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Diffusion Curvature for Estimating Local Curvature in High Dimensional Data.

Dhananjay Bhaskar1, Kincaid MacDonald2, Oluwadamilola Fasina3

  • 1Department of Genetics, Yale University.

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We developed diffusion curvature, a novel method to measure local curvature on point clouds. This technique uses random walk "laziness" to analyze data structure, with applications in geometry and machine learning.

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Area of Science:

  • Computational geometry
  • Data analysis
  • Machine learning

Background:

  • Point cloud data is complex to analyze.
  • Existing curvature measures may not capture intrinsic data structure effectively.

Purpose of the Study:

  • Introduce diffusion curvature, a new intrinsic measure for point cloud local curvature.
  • Extend scalar curvature to a quadratic form using neural networks.
  • Demonstrate applications in diverse datasets.

Main Methods:

  • Utilize diffusion maps and the data diffusion operator to structure point clouds.
  • Define local curvature based on the 'laziness' of random walks.
  • Employ neural network estimations for quadratic form curvature.

Main Results:

  • Diffusion curvature relates to volume comparison in Riemannian geometry.
  • Successfully applied diffusion curvature to toy data, single-cell data, and neural network loss landscapes.
  • Extended the scalar measure to a quadratic form for richer analysis.

Conclusions:

  • Diffusion curvature offers a robust intrinsic measure for point cloud data.
  • The method provides valuable insights into data geometry and structure.
  • Applications demonstrate its versatility across scientific domains.