arXiv subjects

Alexandre Renaux

Publications and source records attributed to Alexandre Renaux.

13 recordsLinked to original sources

arXiv2025

Power-Aware Cognitive Radar Multi-target Tracking Under Unknown Disturbances

This work presents a cognitive radar (CR) framework designed to track multiple aircraft under unknown disturbances using massive multiple-input multiple-output (MMIMO) systems. Since uniform power allocation is suboptimal across varying signal-to-noise ratios (SNRs), we couple an adaptive waveform design driven by Partially Observable Monte Carlo Planning (POMCP). By assigning an independent POMCP tree to each target, the system efficiently predicts target states. These predictions inform a constrained optimization problem that actively directs transmit energy toward weaker targets while maintaining sufficient power for stronger ones. Results confirm that the proposed POMCP method improves the detection probability for low-SNR targets from 0.6 to nearly 0.9, and yields more accurate tracking of the weakest target than a non-adaptive orthogonal waveform or a cognitive uniform-power POMCP baseline.

eess.SP↗

arXiv2024

POMDP-Driven Cognitive Massive MIMO Radar: Joint Target Detection-Tracking In Unknown Disturbances

The joint detection and tracking of a moving target embedded in an unknown disturbance represents a key feature that motivates the development of the cognitive radar paradigm. Building upon recent advancements in robust target detection with multiple-input multiple-output (MIMO) radars, this work explores the application of a Partially Observable Markov Decision Process (POMDP) framework to enhance the tracking and detection tasks in a statistically unknown environment. In the POMDP setup, the radar system is considered as an intelligent agent that continuously senses the surrounding environment, optimizing its actions to maximize the probability of detection $(P_D)$ and improve the target position and velocity estimation, all this while keeping a constant probability of false alarm $(P_{FA})$. The proposed approach employs an online algorithm that does not require any apriori knowledge of the noise statistics, and it relies on a much more general observation model than the traditional range-azimuth-elevation model employed by conventional tracking algorithms. Simulation results clearly show substantial performance improvement of the POMDP-based algorithm compared to the State-Action-Reward-State-Action (SARSA)-based one that has been recently investigated in the context of massive MIMO (MMIMO) radar systems.

cs.LG↗

arXiv2023

Intrinsic Bayesian Cram\'er-Rao Bound with an Application to Covariance Matrix Estimation

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.

math.ST↗

arXiv2023

The Fisher-Rao geometry of CES distributions

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.

stat.ML↗

arXiv2021

Joint Estimation of Location and Scatter in Complex Elliptical Distributions: A robust semiparametric and computationally efficient $R$-estimator of the shape matrix

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).

stat.ME↗

arXiv2020

Robust Semiparametric DOA Estimation in non-Gaussian Environment

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).

eess.SP↗

arXiv2020

Properties of a new $R$-estimator of shape matrices

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.

eess.SP↗

arXiv2020

Robust Semiparametric Efficient Estimators in Elliptical Distributions

Covariance matrices play a major role in statistics, signal processing and machine learning applications. This paper focuses on the \textit{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 \textit{robustness} and (semiparametric) \textit{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 \textit{one-step efficient estimators} and the \textit{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.

eess.SP↗

arXiv2020

A Riemannian Framework for Low-Rank Structured Elliptical Models

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.

math.DG↗

arXiv2016

A Bayesian Lower Bound for Parameter Estimation of Poisson Data Including Multiple Changes (extended)

This paper derives lower bounds for the mean square errors of parameter estimators in the case of Poisson distributed data subjected to multiple abrupt changes. Since both change locations (discrete parameters) and parameters of the Poisson distribution (continuous parameters) are unknown, it is appropriate to consider a mixed Cramer-Rao/Weiss-Weinstein bound for which we derive closed-form expressions and illustrate its tightness by numerical simulations.

cs.IT↗

arXiv2016

A class of Weiss-Weinstein bounds and its relationship with the Bobrovsky-Mayer-Wolf-Zakai bounds

A fairly general class of Bayesian "large-error" lower bounds of the Weiss-Weinstein family, essentially free from regularity conditions on the probability density functions support, and for which a limiting form yields a generalized Bayesian Cramer-Rao bound (BCRB), is introduced. In a large number of cases, the generalized BCRB appears to be the Bobrovsky-Mayer-Wolf-Zakai bound (BMZB). Interestingly enough, a regularized form of the Bobrovsky-Zakai bound (BZB), applicable when the support of the prior is a constrained parameter set, is obtained. Modified Weiss-Weinstein bound and BZB which limiting form is the BMZB are proposed, in expectation of an increased tightness in the threshold region. Some of the proposed results are exemplified with a reference problem in signal processing: the Gaussian observation model with parameterized mean and uniform prior.

cs.IT↗

arXiv2013

Coexistence of Near-Field and Far-Field Sources: the Angular Resolution Limit

Passive source localization is a well known inverse problem in which we convert the observed measurements into information about the direction of arrivals. In this paper we focus on the optimal resolution of such problem. More precisely, we propose in this contribution to derive and analyze the Angular Resolution Limit (ARL) for the scenario of mixed Near-Field (NF) and Far-Field (FF) Sources. This scenario is relevant to some realistic situations. We base our analysis on the Smith's equation which involves the Cramér-Rao Bound (CRB). This equation provides the theoretical ARL which is independent of a specific estimator. Our methodology is the following: first, we derive a closed-form expression of the CRB for the considered problem. Using these expressions, we can rewrite the Smith's equation as a 4-th order polynomial by assuming a small separation of the sources. Finally, we derive in closed-form the analytic ARL under or not the assumption of low noise variance. The obtained expression is compact and can provide useful qualitative informations on the behavior of the ARL.

cs.IT↗

arXiv2012

Some results on the Weiss-Weinstein bound for conditional and unconditional signal models in array processing

In this paper, the Weiss-Weinstein bound is analyzed in the context of sources localization with a planar array of sensors. Both conditional and unconditional source signal models are studied. First, some results are given in the multiple sources context without specifying the structure of the steering matrix and of the noise covariance matrix. Moreover, the case of an uniform or Gaussian prior are analyzed. Second, these results are applied to the particular case of a single source for two kinds of array geometries: a non-uniform linear array (elevation only) and an arbitrary planar (azimuth and elevation) array.

cs.IT↗