Algebraic Analysis of Rotation Data

TL;DR

This paper introduces algebraic tools based on D-modules and noncommutative Gr"obner bases for statistical inference on rotation matrices, enabling efficient maximum likelihood estimation and generalizing to arbitrary Lie groups.

Contribution

It develops a novel algebraic framework for inference on rotation data using D-modules and extends the theory to general Lie groups, with practical algorithms for data analysis.

Findings

Algorithms successfully applied to real-world data

Generalization from SO(3) to arbitrary Lie groups

Efficient maximum likelihood estimation methods

Abstract

We develop algebraic tools for statistical inference from samples of rotation matrices. This rests on the theory of D-modules in algebraic analysis. Noncommutative Gr\"obner bases are used to design numerical algorithms for maximum likelihood estimation, building on the holonomic gradient method of Sei, Shibata, Takemura, Ohara, and Takayama. We study the Fisher model for sampling from rotation matrices, and we apply our algorithms for data from the applied sciences. On the theoretical side, we generalize the underlying equivariant D-modules from SO(3) to arbitrary Lie groups. For compact groups, our D-ideals encode the normalizing constant of the Fisher model.