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International audience; Covariance matrices play a major role in statistics , signal processing and machine learning applications. This paper focuses on the semiparametric covariance/scatter matrix estimation problem in elliptical distributions. The class of elliptical distributions can be seen as a semiparametric model where the finite-dimensional vector of interest is given by the location vector and by the (vectorized) covariance/scatter matrix, while the density generator represents an infinite-dimensional nuisance function. The main aim of this work is then to provide possible estimators of the finite-dimensional parameter vector able to reconcile the two dichotomic concepts of robustness and (semiparametric) efficiency. An R-estimator satisfying these requirements has been recently proposed by Hallin, Oja and Paindaveine for real-valued elliptical data by exploiting the Le Cam's theory of one-step efficient estimators and the rank-based statistics. In this paper, we firstly recall the building blocks underlying the derivation of such real-valued R-estimator, then its extension to complex-valued data is proposed. Moreover, through numerical simulations, its estimation performance and robustness to outliers are investigated in a finite-sample regime.

International audience; Change detection (CD) for remotely sensed images of the Earth has been a popular subject of study in the past decades. With the increase in the number of spatial missions with embedded synthetic aperture radar (SAR) sensors, the amount of readily available observations has now reached the "big data" era. This chapter introduces several families of elliptical distributions that can be used to model multivariate SAR images. It describes several dissimilarity functions based on covariance matrices, which are then compared for CD on the considered datasets. The chapter presents an extension of a statistical detection methodology that allows us to account for low-rank structures in the covariance matrix, whose interest is also illustrated on the real dataset. The Kullback–Leibler divergence is a popular measure between probability density functions, encountered notably in SAR change detection problems. The local covariance matrix of pixel patches appears to be relevant feature to analyze multivariate image time series.

International audience; This paper studies a statistical model for heteroscedastic (i.e., power fluctuating) signals embedded in white Gaussian noise. Using the Riemannian geometry theory, we propose an unified approach to tackle several problems related to this model. The first axis of contribution concerns parameters (signal subspace and power factors) estimation, for which we derive intrinsic Cramér-Rao bounds and propose a flexible Riemannian optimization algorithmic framework in order to compute the maximum likelihood estimator (as well as other cost functions involving the parameters). Interestingly, the obtained bounds are in closed forms and interpretable in terms of problem's dimensions and SNR. The second axis of contribution concerns the problem of clustering data assuming a mixture of heteroscedastic signals model, for which we generalize the Euclidean K-means++ to the considered Riemannian parameter space. We propose an application of the resulting clustering algorithm on the Indian Pines segmentation problem benchmark.

International audience; In this paper, robust mean and covariance matrix estimation are considered in the context of mixed-effects models. Such models are widely used to analyze repeated measures data which arise in several signal processing applications that need to incorporate possible individual variations within a common behavior of individuals. In this context, most algorithms are based on the assumption that the observations follow a Gaussian distribution. Nevertheless, in certain situations in which the data set contains outliers, such assumption is not valid and leads to a dramatic performance loss. To overcome this drawback, we design an expectation-conditional maximization either algorithm in which the heterogeneous component is considered as a part of the complete data. Then, the proposed algorithm is cast into a parallel scheme w.r.t. the individuals in order to mitigate the computational cost and a possible central processor overload. Finally, the proposed algorithm is extended to deal with missing data which refers to the situation where part of the individual responses are unobserved. Numerical simulations are conducted to assess the performance of the proposed algorithm regarding robust regression estimators, probabilistic principal component analysis and its recent robust version.

International audience; In this paper, we propose a new estimator of the covariance matrix parameters when observations follow a mixture of a deterministic Compound-Gaussian (CG) and a white Gaussian noise. In particular, the covariance matrix of the CG contribution is assumed to be expressed as the Kronecker product of two low-rank matrices, which is a structure often involved in MIMO array processing. The proposed estimator is then obtained by maximizing the likelihood of the data with the use of a specifically tailored block Majorization-Minimization (MM) algorithm. Finally, the method is evaluated in terms of adaptive filtering on a MIMO-STAP radar setting, showing important improvements over standard processing.

International audience; The information geometry of the zero-mean tdistribution with Kronecker-product structured covariance matrix is derived. In particular, we obtain the Fisher information metric which shows that this geometry is identifiable to a product manifold of S ++ p (positive definite symmetric matrices) and sS ++ p (positive definite symmetric matrices with unit determinant). From this result, we obtain the geodesics and the Riemannian gradient. Finally, this geometry makes it possible to propose an on-line covariance matrix estimation algorithm well adapted to large datasets. Numerical experiments show that optimal results are obtained for a reasonable number of data.

