Tree-based tensor formats

• Antonio Falcó ^[1] ; Wolfgang Hackbusch ^[2] ; Anthony Nouy ^[3]
  1. [1] Universidad Cardenal Herrera-CEU
  2. [2] Max-Planck-Institut Mathematik in den Naturwissenschaften
  3. [3] Centrale Nantes, LMJL UMR CNRS 6629

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• Localización: SeMA Journal: Boletín de la Sociedad Española de Matemática Aplicada, ISSN-e 2254-3902, ISSN 2254-3902, Vol. 78, Nº. Extra 2, 2021, págs. 159-173
• Idioma: inglés
• DOI: 10.1007/s40324-018-0177-x
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• Resumen

  • 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.