Average Complexity of Matrix Reduction for Clique Filtrations

Barbara Giunti, Michael Kerber, Guillaume Houry

Research output: Chapter in Book/Report/Conference proceedingConference paperpeer-review

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 Erdös-Rényi 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 Erdös-Rényi filtration realising the worst-case.
Original languageEnglish
Title of host publicationProceedings of the 2022 International Symposium on Symbolic and Algebraic Computation
PublisherAssociation of Computing Machinery
Pages187-196
Number of pages9
DOIs
Publication statusPublished - 31 Jul 0005
Event2022 International Symposium on Symbolic and Algebraic Computation: ISSAC '22 - Lille, France
Duration: 4 Jul 20227 Jul 2022

Conference

Conference2022 International Symposium on Symbolic and Algebraic Computation
Abbreviated titleISSAC 2022
Country/TerritoryFrance
CityLille
Period4/07/227/07/22

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