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 towards 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.Comment: Major revision; exposition improved throughout. To appear in
Foundations of Computational Mathematics. 36 page