Tensor decompositions, alternating least squares and other tales

This work was originally motivated by a classification of tensors proposed by Richard Harshman. In particular, we focus on simple and multiple ‘bottlenecks’, and on ‘swamps’. Existing theoretical results are surveyed, some numerical algorithms are described in details, and their numerical complexity...

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Vydané v:	Journal of chemometrics Ročník 23; číslo 7-8; s. 393 - 405
Hlavní autori:	Comon, P., Luciani, X., de Almeida, A. L. F.
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Chichester, UK John Wiley & Sons, Ltd 01.07.2009 Wiley Wiley Subscription Services, Inc
Predmet:	Algorithms canonical decomposition Chemical Sciences Cheminformatics Chemistry Chemometrics Classification computational complexity Computer Science Computer simulation degeneracy Engineering Sciences Environmental Sciences Exact sciences and technology General and physical chemistry General. Nomenclature, chemical documentation, computer chemistry Least squares method Mathematical analysis Numerical analysis PARAFAC Signal and Image Processing Swamps tensor rank Tensors Theory of reactions, general kinetics. Catalysis. Nomenclature, chemical documentation, computer chemistry three-way array canonical decomposition degeneracy Computer simulation Classification PARAFAC Decomposition three-way array Algorithm Chemometrics tensor rank computational complexity Performance measure Levenberg-Marquardt Parafac Tensor rank Canonical Decomposition Three-way Array Candecomp greedy algorithm Indeterminacy compression dimension reduction
ISSN:	0886-9383, 1099-128X, 1099-128X
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Abstract	This work was originally motivated by a classification of tensors proposed by Richard Harshman. In particular, we focus on simple and multiple ‘bottlenecks’, and on ‘swamps’. Existing theoretical results are surveyed, some numerical algorithms are described in details, and their numerical complexity is calculated. In particular, the interest in using the enhanced line search (ELS) enhancement in these algorithms is discussed. Computer simulations feed this discussion. Copyright © 2009 John Wiley & Sons, Ltd. Various aspects of tensor decompositions are addressed: existence, uniqueness and computation. The state of the art is surveyed, by making the difference between conjectures and proved results. Some numerical algorithms are described in details, and their numerical complexity is evaluated. The slowness of numerical algorithms is often due to a form of ill‐conditioning of the tensor to be decomposed. In particular, Richard Harshman called ‘bottleneck’ the fact that two or more factors in a mode are almost collinear.
AbstractList	This work was originally motivated by a classification of tensors proposed by Richard Harshman. In particular, we focus on simple and multiple ''bottlenecks'', and on ''swamps''. Existing theoretical results are surveyed, some numerical algorithms are described in details, and their numerical complexity is calculated. In particular, the interest in using the ELS enhancement in these algorithms is discussed. Computer simulations feed this discussion. This work was originally motivated by a classification of tensors proposed by Richard Harshman. In particular, we focus on simple and multiple 'bottlenecks', and on 'swamps'. Existing theoretical results are surveyed, some numerical algorithms are described in details, and their numerical complexity is calculated. In particular, the interest in using the enhanced line search (ELS) enhancement in these algorithms is discussed. Computer simulations feed this discussion. This work was originally motivated by a classification of tensors proposed by Richard Harshman. In particular, we focus on simple and multiple 'bottlenecks', and on 'swamps'. Existing theoretical results are surveyed, some numerical algorithms are described in details, and their numerical complexity is calculated. In particular, the interest in using the enhanced line search (ELS) enhancement in these algorithms is discussed. Computer simulations feed this discussion. [PUBLICATION ABSTRACT] This work was originally motivated by a classification of tensors proposed by Richard Harshman. In particular, we focus on simple and multiple ‘bottlenecks’, and on ‘swamps’. Existing theoretical results are surveyed, some numerical algorithms are described in details, and their numerical complexity is calculated. In particular, the interest in using the enhanced line search (ELS) enhancement in these algorithms is discussed. Computer simulations feed this discussion. Copyright © 2009 John Wiley & Sons, Ltd. Various aspects of tensor decompositions are addressed: existence, uniqueness and computation. The state of the art is surveyed, by making the difference between conjectures and proved results. Some numerical algorithms are described in details, and their numerical complexity is evaluated. The slowness of numerical algorithms is often due to a form of ill‐conditioning of the tensor to be decomposed. In particular, Richard Harshman called ‘bottleneck’ the fact that two or more factors in a mode are almost collinear.
