Algebraic Homotopy Interleaving Distance

Berkouk, Nicolas

doi:10.1007/978-3-030-80209-7_70

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2023

2024

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Cited by 2 publications

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“…To do so, the TDA community has been mostly using module-theoretic notions, such as the rankinvariant [15,16], the Hilbert function or the graded Betti numbers [24,5,35,31]. From a sheaf-theoretic perspective, a natural numerical invariant to consider is the local Euler characteristic.…”

Section: Introductionmentioning

confidence: 99%

Persistence and the Sheaf-Function Correspondence

Berkouk

2023

Forum of Mathematics, Sigma

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The sheaf-function correspondence identifies the group of constructible functions on a real analytic manifold M with the Grothendieck group of constructible sheaves on M. When M is a finite dimensional real vector space, Kashiwara-Schapira have recently introduced the convolution distance between sheaves of $\mathbf {k}$ -vector spaces on M. In this paper, we characterize distances on the group of constructible functions on a real finite dimensional vector space that can be controlled by the convolution distance through the sheaf-function correspondence. Our main result asserts that such distances are almost trivial: they vanish as soon as two constructible functions have the same Euler integral. We formulate consequences of our result for Topological Data Analysis: there cannot exist nontrivial additive invariants of persistence modules that are continuous for the interleaving distance.

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