Blind channel identification algorithms based on the Parafac decomposition of cumulant tensors: The single and multiuser cases

Fernandes, Carlos Estêvão Rolim; Favier, Gérard; Mota, J.C.M.

doi:10.1016/j.sigpro.2007.12.010

Cited by 37 publications

(37 citation statements)

References 33 publications

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“…However, when the diversity of the observations is not sufficient, one can resort to a second class of tensorbased methods that rely on the multilinearity properties of higherorder statistics (HOS) [14], [18], [35]. A large majority of these methods solves the blind identification problem by means of the CP decomposition of a tensor storing the cumulants of the observations [7], [14], [39], [41], [42], [45]. This is the case, for instance, of FOOBI/FOOBI2 [17], [18], and BIOME [15] algorithms, which capitalize on the triadic decomposition of fourth-and sixth-order output cumulants, respectively.…”

Section: Introductionmentioning

confidence: 99%

CONFAC Decomposition Approach to Blind Identification of Underdetermined Mixtures Based on Generating Function Derivatives

Almeida

Luciani

Stegeman

et al. 2012

IEEE Trans. Signal Process.

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Abstract-This work proposes a new tensor-based approach to solve the problem of blind identification of underdetermined mixtures of complex sources exploiting the cumulant generating function (CGF) of the observations. We show that a collection of secondorder derivatives of the CGF of the observations can be stored in a third-order tensor following a constrained factor (CONFAC) decomposition with known constrained structure. In order to increase the diversity, we combine three derivative types into an extended third-order CONFAC decomposition. A detailed uniqueness study of this decomposition is provided, from which easy-to-check sufficient conditions ensuring the essential uniqueness of the mixing matrix are obtained.

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Section: Introductionmentioning

confidence: 99%

CONFAC Decomposition Approach to Blind Identification of Underdetermined Mixtures Based on Generating Function Derivatives

Almeida

Luciani

Stegeman

et al. 2012

IEEE Trans. Signal Process.

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“…This work has set the basis for several developments. A more general model, taking into account random amplitudes in the CP decomposition would allow to derive an Hybrid CRB and to consider other applications like parameter estimation of Wiener-Hammerstein models from their associated Volterra kernels [19] and blind channel identification using output cumulant tensors [10].…”

Section: Resultsmentioning

confidence: 99%

“…In addition, unlike SVD, it is essentially unique under mild conditions. Therefore, it is naturally well suited for the analysis of data sets constituted by observations of a function of multiple discrete indices, as encountered in signal processing [9,10,11], data mining [12] and biomedical engineering [13] ; see [8,14] for other examples.…”

Section: Introductionmentioning

confidence: 99%

Performance estimation for tensor CP decomposition with structured factors

Boizard

Boyer

Favier

et al. 2015

2015 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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The Canonical Polyadic tensor decomposition (CPD), also known as Candecomp/Parafac, is very useful in numerous scientific disciplines. Structured CPDs, i.e. with Toeplitz, circulant, or Hankel factor matrices, are often encountered in signal processing applications. As subsequently pointed out, specialized algorithms were recently proposed for estimating the deterministic parameters of structured CP decompositions. A closed-form expression of the Cramér-Rao bound (CRB) is derived, related to the problem of estimating CPD parameters, when the observed tensor is corrupted with an additive circular i.i.d. Gaussian noise. This CRB is provided for arbitrary tensor rank and sizes. Finally, the proposed CRB expression is used to asses the statistical efficiency of the existing algorithms by means of simulation results in the cases of third-order tensors having three circulant factors on one hand, and an Hankel factor on the other hand.