Author	Comon, P. de Almeida, A. L. F. Luciani, X.
Author_xml	– sequence: 1 givenname: P. surname: Comon fullname: Comon, P. email: pcomon@unice.fr organization: Laboratory I3S, UMR6070, University of Nice, Algorithmes/Euclide-B, 2000 route des Lucioles, BP 121, F-06903 Sophia-Antipolis Cedex – sequence: 2 givenname: X. surname: Luciani fullname: Luciani, X. organization: Laboratory I3S, UMR6070, University of Nice, Algorithmes/Euclide-B, 2000 route des Lucioles, BP 121, F-06903 Sophia-Antipolis Cedex – sequence: 3 givenname: A. L. F. surname: de Almeida fullname: de Almeida, A. L. F. organization: Laboratory I3S, UMR6070, University of Nice, Algorithmes/Euclide-B, 2000 route des Lucioles, BP 121, F-06903 Sophia-Antipolis Cedex
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Keywords	canonical decomposition degeneracy Computer simulation Classification PARAFAC Decomposition three-way array Algorithm Chemometrics tensor rank computational complexity Performance measure Levenberg-Marquardt Parafac Tensor rank Canonical Decomposition Three-way Array Candecomp greedy algorithm Indeterminacy compression dimension reduction
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PublicationTitle	Journal of chemometrics
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PublicationYear	2009
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References	ten Berge JMF. Partial uniqueness in CANDECOMP/PARAFAC. Jour. Chemometrics 2004; 18: 12-16. Stegeman A, Sidiropoulos ND. On Kruskal's uniqueness condition for the Candecomp/Parafac decomposition. Linear Algebra Appl. 2007; 420: 540-552. Brambilla MC, Ottaviani G. On the Alexander-Hirschowitz theorem. J. Pure Appl. Algebra 2008; 212: 1229-1251. Hitchcock FL. Multiple invariants and generalized rank of a p-way matrix or tensor. J. Math. Phys. 1927; 7 (1): 39-79. Stegeman A. Degeneracy in Candecomp/Parafac and Indscal explained for several three-sliced arrays with a two-valued typical rank. Psychometrika 2007; 72 (4): 601-619. Strassen V. Rank and optimal computation of generic tensors. Linear Algebra Appl. 1983; 52: 645-685. Mitchell BC, Burdick DS. Slowly converging Parafac sequences: Swamps and two-factor degeneracies. J. Chemom. 1994; 8: 155-168. Rao CR, Mitra S. Generalized Inverse of Matrices and Its Applications. New York: Wiley, 1971. Tucker LR. Some mathematical notes for three-mode factor analysis. Psychometrika 1966; 31: 279-311. Tomasi G, Bro R. A comparison of algorithms for fitting the Parafac model. Comp. Stat. Data Anal. 2006; 50: 1700-1734. Golub GH, Van Loan CF. Matrix Computations. Hopkins Univ. Press: Baltimore, 1989. Cox D, Little J, O'Shea D. Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra. Undergraduate Texts in Mathematics. Springer Verlag: New York, 1992; 2nd edition in 1996. de Almeida ALF, Favier G, Mota JCM. PARAFAC-based unified tensor modeling for wireless communication systems with application to blind multiuser equalization. Signal Processing 2007; 87 (2): 337-351. ten Berge JMF. The typical rank of tall three-way arrays. Psychometrika 2000; 65 (5): 525-532. Comon P, Golub G, Lim L-H, Mourrain B. Symmetric tensors and symmetric tensor rank. SIAM J. Matrix Anal. Appl. 2008; 30 (3): 1254-1279. Comon P, Mourrain B. Decomposition