F ¨ Ur Mathematik in Den Naturwissenschaften Leipzig on Minimal Subspaces in Tensor Representations on Minimal Subspaces in Tensor Representations

70 citations recorded by the source corpus.

Subfields: Biomedical Engineering, Computational Mathematics, Computational Mechanics, Pure Mathematics

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Citations in the corpus

A literature survey of low‐rank tensor approximation techniques (2013, neutral): “Tree-structured tensor networks, as the hierarchical Tucker and the TT format, are closed [75, 82, 108] in the sense that tensors with ranks at most r µ form a closed set in n 1 ×···× n d .”
Fast Evaluation of Near-Field Boundary Integralsusing Tensor Approximations (2012, methodology): “Für das Tucker-Format und das hierarchische Format konnte die Existenz von Bestapproximationen im allgemeinen Kontext von Tensor-Banach-Räumen gezeigt werden [20].”
On Local Convergence of Alternating Schemes for Optimization of Convex Problems in the Tensor Train Format (2013, background): “forming a weakly closed set [12] with manifold properties [27], fostering for instance its application in quantum chemistry [4].”
LOCAL CONVERGENCE OF ALTERNATING SCHEMES FOR OPTIMIZATION OF CONVEX PROBLEMS IN THE TT FORMAT (2011, unknown): “An alternative is provided by the { also quite classical { Tucker format [35], with the set of Tucker tensors of bounded rank forming a weakly closed set [11] with manifold properties [25], fostering for instance its application in quantum chemistry [4].”
Proper generalized decomposition for nonlinear convex problems in tensor Banach spaces (2011, unknown): “In [15] it was introduced the following definition.”
Zur Theorie der Niedrigrangapproximation in Tensorprodukten von Hilberträumen (2013, background): “Dafür bildet, wie wir in Kapitel 5 zeigen, die Menge Sr aller Tensoren mit optimalem Unterraumrang genau r eine eingebettete differenzierbare Hilbertuntermannigfaltigkeit in ⊗Uμ, und S≤r ist sogar schwach folgenabgeschlossen [25].”
Variational calculus with sums of elementary tensors of fixed rank (2012, unknown): “As per the more recent TT format, a part of the authors recently obtained some promising practical results for optimization problems in [18]; from the theoretical side, this is complemented by the fact that the TT and HT format share many of the favorable properties of the Tucker format [10,19] whil”
On manifolds of tensors of fixed TT-rank (2012, methodology): “An according decomposition algorithm using hierarchical singular value decompositions and providing a quasi-optimal approximation in the `2-sense has been introduced by Grasedyck [18]; Hackbusch has also recently shown the existence of a best approximation [15].”
DMRG+QTT approach to computation of the ground state for the molecular Schrödinger operator (2010, background): “For more detailed discussion on theoretical aspects of H-Tucker and TT formats, see [11, 7].”
Low-rank methods for high-dimensional approximation and model order reduction (2015, unknown): “inal, that means that there exists a solution to the best approximation problem (3.7) for any u (see Section 1.7). If the norm k·k is not weaker than the injective norm, then Tr is weakly closed (see [37]), and therefore proximinal if Xk·kis reﬂexive (e.g. for X = N ν∈D L p µν (Ξν) for any 1 <p&l”