Deep Centralization for the Circumcentered Reflection Method

TL;DR

The paper introduces ecCRM, a flexible framework for convex feasibility problems that generalizes existing methods by incorporating an operator and parameter, improving convergence and efficiency in large-scale applications.

Contribution

It extends the classical CRM by replacing the centralization step with an admissible operator and parameter, enabling better control over computational cost and step quality.

Findings

Global convergence is retained.

Linear convergence under mild regularity.

Superlinear convergence on smooth manifolds.

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 $\alpha$, 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.