Rank-one approximation of a higher-order tensor by a Riemannian trust-region method

In this paper, we consider a rank-one approximation problem of a higher-order tensor. We treat the problem as an optimization model on a Cartesian product of manifolds and solve this model by using a Riemannian optimization method. We derive the action of the Riemannian Hessian of the objective function on tangent vectors to the Cartesian product of manifolds. A Riemannian trust-region method with block-diagonal Hessian is used to solve this model, and the subproblem is solved by the truncated conjugate gradient method. The convergence analysis of the Riemannian trust-region method has been established in the literature with certain assumptions. We verify those assumptions for the rank-one approximation problem. Numerical experiments illustrate that the proposed model with the method is feasible and effective.