Metrics for Generalized Persistence Modules
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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.
KeywordsPersistent topology Interleaving Stability Sublinear projections Superlinear families Inverse-image persistence
Mathematics Subject Classification55U99 68U05
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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