Orthogonal Tensor Decompositions

By Tamara G. Kolda. SIAM Journal on Matrix Analysis and Applications, 2000

418 citations recorded by the source corpus.

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

View Paper: https://www.semanticscholar.org/paper/62006dabdb449e842a03d3dfd96b6d2485b6726f

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In Computational Mechanics

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Citations in the corpus

Algorithms for Numerical Analysis in High Dimensions (2005, background): “” We refer the reader to [14, 27, 28, 13, 12] and the references therein.”
Stable recovery of sparse overcomplete representations in the presence of noise (2006, methodology): “After [12] appeared, we learned that Kruskal [21] worked with a related notion in the context of fitting trilinear arrays by simple models; this notion has later been called the Kruskal rank [42, 22].”
A least-squares approximation of partial differential equations with high-dimensional random inputs (2009, methodology): “Separated representations and the ALS algorithm Separation-of-variable techniques have been widely used to approximate high-dimensional functions using one-dimensional operations, thus virtually eliminating the curse of dimensionality, see [23,16,15,3,25,13,4] and references therein.”
Zur Theorie der Niedrigrangapproximation in Tensorprodukten von Hilberträumen (2013, background): “Zur Geschichte aus dieser Perspektive verweisen wir auf das schöne Buch von Kroonenberg [56], sowie den Übersichtsartikel von Kolda u. Bader [52].”
The Power of Normalization: Faster Evasion of Saddle Points (2016, methodology): “Thus, tensor decomposition methods have been extensively studied over the years Harshman (1970); Kolda (2001); Anandkumar et al. (2014).”
Proper Generalized Decompositions and Separated Representations for the Numerical Solution of High Dimensional Stochastic Problems (2010, background): “[9,26,25] and the references therein) and in the in nite dimensional case [32].”
Higher-order Web link analysis using multilinear algebra (2005, background): “Unlike the SVD, there is no guarantee that the rank-p PARAFAC approximation will be optimal [25].”
Multilinear image analysis for facial recognition (2002, background): “Unlike the matrix case for which the existence and uniqueness of the SVD is assured, the situation for higher-order tensors is not as simple [5].”
Tensor Rank and the Ill-Posedness of the Best Low-Rank Approximation Problem (2006, neutral): “Having said this, we would like to point out that the ill-posedness of the best rankr approximation problem for high-order tensors is not at all well-known, as is evident from the paragraph quoted earlier as well as other discussions in recent publications [44, 45, 46, 47, 80].”
The Convex Geometry of Linear Inverse Problems (2010, background): “A number of additional technical issues also arise with low-rank tensors, including the nonexistence in general of a singular value decomposition analogous to that for matrices [46], and the difference between the rank of a tensor and its border rank [23].”
Discriminant analysis with tensor representation (2005, methodology): “Before describing the Discriminant Tensor Criterion, we review the terminologies on tensor operations [3][4][11][12].”
Local Convergence of the Alternating Least Squares Algorithm for Canonical Tensor Approximation (2012, background): “For special tensors, such as hyperdiagonal or, more generally, complete orthogonal tensors [10], one can exactly determine the best rank-r approximation by truncation of a complete orthogonal representation.”
Mathematik in den Naturwissenschaften Leipzig Tensor-Product Approximation to Operators and Functions in High Dimensions (2007, background): “Algebraic methods for tensor-product approximations to high-order tensors have been extensively discussed in the literature (see [25, 4, 5, 16, 21, 27] and related references).”
Decompositions of a Higher-Order Tensor in Block Terms - Part II: Definitions and Uniqueness (2008, background): “On the other hand, by imposing orthogonality constraints on PARAFAC one obtains different (approximate) decompositions, with different properties [8, 35, 36, 42].”
Tensor Decompositions and Applications (2009, background): “Kolda [132] provides an example where the best rank-one approximation of a cubic tensor is not a factor in the best rank-two approximation.”
A Link between the Canonical Decomposition in Multilinear Algebra and Simultaneous Matrix Diagonalization (2006, background): “Imposing orthogonality constraints yields a different decomposition that has different properties [20, 29 , 30].”
A Proper Generalized Decomposition for the solution of elliptic problems in abstract form by using a functional Eckart–Young approach (2011, background): “The a posteriori construction of such tensor decompositions, when the function u is known, have been extensively studied over the past years in multilinear algebra community [6, 7, 13, 14, 4, 8] (essentially for finite dimensional vector spaces Vi).”