Gearhart-Koshy Acceleration for Affine Subspaces

TL;DR

This paper introduces an efficient acceleration method for the cyclic projections algorithm in affine subspaces, improving convergence without relying on linearity assumptions, by developing an alternative approach suitable for affine settings.

Contribution

It proposes a new acceleration technique for affine subspace projections that does not depend on linearity, enabling more practical implementations.

Findings

The method accelerates convergence of cyclic projections in affine subspaces.

It provides an explicit implementation for the affine case.

The approach is efficient and does not require the subspaces to be linear.

Abstract

The method of cyclic projections finds nearest points in the intersection of finitely many affine subspaces. To accelerate convergence, Gearhart and Koshy proposed a modification which, in each iteration, performs an exact line search based on minimising the distance to the solution. When the subspaces are linear, the procedure can be made explicit using feasibility of the zero vector. This work studies an alternative approach which does not rely on this fact, thus providing an efficient implementation in the affine setting.