# Accelerating the convergence of the method of alternating projections

Hundal, Hein and Deutsch, F. R. and Park, Sung-Ho and Bauschke, Heinz H. (1999) Accelerating the convergence of the method of alternating projections. [Preprint]

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The powerful von Neumann-Halperin method of alternating projections (MAP) is an algorithm for determining the best approximation to any given point in a Hilbert space from the intersection of a finite number of subspaces. It achieves this by reducing the problem to an iterative scheme which involves only computing best approximations from the {\it individual} subspaces which make up the intersection. The main practical drawback of this algorithm, at least for some applications, is that the method is slowly convergent. In this paper, we consider a general class of iterative methods which includes the MAP as a special case. For such methods, we study an accelerated'' version of this algorithm which was considered earlier by Gubin, Polyak, and Raik [14] and by Gearhart and Koshy [13]. We show that the accelerated algorithm converges faster than the MAP in the case of two subspaces, but is in general {\it not faster} than the MAP for more than two subspaces! However, for a symmetric'' version of the MAP, the accelerated algorithm always converges faster for any number of subspaces. Our proof seems to require the use of the Spectral Theorem for self-adjoint mappings.