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- Springer Journals
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- Copyright © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022. Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
- ISSN
- 2367-1726
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- 2367-1734
- DOI
- 10.1007/s41468-022-00103-8
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Abstract
We use weights on objects in an abelian category to define what we call a path metric. We introduce three special classes of weight: those compatible with short exact sequences; those induced by their path metric; and those which bound their path metric. We prove that these conditions are in fact equivalent, and call such weights exact. As a special case of a path metric, we obtain a distance for generalized persistence modules whose indexing category is a measure space. We use this distance to define Wasserstein distances, which coincide with the previously defined Wasserstein distances for one-parameter persistence modules. For one-parameter persistence modules, we also describe maps to and from an interval module, and we give a matrix reduction for monomorphisms and epimorphisms.
Journal
Journal of Applied and Computational Topology – Springer Journals
Published: Jun 1, 2023
Keywords: Distances for abelian categories; Persistence modules; Wasserstein distance; 55N31; 18E10