On the Best Rank-1 Approximation of Higher-Order Supersymmetric Tensors

By Eleftherios Kofidis, Phillip A. Regalia. 2001

413 citations recorded by the source corpus.

Subfields: Computational Mathematics, Computational Theory and Mathematics, Signal Processing

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Citations in the corpus

Tensor decompositions for learning latent variable models (2012, methodology): “, fourth-order cumulant) as the contrast function, this method reduces to a generalization of the power method for symmetric tensors (Lathauwer et al., 2000; Zhang and Golub, 2001; Kofidis and Regalia, 2002).”
Z-EIGENVALUE INCLUSION THEOREMS FOR TENSORS (2016, background): “The Z-eigenvalue problem plays a fundamental role in best rank-one approximation, which has numerous applications in engineering and higher order statistics [5, 21].”
Eigenvalues and invariants of tensors (2007, background): “Thus, tensors discussed in [ 12 ,14,15,19,21,25] are actually supermatrices in this paper.”
A new eigenvalue inclusion set for tensors and its applications (2015, background): “The tensor A is called supersymmetric [7,18] if ai1···im = aπ(i1···im), ∀π ∈ Πm, where Πm is the permutation group of m indices.”
Eigenvalues of a real supersymmetric tensor (2005, background): “We show that eigenvalues are roots of a one-dimensional polynomial, and when the order of the tensor is even, E-eigenvalues are roots of another one-dimensional polynomial.”
Quasi-Newton Methods on Grassmannians and Multilinear Approximations of Tensors (2009, methodology): “Applications involving symmetric tensors are found in signal processing, independent component analysis, and the analysis of multivariate cumulants in statistics [10, 8, 36, 14, 17, 18, 9, 43, 40].”
Factorization strategies for third-order tensors (2011, background): “Algorithms for finding this (iteratively) can be found in [20, 28, 47].”
Subtracting a best rank-1 approximation may increase tensor rank (2009, methodology): “The latter includes best rank-1 approximation algorithms [9,38,17], conditions under which the best rank-1 approximation is equal to the best symmetric rank-1 approximation [24], and a relation between the best symmetric rank-1 approximation and the notions of eigenvalues and eigenvectors of a symme”
A New Truncation Strategy for the Higher-Order Singular Value Decomposition (2012, methodology): “For instance, the T-HOSVD is used to initialize iterative algorithms to compute the best rank-1 approximation [7, 28, 63] or best approximation of a specified multilinear rank [11, 15, 24, 26, 51].”
An eigenvalue problem for even order tensors with its applications (2016, methodology): “And then, we will explore the low-rank tensor approximation problem [20,23] using the singular value problem.”
A Link between the Canonical Decomposition in Multilinear Algebra and Simultaneous Matrix Diagonalization (2006, background): “The reason is that the best rank-1 approximation of A generally does not correspond to one of the terms in (1.2), and that the residue is in general not of rank R − 1 [13, 28 , 45].”
Nonsingular $H$-tensor and its criteria (2016, background): “For square tensor A, if all the entries Ai1i2···im are invariant under any permutation of their indices {i1i2 · · · im}, then tensor A is called symmetric [15].”
Tensors for Data Mining and Data Fusion (2016, methodology): “The power method for tensors dates back to Koﬁdis and Regalia [2002], wherein the authors propose that method for the symmetric case of x = y = z .”
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.”