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Object-oriented Persistent Homology.

Bao Wang1, Guo-Wei Wei2

  • 1Department of Mathematics Michigan State University, MI 48824, USA.

Journal of Computational Physics
|December 26, 2015
PubMed
Summary
This summary is machine-generated.

This study introduces object-oriented persistent homology using differential geometry for quantitative data analysis. The new method enhances topological feature preservation and enables accurate predictions, like fullerene isomer stability.

Keywords:
Computational topologyFullereneLaplace-Beltrami flowProteinTotal curvature energyVariation

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Area of Science:

  • Computational topology
  • Data science
  • Materials science

Background:

  • Persistent homology (PH) is successful for qualitative data analysis but limited in quantitative modeling.
  • Current PH methods are passive, lacking feature preservation during data filtration.
  • There's a need for proactive PH techniques that enhance data features.

Purpose of the Study:

  • To develop object-oriented persistent homology methods for quantitative analysis.
  • To integrate differential geometry for objective data filtration and feature preservation.
  • To enable predictive modeling using topological persistence.

Main Methods:

  • Constructed an objective functional (surface free energy) using differential geometry.
  • Minimized the functional to derive a Laplace-Beltrami operator for data filtration.
  • Employed cubical complex homology compatible with Laplace-Beltrami flow.
  • Validated the method on proteins, fullerene molecules, and isomer stability prediction.

Main Results:

  • Developed a differential geometry-based object-oriented PH method.
  • Demonstrated consistency between Laplace-Beltrami flow and Euclidean distance filtration.
  • Achieved accurate predictions of fullerene isomer stability correlated with curvature energy.
  • Showcased enhanced topological persistence and feature preservation.

Conclusions:

  • The proposed Laplace-Beltrami flow based PH method enables quantitative modeling and prediction.
  • This work presents the first object-oriented PH approach for feature enhancement and extraction.
  • The method offers a robust tool for analyzing complex molecular structures and predicting properties.