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In this paper, a new classification model based on covariance matrices is built in order to classify buried objects. The inputs of the proposed models are the hyperbola thumbnails obtained with a classical Ground Penetrating Radar (GPR) system. These thumbnails are entered in the first layers of a classical CNN which results in a covariance matrix by using the outputs of the convolutional filters. Next, the covariance matrix is given to a network composed of specific layers to classify Symmetric Positive Definite (SPD) matrices. We show in a large database that our approach outperform shallow networks designed for GPR data and conventional CNNs typically used in computer vision applications, particularly when the number of training data decreases and in the presence of mislabeled data. We also illustrate the interest of our models when training data and test sets are obtained from different weather modes or considerations.

In this study, we consider the realm of covariance matrices in machine learning, particularly focusing on computing Fr\'echet means on the manifold of symmetric positive definite matrices, commonly referred to as Karcher or geometric means. Such means are leveraged in numerous machine-learning tasks. Relying on advanced statistical tools, we introduce a random matrix theory-based method that estimates Fr\'echet means, which is particularly beneficial when dealing with low sample support and a high number of matrices to average. Our experimental evaluation, involving both synthetic and real-world EEG and hyperspectral datasets, shows that we largely outperform state-of-the-art methods.

In this paper, we consider the problem of testing equality of the covariance matrices of L complex Gaussian multivariate time series of dimension $M$ . We study the special case where each of the L covariance matrices is modeled as a rank K perturbation of the identity matrix, corresponding to a signal plus noise model. A new test statistic based on the estimates of the eigenvalues of the different covariance matrices is proposed. In particular, we show that this statistic is consistent and with controlled type I error in the high-dimensional asymptotic regime where the sample sizes $N_1,\ldots,N_L$ of each time series and the dimension $M$ both converge to infinity at the same rate, while $K$ and $L$ are kept fixed. We also provide some simulations on synthetic and real data (SAR images) which demonstrate significant improvements over some classical methods such as the GLRT, or other alternative methods relevant for the high-dimensional regime and the low-rank model., Comment: 16 pages, preprint of the version that will appear in IEEE Transactions on Signal Processing, 2024

Interferometric phase linking (IPL) has become a prominent technique for processing images of areas containing distributed scaterrers in SAR interferometry. Traditionally, IPL consists in estimating consistent phase differences between all pairs of SAR images in a time series from the sample covariance matrix of pixel patches on a sliding window. This paper reformulates this task as a covariance fitting problem: in this setup, IPL appears as a form of projection of an input covariance matrix so that it satisfies the phase closure property. Given this modular formulation, we propose an overview of covariance matrix estimates, regularization options, and matrix distances, that can be of interest when processing multi-temporal SAR data. In particular, we will observe that most of the existing IPL algorithms appear as special instances of this framework. We then present tools to efficiently solve related optimization problems on the torus of phase-only complex vectors: majorization-minimization and Riemannian optimization. We conclude by illustrating the merits of different options on a real-world case study.

We develop the information geometry of scaled Gaussian distributions for which the covariance matrix exhibits a Kronecker product structure. This model and its geometry are then used to propose an online change detection (CD) algorithm for multivariate image times series (MITS). The proposed approach relies mainly on the online estimation of the structured covariance matrix under the null hypothesis, which is performed through a recursive (natural) Riemannian gradient descent. This approach exhibits a practical interest compared to the corresponding offline version, as its computational cost remains constant for each new image added in the time series. Simulations show that the proposed recursive estimators reach the Intrinsic Cram\'er-Rao bound. The interest of the proposed online CD approach is demonstrated on both simulated and real data.

This paper presents a new performance bound for estimation problems where the parameter to estimate lies in a Riemannian manifold (a smooth manifold endowed with a Riemannian metric) and follows a given prior distribution. In this setup, the chosen Riemannian metric induces a geometry for the parameter manifold, as well as an intrinsic notion of the estimation error measure. Performance bound for such error measure were previously obtained in the non-Bayesian case (when the unknown parameter is assumed to deterministic), and referred to as \textit{intrinsic} Cram\'er-Rao bound. The presented result then appears either as: \textit{a}) an extension of the intrinsic Cram\'er-Rao bound to the Bayesian estimation framework; \textit{b}) a generalization of the Van-Trees inequality (Bayesian Cram\'er-Rao bound) that accounts for the aforementioned geometric structures. In a second part, we leverage this formalism to study the problem of covariance matrix estimation when the data follow a Gaussian distribution, and whose covariance matrix is drawn from an inverse Wishart distribution. Performance bounds for this problem are obtained for both the mean squared error (Euclidean metric) and the natural Riemannian distance for Hermitian positive definite matrices (affine invariant metric). Numerical simulation illustrate that assessing the error with the affine invariant metric is revealing of interesting properties of the maximum a posteriori and minimum mean square error estimator, which are not observed when using the Euclidean metric.

When dealing with a parametric statistical model, a Riemannian manifold can naturally appear by endowing the parameter space with the Fisher information metric. The geometry induced on the parameters by this metric is then referred to as the Fisher-Rao information geometry. Interestingly, this yields a point of view that allows for leveragingmany tools from differential geometry. After a brief introduction about these concepts, we will present some practical uses of these geometric tools in the framework of elliptical distributions. This second part of the exposition is divided into three main axes: Riemannian optimization for covariance matrix estimation, Intrinsic Cram\'er-Rao bounds, and classification using Riemannian distances.

