Rank-One Approximation to High Order Tensors

By Tong Zhang, Gene H. Golub. SIAM Journal on Matrix Analysis and Applications, 2001

461 citations recorded by the source corpus.

Subfields: Computational Mathematics, Computational Mechanics, Signal Processing

View Paper: https://www.semanticscholar.org/paper/1a5ff99adceb40fbaeda66a093b90643a542a9ee

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In Computational Mathematics

In Computational Mechanics

In Signal Processing

Citations in the corpus

On the best low multilinear rank approximation of higher-order tensors (2010, methodology): “In [23, 75 , 49], specific algorithms for the best rank-1 approximation have been discussed.”
Tensor decompositions for learning latent variable models (2012, methodology): “The generalized Rayleigh quotient (Zhang and Golub, 2001) for a third-order tensor is u 7→ T (u, u, u) (u⊤u)3/2 ,”
Beating the Perils of Non-Convexity: Guaranteed Training of Neural Networks using Tensor Methods (2017, background): “Note that the convergence guarantees of tensor power iteration for orthogonal tensor decomposition have been developed in the literature (Zhang and Golub, 2001; Anandkumar et al., 2014b).”
Multiscale structural topology optimization with an approximate constitutive model for local material microstructure (2015, unknown): “Iterative solution such as alternated least squares is required to solve this nonlinear minimization problem [46,47].”
Non-negative tensor factorization with applications to statistics and computer vision (2005, methodology): “A least-squares fit of u 1 ⊗ ... ⊗ u n to a given tensor H can be achieved by employing the following " power method " scheme (see (Zhang & Golub, 2001)) summarized in Fig.”
Numerical construction of Green's functions in high dimensional elliptic problems with variable coefficients and analysis of renewable energy data via sparse and separable approximations (2012, background): “To the best of our knowledge, the rank-one update problem and an algorithm to solve it was originally proposed in [152].”
Differential-geometric Newton method for the best rank-(R1, R2, R3) approximation of tensors (2008, methodology): “In [15, 46], specific algorithms for the best rank-1 approximation have been discussed.”
Decompositions of a Higher-Order Tensor in Block Terms - Part II: Definitions and Uniqueness (2008, methodology): “Although Tucker’s decomposition and the best rank-(L,M,N) approximation can be obtained by means of a variational approach [13, 15, 61], the general theory does not fit in the framework of block decompositions.”
Zur Theorie der Niedrigrangapproximation in Tensorprodukten von Hilberträumen (2013, unknown): “Golub [115] unter vergleichbaren Voraussetzungen.”
Unsupervised Multiway Data Analysis: A Literature Survey (2009, background): “For a detailed discussion on rank decompositions and computing best rank approximations, the reader is referred to [40], [41], and [44] .”
Tensors-structured Numerical Methods in Scientific Computing: Survey on Recent Advances (2012, background): “Notice that the orthogonal canonical decomposition is well understood theoretically and allows robust greedy-type computational schemes [103, 65], however, the approximating quality of such representations is rather poor making this tensor format practically inapplicable.”
Local Convergence of the Alternating Least Squares Algorithm for Canonical Tensor Approximation (2012, background): “The rank-one case has also been studied in [27], but within a different framework.”