A limit theorem for persistence diagrams of random filtered complexes built over marked point processes

TL;DR

This paper proves a law of large numbers for persistence diagrams derived from random filtered complexes built over marked point processes, advancing understanding of their asymptotic behavior in Euclidean spaces.

Contribution

It introduces a law of large numbers for persistence diagrams of complexes constructed from marked point processes, a novel theoretical result in topological data analysis.

Findings

Law of large numbers established for persistence diagrams

Applicability to complexes over various set sizes and shapes

Asymptotic behavior characterized as observation window grows

Abstract

We consider random filtered complexes built over marked point processes on Euclidean spaces. Examples of our filtered complexes include a filtration of $\check{\textrm{C}}$ech complexes of a family of sets with various sizes, growths, and shapes. We establish the law of large numbers for persistence diagrams as the size of the convex window observing a marked point process tends to infinity.