Jean-François Cardoso

Equivariant adaptive source separation

Source separation consists of recovering a set of independent signals when only mixtures with unknown coefficients are observed. This paper introduces a class of adaptive algorithms for source separation that implements an adaptive version of equivariant estimation and is henceforth called equivariant adaptive separation via independence (EASI). The EASI algorithms are based on the idea of serial updating. This specific form of matrix updates systematically yields algorithms with a simple structure for both real and complex mixtures. Most importantly, the performance of an EASI algorithm does not depend on the mixing matrix. In particular, convergence rates, stability conditions, and interference rejection levels depend only on the (normalized) distributions of the source signals. Closed-form expressions of these quantities are given via an asymptotic performance analysis. The theme of equivariance is stressed throughout the paper. The source separation problem has an underlying multiplicative structure. The parameter space forms a (matrix) multiplicative group. We explore the (favorable) consequences of this fact on implementation, performance, and optimization of EASI algorithms.

High-order contrasts for independent component analysis

This article considers high-order measures of independence for the independent component analysis problem and discusses the class of Jacobi algorithms for their optimization. Several implementations are discussed. We compare the proposed approaches with gradient-based techniques from the algorithmic point of view and also on a set of biomedical data.

Jacobi Angles for Simultaneous Diagonalization

Simultaneous diagonalization of several matrices can be implemented by a Jacobi-like technique. This note gives the required Jacobi angles in close form.

Infomax and maximum likelihood for blind source separation

Algorithms for the blind separation of sources can be derived from several different principles. This article shows that the infomax (information-maximization) principle is equivalent to the maximum likelihood. The application of the infomax principle to source separation consists of maximizing an output entropy.

Source separation using higher order moments

The author presents a simple algebraic method for the extraction of independent components in multidimensional data. Since statistical independence is a much stronger property than uncorrelation, it is possible, using higher-order moments, to identify source signatures in array data without any a priori model for propagation or reception, that is, without directional vector parameterization, provided that the emitting sources are independent with different probability distributions. The author proposes such a blind identification procedure. Source signatures are directly identified as covariance eigenvectors after data have been orthonormalized and nonlinearly weighted. Potential applications to array processing are illustrated by a simulation consisting of a simultaneous range-bearing estimation with a passive array.<<ETX>>

Multidimensional independent component analysis

This paper proposes to generalize the notion of independent component analysis (ICA) to the notion of multidimensional independent component analysis (MICA). We start from the ICA or blind source separation (BSS) model and show that it can be uniquely identified provided it is properly parameterized in terms of one-dimensional subspaces. From this standpoint, the BSS/ICA model is generalized to multidimensional components. We discuss how ICA standard algorithms can be adapted to MICA decomposition. The relevance of these ideas is illustrated by a MICA decomposition of ECG signals.

Blind source separation-semiparametric statistical approach

The semiparametric statistical model is used to formulate the problem of blind source separation. The method of estimating functions is applied to this problem. It is shown that an estimator of the mixing matrix or its learning version can be described in terms of an estimating function. The statistical efficiencies of these algorithms are studied. The main results are as follows. (1) The space consisting of all the estimating functions is derived. (2) The space is decomposed into the orthogonal sum of the admissible part and a redundant ancillary part. For any estimating function, one can find a better or equally good estimator in the admissible part. (3) The Fisher efficient (that is, asymptotically best) estimating functions are derived. (4) The stability of learning algorithms is studied.

Super-symmetric decomposition of the fourth-order cumulant tensor. Blind identification of more sources than sensors

Ideas for higher-order array processing are introduced, focusing on fourth-order cumulant statistics. They are expressed in an index-free formalism allowing the exploitation of all their symmetry properties. It is shown that, when dealing with 4-index quantities, symmetries are related to rank properties. The rich symmetry structure yields a class of identification algorithms. An algebraic technique for blind identification is included, and a few others are briefly indicated.<<ETX>>

On subspace methods for blind identification of single-input multiple-output FIR systems

Blind identification of single-input multiple-output (SIMO) FIR systems based on second-order statistics has attracted a great deal of research effort. We focus on subspace estimation procedures, which exploit the structure of the range space of certain matrix-valued statistics constructed by arranging in a prescribed order the covariance coefficients of the observations. General subspace identifiability results are obtained, based on properties of minimal polynomial bases of rational subspaces. Several subspace estimation procedures are then derived. These estimators are all based on a weighted least-square solution of an overdetermined system of linear equations. An asymptotic statistical analysis of these estimators is carried out to evaluate the potential of these methods and the impact of the weighting.

Multidetector multicomponent spectral matching and applications for cosmic microwave background data analysis

We present a new method for analysing multidetector maps containing several astrophysical components. Our method, based on matching the data to a model in the spectral domain, permits us to estimate jointly the spatial power spectra of the components and of the noise, as well as their mixing coefficients. It is of particular relevance for analysis of millimetre-wave maps of cosmic microwave background (CMB) anisotropies.