Persistent Topological Laplacians—A Survey

Xiaoqi Wei () and Guo-Wei Wei ()
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Xiaoqi Wei: Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA
Guo-Wei Wei: Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA

Mathematics, 2025, vol. 13, issue 2, 1-29

Abstract: Persistent topological Laplacians constitute a new class of tools in topological data analysis (TDA). They are motivated by the necessity to address challenges encountered in persistent homology when handling complex data. These Laplacians combine multiscale analysis with topological techniques to characterize the topological and geometrical features of functions and data. Their kernels fully retrieve the topological invariants of corresponding persistent homology, while their non-harmonic spectra provide supplementary information. Persistent topological Laplacians have demonstrated superior performance over persistent homology in the analysis of large-scale protein engineering datasets. In this survey, we offer a pedagogical review of persistent topological Laplacians formulated in various mathematical settings, including simplicial complexes, path complexes, flag complexes, digraphs, hypergraphs, hyperdigraphs, cellular sheaves, and N -chain complexes.

Keywords: topological data analysis; topological Laplacians; persistent spectral theory (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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