SeMA Journal

pp 1–15 | Cite as

Tree-based tensor formats

  • Antonio FalcóEmail author
  • Wolfgang Hackbusch
  • Anthony Nouy


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.


Tensor spaces Tree-based tensor format Tree-based rank Best approximation 

Mathematics Subject Classification

15A69 46B28 46A32 


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Copyright information

© Sociedad Española de Matemática Aplicada 2018

Authors and Affiliations

  1. 1.ESI International Chair@CEU-UCH, Departamento de Matemáticas, Física y Ciencias TecnológicasUniversidad Cardenal Herrera-CEU, CEU UniversitiesAlfara del PatriarcaSpain
  2. 2.Max-Planck-Institut Mathematik in den NaturwissenschaftenLeipzigGermany
  3. 3.Centrale Nantes, LMJL UMR CNRS 6629Nantes Cedex 3France

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