The rate of convergence in the method of alternating projections

TL;DR

This paper introduces a generalized cosine parameter for multiple subspaces in Hilbert spaces to analyze the convergence rate of the cyclic alternating projections method, establishing conditions for rapid or slow convergence.

Contribution

It generalizes the Friedrichs angle to multiple subspaces and provides dichotomy theorems for convergence rates in Hilbert and Banach spaces.

Findings

Established conditions for quick uniform convergence.

Identified scenarios leading to arbitrarily slow convergence.

Proposed multiple interpretations of the ASC concept.

Abstract

A generalization of the cosine of the Friedrichs angle between two subspaces to a parameter associated to several closed subspaces of a Hilbert space is given. This parameter is used to analyze the rate of convergence in the von Neumann-Halperin method of cyclic alternating projections. General dichotomy theorems are proved, in the Hilbert or Banach space situation, providing conditions under which the alternative QUC/ASC (quick uniform convergence versus arbitrarily slow convergence) holds. Several meanings for ASC are proposed.