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.

翻译：本文针对由有限点集构成的Vietoris–Rips复形、alpha复形及立方复形滤过结构，给出了零维持久同调与余维一同调在三种操作下的稳定性界：富集（添加新元素）、稀疏化（移除元素）以及网格对齐（均匀离散化元素）。对于富集操作，我们采用重心细分方法；对于稀疏化，引入最小分离距离准则；对于网格对齐，通过对各坐标值除以固定步长取商实现。研究动机源于处理大规模不规则数据集的实际需求，以及持久同调理论在此类数据中的应用拓展。特别地，我们以生态学应用为例，通过以环境变量为维度的高维空间推断观测物种的状态。该“超体积”具有几何特征（体积、凸性）与拓扑特征（连通性、同调性），已知这些特征与物种当前及潜在未来状态相关。我们提出一种具有拓扑保证的超体积分析方法，作为现有统计方法的补充，给出了Vietoris–Rips复形与alpha复形持久图之间的精确边界，以及立方复形的对偶恒等式。相关算法实现（命名为TopoAware）基于GUDHI库开发，提供C++、Python和R语言版本。