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

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.Comment: 8 pages, submitted to GSI 201

Gaussian distributions on Riemannian symmetric spaces, random matrices, and planar Feynman diagrams

Gaussian distributions can be generalized from Euclidean space to a wide class of Riemannian manifolds. Gaussian distributions on manifolds are harder to make use of in applications since the normalisation factors, which we will refer to as partition functions, are complicated, intractable integrals in general that depend in a highly non-linear way on the mean of the given distribution. Nonetheless, on Riemannian symmetric spaces, the partition functions are independent of the mean and reduce to integrals over finite dimensional vector spaces. These are generally still hard to compute numerically when the dimension (more precisely the rank $N$) of the underlying symmetric space gets large. On the space of positive definite Hermitian matrices, it is possible to compute these integrals exactly using methods from random matrix theory and the so-called Stieltjes-Wigert polynomials. In other cases of interest to applications, such as the space of symmetric positive definite (SPD) matrices or the Siegel domain (related to block-Toeplitz covariance matrices), these methods seem not to work quite as well. Nonetheless, it remains possible to compute leading order terms in a large $N$ limit, which provide increasingly accurate approximations as $N$ grows. This limit is inspired by realizing a given partition function as the partition function of a zero-dimensional quantum field theory or even Chern-Simons theory. From this point of view the large $N$ limit arises naturally and saddle-point methods, Feynman diagrams, and certain universalities that relate different spaces emerge

Computing Histogram of Tensor Images using Orthogonal Series Density Estimation and Riemannian Metrics

This paper deals with the computation of the histogram of tensor images, that is, images where at each pixel is given a n by n positive definite symmetric matrix, SPD(n). An approach based on orthogonal series density estimation is introduced, which is particularly useful for the case of measures based on Riemannian metrics. By considering SPD(n) as the space of the covariance matrices of multivariate gaussian distributions, we obtain the corresponding density estimation for the measure of both the Fisher metric and the Wasserstein metric. Experimental results on the application of such histogram estimation to DTI image segmentation, texture segmentation and texture recognition are included