Overview of Constrained PARAFAC Models

In this paper, we present an overview of constrained 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 the 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 $N^{th}$ 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

Fast multilinear Singular Values Decomposition for higher-order Hankel tensors

International audienceThe Higher-Order Singular Value Decomposition (HOSVD) is a possible generalization of the Singular Value Decomposition (SVD) to tensors, which have been successfully applied in various domains. Unfortunately, this decomposition is computationally demanding. Indeed, the HOSVD of a Nth- order tensor involves the computation of the SVD of N matrices. Previous works have shown that it is possible to reduce the complexity of HOSVD for third-order structured tensors. These methods exploit the columns redundancy, which is present in the mode of structured tensors, especially in Hankel tensors. In this paper, we propose to extend these results to fourth order Hankel tensor. We propose two ways to extend Hankel structure to fourth order tensors. For these two types of tensors, a method to build a reordered mode is proposed, which highlights the column redundancy and we derive a fast algorithm to compute their HOSVD. Finally we show the benefit of our algorithms in terms of complexity

Non-iterative solution for PARAFAC with a Toeplitz matrix factor

International audienceRecently, tensor signal processing has received an increased attention, particularly in the context of wireless communication applications. The so-called PARAllel FACtor (PARAFAC) decomposition is certainly the most used tensor tool. In general, the parameter estimation of a PARAFAC decomposition is carried out by means of the iterative ALS algorithm, which exhibits the following main drawbacks: convergence towards local minima, a high number of iterations for convergence, and difficulty to take, optimally, special matrix structures into account. In this paper, we propose a non-iterative parameter estimation method for a PARAFAC decomposition when one matrix factor has a Toeplitz structure, a situation that is commonly encountered in signal processing applications. We illustrate the proposed method by means of simulation results

Estimation récursive des cumulants d'Ordre quatre avec application à l'identification

Les Statistiques d'Ordre Élevé (SOE) sont de plus en plus utilisées dans les applications de traitement du signal. Toutefois, des formules générales d'estimation récursive des cumulants d'ordre supérieur à trois font jusqu'à aujourd'hui défaut. Cet article comble en partie cette lacune en présentant une formule récursive pour l'estimation des cumulants d'ordre quatre. Cette formule est ensuite utilisée dans le cadre de l'identification de modèles paramétriques de type Réponse Impulsionnelle Finie (RIF). Une version moindres carrés de l'algorithme C(Q,k), basée sur cette formule, est proposée. Des résultats de simulation illustrant le comportement de cette méthode d'identification sont présentés

Identification Aveugle de Canaux de Communication Non-linéaires basée sur la décomposition PARAFAC

Dans cet article, nous considérons l'estimation aveugle des noyaux de Volterra associés à un canal non-linéaire structuré en blocs. Pour ce type de structure, il a été montré que la connaissance des coefficients diagonaux des noyaux de Volterra est suffisante pour la caractérisation complète du modèle. Nous proposons donc un précodage de l'entrée permettant de découpler l'effet des coefficients diagonaux et non-diagonaux. Ce précodage induit une représentation tensorielle des signaux mesurés qui admet une décomposition du type PARAFAC dont l'un des facteurs est formé par les coefficients diagonaux recherchés. Nous établissons les conditions d'identifiabilité et illustrons la méthode par quelques simulations

Fusion de données appliquée à la classification de cibles observées à l'aide d'un système optronique passif

Cet article propose une méthode de classification basée sur le formalisme de Dempster-Shafer (DS). Elle est appliquée à la classification d'aéronefs observés à l'aide d'un système optronique passif, donc sans mesure de la distance. L'originalité de l'approche repose sur l'exploitation du formalisme de DS de façon à compenser une absence d'information, ici la distance de la cible, par une décomposition de chaque hypothèse en sous-hypothèses. Dans l'application traitée, cela revient à évaluer la vraisemblance d'hypothèses constituées à la fois de la classe de la cible et de son orientation par rapport au capteur. Les performances de la méthode proposée sont illustrées à l'aide de résultats de simulation

Analysis and power diversity-based cancellation of nonlinear distortions in OFDM systems

International audienceOne of the main drawbacks of orthogonal frequency division multiplexing (OFDM) systems is the high peak-to-average power ratio (PAPR) of the transmitted signals, which can cause the introduction of inter-carrier interference (ICI) due to the presence of nonlinear power amplifiers (PAs). In this paper, a theoretical analysis of ICI in nonlinear OFDM systems with polynomial PAs is made. Contrarily to other works, this analysis provides an exact description of nonlinear ICI. Moreover, three receivers for channel estimation and ICI cancellation in OFDM systems with polynomial PAs are proposed, based on the concept of power diversity that consists in re-transmitting the information symbols several times with a different transmission power each time. The transmission powers that minimize the sum of the residual mean square errors (MSEs) provided by the proposed receivers are derived in the case of a third-degree polynomial PA. An important advantage of the proposed receivers is that the optimal transmission powers do not depend on the channel nor the PA coefficients

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

Accepted for publication in the Proceedings of the European Signal Processing Conference (EUSIPCO) 2015.International audienceThe 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

Performance asymptotique d'estimateurs de modèles CP avec facteurs structurés

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