591 research outputs found
Fast, asymptotically efficient, recursive estimation in a Riemannian manifold
Stochastic optimisation in Riemannian manifolds, especially the Riemannian
stochastic gradient method, has attracted much recent attention. The present
work applies stochastic optimisation to the task of recursive estimation of a
statistical parameter which belongs to a Riemannian manifold. Roughly, this
task amounts to stochastic minimisation of a statistical divergence function.
The following problem is considered : how to obtain fast, asymptotically
efficient, recursive estimates, using a Riemannian stochastic optimisation
algorithm with decreasing step sizes? In solving this problem, several original
results are introduced. First, without any convexity assumptions on the
divergence function, it is proved that, with an adequate choice of step sizes,
the algorithm computes recursive estimates which achieve a fast non-asymptotic
rate of convergence. Second, the asymptotic normality of these recursive
estimates is proved, by employing a novel linearisation technique. Third, it is
proved that, when the Fisher information metric is used to guide the algorithm,
these recursive estimates achieve an optimal asymptotic rate of convergence, in
the sense that they become asymptotically efficient. These results, while
relatively familiar in the Euclidean context, are here formulated and proved
for the first time, in the Riemannian context. In addition, they are
illustrated with a numerical application to the recursive estimation of
elliptically contoured distributions.Comment: updated version of draft submitted for publication, currently under
revie
Higher Order Statistsics of Stokes Parameters in a Random Birefringent Medium
We present a new model for the propagation of polarized light in a random
birefringent medium. This model is based on a decomposition of the higher order
statistics of the reduced Stokes parameters along the irreducible
representations of the rotation group. We show how this model allows a detailed
description of the propagation, giving analytical expressions for the
probability densities of the Mueller matrix and the Stokes vector throughout
the propagation. It also allows an exact description of the evolution of
averaged quantities, such as the degree of polarization. We will also discuss
how this model allows a generalization of the concepts of reduced Stokes
parameters and degree of polarization to higher order statistics. We give some
notes on how it can be extended to more general random media
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
Fast complexified quaternion Fourier transform
A discrete complexified quaternion Fourier transform is introduced. This is a
generalization of the discrete quaternion Fourier transform to the case where
either or both of the signal/image and the transform kernel are complex
quaternion-valued. It is shown how to compute the transform using four standard
complex Fourier transforms and the properties of the transform are briefly
discussed
Conformally Covariant Bi-Differential Operators on a Simple Real Jordan Algebra
For a simple real Jordan algebra a family of bi-differential operators
from to is constructed.
These operators are covariant under the rational action of the conformal group
of They generalize the classical {\em Rankin-Cohen} brackets (case
)
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