International audience; The two basic performance indices characterizing the multi-target detection task in a radar system are the probability of false alarm (PF A) and the probability of detection PD. It is well-known that, when the disturbance model (i.e., clutter and noise) is perfectly known, the Neyman-Pearson (NP) detector provides the best decision strategy, i.e., the detector that maximizes the PD, while keeping a constant PF A. However, in practical scenarios, the a priori knowledge of the statistical model of the disturbance is rarely available. In this paper we investigate the robustness of a reinforcement learning (RL) based Wald-type test to guarantee reliable detection performance even without knowledge of the disturbance distribution. Specifically, the constant false alarm Rate (CFAR) property is obtained by applying tools from misspecified asymptotic statistics, while the PD is maximized by exploiting an RL-based scheme.

The joint estimation of the location vector and the shape matrix of a set of independent and identically Complex Elliptically Symmetric (CES) distributed observations is investigated from both the theoretical and computational viewpoints. This joint estimation problem is framed in the original context of semiparametric models allowing us to handle the (generally unknown) density generator as an \textit{infinite-dimensional} nuisance parameter. In the first part of the paper, a computationally efficient and memory saving implementation of the robust and semiparmaetric efficient $R$-estimator for shape matrices is derived. Building upon this result, in the second part, a joint estimator, relying on the Tyler's $M$-estimator of location and on the $R$-estimator of shape matrix, is proposed and its Mean Squared Error (MSE) performance compared with the Semiparametric Cram\'{e}r-Rao Bound (CSCRB)., Comment: This paper has been submitted to the Special Issue (related to MLSP) of the Journal of Signal Processing Systems

International audience; This paper addresses structured scatter matrix estimation within the non convex set of Kronecker product structure. The latter model usually involves two matrices , which can be themselves linearly constrained, and arises in many applications, such as MIMO communication , MEG/EEG data analysis. Taking this prior knowledge into account generally improves estimation accuracy. In the framework of robust estimation, the t-distribution is particularly suited to model heavy-tailed data. In this context, we introduce an estimator of the scatter matrix, having a Kronecker product structure and potential linear structured factors. In addition, we show that the proposed method yields a consistent and efficient estimate.

A general non-Gaussian semiparametric model is adopted to characterize the measurement vectors, i.e.\ the \textit{snapshots}, collected by a linear array. Moreover, the recently derived \textit{robust semiparametric efficient} $R$-estimator of the data covariance matrix is exploited to implement an original version of the MUSIC estimator. The efficiency of the resulting $R$-MUSIC algorithm is investigated by comparing its Mean Squared Error (MSE) in the estimation of the source spatial frequencies with the relevant Semiparametric Stochastic Cram\'{e}r-Rao Bound (SSCRB)., Comment: This paper has been submitted to 2020 IEEE Radar Conference, Florence, Italy, September 21-25, 2020

International audience; This paper focuses on the joint estimation of the location vector and the shape matrix of a set of Complex Elliptically Symmetric (CES) distributed observations. This well-known estimation problem is framed in the original context of semipara-metric models allowing us to handle the (generally unknown) density generator as an infinite-dimensional nuisance parameter. A joint estimator, relying on the Tyler's M-estimator of location and on a new R-estimator of shape matrix, is proposed and its Mean Squared Error (MSE) performance compared with the Semiparametric Cramér-Rao Bound (CSCRB).

The blind source separation problem is considered through the approach based on non-stationarity and coloration. In both cases, the sources are usually assumed to be Gaussian. In this paper, we extend previous works in order to handle sources drawn from the multivariate Student t-distribution. After studying the structure of the parameter manifold in this case, a new blind source separation criterion based on the log-likelihood of the considered distribution is proposed. To solve the resulting optimization problem, Riemannian optimization on the parameter manifold is leveraged. Practical expressions of the mathematical tools required by first order based Riemmanian optimization methods for this parameter manifold are derived to this end. The performance of the proposed method is illustrated on simulated data.

