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In the domain of Mobility Data Science, the intricate task of interpreting models trained on trajectory data, and elucidating the spatio-temporal movement of entities, has persistently posed significant challenges. Conventional XAI techniques, although brimming with potential, frequently overlook the distinct structure and nuances inherent within trajectory data. Observing this deficiency, we introduced a comprehensive framework that harmonizes pivotal XAI techniques: LIME (Local Interpretable Model-agnostic Explanations), SHAP (SHapley Additive exPlanations), Saliency maps, attention mechanisms, direct trajectory visualization, and Permutation Feature Importance (PFI). Unlike conventional strategies that deploy these methods singularly, our unified approach capitalizes on the collective efficacy of these techniques, yielding deeper and more granular insights for models reliant on trajectory data. In crafting this synthesis, we effectively address the multifaceted essence of trajectories, achieving not only amplified interpretability but also a nuanced, contextually rich comprehension of model decisions. To validate and enhance our framework, we undertook a survey to gauge preferences and reception among various user demographics. Our findings underscored a dichotomy: professionals with academic orientations, particularly those in roles like Data Scientist, IT Expert, and ML Engineer, showcased a profound, technical understanding and often exhibited a predilection for amalgamated methods for interpretability. Conversely, end-users or individuals less acquainted with AI and Data Science showcased simpler inclinations, such as bar plots indicating timestep significance or visual depictions pinpointing pivotal segments of a vessel's trajectory.

In many practical control applications, the performance level of a closed-loop system degrades over time due to the change of plant characteristics. Thus, there is a strong need for redesigning a controller without going through the system modeling process, which is often difficult for closed-loop systems. Reinforcement learning (RL) is one of the promising approaches that enable model-free redesign of optimal controllers for nonlinear dynamical systems based only on the measurement of the closed-loop system. However, the learning process of RL usually requires a considerable number of trial-and-error experiments using the poorly controlled system that may accumulate wear on the plant. To overcome this limitation, we propose a model-free two-step design approach that improves the transient learning performance of RL in an optimal regulator redesign problem for unknown nonlinear systems. Specifically, we first design a linear control law that attains some degree of control performance in a model-free manner, and then, train the nonlinear optimal control law with online RL by using the designed linear control law in parallel. We introduce an offline RL algorithm for the design of the linear control law and theoretically guarantee its convergence to the LQR controller under mild assumptions. Numerical simulations show that the proposed approach improves the transient learning performance and efficiency in hyperparameter tuning of RL.

Many approaches have been proposed to use diffusion models to augment training datasets for downstream tasks, such as classification. However, diffusion models are themselves trained on large datasets, often with noisy annotations, and it remains an open question to which extent these models contribute to downstream classification performance. In particular, it remains unclear if they generalize enough to improve over directly using the additional data of their pre-training process for augmentation. We systematically evaluate a range of existing methods to generate images from diffusion models and study new extensions to assess their benefit for data augmentation. Personalizing diffusion models towards the target data outperforms simpler prompting strategies. However, using the pre-training data of the diffusion model alone, via a simple nearest-neighbor retrieval procedure, leads to even stronger downstream performance. Our study explores the potential of diffusion models in generating new training data, and surprisingly finds that these sophisticated models are not yet able to beat a simple and strong image retrieval baseline on simple downstream vision tasks.

High-dimensional, higher-order tensor data are gaining prominence in a variety of fields, including but not limited to computer vision and network analysis. Tensor factor models, induced from noisy versions of tensor decomposition or factorization, are natural potent instruments to study a collection of tensor-variate objects that may be dependent or independent. However, it is still in the early stage of developing statistical inferential theories for estimation of various low-rank structures, which are customary to play the role of signals of tensor factor models. In this paper, starting from tensor matricization, we aim to ``decode" estimation of a higher-order tensor factor model in the sense that, we recast it into mode-wise traditional high-dimensional vector/fiber factor models so as to deploy the conventional estimation of principle components analysis (PCA). Demonstrated by the Tucker tensor factor model (TuTFaM), which is induced from most popular Tucker decomposition, we summarize that estimations on signal components are essentially mode-wise PCA techniques, and the involvement of projection and iteration will enhance the signal-to-noise ratio to various extend. We establish the inferential theory of the proposed estimations and conduct rich simulation experiments under TuTFaM, and illustrate how the proposed estimations can work in tensor reconstruction, clustering for video and economic datasets, respectively.

