Properties and applications of Fisher distribution on the rotation group

TL;DR

This paper investigates the properties of the Fisher distribution on the rotation group SO(3), introduces computational methods for its normalization and estimation, and compares it with distributions on related manifolds using real data.

Contribution

It applies the holonomic gradient descent and series expansion methods to Fisher distribution on SO(3), enhancing computational techniques for statistical modeling on rotation groups.

Findings

Effective evaluation of the normalizing constant using series expansion.

Application of holonomic gradient descent for maximum likelihood estimation.

Comparison of Fisher distributions on SO(3) and Stiefel manifolds with real data.

Abstract

We study properties of Fisher distribution (von Mises-Fisher distribution, matrix Langevin distribution) on the rotation group SO(3). In particular we apply the holonomic gradient descent, introduced by Nakayama et al. (2011), and a method of series expansion for evaluating the normalizing constant of the distribution and for computing the maximum likelihood estimate. The rotation group can be identified with the Stiefel manifold of two orthonormal vectors. Therefore from the viewpoint of statistical modeling, it is of interest to compare Fisher distributions on these manifolds. We illustrate the difference with an example of near-earth objects data.