This article presents a comprehensive review of recent advances in intrinsic Cramér-Rao bounds (ICRBs) for Lie groups (LGs), which play a pivotal role in addressing estimation problems involving parameters and/or observations constrained by geometric structures. The review encompasses both deterministic and Bayesian frameworks, with a detailed examination of their formulation, derivation, and theoretical foundations. Furthermore, we underscore significant theoretical contributions and extend the discussion to practical estimation challenges, offering insights into their applicability. Emphasis is placed on methodologies for validating these bounds, providing a robust framework for performance evaluation across a variety of estimation problems in engineering and applied sciences.

Diffusion models have emerged as a powerful tool in generative modeling, demonstrating remarkable capabilities in synthesizing high-fidelity data across various domains. These models transform an initial simple distribution into a more complex one through a denoising process, making them particularly effective for generating detailed and realistic data. In this seminar, we will explore how to integrate diffusion models within a mathematical framework to solve inverse problems, that is, to reconstruct data from partial observations. By leveraging diffusion models as prior knowledge of the data, we introduce a new approach that enables precise generation of conditional data under various noise and artifact conditions. We validate our approach through extensive experiments using various public image datasets, demonstrating its versatility and effectiveness. Furthermore, we demonstrate the practical implications of our method by applying it to reconstruct electrocardiograms (ECGs), where it enhances the quality and reliability of ECG signals, paving the way for broader applications in medical diagnostics and real-time health monitoring.

Cette présentation a pour but d’introduire le radar météorologique et de présenter son fonctionnement global.

In this talk, Colin Cros will focus on the problem of cooperative positioning in the context of GNSS (Global Navigation Satellite Systems). The presentation is divided into two parts. The first examines the solvability of the problem from a theoretical point of view, where the specificity comes from the type of measurements made: pseudo-distances. The approach adopted is based on a study of the measurement graph and the theory of rigidity. The second part deals with practical aspects, presenting how to integrate a cooperative measurement into a Kalman-type navigation filter. The difficulty arises from the lack of knowledge of the correlations between the agents' errors, which means that so-called conservative filters have to be used. This presentation is based on my doctoral thesis, which is available at: https://theses.fr/2024GRALT032

Fuzz testing is a method used in software testing that involves inputting random or unexpected data into a system to identify vulnerabilities. Unlike deterministic methods, which test performance under controlled and predictable conditions, fuzz testing introduces variability to uncover hidden issues. This variability simulates real-world scenarios, uncovering weaknesses that might otherwise remain unnoticed. For instance, fuzz testing can effectively reveal how GNSS receivers respond to rapid signal fluctuations and other anomalous behaviors, situations often overlooked by standard tests. Unlike traditional methods that rely on predefined inputs, Collins Aerospace works on a new fuzz testing framework for GNSS, which employs advanced techniques such as automated input generation and real-time response monitoring. This approach not only facilitates a comprehensive assessment of receiver resilience but also allows for the dynamic adaptation of test scenarios in real-time, ensuring that a wide range of operational conditions is explored. The navigation equipment minimum testing procedures must be defined and need scenarios definitions as well as test steps and pass/fail criteria to provide minimum guidance to manufacturers for future equipment certification. The limitations of current testing methods further highlight the necessity of adopting fuzz testing. These methods predominantly rely on deterministic approaches, which do not effectively simulate the unpredictable nature of real-world signal degradation or complex interference scenarios posed by advanced spoofing techniques. As technology advances, the techniques utilized by malevolent actors likewise evolve, emphasizing the necessity for adaptive testing methodologies capable of responding to these changes. By introducing randomness and variability, fuzz testing plays a critical role in bolstering the reliability and operational integrity of GNSS systems by rigorously assessing their ability to withstand both known and unknown threats. The anticipated results from this fuzz testing framework are expected to identify vulnerabilities and enhance the resilience of GNSS receivers, suggesting that fuzz testing can play a transformative role in GNSS validation.

If P1, . . . , Pn and Q1, . . . , Qn are probability measures on Rd and P1 ∗ · · · ∗ Pn and Q1 ∗ · · · ∗ Qn are their respective convolutions, the Rényi divergence Dλ of order λ ∈ (0, 1] satisfies Dλ(P1 ∗ · · · ∗ Pn||Q1 ∗ · · · ∗ Qn) ≤ ni=1 Dλ(Pi ||Qi ). When Pi belongs to the natural exponential family generated by Qi , with the same natural parameter θ for any i = 1, . . . , n, the equality sign holds. The present note tackles the inverse problem, namely “does the equality Dλ(P1 ∗ · · · ∗ Pn||Q1 ∗ · · · ∗ Qn) = ni=1 Dλ(Pi ||Qi ) imply that Pi belongs to the natural exponential family generated by Qi for every i = 1, . . . , n?” The answer is not always positive and depends on the set of solutions of a generalization of the celebrated Cauchy functional equation. We discuss in particular the case P1 = · · · = Pn = P and Q1 = · · · = Qn = Q, with n = 2 and n = ∞, the latter meaning that the equality holds for all n. Our analysis is mainly devoted to P and Q concentrated on non-negative integers, and P and Q with densities with respect to the Lebesgue measure. The results cover the Kullback– Leibler divergence (KL), this being the Rényi divergence for λ = 1. We also show that the only f -divergences such that Df (P∗2||Q∗2) = 2Df (P||Q), for P and Q in the same exponential family, are mixtures of KL divergence and its dual.

This article addresses the problem of computing a Cramér-Rao bound when the likelihood of Euclidean observations is parameterized by both unknown Lie group (LG) parameters and covariance matrix. To achieve this goal, we leverage the LG structure of the space of positive definite matrices. In this way, we can assemble a global LG parameter that lies on the product of the two groups, on which LG's intrinsic tools can be applied. From this, we derive an inequality on the intrinsic error, which can be seen as the equivalent of the Slepian-Bangs formula on LGs. Subsequently, we obtain a closed-form expression of this formula for Euclidean observations. The proposed bound is computed and implemented on two real-world problems involving observations lying in $\mathbb{R}^{p}$, dependent on an unknown LG parameter and an unknown noise covariance matrix: the Wahba's estimation problem on $SE(3)$, and the inference of the pose in $SE(3)$ of a camera from pixel detections.

Nonlinear regression plays a crucial role in various engineering applications. For the sake of mathematical tractability and ease of implementation, most of the existing inference procedures are derived under the assumption of independent and identically distributed (i.i.d.) Gaussian-distributed data. However, real-world situations often deviate from this assumption, with the true data generating process being a correlated, heavy-tailed and non-Gaussian one. The paper aims at providing the Misspecified Cramér–Rao Bound (MCRB) on the Mean Squared Error (MSE) of any unbiased (in a proper sense) estimator of the parameters of a nonlinear regression model derived under the i.i.d. Gaussian assumption in the place of the actual correlated, non-Gaussian data generating process. As a special case, the MCRB for an uncorrelated, i.i.d. Complex Elliptically Symmetric (CES) data generating process under Gaussian assumption is also provided. Consistency and asymptotic normality of the related Mismatched Maximum Likelihood Estimator (MMLE) will be discussed along with its connection with the Nonlinear Least Square Estimator (NLLSE) inherent to the nonlinear regression model. Finally, the derived theoretical findings will be applied in the well-known problem of time-delay and Doppler estimation for GNSS.