Optimal rates of linear convergence of the averaged alternating modified reflections method for two subspaces
Provides theoretical convergence guarantees for a projection algorithm, benefiting researchers in optimization and numerical analysis, but the result is incremental as it extends known analysis to a specific method.
The paper derives optimal linear convergence rates for the averaged alternating modified reflections (AAMR) method applied to two subspaces, showing it outperforms other projection methods. The rate is expressed in terms of the Friedrichs angle and optimized parameters.
The averaged alternating modified reflections (AAMR) method is a projection algorithm for finding the closest point in the intersection of convex sets to any arbitrary point in a Hilbert space. This method can be seen as an adequate modification of the Douglas--Rachford method that yields a solution to the best approximation problem. In this paper we consider the particular case of two subspaces in a Euclidean space. We obtain the rate of linear convergence of the AAMR method in terms of the Friedrichs angle between the subspaces and the parameters defining the scheme, by studying the linear convergence rates of the powers of matrices. We further optimize the value of these parameters in order to get the minimal convergence rate, which turns out to be better than the one of other projection methods. Finally, we provide some numerical experiments that demonstrate the theoretical results.