On the consistency and asymptotic normality of multiparameter persistent Betti numbers

TL;DR

This paper proves the strong consistency and asymptotic normality of multiparameter persistent Betti numbers in growing domains, enabling statistical inference in topological data analysis with multiple filtration parameters.

Contribution

It establishes the asymptotic properties of multiparameter persistent Betti numbers for the first time, covering both the ch bifiltration and multicover bifiltration models.

Findings

Asymptotic normality holds for multiparameter persistent Betti numbers.

The results enable goodness-of-fit tests for complex spatial point processes.

Simulation demonstrates the statistical power of the proposed tests.

Abstract

The persistent Betti numbers are used in topological data analysis to infer the scales at which topological features appear and disappear in the filtration of a topological space. Most commonly by means of the corresponding barcode or persistence diagram. While this approach to data science has been very successful, it suffers from sensitivity to outliers, and it does not allow for additional filtration parameters. Such parameters naturally appear when a cloud of data points comes together with additional measurements taken at the locations of the data. For these reasons, multiparameter persistent homology has recently received significant attention. In particular, the multicover and \v{C}ech bifiltration have been introduced to overcome the aforementioned shortcomings. In this work, 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 \v{C}ech 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.