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Activity Number: 47 - Geometric and Topological Information in Data Analysis
Type: Topic-Contributed
Date/Time: Sunday, August 8, 2021 : 3:30 PM to 5:20 PM
Sponsor: IMS
Abstract #317251
Title: Convergence of Persistence Diagram in the Subcritical Regime
Author(s): Takashi Owada*
Companies: Purdue University, Department of Statistics
Keywords: stochastic topology; persistent homology; persistence diagram; persistent Betti number; point process; extreme value theory
Abstract:

The objective of this study is to examine the asymptotic behavior of persistence diagrams associated with \v{C}ech filtration. A persistence diagram is a graphical descriptor of a topological and algebraic structure of geometric objects. We consider \v{C}ech 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 subcritical regime: $nr_n^d\to0$, $n\to\infty$. 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 convergence will be treated in terms of the $\mathcal M_0$-topology


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