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Computing the Multicover Bifiltration.

René Corbet1, Michael Kerber2, Michael Lesnick3

  • 1Department of Mathematics, KTH Royal Institute of Technology, Lindstedtsvägen 25, 11428 Stockholm, Sweden.

Discrete & Computational Geometry
|August 15, 2023
PubMed
Summary
This summary is machine-generated.

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Researchers introduce a smaller, computationally efficient multicover bifiltration using combinatorial methods. This new approach simplifies homology computation for complex spatial data, outperforming previous Čech-based models.

Area of Science:

  • Computational Topology
  • Geometric Analysis
  • Data Science

Background:

  • The multicover bifiltration models spatial data by considering points within a certain distance (r) to a minimum number of data points (k).
  • Existing Čech-based models for this bifiltration are computationally intensive and large.
  • Efficient computation of topological features for evolving spatial datasets is a significant challenge.

Purpose of the Study:

  • To develop a computationally efficient and topologically equivalent alternative to the multicover bifiltration.
  • To introduce novel combinatorial constructions (polyhedral and simplicial) for the multicover bifiltration.
  • To facilitate the computation of homology for the multicover bifiltration.

Main Methods:

  • Introduction of a polyhedral bifiltration based on the rhomboid tiling, utilizing a modified algorithm for efficient computation.
Keywords:
BifiltrationsDenoisingHigher-order Delaunay complexesMultiparameter persistent homologyNervesRhomboid tiling

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  • Development of a related simplicial bifiltration to aid in understanding and validating the polyhedral construction.
  • Implementation and experimental evaluation of the constructions in dimensions 2 and 3.

    • Main Results:

    • The proposed polyhedral and simplicial bifiltrations are topologically equivalent to the multicover bifiltration.
    • These combinatorial constructions are significantly smaller and more computationally efficient than previous Čech-based models.
    • Experimental results in dimensions 2 and 3 demonstrate the practical applicability and efficiency of the new methods.

    Conclusions:

    • The novel combinatorial bifiltrations offer a more efficient approach to computing the homology of the multicover bifiltration.
    • These methods provide a valuable tool for analyzing complex spatial data and topological structures.
    • The research advances computational topology by offering practical and scalable solutions for data analysis.