Persistent Magnitude

Dejan Govc* (Corresponding Author), Richard Hepworth

*Corresponding author for this work

Research output: Contribution to journalArticle

Abstract

In this paper we introduce the persistent magnitude, a new numerical invariant of (sufficiently nice) graded persistence modules. It is a weighted and signed count of the bars of the persistence module, in which a bar of the form [a, b) in degree d is counted with weight (e−a −e −b ) and sign (−1)d. Persistent magnitude has good formal properties, such as additivity with respect to exact sequences and compatibility with tensor products, and has interpretations in terms of both the associated graded functor, and the Laplace transform. Our definition is inspired by Otter’s notion of blurred magnitude homology: we show that the magnitude of a finite metric space is precisely the persistent magnitude of its blurred magnitude homology. Turning this result on its head, we obtain a strategy for turning existing persistent homology theories into new numerical invariants by applying the persistent magnitude. We explore this strategy in detail in the case of persistent homology of Morse functions, and in the case of Rips homology.
Original languageEnglish
Article number106517
Number of pages40
JournalJournal of Pure and Applied Algebra
Volume225
Issue number3
Early online date11 Aug 2020
DOIs
Publication statusE-pub ahead of print - 11 Aug 2020

Keywords

  • COMPLEXES
  • HOMOLOGY

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