Homology, Homotopy and Applications

Volume 26 (2024)

Number 2

The algebraic stability for persistent Laplacians

Pages: 297 – 323

DOI: https://dx.doi.org/10.4310/HHA.2024.v26.n2.a15

Authors

Jian Liu (Mathematical Science Research Center, Chongqing University of Technology, Chongqing, China; and Beijing Institute of Mathematical Sciences and Applications, Beijing, China)

Jingyan Li (Beijing Institute of Mathematical Sciences and Applications, Beijing, China)

Jie Wu (Beijing Institute of Mathematical Sciences and Applications, Beijing, China)

Abstract

The stability of topological persistence is one of the fundamental issues in topological data analysis. Numerous methods have been proposed to address the stability of persistent modules or persistence diagrams. Recently, the concept of persistent Laplacians has emerged as a novel approach to topological persistence, attracting significant attention and finding applications in various fields. In this paper, we investigate the stability of persistent Laplacians. We introduce the notion of “Laplacian trees”, which captures the collection of persistent Laplacians that persist from a given parameter. To formalize our study, we construct the category of Laplacian trees and establish an algebraic stability theorem for persistent Laplacian trees. Notably, our stability theorem is applied to the real-valued functions on simplicial complexes and digraphs.

Keywords

persistent homology, persistent Laplacian, algebraic stability, inner product space, persistence harmonic space

2010 Mathematics Subject Classification

15A63, 18A25

Received 25 July 2023

Received revised 14 December 2023

Accepted 18 December 2023

Published 9 October 2024