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The Theory of the Interleaving Distance on Multidimensional Persistence Modules

The Theory of the Interleaving Distance on Multidimensional Persistence Modules In 2009, Chazal et al. introduced $$\epsilon $$ ϵ -interleavings of persistence modules. $$\epsilon $$ ϵ -interleavings induce a pseudometric $$d_\mathrm{I}$$ d I on (isomorphism classes of) persistence modules, the interleaving distance. The definitions of $$\epsilon $$ ϵ -interleavings and $$d_\mathrm{I}$$ d I generalize readily to multidimensional persistence modules. In this paper, we develop the theory of multidimensional interleavings, with a view toward applications to topological data analysis. We present four main results. First, we show that on 1-D persistence modules, $$d_\mathrm{I}$$ d I is equal to the bottleneck distance $$d_\mathrm{B}$$ 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 $$\epsilon $$ ϵ -interleaving relation on multidimensional persistence modules. This expresses transparently the sense in which two $$\epsilon $$ ϵ -interleaved modules are algebraically similar. Third, using this characterization, we show that when we define our persistence modules over a prime field, $$d_\mathrm{I}$$ d I satisfies a universality property. This universality result is the central result of the paper. It says that $$d_\mathrm{I}$$ d I satisfies a stability property generalizing one which $$d_\mathrm{B}$$ d B is known to satisfy, and that in addition, if $$d$$ d is any other pseudometric on multidimensional persistence modules satisfying the same stability property, then $$d\le d_\mathrm{I}$$ d ≤ d I . We also show that a variant of this universality result holds for $$d_\mathrm{B}$$ d B , over arbitrary fields. Finally, we show that $$d_\mathrm{I}$$ d I restricts to a metric on isomorphism classes of finitely presented multidimensional persistence modules. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Foundations of Computational Mathematics Springer Journals

The Theory of the Interleaving Distance on Multidimensional Persistence Modules

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References (50)

Publisher
Springer Journals
Copyright
Copyright © 2015 by SFoCM
Subject
Mathematics; Numerical Analysis; Economics general; Applications of Mathematics; Linear and Multilinear Algebras, Matrix Theory; Math Applications in Computer Science; Computer Science, general
ISSN
1615-3375
eISSN
1615-3383
DOI
10.1007/s10208-015-9255-y
Publisher site
See Article on Publisher Site

Abstract

In 2009, Chazal et al. introduced $$\epsilon $$ ϵ -interleavings of persistence modules. $$\epsilon $$ ϵ -interleavings induce a pseudometric $$d_\mathrm{I}$$ d I on (isomorphism classes of) persistence modules, the interleaving distance. The definitions of $$\epsilon $$ ϵ -interleavings and $$d_\mathrm{I}$$ d I generalize readily to multidimensional persistence modules. In this paper, we develop the theory of multidimensional interleavings, with a view toward applications to topological data analysis. We present four main results. First, we show that on 1-D persistence modules, $$d_\mathrm{I}$$ d I is equal to the bottleneck distance $$d_\mathrm{B}$$ 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 $$\epsilon $$ ϵ -interleaving relation on multidimensional persistence modules. This expresses transparently the sense in which two $$\epsilon $$ ϵ -interleaved modules are algebraically similar. Third, using this characterization, we show that when we define our persistence modules over a prime field, $$d_\mathrm{I}$$ d I satisfies a universality property. This universality result is the central result of the paper. It says that $$d_\mathrm{I}$$ d I satisfies a stability property generalizing one which $$d_\mathrm{B}$$ d B is known to satisfy, and that in addition, if $$d$$ d is any other pseudometric on multidimensional persistence modules satisfying the same stability property, then $$d\le d_\mathrm{I}$$ d ≤ d I . We also show that a variant of this universality result holds for $$d_\mathrm{B}$$ d B , over arbitrary fields. Finally, we show that $$d_\mathrm{I}$$ d I restricts to a metric on isomorphism classes of finitely presented multidimensional persistence modules.

Journal

Foundations of Computational MathematicsSpringer Journals

Published: Mar 24, 2015

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