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 Gröbner 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 to data from the applied sciences. On the theoretical side, we generalize the underlying equivariant $D$-modules from $SO\left(3\right)$ to arbitrary Lie groups. For compact groups, our $D$-ideals encode the normalizing constant of the Fisher model.

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