Optimization problems in contracted tensor networks

By M. Espig, Wolfgang Hackbusch, Stefan Handschuh, Reinhold Schneider. 2011

83 citations recorded by the source corpus.

Subfields: Computational Mathematics, Computational Theory and Mathematics, Statistical and Nonlinear Physics

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In Computational Mathematics

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Citations in the corpus

A literature survey of low‐rank tensor approximation techniques (2013, neutral): “Possibly for this reason, more general networks have not yet been considered to a large extend in the numerical analysis community for, e.g., the solution of high-dimensional PDEs, but see [75,121] for some recent mathematically oriented work.”
Efficient low-rank approximation of the stochastic Galerkin matrix in tensor formats (2014, background): “The last two tensor formats are so called tensor networks, we refer to [6] for a mathematical description of tensor networks.”
Algorithm 941 (2014, unknown): “This includes the tensor train decomposition [Khoromskij and Oseledets 2011], the closely related but somewhat more general HTD (hierarchical Tucker decomposition) [Grasedyck 2010b; Hackbusch and Kühn 2009], and even more general tensor networks [Espig et al. 2011].”
Tensor product methods and entanglement optimization for ab initio quantum chemistry (2014, unknown): “×rd is the only virtual component and Ai(αi,mi) = 〈φ{i} αi |ξ{i} mi 〉 (65)”
Changing the topology of Tensor Networks (2012, methodology): “Another general approach is to ﬁx the resulting format and use approximation algorithms such as ALS, DMRG (both are non linear block Gauss-Seidel methods, see [8]).”
On Local Convergence of Alternating Schemes for Optimization of Convex Problems in the Tensor Train Format (2013, background): “Our general idea of proof as pursued in this work extends [43] to the (classical and hierarchical) Tucker format and, more generally, to those tensor networks [14] for which the redundancy of the respective parametrization can be characterized explicitly as exemplified in this work for the TT format”
T-Basis: a Compact Representation for Neural Networks (2020, background): “Another structure of TNs, with arguably better generalization capabilities (Wang et al., 2017), is Tensor-Ring (TR) (Perez-Garcia et al., 2007; Khoromskij, 2011; Espig et al., 2011; Zhao et al., 2016).”
Tensor Ring Decomposition (2016, methodology): “For example, TT-format has be successfully applied to different kinds of large-scale problems in numerical analysis, which includes the optimization of Rayleigh quotient, e.g., density matrix renormalization group (DMRG) [37], [38], the eigenvalue or singular value problem [39], [40], and the…”
Two-level Tucker-TT-QTT format for optimized tensor calculus (2012, methodology): “The format proposed forms a closed manifold, as the Tucker, TT and HT formats [37] [38], hence it allows stable solution of optimization problems.”