Limit Theorems for Multiparameter Persistent Betti Numbers

Abstract

Multiparameter persistence provides a natural framework for data sets with several relevant filtration parameters, such as geometric scale, density, or coverage depth. Unlike one-parameter persistence, it does not generally admit a barcode decomposition, so one instead studies computable summaries such as the rank invariant and persistent Betti numbers. In this talk, I will describe laws of large numbers and functional central limit theorems for persistent Betti numbers of marked Čech and multicover bifiltrations constructed from random point clouds. I will also discuss their use in goodness-of-fit testing and conclude with an outlook on non-monotone, time-dependent models and zigzag persistence.

Joint (ongoing) work with M. Botnan, R. Davidsen, and N. Lundbye.