Geometric structures in tensor representations : final release

Abstract

The main goal of this paper is to study the geometric structures associated with the representation of tensors in subspace based formats. To do this we use a property of the so-called 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, in the underlying algebraic tensor space, 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, we show that the set of tensors in the tree-based format with bounded (respectively, fixed) tree-based rank of an algebraic tensor product of normed vector spaces is an analytic Banach manifold. Indeed, the manifold geometry for the set of tensors with fixed tree-based rank is induced by a fibre bundle structure and the manifold geometry for the set of tensors with bounded tree-based rank is given by a finite union of connected components where each of them is a manifold of tensors in the tree-based format with a fixed tree-based rank. The local chart representation of these manifolds is often crucial for an algorithmic treatment of high-dimensional PDEs and minimization problems. In order to describe the relationship between these manifolds and the natural ambient space, we introduce the definition of topological tensor spaces in the tree-based format. We prove under natural conditions that any tensor of the topological tensor space under consideration admits best approximations in the manifold of tensors in the tree-based format with bounded tree-based rank. In this framework, we also show 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 bounded (respectively, fixed) tree-based rank is an immersed submanifold. This fact allows us to extend the Dirac-Frenkel variational principle in the bodywork of topological tensor spaces.