Deep Centralization for the Circumcentered Reflection Method

Abstract: We introduce the extended centralized circumcentered reflection method (ecCRM), a framework for two-set convex feasibility that encompasses the classical centralized CRM (cCRM) of Behling, Bello-Cruz, Iusem and Santos as a special case. Our method replaces the fixed centralization step of cCRM with an admissible operator $T$ and a parameter $α$, allowing control over computational cost and step quality. We show that ecCRM retains global convergence, linear rates under mild regularity, and superlinearity for smooth manifolds. Numerical experiments on large-scale matrix completion indicate that deeper operators can dramatically reduce overall runtime, and tests on high-dimensional ellipsoids show that vanishing step sizes can yield significant acceleration, validating the practical utility of both algorithmic components of ecCRM.