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Alternating Directional Gradient Algorithms and Tensor Completion in Hierachical Tensor Formats

Banff International Research Station for Mathematical Innovation and Discovery

Hierarchical tensor representation , e.g. Tucker tensor format (Hackbusch), Multi-layer TDMCH (Meyer et al.), or tree tensor network states (G. Chan et al.) and Tensor Trains (TT) (Oseledets) or Matrix product states (MPS) offer stable and robust approximation by a low order cost . We will discuss tensor recovery, in particular tensor completion for hierarchical tensors (resp. tree tensor networks) in analogy to matrix completion. The goal is to recover or to approximate a low rank rank tensor from few samples or measurements. A typical application can be the approximation of the potential energy surface. For this purpose, we will discuss ALS (one site DMRG) approach and a new ADF and an alternating directional gradient method which has a better scaling than ALS. However the ADF approach can also be used for eigenvalue computation (e.g. ground state) and time evaluation. We will investigate the intimate relationship to Riemannian gradient optimization techniques and Dirac Frenkel principle.

Mathematics; Quantum theory; Partial differential equations; Applied computer science

video/mp4

Author affiliation: Technische Universitat Berlin (Germany)

2019-05-30

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/70416

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/