The Theory of the Interleaving Distance on Multidimensional Persistence Modules

@article{Lesnick2015TheTO,
  title={The Theory of the Interleaving Distance on Multidimensional Persistence Modules},
  author={Michael Lesnick},
  journal={Foundations of Computational Mathematics},
  year={2015},
  volume={15},
  pages={613-650}
}
  • M. Lesnick
  • Published 26 June 2011
  • Mathematics
  • Foundations of Computational Mathematics
In 2009, Chazal et al. introduced $$\epsilon $$ϵ-interleavings of persistence modules. $$\epsilon $$ϵ-interleavings induce a pseudometric $$d_\mathrm{I}$$dI on (isomorphism classes of) persistence modules, the interleaving distance. The definitions of $$\epsilon $$ϵ-interleavings and $$d_\mathrm{I}$$dI 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… 
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In topological data analysis persistence modules are used to distinguish the legitimate topological features of a finite data set from noise. Interleavings between persistence modules feature
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A polynomial time algorithm is presented to compute the bottleneck distance for modules from indecomposables, which bounds the interleaving distance from above, and another algorithm is given to compute a new distance called {\em dimension distance} that bounds it from below.
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We characterize the class of persistence modules indexed over $$\mathbb {R}^2$$ R 2 that are decomposable into summands whose supports have the shape of a block —i.e. a horizontal band, a vertical
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TLDR
A polynomial time algorithm is presented to compute the bottleneck distance for modules from indecomposables, which bounds the interleaving distance from above, and another algorithm is given to compute a new distance called {\em dimension distance} that bounds it from below.
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TLDR
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TLDR
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  • Mathematics
    J. Appl. Comput. Topol.
  • 2018
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Theory of interleavings on categories with a flow
The interleaving distance was originally defined in the field of Topological Data Analysis (TDA) by Chazal et al. as a metric on the class of persistence modules parametrized over the real line.
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