Perturbation Theory and Optimality Conditions for the Best Multilinear Rank Approximation of a Tensor

The problem of computing the best rank-(p,q,r) approximation of a third order tensor is considered. First the problem is reformulated as a maximization problem on a product of three Grassmann manifolds. Then expressions for the gradient and the Hessian are derived in a local coordinate system at a stationary point, and conditions for a local maximum are given. A first order perturbation analysis is performed using the Grassmann manifold framework. The analysis is illustrated in a few examples, and it is shown that the perturbation theory for the singular value decomposition is a special case of the tensor theory.funding agencies|Swedish Research Council||Institute for Computational Engineering and Sciences at The University of Texas at Austin||| Dnr 2008-7145

Perturbation Theory and Optimality Conditions for the Best Multilinear Rank Approximation of a Tensor

The problem of computing the best rank-(p,q,r) approximation of a third order tensor is considered. First the problem is reformulated as a maximization problem on a product of three Grassmann manifolds. Then expressions for the gradient and the Hessian are derived in a local coordinate system at a stationary point, and conditions for a local maximum are given. A first order perturbation analysis is performed using the Grassmann manifold framework. The analysis is illustrated in a few examples, and it is shown that the perturbation theory for the singular value decomposition is a special case of the tensor theory.funding agencies|Swedish Research Council||Institute for Computational Engineering and Sciences at The University of Texas at Austin||| Dnr 2008-7145