A Constructive Cayley Representation of Orthogonal Matrices and Applications to Optimization

TL;DR

The paper solves the problem that the Cayley transform is not defined on all orthogonal matrices by introducing a constructive pivoting via a diagonal signature matrix $D$ so that $C(DU)$ is well defined for any $U\in O(n)$. It develops a representation $U = D(I-S)(I+S)^{-1}$ with a skew-symmetric generator $S$ and provides a Gaussian-elimination based $O(n^3)$ algorithm to compute $D$ and bound $S$. It proves a uniform bound $\rho(C(DU))\le 1+2^n$ on the pivoted transform, enabling re-centering to avoid $-1$ singularities and enabling stable optimization on $SO(n)$, while also compressing orthogonal matrices to $n(n-1)/2$ real parameters plus $n$ bits. The work thus yields both practical optimization safeguards and storage-efficient encodings, with open questions about achieving a constructive uniform bound on entries comparable to nonconstructive results.

Abstract

It is known that every real orthogonal matrix can be brought into the domain of the Cayley transform by multiplication with a suitable diagonal signature matrix. In this paper we provide a constructive and numerically efficient algorithm that, given a real orthogonal matrix $U$, computes a diagonal matrix $D$ with entries in $\{\pm1\}$ such that the Cayley transform of $DU$ is well defined. This yields a representation of $U$ in the form \[ U = D(I-S)(I+S)^{-1}, \] where $S$ is a skew-symmetric matrix. The proposed algorithm requires $O(n^{3})$ arithmetic operations and produces an explicit quantitative bound on the associated skew-symmetric generator. As an application, we show how this construction can be used to control singularities in Cayley-transform-based optimization methods on the orthogonal group.

Theorems & Definitions (10)

• Theorem 1: Bib
• Theorem 2
• proof
• Corollary 1
• Remark 1: Compression of orthogonal matrices
• Theorem 3
• proof
• Remark 2
• Remark 3
• Remark 4: Quantitative control of Cayley re-centering