On the consistency and asymptotic normality of multiparameter persistent Betti numbers

Abstract

The persistent Betti numbers are used in topological data analysis (TDA) to infer the scales at which topological features appear and disappear in the filtration of a topological space. Understanding the statistical foundations of these descriptors, and their corresponding barcodes, is thus an important problem that has received a significant amount of attention. There are, however, many situations for which it is natural to simultaneously consider multiple filtration parameters, e.g. when a point cloud comes equipped with additional measurements taken at the locations of the data. Multiparameter persistent homology (MPH) was introduced to accommodate such multifiltrations, and it has become one of the most active areas of research within TDA, with exciting progress on multiple fronts. The present work offers a first step towards a rigorous statistical foundation of MPH. Notably, we establish the strong consistency and asymptotic normality of the multiparameter persistent Betti numbers in growing domains. Our asymptotic results are established for a general framework encompassing both the marked Čech bifiltration, as well as the multicover bifiltration constructed on the null model of an independently marked Poisson point process. In a simulation study, we explain how the asymptotic normality can be used to derive tests for the goodness of fit. The statistical power of such tests is illustrated through different alternatives exhibiting more clustering, or more repulsion than the null model.