The Representation and Parametrization of Orthogonal Matrices

Abstract

Four representations and parametrizations of orthogonal matrices Q ∈ R(m×n) in terms of the minimal number of essential parameters {φ} are discussed: the exponential representation, the Householder reflector representation, the Givens rotation representation, and the rational Cayley transform representation. Both square n = m and rectangular n < m situations are considered. Two separate kinds of parametrizations are considered: one in which the individual columns of Q are distinct, the Stiefel manifold, and the other in which only span(Q) is significant, the Grassmann manifold. The practical issues of numerical stability, continuity, and uniqueness are discussed. The computation of Q in terms of the essential parameters {φ}, and also the extraction of {φ} for a given Q are considered for all of the parametrizations. The transformation of gradient arrays between the Q and {φ} variables is discussed for all representations. It is our hope that developers of new methods will benefit from this comparative presentation of an important but rarely analyzed subject.

Bayesian Inference over the Stiefel Manifold via the Givens Representation

“…we can consider any orthogonal matrix as a function, Y (Θ), of these angles, effectively parameterizing the Stiefel manifold and yielding the Givens representation. The Givens representation is a smooth representation with respect to the angles Θ (Shepard et al, 2015), and lines up with our geometric insight discussed in the previous subsection. We also note that the number of angles in the Givens representation corresponds exactly to the inherent dimensionality, d, of the Stiefel manifold.…”

“…While appealing, the transformation approach can pose its own challenges related to the change in measure, topology, and parameterization. While there are many possible parameterizations of orthogonal matrices (Anderson et al, 1987;Shepard et al, 2015), we seek smooth continuously differentiable representations that can be used readily in inference methods such as Hamiltonian Monte Carlo (HMC). It is also important to consider for transformed random variables the change-of-measure adjustment term which needs to be computable efficiently.…”

“…We introduce an approach to posterior inference for statistical models with orthogonal matrix parameters based on Givens representation of orthogonal matrices (Shepard et al, 2015). Our approach addresses several of the concerns of using a transformationbased approach.…”

Re-parametrising Cost Matrices for Tuning Model Predictive Controllers

“…where P * is fixed to be the solution of the discrete-time algebraic Riccati equation (9). Theorem 1: Under Assumptions 1, 2, 3, 4, the closed-loop system using any MPC with cost matrices {Q, P, R} with augmented formulation (11) found through meta-cost tuning (7) with the re-parametrisation approach in Section III is exponentially stable with region of attraction X r .…”

“…where W Q , W R are orthogonal matrices and D Q , D R are diagonal matrices. The orthogonal matrix W Q may be minimally parametrised by n 2 − n /2 parameters (see [11] for a survey covering various approaches to parametrise orthogonal matrices), and likewise W R may be minimally parametrised by m 2 − m /2 parameters. In this paper, we choose the method of parametrisation to be the Givens rotations, which are a generalisation of the Euler rotations.…”