On limit theorems for persistent Betti numbers from dependent data

2019/05/10 by Krebs, Johannes · 2 citations
#FOS: Mathematics #Probability (math.PR) #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.1905.04045

Abstract

We study persistent Betti numbers and persistence diagrams obtained from time series and random fields. It is well known that the persistent Betti function is an efficient descriptor of the topology of a point cloud. So far, convergence results for the (r,s)-persistent Betti number of the qth homology group, β^r,s_q, were mainly considered for finite-dimensional point cloud data obtained from i.i.d. observations or stationary point processes such as a Poisson process. In this article, we extend these considerations. We derive limit theorems for the pointwise convergence of persistent Betti numbers β^r,s_q in the critical regime under quite general dependence settings.

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• Persistent reachability homology in machine learning applications
• Limit Theorems for Verbose Persistence Diagrams

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