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Understanding Higher-Order Interactions in Information Space.

Herbert Edelsbrunner1, Katharina Ölsböck1, Hubert Wagner2

  • 1ISTA (Institute of Science and Technology Austria), 3400 Klosterneuburg, Austria.

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|August 29, 2024
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Summary
This summary is machine-generated.

This study introduces novel algorithms and software for geometric and topological data analysis in information spaces, enabling exploration of higher-order interactions in data beyond traditional metric spaces.

Keywords:
Bregman divergenceBregman geometryShannon entropyalpha shapehigher-order interactionsinformation theorynon-Euclidean geometrypersistent homologyrelative entropysimplicial complextopological data analysiswrap complex

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Area of Science:

  • Computational topology
  • Information geometry
  • Data science

Background:

  • Topological data analysis (TDA) effectively captures complex interactions in metric spaces.
  • Extending TDA to information spaces, such as probability distributions with Kullback-Leibler divergence, presents unique challenges.
  • Existing theoretical frameworks lack practical implementations for analyzing information spaces.

Purpose of the Study:

  • To develop the first robust algorithms and software for geometric and topological data analysis specifically designed for information spaces.
  • To facilitate the exploration and comparison of topological and geometric properties within various information spaces.
  • To bridge the gap between theoretical TDA in information spaces and practical data analysis applications.

Main Methods:

  • Development of novel algorithms for TDA in information spaces, leveraging Bregman divergences.
  • Adaptation and reuse of existing robust libraries from Euclidean TDA, demonstrating surprising compatibility.
  • Implementation of software tools for practical application of these new methods.

Main Results:

  • Successful creation of the first robust software for geometric and topological data analysis in information spaces.
  • Demonstration that Euclidean TDA libraries can be effectively adapted for information spaces using Bregman divergences.
  • Initial insights into the geometric-topological structure of information spaces, including comparisons with Euclidean and Fisher metrics.

Conclusions:

  • The developed algorithms and software provide essential tools for advancing TDA in information spaces.
  • This work opens new avenues for understanding the geometry and topology of complex data distributions.
  • The findings suggest a promising synergy between information theory, topology, and practical data analysis.