Deep Centralization for the Circumcentered Reflection Method
Pablo Barros
TL;DR
This work extends centralized CRM to ecCRM by incorporating an admissible operator $T$ and a relaxation schedule $\alpha_k$ for two-set CFPs, preserving global convergence and achieving linear rates with potential for superlinear speed on smooth intersections. The framework reduces computation to a small number of projections per iteration and allows modular kernel choices, including deep kernels that amplify contraction. Theoretical results establish Fejér monotonicity and linear convergence under mild regularity, with superlinear behavior under favorable geometry and kernel choices. Empirical tests on matrix completion and ellipsoids demonstrate meaningful runtime and iteration reductions from deeper centralization and vanishing-step strategies, highlighting practical gains in large-scale convex feasibility problems.
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.