# Riemannian Gaussian distributions on the space of positive-definite quaternion matrices

## TL;DR

This paper extends Riemannian Gaussian distributions to positive-definite quaternion matrices by developing their geometric properties, providing formulas, sampling methods, and inference techniques for this new space.

## Contribution

It introduces the Riemannian geometry of positive-definite quaternion matrices and formulates Gaussian distributions on this space, including density, sampling, and inference methods.

## Key findings

- Derived the Riemannian metric and geodesics for quaternion matrices
- Provided explicit formulas for Riemannian Gaussian densities
- Developed sampling algorithms and statistical inference methods

## Abstract

Recently, Riemannian Gaussian distributions were defined on spaces of positive-definite real and complex matrices. The present paper extends this definition to the space of positive-definite quaternion matrices. In order to do so, it develops the Riemannian geometry of the space of positive-definite quaternion matrices, which is shown to be a Riemannian symmetric space of non-positive curvature. The paper gives original formulae for the Riemannian metric of this space, its geodesics, and distance function. Then, it develops the theory of Riemannian Gaussian distributions, including the exact expression of their probability density, their sampling algorithm and statistical inference.