On manifolds of tensors of fixed TT-rank

Recently, the format of TT tensors (Hackbusch and Kuhn 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 U is an element of R-n1x ... xnd and for the manifold of tensors of TT-rank (r) under bar. 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 separation 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 T of TT tensors of fixed rank (r) under bar locally forms an embedded manifold in R-n1x ... xnd, therefore preserving the essential theoretical properties of the Tucker format, but often showing an improved scaling behaviour. Extending a similar approach for matrices (Conte and Lubich in M2AN 44: 759, 2010), we introduce certain gauge conditions to obtain a unique representation of the tangent space TUT of T and deduce a local parametrization of the TT manifold. The parametrisation of TUT is often crucial for an algorithmic treatment of high-dimensional time-dependent PDEs and minimisation problems (Lubich in From quantum to classical molecular dynamics: reduced methods and numerical analysis, 2008). We conclude with remarks on those applications and present some numerical examples.