PARAFAC: Parallel factor analysis. We review the method of Parallel Factor Analysis, which simultaneously fits multiple two-way arrays or ‘slices’ of a three-way array in terms of a common set of factors with differing relative weights in each ‘slice’. Mathematically, it is a straightforward generalization of the bilinear model of factor (or component) analysis (xij = ΣRr = 1airbjr) to a trilinear model (xijk = ΣRr = 1airbjrckr). Despite this simplicity, it has an important property not possessed by the two-way model: if the latent factors show adequately distinct patterns of three-way variation, the model is fully identified; the orientation of factors is uniquely determined by minimizing residual error, eliminating the need for a separate ‘rotation’ phase of analysis. The model can be used several ways. It can be directly fit to a three-way array of observations with (possibly incomplete) factorial structure, or it can be indirectly fit to the original observations by fitting a set of covariance matrices computed from the observations, with each matrix corresponding to a two-way subset of the data. Even more generally, one can simultaneously analyze covariance matrices computed from different samples, perhaps corresponding to different treatment groups, different kinds of cases, data from different studies, etc. To demonstrate the method we analyze data from an experiment on right vs. left cerebral hemispheric control of the hands during various tasks. The factors found appear to correspond to the causal influences manipulated in the experiment, revealing their patterns of influence in all three ways of the data. Several generalizations of the parallel factor analysis model are currently under development, including ones that combine parallel factors with Tucker-like factor ‘interactions’. Of key importance is the need to increase the method’s robustness against nonstationary factor structures and qualitative (nonproportional) factor change.

References in zbMATH (referenced in 24 articles , 1 standard article )

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  1. Hawkins, Cole; Liu, Xing; Zhang, Zheng: Towards compact neural networks via end-to-end training: a Bayesian tensor approach with automatic rank determination (2022)
  2. Boutalbi, Rafika; Labiod, Lazhar; Nadif, Mohamed: Implicit consensus clustering from multiple graphs (2021)
  3. Alexandrov, Boian S.; Stanev, Valentin G.; Vesselinov, Velimir V.; Rasmussen, Kim Ø.: Nonnegative tensor decomposition with custom clustering for microphase separation of block copolymers (2019)
  4. Vesselinov, V. V.; Mudunuru, M. K.; Karra, S.; O’Malley, D.; Alexandrov, B. S.: Unsupervised machine learning based on non-negative tensor factorization for analyzing reactive-mixing (2019)
  5. Benson, Austin R.; Gleich, David F.; Lim, Lek-Heng: The spacey random walk: a stochastic process for higher-order data (2017)
  6. Cem Bassoy: TLib: A Flexible C++ Tensor Framework for Numerical Tensor Calculus (2017) arXiv
  7. Domanov, Ignat; De Lathauwer, Lieven: Canonical polyadic decomposition of third-order tensors: relaxed uniqueness conditions and algebraic algorithm (2017)
  8. Mendes, Susana; Fernández-Gómez, M. José; Marques, Sónia Cotrim; Pardal, Miguel Ângelo; Azeiteiro, Ulisses Miranda; Galindo-Villardón, M. Purificación: CO-Tucker: a new method for the simultaneous analysis of a sequence of paired tables (2017)
  9. Khan, Suleiman A.; Leppäaho, Eemeli; Kaski, Samuel: Bayesian multi-tensor factorization (2016)
  10. Reynolds, Matthew J.; Doostan, Alireza; Beylkin, Gregory: Randomized alternating least squares for canonical tensor decompositions: application to a PDE with random data (2016)
  11. Domanov, Ignat; De Lathauwer, Lieven: Generic uniqueness conditions for the canonical polyadic decomposition and INDSCAL (2015)
  12. Albers, Casper J.; Gower, John C.: A contribution to the visualisation of three-way arrays (2014)
  13. Bordes, Antoine; Glorot, Xavier; Weston, Jason; Bengio, Yoshua: A semantic matching energy function for learning with multi-relational data (2014)
  14. Domanov, Ignat; De Lathauwer, Lieven: Canonical polyadic decomposition of third-order tensors: reduction to generalized eigenvalue decomposition (2014)
  15. Stegeman, Alwin: Finding the limit of diverging components in three-way Candecomp/Parafac -- a demonstration of its practical merits (2014)
  16. Karfoul, Ahmad; Albera, Laurent; De Lathauwer, Lieven: Iterative methods for the canonical decomposition of multi-way arrays: application to blind underdetermined mixture identification (2011)
  17. Brachat, Jerome; Comon, Pierre; Mourrain, Bernard; Tsigaridas, Elias: Symmetric tensor decomposition (2010)
  18. Derado, Gordana; Bowman, F. DuBois; Ely, Timothy D.; Kilts, Clinton D.: Evaluating functional autocorrelation within spatially distributed neural processing networks (2010)
  19. Badeau, Roland; Boyer, Rémy: Fast multilinear singular value decomposition for structured tensors (2008)
  20. De Lathauwer, Lieven; De Moor, Bart; Vandewalle, Joos: Computation of the canonical decomposition by means of a simultaneous generalized Schur decomposition (2004)

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