Stability of 0-dimensional persistent homology in enriched and sparsified point clouds

TL;DR

This paper establishes bounds and stability results for 0-dimensional and codimension 1 persistent homology of various complexes derived from finite point clouds under operations like enrichment, sparsification, and grid alignment, with applications to ecological hypervolume analysis.

Contribution

It provides new theoretical bounds and stability guarantees for persistent homology under data modifications, with practical implementation in TopoAware for ecological data analysis.

Findings

Bounds for persistent homology under enrichment, sparsification, and grid alignment.

Duality identity for cubical complexes.

Implementation available in C++, Python, and R.

Abstract

We give bounds for dimension 0 persistent homology and codimension 1 homology of Vietoris--Rips, alpha, and cubical complex filtrations from finite sets related by enrichment (adding new elements), sparsification (removing elements), and aligning to a grid (uniformly discretizing elements). For enrichment we use barycentric subdivision, for sparsification we use a minimum separating distance, and for aligning to a grid we take the quotient when dividing each coordinate value by a fixed step size. We are motivated by applications presenting large and irregular datasets, and the development of persistent homology to better work with them. In particular, we consider an application to ecology, in which the state of an observed species is inferred through a high-dimensional space with environmental variables as dimensions. This ``hypervolume'' has geometry (volume, convexity) and topology (connectedness, homology), which are known to be related to the current and potentially future status of the species. We offer an approach for the analysis of hypervolumes with topological guarantees, complementary to current statistical methods, giving precise bounds between persistence diagrams of Vietoris--Rips and alpha complexes, and a duality identity for cubical complexes. Implementation of our methods, called TopoAware, is made available in C++, Python, and R, building upon the GUDHI library.