Random Čech complexes on ℝ^d: decrackling the noise with local scalings

2019/12/22 by Henry‐Louis de Kergorlay, de Kergorlay, Henry-Louis
Computer Science · #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.1912.10548

Abstract

We investigate the homology of an unbounded noisy sample on ℝ^d, under various assumptions on the sampling density. This investigation is based on previous results by Adler, Bobrowski, and Weinberger (\citecrackle), and Owada and Adler (\citetopoCrackle). There, it was found that unbounded noise generally introduces non-vanishing homology, a phenomenon called topological crackle, unless the density has superexponential decay on ℝ^d. We show how some well-chosen 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 decrackling the noise.

Related

• Crackle: The Persistent Homology of Noise
• Strange Random Topology of the Circle
• Random Čech Complexes on Manifolds with Boundary
• On the Vanishing of Homology in Random Čech Complexes
• Homological Connectivity in Random Čech Complexes