The Theory of the Interleaving Distance on Multidimensional Persistence Modules release_644eu54uyfek3fc2fiyzgsmcm4

by Michael Lesnick

Released as a article .

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.
In text/plain format

Archived Files and Locations

application/pdf  543.7 kB
file_6iqtuo5kerhozhgdothuj3n3ya
arxiv.org (repository)
web.archive.org (webarchive)
Read Archived PDF
Preserved and Accessible
Type  article
Stage   submitted
Date   2011-06-27
Version   v1
Language   en ?
arXiv  1106.5305v1
Work Entity
access all versions, variants, and formats of this works (eg, pre-prints)
Catalog Record
Revision: 016e2d50-7ecb-4658-a9c6-a97f89bb9f23
API URL: JSON