of quantics in sums of powers of linear forms. Signal Processing, Elsevier 1996; 53 (2): 93-107; Special Issue on High-Order Statistics. Terracini A. Sulla rappresentazione delle forme quaternarie mediante somme di potenze di forme lineari. Atti della R. Acc. delle Scienze di Torino 1916; 51: 107-117. Chiantini L, Ciliberto C. On the concept of k-sectant order of a variety. J. London Math. Soc. 2006; 73 (2): 436-454. Stegeman A. Low-rank approximation of generic p× q× 2 arrays and diverging components in the Candecomp/Parafac model. SIAM J. Matrix Anal. Appl. 2008; 30 (3): 988-1007. Paatero P. Construction and analysis of degenerate Parafac models. J. Chemom. 2000; 14: 285-299. DeLathauwer L, de Moor B, Vandewalle J. A multilinear singular value decomposition. SIAM J. Matrix Anal. Appl. 2000; 21 (4): 1253-1278. Chan TF. An improved algorithm for computing the singular value decomposition. ACM Trans. Math. Soft. 1982; 8 (1): 72-83. Lasker E. Kanonische formen. Math. Annal. 1904; 58: 434-440. Smilde A, Bro R, Geladi P. Multi-Way Analysis. Wiley: Chichester, 2004. Cayley A. On the theory of linear transformation. Cambridge Math. J. 1845; 4: 193-209. Comon P, ten Berge JMF, DeLathauwer L, Castaing J. Generic and typical ranks of multi-way arrays. Linear Algebra Appl., 2009, DOI: 10.1016/j.laa.2009.01.014 DeLathauwer L, Castaing J. Tensor-based techniques for the blind separation of DS-CDMA signals. Signal Processing 2007; 87 (2): 322-336. Rao CR. Linear Statistical Inference and its Applications. Probability and Statistics. Wiley: New York, 1965. Kruskal JB. Three-way arrays: Rank and uniqueness of trilinear decompositions. Linear Algebra Appl. 1977; 18: 95-138. ten Berge JMF. Kruskal's polynomial for 2×2×2 arrays and a generalization to 2×n×n arrays. Psychometrika 1991; 56: 631-636. Carroll JD, Chang JJ. Analysis of individual differences in multidimensional scaling via n-way generalization of Eckart-Young decomposition. Psychometrika 1970; 35 (3): 283-319. Krijnen WP, Dijkstra TK, Stegeman A. On the non-existence of optimal solutions and the occurrence of degeneracy in the Candecomp/Parafac model. Psychometrika 2008; 73 (3): 431-439. Mella M. Singularities of linear systems and the Waring problem. Trans. Am. Math. Soc. 2006; 358 (12): 5523-5538. Harshman RA. Foundations of the Parafac procedure: Models and conditions for an explanatory multimodal factor analysis. UCLA Working Papers in Phonetics 1970; 16: 1-84. http://www.publish.uwo.ca/harshman. Comon P, Golub GH. Tracking of a few extreme singular values and vectors in signal processing. Proceedings of the IEEE 1990; 78 (8): 1327-1343 (published from Stanford report 78NA-89-01, February 1989). Bürgisser P, Clausen M, Shokrollahi MA. Algebraic Complexity Theory, vol. 315. Springer: Berlin, 1997. ten Berge JMF, Stegeman A. Symmetry transformations for square sliced three way arrays, with applications to their typical rank. Linear Algebra Appl. 2006; 418: 215-224. De Silva V, Lim L-H. Tensor rank and the ill-posedness of the best low-rank approximation problem. SIAM J. Matrix Anal. Appl. 2008; 30 (3): 1084-1127. Lim L-H, Comon P. Nonnegative approximations of nonnegative tensors. J. Chemom., 2009 , this issue. Sidiropoulos ND, Bro R. On the uniqueness of multilinear decomposition of N-way arrays. J. Chemom. 