Principal Component Analysis is a key technique for reducing the complexity of high-dimensional data while preserving its fundamental data structure, ensuring models remain stable and interpretable. This is achieved by transforming the original variables into a new set of uncorrelated variables (principal components) based on the covariance structure of the original variables. However, since the traditional maximum likelihood covariance estimator does not accurately converge to the true covariance matrix, the standard principal component analysis performs poorly as a dimensionality reduction technique in high-dimensional scenarios $n

Intercepting dynamic objects in uncertain environments involves a significant unresolved challenge in modern robotic systems. Current control approaches rely solely on estimated information, and results lack guarantees of robustness and feasibility. In this work, we introduce a novel method to tackle the interception of targets whose motion is affected by known and bounded uncertainty. Our approach introduces new techniques of reachability analysis for rigid bodies, leveraged to guarantee feasibility of interception under uncertain conditions. We then propose a Reachability-Guaranteed Optimal Control Problem, ensuring robustness and guaranteed reachability to a target set of configurations. We demonstrate the methodology in the case study of an interception maneuver of a tumbling target in space.

Artificial intelligence systems based on large language models (LLMs) are increasingly used as agents that interact with users and with the world. To do so successfully, LLMs need to construct internal representations of the world and form probabilistic beliefs about those representations. To provide a user with personalized recommendations, for example, the LLM needs to gradually infer the user's preferences, over the course of multiple interactions. To evaluate whether contemporary LLMs are able to do so, we use the Bayesian inference framework from probability theory, which lays out the optimal way to update an agent's beliefs as it receives new information. We first show that the LLMs do not update their beliefs as expected from the Bayesian framework, and that consequently their predictions do not improve as expected as more information becomes available, even less so than we find is the case for humans. To address this issue, we teach the LLMs to reason in a Bayesian manner by training them to mimic the predictions of an optimal Bayesian model. We find that this approach not only significantly improves the LLM's performance on the particular recommendation task it is trained on, but also enables generalization to other tasks. This suggests that this method endows the LLM with broader Bayesian reasoning skills. More generally, our results indicate that LLMs can learn about reasoning strategies effectively and generalize those skills to new domains, which in part explains LLMs' empirical success.

Current end-to-end (E2E) and plug-and-play (PnP) image reconstruction algorithms approximate the maximum a posteriori (MAP) estimate but cannot offer sampling from the posterior distribution, like diffusion models. By contrast, it is challenging for diffusion models to be trained in an E2E fashion. This paper introduces a Deep End-to-End Posterior ENergy (DEEPEN) framework, which enables MAP estimation as well as sampling. We learn the parameters of the posterior, which is the sum of the data consistency error and the negative log-prior distribution, using maximum likelihood optimization in an E2E fashion. The proposed approach does not require algorithm unrolling, and hence has a smaller computational and memory footprint than current E2E methods, while it does not require contraction constraints typically needed by current PnP methods. Our results demonstrate that DEEPEN offers improved performance than current E2E and PnP models in the MAP setting, while it also offers faster sampling compared to diffusion models. In addition, the learned energy-based model is observed to be more robust to changes in image acquisition settings.

Ensuring trustworthiness in machine learning (ML) systems is crucial as they become increasingly embedded in high-stakes domains. This paper advocates for integrating causal methods into machine learning to navigate the trade-offs among key principles of trustworthy ML, including fairness, privacy, robustness, accuracy, and explainability. While these objectives should ideally be satisfied simultaneously, they are often addressed in isolation, leading to conflicts and suboptimal solutions. Drawing on existing applications of causality in ML that successfully align goals such as fairness and accuracy or privacy and robustness, this paper argues that a causal approach is essential for balancing multiple competing objectives in both trustworthy ML and foundation models. Beyond highlighting these trade-offs, we examine how causality can be practically integrated into ML and foundation models, offering solutions to enhance their reliability and interpretability. Finally, we discuss the challenges, limitations, and opportunities in adopting causal frameworks, paving the way for more accountable and ethically sound AI systems.

