Providing non-conservative uncertainty quantification for function estimates derived from noisy observations remains a fundamental challenge in statistical machine learning, particularly for applications in safety-critical domains. In this work, we propose novel non-asymptotic probabilistic uniform error bounds for kernel-based regression. Compared to related bounds in the literature that are restricted to (conditionally) independent sub-Gaussian noise, our bounds allow to consider a broad class of non-Gaussian distributions, such as sub-Gaussian, bounded, sub-exponential, and variance/moment-bounded noise. Moreover, our results apply to correlated and uncorrelated noise. We compare our proposed error bounds with existing results in terms of the induced uncertainty region and their performance in safe control, demonstrating the tightness of the proposed bounds.

We propose GaussDetect-LiNGAM, a novel approach for bivariate causal discovery that eliminates the need for explicit Gaussianity tests by leveraging a fundamental equivalence between noise Gaussianity and residual independence in the reverse regression. Under the standard LiNGAM assumptions of linearity, acyclicity, and exogeneity, we prove that the Gaussianity of the forward-model noise is equivalent to the independence between the regressor and residual in the reverse model. This theoretical insight allows us to replace fragile and sample-sensitive Gaussianity tests with robust kernel-based independence tests. Experimental results validate the equivalence and demonstrate that GaussDetect-LiNGAM maintains high consistency across diverse noise types and sample sizes, while reducing the number of tests per decision (TPD). Our method enhances both the efficiency and practical applicability of causal inference, making LiNGAM more accessible and reliable in real-world scenarios.

Sliding window-factor graph optimization (SW-FGO) has gained more and more attention in navigation research due to its robust approximation to non-Gaussian noises and nonlinearity of measuring models. There are lots of works focusing on its application performance compared to extended Kalman filter (EKF) but there is still a myth at the theoretical relationship between the SW-FGO and EKF. In this paper, we find the necessarily fair condition to connect SW-FGO and Kalman filter variants (KFV) (e.g., EKF, iterative EKF (IEKF), robust EKF (REKF) and robust iterative EKF (RIEKF)). Based on the conditions, we propose a recursive FGO (Re-FGO) framework to represent KFV under SW-FGO formulation. Under explicit conditions (Markov assumption, Gaussian noise with L2 loss, and a one-state window), Re-FGO regenerates exactly to EKF/IEKF/REKF/RIEKF, while SW-FGO shows measurable benefits in nonlinear, non-Gaussian regimes at a predictable compute cost. Finally, after clarifying the connection between them, we highlight the unique advantages of SW-FGO in practical phases, especially on numerical estimation and deep learning integration. The code and data used in this work is open sourced at https://github.com/Baoshan-Song/KFV-FGO-Comparison.

State estimation remains a fundamental challenge across numerous domains, from autonomous driving, aircraft tracking to quantum system control. Although Bayesian filtering has been the cornerstone solution, its classical model-based paradigm faces two major limitations: it struggles with inaccurate state space model (SSM) and requires extensive prior knowledge of noise characteristics. We present TrackDiffuser, a generative framework addressing both challenges by reformulating Bayesian filtering as a conditional diffusion model. Our approach implicitly learns system dynamics from data to mitigate the effects of inaccurate SSM, while simultaneously circumventing the need for explicit measurement models and noise priors by establishing a direct relationship between measurements and states. Through an implicit predict-and-update mechanism, TrackDiffuser preserves the interpretability advantage of traditional model-based filtering methods. Extensive experiments demonstrate that our framework substantially outperforms both classical and contemporary hybrid methods, especially in challenging non-linear scenarios involving non-Gaussian noises. Notably, TrackDiffuser exhibits remarkable robustness to SSM inaccuracies, offering a practical solution for real-world state estimation problems where perfect models and prior knowledge are unavailable.

Weaknesses: - Some of the points I'll make here are more conceptual and would just like to hear from the authors what their thoughts are. However, there is another school of thought related to the Nonparanormal distribution that says everything can be transformed into something that looks Gaussian. In practice, either in applications or real-world analyses the authors have undertaken, what has been their experience in the usage of Gaussian vs non-Gaussian methods for structure learning. Is the first method they recommend a non-Gaussian method or one that relies on Gaussianity assumptions? In that sense, it is closer to algorithms like PC/FCI that are in theory nonparametric.

