Fast Jacobi algorithm for non-orthogonal joint diagonalization of non-symmetric third-order tensors

Maurandi, Victor; Moreau, Éric

doi:10.1109/eusipco.2015.7362596

Cited by 6 publications

(5 citation statements)

References 18 publications

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“…The plane rotation G (i,j,Ψ) was used very often in Jacobi-type algorithms for joint approximate diagonalization of matrices or tensors by orthogonal or non-orthogonal transformations [15,22,40,4,5,37]. The upper triangular rotation U (i,j,Ψ) and lower triangular rotation L (i,j,Ψ) also appeared many times in the Jacobi-type algorithms on special linear group SL m (C) or SL m (R) [4,5,31,32,33]. In the real case, the diagonal rotation D (i,j,Ψ) was used in [38].…”

Section: Line Search Descent Methods On St(m N C)mentioning

confidence: 99%

“…Denote Λ = Λ(X), which is defined as in Section 2.4. Let P T i,j : C 2×2 → C m×m be the adjoint operator of projection operator P i,j defined in (31), i.e.,…”

Section: Line Search Descent Methods On St(m N C)mentioning

confidence: 99%

“…To solve the JADM-H problem, Jacobi-type algorithms were proposed based on the LU decomposition in [32,33], and based on the QL decomposition in [37]. To solve the JADM-CS problem, a Jacobi-type algorithm was proposed based on the LU decomposition in [31,32]. However, to our knowledge, there was no theoretical result about the convergence of these Jacobi-type algorithms in the literature.…”

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confidence: 99%

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Gradient based block coordinate descent algorithms for joint approximate diagonalization of matrices

Li¹,

Usevich²,

Comon³

2020

Preprint

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In this paper, we propose a gradient based block coordinate descent (BCD-G) framework to solve the joint approximate diagonalization of matrices defined on the product of the complex Stiefel manifold and the special linear group. Instead of the cyclic fashion, we choose the block for optimization in a way based on the Riemannian gradient. To update the first block variable in the complex Stiefel manifold, we use the well-known line search descent method. To update the second block variable in the special linear group, based on four different kinds of elementary rotations, we construct two classes: Jacobi-GLU and Jacobi-GLQ, and then get two BCD-G algorithms: BCD-GLU and BCD-GLQ. We establish the weak convergence and global convergence of these two algorithms using the Lojasiewicz gradient inequality under the assumption that the iterates are bounded. In particular, the problem we focus on in this paper includes as special cases the well-known joint approximate diagonalization of Hermitian (or complex symmetric) matrices by invertible transformations in blind source separation, and our algorithms specialize as the gradient based Jacobi-type algorithms. All the algorithms and convergence results in this paper also apply to the real case.

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Section: Line Search Descent Methods On St(m N C)mentioning

confidence: 99%

“…Denote Λ = Λ(X), which is defined as in Section 2.4. Let P T i,j : C 2×2 → C m×m be the adjoint operator of projection operator P i,j defined in (31), i.e.,…”

Section: Line Search Descent Methods On St(m N C)mentioning

confidence: 99%

mentioning

confidence: 99%

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Gradient based block coordinate descent algorithms for joint approximate diagonalization of matrices

Li¹,

Usevich²,

Comon³

2020

Preprint

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“…Complex matrix problems frequently arise in mathematics and engineering, since complex matrices are widely applied in signal processing ( Liu et al, 2014 ), image quality assessment ( Wang, 2012 ), joint diagonalization ( Maurandi et al, 2013 ), and robot path tracking ( Guo et al, 2019 , 2020 ; Jin et al, 2020 , 2022a ,c,e; Shi et al, 2021 , 2022a ; Liu et al, 2022 ). Various numerical algorithms have been presented to solve the complex matrix problems, such as the Newton iterative method ( Rajbenbach et al, 1987 ) and the Greville recursive method ( Gan and Ling, 2008 ).…”

Section: Introductionmentioning

confidence: 99%

A robust zeroing neural network and its applications to dynamic complex matrix equation solving and robotic manipulator trajectory tracking

Zhao

Chen

2022

Front. Neurorobot.

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Dynamic complex matrix equation (DCME) is frequently encountered in the fields of mathematics and industry, and numerous recurrent neural network (RNN) models have been reported to effectively find the solution of DCME in no noise environment. However, noises are unavoidable in reality, and dynamic systems must be affected by noises. Thus, the invention of anti-noise neural network models becomes increasingly important to address this issue. By introducing a new activation function (NAF), a robust zeroing neural network (RZNN) model for solving DCME in noisy-polluted environment is proposed and investigated in this paper. The robustness and convergence of the proposed RZNN model are proved by strict mathematical proof and verified by comparative numerical simulation results. Furthermore, the proposed RZNN model is applied to manipulator trajectory tracking control, and it completes the trajectory tracking task successfully, which further validates its practical applied prospects.

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“…An algorithm for two-sided diagonalization of order-3 tensor was proposed in a conference paper [43]. An algorithm for three-sided diagonalization of order-3 tensor was proposed in [44]. This paper presents a different approach to the same problem.…”

mentioning

confidence: 99%

Non-orthogonal tensor diagonalization

2017

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Tensor diagonalization means transforming a given tensor to an exactly or nearly diagonal form through multiplying the tensor by non-orthogonal invertible matrices along selected dimensions of the tensor. It is generalization of approximate joint diagonalization (AJD) of a set of matrices. In particular, we derive (1) a new algorithm for symmetric AJD, which is called two-sided symmetric diagonalization of order-three tensor, (2) a similar algorithm for non-symmetric AJD, also called general two-sided diagonalization of an order-3 tensor, and (3) an algorithm for three-sided diagonalization of order-3 or order-4 tensors. The latter two algorithms may serve for canonical polyadic (CP) tensor decomposition, and they can outperform other CP tensor decomposition methods in terms of computational speed under the restriction that the tensor rank does not exceed the tensor multilinear rank. Finally, we propose (4) similar algorithms for tensor block diagonalization, which is related to the tensor block-term decomposition.

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Fast Jacobi algorithm for non-orthogonal joint diagonalization of non-symmetric third-order tensors

Cited by 6 publications

References 18 publications

Gradient based block coordinate descent algorithms for joint approximate diagonalization of matrices

Gradient based block coordinate descent algorithms for joint approximate diagonalization of matrices

A robust zeroing neural network and its applications to dynamic complex matrix equation solving and robotic manipulator trajectory tracking

Non-orthogonal tensor diagonalization

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