The circumcentered-reflection method achieves better rates than alternating projections

TL;DR

This paper demonstrates that the Circumcentered-Reflection Method (CRM) converges faster than the Method of Alternating Projections (MAP) in convex feasibility problems, showing linear, superlinear, and improved convergence rates under various conditions.

Contribution

It provides the first convergence rate comparison between CRM and MAP, proving CRM's superior asymptotic constants and convergence behavior in different geometric scenarios.

Findings

CRM has strictly better convergence constants than MAP.

CRM maintains linear convergence even when MAP is sublinear.

CRM achieves superlinear convergence in certain cases where MAP converges linearly.

Abstract

We study the convergence rate of the Circumcentered-Reflection Method (CRM) for solving the convex feasibility problem and compare it with the Method of Alternating Projections (MAP). Under an error bound assumption, we prove that both methods converge linearly, with asymptotic constants depending on a parameter of the error bound, and that the one derived for CRM is strictly better than the one for MAP. Next, we analyze two classes of fairly generic examples. In the first one, the angle between the convex sets approaches zero near the intersection, so that the MAP sequence converges sublinearly, but CRM still enjoys linear convergence. In the second class of examples, the angle between the sets does not vanish and MAP exhibits its standard behavior, i.e., it converges linearly, yet, perhaps surprisingly, CRM attains superlinear convergence.