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.

翻译：持久性同系物是各种应用中所使用的一种地形特征,例如生成数据分析和惩罚优化问题的特点。我们制定了一种方法,以加速在许多类似的过滤式表层空间进行的持久性同系物计算,该方法以更新相关的矩阵因子化为基础。我们的方法改进了Cohen-Steiner、Edelsbrunner和Morozov的更新计划,通过在过滤式表层空间中额外处理和删除细胞,并通过处理单个批次的改变来改变。我们的方法复杂,对过滤法进行了基本修改,结果往往比完全的持久性同系物计算费用低。最后,我们进行了计算实验,表明在若干情况下实际加快速度,其中包括在持续同质学指导下生成和优化特性。