Trust Region Method On Manifolds For Solving The Unbalanced Procrustes Problem

Procrustes Problem is widely used in rigid body movements, factor analysis, glob-al positioning system and so on. Results from earlier work are iterative algorithms that make use of balanced procrustes problem and linear algebra. These algorithms need to compute the singular value decomposition, thus can be time-consuming handle with large scaled matrices. In this paper, we attempt to solve the unbalanced procrustes prob-lem using Riemannian trust region method. And we solve the trust region subproblem using truncated conjugate method to guarantee the efficiency of our algorithm. In our process of forming the whole algorithm, we first define the gradient and hessian of the objective function, then we use retraction to combine the tangent space of Stiefel Man-ifold which is similar to Euclidean space with Stiefel Manifold. We also present the global and local convergence of our trust region algorithm. At last, we conduct some numerical experiments to verify the feasibility and convergence of our algorithm. We also compared our algorithm with the existing algorithms.