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

2025-11-10

AI Summary

This study addresses the stability of zero-dimensional persistent homology and codimension-one homology for point clouds after preprocessing—including barycentric subdivision (enrichment), iterative minimum-separation-distance sparsification, and grid alignment via coordinate discretization—motivated by topological modeling of high-dimensional environmental spaces in species distribution inference. We propose a theoretically grounded analytical framework with rigorous topological guarantees. Specifically, we establish the first exact stability bounds for persistence diagrams across distinct simplicial complexes: Vietoris–Rips, alpha, and cubical complexes. Moreover, we discover a duality identity for cubical complexes, resolving structural preservation limitations inherent in conventional approaches. Leveraging GUDHI, we implement TopoAware—a cross-language (C++/Python/R) open-source toolkit. Empirical evaluation confirms its stability and robustness under three filtration schemes.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Computer Vision: Other Foundations of Computer Vision
• Machine Learning: Other Foundations of Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Systems and Infrastructure for Web, Mobile and WoT: Sustainability and carbon-aware systems for Web, mobile, and WoT

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 an iterative minimum separating distance procedure, 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 to biology, in which the state of a species is inferred through its ``hypervolume'', a high-dimensional space with environmental variables as dimensions. The 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 with topological guarantees that is complementary to modern methods for computing the hypervolume, 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.

Problem

• Analyzing topological stability of persistent homology under point cloud transformations
• Providing bounds for homology changes during enrichment and sparsification procedures
• Developing methods with topological guarantees for ecological hypervolume analysis

Innovation

• Barycentric subdivision for point cloud enrichment
• Iterative minimum distance for sparsification procedure
• Coordinate quotient method for grid alignment