TY - JOUR

T1 - Persistent Magnitude

AU - Govc, Dejan

AU - Hepworth, Richard

N1 - 2010 Mathematics Subject Classification. Primary 55N99; Secondary 55N35, 51F99,
11A25. DG was supported by EPSRC grant EP/P025072/1.

PY - 2021/3/31

Y1 - 2021/3/31

N2 - 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.

AB - 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.

KW - COMPLEXES

KW - HOMOLOGY

UR - http://www.scopus.com/inward/record.url?scp=85089484448&partnerID=8YFLogxK

U2 - 10.1016/j.jpaa.2020.106517

DO - 10.1016/j.jpaa.2020.106517

M3 - Article

VL - 225

JO - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

SN - 0022-4049

IS - 3

M1 - 106517

ER -