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Persistent Topological Laplacians-A Survey.

Xiaoqi Wei1, Guo-Wei Wei1,2,3

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Summary
This summary is machine-generated.

Persistent topological Laplacians offer advanced tools for topological data analysis (TDA), outperforming persistent homology in complex datasets like protein engineering. This review explores their mathematical formulations and applications.

Keywords:
persistent spectral theorytopological Laplacianstopological data analysis

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Area of Science:

  • Topological Data Analysis (TDA)
  • Computational Topology
  • Applied Mathematics

Background:

  • Persistent homology faces challenges with complex datasets.
  • New tools are needed to analyze topological and geometrical features.
  • Multiscale analysis combined with topology is a promising approach.

Purpose of the Study:

  • Introduce persistent topological Laplacians as a novel TDA tool.
  • Review their mathematical formulations across diverse settings.
  • Highlight their advantages over persistent homology.

Main Methods:

  • Formulating Laplacians on various mathematical structures (simplicial complexes, digraphs, sheaves, etc.).
  • Analyzing the kernels for topological invariants.
  • Examining non-harmonic spectra for supplementary information.

Main Results:

  • Kernels fully retrieve topological invariants.
  • Non-harmonic spectra offer additional insights.
  • Demonstrated superior performance in protein engineering data analysis.

Conclusions:

  • Persistent topological Laplacians are powerful TDA tools.
  • They offer enhanced capabilities for complex data analysis.
  • Further exploration across various mathematical settings is warranted.