Characterizing arbitrarily slow convergence in the method of alternating projections

TL;DR

This paper revisits a 1997 theorem on the slow convergence of the method of alternating projections, correcting errors in the original proof and extending its validity to complex Hilbert spaces.

Contribution

It provides a new proof of the trichotomy theorem using spectral theory and confirms its correctness in both real and complex Hilbert spaces.

Findings

The original theorem is correct despite proof errors.

A new proof using spectral theorem is provided.

The theorem applies to complex Hilbert spaces as well.

Abstract

In 1997, Bauschke, Borwein, and Lewis have stated a trichotomy theorem that characterizes when the convergence of the method of alternating projections can be arbitrarily slow. However, there are two errors in their proof of this theorem. In this note, we show that although one of the errors is critical, the theorem itself is correct. We give a different proof that uses the multiplicative form of the spectral theorem, and the theorem holds in any real or complex Hilbert space, not just in a real Hilbert space.