  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.
