Preconditioned optimization algorithms solving the problem of the non unitary joint block diagonalization: application to blind separation of convolutive mixtures

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

doi:10.1007/s11045-017-0506-8

“…Approximate S-JBD. Optimization based schemes for the approximate S-JBD problem are discussed in the recent paper [6] (see also [5,21,35] and references therein). The authors of [5] proposed a variant of Algorithm 1.1 in which the null Algorithm 1.1 Computation of S-JBD problem (1.6) under the conditions in Theorem 1.10 Input: K × K symmetric matrices V 1 , .…”

Section: Introductionmentioning

confidence: 99%

On Uniqueness and Computation of the Decomposition of a Tensor into Multilinear Rank-$(1,L_r,L_r)$ Terms

Domanov¹,

Lathauwer²

2020

SIAM J. Matrix Anal. Appl.

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Canonical Polyadic Decomposition (CPD) represents a third-order tensor as the minimal sum of rank-1 terms. Because of its uniqueness properties the CPD has found many concrete applications in telecommunication, array processing, machine learning, etc. On the other hand, in several applications the rank-1 constraint on the terms is too restrictive. A multilinear rank-(M, N, L) constraint (where a rank-1 term is the special case for which M = N = L = 1) could be more realistic, while it still yields a decomposition with attractive uniqueness properties.In this paper we focus on the decomposition of a tensor T into a sum of multilinear rank-(1, Lr, Lr) terms, r = 1, . . . , R. This particular decomposition type has already found applications in wireless communication, chemometrics and the blind signal separation of signals that can be modelled as exponential polynomials and rational functions. We find conditions on the terms which guarantee that the decomposition is unique and can be computed by means of the eigenvalue decomposition of a matrix even in the cases where none of the factor matrices has full column rank. We consider both the case where the decomposition is exact and the case where the decomposition holds only approximately. We show that in both cases the number of the terms R and their "sizes" L 1 , . . . , L R do not have to be known a priori and can be estimated as well. The conditions for uniqueness are easy to verify, especially for terms that can be considered "generic". In particular, we obtain the following two generalizations of a well known result on generic uniqueness of the CPD (i.e., the case L 1 = • • • = L R = 1): we show that the multilinear rank-(1, Lr, Lr) decomposition of an I × J × K tensor is generically unique if i) L 1 = • • • = L R =: L and R ≤ min((J − L)(K − L), I) or if ii) L R ≤ min((I − 1)(J − 1), K) and J ≥ max(L i + L j ).

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“…Blindly and via the BSS, the principle of the solution is based on hypotheses concerning the mixing system as well as hypotheses on the nature of the source signals. Related to the type of mixing system we find in the literature proposals that deal with the linear problem for the instantaneous case [11] and also others that deal with the convolutive case [12][13][14] as we also find solutions that deal with the nonlinear problem [15]. Regarding the assumptions of the source signals, the major tool applied to solve the BSS problem is called independent component analysis (ICA) where the solutions that apply this technique, are based on the concept of statistical independence of the signals, to recover the original sources.…”

Section: Introductionmentioning

confidence: 99%

Source recovery by analytical maximization of phase-shifted kurtosis for the mixtures of noisy and noiseless signals

SMATTI

¹

,

Arar

²

2023

Preprint

0

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This manuscript presents a work that provides a study as well as a simple analytical solution for solving the blind source separation problem (BSS) for noiseless and noisy linear mixing of statistically independent stationary and nonstationary signals. The study is based on the exploitation of the probabilistic characteristics of the mixed signals by using the statistics of the second order and the fourth order for the completion of the separation. The proposed solution consists mainly of two steps based on the concept of the geometric solution. For the case of the mixture of two sources (2×2), the first step aims to transform the dependent signals into orthogonal signals (whitening) via the principal component analysis (PCA) principle. After the application of the PCA and in order to complete the statistical independence of the two uncorrelated signals, the second step aims to determine an adequate rotating angle that leads directly to the separation, and this angle is determined in this work analytically by the simple calculation of the phase shift of a sinusoidal objective function based on the sum of the kurtosis of the whitened signals. In the case of several sources (n×n), the solution (2×2) can be applied by a simple generalization which leads to the global separation. Whether for the noisy or noiseless case, the results obtained prove the reliability and efficiency by applying this analytical solution to achieve the desired objective, in particular by comparing the proposed algorithm with the application of two other separation algorithms, one of which involves the application of optimization techniques

show abstract

Source recovery by analytical maximization of phase-shifted kurtosis for the mixtures of noisy and noiseless signals

SMATTI

¹

,

Arar

²

2023

Preprint

0

View full text Add to dashboard Cite

This manuscript presents a work that provides a study as well as a simple analytical solution for solving the blind source separation problem (BSS) for noiseless and noisy linear mixing of statistically independent stationary and nonstationary signals. The study is based on the exploitation of the probabilistic characteristics of the mixed signals by using the statistics of the second order and the fourth order for the completion of the separation. The proposed solution consists mainly of two steps based on the concept of the geometric solution. For the case of the mixture of two sources (2×2), the first step aims to transform the dependent signals into orthogonal signals (whitening) via the principal component analysis (PCA) principle. After the application of the PCA and in order to complete the statistical independence of the two uncorrelated signals, the second step aims to determine an adequate rotating angle that leads directly to the separation, and this angle is determined in this work analytically by the simple calculation of the phase shift of a sinusoidal objective function based on the sum of the kurtosis of the whitened signals. In the case of several sources (n×n), the solution (2×2) can be applied by a simple generalization which leads to the global separation. Whether for the noisy or noiseless case, the results obtained prove the reliability and efficiency by applying this analytical solution to achieve the desired objective, in particular by comparing the proposed algorithm with the application of two other separation algorithms, one of which involves the application of optimization techniques

show abstract

Preconditioned optimization algorithms solving the problem of the non unitary joint block diagonalization: application to blind separation of convolutive mixtures

Cited by 3 publications

References 35 publications

On Uniqueness and Computation of the Decomposition of a Tensor into Multilinear Rank-$(1,L_r,L_r)$ Terms

On Uniqueness and Computation of the Decomposition of a Tensor into Multilinear Rank-$(1,L_r,L_r)$ Terms

Source recovery by analytical maximization of phase-shifted kurtosis for the mixtures of noisy and noiseless signals

Source recovery by analytical maximization of phase-shifted kurtosis for the mixtures of noisy and noiseless signals

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