The Theory of the Interleaving Distance on Multidimensional Persistence Modules

In 2009, Chazal et al. introduced ϵ-interleavings of persistence modules. ϵ-interleavings induce a pseudometric d_I on (isomorphism classes of) persistence modules, the interleaving distance. The definitions of ϵ-interleavings and d_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_I is equal to the bottleneck distance d_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 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, d_I satisfies a universality property. This universality result is the central result of the paper. It says that d_I satisfies a stability property generalizing one which d_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≤d_I. We also show that a variant of this universality result holds for d_B, over arbitrary fields. Finally, we show that d_I restricts to a metric on isomorphism classes of finitely presented multidimensional persistence modules.