The study examines the algorithmic complexity of computing persistent homology through matrix reduction in ˇCech, Vietoris–Rips, and Erd˝os–R´enyi filtrations. Results show that reduced matrices are sparser than worst-case scenarios, with bounds on fill-in and runtime. The analysis provides formal evidence supporting the hypothesis that typical performance is better than worst-case predictions.
The research delves into random models for boundary matrices based on different filtration types. It establishes a connection between Betti numbers and fill-in, demonstrating expected fill-in and cost reductions for various models. The paper also discusses good order properties in filtrations and probabilistic bounds on non-trivial homology occurrences.
To Another Language
from source content
arxiv.org
Deeper Inquiries