Convergence of persistence diagram in the sparse regime

Abstract

The objective of this paper is to examine the asymptotic behavior of persistence diagrams associated with Čech filtration. A persistence diagram is a graphical descriptor of a topological and algebraic structure of geometric objects. We consider Čech filtration over a scaled random sample $r_n^{-1}{\mathcal{X}}_n=\{r_n^{-1}{X}_1,\dots ,r_n^{-1}{X}_n\}$, such that $r_n\to 0$ as $n\to \infty$. We treat persistence diagrams as a point process and establish their limit theorems in the sparse regime: $n{r}_n^d\to 0$, $n\to \infty$. In this setting, we show that the asymptotics of the $k$th persistence diagram depends on the limit value of the sequence $n^{k+2}{r}_n^{d(k+1)}$. If $n^{k+2}{r}_n^{d(k+1)}\to \infty$, the scaled persistence diagram converges to a deterministic Radon measure almost surely in the vague metric. If $r_n$ decays faster so that $n^{k+2}{r}_n^{d(k+1)}\to c\in (0,\infty )$, the persistence diagram weakly converges to a limiting point process without normalization. Finally, if $n^{k+2}{r}_n^{d(k+1)}\to 0$, the sequence of probability distributions of a persistence diagram should be normalized, and the resulting convergence will be treated in terms of the ${\mathcal{M}}_0$-topology.