Expected Complexity of Barcode Computation via Matrix Reduction

Abstract: We study the algorithmic complexity of computing persistent homology of a randomly generated filtration. We prove upper bounds for the average fill-in (number of non-zero entries) of the boundary matrix on \v{C}ech, Vietoris--Rips and Erd\H{o}s--R\'enyi filtrations after matrix reduction, which in turn provide bounds on the expected complexity of the barcode computation. Our method is based on previous results on the expected Betti numbers of the corresponding complexes, which we link to the fill-in of the boundary matrix. Our fill-in bounds for \v{C}ech and Vietoris--Rips complexes are asymptotically tight up to a logarithmic factor. In particular, both our fill-in and computation bounds are better than the worst-case estimates. We also provide an Erd\H{o}s--R\'enyi filtration realising the worst-case fill-in and computation.