2021/03/24 by Takashi Owada, Owada, Takashi · 1 citation
Computer Science · Mathematics · Medicine · #Advanced Neuroimaging Techniques and Applications #Algebraic Topology (math.AT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Probability (math.PR) #Topological and Geometric Data Analysis #math.AT #math.PR
paper · pdf · doi:10.48550/arxiv.2103.12943
openalex publication_date 2021/03/24 · arxiv created 2021/09/14 · arxiv updated 2021/09/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 rn-1\mathcal Xn = \ rn-1X1,…, rn-1Xn \, such that rn→ 0 as n→∞. We treat persistence diagrams as a point process and establish their limit theorems in the sparse regime: nrnd→0, n→∞. In this setting, we show that the asymptotics of the kth persistence diagram depends on the limit value of the sequence nk+2rnd(k+1). If nk+2rnd(k+1) → ∞, the scaled persistence diagram converges to a deterministic Radon measure almost surely in the vague metric. If rn decays faster so that nk+2rnd(k+1) → c∈ (0,∞), the persistence diagram weakly converges to a limiting point process without normalization. Finally, if nk+2rnd(k+1) → 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 M0-topology.