# The Block-wise Circumcentered-Reflection Method

## TL;DR

This paper introduces a block-wise circumcentered-reflection method that accelerates projection algorithms onto affine subspaces, generalizes the method of alternating projections, and guarantees linear convergence with optimal solutions in certain cases.

## Contribution

It presents a novel block-wise circumcentered-reflection framework that generalizes existing methods and proves linear convergence, with special cases achieving optimal solutions in one step.

## Key findings

- The method accelerates convergence compared to classical approaches.
- Linear convergence is theoretically established and numerically demonstrated.
- In special cases, the method finds the best approximation in a single step.

## Abstract

The elementary Euclidean concept of circumcenter has recently been employed to improve two aspects of the classical Douglas--Rachford method for projecting onto the intersection of affine subspaces. The so-called circumcentered-reflection method is able to both accelerate the average reflection scheme by the Douglas--Rachford method and cope with the intersection of more than two affine subspaces. We now introduce the technique of circumcentering in blocks, which, more than just an option over the basic algorithm of circumcenters, turns out to be an elegant manner of generalizing the method of alternating projections. Linear convergence for this novel block-wise circumcenter framework is derived and illustrated numerically. Furthermore, we prove that the original circumcentered-reflection method essentially finds the best approximation solution in one single step if the given affine subspaces are hyperplanes.

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Source: https://tomesphere.com/paper/1902.10866