Canonical decomposition of semi-symmetric semi-nonnegative three-way arrays

In this paper, we focus on a special case of canonical decomposition, say the canonical decomposition of three-way arrays, which have two equal and nonnegative loading matrices. This problem has a great interest in blind source separation, particularly in magnetic resonance spectroscopy, where each observation spectrum is a nonnegative linear combination of different constituent spectra. In order to achieve the semi-symmetric semi-nonnegative canonical decomposition of a given three-way array, we propose a novel technique named ELS-ALSsym+, which optimizes an unconstrained problem obtained by means of a square change of variable. The method is compared with a Levenberg-Marquardt-like approach recently submitted for publication in Linear Algebra and its applications and classical methods, which uses no a priori about the considered array, such as the ELS-ALS technique. Such a comparison is made in terms of performance and numerical complexity on random synthetic arrays.

[1]  Pierre Comon,et al.  Enhanced Line Search: A Novel Method to Accelerate PARAFAC , 2008, SIAM J. Matrix Anal. Appl..

[2]  P. Comon,et al.  Blind Identification of Overcomplete MixturEs of sources (BIOME) , 2004 .

[3]  Lotfi Senhadji,et al.  Decomposition of Semi-Nonnegative Semi-Symmetric Three-Way Tensors Based on Lu Matrix Factorization. , 2012 .

[4]  Laurent Albera,et al.  Décomposition de tableaux d'ordre trois semi-nonnégatifs et semi-symétriques , 2011 .

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

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

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

[8]  Laurent Albera,et al.  Iterative methods for the canonical decomposition of multi-way arrays: Application to blind underdetermined mixture identification , 2011, Signal Process..

[9]  P. Comon,et al.  Tensor decompositions, alternating least squares and other tales , 2009 .

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

[11]  P. Paatero,et al.  Positive matrix factorization: A non-negative factor model with optimal utilization of error estimates of data values† , 1994 .

[12]  Theodoros N. Arvanitis,et al.  A comparative study of feature extraction and blind source separation of independent component analysis (ICA) on childhood brain tumour 1H magnetic resonance spectra , 2009, NMR in biomedicine.

[13]  Arie Yeredor,et al.  Non-orthogonal joint diagonalization in the least-squares sense with application in blind source separation , 2002, IEEE Trans. Signal Process..

[14]  Rasmus Bro,et al.  A comparison of algorithms for fitting the PARAFAC model , 2006, Comput. Stat. Data Anal..

[15]  Joe Brewer,et al.  Kronecker products and matrix calculus in system theory , 1978 .