A new Riemannian geometry for the Compound Gaussian distribution is proposed. In particular, the Fisher information metric is obtained, along with corresponding geodesics and distance function. This new geometry is applied on a change detection problem on Multivariate Image Times Series: a recursive approach based on Riemannian optimization is developed. As shown on simulated data, it allows to reach optimal performance while being computationally more efficient., 13 pages

International audience; This paper aims at providing an original Riemannian geometry to derive robust covariance matrix estimators in spiked models (i.e. when the covariance matrix has a low-rank plus identity structure). The considered geometry is the one induced by the product of the Stiefel manifold and the manifold of Hermitian positive definite matrices, quotiented by the uni-tary group. One of the main contributions is to consider a Riemannian metric related to the Fisher information metric of elliptical distributions, leading to new representations for the tangent spaces and a new retraction. A new robust covari-ance matrix estimator is then obtained as the minimizer of Tyler's cost function, redefined directly on the set of low-rank plus identity matrices, and computed with the aforementioned tools. The main interest of this approach is that it appears well suited to the cases where the sample size is lower than the dimension , as illustrated by numerical experiments.

International audience; We consider the optimal performance of blind separation of Gaussian sources. In practice, this estimation problem is solved by a two-step procedure: estimation of a set of co-variance matrices from the observed data and approximate joint diagonalization of this set to find the unmixing matrix. Rather than studying the theoretical performance of a specific method, we are interested in the optimal attainable performance of any estimator. To do so, we consider the so-called intrinsic Cramér-Rao bound, which exploits the geometry of the parameters of the model. Unlike previous works developing a Cramér-Rao bound in this context, our solution does not require any additional hypotheses. To obtain our bound, we define and study a new Riemannian manifold holding the parameters of interest. An original estimation error measure is defined with the help of our Riemannian distance function. The corresponding Fisher information matrix is then obtained from the Fisher information metric and orthonormal bases on the tangent spaces of the manifold. Finally, our theoretical results are validated on simulated data.

International audience; This paper proposes an original Riemmanian geometry for low-rank structured elliptical models, i.e., when samples are elliptically distributed with a covariance matrix that has a low-rank plus identity structure. The considered geometry is the one induced by the product of the Stiefel manifold and the manifold of Hermitian positive definite matrices, quotiented by the unitary group. One of the main contribution is to consider an original Riemannian metric, leading to new representations of tangent spaces and geodesics. From this geometry, we derive a new Riemannian optimization framework for robust covariance estimation, which is leveraged to minimize the popular Tyler's cost function on the considered quotient manifold. We also obtain a new divergence function, which is exploited to define a geometrical error measure on the quotient, and the corresponding intrinsic Cramér-Rao lower bound is derived. Thanks to the structure of the chosen parametrization, we further consider the subspace estimation error on the Grassmann manifold and provide its intrinsic Cramér-Rao lower bound. Our theoretical results are illustrated on some numerical experiments, showing the interest of the proposed optimization framework and that performance bounds can be reached.

This paper aims at presenting a simulative analysis of the main properties of a new $R$-estimator of shape matrices in Complex Elliptically Symmetric (CES) distributed observations. First proposed by Hallin, Oja and Paindaveine for the real-valued case and then extended to the complex field in our recent work, this $R$-estimator has the remarkable property to be, at the same time, \textit{distributionally robust} and \textit{semiparametric efficient}. Here, the efficiency of different possible configurations of this $R$-estimator are investigated by comparing the resulting Mean Square Error (MSE) with the Constrained Semiparametric Cram\'{e}r-Rao Bound (CSCRB). Moreover, its robustness to outliers is assessed and compared with the one of the celebrated Tyler's estimator., Comment: This paper has been accepted to the 28th European Signal Processing Conference, EUSIPCO 2020 (5 pages, 5 figures)

International audience; In this paper, the performance of a maximum likelihood estimator (MLE) for a signal model accounting for possible coherence of the signal sources is studied. It is done by combining a dynamical evolution model of the source amplitude, namely a Gaussian random walk, with the usual observation model, assumed to be Gaussian. This implies an underlying relation between the signal source parameters and both the mean and the covariance matrix of the observations, and hence leads to a more general model than those usually used in array processing, such as the conditional signal model or the unconditional one. The resulting so-called "generalized conditional MLE" can be expressed both in batch form and in recursive form, which is updated since a term was missing in a previous paper. In addition, we derive the Cramér-Rao bounds associated with a model of this type, accounting for the fluctuations of a radar target's backscattering coefficient. Simulation results highlight a non-standard behavior of the estimators: consistent but not efficient for frequencies, efficient but non-consistent for amplitudes.

