Dual Algorithm for Orthogonal Procrustes Rotations
Alexander Shapiro, J.D. Botha · SIAM Journal on Matrix Analysis and Applications · 1988
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 $m = 2$ 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 $m > 2$ 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 of the sum of k largest eigenvalues of a 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.