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.

翻译：我们针对由增强（添加新元素）、稀疏化（移除元素）及网格对齐（均匀离散化元素）相关的有限集，给出了Vietoris–Rips复形、Alpha复形及立方复形滤链的0维持续同调与余维1同调界的估计。对于增强操作，我们采用重心细分方法；对于稀疏化操作，我们使用迭代最小分离距离算法；对于网格对齐操作，我们通过对每个坐标值除以固定步长取商实现。本研究受生物学应用驱动，其中物种状态通过其“超体积”推断——即以环境变量为维度的高维空间。超体积具有几何特性（体积、凸性）与拓扑特性（连通性、同调性），已知这些特性与物种当前及潜在未来状态相关。我们提出了一种具有拓扑保证的方法，作为现代超体积计算方法的补充，给出了Vietoris–Rips复形与Alpha复形持续图之间的精确界，以及立方复形的对偶恒等式。我们基于GUDHI库开发了名为TopoAware的方法实现，并提供C++、Python和R语言版本。