Move Schedules: Fast persistence computations in sparse dynamic settings

The standard procedure for computing the persistent homology of a filtered simplicial complex is the matrix reduction algorithm. Its output is a particular decomposition of the total boundary matrix, from which the persistence diagrams and generating cycles can be derived. Persistence diagrams are known to vary continuously with respect to their input; this motivates the algorithmic study of persistence computations for time-varying filtered complexes. Computationally, simulating persistence dynamically can be reduced to maintaining a valid decomposition under adjacent transpositions in the filtration order. In practice, the quadratic scaling in the number of transpositions often makes this maintenance procedure slower than simply computing the decomposition from scratch, effectively limiting the application of dynamic persistence to relatively small data sets. In this work, we propose a coarser strategy for maintaining the decomposition over a discrete 1-parameter family of filtrations. Our first result is an analysis of a simple linear-time strategy for reducing the number of column operations needed to simulate persistence across a fixed homotopy by at most a factor of 2. We then show a modification of this technique which maintains only a sublinear number of valid states, as opposed to a quadratic number of states, and we provide tight lower bounds for this technique. Finally, we provide empirical results suggesting that the decrease in operations needed to compute diagrams across a family of filtrations is proportional to the difference between the expected quadratic number of states, and the proposed sublinear coarsening.