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 23 articles )

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  1. Wang, Liqi; Chu, Moody T.; Yu, Bo: Orthogonal low rank tensor approximation: alternating least squares method and its global convergence (2015)
  2. Dong, Bo; Lin, Matthew M.; Chu, Moody T.: Nonnegative rank factorization -- a heuristic approach via rank reduction (2014)
  3. Kindermann, Stefan; Navasca, Carmeliza: News algorithms for tensor decomposition based on a reduced functional (2014)
  4. Özay, Evrim Korkmaz; Demiralp, Metin: Reductive enhanced multivariance product representation for multi-way arrays (2014)
  5. Arora, Raman; Gupta, Maya R.; Kapila, Amol; Fazel, Maryam: Similarity-based clustering by left-stochastic matrix factorization (2013)
  6. Liu, Hongwei; Li, Xiangli; Zheng, Xiuyun: Solving non-negative matrix factorization by alternating least squares with a modified strategy (2013)
  7. Espig, Mike; Hackbusch, Wolfgang: A regularized Newton method for the efficient approximation of tensors represented in the canonical tensor format (2012)
  8. Xu, Yangyang; Yin, Wotao; Wen, Zaiwen; Zhang, Yin: An alternating direction algorithm for matrix completion with nonnegative factors (2012)
  9. Lin, Matthew M.: Discrete Eckart-Young theorem for integer matrices (2011)
  10. Royer, Jean-Philip; Thirion-Moreau, Nadège; Comon, Pierre: Computing the polyadic decomposition of nonnegative third order tensors (2011)
  11. Brachat, Jerome; Comon, Pierre; Mourrain, Bernard; Tsigaridas, Elias: Symmetric tensor decomposition (2010)
  12. Kolda, Tamara G.; Bader, Brett W.: Tensor decompositions and applications (2009)
  13. 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)
  14. Rajih, Myriam; Comon, Pierre; Harshman, Richard A.: Enhanced line search: a novel method to accelerate PARAFAC (2008)
  15. Bader, Brett W.; Kolda, Tamara G.: Efficient MATLAB computations with sparse and factored tensors (2007)
  16. Berry, Michael W.; Browne, Murray; Langville, Amy N.; Pauca, V.Paul; Plemmons, Robert J.: Algorithms and applications for approximate nonnegative matrix factorization (2007)
  17. De Vos, Maarten; De Lathauwer, Lieven; Vanrumste, Bart; Van Huffel, Sabine; Van Paesschen, W.: Canonical decomposition of ictal scalp EEG and accurate source localisation: principles and simulation study (2007)
  18. Lin, Chih-Jen: Projected gradient methods for nonnegative matrix factorization (2007)
  19. Stegeman, Alwin: Degeneracy in Candecomp/Parafac and Indscal explained for several three-sliced arrays with a two-valued typical rank (2007)
  20. De Lathauwer, Lieven: A link between the canonical decomposition in multilinear algebra and simultaneous matrix diagonalization (2006)

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