Limit theorems for Betti numbers of extreme sample clouds with application to persistence barcodes

DOI10.1214/17-AAP1375zbMath1402.60065MaRDI QIDQ1617144

Takashi Owada

Publication date: 7 November 2018

Published in: The Annals of Applied Probability (Search for Journal in Brave)

Full work available at URL: https://projecteuclid.org/euclid.aoap/1535443235

zbMATH Keywords

central limit theoremextreme value theoryBetti numberpersistent homologyPoisson limit theoremrandom topology

Mathematics Subject Classification ID

Geometric probability and stochastic geometry (60D05) Extreme value theory; extremal stochastic processes (60G70) Interacting random processes; statistical mechanics type models; percolation theory (60K35) Simplicial sets and complexes in algebraic topology (55U10)

Related Items (12)

Geometric and probabilistic limit theorems in topological data analysis ⋮ Functional strong laws of large numbers for Euler characteristic processes of extreme sample clouds ⋮ Convergence of persistence diagrams for topological crackle ⋮ Bootstrapping persistent Betti numbers and other stabilizing statistics ⋮ Functional central limit theorems for local statistics of spatial birth-death processes in the thermodynamic regime ⋮ Topological data analysis of noise: uniform unimodal distributions ⋮ Thresholds for vanishing of `isolated' faces in random Čech and Vietoris-Rips complexes ⋮ Limit theorems for process-level Betti numbers for sparse and critical regimes ⋮ Topological crackle of heavy-tailed moving average processes ⋮ On limit theorems for persistent Betti numbers from dependent data ⋮ Functional strong law of large numbers for Betti numbers in the tail ⋮ Convergence of persistence diagram in the sparse regime

Uses Software

• ismev

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Confidence sets for persistence diagrams
• Erratum to: ``Limit theorems for Betti numbers of random simplicial complexes
• Limit theorems for Betti numbers of random simplicial complexes
• Random geometric complexes in the thermodynamic regime
• Functional central limit theorem for subgraph counting processes
• Random geometric complexes
• The scaling limit of Poisson-driven order statistics with applications in geometric probability
• Functional Poisson approximation in Kantorovich-Rubinstein distance with applications to \(U\)-statistics and stochastic geometry
• On the topology of random complexes built over stationary point processes
• Meta densities and the shape of their sample clouds
• High risk scenarios and extremes. A geometric approach
• Extremes and related properties of random sequences and processes
• Two moments suffice for Poisson approximations: The Chen-Stein method
• Maximally persistent cycles in random geometric complexes
• Poisson limits for \(U\)-statistics.
• Computing persistent homology
• Topological persistence and simplification
• Limit theorems for point processes under geometric constraints (and topological crackle)
• The topology of probability distributions on manifolds
• Coverage in sensor networks via persistent homology
• Finding the homology of submanifolds with high confidence from random samples
• Crackle: the homology of noise
• The Shape of Asymptotic Dependence
• Topology of random simplicial complexes: a survey
• Random Geometric Graphs
• Topological pattern recognition for point cloud data
• Minimum spanning acycle and lifetime of persistent homology in the Linial-Meshulam process
• Heavy-Tail Phenomena
• Statistical topological data analysis using persistence landscapes
• Barcodes: The persistent topology of data
• An introduction to statistical modeling of extreme values

This page was built for publication: Limit theorems for Betti numbers of extreme sample clouds with application to persistence barcodes