Abstract of Vladimir Kazeev

Structure of the Hessian and economical realization of Newton Trust Region method for the problem of low-rank canonical tensor approximation
Given a tensor in some canonical representation, we consider a problem of approximation with a fixed number of summands. To solve this problem, one can naturally try to adopt some general minimization techniques. One successful method in this line was recently proposed by M.Espig. This talk contributes to the same line in the two ways. First, we discover a more detailed structure in the Hessian and show that all auxiliary matrices of quadratic model can be obtained with the cost quadratic in the dimensionality (instead of cubic in the previous works). Second, we develop a successful version of a Newton Trust Region-based algorithm. This algorithm effectively exploits stucture of the Hessian to minimize the objective function. To solve the Newton minimization problem with trust-region constraints at each step, we use the preconditioned conjugate gradients. When the Hessian becomes indefinite, we apply the strategy proposed by Steihaug. Convergence to a local minimum is global superlinear and local quadratic.


Lars Grasedyck (MPI Leipzig, Germany)
Wolfgang Hackbusch (MPI Leipzig, Germany)
Boris Khoromskij (MPI Leipzig, Germany)