SIAM Review, Volume 63, Issue 1, Page 67-99, January 2021.
A crucial step in the analysis of persistent homology is the transformation of data into an appropriate topological object (which, in our case, is a simplicial complex). Software packages for computing persistent homology typically construct Vietoris--Rips or other distance-based simplicial complexes on point clouds because they are relatively easy to compute. We investigate alternative methods of constructing simplicial complexes and the effects of making associated choices during simplicial-complex construction on the output of persistent-homology algorithms. We present two new methods for constructing simplicial complexes from two-dimensional geospatial data (such as maps). We apply these methods to a California precinct-level voting data set, and we thereby demonstrate that our new constructions can capture geometric characteristics that are missed by distance-based constructions. Our new constructions can thus yield more interpretable persistence modules and barcodes for geospatial data. In particular, they are able to distinguish short-persistence features that occur only for a narrow range of distance scales (e.g., voting patterns in densely populated cities) from short-persistence noise by incorporating information about other spatial relationships between regions.

中文翻译：

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地理空间数据的持久同源性：投票案例研究

SIAM 评论，第 63 卷，第 1 期，第 67-99 页，2021 年 1 月。
分析持久同源性的一个关键步骤是将数据转换为适当的拓扑对象（在我们的例子中，它是一个单纯复形）。用于计算持久同源性的软件包通常在点云上构建 Vietoris--Rips 或其他基于距离的单纯复形，因为它们相对容易计算。我们研究了构造单纯复形的替代方法以及在单纯复形构造过程中做出相关选择对持久同源算法输出的影响。我们提出了两种从二维地理空间数据（例如地图）构造单纯复形的新方法。我们将这些方法应用于加利福尼亚选区级别的投票数据集，因此，我们证明了我们的新构造可以捕获基于距离的构造所遗漏的几何特征。因此，我们的新结构可以为地理空间数据产生更多可解释的持久性模块和条形码。特别是，它们能够通过结合有关区域之间其他空间关系的信息，将仅在较窄距离范围内（例如，人口稠密城市中的投票模式）出现的短持久特征与短持久噪声区分开来。