3,107 research outputs found
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 ) 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
limit, which provide increasingly accurate approximations as 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 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
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