2000; 14: 229-239. ten Berge JMF, Kiers HAL. Simplicity of core arrays in three-way principal component analysis and the typical rank of p× q× 2 arrays. Linear Algebra Appl. 1999; 294: 169-179. Alexander J, Hirschowitz A. La methode d'Horace eclatee: application a l'interpolation en degre quatre. Invent. Math. 1992; 107 (3): 585-602. Paatero P. The multilinear engine: a table-driven, least squares program for solving multilinear problems, including the n-way parallel factor analysis model. J. Comput. Graph. Stat. 1999; 8 (4): 854-888. DeLathauwer L, de Moor B, Vandewalle J. On the best rank-1 and rank-(R1,R2,/,/ldots/,RN) approximation of high-order tensors. SIAM J. Matrix Anal. Appl. 2000; 21 (4): 1324-1342. Kroonenberg P. Three Mode Principal Component Analysis. SWO Press: Leiden, 1983. Ehrenborg R. Canonical forms of two by two matrices. Jour. of Algebra 1999; 213: 195-224. Lickteig T. Typical tensorial rank. Linear Algebra Appl. 1985; 69: 95-120. DeLathauwer L. A link between canonical decomposition in multilinear algebra and simultaneous matrix diagonalization. SIAM J. Matrix Anal. 2006; 28 (3): 642-666. Bro R. Parafac, tutorial and applications. Chemom. Intel. Lab. Syst. 1997; 38: 149-171. Stegeman A. Degeneracy in Candecomp/Parafac explained for p× p× 2 arrays of rank p+1 or higher. Psychometrika 2006; 71 (3): 483-501. Rayens WS, Mitchell BC. Two-factor degeneracies and a stabilization of Parafac. Chemom. Intell. Lab. Syst. 1997; 38: 173-181. Harshman RA, Lundy ME. Parafac: Parallel factor analysis. Computational Statistics and Data Analysis 1994; 39-72. Sidiropoulos ND, Giannakis GB, Bro R. Blind PARAFAC receivers for DS-CDMA systems. Trans. Sig. Proc. 2000; 48 (3): 810-823. Comon P, Rajih M. Blind identification of under-determined mixtures based on the characteristic function. Signal Processing 2006; 86 (9): 2271-2281. Nion D, DeLathauwer L. An enhanced line search scheme for complex-valued tensor decompositions. application in DS-CDMA. Signal Processing 2008; 88 (3): 749-755. Bini D. Border rank of m× n×(mn-q) tensors. Linear Algebra Appl. 1986; 79: 45-51. Rajih M, Comon P, Harshman R. Enhanced line search: a novel method to accelerate PARAFAC. SIAM J. Matrix Anal. Appl. 2008; 30 (3): 1148-1171. Bro R, Andersson CA. Improving the speed of multiway algorithms. part ii: Compression. Chemom. Intell. Lab. Syst. 1998; 42 (1-1): 105-113. 2006; 71 2007; 420 1997; 315 2006; 418 2006; 73 1991; 56 2000; 48 1980; 85 1986; 79 1976 1983; 52 2007; 72 1971 2008; 30 2008; 73 1998; 42 1970; 35 1985; 69 2000; 14 2006; 28 1845; 4 1999; 294 1982; 8 1983 1927; 7 1999; 213 1989 1990; 78 2006; 50 2000; 21 2000; 65 2009 2008 1992; 107 2006 1994 2005 2004 1992 2002 2006; 358 1999; 8 1970; 16 1996; 53 1966; 31 1916; 51 1994; 8 2006; 86 2004; 18 1904; 58 1977; 18 1965 1997; 38 2008; 88 2008; 212 2007; 87 Terracini A (e_1_2_1_44_2) 1916; 51 e_1_2_1_64_2 e_1_2_1_66_2 e_1_2_1_22_2 e_1_2_1_45_2 e_1_2_1_60_2 e_1_2_1_20_2 e_1_2_1_43_2 e_1_2_1_49_2 e_1_2_1_24_2 e_1_2_1_47_2 e_1_2_1_68_2 e_1_2_1_28_2 Harshman RA (e_1_2_1_33_2) 1970; 16 Hitchcock FL (e_1_2_1_31_2) 1927; 7 e_1_2_1_54_2 e_1_2_1_4_2 Kruskal JB (e_1_2_1_13_2) 1989 e_1_2_1_56_2 e_1_2_1_2_2 e_1_2_1_12_2 e_1_2_1_50_2 e_1_2_1_71_2 