Multilinear Engine

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.


References in zbMATH (referenced in 27 articles )

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  1. Bi, Xuan; Qu, Annie; Shen, Xiaotong: Multilayer tensor factorization with applications to recommender systems (2018)
  2. Breiding, Paul; Vannieuwenhoven, Nick: A Riemannian trust region method for the canonical tensor rank approximation problem (2018)
  3. Gong, Xue; Mohlenkamp, Martin J.; Young, Todd R.: The optimization landscape for fitting a rank-2 tensor with a rank-1 tensor (2018)
  4. De Sterck, Hans; Winlaw, Manda: A nonlinearly preconditioned conjugate gradient algorithm for rank-(R) canonical tensor approximation. (2015)
  5. Wang, Liqi; Chu, Moody T.; Yu, Bo: Orthogonal low rank tensor approximation: alternating least squares method and its global convergence (2015)
  6. Dong, Bo; Lin, Matthew M.; Chu, Moody T.: Nonnegative rank factorization -- a heuristic approach via rank reduction (2014)
  7. Kindermann, Stefan; Navasca, Carmeliza: News algorithms for tensor decomposition based on a reduced functional (2014)
  8. Özay, Evrim Korkmaz; Demiralp, Metin: Reductive enhanced multivariance product representation for multi-way arrays (2014)
  9. Arora, Raman; Gupta, Maya R.; Kapila, Amol; Fazel, Maryam: Similarity-based clustering by left-stochastic matrix factorization (2013)
  10. Liu, Hongwei; Li, Xiangli; Zheng, Xiuyun: Solving non-negative matrix factorization by alternating least squares with a modified strategy (2013)
  11. Espig, Mike; Hackbusch, Wolfgang: A regularized Newton method for the efficient approximation of tensors represented in the canonical tensor format (2012)
  12. Xu, Yangyang; Yin, Wotao; Wen, Zaiwen; Zhang, Yin: An alternating direction algorithm for matrix completion with nonnegative factors (2012)
  13. Lin, Matthew M.: Discrete Eckart-Young theorem for integer matrices (2011)
  14. Royer, Jean-Philip; Thirion-Moreau, Nadège; Comon, Pierre: Computing the polyadic decomposition of nonnegative third order tensors (2011)
  15. Brachat, Jerome; Comon, Pierre; Mourrain, Bernard; Tsigaridas, Elias: Symmetric tensor decomposition (2010)
  16. Kolda, Tamara G.; Bader, Brett W.: Tensor decompositions and applications (2009)
  17. Krijnen, Wim P.; Dijkstra, Theo K.; Stegeman, Alwin: On the non-existence of optimal solutions and the occurrence of “degeneracy” in the CANDECOMP/PARAFAC model (2008)
  18. Rajih, Myriam; Comon, Pierre; Harshman, Richard A.: Enhanced line search: a novel method to accelerate PARAFAC (2008)
  19. Bader, Brett W.; Kolda, Tamara G.: Efficient MATLAB computations with sparse and factored tensors (2007)
  20. Berry, Michael W.; Browne, Murray; Langville, Amy N.; Pauca, V. Paul; Plemmons, Robert J.: Algorithms and applications for approximate nonnegative matrix factorization (2007)

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