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A charge distribution has spherical symmetry if the density of charge depends only on the distance from a point in space and not on the direction. In other words, if the system is rotated, it doesn't look different. For instance, if a sphere of radius R is uniformly charged with charge density ρ0, then the distribution has spherical symmetry. On the other hand, if a sphere of radius R is charged so that the top half of the sphere has a uniform charge density ρ1 and the bottom half has...
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Geodesic Monte Carlo on Embedded Manifolds.

Simon Byrne1, Mark Girolami1

  • 1Department of Statistical Science, University College London.

Scandinavian Journal of Statistics, Theory and Applications
|October 14, 2014
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Summary
This summary is machine-generated.

This study introduces novel Markov chain Monte Carlo (MCMC) methods for simulating probability distributions on manifolds. These advanced techniques leverage differential geometry and geodesic flows for enhanced sampling efficiency.

Keywords:
Riemannian manifoldStiefel manifolddirectional statisticsgeodesic, Hamiltonian Monte Carlo

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Area of Science:

  • Computational Statistics
  • Differential Geometry
  • Machine Learning

Background:

  • Markov chain Monte Carlo (MCMC) methods are foundational for statistical inference.
  • Existing MCMC methods are being extended to operate directly on the manifold of probability distributions.
  • These methods utilize diffusions and geodesic flows within the Hamilton-Jacobi framework.

Purpose of the Study:

  • To extend differential geometric MCMC methods to simulate from probability distributions defined on manifolds.
  • To address the simulation of directional statistics and other manifold-supported distributions.
  • To develop new proposal mechanisms based on geodesic flows.

Main Methods:

  • Development of MCMC algorithms with proposal mechanisms derived from geodesic flows on manifolds.
  • Application of differential geometric principles to construct simulation methods.
  • Illustrative examples on the hypersphere and Stiefel manifold.

Main Results:

  • Demonstration of novel MCMC methods for distributions on manifolds.
  • Successful application of geodesic flow-based proposal mechanisms.
  • Validation through examples on complex manifolds like the hypersphere and Stiefel manifold.

Conclusions:

  • The proposed methods offer a powerful new approach for sampling from manifold-supported probability distributions.
  • This work advances the integration of differential geometry into MCMC methodologies.
  • The techniques are particularly relevant for directional statistics and high-dimensional data analysis.