Performance estimation for tensor CP decomposition with structured factors

Boizard, Maxime; Boyer, Rémy; Favier, Gérard; Cohen, Jérémy E.; Comon, Pierre

doi:10.1109/icassp.2015.7178618

Cited by 12 publications

(21 citation statements)

References 23 publications

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“…zero-mean Gaussian entries of variance σ 2 . By exploiting (10) together with (5), we can conveniently express the SCPD of X under its vectorized form as (1) .…”

Section: A Tensors and The Cp Decompositionmentioning

confidence: 99%

“…Without loss of generality, we assume θ (2) = θ (3) . The parameter vector is thus written as η = [θ (1) ,θ (2) , λ] T . The Jacobian is then given by…”

Section: Propositionmentioning

confidence: 99%

“…Hence we have matrix Ω = A (2) A (1) up to scaling. If ω r denotes the rth column of matrix Ω, the rth column of matrices A (1) and A (2) can eventually be obtained by computing the best rank-one approximation of the I 1 × I 2 matrix Unvec(ω r ). The appropriate normalization of columns of A (1) and A (2) finally gives λ.…”

Section: B Subspace-based Closed Form Solutionsmentioning

confidence: 99%

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Tensor CP Decomposition With Structured Factor Matrices: Algorithms and Performance

Goulart

Boizard

Boyer

et al. 2016

IEEE J. Sel. Top. Signal Process.

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The canonical polyadic decomposition (CPD) of high-order tensors, also known as Candecomp/Parafac, is very useful for representing and analyzing multidimensional data. This paper considers a CPD model having structured matrix factors, as e.g. Toeplitz, Hankel or circulant matrices, and studies its associated estimation problem. This model arises in signal processing applications such as Wiener-Hammerstein system identification and cumulant-based wireless communication channel estimation. After introducing a general formulation of the considered structured CPD (SCPD), we derive closed-form expressions for the Cramér-Rao bound (CRB) of its parameters under the presence of additive white Gaussian noise. Formulas for special cases of interest, as when the CPD contains identical factors, are also provided. Aiming at a more relevant statistical evaluation from a practical standpoint, we discuss the application of our formulas in a Bayesian context, where prior distributions are assigned to the model parameters. Three existing algorithms for computing SCPDs are then described: a constrained alternating least squares (CALS) algorithm, a subspace-based solution and an algebraic solution for SCPDs with circulant factors. Subsequently, we present three numerical simulation scenarios, in which several specialized estimators based on these algorithms are proposed for concrete examples of SCPD involving circulant factors. In particular, the third scenario concerns the identification of a Wiener-Hammerstein system via the SCPD of an associated Volterra kernel. The statistical performance of the proposed estimators is assessed via Monte Carlo simulations, by comparing their Bayesian mean-square error with the expected CRB.

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“…zero-mean Gaussian entries of variance σ 2 . By exploiting (10) together with (5), we can conveniently express the SCPD of X under its vectorized form as (1) .…”

Section: A Tensors and The Cp Decompositionmentioning

confidence: 99%

“…Without loss of generality, we assume θ (2) = θ (3) . The parameter vector is thus written as η = [θ (1) ,θ (2) , λ] T . The Jacobian is then given by…”

Section: Propositionmentioning

confidence: 99%

Section: B Subspace-based Closed Form Solutionsmentioning

confidence: 99%

See 1 more Smart Citation

Tensor CP Decomposition With Structured Factor Matrices: Algorithms and Performance

Goulart

Boizard

Boyer

et al. 2016

IEEE J. Sel. Top. Signal Process.

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“…In order to identify the contribution of w and h in CRB(w k ) and CRB(h r ), we propose to extend the results presented in [13] by using oblique projection. This is the purpose of the following proposition.…”

Section: Closed-form Expression For the Crbmentioning

confidence: 99%

“…Regarding the structured case, the CRB for the estimation of a complex CPD with a particular Vandermonde factor has been derived in [12], motivated by the problem of estimating the directions of arrival of multiple source signals. Also, [13] has provided a closed-form expression for the CRB associated with the estimation of a CPD having Hankel and/or Toeplitz factors. This paper addresses the statistical evaluation of algorithms specialized in computing a CPD having banded circulant factors when applied to estimate the parameters of a Wiener-Hammerstein (WH) model, which is a well-known block-oriented model used for representing nonlinear dynamical systems [14].…”

Section: Introductionmentioning

confidence: 99%

Statistical efficiency of structured CPD estimation applied to Wiener-Hammerstein modeling

Goulart

Boizard

Boyer

et al. 2015

2015 23rd European Signal Processing Conference (EUSIPCO)

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The computation of a structured canonical polyadic decomposition (CPD) is useful to address several important modeling problems in real-world applications. In this paper, we consider the identification of a nonlinear system by means of a Wiener-Hammerstein model, assuming a high-order Volterra kernel of that system has been previously estimated. Such a kernel, viewed as a tensor, admits a CPD with banded circulant factors which comprise the model parameters. To estimate them, we formulate specialized estimators based on recently proposed algorithms for the computation of structured CPDs. Then, considering the presence of additive white Gaussian noise, we derive a closed-form expression for the Cramer-Rao bound (CRB) associated with this estimation problem. Finally, we assess the statistical performance of the proposed estimators via Monte Carlo simulations, by comparing their mean-square error with the CRB.

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Constrained Cramér–Rao bounds for reconstruction problems formulated as coupled canonical polyadic decompositions

et al. 2022

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Performance estimation for tensor CP decomposition with structured factors

Cited by 12 publications

References 23 publications

Tensor CP Decomposition With Structured Factor Matrices: Algorithms and Performance

Tensor CP Decomposition With Structured Factor Matrices: Algorithms and Performance

Statistical efficiency of structured CPD estimation applied to Wiener-Hammerstein modeling

Constrained Cramér–Rao bounds for reconstruction problems formulated as coupled canonical polyadic decompositions

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