Tree-based tensor formats
For researchers in tensor approximation, this work offers theoretical foundations for tree-based tensor formats, though it is incremental in extending existing characterizations and existence results.
The paper studies topological properties of tensors in tree-based Tucker format, providing a new characterization of minimal subspaces and proving existence of best approximations from sets of tensors with bounded tree-based rank under weaker norm assumptions.
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.