Expected Complexity of Persistent Homology Computation via Matrix Reduction

We study the algorithmic complexity of computing persistent homology of a randomly generated filtration. Specifically, we prove upper bounds for the average fill-in (number of non-zero entries) of the boundary matrix on Čech, Vietoris–Rips and Erdős–Rényi filtrations after matrix reduction. Our bounds show that the reduced matrix is expected to be significantly sparser than what the general worst-case predicts. Our method is based on previous results on the expected Betti numbers of the corresponding complexes. We establish a link between these results and the fill-in of the boundary matrix. In the 1-dimensional case, our bound for Čech and Vietoris–Rips complexes is asymptotically tight up to a logarithmic factor. We also provide an Erdős–Rényi filtration realising the worst-case.