Persistent Topological Laplacians—A Survey

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 multi-scale 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.