Persistent Magnitude

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.

arXiv comment43 pages, 9 figures. v2: Modifications in response to referee comments. Final version, to appear in Journal of Pure and Applied Algebra

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Cited by in corpus (7)

• Homological Algebra for Persistence Modules
• Magnitude homology of enriched categories and metric spaces
• Hybrid transforms of constructible functions
• Entropy and Diversity: The Axiomatic Approach
• Alpha magnitude
• Semiclassical analysis of a nonlocal boundary value problem related to magnitude
• Convolution of Persistence Modules