Safety-critical cyber-physical systems (CPS), such as quadrotor UAVs, are particularly prone to cyber attacks, which can result in significant consequences if not detected promptly and accurately. During outdoor operations, the nonlinear dynamics of UAV systems, combined with non-Gaussian noise, pose challenges to the effectiveness of conventional statistical and machine learning methods. To overcome these limitations, we present QUADFormer, an advanced attack detection framework for quadrotor UAVs leveraging a transformer-based architecture. This framework features a residue generator that produces sequences sensitive to anomalies, which are then analyzed by the transformer to capture statistical patterns for detection and classification. Furthermore, an alert mechanism ensures UAVs can operate safely even when under attack. Extensive simulations and experimental evaluations highlight that QUADFormer outperforms existing state-of-the-art techniques in detection accuracy.

Physics-informed neural networks (PINNs) constitute a flexible approach to both finding solutions and identifying parameters of partial differential equations. Most works on the topic assume noiseless data, or data contaminated with weak Gaussian noise. We show that the standard PINN framework breaks down in case of non-Gaussian noise. We give a way of resolving this fundamental issue and we propose to jointly train an energy-based model (EBM) to learn the correct noise distribution. We illustrate the improved performance of our approach using multiple examples.

The present study proposes incorporating non-parametric knowledge into the diffusion least-mean-squares algorithm in the framework of a maximum a posteriori (MAP) estimation. The proposed algorithm leads to a robust estimation of an unknown parameter vector in a group of cooperative estimators. Utilizing kernel density estimation and buffering some intermediate estimations, the prior distribution and conditional likelihood of the parameters vector in each node are calculated. Pseudo Huber loss function is used for designing the likelihood function. Also, an error thresholding function is defined to reduce the computational overhead as well as more relaxation against noise, which stops the update every time an error is less than a predefined threshold. The performance of the proposed algorithm is examined in the stationary and non-stationary scenarios in the presence of Gaussian and non-Gaussian noise. Results show the robustness of the proposed algorithm in the presence of different noise types.

The connection between diffusion processes and homogeneous Markov chains has been investigated for a long time Skorokhod [1963]. If we need to approximate the given diffusion by some homogeneous Markov chain, it is easy to realize because we are free to construct the chain nicely, meaning that we can choose terms and properties of MC, e.g., as it was shown in Raginsky et al. [2017]. However, often the inverse problem arises, namely, we have the a priori given chain, and the goal is to study it via the corresponding diffusion approximation. This task is an increasingly popular and hot research topic. Indeed, it is used to investigate different sampling techniques Orvieto and Lucchi [2018], to describe the behavior of optimization methods Raginsky et al. [2017] and to understand the convergence of boosting algorithms Ustimenko and Prokhorenkova [2021]. From practical experience, the given Markov chain may not have good properties that are easy to analyze in theory. Thus, the aim of our work is to study when diffusion approximation holds for as broad as the possible class of homogeneous Markov chains, i.e., we want to consider the maximally general chain and place the broadest possible assumptions on it whilst obtaining diffusion approximation guarantee.

A nonlinearity is required before matched filtering in mInimum error receivers when additive noise is present which is impulsive and highly non-Gaussian. Experiments were performed to determine whether the correct clipping nonlinearity could be provided by a single-input single(cid:173) output multi-layer perceptron trained with back propagation. It was found that a multi-layer perceptron with one input and output node, 20 nodes in the first hidden layer, and 5 nodes in the second hidden layer could be trained to provide a clipping nonlinearity with fewer than 5,000 presentations of noiseless and corrupted waveform samples. A network trained at a relatively high signal-to-noise (SIN) ratio and then used as a front end for a linear matched filter detector greatly reduced the probability of error. The clipping nonlinearity formed by this network was similar to that used in current receivers designed for impulsive noise and provided similar substantial improvements in performance.