The Theory of the Interleaving Distance on Multidimensional Persistence Modules

TL;DR

This paper develops the theory of the interleaving distance for multidimensional persistence modules, establishing its properties, universality, and relation to the bottleneck distance, with implications for topological data analysis.

Contribution

It extends the interleaving distance theory to multidimensional modules, proves its universality, and relates it to the bottleneck distance, advancing topological data analysis tools.

Findings

$d_I$ equals the bottleneck distance $d_B$ on 1-D modules.

Characterization of $ extepsilon$-interleavings for multidimensional modules.

$d_I$ satisfies a universality property among stable pseudometrics.

Abstract

In 2009, Chazal et al. introduced $\epsilon$-interleavings of persistence modules. $\epsilon$-interleavings induce a pseudometric $d_I$ on (isomorphism classes of) persistence modules, the interleaving distance. The definitions of $\epsilon$-interleavings and $d_I$ generalize readily to multidimensional persistence modules. In this paper, we develop the theory of multidimensional interleavings, with a view towards 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 $\epsilon$-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, $d_I$ 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 $d\leq d_I$. 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.