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On the asymptotic normality of persistent Betti numbers

Johannes Krebs, Wolfgang Polonik

Abstract

Persistent Betti numbers are a major tool in persistent homology, a subfield of topological data analysis. Many tools in persistent homology rely on the properties of persistent Betti numbers considered as a two-dimensional stochastic process $ (r,s) \mapsto n^{-1/2} (β^{r,s}_q ( \mathcal{K}(n^{1/d} \mathcal{X}_n))-\mathbb{E}[β^{r,s}_q ( \mathcal{K}( n^{1/d} \mathcal{X}_n))])$. So far, pointwise limit theorems have been established in different settings. In particular, the pointwise asymptotic normality of (persistent) Betti numbers has been established for stationary Poisson processes and binomial processes with constant intensity function in the so-called critical (or thermodynamic) regime, see Yogeshwaran et al. [2017] and Hiraoka et al. [2018]. In this contribution, we derive a strong stabilization property (in the spirit of Penrose and Yukich [2001] of persistent Betti numbers and generalize the existing results on the asymptotic normality to the multivariate case and to a broader class of underlying Poisson and binomial processes. Most importantly, we show that the multivariate asymptotic normality holds for all pairs $(r,s)$, $0\le r\le s<\infty$, and that it is not affected by percolation effects in the underlying random geometric graph.

On the asymptotic normality of persistent Betti numbers

Abstract

Persistent Betti numbers are a major tool in persistent homology, a subfield of topological data analysis. Many tools in persistent homology rely on the properties of persistent Betti numbers considered as a two-dimensional stochastic process . So far, pointwise limit theorems have been established in different settings. In particular, the pointwise asymptotic normality of (persistent) Betti numbers has been established for stationary Poisson processes and binomial processes with constant intensity function in the so-called critical (or thermodynamic) regime, see Yogeshwaran et al. [2017] and Hiraoka et al. [2018]. In this contribution, we derive a strong stabilization property (in the spirit of Penrose and Yukich [2001] of persistent Betti numbers and generalize the existing results on the asymptotic normality to the multivariate case and to a broader class of underlying Poisson and binomial processes. Most importantly, we show that the multivariate asymptotic normality holds for all pairs , , and that it is not affected by percolation effects in the underlying random geometric graph.

Paper Structure

This paper contains 10 sections, 14 theorems, 33 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Main results
  4. Related results
  5. Background for the strong stabilization
  6. Technical results
  7. Stabilization
  8. The asymptotic normality for the Poisson process
  9. The asymptotic normality for the binomial process
  10. Appendix

Key Result

Theorem 1

Let $\lambda>0$, $(r,s)\in\Delta$ and $q \in \{0,\ldots,d-1\}$. There is an $\mathcal{F}$-measurable random variable $S^{(r,s)}_q \coloneqq S^{(r,s)}_q(\mathcal{P}(\lambda))$ which is $a.s.$ finite such that for all finite sets $A \subseteq \mathbb{R}^d\setminus B(0, S^{(r,s)}_q)$, the add one cost

Figures (2)

  • Figure 1: $\beta^{r,s}_q(\mathcal{K}(P))$ equals the number of points in the gray-shaded rectangle; the point on the dashed red line is not counted whereas the point on the solid red line is.
  • Figure 3: Illustration of encounters of maximal chains in two dimensions (in a reduced set-up and not true to scale). The left figure depicts several encounters inside boxes $Q(y,m)$ (for certain $y\in\mathbb{Z}^d$): The (blue) central boxes are located inside the (black) encounter boxes which are part of the (black) lattice which partitions the plane. Each (green) encounter configuration merges three branches (red and partly in green and orange) through the corresponding (orange) intermediate configuration. The right figure considers a specific central box $Q(y,n)$ (blue). Here a suitable configuration (violet) inside the central box converts the corresponding branches to maximal chains.

Theorems & Definitions (14)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 1
  • Proposition 2: Pointwise normality of (persistent) Betti numbers
  • Theorem 4
  • Lemma 1
  • Theorem 5: Uniform stabilization
  • Lemma 2: Corollary of hiraoka2018limit Lemma 2.11
  • Proposition 3: Encounters of maximal chains
  • ...and 4 more