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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Notes
By using the notion of edge, that is, the connection between one vertex to another, then our definition of depth coincides with the classical definition of height, i.e. the longest downward path between the root and a leaf.
It suffices to have in (8) the terms \(n=0\) and \(n=N.\) The derivatives are to be understood as weak derivatives.
Recall that a multilinear map T from
equipped with the product topology to a normed space \((W,\Vert \cdot \Vert )\) is continuous if and only if \(\Vert T\Vert <\infty \), with
$$\begin{aligned} \Vert T \Vert \!:=\! \sup _{\begin{array}{c} (v_1,\ldots ,v_d) \\ \Vert (v_1,\ldots ,v_d)\Vert \le 1 \end{array}} \Vert T(v_{1},\ldots ,v_{d})\Vert&\!=\! \!\sup _{\begin{array}{c} (v_1,\ldots ,v_d) \\ \Vert v_1\Vert _1\le 1,\ldots , \Vert v_d\Vert _d\le 1 \end{array}} \Vert T(v_{1},\ldots ,v_{d})\Vert \!=\! \sup _{{ (v_1,\ldots ,v_d)}} \frac{\Vert T (v_{1},\ldots ,v_{d})\Vert }{\Vert v_1\Vert _1\ldots \Vert v_d\Vert _d}. \end{aligned}$$
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Falcó, A., Hackbusch, W. & Nouy, A. Tree-based tensor formats. SeMA 78, 159–173 (2021). https://doi.org/10.1007/s40324-018-0177-x
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DOI: https://doi.org/10.1007/s40324-018-0177-x