Linear algebra for tensor problems

DOI10.1007/s00607-009-0047-6zbMath1172.65018OpenAlexW2022179510MaRDI QIDQ841137

Evgenij E. Tyrtyshnikov, Dmitry V. Savostyanov, Ivan V. Oseledets

Publication date: 14 September 2009

Published in: Computing (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/s00607-009-0047-6

zbMATH Keywords

algorithmsdata compressionmatrix inversiondimensionality reductionlow rank approximationTucker decompositiontensor approximationmultidimensional arraysnumerical illustrationsBLAS/LAPACK routinesdata-sparse methodslarge scale matricesskeleton decompositions

Mathematics Subject Classification ID

Related Items (19)

Tensor rank bounds for point singularities in \(\mathbb{R}^3\) ⋮ Computations in quantum tensor networks ⋮ Kronecker Product Approximation of Operators in Spectral Norm via Alternating SDP ⋮ Tensor product approach to modelling epidemics on networks ⋮ Numerical solution of the Boltzmann equation with S-model collision integral using tensor decompositions ⋮ Fast stray field computation on tensor grids ⋮ A low-rank isogeometric solver based on Tucker tensors ⋮ Exploiting Efficient Representations in Large-Scale Tensor Decompositions ⋮ Fast tensor product solvers for optimization problems with fractional differential equations as constraints ⋮ Fast truncation of mode ranks for bilinear tensor operations ⋮ Numerical tensor calculus ⋮ Fast and accurate tensor approximation of a multivariate convolution with linear scaling in dimension ⋮ Grid-based electronic structure calculations: the tensor decomposition approach ⋮ Superfast solution of linear convolutional Volterra equations using QTT approximation ⋮ Cross approximation in tensor electron density computations ⋮ Approximate solution of linear systems with Laplace-like operators via cross approximation in the frequency domain ⋮ Iterative algorithms for the post-processing of high-dimensional data ⋮ Fast Multidimensional Convolution in Low-Rank Tensor Formats via Cross Approximation ⋮ A literature survey of low-rank tensor approximation techniques

Uses Software

• LAPACK
• BLAS
• Algorithm 862

Cites Work

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• Approximate iterations for structured matrices
• On cross approximation of multi-index arrays
• Matrix inversion cases with size-independent tensor rank estimates
• A theory of pseudoskeleton approximations
• Sequentiality, monadic second-order logic and tree automata.
• Analysis of individual differences in multidimensional scaling via an \(n\)-way generalization of ``Eckart-Young decomposition
• Algorithm 862
• A Newton–Grassmann Method for Computing the Best Multilinear Rank-$(r_1,$ $r_2,$ $r_3)$ Approximation of a Tensor
• Fast Simultaneous Orthogonal Reduction to Triangular Matrices
• Combining Kronecker Product Approximation with Discrete Wavelet Transforms to Solve Dense, Function-Related Linear Systems
• A Multilinear Singular Value Decomposition
• On the Best Rank-1 and Rank-(R₁ ,R₂ ,. . .,R_N) Approximation of Higher-Order Tensors
• Numerical operator calculus in higher dimensions
• Computation of the Canonical Decomposition by Means of a Simultaneous Generalized Schur Decomposition
• Tucker Dimensionality Reduction of Three-Dimensional Arrays in Linear Time
• Hierarchical Kronecker tensor-product approximations

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