Non-unitary joint zero-block diagonalization of matrices using a conjugate gradient algorithm

Cherrak, Omar; Ghennioui, Hicham; Abarkan, El Houssein; Thirion-Moreau, Nadège; Moreau, Éric

doi:10.1109/eusipco.2015.7362710

Cited by 4 publications

(9 citation statements)

References 20 publications

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“…All the numerical examples were carried out using MATLAB R2014b, with machine ǫ = 2.2 × 10 −16 . We compare the performance of our algorithms with four other algorithms for the geanojbd problem, namely, JBD-OG, JBD-ORG [19], JBD-LM [6] and JBD-NCG [26]. For the JBD-OG method and the JBD-ORG method the stopping criteria are W k+1 − W k F < 10 −12 , or φ k − φ k+1 φ k < 10 −8 for successive 5 steps, or the maximum number of iterations, which is set as 2000, exceeded.…”

Section: Numericalmentioning

confidence: 99%

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An Algebraic Approach to Nonorthogonal General Joint Block Diagonalization

Cai¹,

Liu²

2017

SIAM J. Matrix Anal. & Appl.

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The exact/approximate non-orthogonal general joint block diagonalization (nogjbd) problem of a given real matrix set A = {A i } m i=1 is to find a nonsingular matrix W ∈ R n×n (diagonalizer) such that W T A i W for i = 1, 2, . . . , m are all exactly/approximately block diagonal matrices with the same diagonal block structure and with as many diagonal blocks as possible. In this paper, we show that a solution to the exact/approximate nogjbd problem can be obtained by finding the exact/approximate solutions to the system of linear equations A i Z = Z T A i for i = 1, . . . , m, followed by a block diagonalization of Z via similarity transformation. A necessary and sufficient condition for the equivalence of the solutions to the exact nogjbd problem is established. Two numerical methods are proposed to solve the nogjbd problem, and numerical examples are presented to show the merits of the proposed methods.Key words. joint block diagonalization, tensor decomposition, independent component analysis AMS subject classifications. 15A21,15A69, 65F30.1. Introduction. The joint block diagonalization problem, also called the simultaneous block diagonalization problem, is a particular case of the block term decomposition (BTD) of a third order tensor [10,11,14,26]. Such problem has found many applications in independent subspace analysis (e.g., [4,13,30,31]) and semidefinite programming (e.g., [18,8,2,9]). To specify the problem, we name the problem by nine capital letters wwxyyzzzz. The first two letters, ww, indicate the type of the matrices in the matrix set, sy/he for real symmetric/complex Hermitian matrices, ge for general matrices. The second letter, x, indicates that the problem is solved in the exact sense or the approximate sense, e for the former and a for the latter. The next two letters, yy, indicate the type of the diagonalizer, no/nu for non-orthogonal/non-unitary matrix, o/u for orthogonal/unitary matrix(often left as blank). The last four letters, zzzz, indicate the computation performed, jd for joint diagonalization, jbd for joint block diagonalization, gjbd for general joint block diagonalization. Next, we first give some definitions, then formulate the wweyyjbd problem and the wweyygjbd problem mathematically.Definition 1.1. We call τ n = (n 1 , . . . , n t ) a partition of positive integer n if n 1 , n 2 , . . . , n t are all positive integers and the sum of them is n, i.e., t i=1 n i = n. The integer t is called the cardinality of the partition τ n , denoted by card(τ n ). The set of all partitions of n is denoted by T n .Definition 1.2. Given a partition τ n = (n 1 , . . . , n t ), for any matrix A of order n, define its block diagonal part and off-block-diagonal part associated with τ n as Bdiag τn (A) = diag(A 11 , . . . , A tt ), OffBdiag τn (A) = A − Bdiag τn (A), respectively, where A ii is of order n i for i = 1, . . . , t. A matrix A is referred to as a τ n -block diagonal matrix if OffBdiag τn (A) = 0.

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Section: Numericalmentioning

confidence: 99%

“…Great efforts has been devoted to solving the wwayyjbd problem and numerous algorithms are proposed. For example, the JBD-OG/ORG method by H. Ghennioui et al [19], the JBD-LM method by O. Cherrak et al [6], the JBD-NCG method by D. Nion [26]. For more methods, we refer the readers to [12,5,33] and reference therein.…”

mentioning

confidence: 99%

An Algebraic Approach to Nonorthogonal General Joint Block Diagonalization

Cai¹,

Liu²

2017

SIAM J. Matrix Anal. & Appl.

