Random \v{C}ech complexes on $\mathbb{R}^d$: decrackling the noise with local scalings

TL;DR

This paper explores how local scalings in random Čech complexes can reduce topological noise (crackle) caused by unbounded sampling noise, extending the class of densities where noise doesn't affect homology.

Contribution

It introduces variable bandwidth constructions that mitigate topological crackle for a broader class of light tail densities, advancing noise reduction techniques in topological data analysis.

Findings

Variable bandwidths can prevent crackle in light tail densities.

Decrackling extends noise robustness beyond superexponential decay densities.

Homology remains stable under new local scaling methods.

Abstract

We investigate the homology of an unbounded noisy sample on $\mathbb{R}^d$, under various assumptions on the sampling density. This investigation is based on previous results by Adler, Bobrowski, and Weinberger (\cite{crackle}), and Owada and Adler (\cite{topoCrackle}). There, it was found that unbounded noise generally introduces non-vanishing homology, a phenomenon called \textit{topological crackle}, unless the density has superexponential decay on $\mathbb{R}^d$. We show how some well-chosen \textit{non-trivial} variable bandwidth constructions can extend the class of densities where crackle doesn't occur to any light tail density with mild assumptions, what we call \textit{decrackling the noise}.