Hierarchical tensor approximation or tensor network states by convex optimization
Talk by Reinhold Schneider
Date: 15.10.14 Time: 16.15 - 17.15 Room: Y27H25
Hierarchical Tucker tensor format (Hackbusch) and a particular case Tensor Trains (TT) (Tyrtyshnikov) have been introduced for high dimensional problems. The parametrization has been known in quantum physics as tensor network states. There are several ways to cast an approximate numerical solution into a variational framework. The Ritz Galerkin ansatz leads to an optimization problem on Riemannian manifolds. With this techniques, or in simplyfied form with a one-site DMRG one can be easily trapped into local minima. Although this problem can be avoided in combination with greedy techniques, we pursue a variational framework allowing concepts of compressive sensing. We obtain a soft shrinkage iteration scheme with in a Hierarchical SVD (HSVD) (or Vidal decomposition). We show that the iterates converge to the unique minimum of a convex optimization problem, even if the problem is ill-conditioned. We show a quasi-optimal error rate for the solution. The talk presents ongoing joint work with M. Bachmayr (RWTH Aachen).