Alapfogalmak
The author analyzes the expected complexity of computing persistent homology using matrix reduction, showing that the reduced matrix is sparser than worst-case predictions. The study links Betti numbers to the fill-in of the boundary matrix.
Kivonat
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.
Statisztikák
Our main result is that the expected fill-in is given by O(n2k log2(n)) and the expected cost of matrix reduction is bounded by O(n3k+2 log2(n)).
Lemma 3.1: #D′ ≥ n^(k+1) - n^k = Ω(n^(k+1))
Lemma 4.1: For a matrix M with c columns, let M' denote its reduced matrix. Then cost(M) ≤ c * #M'