinneapolisUSA

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