CONVERGENCE OF PERSISTENCE DIAGRAM IN THE SPARSE REGIME

• 成果类型: Article
• 署名作者: Owada, Takashi
• 署名单位: Purdue University System; Purdue University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/22-AAP1800
• 发表日期: 2022
• 页码: 4706-4736
• 关键词: limit-theorems; betti numbers; Regular Variation; U-statistics; TOPOLOGY; COHOMOLOGY; complexes; HOMOLOGY

摘要

The objective of this paper is to examine the asymptotic behavior of persistence diagrams associated with Cech filtration. A persistence diagram is a graphical descriptor of a topological and algebraic structure of geometric objects. We consider Cech filtration over a scaled random sample r(n)(-1) chi(n) = {r(n)(-1) x(1), ..., r(n)(-1) X-n}, such that r(n) -> 0 as n -> infinity. We treat per- sistence diagrams as a point process and establish their limit theorems in the sparse regime: nr(n)(d) -> 0, n -> infinity. In this setting, we show that the asymptotics of the kth persistence diagram depends on the limit value of the se- quence n(k+2)r(n)(d(k+1)). If n(k+2)r(n)(d(k+1)) -> infinity, 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)) -> c is an element of (0, infinity), the persistence diagram weakly converges to a limiting point process without normalization. Finally, if n(k+2)r(n)(d(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 M-0-topology.