Expected Complexity of Persistence Barcode Computation via Matrix Reduction

TL;DR

This paper analyzes the expected computational complexity of persistence barcode calculation for random filtrations, providing bounds that outperform worst-case estimates and applying to common complexes like Čech and Vietoris–Rips.

Contribution

It introduces a technique to bound the expected complexity of matrix reduction in persistent homology, with tight bounds for certain filtrations and a worst-case example.

Findings

Bounds on average fill-in for Čech and Vietoris–Rips complexes.

Asymptotically tight bounds up to a logarithmic factor.

Worst-case fill-in example for Erdős–Rényi filtrations.

Abstract

We study the algorithmic complexity of computing the persistence barcode of a randomly generated filtration. We provide a general technique to bound the expected complexity of reducing the boundary matrix in terms of the density of its reduced form. We apply this technique finding 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, thus obtaining 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. 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.