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“…We show, here, how the algorithms proposed in [16], [17] adresses the problem of the separation of instantaneous mixtures of S-AIS data. The principle of the proposed methods are based on three main steps: first, the SGMAF of the observations across the array is constructed.…”

Section: Principle Of the Proposed Methods Based On The Spatial Genermentioning

confidence: 99%

“…Hence, two BSS methods can be derived. The first called JZD CG DD algorithm based on conjugate gradient approach [16]. The second JZD LM DD algorithm based on Levenbreg-Marquardt scheme [17].…”

Section: Non-unitary Joint Zero-(block) Diagonalization Algorithms (Nmentioning

confidence: 99%

Blind separation of complex-valued satellite-AIS data for marine surveillance: a spatial quadratic time-frequency domain approach

Cherrak

Ghennioui

Thirion-Moreau

et al. 2019

IJECE

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<p>In this paper, the problem of the blind separation of complex-valued Satellite-AIS data for marine surveillance is addressed. Due to the specific properties of the sources under consideration: they are cyclo-stationary signals with two close cyclic frequencies, we opt for spatial quadratic time-frequency domain methods. The use of an additional diversity, the time delay, is aimed at making it possible to undo the mixing of signals at the multi-sensor receiver. The suggested method involves three main stages. First, the spatial generalized mean Ambiguity function of the observations across the array is constructed. Second, in the Ambiguity plane, Delay-Doppler regions of high magnitude are determined and Delay-Doppler points of peaky values are selected. Third, the mixing matrix is estimated from these Delay-Doppler regions using our proposed non-unitary joint zero-(block) diagonalization algorithms as to perform separation.</p>

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“…All the numerical examples were carried out on a quad-core Intel Core i5-6300HQ running at 2.30 GHz with 3.87-GB RAM, using MATLAB R2014a with machine = 2.2 × 10 −16 . We compare the performance of PEAR (with and without refinement) with the second GJBD algorithm in the work of Cai and Liu, 21 namely, ⋆-commuting-based method with a conservative strategy, SCMC for short, and two algorithms for the JBD problem, namely, JBD-LM 40 and JBD-NCG. 4 For PEAR, three refinement loops are used to improve the quality of the diagonalizer.…”

Section: Numerical Examplesmentioning

confidence: 99%

Solving the general joint block diagonalization problem via linearly independent eigenvectors of a matrix polynomial

Cai

Cheng

Shi

2019

Numerical Linear Algebra App

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Summary In this paper, we consider the exact/approximate general joint block diagonalization (GJBD) problem of a matrix set false{Aifalse}i=0p ( p ≥ 1), where a nonsingular matrix W (often referred to as a diagonalizer) needs to be found such that the matrices W HAiW 's are all exactly/approximately block‐diagonal matrices with as many diagonal blocks as possible. We show that the diagonalizer of the exact GJBD problem can be given by W = [x1,x2,…,xn]Π, where Π is a permutation matrix and xi's are eigenvectors of the matrix polynomial Pfalse(λfalse)=∑i=0pλiAi, satisfying that [x1,x2,…,xn] is nonsingular and where the geometric multiplicity of each λi corresponding with xi is equal to 1. In addition, the equivalence of all solutions to the exact GJBD problem is established. Moreover, a theoretical proof is given to show why the approximate GJBD problem can be solved similarly to the exact GJBD problem. Based on the theoretical results, a three‐stage method is proposed, and numerical results show the merits of the method.

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Non-unitary joint zero-block diagonalization of matrices using a conjugate gradient algorithm

Cited by 4 publications

References 20 publications

An Algebraic Approach to Nonorthogonal General Joint Block Diagonalization

An Algebraic Approach to Nonorthogonal General Joint Block Diagonalization

Blind separation of complex-valued satellite-AIS data for marine surveillance: a spatial quadratic time-frequency domain approach

Solving the general joint block diagonalization problem via linearly independent eigenvectors of a matrix polynomial

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