Nonnegative Joint Diagonalization by Congruence Based on LU Matrix Factorization

Wang, Lu; Albera, Laurent; Kachenoura, Amar; Hu, Shu; Senhadji, Lotfi

doi:10.1109/lsp.2013.2267797

Cited by 11 publications

(12 citation statements)

References 19 publications

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“…We also evaluate the proposed methods for estimating a nonsquare matrix A. The proposed algorithms are compared with four classical nonorthogonal JDC methods, namely ACDC [23], FFDIAG [25], LUJ1D [26], QRJ1D [26], and the nonnegative JDC method ACDC + LU [41]. In the second part, the source separation ability of the proposed algorithms is studied through a BSS application.…”

Section: Simulation Resultsmentioning

confidence: 99%

“…Four classical nonorthogonal JDC methods without nonnegativity constraints including ACDC [23], FFDIAG [25], LUJ1D [26], and QRJ1D [26] and one nonnegative JDC method ACDC + LU [41] are tested as reference methods. The performance is assessed in terms of the matrix estimation accuracy, the numerical complexity, and the CPU time.…”

Section: Discussionmentioning

confidence: 99%

“…Existing semi-nonnegative INDSCAL algorithms are based on the minimization of the cost function (4) [40,41]. They are able to achieve a better estimation of A than ACDC when the data satisfies the semi-nonnegative INDSCAL model at the cost of a higher computational complexity.…”

Section: Problem Reformulationmentioning

confidence: 99%

“…For both algorithms, the inverses [41], as well as those of four classical JDC algorithms, namely ACDC [23], FFDIAG [25], LUJ1D [26], and QRJ1D [26], are listed in …”

Section: Numerical Complexitymentioning

confidence: 99%

“…Coloigner et al proposed a family of algorithms based on line search and trust region strategies [40]. Wang et al developed an alternating minimization scheme [41]. Those methods appear to depend on initialization, and therefore in practice require a multi-initialization procedure, leading to an increase of numerical complexity.…”

Section: Introductionmentioning

confidence: 99%

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Canonical polyadic decomposition of third-order semi-nonnegative semi-symmetric tensors using LU and QR matrix factorizations

Wang

Albera

Kachenoura

et al. 2014

EURASIP J. Adv. Signal Process.

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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.

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Section: Simulation Resultsmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Problem Reformulationmentioning

confidence: 99%

“…For both algorithms, the inverses [41], as well as those of four classical JDC algorithms, namely ACDC [23], FFDIAG [25], LUJ1D [26], and QRJ1D [26], are listed in …”

Section: Numerical Complexitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Canonical polyadic decomposition of third-order semi-nonnegative semi-symmetric tensors using LU and QR matrix factorizations

Wang

Albera

Kachenoura

et al. 2014

EURASIP J. Adv. Signal Process.

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A Descent Algorithm-Based Parallel Variable Dual Matrix Model and Application to Blind Source Separation

Zeng

Feng

2014

Circuits Syst Signal Process

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In this paper, we focus on the problem of blind source separation (BSS). To solve the problem efficiently, a new algorithm is proposed. First, a parallel variable dual-matrix model (PVDMM) that considers all the numerical relations between a mixing matrix and a separating matrix is proposed. Different constrained terms are used to construct the cost function for every sub-algorithm. These constrained terms reflect a numerical relation. Therefore, a number of undesired solutions are excluded, the search region is reduced, and the convergence efficiency of the algorithm is ultimately improved. Parallel sub-algorithms are proven to converge to a separable matrix only if the cost function approaches zero. Second, a descent algorithm (DA) as a PVDMM optimizing approach is proposed in this paper as well. Apparently, the efficiency of the algorithm depends completely on the DA. Unlike the traditional descent method, the DA defines step length by solving inequality instead of merely utilizing the Wolfeor Armijo-type search rule. Stimulation results indicate that the DA can improve computational efficiency. Under mild conditions, the DA has been proven to have strong convergence properties. Numerical results also show that the method is very efficient and robust. Finally, we applied the combinative algorithm to the BSS problem. Computer simulations illustrate its good performance.

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Fast non-unitary simultaneous diagonalization of third-order tensors

Maurandi

Moreau

2014

2014 48th Asilomar Conference on Signals, Systems and Computers

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Nonnegative Joint Diagonalization by Congruence Based on LU Matrix Factorization

Cited by 11 publications

References 19 publications

Canonical polyadic decomposition of third-order semi-nonnegative semi-symmetric tensors using LU and QR matrix factorizations

Canonical polyadic decomposition of third-order semi-nonnegative semi-symmetric tensors using LU and QR matrix factorizations

A Descent Algorithm-Based Parallel Variable Dual Matrix Model and Application to Blind Source Separation

Fast non-unitary simultaneous diagonalization of third-order tensors

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