Quasi-Newton Methods on Grassmannians and Multilinear Approximations of Tensors

In this paper we proposed quasi-Newton and limited memory quasi-Newton methods for objective functions defined on Grassmann manifolds or a product of Grassmann manifolds. Specifically we defined bfgs and l-bfgs updates in local and global coordinates on Grassmann manifolds or a product of these. We proved that, when local coordinates are used, our bfgs updates on Grassmann manifolds share the same optimality property as the usual bfgs updates on Euclidean spaces. When applied to the best multilinear rank approximation problem for general and symmetric tensors, our approach yields fast, robust, and accurate algorithms that exploit the special Grassmannian structure of the respective problems, and which work on tensors of large dimensions and arbitrarily high order. Extensive numerical experiments are included to substantiate our claims.