Tree-based tensor formats

TL;DR

This paper investigates the topological and algebraic properties of tensors in tree-based Tucker formats, providing new characterizations and existence results for best tensor approximations with bounded ranks.

Contribution

It introduces a new characterization of minimal subspaces and establishes the existence of best approximations in topological tensor spaces with weaker norm assumptions.

Findings

New characterization of minimal subspaces

Existence of best tensor approximations proven

Extension of topological tensor space definitions

Abstract

The main goal of this paper is to study the topological properties of tensors in tree-based Tucker format. These formats include the Tucker format and the Hierarchical Tucker format. A property of the so-called minimal subspaces is used for obtaining a representation of tensors with either bounded or fixed tree-based rank in the underlying algebraic tensor space. We provide a new characterisation of minimal subspaces which extends the existing characterisations. We also introduce a definition of topological tensor spaces in tree-based format, with the introduction of a norm at each vertex of the tree, and prove the existence of best approximations from sets of tensors with bounded tree-based rank, under some assumptions on the norms weaker than in the existing results.