Persistent homology is a topological feature used in a variety of applications such as generating features for data analysis and penalizing optimization problems. We develop an approach to accelerate persistent homology computations performed on many similar filtered topological spaces which is based on updating associated matrix factorizations. Our approach improves the update scheme of Cohen-Steiner, Edelsbrunner, and Morozov for permutations by additionally handling addition and deletion of cells in a filtered topological space and by processing changes in a single batch. We show that the complexity of our scheme scales with the number of elementary changes to the filtration which as a result is often less expensive than the full persistent homology computation. Finally, we perform computational experiments demonstrating practical speedups in several situations including feature generation and optimization guided by persistent homology.

中文翻译：

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通过热启动加速迭代持久同源计算

持久同源性是一种拓扑特征，可用于多种应用，例如生成用于数据分析的特征和惩罚优化问题。我们开发了一种方法来加速在许多相似的过滤拓扑空间上执行的持久同源计算，该计算基于更新相关的矩阵分解。我们的方法通过额外处理过滤拓扑空间中单元的添加和删除以及处理单个批次中的更改，改进了 Cohen-Steiner、Edelsbrunner 和 Morozov 的排列更新方案。我们表明，我们方案的复杂性随着过滤的基本变化的数量而变化，因此通常比完整的持久同源计算更便宜。最后，我们进行了计算实验，证明了在几种情况下的实际加速，包括由持久同源性引导的特征生成和优化。