Offline optimization is a fundamental challenge in science and engineering, where the goal is to optimize black-box functions using only offline datasets. This setting is particularly relevant when querying the objective function is prohibitively expensive or infeasible, with applications spanning protein engineering, material discovery, neural architecture search, and beyond. The main difficulty lies in accurately estimating the objective landscape beyond the available data, where extrapolations are fraught with significant epistemic uncertainty. This uncertainty can lead to objective hacking(reward hacking), exploiting model inaccuracies in unseen regions, or other spurious optimizations that yield misleadingly high performance estimates outside the training distribution. Recent advances in model-based optimization(MBO) have harnessed the generalization capabilities of deep neural networks to develop offline-specific surrogate and generative models. Trained with carefully designed strategies, these models are more robust against out-of-distribution issues, facilitating the discovery of improved designs. Despite its growing impact in accelerating scientific discovery, the field lacks a comprehensive review. To bridge this gap, we present the first thorough review of offline MBO. We begin by formalizing the problem for both single-objective and multi-objective settings and by reviewing recent benchmarks and evaluation metrics. We then categorize existing approaches into two key areas: surrogate modeling, which emphasizes accurate function approximation in out-of-distribution regions, and generative modeling, which explores high-dimensional design spaces to identify high-performing designs. Finally, we examine the key challenges and propose promising directions for advancement in this rapidly evolving field including safe control of superintelligent systems.

A representation of the cause-effect mechanism is needed to enable artificial intelligence to represent how the world works. Bayesian Networks (BNs) have proven to be an effective and versatile tool for this task. BNs require constructing a structure of dependencies among variables and learning the parameters that govern these relationships. These tasks, referred to as structural learning and parameter learning, are actively investigated by the research community, with several algorithms proposed and no single method having established itself as standard. A wide range of software, tools, and packages have been developed for BNs analysis and made available to academic researchers and industry practitioners. As a consequence of having no one-size-fits-all solution, moving the first practical steps and getting oriented into this field is proving to be challenging to outsiders and beginners. In this paper, we review the most relevant tools and software for BNs structural and parameter learning to date, providing our subjective recommendations directed to an audience of beginners. In addition, we provide an extensive easy-to-consult overview table summarizing all software packages and their main features. By improving the reader understanding of which available software might best suit their needs, we improve accessibility to the field and make it easier for beginners to take their first step into it.

We introduce a new framework that employs Malliavin calculus to derive explicit expressions for the score function -- i.e., the gradient of the log-density -- associated with solutions to stochastic differential equations (SDEs). Our approach integrates classical integration-by-parts techniques with modern tools, such as Bismut's formula and Malliavin calculus, to address linear and nonlinear SDEs. In doing so, we establish a rigorous connection between the Malliavin derivative, its adjoint (the Malliavin divergence or the Skorokhod integral), Bismut's formula, and diffusion generative models, thus providing a systematic method for computing $\nabla \log p_t(x)$. For the linear case, we present a detailed study proving that our formula is equivalent to the actual score function derived from the solution of the Fokker--Planck equation for linear SDEs. Additionally, we derive a closed-form expression for $\nabla \log p_t(x)$ for nonlinear SDEs with state-independent diffusion coefficients. These advancements provide fresh theoretical insights into the smoothness and structure of probability densities and practical implications for score-based generative modelling, including the design and analysis of new diffusion models. Moreover, our findings promote the adoption of the robust Malliavin calculus framework in machine learning research. These results directly apply to various pure and applied mathematics fields, such as generative modelling, the study of SDEs driven by fractional Brownian motion, and the Fokker--Planck equations associated with nonlinear SDEs.

Whether large language models (LLMs) process language similarly to humans has been the subject of much theoretical and practical debate. We examine this question through the lens of the production-interpretation distinction found in human sentence processing and evaluate the extent to which instruction-tuned LLMs replicate this distinction. Using an empirically documented asymmetry between production and interpretation in humans for implicit causality verbs as a testbed, we find that some LLMs do quantitatively and qualitatively reflect human-like asymmetries between production and interpretation. We demonstrate that whether this behavior holds depends upon both model size - with larger models more likely to reflect human-like patterns and the choice of meta-linguistic prompts used to elicit the behavior.

Children acquire language despite being exposed to several orders of magnitude less data than large language models require. Meta-learning has been proposed as a way to integrate human-like learning biases into neural-network architectures, combining both the structured generalizations of symbolic models with the scalability of neural-network models. But what does meta-learning exactly imbue the model with? We investigate the meta-learning of formal languages and find that, contrary to previous claims, meta-trained models are not learning simplicity-based priors when meta-trained on datasets organised around simplicity. Rather, we find evidence that meta-training imprints neural mechanisms (such as counters) into the model, which function like cognitive primitives for the network on downstream tasks. Most surprisingly, we find that meta-training on a single formal language can provide as much improvement to a model as meta-training on 5000 different formal languages, provided that the formal language incentivizes the learning of useful neural mechanisms. Taken together, our findings provide practical implications for efficient meta-learning paradigms and new theoretical insights into linking symbolic theories and neural mechanisms.