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

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}.