International audience; Covariance matrix estimation is a ubiquitous problem in signal processing. In most modern signal processing applications, data are generally modeled by non-Gaussian distributions with covariance matrices exhibiting a particular structure. Taking into account this structure and the non-Gaussian behavior improve drastically the estimation accuracy. In this paper, we consider the estimation of structured scatter matrix for complex elliptically distributed observations, where the assumed model can differ from the actual distribution of the observations. Specifically, we tackle this problem, in a mismatched framework, by proposing a novel estimator, named StructurEd ScAtter Matrix Estimator (SESAME), which is based on a two-step estimation procedure. We conduct theoretical analysis on the unbiasedness and the asymptotic efficiency and Gaussianity of SESAME. In addition, we derive a recur-sive estimation procedure that iteratively applies the SESAME method, called Recursive-SESAME (R-SESAME), reaching with improved performance at lower sample support the (Mismatched) Cramér-Rao Bound. Furthermore, we show that some special cases of the proposed method allow to retrieve preexisting methods. Finally, numerical results corroborate the theoretical analysis and assess the usefulness of the proposed algorithms.

International audience; This paper addresses robust estimation of structured shape (normalized covariance) matrices. Shape matrices most often own a particular structure depending on the application of interest and taking this structure into account improves estimation accuracy. In the framework of robust estimation, we introduce a recursive robust shape matrix estimation technique based on Tyler's M-estimate for convexly structured shape matrices. We prove that the proposed estimator is consistent, asymptotically efficient and Gaussian distributed and we notice that it reaches its asymptotic regime faster as the number of recursions increases. Finally, in the particular wide spreaded case of Hermitian persymmetric structure, we study the convergence of the recursions of the proposed algorithm.

Robust scatter matrix estimators are often obtained up to a scaling factor. Since most of the adaptive processes are invariant to this scaling ambiguity, we are mostly interested in estimating a normalized scatter matrix. In the context of complex elliptical symmetric distributions, and the framework of [1], we propose to develop the intrinsic Cramér-Rao bound for this problem. First, the Fisher information metric is derived for Hermitian positive definite matrices of unit determinant. This metric allows to bound two performance criteria, that are the Euclidian and Riemmanian distances., Dans un processus d'estimation robuste, les estimateurs de la matrice de dispersion sont souvent obtenusà un facteur d'échelle près. La plupart des traitements adaptatifsétant invariantsà ce facteur d'échelle, on s'intéresse principalementà l'estimation de la dispersion normalisée. Ainsi, dans le contexte des distributions elliptiques complexes, et en se plaçant dans le cadre introduit par [1], nous proposons dans cette communication une dérivation de la borne de Cramér-Rao intrinsèque pour ce problème. Tout d'abord, la métrique de Fisher est dérivée pour les matrices matrices symétriques définies positives de déterminant unitaire. Cette métrique permet ensuite d'établir une borne d'estimation pour deux mesures de performance : la distance euclidienne et la distance riemanienne entre les matrices Hermitiennes positives définies.

Ce papier porte sur l'estimation robuste de matrices de forme (covariance normalisée) structurées. Dans un contexte d'estimation robuste, nous introduisons un estimateur robuste et récursif de matrice de forme basé sur l'estimateur de Tyler. Nous prouvons que cet estimateur est consistant, asymptotiquement efficace et gaussien. Par ailleurs, nous constatons qu'il atteint son régime asymptotique d'autant plus rapidement que le nombre d'itérations augmente. Finalement, nous étudions la convergence des récursions pour la structure persymétrique hermitienne., This paper addresses robust estimation of structured shape (normalized covariance) matrices. In the framework of robust estimation, we introduce a recursive robust shape matrix estimation technique based on Tyler's M-estimate for convexly structured shape matrices. We prove that the proposed estimator is consistent, asymptotically efficient and Gaussian distributed and we notice that it reaches its asymptotic regime faster as the number of recursions increases. Finally, in the case of Hermitian persymmetric structure, we study the convergence of the recursions of the proposed algorithm.

International audience; The joint estimation of means and scatter matrices is often a core problem in multivariate analysis. In order to overcome robustness issues, such as outliers from Gaussian assumption, M-estimators are now preferred to the traditional sample mean and sample covariance matrix. These estimators are well established and studied in the real case since the sev-enties. Their extension to the complex case has drawn recent interest. In this letter, we derive the asymptotic performance of complex M-estimators for multivariate location and scatter matrix estimation.