e_1_2_1_52_2 e_1_2_1_16_2 e_1_2_1_37_2 e_1_2_1_14_2 e_1_2_1_58_2 e_1_2_1_8_2 e_1_2_1_18_2 e_1_2_1_39_2 Cayley A (e_1_2_1_41_2) 1845; 4 Comon P (e_1_2_1_6_2) 2002 Smilde A (e_1_2_1_10_2) 2004 e_1_2_1_40_2 e_1_2_1_65_2 e_1_2_1_67_2 e_1_2_1_23_2 e_1_2_1_61_2 e_1_2_1_21_2 e_1_2_1_42_2 e_1_2_1_27_2 e_1_2_1_48_2 Rao CR (e_1_2_1_63_2) 1971 e_1_2_1_25_2 e_1_2_1_46_2 e_1_2_1_29_2 Golub GH (e_1_2_1_69_2) 1989 e_1_2_1_70_2 e_1_2_1_30_2 e_1_2_1_53_2 e_1_2_1_7_2 e_1_2_1_55_2 Rao CR (e_1_2_1_62_2) 1965 e_1_2_1_5_2 e_1_2_1_11_2 Bini D (e_1_2_1_26_2) 1980 e_1_2_1_34_2 e_1_2_1_3_2 Kroonenberg P (e_1_2_1_35_2) 1983 e_1_2_1_32_2 e_1_2_1_51_2 e_1_2_1_15_2 e_1_2_1_38_2 e_1_2_1_36_2 e_1_2_1_19_2 e_1_2_1_57_2 e_1_2_1_17_2 e_1_2_1_59_2 e_1_2_1_9_2
References_xml	– reference: Rao CR, Mitra S. Generalized Inverse of Matrices and Its Applications. New York: Wiley, 1971. – reference: Paatero P. Construction and analysis of degenerate Parafac models. J. Chemom. 2000; 14: 285-299. – reference: ten Berge JMF, Stegeman A. Symmetry transformations for square sliced three way arrays, with applications to their typical rank. Linear Algebra Appl. 2006; 418: 215-224. – reference: Carroll JD, Chang JJ. Analysis of individual differences in multidimensional scaling via n-way generalization of Eckart-Young decomposition. Psychometrika 1970; 35 (3): 283-319. – reference: Cox D, Little J, O'Shea D. Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra. Undergraduate Texts in Mathematics. Springer Verlag: New York, 1992; 2nd edition in 1996. – reference: Chiantini L, Ciliberto C. On the concept of k-sectant order of a variety. J. London Math. Soc. 2006; 73 (2): 436-454. – reference: Mella M. Singularities of linear systems and the Waring problem. Trans. Am. Math. Soc. 2006; 358 (12): 5523-5538. – reference: Kroonenberg P. Three Mode Principal Component Analysis. SWO Press: Leiden, 1983. – reference: Bro R, Andersson CA. Improving the speed of multiway algorithms. part ii: Compression. Chemom. Intell. Lab. Syst. 1998; 42 (1-1): 105-113. – reference: Rajih M, Comon P, Harshman R. Enhanced line search: a novel method to accelerate PARAFAC. SIAM J. Matrix Anal. Appl. 2008; 30 (3): 1148-1171. – reference: Comon P, ten Berge JMF, DeLathauwer L, Castaing J. Generic and typical ranks of multi-way arrays. Linear Algebra Appl., 2009, DOI: 10.1016/j.laa.2009.01.014/ – reference: Golub GH, Van Loan CF. Matrix Computations. Hopkins Univ. Press: Baltimore, 1989. – reference: Stegeman A, Sidiropoulos ND. On Kruskal's uniqueness condition for the Candecomp/Parafac decomposition. Linear Algebra Appl. 2007; 420: 540-552. – reference: Strassen V. Rank and optimal computation of generic tensors. Linear Algebra Appl. 1983; 52: 645-685. – reference: Hitchcock FL. Multiple invariants and generalized rank of a p-way matrix or tensor. J. Math. Phys. 1927; 7 (1): 39-79. – reference: Lickteig T. Typical tensorial rank. Linear Algebra Appl. 1985; 69: 95-120. – reference: Lim L-H, Comon P. Nonnegative approximations of nonnegative tensors. J. Chemom., 2009 , this issue. – reference: Harshman RA. Foundations of the Parafac procedure: Models and conditions for an explanatory multimodal factor analysis. UCLA Working Papers in Phonetics 1970; 16: 1-84. http://www.publish.uwo.ca/harshman. – reference: Bini D. Border rank of m× n×(mn-q) tensors. Linear Algebra Appl. 1986; 79: 45-51. – reference: Sidiropoulos ND, Bro R. On the uniqueness of multilinear decomposition of N-way arrays. J. Chemom. 