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 language | English |
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Article number | 106517 |
Number of pages | 40 |
Journal | Journal of Pure and Applied Algebra |
Volume | 225 |
Issue number | 3 |
Early online date | 11 Aug 2020 |
DOIs | |
Publication status | Published - 31 Mar 2021 |
Bibliographical note
2010 Mathematics Subject Classification. Primary 55N99; Secondary 55N35, 51F99,11A25. DG was supported by EPSRC grant EP/P025072/1.
Keywords
- COMPLEXES
- HOMOLOGY