COMON , Pierre

The first goal of this tutorial is to show that tensor decompositions are of interest in various quite different fields. In particular, the talk addresses communities of Signal Processing, Algebraic Geometry, and Chemometrics. The second goal is to introduce and explain the vocabulary used in the latter fields, so that people can understand each other.

In Engineering, the identification of a linear statistical model is omnipresent (antenna array processing, factor analysis, etc). This problem has been extensively addressed since the fifties, refer e.g. to early contributions of Darmois and Skitovich. It consists of estimating a mixing matrix from observed realizations, under the assumption...

International audience

A matrix (and any associated linear system) will be referred to as structured if it has a small displacement rank. It is known that the inverse of a structured matrix is structured, which allows fast inversion (or solution), and reduced storage requirements. According to two definitions of displacement structure of practical interest, it is...

The first goal of this tutorial is to show that tensor decompositions are of interest in various quite different fields. In particular, the talk addresses communities of Signal Processing, Algebraic Geometry, and Chemometrics. The second goal is to introduce and explain the vocabulary used in the latter fields, so that people can understand each other.

Tensors appear more and more often in signal processing problems, and especially spatial processing, which typically involves multichannel modeling. Even if it is not always obvious that tensor algebra is the best framework to address a problem, there are cases where no choice is left. Blind identification of multichannel non monic MA models is...

Quelques développements récents en traitement du signal

In this paper, we present a partial survey of the tools borrowed from tensor algebra, which have been utilized recently in Statistics and Signal Processing. It is shown why the decompositions well known in linear algebra can hardly be extended to tensors. The concept of rank is itself difficult to define, and its calculation raises...

The algorithm proposed aims at identifying moving average coefficient matrices of an MA process, not necessarily minimum-phase, driven by an unobserved non-Gaussian input. It is assumed that the observation available is of limited duration, and coefficients are estimated from the set of fourth order output cumulants. It is shown that much more...

The Canonical Polyadic (CP) decomposition of a tensor is difficult to compute. Even algorithms computing the best rank-one approximation are not entirely satisfactory. And deflation approaches (successive rank-1 tensor approximations) do not work for tensors. However, there are cases where successive rank-1 matrix approximations can help in...

Les variables aléatoires complexes rencontrées en traitement du signal proviennent souvent de la transformée de Fourier de signaux réels. De ce fait, elles ne sont pas des variables complexes quelconques, mais jouissent de la propriété dite de circularité. Après avoir résumé quelques définitions et introduit les variables aléatoires complexes,...

Tensor decompositions have been increasingly used in Signal Processing during the last decade. In particular, Tucker and HOSVD decompositions are attractive for compression purposes, and the Canonical Polyadic decomposition (CP), sometimes referred to as Parafac, allows to restore identifiability in some Blind Identification problems, as...

Tensor decompositions permit to estimate in a deterministic way the parameters in a multi-linear model. Applications have been already pointed out in antenna array processing and digital communications [1], among others, and are extremely attractive provided some diversity at the receiver is available. In addition, they often involve structured...

It is often admitted that a static system with more inputs (sources) than outputs (sensors, or channels) cannot be blindly identified, that is, identified only from the observation of its outputs, and without any a priori knowledge on the source statistics but their independence. By resorting to High-Order Statistics, it turns out that static...

Tensor decompositions have been increasingly used in Signal Processing during the last decade. In particular, Tucker and HOSVD decompositions are attractive for compression purposes, and the Canonical Polyadic decomposition (CP), sometimes referred to as Parafac, allows to restore identifiability in some Blind Identification problems, as...

The problem of identifying linear mixtures of independent random variables only from outputs can be traced back to 1953 with the works of Darmois or Skitovich. They pointed out that when data are non Gaussian, a lot more can be said about the mixture. In practice, Blind Identification of linear mixtures is useful especially in Factor Analysis,...

Les variables aléatoires complexes rencontrées en traitement du signal proviennent souvent de la transformée de Fourier de signaux réels. De ce fait, elles ne sont pas des variables complexes quelconques, mais jouissent de la propriété dite de circularité. Après avoir résumé quelques définitions et introduit les variables aléatoires complexes,...

ISBN 978-2-296-12827-9

The independent component analysis (ICA) of a random vector consists of searching for a linear transformation that minimizes the statistical dependence between its components. In order to define suitable search criteria, the expansion of mutual information is utilized as a function of cumulants of increasing orders. An efficient algorithm is...

A general definition of contrast criteria is proposed, which induces the concept of trivial filters. These optimization criteria enjoy identifiability properties, and aim at delivering outputs satisfying specific properties, such as statistical independence or a discrete character. Several ways of building new contrast criteria are described....

The Canonical Polyadic (CP) decomposition of a tensor is difficult to compute. Even algorithms computing the best rank-one approximation are not entirely satisfactory. And deflation approaches (successive rank-1 tensor approximations) do not work for tensors. However, there are cases where successive rank-1 matrix approximations can help in...

Since the nineties, tensors are increasingly used in Signal Processing and Data Analysis. There exist striking differences between tensors and matrices, some being advantages, and others raising difficulties. These differences are pointed out in this paper while briefly surveying the state of the art. The conclusion is that tensors are...

Tensors of order r are implicitly used for a long time in Engineering, since derivatives of order r of multivariate scalar functions are indeed tensors. For instance, cumulants of order r of an n-dimensional random variable are related to rth derivatives of the joint characteristic function. As such, they form a symmetric tensor of order r and...

Tensors of order r are implicitly used for a long time in Engineering, since derivatives of order r of multivariate scalar functions are indeed tensors. For instance, cumulants of order r of an n-dimensional random variable are related to rth derivatives of the joint characteristic function. As such, they form a symmetric tensor of order r and...

Finite Impulse Responses (FIR) of Single-Input Single-Output (SISO) channels can be blindly identified from second order statistics of transformed data, for instance when the channel is excited by Binary Phase Shift Keying (BPSK), Minimum Shift Keying (MSK) or Quadrature Phase Shift Keying (QPSK) inputs. Identifiability conditions are derived...