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

TL;DR

This paper establishes rigorous stability guarantees for 0-dimensional persistent homology under three data-modification operations—enrichment via barycentric subdivision, sparsification, and grid alignment—for Vietoris–Rips, alpha, and cubical filtrations. It derives explicit bottleneck-distance bounds, and introduces a duality identity enabling computation of codimension-1 cubical homology from 0-dimensional data, all implemented in the open-source TopoAware toolkit built on GUDHI. Through synthetic experiments and comparisons with a statistical niche-modeling approach, the work demonstrates that topology-based modifications preserve essential features while reducing computational burden, with direct implications for analyzing large, irregular ecological datasets (hypervolumes). Overall, the contributions provide practical, provable guarantees for topology-preserving data reduction and discretization, supporting scalable, reliable hypervolume analysis and similar applications in computational ecology and beyond.

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.

Figures (13)

• Figure 1: An overview of the simplicial and cubical complexes used here and their filtrations. From left to right: Vietoris--Rips complexes at radii $r<r'$, alpha complexes at radii $r<r'$, and cubical complexes at filtration values $r=5<r'=9$ (by the ordering on the 0-cubes). On the left, 0-cubes in white are not included in the cubical complex.
• Figure 2: An ordered set $X\subseteq \mathbf{R}^2$ (left) and the persistence diagram $D_X$ of its Vietoris--Rips filtration in degree 0 (right). The first two elements $x_0,x_1\in X$ and the elements $x_0',x_1'$ whose distance defines the second longest element of $D_X$ are emphasized.
• Figure 3: An overview of point cloud modifications. From left to right: A set $X\subseteq \mathbf{R}^2$, its Vietoris--Rips complex, its barycentric subdivision, the sparsification of its barycentric subdivision, and the alpha complex of the sparsification of its barycentric subdivision. The values chosen for this example are $\delta = .3\cdot \mathrm{diam}(X)$ and $\varepsilon=.06\cdot \mathrm{diam}(X)$.
• Figure 4: An overview of constructions on grids. From left to right: A set $X\subseteq \mathbf{R}^2$ (taken from \ref{['fig_barysparseex']}, second right) overlaid by the grid $G(X,\mu)$, the cubical complex on the grid, the complement of the grid, and the cubical complex on the complement.
• Figure 5: From left to right: A subset $G(X,\mu)$ of a $\mu$-grid (taken from \ref{['fig_gridcompex']}, left), the subdivision of the subset, and the thickening of the subset.

Theorems & Definitions (16)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Remark 2.5
• Remark 2.6
• Remark 2.7
• Theorem 3.1
• proof
• Theorem 3.2