In inverse scattering problems, a model that allows for the simultaneous recovery of both the domain shape and an impedance boundary condition covers a wide range of problems with impenetrable domains, including recovering the shape of sound-hard and sound-soft obstacles and obstacles with thin coatings. This work develops an optimization framework for recovering the shape and material parameters of a penetrable, dissipative obstacle in the multifrequency setting, using a constrained class of curvature-dependent impedance function models proposed by Antoine, Barucq, and Vernhet. We find that this constrained model improves the robustness of the recovery problem, compared to more general models, and provides meaningfully better obstacle recovery than simpler models. We explore the effectiveness of the model for varying levels of dissipation, for noise-corrupted data, and for limited aperture data in the numerical examples.

Evaluating the reliability of complex technical networks, such as those in energy distribution, logistics, and transportation systems, is of paramount importance. These networks are often represented as multistate flow networks (MFNs). While there has been considerable research on assessing MFN reliability, many studies still need to pay more attention to a critical factor: transmission distance constraints. These constraints are typical in real-world applications, such as Internet infrastructure, where controlling the distances between data centers, network nodes, and end-users is vital for ensuring low latency and efficient data transmission. This paper addresses the evaluation of MFN reliability under distance constraints. Specifically, it focuses on determining the probability that a minimum of $d$ flow units can be transmitted successfully from a source node to a sink node, using only paths with lengths not exceeding a predefined distance limit of $\lambda $. We introduce an effective algorithm to tackle this challenge, provide a benchmark example to illustrate its application and analyze its computational complexity.

We study the numerical approximation of multidimensional stochastic differential equations (SDEs) with distributional drift, driven by a fractional Brownian motion. We work under the Catellier-Gubinelli condition for strong well-posedness, which assumes that the regularity of the drift is strictly greater than $1-1/(2H)$, where $H$ is the Hurst parameter of the noise. The focus here is on the case $H<1/2$, allowing the drift $b$ to be a distribution. We compare the solution $X$ of the SDE with drift $b$ and its tamed Euler scheme with mollified drift $b^n$, to obtain an explicit rate of convergence for the strong error. This extends previous results where $b$ was assumed to be a bounded measurable function. In addition, we investigate the limit case when the regularity of the drift is equal to $1-1/(2H)$, and obtain a non-explicit rate of convergence. As a byproduct of this convergence, there exists a strong solution that is pathwise unique in a class of H\"older continuous solutions. The proofs rely on stochastic sewing techniques, especially to deduce new regularising properties of the discrete-time fractional Brownian motion. In the limit case, we introduce a critical Gr\"onwall-type lemma to quantify the error. We also present several examples and numerical simulations that illustrate our results.

We prove closed-form equations for the exact high-dimensional asymptotics of a family of first order gradient-based methods, learning an estimator (e.g. M-estimator, shallow neural network, ...) from observations on Gaussian data with empirical risk minimization. This includes widely used algorithms such as stochastic gradient descent (SGD) or Nesterov acceleration. The obtained equations match those resulting from the discretization of dynamical mean-field theory (DMFT) equations from statistical physics when applied to gradient flow. Our proof method allows us to give an explicit description of how memory kernels build up in the effective dynamics, and to include non-separable update functions, allowing datasets with non-identity covariance matrices. Finally, we provide numerical implementations of the equations for SGD with generic extensive batch-size and with constant learning rates.

Many generalised distributions exist for modelling data with vastly diverse characteristics. However, very few of these generalisations of the normal distribution have shape parameters with clear roles that determine, for instance, skewness and tail shape. In this chapter, we review existing skewing mechanisms and their properties in detail. Using the knowledge acquired, we add a skewness parameter to the body-tail generalised normal distribution \cite{BTGN}, that yields the \ac{FIN} with parameters for location, scale, body-shape, skewness, and tail weight. Basic statistical properties of the \ac{FIN} are provided, such as the \ac{PDF}, cumulative distribution function, moments, and likelihood equations. Additionally, the \ac{FIN} \ac{PDF} is extended to a multivariate setting using a student t-copula, yielding the \ac{MFIN}. The \ac{MFIN} is applied to stock returns data, where it outperforms the t-copula multivariate generalised hyperbolic, Azzalini skew-t, hyperbolic, and normal inverse Gaussian distributions.

We study the optimal sample complexity of neighbourhood selection in linear structural equation models, and compare this to best subset selection (BSS) for linear models under general design. We show by example that -- even when the structure is \emph{unknown} -- the existence of underlying structure can reduce the sample complexity of neighbourhood selection. This result is complicated by the possibility of path cancellation, which we study in detail, and show that improvements are still possible in the presence of path cancellation. Finally, we support these theoretical observations with experiments. The proof introduces a modified BSS estimator, called klBSS, and compares its performance to BSS. The analysis of klBSS may also be of independent interest since it applies to arbitrary structured models, not necessarily those induced by a structural equation model. Our results have implications for structure learning in graphical models, which often relies on neighbourhood selection as a subroutine.