Scholarlink

<p>In topological data analysis, the notions of persistent homology, birthtime, lifetime, and deathtime are used to assign and capture relevant cycles (i.e., topological features) of a point cloud, such as loops and cavities. In particular, cycles with a large lifetime are of special interest. In this paper, we study such large-lifetime cycles when the point cloud is modeled as a Poisson point process. First, we consider the case with no bound on the deathtime, where we establish Poisson convergence of the centers of large-lifetime cycles on the 2-dimensional flat torus. Afterwards, by imposing a bound on the deathtime, we enter a sparse connectivity regime, and we prove joint Poisson convergence of the centers, lifetimes, and deathtimes in dimensions <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\varvec{d \geqslant 2}\)</EquationSource> </InlineEquation> under suitable model conditions.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Poisson approximation of large-lifetime cycles

  • 期刊论文

  • 出版模式：

    开放获取

  • 发表日期：

    2026年2月26日

查看全文
  • Christian Hirsch,
  • Nikolaj N. Lundbye,
  • Moritz Otto

摘要

In topological data analysis, the notions of persistent homology, birthtime, lifetime, and deathtime are used to assign and capture relevant cycles (i.e., topological features) of a point cloud, such as loops and cavities. In particular, cycles with a large lifetime are of special interest. In this paper, we study such large-lifetime cycles when the point cloud is modeled as a Poisson point process. First, we consider the case with no bound on the deathtime, where we establish Poisson convergence of the centers of large-lifetime cycles on the 2-dimensional flat torus. Afterwards, by imposing a bound on the deathtime, we enter a sparse connectivity regime, and we prove joint Poisson convergence of the centers, lifetimes, and deathtimes in dimensions \(\varvec{d \geqslant 2}\) under suitable model conditions.