Persistence by Parts: Multiscale Feature Detection via Distributed Persistent Homology

Abstract: A method is presented for the distributed computation of persistent homology, based on an extension of the generalized Mayer-Vietoris principle to filtered spaces. Cellular cosheaves and spectral sequences are used to compute global persistent homology based on local computations indexed by a scalar field. These techniques permit computation localized not merely by geography, but by other features of data points, such as density. As an example of the latter, the construction is used in the multi-scale analysis of point clouds to detect features of varying sizes that are overlooked by standard persistent homology.