  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.
