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Explicit Geodesic Projection Distance on the Statistical Manifold of Multivariate Elliptical Distributions.

Xiangbing Chen1, Yingying Wang1, Jihong Xiao2

  • 1School of Mathematics and Statistics, Kashi University, Kashi 844000, China.

Entropy (Basel, Switzerland)
|June 26, 2026
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Summary
This summary is machine-generated.

This study introduces geodesic projection distances for multivariate elliptical distributions (MEDs), extending beyond Gaussian models. A novel symmetry technique provides explicit solutions for complex MED manifolds.

Keywords:
fisher metricgeodesic projection distanceinformation geometrymultivariate elliptical distribution

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Area of Science:

  • Statistics
  • Information Geometry
  • Machine Learning

Background:

  • Geodesic projection distances are crucial in statistics and machine learning.
  • Current solutions are limited to Gaussian manifolds.
  • Multivariate elliptical distributions (MEDs) offer broader applicability but lack explicit geodesic solutions.

Purpose of the Study:

  • Derive explicit geodesic projection distances for MED manifolds.
  • Generalize existing methods from Gaussian to MEDs.
  • Introduce a novel technique for complex geodesic equations.

Main Methods:

  • Developed a novel symmetry-exploitation technique.
  • Applied the technique to derive geodesic equations for MEDs.
  • Calculated explicit geodesic projection distances onto a fixed mean submanifold.

Main Results:

  • Achieved the first explicit geodesic projection distance solution for MEDs.
  • Demonstrated a method to solve complex, nonlinear geodesic equations.
  • Extended the applicability of geodesic distances to heavy-tailed distributions.

Conclusions:

  • The derived geodesic projection distances significantly advance statistical manifold theory.
  • The novel methodology is adaptable to other statistical manifolds.
  • This work broadens the scope of geodesic analysis in statistics and data science.