Homotopy continuation for the spectra of persistent Laplacians
type: Item
instance of: scholarly article
title: Homotopy continuation for the spectra of persistent Laplacians
zbMATH Open document ID: 1494.55011
author: Xiaoqi Wei
author: G. W. Wei
published in: Foundations of Data Science
publication date: 2022-01-26T00:00:00Z
review text: The Laplacian is a critical concept in network analysis and shape tasks. Given its limitation to a single scale of the data set, it was recently imbued with ideas from persistent homology, leading to the \(p\)-persistent \(q\)-combinatorial Laplacian of a simplicial complex. This construction, colloquially referred to as a \textit{persistent Laplacian} can be shown to be intricately linked to multi-scale topological information, such as the persistent Betti numbers of a data set. Motivated by the utility of persistent Laplacians, this paper presents a new way of calculating their spectral information, specifically, the roots of their characteristic polynomials, by homotopy continuation. Homotopy continuation addresses the problem of solving a system~\(f\) of polynomial equations by starting with an easy-to-solve system~\(g\), building a homotopy between~\(f\) and~\(g\), and, finally, tracking the roots of~\(g\) to those of~\(f\). The authors demonstrate the feasibility of this continuation method and provide empirical evidence of the utility of using spectral information from persistent Laplacians to characterise aromatic molecules, for instance.
reviewed by: Bastian Rieck
Mathematics Subject Classification ID: 55N31
Mathematics Subject Classification ID: 55U10
Mathematics Subject Classification ID: 65H14
zbMATH DE Number: 7464718
zbMATH Keywords: persistent Laplacian
zbMATH Keywords: homotopy continuation
zbMATH Keywords: persistent homology
zbMATH Keywords: algebraic topology
zbMATH Keywords: combinatorial graph
zbMATH Keywords: numerical algebraic geometry
cites work: HERMES
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cites work: HERMES: persistent spectral graph software
describes a project that uses: HERMES
describes a project that uses: HomotopyContinuation
describes a project that uses: NumPy
describes a project that uses: PHCpack
describes a project that uses: Gudhi
describes a project that uses: Bertini
describes a project that uses: Hom4PS-3
MaRDI profile type: Publication
full work available at URL: fods.2021017
OpenAlex ID: W3185736110
DOI: 10.3934/FODS.2021017
description: scientific article; zbMATH DE number 7464718