We consider the problem of detecting a known $M$ -dimensional target signature vector from an observation corrupted by an additive noise with unknown covariance matrix. In that case, standard statistical methods of detection usually assume that $N$ - “target free” observations are available to perform estimation of the noise covariance matrix. However, in several applications, the target signal may contaminate the training data, resulting in a deviation of the expected performance of the detectors. In this paper, we consider the performance analysis of two low-rank detectors under the assumption that $N_{c}$ elements of the training data are contaminated by the target signal. More precisely, we derive the asymptotic false alarm and detection probabilities in the high dimensional regime in which both the dimension $M$ , the number of training data $N$ and contaminated data $N_{c}$ converge to infinity at the same rate. Numerical simulations illustrate the fact that, despite the asymptotic nature of the analysis, the results obtained are accurate for reasonable values of $M, N$ and $N_{c}$ .

International audience; Detecting targets embedded in a noisy environment is an important topic in adaptive array processing. In the traditional statistical framework, this problem is addressed through a binary hypothesis test, which usually requires the estimation of side parameters from secondary data. The latter are assumed to be homogeneous and target-free, which is in practice questionable. Indeed, secondary data are usually corrupted by radar clutters and/or jammers which can be non-stationary and locally low rank. Fortunately, the latter behaviors can be well acknowledged by a union-of-subspaces model. In this work, we propose a modified subspace clustering model which can be solved using convex optimization algorithms. In the context of multiple sparse target detection and localization, a comparison is performed with various robust detection methods exhibiting advantages and drawbacks of the proposed one.

In most modern signal processing applications, observations are generally modeled by non-Gaussian distributions with covariance matrices exhibiting a particular structure. Taking these properties into account in the estimation scheme improves drastically the estimation accuracy. In this paper, we consider the estimation of structured scatter matrix, where the assumed model can differ from the true model of the data. Specifically, we propose a novel class of estimators, named StructurEd ScAtter Matrix Estimator (SESAME) in the mismatched framework. We also conduct a theoretical analysis of its asymptotic performance., Dans la plupart des méthodes récentes en traitement du signal, les données sont modélisées par des distributions non gaussiennes, dont la matrice de covariance possède une structure particulière. Prendre ces propriétés en compte dans le processus d'estimation améliore nettement la qualité des estimées. Dans ce papier, nous considérons l'estimation de matrices de dispersion structurées, où le modèle supposé peut différer du vrai modèle des données. Plus précisément, nous proposons une nouvelle classe d'estimateurs dénommée StructurEd ScAtter Matrix Estimator (SESAME) dans un contexte de modèles mal spécifiés. Nous menons aussi une analyse théorique de leurs performances asymptotiques.

International audience; The spherically invariant random process (SIRP) clutter model is commonly used in scenarios where the radar clutter cannot be correctly modeled as a Gaussian process. In this short communication, we devise a novel Maximum-Likelihood (ML)-based iterative estimator for direction-of-departure and direction-of-arrival estimation in the Multiple-input multiple-output (MIMO) radar context in the presence of SIRP clutter. The proposed estimator employs a stepwise numerical concentration approach w.r.t. the objective function related to the marginal likelihood of the observation data. Our estimator leads to superior performance, as our simulations show, w.r.t. to the existing likelihood based methods, namely, the conventional, the conditional and the joint likelihood based estimators, and w.r.t. the robust subspace decomposition based methods. Finally, interconnections and comparison between the Iterative Marginal ML Estimator (IMMLE), Iterative Joint ML Estimator (IJMLE) and Iterative Conditional ML Estimator (ICdMLE) are provided.

International audience; In this paper, we study the behavior of the maximum likelihood estimator in the framework of low noise level (or high signal-to-noise ratio), when the data follow an unimodal elliptical distribution. The maximum likelihood estimator appears to be the same as in the Gaussian context, regardless the noise distribution. We also show that the asymptotic distribution of this estimator is unimodal elliptical, where the law is intimately linked to that of the noise distribution. Additionally, this estimator is shown to be not efficient, except in the Gaussian noise case. Finally, we validate our analytic results by some simulations. Index Terms-Maximum likelihood estimator statistics, high signal-to-noise ratio regime, elliptically symmetric distribution.

International audience; This paper addresses structured covariance matrix estimation under t-distribution. Covariance matrices frequently reveal a particular structure due to the considered application and taking into account this structure usually improves estimation accuracy. In the framework of robust estimation, the t-distribution is particularly suited to describe heavy-tailed observation. In this context, we propose an efficient estimation procedure for covariance matrices with convex structure under t-distribution. Numerical examples for Hermitian Toeplitz structure corroborate the theoretical analysis.