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arXiv · 1810.01262

Tree-based tensor formats

Antonio Falco·Wolfgang Hackbusch·Anthony Nouy
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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.

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Antonio Falco, Wolfgang Hackbusch, Anthony Nouy. 2018-09-29. Tree-based tensor formats. https://doi.org/10.1007/s40324-018-0177-x

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