2020/11/17 by Antonio Falcó, Falcó, Antonio, Wolfgang Hackbusch +3 · 1 citation
Engineering · Mathematics · Medicine · #15A69 #46A32 #46B28 #Advanced Neuroimaging Techniques and Applications #Composite Material Mechanics #Differential Geometry (math.DG) #FOS: Mathematics #Functional Analysis (math.FA) #Numerical Analysis (math.NA) #Tensor decomposition and applications
paper · doi:10.48550/arxiv.2011.08466
openalex publication_date 2020/11/17 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
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.