Canonical polyadic decomposition of 3rd order semi-nonnegative semi-symmetric tensors using LU and QR matrix factorizationsReport as inadecuate

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1 LTSI - Laboratoire Traitement du Signal et de l-Image 2 CRIBS - Centre de Recherche en Information Biomédicale sino-français 3 PANAMA - Parcimonie et Nouveaux Algorithmes pour le Signal et la Modélisation Audio Inria Rennes – Bretagne Atlantique , IRISA-D5 - SIGNAUX ET IMAGES NUMÉRIQUES, ROBOTIQUE 4 LIST - Laboratory of Image Science and Technology Nanjing

Abstract : Semi-symmetric three-way arrays are essential tools in Blind Source Separation BSS particularly in Independent Component Analysis ICA. These arrays can be built by resorting to higher order statistics of the data. The Canonical Polyadic CP decomposition of such semi-symmetric three-way arrays allows us to identify the so-called mixing matrix, which contains the information about the intensities of some latent source signals present in the observation channels. In addition, in many applications, such as the Magnetic Resonance Spectroscopy MRS, the columns of the mixing matrix are viewed as relative concentrations of the spectra of the chemical components. Therefore, the two loading matrices of the three-way array, which are equal to the mixing matrix, are nonnegative. Most existing CP algorithms handle the symmetry and the nonnegativity separately. Up to now, very few of them consider both the semi-nonnegativity and the semi-symmetry structure of the three-way array. Nevertheless, like all the methods based on line search, trust region strategies and alternating optimization, they appear to be dependent on initialization, requiring in practice a multi-initialization procedure. In order to overcome this drawback, we propose two new methods, called JD+LU and JD+QR, to solve the problem of CP decomposition of semi-nonnegative semi-symmetric three-way arrays. Firstly, we rewrite the constrained optimization problem as an unconstrained one. In fact, the nonnegativity constraint of the two symmetric modes is ensured by means of a square change of variable. Secondly, a Jacobi-like optimization procedure is adopted because of its good convergence property. More precisely, the two new methods use LU and QR matrix factorizations, respectively, which consist in formulating high-dimensional optimization problems into several sequential polynomial and rational subproblems. By using both LU and QR matrix factorizations we aim at studying the influence of the used matrix factorization. Numerical experiments on simulated arrays emphasize the advantages of the proposed methods especially the one based on LU factorization, in the presence of high-variance model error and of degeneracies such as bottlenecks. A BSS application on MRS data confirms the validity and improvement of the proposed methods.

Keywords : Canonical polyadic decomposition semi-nonnegative semi-symmetric tensor joint diagonalization by congruence individuals differences in scaling analysis blind source separation independent component analysis magnetic resonance spectroscopy

Author: Lu Wang - Laurent Albera - Amar Kachenoura - Huazhong Shu - Lotfi Senhadji -



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