The detection of multiple targets in an enclosed scene, from its outside, is a challenging topic of research addressed by Through-the-Wall Radar Imaging (TWRI). Traditionally, TWRI methods operate in two steps: first the removal of wall clutter then followed by the recovery of targets positions. Recent approaches manage in parallel the processing of the wall and targets via low rank plus sparse matrix decomposition and obtain better performances. In this paper, we reformulate this precisely via a RPCA-type problem, where the sparse vector appears in a Kronecker product. We extend this approach by adding a robust distance with flexible structure to handle heterogeneous noise and outliers, which may appear in TWRI measurements. The resolution is achieved via the Alternating Direction Method of Multipliers (ADMM) and variable splitting to decouple the constraints. The removal of the front wall is achieved via a closed-form proximal evaluation and the recovery of targets is possible via a tailored Majorization-Minimization (MM) step. The analysis and validation of our method is carried out using Finite-Difference Time-Domain (FDTD) simulated data, which show the advantage of our method in detection performance over complex scenarios.

Multi-array systems are widely used in sonar and radar applications. They can improve communication speeds, target discrimination, and imaging. In the case of a multibeam sonar system that can operate two receiving arrays, we derive new adaptive to improve detection capabilities compared to traditional sonar detection approaches. To do so, we more specifically consider correlated arrays, whose covariance matrices are estimated up to scale factors, and an impulsive clutter. In a partially homogeneous environment, the 2-step Generalized Likelihood ratio Test (GLRT) and Rao approach lead to a generalization of the Adaptive Normalized Matched Filter (ANMF) test and an equivalent numerically simpler detector with a well-established texture Constant False Alarm Rate (CFAR) behavior. Performances are discussed and illustrated with theoretical examples, numerous simulations, and insights into experimental data. Results show that these detectors outperform their competitors and have stronger robustness to environmental unknowns.

This paper studies the statistical model of the non-centered mixture of scaled Gaussian distributions (NC-MSG). Using the Fisher-Rao information geometry associated to this distribution, we derive a Riemannian gradient descent algorithm. This algorithm is leveraged for two minimization problems. The first one is the minimization of a regularized negative log-likelihood (NLL). The latter makes the trade-off between a white Gaussian distribution and the NC-MSG. Conditions on the regularization are given so that the existence of a minimum to this problem is guaranteed without assumptions on the samples. Then, the Kullback-Leibler (KL) divergence between two NC-MSG is derived. This divergence enables us to define a minimization problem to compute centers of mass of several NC-MSGs. The proposed Riemannian gradient descent algorithm is leveraged to solve this second minimization problem. Numerical experiments show the good performance and the speed of the Riemannian gradient descent on the two problems. Finally, a Nearest centroid classifier is implemented leveraging the KL divergence and its associated center of mass. Applied on the large scale dataset Breizhcrops, this classifier shows good accuracies as well as robustness to rigid transformations of the test set., Comment: Accepted at IEEE Transactions on Signal Processing

This paper proposes new algorithms for the metric learning problem. We start by noticing that several classical metric learning formulations from the literature can be viewed as modified covariance matrix estimation problems. Leveraging this point of view, a general approach, called Robust Geometric Metric Learning (RGML), is then studied. This method aims at simultaneously estimating the covariance matrix of each class while shrinking them towards their (unknown) barycenter. We focus on two specific costs functions: one associated with the Gaussian likelihood (RGML Gaussian), and one with Tyler's M -estimator (RGML Tyler). In both, the barycenter is defined with the Riemannian distance, which enjoys nice properties of geodesic convexity and affine invariance. The optimization is performed using the Riemannian geometry of symmetric positive definite matrices and its submanifold of unit determinant. Finally, the performance of RGML is asserted on real datasets. Strong performance is exhibited while being robust to mislabeled data., Comment: Published in EUSIPCO 2022. Best student paper award

This paper tackles the problem of robust covariance matrix estimation when the data is incomplete. Classical statistical estimation methodologies are usually built upon the Gaussian assumption, whereas existing robust estimation ones assume unstructured signal models. The former can be inaccurate in real-world data sets in which heterogeneity causes heavy-tail distributions, while the latter does not profit from the usual low-rank structure of the signal. Taking advantage of both worlds, a covariance matrix estimation procedure is designed on a robust (mixture of scaled Gaussian) low-rank model by leveraging the observed-data likelihood function within an expectation-maximization algorithm. It is also designed to handle general pattern of missing values. The proposed procedure is first validated on simulated data sets. Then, its interest for classification and clustering applications is assessed on two real data sets with missing values, which include multispectral and hyperspectral time series., Comment: 35 pages, 10 figures

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., Comment: 13 pages

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 Cram\'er-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 Cram\'er-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 addresses the problem of deriving the asymptotic performance of adaptive Low Rank (LR) filters used in target detection embedded in a disturbance composed of a LR Gaussian noise plus a white Gaussian noise. In this context, we use the Signal to Interference to Noise Ratio (SINR) loss as performance measure which is a function of the estimated projector onto the LR noise subspace. However, although the SINR loss can be determined through Monte-Carlo simulations or real data, this process remains quite time consuming. Thus, this paper proposes to predict the SINR loss behavior in order to not depend on the data anymore and be quicker. To derive this theoretical result, previous works used a restrictive hypothesis assuming that the target is orthogonal to the LR noise. In this paper, we propose to derive this theoretical performance by relaxing this hypothesis and using Random Matrix Theory (RMT) tools. These tools will be used to present the convergences of simple quadratic forms and perform new RMT convergences of structured quadratic forms and SINR loss in the large dimensional regime, i.e. the size and the number of the data tend to infinity at the same rate. We show through simulations the interest of our approach compared to the previous works when the restrictive hypothesis is no longer verified.