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“…• Application of the formula (38) to the constrained PARAFAC model (58), with the matrix factors (A, B, F, D…”

Section: Remarksmentioning

confidence: 99%

“…Block structured nonlinear systems like Wiener, Hammerstein, and parallel-cascade Wiener systems can be identified from their associated Volterra kernels that admit symmetric PARAFAC decompositions with Toeplitz factors [36,37]. Symmetric PARAFAC models with Hankel factors and symmetric block PARAFAC models with block Hankel factors are encountered for http://asp.eurasipjournals.com/content/2014/1/142 blind identification of multiple-input multiple-output (MIMO) linear channels using fourth-order cumulant tensors, in the cases of memoryless and convolutive channels, respectively [38,39]. In the presence of structural constraints, specific estimation algorithms can be derived as it is the case for symmetric CP decompositions [40], CP decompositions with Toeplitz factors (in [41], an iterative solution was proposed, whereas in [42], a non-iterative algorithm was developed), Vandermonde factors [43], circulant factors [44], banded and/or structured matrix factors [45,46], and also for Hankel and Vandermonde structured core tensors [33].…”

Section: Introductionmentioning

confidence: 99%

Overview of constrained PARAFAC models

Favier

Almeida

2014

EURASIP J. Adv. Signal Process.

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In this paper, we present an overview of constrained parallel factor (PARAFAC) models where the constraints model linear dependencies among columns of the factor matrices of the tensor decomposition or, alternatively, the pattern of interactions between different modes of the tensor which are captured by the equivalent core tensor. Some tensor prerequisites with a particular emphasis on mode combination using Kronecker products of canonical vectors that makes easier matricization operations, are first introduced. This Kronecker product-based approach is also formulated in terms of an index notation, which provides an original and concise formalism for both matricizing tensors and writing tensor models. Then, after a brief reminder of PARAFAC and Tucker models, two families of constrained tensor models, the co-called PARALIND/CONFAC and PARATUCK models, are described in a unified framework, for Nth-order tensors. New tensor models, called nested Tucker models and block PARALIND/CONFAC models, are also introduced. A link between PARATUCK models and constrained PARAFAC models is then established. Finally, new uniqueness properties of PARATUCK models are deduced from sufficient conditions for essential uniqueness of their associated constrained PARAFAC models.Keywords: Constrained PARAFAC; PARALIND/CONFAC; PARATUCK; Tensor models; Tucker models 1 Review IntroductionTensor calculus was introduced in differential geometry, at the end of the nineteenth century, and then tensor analysis was developed in the context of Einstein's theory of general relativity, with the introduction of index notation, the so-called Einstein summation convention, at the beginning of the twentieth century, which allows to simplify and shorten physics equations involving tensors. Index notation is also useful for simplifying multivariate statistical calculations, particularly those involving cumulant tensors [1]. Generally speaking, tensors are used in physics and differential geometry for characterizing the properties of a physical system, representing fundamental laws of physics, and defining geometrical objects whose components are functions. When these functions are defined over a continuum of points of a mathematical space, the tensor forms what is called a tensor field, a generalization of vector field used to solve problems involving curved surfaces or spaces, as it is the case of After the first tensor developments by mathematicians and physicists, the need of analyzing collections of data matrices that can be seen as three-way data arrays gave rise to three-way models for data analysis, with the pioneering works of Tucker in psychometrics [3], and Harshman in phonetics [4], who proposed what is now referred to as the Tucker and parallel factor (PARAFAC) decompositions, respectively. The PARAFAC decomposition was independently proposed by Carroll and Chang [5] under the name canonical decomposition (CANDE-COMP) and then called CANDECOMP/PARAFAC (CP) in [6]. For a history of the development of multi-way models in the context ...

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Blind channel identification algorithms based on the Parafac decomposition of cumulant tensors: The single and multiuser cases

Cited by 37 publications

References 33 publications

CONFAC Decomposition Approach to Blind Identification of Underdetermined Mixtures Based on Generating Function Derivatives

CONFAC Decomposition Approach to Blind Identification of Underdetermined Mixtures Based on Generating Function Derivatives

Performance estimation for tensor CP decomposition with structured factors

Overview of constrained PARAFAC models

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