The Fisher-Rao Distance between Multivariate Normal Distributions: Special Cases, Bounds and Applications

Area of Science:

• Statistics
• Information Geometry
• Machine Learning

Background:

• The Fisher-Rao distance is a fundamental metric in information geometry, measuring dissimilarity between probability distributions.
• It possesses unique properties, including invariance under Markov morphisms, and relates to Shannon entropy.
• Applications span image processing, radar systems, and morphological classification.

Purpose of the Study:

• To derive explicit expressions for the Fisher-Rao distance within the statistical model of multivariate normal distributions.
• To extend the applicability of the Fisher-Rao distance by addressing cases with identical and mirrored covariance matrices.
• To demonstrate a practical application in simplifying Gaussian mixtures using hierarchical clustering.

Main Methods:

• Leveraging known results for submanifolds and bounds of the Fisher-Rao distance.
• Developing analytical expressions for specific cases of multivariate normal distributions.
• Implementing hierarchical clustering algorithms for Gaussian mixture simplification.

Main Results:

• Derived expressions for the Fisher-Rao distance between multivariate normal distributions with identical covariance matrices.
• Derived expressions for the Fisher-Rao distance between multivariate normal distributions with mirrored covariance matrices.
• Successfully applied the Fisher-Rao distance to simplify Gaussian mixtures, demonstrating its utility.

Conclusions:

• The study provides valuable analytical tools for computing the Fisher-Rao distance in multivariate normal distributions.
• The derived expressions and demonstrated application enhance the practical use of this metric in statistical analysis and machine learning.
• This work contributes to a deeper understanding and broader application of information-theoretic measures.