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Article Dans Une Revue Annali di Matematica Pura ed Applicata Année : 2023

Geometry of tree-based tensor formats in tensor Banach spaces

Résumé

In the paper `On the Dirac-Frenkel Variational Principle on Tensor Banach Spaces', we provided a geometrical description of manifolds of tensors in Tucker format with fixed multilinear (or Tucker) rank in tensor Banach spaces, that allowed to extend the Dirac-Frenkel variational principle in the framework of topological tensor spaces. The purpose of this note is to extend these results to more general tensor formats. More precisely, we provide a new geometrical description of manifolds of tensors in tree-based (or hierarchical) format, also known as tree tensor networks, which are intersections of manifolds of tensors in Tucker format associated with different partitions of the set of dimensions. The proposed geometrical description of tensors in tree-based format is compatible with the one of manifolds of tensors in Tucker format.

Dates et versions

hal-03919920 , version 1 (03-01-2023)

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Citer

Antonio Falcó, Wolfgang Hackbusch, Anthony Nouy. Geometry of tree-based tensor formats in tensor Banach spaces. Annali di Matematica Pura ed Applicata, 2023, 202 (5), pp.2127-2144. ⟨10.1007/s10231-023-01315-0⟩. ⟨hal-03919920⟩
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