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Persistent magnitude. (English) Zbl 07251293
Summary: 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.
##### MSC:
 55N31 Persistent homology and applications, topological data analysis 51F99 Metric geometry 11A25 Arithmetic functions; related numbers; inversion formulas
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