Random orthogonal matrices and the Cayley transform

10/05/2018
by   Michael Jauch, et al.
0

Random orthogonal matrices play an important role in probability and statistics, arising in multivariate analysis, directional statistics, and models of physical systems, among other areas. Calculations involving random orthogonal matrices are complicated by their constrained support. Accordingly, we parametrize the Stiefel and Grassmann manifolds, represented as subsets of orthogonal matrices, in terms of Euclidean parameters using the Cayley transform. We derive the necessary Jacobian terms for change of variables formulas. Given a density defined on the Stiefel or Grassmann manifold, these allow us to specify the corresponding density for the Euclidean parameters, and vice versa. As an application, we describe and illustrate through examples a Markov chain Monte Carlo approach to simulating from distributions on the Stiefel and Grassmann manifolds. Finally, we establish an asymptotic independent normal approximation for the distribution of the Euclidean parameters which corresponds to the uniform distribution on the Stiefel manifold. This result contributes to the growing literature on normal approximations to the entries of random orthogonal matrices or transformations thereof.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
06/18/2019

Monte Carlo simulation on the Stiefel manifold via polar expansion

Motivated by applications to Bayesian inference for statistical models w...
research
09/30/2022

Generalized Fiducial Inference on Differentiable Manifolds

We introduce a novel approach to inference on parameters that take value...
research
04/11/2021

A symmetric matrix-variate normal local approximation for the Wishart distribution and some applications

The noncentral Wishart distribution has become more mainstream in statis...
research
09/04/2020

Density estimation and modeling on symmetric spaces

In many applications, data and/or parameters are supported on non-Euclid...
research
01/23/2019

Hamiltonian Monte-Carlo for Orthogonal Matrices

We consider the problem of sampling from posterior distributions for Bay...
research
03/02/2017

The Unreasonable Effectiveness of Structured Random Orthogonal Embeddings

We examine a class of embeddings based on structured random matrices wit...
research
03/12/2021

Differentiating densities on smooth manifolds

Lebesgue integration of derivatives of strongly-oscillatory functions is...

Please sign up or login with your details

Forgot password? Click here to reset