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On manifolds of tensors of fixed TT-rank
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In this talk, we present some recent results on the TT format which has recently turned out to be a promising format for the representation and approximation of high dimensional tensors $U \in \R^{n_1\times \ldots\times n_d}$ [1-4].
As a first result, we prove that the TT (or compression) ranks $r_i$ of a tensor $U$ are unique and equal to the respective seperation ranks of $U$ if the components of the TT decomposition are required to fulfil a certain maximal rank condition. We then show that the set $\bbT$ of TT tensors of fixed rank $\underline{r}$ forms an embedded manifold in $\R^{n^d}$, therefore preserving the essential theoretical properties of the Tucker format, but often showing an improved scaling behaviour. Extending a similar approach for matrices [5], we introduce certain gauge conditions to obtain a unique representation of the tangent space $\cT_U\bbT$ of $\bbT$ and deduce a local parametrization of the TT manifold. The parametrisation of $\cT_{U}\bbT$ is often crucial for an algorithmic treatment of high-dimensional time-dependent PDEs and minimisation problems [6]. We conclude with some numerical examples.

[1] W. Hackbusch, S. K\"uhn, \emph{A new scheme for the tensor representation}, Preprint 2, MPI MIS, 2009.
[2] Breaking the curse of dimensionality, or how to use SVD in many dimensions, ICM HKBU Research
Report 09-03, February 2009 (www.math.hkbu.edu.hk/ICM); SIAM J. Sci. Comput., 2009, to appear.
[3] I. Oseledets, \emph{On a new tensor decomposition}, Doklady Math. 427, no. 2, 2009.
[4]I.V. Oseledets; E.E. Tyrtyshnikov, \emph{Tensor tree decomposition does not need a tree}, Submitted to Linear Algebra Appl., 2009.
[5] D. Conte, C. Lubich, \emph{An error analysis of the multi-configuration time-dependent Hartree method of quantum dynamics}, M2AN 44, p. 759, 2010.
[6] C. Lubich, \emph{From Quantum to Classical Molecular Dynamics: Reduced methods and Numerical Analysis}, Z\"urich Lectures in advanced mathematics, EMS, 2008.