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.

Multifractal analysis (MFA) provides a framework for the global characterization of image textures by describing the spatial fluctuations of their local regularity based on the multifractal spectrum. Several works have shown the interest of using MFA for the description of homogeneous textures in images. Nevertheless, natural images can be composed of several textures and, in turn, multifractal properties associated with those textures. This paper introduces an unsupervised Bayesian multifractal segmentation method to model and segment multifractal textures by jointly estimating the multifractal parameters and labels on images, at the pixel-level. For this, a computationally and statistically efficient multifractal parameter estimation model for wavelet leaders is firstly developed, defining different multifractality parameters for different regions of an image. Then, a multiscale Potts Markov random field is introduced as a prior to model the inherent spatial and scale correlations (referred to as cross-scale correlations) between the labels of the wavelet leaders. A Gibbs sampling methodology is finally used to draw samples from the posterior distribution of the unknown model parameters. Numerical experiments are conducted on synthetic multifractal images to evaluate the performance of the proposed segmentation approach. The proposed method achieves superior performance compared to traditional unsupervised segmentation techniques as well as modern deep learning-based approaches, showing its effectiveness for multifractal image segmentation.

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.

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.