arXiv subjects

Matthias Söls

Publications and source records attributed to Matthias Söls.

3 recordsLinked to original sources

arXiv2026

Fast free resolutions of bifiltered chain complexes

In a $k$-critical bifiltration, every simplex enters along a staircase with at most $k$ steps. Examples with $k>1$ include degree-Rips bifiltrations and models of the multicover bifiltration. We consider the problem of converting a $k$-critical bifiltration into a $1$-critical (i.e. free) chain complex with equivalent homology. This is known as computing a free resolution of the underlying chain complex and is a first step toward post-processing such bifiltrations. We present two algorithms. The first one computes free resolutions corresponding to path graphs and assembles them to a chain complex by computing additional maps. The simple combinatorial structure of path graphs leads to good performance in practice, as demonstrated by extensive experiments. However, its worst-case bound is quadratic in the input size because long paths might yield dense boundary matrices in the output. Our second algorithm replaces the simplex-wise path graphs with ones that maintain short paths which leads to almost linear runtime and output size. We demonstrate that pre-computing a free resolution speeds up the task of computing a minimal presentation of the homology of a $k$-critical bifiltration in a fixed dimension. Furthermore, our findings show that a chain complex that is minimal in terms of generators can be asymptotically larger than the non-minimal output complex of our second algorithm in terms of description size.

math.AT↗

arXiv2025

Expected Complexity of Persistence Barcode Computation via Matrix Reduction

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 Čech, Vietoris--Rips and Erdős--Rényi 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 Čech 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 Erdős--Rényi filtration realising the worst-case fill-in and computation.

math.AT↗

arXiv2023

The Localized Union-of-Balls Bifiltration

We propose an extension of the classical union-of-balls filtration of persistent homology: fixing a point $q$, we focus our attention to a ball centered at $q$ whose radius is controlled by a second scale parameter. We discuss an absolute variant, where the union is just restricted to the $q$-ball, and a relative variant where the homology of the $q$-ball relative to its boundary is considered. Interestingly, these natural constructions lead to bifiltered simplicial complexes which are not $k$-critical for any finite $k$. Nevertheless, we demonstrate that these bifiltrations can be computed exactly and efficiently, and we provide a prototypical implementation using the CGAL library. We also argue that some of the recent algorithmic advances for $2$-parameter persistence (which usually assume $k$-criticality for some finite $k$) carry over to the $\infty$-critical case.

cs.CG↗