Geometry of tree-based tensor formats in tensor Banach spaces

TL;DR

This paper extends the geometric framework of tensor Banach spaces to include tree-based tensor formats, providing a unified description that encompasses hierarchical tensor networks and Tucker formats.

Contribution

It introduces a new geometric description of manifolds of tensors in tree-based formats, generalizing previous Tucker format results within tensor Banach spaces.

Findings

Unified geometric framework for tree-based tensor formats

Compatibility with Tucker format manifolds

Extension of Dirac-Frenkel variational principle

Abstract

In the paper `On the Dirac-Frenkel Variational Principle on Tensor Banach Spaces', we provided a geometrical description of manifolds of tensors in Tucker format with fixed multilinear (or Tucker) rank in tensor Banach spaces, that allowed to extend the Dirac-Frenkel variational principle in the framework of topological tensor spaces. The purpose of this note is to extend these results to more general tensor formats. More precisely, we provide a new geometrical description of manifolds of tensors in tree-based (or hierarchical) format, also known as tree tensor networks, which are intersections of manifolds of tensors in Tucker format associated with different partitions of the set of dimensions. The proposed geometrical description of tensors in tree-based format is compatible with the one of manifolds of tensors in Tucker format.