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

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.Comment: 16 pages, 3 figures (Corrected minor typos in Remark 2.2, Algorithm 2.5, proof of Theorem 3.12, as well as elaborated on certain proof