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Discovering latent variables in high dimensional high-order data.

Tensor Factorization Models

  • Tucker Decomposition
“Tucker

  • Unconstrained Tucker decomposition has rotational ambiguity.
  • By imposing proper constraints on all/partial factors, we are able to extract multiple sets of unique components with specific physical meaning, which is also referred to as MBSS [PDF].
  • As most existing nonnegative Tucker decomposition algorithms are very slow, we developed a quite fast one by incorporating dimensionality reduction techniques. [PDF][MATLAB code]

  • CP Decomposition
“cp

  • Under mild conditions the decomposition is essentially unique.
  • Once at least one factor matrix has been correctly estimated, all the other factors can often be uniquely computed [PDF].
  • By imposing proper constraints on all/partial factors, the property of uniqueness may be improved.


Nonnegative Tensor Factorizations (Tucker model and CP model)

  • G. Zhou, A. Cichocki, Q. Zhao, S. Xie, Efficient Nonnegative Tucker Decompositions: Algorithms and Uniqueness, [ArXiv e-prints]: http://arxiv.org/abs/1404.4412

  • G. Zhou, A. Cichocki, Q. Zhao, S. Xie, Nonnegative Matrix and Tensor Factorizations: An algorithmic perspective, IEEE Signal Processing Magazine, vol.31, no.3, pp.54--65, May 2014. [Pseudo-code]

  • Nonnegative Matrix Factorization (NMF) based on Low-rank Approximation(LRA), including the Multiplicative Update (MU), the Accelerated Proximal Gradient (APG), and the Hierarchical ALS (HALS) method.
  • Nonnegative Tensor Factorization (NTF, i.e. CP Decompositions with Nonnegativity Constraints), include: [1] CP_HALS performs CPD with or without nonnegativity constraints; [2] FastNTFAPG performs NTF based on the LRA and accelerated proximal gradient methods; [3] lraNTF allows to select the update rule from MU/HALS/AGP.
  • Nonnegative Tucker Decomposition (NTD) based on Low-rank Approximation by Zhou [New!].




Software

MATLAB Toolbox for Tensor Decomposition and Analysis [TDALAB] ver1.1 by Zhou & Cichocki.

TensorLab: MATLAB toolbox for tensor computations by Laurent Sorber, Marc Van Barel, and Lieven De Lathauwer.

MATLAB Tensor Toolbox by Brett W. Bader, Tamara G. Kolda and others.
The N-way toolbox for MATLAB by Rasmus Bro and Claus A. Andersson.