# Riemannian Gaussian distributions, random matrix ensembles and diffusion kernels

arXiv: https://arxiv.org/abs/2011.13680 · HTML: https://mlsanity.com/paper/2011.13680/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Nov 2020.

## Built on by (most cited first, 3 of 3)

- [Online learning of Riemannian hidden Markov models in homogeneous Hadamard spaces](https://mlsanity.com/paper/2102.07771/) (2021)
- [Geometric Learning of Hidden Markov Models via a Method of Moments Algorithm](https://mlsanity.com/paper/2207.00818/) (2022)
- [Riemannian statistics meets random matrix theory: towards learning from high-dimensional covariance matrices](https://mlsanity.com/paper/2203.00204/) (2022)

## Abstract (arXiv)

We show that the Riemannian Gaussian distributions on symmetric spaces, introduced in recent years, are of standard random matrix type. We exploit this to compute analytically marginals of the probability density functions. This can be done fully, using Stieltjes-Wigert orthogonal polynomials, for the case of the space of Hermitian matrices, where the distributions have already appeared in the physics literature. For the case when the symmetric space is the space of m × m symmetric positive definite matrices, we show how to efficiently compute by evaluating Pfaffians at specific values of m. Equivalently, we can obtain the same result by constructing specific skew orthogonal polynomials with regards to the log-normal weight function (skew Stieltjes-Wigert polynomials). Other symmetric spaces are studied and the same type of result is obtained for the quaternionic case. Moreover, we show how the probability density functions are a particular case of diffusion reproducing kernels of the Karlin-McGregor type, describing non-intersecting Brownian motions, which are also diffusion processes in the Weyl chamber of Lie groups.