The Theory of the Interleaving Distance on Multidimensional Persistence
Modules
release_644eu54uyfek3fc2fiyzgsmcm4
by
Michael Lesnick
2011
Abstract
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 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 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.
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