Persistent Topological Laplacians -- a Survey

TL;DR

Persistent topological Laplacians are advanced tools in topological data analysis that enhance the characterization of data's topological features, outperforming traditional methods in large-scale biological datasets.

Contribution

This survey provides a comprehensive review of persistent topological Laplacians across various mathematical frameworks, highlighting their advantages over persistent homology.

Findings

Superior performance in protein data analysis

Kernel fully retrieves topological invariants

Non-harmonic spectra offer additional insights

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 combines 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 analyzing 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, as well as $N$-chain complexes.