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Metrics for Generalized Persistence Modules

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Abstract

We consider the question of defining interleaving metrics on generalized persistence modules over arbitrary preordered sets. Our constructions are functorial, which implies a form of stability for these metrics. We describe a large class of examples, inverse-image persistence modules, which occur whenever a topological space is mapped to a metric space. Several standard theories of persistence and their stability can be described in this framework. This includes the classical case of sublevelset persistent homology. We introduce a distinction between ‘soft’ and ‘hard’ stability theorems. While our treatment is direct and elementary, the approach can be explained abstractly in terms of monoidal functors.

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Notes

  1. The word ‘diagram’ is used in two senses here, for unavoidable historical reasons.

  2. In the earlier work [3], the target category is arbitrary and the indexing category is \(\mathbf {R}\).

  3. Lawvere spaces are also called extended quasi-pseudometric spaces.

  4. For preordered sets, the statement is that there exist natural isomorphisms between the functors \(\Omega _0^2\) and \(\Omega _0\) and between the functors \(\Omega _0 \Omega _{\varepsilon }\), \(\Omega _{\varepsilon } \Omega _0\), and \(\Omega _{\varepsilon }\).

  5. A metric space is ‘proper’ if its closed bounded subsets are compact.

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Acknowledgments

The first author gratefully acknowledges the support of AFOSR Grant FA9550-13-1-0115. The second author thanks his home institution, Pomona College, for a sabbatical leave of absence in 2013–2014. The sabbatical was partially supported by the Simons Foundation (Grant #267571) and hosted by the Institute for Mathematics and its Applications, University of Minnesota, with funds provided by the National Science Foundation.

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Authors and Affiliations

  1. Department of Mathematics, Cleveland State University, Cleveland, OH, USA

    Peter Bubenik & Jonathan Scott

  2. Department of Mathematics, Pomona College, Claremont, CA, USA

    Vin de Silva

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  1. Peter Bubenik
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  2. Vin de Silva
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  3. Jonathan Scott
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Correspondence to Peter Bubenik.

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Communicated by Gunnar Carlsson.

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Bubenik, P., de Silva, V. & Scott, J. Metrics for Generalized Persistence Modules. Found Comput Math 15, 1501–1531 (2015). https://doi.org/10.1007/s10208-014-9229-5

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