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Persistence diagrams from random point-clouds follow a universal probability law when normalized. This finding enables a new framework for assessing the significance of topological features in data.

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Area of Science:

  • Topological Data Analysis
  • Computational Topology
  • Data Science

Background:

  • Understanding persistence diagram distributions is a key challenge in topological data analysis.
  • Current methods struggle to characterize the statistical properties of these diagrams.
  • This limits the quantitative assessment of topological features in complex datasets.

Purpose of the Study:

  • To investigate the statistical distribution of persistence diagrams from random point-clouds.
  • To identify a universal probability law governing these distributions.
  • To develop a novel hypothesis testing framework for feature significance.

Main Methods:

  • Extensive experimentation on simulated and real-world point-cloud data.
  • Normalization of persistence diagrams to identify underlying statistical patterns.
  • Statistical analysis to determine a candidate universal probability distribution.
  • Development of a hypothesis testing framework based on the discovered law.

Main Results:

  • A surprising universal probability law governs normalized persistence diagrams from random point-clouds.
  • The discovered law holds across diverse data geometries, topologies, and distributions.
  • An explicit, well-known distribution is proposed as a candidate for this universal law.
  • A new hypothesis testing framework for computing significance values of topological features was developed.

Conclusions:

  • The statistical distribution of persistence diagrams is not arbitrary but follows a universal law.
  • This discovery provides a quantitative method for assessing the significance of topological structures.
  • The proposed framework advances topological data analysis by offering robust statistical significance testing.