Convergence of persistence diagrams for topological crackle

Abstract

In this paper we study the persistent homology associated with topological crackle generated by distributions with an unbounded support. Persistent homology is a topological and algebraic structure that tracks the creation and destruction of homological cycles (generalizations of loops or holes) in different dimensions. Topological crackle is a term that refers to homological cycles generated by "noisy" samples where the support is unbounded. We aim to establish weak convergence results for persistence diagramsa point process representation for persistent homology, where each homological cycle is represented by its (birth, death) coordinates. In this work we treat persistence diagrams as random closed sets, so that the resulting weak convergence is defined in terms of the Fell topology. In this framework we show that the limiting persistence diagrams can be divided into two parts. The first part is a deterministic limit containing a densely-growing number of persistence pairs with a short lifespan. The second part is a two-dimensional Poisson process, representing persistence pairs with a longer lifespan.2010 Mathematics Subject Classification. Primary 60G70. Secondary 55U10, 60F05, 60G55.

Random Simplicial Complexes: Models and Phenomena

“…A different approach was taken in [103], where persistence diagrams were studied from the point of view of set-topology. In that case it was shown that each persistence diagram can be roughly split into three different areas.…”

“…• Functional limit theorems: One can examine functionals such as the Betti numbers and the Euler characteristic in a dynamic setting, and seek a limit in the form of a stochastic (Gaussian) process. In [103,108], geometric complexes were studied, and the dynamics was the growing connectivity radius in the complex. The results show that the limiting process is indeed Gaussian.…”

Functional strong law of large numbers for Betti numbers in the tail

“…After the pioneering paper of [1], the layered structure in Figure 2 has been intensively studied via the behavior of various topological invariants [16,14,15,17,22]. In particular, from the viewpoints of (1.4), we have in an asymptotic sense,…”

“…As expected from Figure 2, it is not surprising that the stochastic features of (1.4) drastically vary, depending on how rapidly R n diverges. Among many of the related studies [16,14,15,17,22], the all consist of k + 2 points). If R n diverges even more slowly, so that R n = R d,n = (Cn) 1/α , then U (t) becomes highly connected in the area close to a boundary of a weak core.…”

“…In the context of topological crackle, however, the limit theorems for topological invariants heavily depend on the decay rate of a probability density. For instance, according to [1,16,17], the layered structure in Figure 2 appears only when the distribution for X n has a tail at least as heavy as that of an exponential distribution. Therefore, if X n is drawn from a Gaussian distribution, for each k ≥ 1, β k,n (t) in (1.4) simply vanishes as n → ∞.…”