Thorsten Rohwedder

The Alternating Linear Scheme for Tensor Optimization in the Tensor Train Format

Recent achievements in the field of tensor product approximation provide promising new formats for the representation of tensors in form of tree tensor networks. In contrast to the canonical

Dynamical Approximation By Hierarchical Tucker And Tensor-Train Tensors

r

On manifolds of tensors of fixed TT-rank

-term representation (CANDECOMP, PARAFAC), these new formats provide stable representations, while the amount of required data is only slightly larger. The tensor train (TT) format [SIAM J. Sci. Comput., 33 (2011), pp. 2295-2317], a simple special case of the hierarchical Tucker format [J. Fourier Anal. Appl., 5 (2009), p. 706], is a useful prototype for practical low-rank tensor representation. In this article, we show how optimization tasks can be treated in the TT format by a generalization of the well-known alternating least squares (ALS) algorithm and by a modified approach (MALS) that enables dynamical rank adaptation. A formulation of the component equations in terms of so-called retraction operators helps to show that many structural properties of the original problems transfer to the micro-iterations, giving what is to our knowledge the first stable generic algorithm for the treatment of optimization tasks in the tensor format. For the examples of linear equations and eigenvalue equations, we derive concrete working equations for the micro-iteration steps; numerical examples confirm the theoretical results concerning the stability of the TT decomposition and of ALS and MALS but also show that in some cases, high TT ranks are required during the iterative approximation of low-rank tensors, showing some potential of improvement.

Variational calculus with sums of elementary tensors of fixed rank

We extend results on the dynamical low-rank approximation for the treatment of time-dependent matrices and tensors (Koch and Lubich; see [SIAM J. Matrix Anal. Appl., 29 (2007), pp. 434--454], [SIAM J. Matrix Anal. Appl., 31 (2010), pp. 2360--2375]) to the recently proposed hierarchical Tucker (HT) tensor format (Hackbusch and Kuhn; see [J. Fourier Anal. Appl., 15 (2009), pp. 706--722]) and the tensor train (TT) format (Oseledets; see [SIAM J. Sci. Comput., 33 (2011), pp. 2295--2317]), which are closely related to tensor decomposition methods used in quantum physics and chemistry. In this dynamical approximation approach, the time derivative of the tensor to be approximated is projected onto the time-dependent tangent space of the approximation manifold along the solution trajectory. This approach can be used to approximate the solutions to tensor differential equations in the HT or TT format and to compute updates in optimization algorithms within these reduced tensor formats. By deriving and analyzing th...

Adaptive eigenvalue computation: complexity estimates

Recently, the format of TT tensors (Hackbusch and Kühn in J Fourier Anal Appl 15:706–722, 2009; Oseledets in SIAM J Sci Comput 2009, submitted; Oseledets and Tyrtyshnikov in SIAM J Sci Comput 31:5, 2009; Oseledets and Tyrtyshnikov in Linear Algebra Appl 2009, submitted) has turned out to be a promising new format for the approximation of solutions of high dimensional problems. In this paper, we prove some new results for the TT representation of a tensor

Adaptive Methods in Quantum Chemistry

{U \in \mathbb{R}^{n_1\times \cdots\times n_d}}

Error estimates for the Coupled Cluster method

and for the manifold of tensors of TT-rank