Deflation method for CANDECOMP/PARAFAC tensor decomposition

CANDECOMP/PARAFAC tensor decomposition (CPD) approximates multiway data by rank-1 tensors. Unlike matrix decomposition, the procedure which estimates the best rank-R tensor approximation through R sequential best rank-1 approximations does not work for tensors, because the deflation does not always reduce the tensor rank. In this paper we propose a novel deflation method for the problem in which rank R does not exceed the tensor dimensions. A rank-R CPD can be performed through (R - 1) rank-1 reductions. At each deflation stage, the residue tensor is constrained to have a reduced multilinear rank.

[1]  Vin de Silva,et al.  Tensor rank and the ill-posedness of the best low-rank approximation problem , 2006, math/0607647.

[2]  A. Stegeman,et al.  On the Non-Existence of Optimal Solutions and the Occurrence of “Degeneracy” in the CANDECOMP/PARAFAC Model , 2008, Psychometrika.

[3]  Lieven De Lathauwer,et al.  Decompositions of a Higher-Order Tensor in Block Terms - Part I: Lemmas for Partitioned Matrices , 2008, SIAM J. Matrix Anal. Appl..

[4]  F. Chinesta,et al.  Recent advances on the use of separated representations , 2009 .

[5]  Phillip A. Regalia,et al.  On the Best Rank-1 Approximation of Higher-Order Supersymmetric Tensors , 2001, SIAM J. Matrix Anal. Appl..

[6]  Tamir Hazan,et al.  Non-negative tensor factorization with applications to statistics and computer vision , 2005, ICML.

[7]  Laurent Albera,et al.  Multi-way space-time-wave-vector analysis for EEG source separation , 2012, Signal Process..

[8]  Nicholas D. Sidiropoulos,et al.  Parafac techniques for signal separation , 2000 .

[9]  A. Atsawarungruangkit,et al.  Generating Correlation Matrices Based on the Boundaries of Their Coefficients , 2012, PloS one.

[10]  Alwin Stegeman,et al.  Candecomp/Parafac: From Diverging Components to a Decomposition in Block Terms , 2012, SIAM J. Matrix Anal. Appl..

[11]  Joos Vandewalle,et al.  On the Best Rank-1 and Rank-(R1 , R2, ... , RN) Approximation of Higher-Order Tensors , 2000, SIAM J. Matrix Anal. Appl..

[12]  A. Cichocki,et al.  Tensor decompositions for feature extraction and classification of high dimensional datasets , 2010 .

[13]  Zbynek Koldovský,et al.  Cramér-Rao-Induced Bounds for CANDECOMP/PARAFAC Tensor Decomposition , 2012, IEEE Transactions on Signal Processing.

[14]  Pierre Comon,et al.  Subtracting a best rank-1 approximation may increase tensor rank , 2009, 2009 17th European Signal Processing Conference.

[15]  Rasmus Bro,et al.  MULTI-WAY ANALYSIS IN THE FOOD INDUSTRY Models, Algorithms & Applications , 1998 .

[16]  Gene H. Golub,et al.  A Rank-One Reduction Formula and Its Applications to Matrix Factorizations , 1995, SIAM Rev..

[17]  Pierre Comon,et al.  Blind Multilinear Identification , 2012, IEEE Transactions on Information Theory.

[18]  Zbynek Koldovský,et al.  Weight Adjusted Tensor Method for Blind Separation of Underdetermined Mixtures of Nonstationary Sources , 2011, IEEE Transactions on Signal Processing.

[19]  Andrzej Cichocki,et al.  CANDECOMP/PARAFAC Decomposition of High-Order Tensors Through Tensor Reshaping , 2012, IEEE Transactions on Signal Processing.

[20]  Andrzej Cichocki,et al.  Tensor diagonalization - a new tool for PARAFAC and block-term decomposition , 2014, ArXiv.

[21]  Lieven De Lathauwer,et al.  Decompositions of a Higher-Order Tensor in Block Terms - Part III: Alternating Least Squares Algorithms , 2008, SIAM J. Matrix Anal. Appl..

[22]  Zbynek Koldovský,et al.  Stability of CANDECOMP-PARAFAC tensor decomposition , 2011, 2011 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[23]  Andrzej Cichocki,et al.  Fast Alternating LS Algorithms for High Order CANDECOMP/PARAFAC Tensor Factorizations , 2013, IEEE Transactions on Signal Processing.

[24]  Andrzej Cichocki,et al.  Nonnegative Matrix and Tensor Factorization T , 2007 .

[25]  J. Chang,et al.  Analysis of individual differences in multidimensional scaling via an n-way generalization of “Eckart-Young” decomposition , 1970 .

[26]  R. Bro,et al.  Practical aspects of PARAFAC modeling of fluorescence excitation‐emission data , 2003 .

[27]  Liqun Qi,et al.  On the successive supersymmetric rank‐1 decomposition of higher‐order supersymmetric tensors , 2007, Numer. Linear Algebra Appl..

[28]  Richard A. Harshman,et al.  Foundations of the PARAFAC procedure: Models and conditions for an "explanatory" multi-model factor analysis , 1970 .