@inproceedings{4bbd3d9a1df043b6a4d1088ab28c263f,
title = "Average Complexity of Matrix Reduction for Clique Filtrations",
abstract = "We study the algorithmic complexity of computing persistent homology of a randomly chosen filtration. Specifically, we prove upper bounds for the average fill-up (number of non-zero entries) of the boundary matrix on Erd{\"o}s-R{\'e}nyi and Vietoris-Rips filtrations after matrix reduction. Our bounds show that, in both cases, the reduced matrix is expected to be significantly sparser than what the general worst-case predicts. Our method is based on a link between the fillup of the boundary matrix and expected Betti numbers of random filtrations. Our bound for Vietoris-Rips complexes is asymptotically tight up to logarithmic factors. We also provide an Erd{\"o}s-R{\'e}nyi filtration realising the worst-case.",
keywords = "average complexity, matrix reduction, persistence algorithm",
author = "Barbara Giunti and Guillaume Houry and Michael Kerber",
year = "2022",
month = jul,
day = "4",
doi = "10.1145/3476446.3535474",
language = "English",
series = "Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC",
publisher = "Association of Computing Machinery",
pages = "187--196",
editor = "Amir Hashemi",
booktitle = "ISSAC 2022 - Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation47th International Symposium on Symbolic and Algebraic Computation, ISSAC 2022",
address = "United States",
note = "2022 International Symposium on Symbolic and Algebraic Computation : ISSAC 2022, ISSAC '22 ; Conference date: 04-07-2022 Through 07-07-2022",
}