Optimal Convergence Rates for Generalized Alternating Projections

TL;DR

This paper derives optimal convergence rates for generalized alternating projections when applied to two linear subspaces, showing improved performance over existing methods and proposing an adaptive scheme for parameter tuning.

Contribution

It provides a method to select algorithm parameters optimally based on the Friedrichs angle, enhancing convergence rates for two subspace projections.

Findings

Optimal parameter selection improves convergence rate.

The convergence rate depends on the Friedrichs angle.

Adaptive scheme estimates angle and updates parameters online.

Abstract

Generalized alternating projections is an algorithm that alternates relaxed projections onto a finite number of sets to find a point in their intersection. We consider the special case of two linear subspaces, for which the algorithm reduces to a matrix teration. For convergent matrix iterations, the asymptotic rate is linear and decided by the magnitude of the subdominant eigenvalue. In this paper, we show how to select the three algorithm parameters to optimize this magnitude, and hence the asymptotic convergence rate. The obtained rate depends on the Friedrichs angle between the subspaces and is considerably better than known rates for other methods such as alternating projections and Douglas-Rachford splitting. We also present an adaptive scheme that, online, estimates the Friedrichs angle and updates the algorithm parameters based on this estimate. A numerical example is provided that supports our theoretical claims and shows very good performance for the adaptive method.