A Link between the Canonical Decomposition in Multilinear Algebra and Simultaneous Matrix Diagonalization

By Lieven De Lathauwer. SIAM Journal on Matrix Analysis and Applications, 2006

453 citations recorded by the source corpus.

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

View Paper: https://www.semanticscholar.org/paper/8ac98d52fac0929bddee0704329d4cd55253a1e6

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In Computational Mathematics

In Computational Theory and Mathematics

In Signal Processing

Citations in the corpus

Fourth-Order Cumulant-Based Blind Identification of Underdetermined Mixtures (2007, background): “This is a decomposition of a symmetric fourth-order tensor in a sum of symmetric rank-1 terms, cf. [9], [14], [16], [20], [ 21 ], [30], and the references therein.”
Canonical polyadic decomposition of third-order tensors: relaxed uniqueness conditions and algebraic algorithm (2015, methodology): “tioned earlier, while (15) and (16) reduce to R(J 1 2 )(I 1 2 )= p 2 + 1 and RI+ J 1, respectively. 5 2.6 Our contribution and organization of the paper In this paper we further extend results from [6, 7, 8, 9], narrowing the gap with what is known from algebraic geometry. Namely, we present new det”
On Kruskal's uniqueness condition for the Candecomp/Parafac decomposition (2007, methodology): “Moreover, for random component matrices A, B and C, [6] derives a condition for “uniqueness with probability 1” in the form of a dimensionality constraint.”
Tensor Algebra and Multidimensional Harmonic Retrieval in Signal Processing for MIMO Radar (2010, background): “The latter spectra being computed for every pair of angles of interest, complexity is significant.”
Enhanced Line Search: A novel method to accelerate Parafac (2005, background): “Progress in this area is ongoing— for example, in the case of “tall” arrays, a significantly more relaxed condition has been derived in [19].”
Tensor Decompositions for Signal Processing Applications: From two-way to multiway component analysis (2014, background): “For instance, a low multilinear rank approximation achieved through Tucker decomposition may yield a higher Signal-to-Noise Ratio (SNR) than the SNR in the original raw data tensor, making Tucker decomposition a very natural tool for compression and signal enhancement [7,8,24].”
Tensor Factorization via Matrix Factorization (2015, unknown): “In the orthogonal setting, we compare our algorithms (OJD0, which uses random projections, and OJD1 which uses with plug-in) with the tensor power method (TPM), alternating least squares (ALS), and with the method of de Lathauwer [23].”
On uniqueness conditions for Candecomp/Parafac and Indscal with full column rank in one mode (2009, methodology): “We analyzed the relation between Kruskal’s [11] uniqueness condition and the recent uniqueness conditions of Jiang and Sidiropoulos [6] and De Lathauwer [2].”
On the best low multilinear rank approximation of higher-order tensors (2010, background): “The tensor corresponding to S can be made as diagonal as possible (in a least squares sense) under orthogonal transformations [12, 24, 56, 10], or the original tensor can be decomposed in a minimal number of rank-1 terms (CANDECOMP/PARAFAC) [13, 37, 9, 25, 17 ], on which orthogonal [50] or symmetry”
On Fast Computation of Gradients for CANDECOMP/PARAFAC Algorithms (2012, background): “Another approach is to consider the CP decomposition as a joint diagonalization problem [11,21,22,28,28].”
An Adaptive Algebraic Multigrid Algorithm for Low-Rank Canonical Tensor Decomposition (2011, unknown): “Many algorithms have been proposed for computing the CP decomposition [20, 25, 22, 34, 1].”
Tensor Decompositions and Applications (2009, methodology): “More recently, De Lathauwer [55] developed a method based on simultaneous matrix diagonalization in the case that, for an N th-order tensor X ∈ R I 1 × I 2 × ··· × I N , max I n : n = 1 , . . . , sparse N } ≥ rank( X ).”
Decompositions of a Higher-Order Tensor in Block Terms - Part II: Definitions and Uniqueness (2008, methodology): “In [17] we derived a more relaxed uniqueness condition that applies when T is tall in one mode (meaning that, for instance, K R).”