On manifolds of tensors of fixed TT-rank

Abstract

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 ${U \in \mathbb{R}^{n_1\times \cdots\times n_d}}$ and for the manifold of tensors of TT-rank ${\underline{r}}$ . 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 ${\mathbb{T}}$ of TT tensors of fixed rank ${\underline{r}}$ locally forms an embedded manifold in ${\mathbb{R}^{n_1\times\cdots\times n_d}}$ , therefore preserving the essential theoretical properties of the Tucker format, but often showing an improved scaling behaviour. Extending a similar approach for matrices (Conte and Lubich in M2AN 44:759, 2010), we introduce certain gauge conditions to obtain a unique representation of the tangent space ${\mathcal{T}_U\mathbb{T}}$ of ${\mathbb{T}}$ and deduce a local parametrization of the TT manifold. The parametrisation of ${\mathcal{T}_{U}\mathbb{T}}$ 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.