PARAFAC-based unified tensor modeling for wireless communication systems with application to blind multiuser equalization

Almeida, André L. F. de; Favier, Gérard; Mota, J.C.M.

doi:10.1016/j.sigpro.2005.12.014

Cited by 93 publications

(105 citation statements)

References 21 publications

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“…Block constrained PARAFAC models were used in [70][71][72] for modeling different types of multiuser wireless communication systems. Block constrained Tucker models were used for space-time multiplexing MIMO-OFDM systems [73] and for blind beamforming [74].…”

Section: Block Paralind/confac Modelsmentioning

confidence: 99%

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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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Section: Block Paralind/confac Modelsmentioning

confidence: 99%

“…, showing the equivalence of the two core tensor expressions (72) and (54). N −2, N) and constrained PARAFAC-N models Let us consider the PARATUCK- (N 1 , N) model (52) in the case N 1 = N − 2…”

Section: Remarksmentioning

confidence: 99%

Overview of constrained PARAFAC models

Favier

Almeida

2014

EURASIP J. Adv. Signal Process.

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“…The low rank Canonical Polyadic model is the most famous one [1]. It is widely used in signal processing [2] [3], image processing [4] and brain source imaging [5] [6]. Since a few decades and Harshman's pioneer work [7], many algorithms have been proposed to compute the CP decomposition.…”

Section: Introductionmentioning

confidence: 99%

Low rank canonical polyadic decomposition of tensors based on group sparsity

Han¹,

Albera²,

Kachenoura³

et al. 2017

2017 25th European Signal Processing Conference (EUSIPCO)

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Abstract-A new and robust method for low rank Canonical Polyadic (CP) decomposition of tensors is introduced in this paper. The proposed method imposes the Group Sparsity of the coefficients of each Loading (GSL) matrix under orthonormal subspace. By this way, the low rank CP decomposition problem is solved without any knowledge of the true rank and without using any nuclear norm regularization term, which generally leads to computationally prohibitive iterative optimization for largescale data. Our GSL-CP technique can be then implemented using only an upper bound of the rank. It is compared in terms of performance with classical methods, which require to know exactly the rank of the tensor. Numerical simulated experiments with noisy tensors and results on fluorescence data show the advantages of the proposed GSL-CP method in comparison with classical algorithms.

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“…However, practical problems are encountered where the factor matrices have a structure [15,16,17] such as Toeplitz, circulant, Hankel or Vandermonde. In order to fill this gap, several algorithms have been proposed, including non iterative [16,18] or iterative, e.g.…”

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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PARAFAC-based unified tensor modeling for wireless communication systems with application to blind multiuser equalization

Cited by 93 publications

References 21 publications

Overview of constrained PARAFAC models

Overview of constrained PARAFAC models

Low rank canonical polyadic decomposition of tensors based on group sparsity

Performance estimation for tensor CP decomposition with structured factors

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