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 34 articles )

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  1. Khouja, Rima; Khalil, Houssam; Mourrain, Bernard: Riemannian Newton optimization methods for the symmetric tensor approximation problem (2022)
  2. Khouja, Rima; Mattei, Pierre-Alexandre; Mourrain, Bernard: Tensor decomposition for learning Gaussian mixtures from moments (2022)
  3. Mohlenkamp, Martin; Young, Todd R.; Bárány, Balázs: Transient dynamics of block coordinate descent in a valley (2020)
  4. Karaev, Sanjar; Miettinen, Pauli: Algorithms for approximate subtropical matrix factorization (2019)
  5. Mohlenkamp, Martin J.: The dynamics of swamps in the canonical tensor approximation problem (2019)
  6. Bi, Xuan; Qu, Annie; Shen, Xiaotong: Multilayer tensor factorization with applications to recommender systems (2018)
  7. Breiding, Paul; Vannieuwenhoven, Nick: A Riemannian trust region method for the canonical tensor rank approximation problem (2018)
  8. Cherrak, Omar; Ghennioui, Hicham; Thirion-Moreau, Nadège; Abarkan, El Hossain: Preconditioned optimization algorithms solving the problem of the non unitary joint block diagonalization: application to blind separation of convolutive mixtures (2018)
  9. Gong, Xue; Mohlenkamp, Martin J.; Young, Todd R.: The optimization landscape for fitting a rank-2 tensor with a rank-1 tensor (2018)
  10. De Sterck, Hans; Winlaw, Manda: A nonlinearly preconditioned conjugate gradient algorithm for rank-(R) canonical tensor approximation. (2015)
  11. Wang, Liqi; Chu, Moody T.; Yu, Bo: Orthogonal low rank tensor approximation: alternating least squares method and its global convergence (2015)
  12. Dong, Bo; Lin, Matthew M.; Chu, Moody T.: Nonnegative rank factorization -- a heuristic approach via rank reduction (2014)
  13. Kindermann, Stefan; Navasca, Carmeliza: News algorithms for tensor decomposition based on a reduced functional (2014)
  14. Özay, Evrim Korkmaz; Demiralp, Metin: Reductive enhanced multivariance product representation for multi-way arrays (2014)
  15. Arora, Raman; Gupta, Maya R.; Kapila, Amol; Fazel, Maryam: Similarity-based clustering by left-stochastic matrix factorization (2013)
  16. Liu, Hongwei; Li, Xiangli; Zheng, Xiuyun: Solving non-negative matrix factorization by alternating least squares with a modified strategy (2013)
  17. Espig, Mike; Hackbusch, Wolfgang: A regularized Newton method for the efficient approximation of tensors represented in the canonical tensor format (2012)
  18. Xu, Yangyang; Yin, Wotao; Wen, Zaiwen; Zhang, Yin: An alternating direction algorithm for matrix completion with nonnegative factors (2012)
  19. Lin, Matthew M.: Discrete Eckart-Young theorem for integer matrices (2011)
  20. Royer, Jean-Philip; Thirion-Moreau, Nadège; Comon, Pierre: Computing the polyadic decomposition of nonnegative third order tensors (2011)

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