2000; 14: 229-239. – reference: Stegeman A. Degeneracy in Candecomp/Parafac explained for p× p× 2 arrays of rank p+1 or higher. Psychometrika 2006; 71 (3): 483-501. – reference: Tucker LR. Some mathematical notes for three-mode factor analysis. Psychometrika 1966; 31: 279-311. – reference: ten Berge JMF. Kruskal's polynomial for 2×2×2 arrays and a generalization to 2×n×n arrays. Psychometrika 1991; 56: 631-636. – reference: Chan TF. An improved algorithm for computing the singular value decomposition. ACM Trans. Math. Soft. 1982; 8 (1): 72-83. – reference: Alexander J, Hirschowitz A. La methode d'Horace eclatee: application a l'interpolation en degre quatre. Invent. Math. 1992; 107 (3): 585-602. – reference: DeLathauwer L. A link between canonical decomposition in multilinear algebra and simultaneous matrix diagonalization. SIAM J. Matrix Anal. 2006; 28 (3): 642-666. – reference: Mitchell BC, Burdick DS. Slowly converging Parafac sequences: Swamps and two-factor degeneracies. J. Chemom. 1994; 8: 155-168. – reference: Comon P, Golub G, Lim L-H, Mourrain B. Symmetric tensors and symmetric tensor rank. SIAM J. Matrix Anal. Appl. 2008; 30 (3): 1254-1279. – reference: Sidiropoulos ND, Giannakis GB, Bro R. Blind PARAFAC receivers for DS-CDMA systems. Trans. Sig. Proc. 2000; 48 (3): 810-823. – reference: Comon P, Golub GH. Tracking of a few extreme singular values and vectors in signal processing. Proceedings of the IEEE 1990; 78 (8): 1327-1343 (published from Stanford report 78NA-89-01, February 1989). – reference: Stegeman A. Low-rank approximation of generic p× q× 2 arrays and diverging components in the Candecomp/Parafac model. SIAM J. Matrix Anal. Appl. 2008; 30 (3): 988-1007. – reference: Terracini A. Sulla rappresentazione delle forme quaternarie mediante somme di potenze di forme lineari. Atti della R. Acc. delle Scienze di Torino 1916; 51: 107-117. – reference: DeLathauwer L, Castaing J. Tensor-based techniques for the blind separation of DS-CDMA signals. Signal Processing 2007; 87 (2): 322-336. – reference: Comon P, Mourrain B. Decomposition of quantics in sums of powers of linear forms. Signal Processing, Elsevier 1996; 53 (2): 93-107; Special Issue on High-Order Statistics. – reference: de Almeida ALF, Favier G, Mota JCM. PARAFAC-based unified tensor modeling for wireless communication systems with application to blind multiuser equalization. Signal Processing 2007; 87 (2): 337-351. – reference: Rayens WS, Mitchell BC. Two-factor degeneracies and a stabilization of Parafac. Chemom. Intell. Lab. Syst. 1997; 38: 173-181. – reference: DeLathauwer L, de Moor B, Vandewalle J. On the best rank-1 and rank-(R1,R2,/,/ldots/,RN) approximation of high-order tensors. SIAM J. Matrix Anal. Appl. 2000; 21 (4): 1324-1342. – reference: ten Berge JMF, Kiers HAL. Simplicity of core arrays in three-way principal component analysis and the typical rank of p× q× 2 arrays. Linear Algebra Appl. 1999; 294: 169-179. – reference: De Silva V, Lim L-H. Tensor rank and the ill-posedness of the best low-rank approximation