Confidence Bands for Multiparameter Persistence Landscapes

TL;DR

The paper develops statistical inference for MPH landscapes by proving a functional CLT and constructing bootstrap confidence bands. It presents a functional CLT showing that sqrt(n)(avg - mean) converges to a Gaussian process with covariance $\kappa(x,y) = E[\lambda(x)\lambda(y)] - E[\lambda(x)] E[\lambda(y)]$. An algorithm for confidence bands is implemented and demonstrated on synthetic data with 2D filtrations (Vietoris-Rips and kernel-density superlevel sets), using the first landscape of H1 and a maximum-depth-band classifier. The study highlights the value of MPH-based inference in noisy data analysis while noting computational bottlenecks that challenge real-data scalability.

Abstract

Multiparameter persistent homology is a generalization of classical persistent homology, a central and widely-used methodology from topological data analysis, which takes into account density estimation and is an effective tool for data analysis in the presence of noise. Similar to its classical single-parameter counterpart, however, it is challenging to compute and use in practice due to its complex algebraic construction. In this paper, we study a popular and tractable invariant for multiparameter persistent homology in a statistical setting: the multiparameter persistence landscape. We derive a functional central limit theorem for multiparameter persistence landscapes, from which we compute confidence bands, giving rise to one of the first statistical inference methodologies for multiparameter persistence landscapes. We provide an implementation of confidence bands and demonstrate their application in a machine learning task on synthetic data.

Figures (1)

• Figure 1: Input point clouds and average landscapes with the standard bootstrap confidence bands.Above: Samples with $N=500$ points over a sphere of radius $R=3$ (left), a torus with radii $R=3$ and $r=0.7$ (center), and a Klein bottle (right) with Gaussian and salt and pepper noise added. Points are colored by the value of the Kernel density estimation. Below: Average sample landscapes for $n=100$ samples and confidence bands with Standard Bootstrap.

Theorems & Definitions (4)

• Definition 2.1: Multiparameter Persistence Landscape Bubenik2015Vipond2020
• Theorem 3.1: Uniform Convergence of Multiparameter Persistence Landscapes
• Lemma 3.2
• proof : Theorem \ref{['thm:fclt']}