Circumcentered methods induced by isometries

TL;DR

This paper investigates the properties and convergence behavior of circumcentered methods induced by isometries, including their linear convergence and performance in approximation problems.

Contribution

It establishes weak and linear convergence results for circumcentered isometry methods, extending existing methods like DRM and reflection methods.

Findings

Weak convergence of circumcentered isometry methods.

Conditions for linear convergence.

Performance comparison of methods on approximation tasks.

Abstract

Motivated by the circumcentered Douglas--Rachford method recently introduced by Behling, Bello Cruz and Santos to accelerate the Douglas--Rachford method, we study the properness of the circumcenter mapping and the circumcenter method induced by isometries. Applying the demiclosedness principle for circumcenter mappings, we present weak convergence results for circumcentered isometry methods, which include the Douglas--Rachford method (DRM) and circumcentered reflection methods as special instances. We provide sufficient conditions for the linear convergence of circumcentered isometry/reflection methods. We explore the convergence rate of circumcentered reflection methods by considering the required number of iterations and as well as run time as our performance measures. Performance profiles on circumcentered reflection methods, DRM and method of alternating projections for finding the best approximation to the intersection of linear subspaces are presented.