Convergence of an algorithm for finding the distance between a ball in a subspace and a sum of subspaces

Let S₀,S₁,...,S_k be all subspaces of a Hilbert space, and let S = S₁⊕,..., ⊕S_k. An algorithm is investigated for finding members of S₀ and S with the minimal angle between them. The algorithm is a modification of the alternating projection algorithm of von Neumann [Ann. of Math., 50(1949), 401-485; Functional Operators, The Geometry of Orthogonal Spaces, Ann. of Math Stud., 1950]. It is similar to the algorithm suggested by Brieman and Friedman [J. Amer. Statist. Assoc., 17(1985), pp. 580-598] without a proof. The convergence of the algorithm is proved to be exponentially fast.