The orthogonally constrained regression revisited

The Penrose regression problem, including the orthonormal Procrustes problem and rotation problem to a partially specified target, is an important class of data matching problems arising frequently in multivariate analysis, yet its optimality conditions have never been clearly understood. This work offers a way to calculate the projected gradient and the, projected Hessian explicitly. One consequence of this calculation is the complete characterization of the first order and the second order necessary and sufficient optimality conditions for this problem. Another application is the natural formulation of a continuous steepest descent flow that can serve as a globally convergent numerical method. Applications to the orthonormal Procrustes problem and Penrose regression problem with partially specified target are demonstrated in this article. Finally, some numerical results are reported and commented.