problem. SIAM J. Matrix Anal. Appl. 2008; 30 (3): 1084-1127. – reference: Stegeman A. Degeneracy in Candecomp/Parafac and Indscal explained for several three-sliced arrays with a two-valued typical rank. Psychometrika 2007; 72 (4): 601-619. – reference: Nion D, DeLathauwer L. An enhanced line search scheme for complex-valued tensor decompositions. application in DS-CDMA. Signal Processing 2008; 88 (3): 749-755. – reference: Kruskal JB. Three-way arrays: Rank and uniqueness of trilinear decompositions. Linear Algebra Appl. 1977; 18: 95-138. – reference: ten Berge JMF. Partial uniqueness in CANDECOMP/PARAFAC. Jour. Chemometrics 2004; 18: 12-16. – reference: Paatero P. The multilinear engine: a table-driven, least squares program for solving multilinear problems, including the n-way parallel factor analysis model. J. Comput. Graph. Stat. 1999; 8 (4): 854-888. – reference: Smilde A, Bro R, Geladi P. Multi-Way Analysis. Wiley: Chichester, 2004. – reference: Bro R. Parafac, tutorial and applications. Chemom. Intel. Lab. Syst. 1997; 38: 149-171. – reference: Harshman RA, Lundy ME. Parafac: Parallel factor analysis. Computational Statistics and Data Analysis 1994; 39-72. – reference: Tomasi G, Bro R. A comparison of algorithms for fitting the Parafac model. Comp. Stat. Data Anal. 2006; 50: 1700-1734. – reference: Krijnen WP, Dijkstra TK, Stegeman A. On the non-existence of optimal solutions and the occurrence of degeneracy in the Candecomp/Parafac model. Psychometrika 2008; 73 (3): 431-439. – reference: Comon P, Rajih M. Blind identification of under-determined mixtures based on the characteristic function. Signal Processing 2006; 86 (9): 2271-2281. – reference: Brambilla MC, Ottaviani G. On the Alexander-Hirschowitz theorem. J. Pure Appl. Algebra 2008; 212: 1229-1251. – reference: Lasker E. Kanonische formen. Math. Annal. 1904; 58: 434-440. – reference: Cayley A. On the theory of linear transformation. 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Snippet	This work was originally motivated by a classification of tensors proposed by Richard Harshman. In particular, we focus on simple and multiple ‘bottlenecks’,... This work was originally motivated by a classification of tensors proposed by Richard Harshman. In particular, we focus on simple and multiple 'bottlenecks',... This work was originally motivated by a classification of tensors proposed by Richard Harshman. In particular, we focus on simple and multiple ''bottlenecks'',...
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SubjectTerms	Algorithms canonical decomposition Chemical Sciences Cheminformatics Chemistry Chemometrics Classification computational complexity Computer Science Computer simulation degeneracy Engineering Sciences Environmental Sciences Exact sciences and technology General and physical chemistry General. Nomenclature, chemical documentation, computer chemistry Least squares method Mathematical analysis Numerical analysis PARAFAC Signal and Image Processing Swamps tensor rank Tensors Theory of reactions, general kinetics. Catalysis. Nomenclature, chemical documentation, computer chemistry three-way array
Title	Tensor decompositions, alternating least squares and other tales
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