DUAL ALGORITHMS FOR ORTHOGONAL PROCRUSTES ROTATIONS *

This paper considers a problem of rotating m matrices toward a best least-squares fit. The problem is known as the orthogonal Procrustes problem. For rn 2 the solution of this problem is known and can be given in a closed form using the singular value decomposition. It appears that the general case of rn > 2 cannot be solved explicitly and an iterative procedure is required. The authors discuss a dual approach to the Procrustes problem where the maximal value of the objective function is approximated from above. This involves minimization ofthe sum ofk largest eigenvalues ofa symmetric matrix. It will be shown that under certain conditions ensuring differentiability of the obtained function at the minimum, this method gives the global solution of the Procrustes problem.