Geometric structures in tensor representations (release 2

In this paper we introduce a tensor subspace based format for the representation of a tensor in a tensor space. To do this we use a property of minimal subspaces which allows us to describe the tensor representation by means of a rooted tree. By using the tree structure and the dimensions of the associated minimal subspaces, we introduce the set of tensors in a tree based format with either bounded or fixed tree based rank. This class contains the Tucker format and the Hierarchical Tucker format (including the Tensor Train format). In particular, any tensor of the topological tensor space under consideration admits best approximations in the set of tensors in the tree based format with bounded tree based rank. Moreover, we show that the set of tensors in the tree based format with fixed tree based rank is an analytic Banach manifold. The local chart representation of the manifold is often crucial for an algorithmic treatment of high-dimensional time-dependent PDEs and minimisation problems. We also show, under some natural assumptions, that the tangent (Banach) space at a given tensor is a complemented subspace in the natural ambient tensor Banach space and hence the set of tensors in the tree based format with fixed tree based rank is an immersed submanifold. This fact allows us to extend the Dirac-Frenkel variational principle in the framework of topological tensor spaces