Random orthogonal matrices and the Cayley transform

TL;DR

This paper introduces a method to parametrize and analyze random orthogonal matrices using the Cayley transform, facilitating statistical modeling and simulation on manifolds with asymptotic normality results.

Contribution

It provides a Euclidean parametrization of Stiefel and Grassmann manifolds via the Cayley transform, including Jacobian calculations and an MCMC sampling approach.

Findings

Derived Jacobian for change of variables on manifolds

Developed MCMC method for sampling distributions

Proved asymptotic normal approximation for Euclidean parameters

Abstract

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.