A topological method for shape comparison

Ishkhanov, Tigran

doi:10.1109/cvprw.2008.4563076

Cited by 1 publication

(1 citation statement)

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“…A number of papers have introduced pseudometrics on multidimensional persistence modules [12,15,34,38], and several of these have presented stability results for the metrics they introduce [12,15,34]. The multi-dimensional matching distance of [15] is unique amongst the pseudometrics introduced by these papers in that it is a generalization of d B .…”

Section: Distances On Multidimensional Persistence Modulesmentioning

confidence: 99%

The Theory of the Interleaving Distance on Multidimensional Persistence Modules

Lesnick¹

2015

Found Comput Math

182

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In 2009, Chazal et al. introduced -interleavings of persistence modules.interleavings induce a pseudometric d I on (isomorphism classes of) persistence modules, the interleaving distance. The definitions of -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 -interleaving relation on multidimensional persistence modules. This expresses transparently the sense in which two -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. * mlesnick@ima.umn.edu.

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