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Graphical models and factor analysis are well-established tools in multivariate statistics. While these models can be both linked to structures exhibited by covariance and precision matrices, they are generally not jointly leveraged within graph learning processes. This paper therefore addresses this issue by proposing a flexible algorithmic framework for graph learning under low-rank structural constraints on the covariance matrix. The problem is expressed as penalized maximum likelihood estimation of an elliptical distribution (a generalization of Gaussian graphical models to possibly heavy-tailed distributions), where the covariance matrix is optionally constrained to be structured as low-rank plus diagonal (low-rank factor model). The resolution of this class of problems is then tackled with Riemannian optimization, where we leverage geometries of positive definite matrices and positive semi-definite matrices of fixed rank that are well suited to elliptical models. Numerical experiments on real-world data sets illustrate the effectiveness of the proposed approach.

Distributionally Robust Optimal Control (DROC) is a technique that enables robust control in a stochastic setting when the true distribution is not known. Traditional DROC approaches require given ambiguity sets or a KL divergence bound to represent the distributional uncertainty. These may not be known a priori and may require hand-crafting. In this paper, we lift this assumption by introducing a data-driven technique for estimating the uncertainty and a bound for the KL divergence. We call this technique D3ROC. To evaluate the effectiveness of our approach, we consider a navigation problem for a car-like robot with unknown noise distributions. The results demonstrate that D3ROC provides robust and efficient control policies that outperform the iterative Linear Quadratic Gaussian (iLQG) control. The results also show the effectiveness of our proposed approach in handling different noise distributions.

This paper presents novel technology and methodology aimed at enhancing crowd management in both the planning and operational phases. The approach encompasses innovative data collection techniques, data integration, and visualization using a 3D Digital Twin, along with the incorporation of artificial intelligence (AI) tools for risk identification. The paper introduces the Bowtie model, a comprehensive framework designed to assess and predict risk levels. The model combines objective estimations and predictions, such as traffic flow operations and crowdedness levels, with various aggravating factors like weather conditions, sentiments, and the purpose of visitors, to evaluate the expected risk of incidents. The proposed framework is applied to the Crowd Safety Manager project in Scheveningen, where the DigiTwin is developed based on a wealth of real-time data sources. One noteworthy data source is Resono, offering insights into the number of visitors and their movements, leveraging a mobile phone panel of over 2 million users in the Netherlands. Particular attention is given to the left-hand side of the Bowtie, which includes state estimation, prediction, and forecasting. Notably, the focus is on generating multi-day ahead forecasts for event-planning purposes using Resono data. Advanced machine learning techniques, including the XGBoost framework, are compared, with XGBoost demonstrating the most accurate forecasts. The results indicate that the predictions are adequately accurate. However, certain locations may benefit from additional input data to further enhance prediction quality. Despite these limitations, this work contributes to a more effective crowd management system and opens avenues for further advancements in this critical field.

A key challenge in off-road navigation is that even visually similar terrains or ones from the same semantic class may have substantially different traction properties. Existing work typically assumes no wheel slip or uses the expected traction for motion planning, where the predicted trajectories provide a poor indication of the actual performance if the terrain traction has high uncertainty. In contrast, this work proposes to analyze terrain traversability with the empirical distribution of traction parameters in unicycle dynamics, which can be learned by a neural network in a self-supervised fashion. The probabilistic traction model leads to two risk-aware cost formulations that account for the worst-case expected cost and traction. To help the learned model generalize to unseen environment, terrains with features that lead to unreliable predictions are detected via a density estimator fit to the trained network's latent space and avoided via auxiliary penalties during planning. Simulation results demonstrate that the proposed approach outperforms existing work that assumes no slip or uses the expected traction in both navigation success rate and completion time. Furthermore, avoiding terrains with low density-based confidence score achieves up to 30% improvement in success rate when the learned traction model is used in a novel environment.

Simulation is an integral part in the process of developing autonomous vehicles and advantageous for training, validation, and verification of driving functions. Even though simulations come with a series of benefits compared to real-world experiments, various challenges still prevent virtual testing from entirely replacing physical test-drives. Our work provides an overview of these challenges with regard to different aspects and types of simulation and subsumes current trends to overcome them. We cover aspects around perception-, behavior- and content-realism as well as general hurdles in the domain of simulation. Among others, we observe a trend of data-driven, generative approaches and high-fidelity data synthesis to increasingly replace model-based simulation.

