TT-cross approximation for multidimensional arrays

By Ivan V. Oseledets, E. E. Tyrtyshnikov. 2010

709 citations recorded by the source corpus.

Subfields: Computational Mathematics, Computational Theory and Mathematics, Numerical Methods & Scientific Computing

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Citations in the corpus

A literature survey of low‐rank tensor approximation techniques (2013, neutral): “A cross approximation method for the TT decomposition has been proposed in [214,231].”
Tensors-structured Numerical Methods in Scientific Computing: Survey on Recent Advances (2012, methodology): “As the important step, in [80] the TT factorization is extended to the case of incomplete input data by using the TT-Cross approximation via interpolation on the entries of a TT-cross.”
Multilevel preconditioning and low‐rank tensor iteration for space–time simultaneous discretizations of parabolic PDEs (2015, methodology): “As an alternative to full pointwise inversion of g with subsequent truncation, we investigate the approximation of the tensor h by means of black box approximation [43, 44, 45, 46, 47].”
Efficient High-Dimensional Stochastic Optimal Motion Control using Tensor-Train Decomposition (2015, methodology): “A. Interpolation approach The first approach relies on interpolation using TT-cross, and it provides a general solution to the problem defined by (1) and (2).”
Low-rank methods for high-dimensional approximation and model order reduction (2015, neutral): “...,iD|. A practical implementation of the truncation operator Πǫ can be based on the Empirical Interpolation Method [5, 58] or higherorder extensions of Adaptive Cross Approximation (ACA) algorithms [72]. Remark 7.1 For the solution of parameter-dependent equations with interpolation methods, simpl”
Tensor Decompositions for Signal Processing Applications: From two-way to multiway component analysis (2014, background): “been derived in [99]–[101] and are illustrated in Figure 11, while cross-approximation for the TT format has been derived in [102].”
Fast Evaluation of Near-Field Boundary Integralsusing Tensor Approximations (2012, methodology): “12), a black box algorithm has been introduced in [41] which uses ideas from [22] for the approximation of the matricisations of a tensor.”
Fast evaluation of singular BEM integrals based on tensor approximations (2012, methodology): “In [11], they also suggest a method for the approximation of a tensor in (TT) format in a black box fashion.”
An Introduction to Hierarchical (H-) Rank and TT-Rank of Tensors with Examples (2011, methodology): “In this case the complexity for the storage (representation) coincides with that in H -Tucker format. rank k is In [14] the approximation error for the projection to nodewise µ given as which is a corollary of the general H -Tucker estimate when one leaves out the projection in the d singletons…”
Tensor-Train Decomposition (2011, neutral): “2 two corollaries immediately follow [29].”
The ITensor Software Library for Tensor Network Calculations (2020, background): “from studying holographic dualities between physical theories [59, 60] to computing highdimensional integrals in applied mathematics [38,61].”
Alternating Minimal Energy Methods for Linear Systems in Higher Dimensions (2013, neutral): “For ALS there is also a lack of convergence estimates both in the classical papers [16, 4], and in the recent ones, where ALS was applied to the Tucker model [5, 33], tensor trains [29], hierarchical Tucker format [25] and high–dimensional interpolation [34].”
Black box approximation of tensors in hierarchical Tucker format (2013, methodology): “A similar result is obtained by Oseledets and Tyrtyshnikov [10] for the TT format.”
Fast Quadrature Techniques for Retarded Potentials Based on TT/QTT Tensor Approximation (2011, methodology): “Moreover, assuming the existence of low-rank TT representation the rank-r TT approximation can be computed by the heuristic algorithm called TT-cross approximation [24] avoiding the curse of dimensionality (see the numerical example below).”