Limit theorems for persistence diagrams

Yasuaki Hiraoka, Tomoyuki Shirai, Khanh Duy Trinh

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

The persistent homology of a stationary point process on RN is studied in this paper. As a generalization of continuum percolation theory, we study higher dimensional topological features of the point process such as loops, cavities, etc. in a multiscale way. The key ingredient is the persistence diagram, which is an expression of the persistent homology. We prove the strong law of large numbers for persistence diagrams as the window size tends to infinity and give a sufficient condition for the support of the limiting persistence diagram to coincide with the geometrically realizable region. We also discuss a central limit theorem for persistent Betti numbers.

Original languageEnglish
Pages (from-to)2740-2780
Number of pages41
JournalAnnals of Applied Probability
Volume28
Issue number5
DOIs
Publication statusPublished - Oct 1 2018

Fingerprint

Limit Theorems
Persistence
Diagram
Point Process
Homology
Continuum Percolation
Percolation Theory
Strong law of large numbers
Betti numbers
Stationary point
Stationary Process
Central limit theorem
Cavity
High-dimensional
Limiting
Infinity
Tend
Sufficient Conditions
Limit theorems
Diagrams

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Cite this

Limit theorems for persistence diagrams. / Hiraoka, Yasuaki; Shirai, Tomoyuki; Trinh, Khanh Duy.

In: Annals of Applied Probability, Vol. 28, No. 5, 01.10.2018, p. 2740-2780.

Research output: Contribution to journalArticle

Hiraoka, Yasuaki ; Shirai, Tomoyuki ; Trinh, Khanh Duy. / Limit theorems for persistence diagrams. In: Annals of Applied Probability. 2018 ; Vol. 28, No. 5. pp. 2740-2780.
@article{2d3788d89d9342c1a68245038705fdda,
title = "Limit theorems for persistence diagrams",
abstract = "The persistent homology of a stationary point process on RN is studied in this paper. As a generalization of continuum percolation theory, we study higher dimensional topological features of the point process such as loops, cavities, etc. in a multiscale way. The key ingredient is the persistence diagram, which is an expression of the persistent homology. We prove the strong law of large numbers for persistence diagrams as the window size tends to infinity and give a sufficient condition for the support of the limiting persistence diagram to coincide with the geometrically realizable region. We also discuss a central limit theorem for persistent Betti numbers.",
author = "Yasuaki Hiraoka and Tomoyuki Shirai and Trinh, {Khanh Duy}",
year = "2018",
month = "10",
day = "1",
doi = "10.1214/17-AAP1371",
language = "English",
volume = "28",
pages = "2740--2780",
journal = "Annals of Applied Probability",
issn = "1050-5164",
publisher = "Institute of Mathematical Statistics",
number = "5",

}

TY - JOUR

T1 - Limit theorems for persistence diagrams

AU - Hiraoka, Yasuaki

AU - Shirai, Tomoyuki

AU - Trinh, Khanh Duy

PY - 2018/10/1

Y1 - 2018/10/1

N2 - The persistent homology of a stationary point process on RN is studied in this paper. As a generalization of continuum percolation theory, we study higher dimensional topological features of the point process such as loops, cavities, etc. in a multiscale way. The key ingredient is the persistence diagram, which is an expression of the persistent homology. We prove the strong law of large numbers for persistence diagrams as the window size tends to infinity and give a sufficient condition for the support of the limiting persistence diagram to coincide with the geometrically realizable region. We also discuss a central limit theorem for persistent Betti numbers.

AB - The persistent homology of a stationary point process on RN is studied in this paper. As a generalization of continuum percolation theory, we study higher dimensional topological features of the point process such as loops, cavities, etc. in a multiscale way. The key ingredient is the persistence diagram, which is an expression of the persistent homology. We prove the strong law of large numbers for persistence diagrams as the window size tends to infinity and give a sufficient condition for the support of the limiting persistence diagram to coincide with the geometrically realizable region. We also discuss a central limit theorem for persistent Betti numbers.

UR - http://www.scopus.com/inward/record.url?scp=85052699592&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85052699592&partnerID=8YFLogxK

U2 - 10.1214/17-AAP1371

DO - 10.1214/17-AAP1371

M3 - Article

VL - 28

SP - 2740

EP - 2780

JO - Annals of Applied Probability

JF - Annals of Applied Probability

SN - 1050-5164

IS - 5

ER -