Golden-section search and bisection search are the two main principled algorithms for 1d minimization of quasiconvex (unimodal) functions. The first one only uses function queries, while the second one also uses gradient queries. Other algorithms exist under much stronger assumptions, such as Newton's method. However, to the best of our knowledge, there is no principled exact line search algorithm for general convex functions -- including piecewise-linear and max-compositions of convex functions -- that takes advantage of convexity. We propose two such algorithms: $\Delta$-Bisection is a variant of bisection search that uses (sub)gradient information and convexity to speed up convergence, while $\Delta$-Secant is a variant of golden-section search and uses only function queries. While bisection search reduces the $x$ interval by a factor 2 at every iteration, $\Delta$-Bisection reduces the (sometimes much) smaller $x^*$-gap $\Delta^x$ (the $x$ coordinates of $\Delta$) by at least a factor 2 at every iteration. Similarly, $\Delta$-Secant also reduces the $x^*$-gap by at least a factor 2 every second function query. Moreover, the $y^*$-gap $\Delta^y$ (the $y$ coordinates of $\Delta$) also provides a refined stopping criterion, which can also be used with other algorithms. Experiments on a few convex functions confirm that our algorithms are always faster than their quasiconvex counterparts, often by more than a factor 2. We further design a quasi-exact line search algorithm based on $\Delta$-Secant. It can be used with gradient descent as a replacement for backtracking line search, for which some parameters can be finicky to tune -- and we provide examples to this effect, on strongly-convex and smooth functions. We provide convergence guarantees, and confirm the efficiency of quasi-exact line search on a few single- and multivariate convex functions.

We study parametric inference for hypo-elliptic Stochastic Differential Equations (SDEs). Existing research focuses on a particular class of hypo-elliptic SDEs, with components split into `rough'/`smooth' and noise from rough components propagating directly onto smooth ones, but some critical model classes arising in applications have yet to be explored. We aim to cover this gap, thus analyse the highly degenerate class of SDEs, where components split into further sub-groups. Such models include e.g.~the notable case of generalised Langevin equations. We propose a tailored time-discretisation scheme and provide asymptotic results supporting our scheme in the context of high-frequency, full observations. The proposed discretisation scheme is applicable in much more general data regimes and is shown to overcome biases via simulation studies also in the practical case when only a smooth component is observed. Joint consideration of our study for highly degenerate SDEs and existing research provides a general `recipe' for the development of time-discretisation schemes to be used within statistical methods for general classes of hypo-elliptic SDEs.

Optimization is offered as an objective approach to resolving complex, real-world decisions involving uncertainty and conflicting interests. It drives business strategies as well as public policies and, increasingly, lies at the heart of sophisticated machine learning systems. A paradigm used to approach potentially high-stakes decisions, optimization relies on abstracting the real world to a set of decision(s), objective(s) and constraint(s). Drawing from the modeling process and a range of actual cases, this paper describes the normative choices and assumptions that are necessarily part of using optimization. It then identifies six emergent problems that may be neglected: 1) Misspecified values can yield optimizations that omit certain imperatives altogether or incorporate them incorrectly as a constraint or as part of the objective, 2) Problematic decision boundaries can lead to faulty modularity assumptions and feedback loops, 3) Failing to account for multiple agents' divergent goals and decisions can lead to policies that serve only certain narrow interests, 4) Mislabeling and mismeasurement can introduce bias and imprecision, 5) Faulty use of relaxation and approximation methods, unaccompanied by formal characterizations and guarantees, can severely impede applicability, and 6) Treating optimization as a justification for action, without specifying the necessary contextual information, can lead to ethically dubious or faulty decisions. Suggestions are given to further understand and curb the harms that can arise when optimization is used wrongfully.

This PhD thesis contains several contributions to the field of statistical causal modeling. Statistical causal models are statistical models embedded with causal assumptions that allow for the inference and reasoning about the behavior of stochastic systems affected by external manipulation (interventions). This thesis contributes to the research areas concerning the estimation of causal effects, causal structure learning, and distributionally robust (out-of-distribution generalizing) prediction methods. We present novel and consistent linear and non-linear causal effects estimators in instrumental variable settings that employ data-dependent mean squared prediction error regularization. Our proposed estimators show, in certain settings, mean squared error improvements compared to both canonical and state-of-the-art estimators. We show that recent research on distributionally robust prediction methods has connections to well-studied estimators from econometrics. This connection leads us to prove that general K-class estimators possess distributional robustness properties. We, furthermore, propose a general framework for distributional robustness with respect to intervention-induced distributions. In this framework, we derive sufficient conditions for the identifiability of distributionally robust prediction methods and present impossibility results that show the necessity of several of these conditions. We present a new structure learning method applicable in additive noise models with directed trees as causal graphs. We prove consistency in a vanishing identifiability setup and provide a method for testing substructure hypotheses with asymptotic family-wise error control that remains valid post-selection. Finally, we present heuristic ideas for learning summary graphs of nonlinear time-series models.

Graph Neural Networks (GNNs) have been shown to be effective models for different predictive tasks on graph-structured data. Recent work on their expressive power has focused on isomorphism tasks and countable feature spaces. We extend this theoretical framework to include continuous features - which occur regularly in real-world input domains and within the hidden layers of GNNs - and we demonstrate the requirement for multiple aggregation functions in this context. Accordingly, we propose Principal Neighbourhood Aggregation (PNA), a novel architecture combining multiple aggregators with degree-scalers (which generalize the sum aggregator). Finally, we compare the capacity of different models to capture and exploit the graph structure via a novel benchmark containing multiple tasks taken from classical graph theory, alongside existing benchmarks from real-world domains, all of which demonstrate the strength of our model. With this work, we hope to steer some of the GNN research towards new aggregation methods which we believe are essential in the search for powerful and robust models.