Understanding Higher-Order Interactions in Information Space

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.