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 ${\mathit{r}}_{\mathit{n}}^{-1}{\mathcal{X}}_{\mathit{n}}=\{{\mathit{r}}_{\mathit{n}}^{-1}{\mathit{X}}_1,\dots ,{\mathit{r}}_{\mathit{n}}^{-1}{\mathit{X}}_{\mathit{n}}\}$, such that ${\mathit{r}}_{\mathit{n}}\to 0$ as $\mathit{n}\to \infty$. We treat persistence diagrams as a point process and establish their limit theorems in the sparse regime: $\mathit{n}{\mathit{r}}_{\mathit{n}}^{\mathit{d}}\to 0$, $\mathit{n}\to \infty$. In this setting, we show that the asymptotics of the kth persistence diagram depends on the limit value of the sequence ${\mathit{n}}^{\mathit{k}+2}{\mathit{r}}_{\mathit{n}}^{\mathit{d}(\mathit{k}+1)}$. If ${\mathit{n}}^{\mathit{k}+2}{\mathit{r}}_{\mathit{n}}^{\mathit{d}(\mathit{k}+1)}\to \infty$, the scaled persistence diagram converges to a deterministic Radon measure almost surely in the vague metric. If ${\mathit{r}}_{\mathit{n}}$ decays faster so that ${\mathit{n}}^{\mathit{k}+2}{\mathit{r}}_{\mathit{n}}^{\mathit{d}(\mathit{k}+1)}\to \mathit{c}\in (0,\infty )$, the persistence diagram weakly converges to a limiting point process without normalization. Finally, if ${\mathit{n}}^{\mathit{k}+2}{\mathit{r}}_{\mathit{n}}^{\mathit{d}(\mathit{k}+1)}\to 0$, 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.