Multilinear Engine swMATH ID: 14430 Software Authors: Pentti Paatero Description: The Multilinear Engine: A Table-Driven, Least Squares Program for Solving Multilinear Problems, including the n-Way Parallel Factor Analysis Model. A technique for fitting multilinear and quasi-multilinear mathematical expressions or models to two-, three-, and many-dimensional data arrays is described. Principal component analysis and three-way PARAFAC factor analysis are examples of bilinear and trilinear least squares fit. This work presents a technique for specifying the problem in a structured way so that one program (the Multilinear Engine) may be used for solving widely different multilinear problems. The multilinear equations to be solved are specified as a large table of integer code values. The end user creates this table by using a small preprocessing program. For each different case, an individual structure table is needed. The solution is computed by using the conjugate gradient algorithm. Non-negativity constraints are implemented by using the well-known technique of preconditioning in opposite way for slowing down changes of variables that are about to become negative. The iteration converges to a minimum that may be local or global. Local uniqueness of the solution may be determined by inspecting the singular values of the Jacobian matrix. A global solution may be searched for by starting the iteration from different pseudorandom starting points. Application examples are discussed—for example, n-way PARAFAC, PARAFAC2, Linked mode PARAFAC, blind deconvolution, and nonstandard variants of these. Homepage: http://www.jstor.org/stable/1390831?seq=1#page_scan_tab_contents Related Software: TensorToolbox; Matlab; N-way Toolbox; Tensorlab; NMFLAB; PRMLT; pdfCluster; MBCbook; Rmixmod; MNIST; UCI-ml; mclust; Complex Optimization Toolbox; NMF MATLAB Toolbox; UMDES; SparseMatrix; Eigentaste; mctoolbox; Anderson; Poblano Cited in: 33 Publications all top 5 Cited by 66 Authors 3 Comon, Pierre 3 Mohlenkamp, Martin J. 3 Mourrain, Bernard 3 Stegeman, Alwin 2 Bader, Brett W. 2 Chu, Moody T. 2 Khouja, Rima 2 Kolda, Tamara Gibson 2 Lin, Matthew M. 2 Thirion-Moreau, Nadège 2 Young, Todd R. 1 Abarkan, El Hossain 1 Arora, Raman 1 Bárány, Balázs 1 Berry, Michael W. 1 Bi, Xuan 1 Biswas, Atanu 1 Brachat, Jerome 1 Breiding, Paul 1 Browne, Murray 1 Cherrak, Omar 1 De Lathauwer, Lieven 1 De Sterck, Hans 1 Demiralp, Metin 1 Dhillon, Inderjit S. 1 Dijkstra, Theo K. 1 Dong, Bo 1 Espig, Mike 1 Fazel, Maryam 1 Ghennioui, Hicham 1 Gong, Xue 1 Gupta, Maya R. 1 Hackbusch, Wolfgang 1 Harshman, Richard A. 1 Kapila, Amol 1 Karaev, Sanjar 1 Khalil, Houssam 1 Kim, Dongmin 1 Kindermann, Stefan 1 Krijnen, Wim P. 1 Langville, Amy N. 1 Li, Xiangli 1 Lin, Chih-Jen 1 Liu, Hongwei 1 Mattei, Pierre-Alexandre 1 Miettinen, Pauli 1 Navasca, Carmeliza 1 Özay, Evrim Korkmaz 1 Pauca, V. Paul 1 Plemmons, Robert James 1 Qu, Annie 1 Rajih, Myriam 1 Royer, Jean-Philip 1 Shen, Xiaotong 1 Sidiropoulos, Nikolaos D. 1 Sra, Suvrit 1 ten Berge, Jos M. F. 1 Tsigaridas, Elias P. 1 Vannieuwenhoven, Nick 1 Wang, Liqi 1 Wen, Zaiwen 1 Winlaw, Manda 1 Xu, Yangyang 1 Yin, Wotao 1 Zhang, Yin 1 Zheng, Xiuyun all top 5 Cited in 23 Serials 4 Psychometrika 4 SIAM Journal on Matrix Analysis and Applications 2 Linear Algebra and its Applications 2 Numerical Linear Algebra with Applications 2 Data Mining and Knowledge Discovery 2 SIAM Journal on Applied Dynamical Systems 1 The Annals of Statistics 1 Numerische Mathematik 1 Statistics & Probability Letters 1 Journal of Symbolic Computation 1 Signal Processing 1 Multidimensional Systems and Signal Processing 1 Neural Computation 1 Numerical Algorithms 1 SIAM Review 1 Computational Statistics and Data Analysis 1 SIAM Journal on Optimization 1 SIAM Journal on Scientific Computing 1 Journal of Mathematical Chemistry 1 Journal of Machine Learning Research (JMLR) 1 International Journal of Numerical Analysis and Modeling 1 Frontiers of Mathematics in China 1 Statistical Analysis and Data Mining all top 5 Cited in 12 Fields 22 Linear and multilinear algebra; matrix theory (15-XX) 22 Numerical analysis (65-XX) 12 Statistics (62-XX) 6 Operations research, mathematical programming (90-XX) 5 Computer science (68-XX) 3 Algebraic geometry (14-XX) 3 Dynamical systems and ergodic theory (37-XX) 2 Approximations and expansions (41-XX) 2 Differential geometry (53-XX) 1 Calculus of variations and optimal control; optimization (49-XX) 1 Probability theory and stochastic processes (60-XX) 1 Information and communication theory, circuits (94-XX) Citations by Year