The Theory of the Interleaving Distance on Multidimensional Persistence Modules

Abstract

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.