Circumcentered methods induced by isometries

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.