Accelerating the alternating projection algorithm for the case of affine subspaces using supporting hyperplanes

TL;DR

This paper introduces acceleration schemes for the von Neumann-Halperin alternating projection method onto affine subspaces, leveraging hyperplanes to improve convergence, supported by theoretical conditions and numerical experiments.

Contribution

It proposes novel acceleration techniques using hyperplanes for the alternating projection algorithm on affine subspaces, with proven convergence conditions.

Findings

Accelerations improve convergence speed in matrix model updating.

The proposed methods converge strongly under certain conditions.

Numerical results demonstrate enhanced performance over standard methods.

Abstract

The von Neumann-Halperin method of alternating projections converges strongly to the projection of a given point onto the intersection of finitely many closed affine subspaces. We propose acceleration schemes making use of two ideas: Firstly, each projection onto an affine subspace identifies a hyperplane of codimension 1 containing the intersection, and secondly, it is easy to project onto a finite intersection of such hyperplanes. We give conditions for which our accelerations converge strongly. Finally, we perform numerical experiments to show that these accelerations perform well for a matrix model updating problem.