Sparse regularisation has proven its worth and effectiveness in many fields, such as medical imaging. In this sense, nuclear magnetic resonance spectroscopy (MRS) is one of the modalities that could greatly benefit from sparse regularisation. This paper introduces a novel Bayesian approach for MRS restoration that accounts for possible errors in the observation linear operator. The algorithm is tailored to the complex nature of MRS data, incorporating both real and imaginary parts of the spectrum. An MCMC (Markov chain Monte Carlo) inference is conducted using a Gibbs sampler strategy. The method has been successfully validated on both synthetic and clinical data of high-grade brain tumor glioblastoma (GBM) patients. This study will enable further analysis of metabolites of interest not conventionally considered in clinics because of their undetectable concentration.

The estimation of high resolution spectrometer Instrument Spectral Response Functions (ISRFs) is crucial because an imperfect knowledge of these functions can induce errors in the measurements. The state-of-the-art for this problem currently relies on the use of parametric models, which frequently lack flexibility to accurately model real-world ISRFs. To address this limitation, this paper proposes and investigates the use of sparse representations for modeling and estimating ISRFs, where the ISRFs are decomposed in a fixed dictionary of atoms. To estimate the sparse coefficient vector, a novel sparsity inducing regularization of the problem based on quadratic envelopes is studied and compared to the classical LASSO estimator and to a greedy method based on the Orthogonal Matching Pursuit (OMP) algorithm. Results for simulated ISRFs from the MicroCarb mission indicate that the proposed spectral representations yield excellent ISRF estimates, and that the use of quadratic envelopes can yield significantly better precision than competing methods.

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

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.

High resolution spectrometers are composed of different optical elements and detectors that must be modeled as accurately as possible. Specifically, accurate estimates of Instrument Spectral Response Functions (ISRFs) are critical in order not to compromise the retrieval of trace gas concentrations from spectral measurements. Currently, parametric models are used to estimate these response functions. However, these models cannot always take into account the diversity of ISRF shapes that are encountered in practical applications. This paper studies a new ISRF estimation method based on a sparse representation of the ISRF in a dictionary. The proposed method is shown to be very competitive when compared to parametric models, yielding up to one order of magnitude smaller normalized ISRF estimation errors. The method is applied to different high-resolution spectrometers, demonstrating its reproducibility for multiple remote sensing missions.

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.

In this Ph.D. thesis, we explore machine learning-based solutions for channel decoding in Machine-to-Machine type communications, where achieving ultra-reliable lowlatency communications (URLLC) is essential. Their primary issue arises from the exponential growth in the decoder’s complexity as the packet size increases. This curse of dimensionality manifests itself in three different aspects: i) the number of correctable noise patterns, ii) the codeword space to be explored, and iii) the number of trainable parameters in the models. To address the first limitation, we explore solutions based on a Support Vector Machine (SVM) framework and suggest a bitwise SVM approach that significantly reduces the complexity of existing SVM-based solutions. To tackle the second limitation, we investigate syndromebased neural decoders and introduce a novel message-oriented decoder, which improves on existing schemes both in the decoder architecture and in the choice of the parity check matrix. Regarding the neural network size, we develop a recurrent version of a transformer-based decoder, which reduces the number of parameters while maintaining efficiency, compared to previous neural-based solutions. Lastly, we extend the proposed decoder to support higherorder modulations through Bit-Interleaved and generic Coded Modulations (BICM and CM, respectively), aiding its application in more realistic communication environments.

In this Ph.D. thesis, we explore machine learning-based solutions for channel decoding in Machine-to-Machine type communications, where achieving ultra-reliable lowlatency communications (URLLC) is essential. Their primary issue arises from the exponential growth in the decoder’s complexity as the packet size increases. This curse of dimensionality manifests itself in three different aspects: i) the number of correctable noise patterns, ii) the codeword space to be explored, and iii) the number of trainable parameters in the models. To address the first limitation, we explore solutions based on a Support Vector Machine (SVM) framework and suggest a bitwise SVM approach that significantly reduces the complexity of existing SVM-based solutions. To tackle the second limitation, we investigate syndromebased neural decoders and introduce a novel message-oriented decoder, which improves on existing schemes both in the decoder architecture and in the choice of the parity check matrix. Regarding the neural network size, we develop a recurrent version of a transformer-based decoder, which reduces the number of parameters while maintaining efficiency, compared to previous neural-based solutions. Lastly, we extend the proposed decoder to support higherorder modulations through Bit-Interleaved and generic Coded Modulations (BICM and CM, respectively), aiding its application in more realistic communication environments.