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Persistent homology is a topological feature used in a variety ofapplications such as generating features for data analysis and penalizingoptimization problems. We develop an approach to accelerate persistent homologycomputations performed on many similar filtered topological spaces which isbased on updating associated matrix factorizations. Our approach improves theupdate scheme of Cohen-Steiner, Edelsbrunner, and Morozov for permutations byadditionally handling addition and deletion of cells in a filtered topologicalspace and by processing changes in a single batch. We show that the complexityof our scheme scales with the number of elementary changes to the filtrationwhich as a result is often less expensive than the full persistent homologycomputation. Finally, we perform computational experiments demonstratingpractical speedups in several situations including feature generation andoptimization guided by persistent homology.

Accelerating Iterated Persistent Homology Computations with Warm Starts