Kernel mean embeddings -- integrals of a kernel with respect to a probability distribution -- are essential in Bayesian quadrature, but also widely used in other computational tools for numerical integration or for statistical inference based on the maximum mean discrepancy. These methods often require, or are enhanced by, the availability of a closed-form expression for the kernel mean embedding. However, deriving such expressions can be challenging, limiting the applicability of kernel-based techniques when practitioners do not have access to a closed-form embedding. This paper addresses this limitation by providing a comprehensive dictionary of known kernel mean embeddings, along with practical tools for deriving new embeddings from known ones. We also provide a Python library that includes minimal implementations of the embeddings.

Integrating various data modalities brings valuable insights into underlying phenomena. Multimodal factor analysis (FA) uncovers shared axes of variation underlying different simple data modalities, where each sample is represented by a vector of features. However, FA is not suited for structured data modalities, such as text or single cell sequencing data, where multiple data points are measured per each sample and exhibit a clustering structure. To overcome this challenge, we introduce FACTM, a novel, multi-view and multi-structure Bayesian model that combines FA with correlated topic modeling and is optimized using variational inference. Additionally, we introduce a method for rotating latent factors to enhance interpretability with respect to binary features. On text and video benchmarks as well as real-world music and COVID-19 datasets, we demonstrate that FACTM outperforms other methods in identifying clusters in structured data, and integrating them with simple modalities via the inference of shared, interpretable factors.

We study regret minimization under privacy constraints in episodic inhomogeneous linear Markov Decision Processes (MDPs), motivated by the growing use of reinforcement learning (RL) in personalized decision-making systems that rely on sensitive user data. In this setting, both transition probabilities and reward functions are assumed to be linear in a feature mapping $\phi(s, a)$, and we aim to ensure privacy through joint differential privacy (JDP), a relaxation of differential privacy suited to online learning. Prior work has established suboptimal regret bounds by privatizing the LSVI-UCB algorithm, which achieves $\widetilde{O}(\sqrt{d^3 H^4 K})$ regret in the non-private setting. Building on recent advances that improve this to near minimax optimal regret $\widetilde{O}(d\sqrt{H^{3}K})$ via LSVI-UCB++ with Bernstein-style bonuses, we design a new differentially private algorithm by privatizing LSVI-UCB++ and adapting techniques for variance-aware analysis from offline RL. Our algorithm achieves a regret bound of $\widetilde{O}(d \sqrt{H^3 K} + H^{15/4} d^{7/6} K^{1/2} / \epsilon)$, improving over previous private methods. Empirical results show that our algorithm retains near-optimal utility compared to non-private baselines, indicating that privacy can be achieved with minimal performance degradation in this setting.

Probabilities or confidence values produced by artificial intelligence (AI) and machine learning (ML) models often do not reflect their true accuracy, with some models being under or over confident in their predictions. For example, if a model is 80% sure of an outcome, is it correct 80% of the time? Probability calibration metrics measure the discrepancy between confidence and accuracy, providing an independent assessment of model calibration performance that complements traditional accuracy metrics. Understanding calibration is important when the outputs of multiple systems are combined, for assurance in safety or business-critical contexts, and for building user trust in models. This paper provides a comprehensive review of probability calibration metrics for classifier and object detection models, organising them according to a number of different categorisations to highlight their relationships. We identify 82 major metrics, which can be grouped into four classifier families (point-based, bin-based, kernel or curve-based, and cumulative) and an object detection family. For each metric, we provide equations where available, facilitating implementation and comparison by future researchers.

We study the problem of distributed multi-view representation learning. In this problem, $K$ agents observe each one distinct, possibly statistically correlated, view and independently extracts from it a suitable representation in a manner that a decoder that gets all $K$ representations estimates correctly the hidden label. In the absence of any explicit coordination between the agents, a central question is: what should each agent extract from its view that is necessary and sufficient for a correct estimation at the decoder? In this paper, we investigate this question from a generalization error perspective. First, we establish several generalization bounds in terms of the relative entropy between the distribution of the representations extracted from training and "test" datasets and a data-dependent symmetric prior, i.e., the Minimum Description Length (MDL) of the latent variables for all views and training and test datasets. Then, we use the obtained bounds to devise a regularizer; and investigate in depth the question of the selection of a suitable prior. In particular, we show and conduct experiments that illustrate that our data-dependent Gaussian mixture priors with judiciously chosen weights lead to good performance. For single-view settings (i.e., $K=1$), our experimental results are shown to outperform existing prior art Variational Information Bottleneck (VIB) and Category-Dependent VIB (CDVIB) approaches. Interestingly, we show that a weighted attention mechanism emerges naturally in this setting. Finally, for the multi-view setting, we show that the selection of the joint prior as a Gaussians product mixture induces a Gaussian mixture marginal prior for each marginal view and implicitly encourages the agents to extract and output redundant features, a finding which is somewhat counter-intuitive.

We fully characterize a large class of feedforward neural networks in terms of truncation maps. As an application, we show how a ReLU neural network can implement a feature map which separates concentric data.

Reliable density estimation is fundamental for numerous applications in statistics and machine learning. In many practical scenarios, data are best modeled as mixtures of component densities that capture complex and multimodal patterns. However, conventional density estimators based on uniform histograms often fail to capture local variations, especially when the underlying distribution is highly nonuniform. Furthermore, the inherent discontinuity of histograms poses challenges for tasks requiring smooth derivatives, such as gradient-based optimization, clustering, and nonparametric discriminant analysis. In this work, we present a novel non-parametric approach for multivariate probability density function (PDF) estimation that utilizes minimum description length (MDL)-based binning with quantile cuts. Our approach builds upon tensor factorization techniques, leveraging the canonical polyadic decomposition (CPD) of a joint probability tensor. We demonstrate the effectiveness of our method on synthetic data and a challenging real dry bean classification dataset.

Traditional gradient-based sampling methods, like standard Hamiltonian Monte Carlo, require that the desired target distribution is continuous and differentiable. This limits the types of models one can define, although the presented models capture the reality in the observations better. In this project, Generalized Randomized Hamiltonian Monte Carlo (GRHMC) processes for sampling continuous densities with discontinuous gradient and piecewise smooth targets are proposed. The methods combine the advantages of Hamiltonian Monte Carlo methods with the nature of continuous time processes in the form of piecewise deterministic Markov processes to sample from such distributions. It is argued that the techniques lead to GRHMC processes that admit the desired target distribution as the invariant distribution in both scenarios. Simulation experiments verifying this fact and several relevant real-life models are presented, including a new parameterization of the spike and slab prior for regularized linear regression that returns sparse coefficient estimates and a regime switching volatility model.

In social learning, a network of agents assigns probability scores (beliefs) to some hypotheses of interest, which rule the generation of local streaming data observed by each agent. Belief formation takes place by means of an iterative two-step procedure where: i) the agents update locally their beliefs by using some likelihood model; and ii) the updated beliefs are combined with the beliefs of the neighboring agents, using a pooling rule. This procedure can fail to perform well in the presence of dynamic drifts, leading the agents to incorrect decision making. Here, we focus on the fully online setting where both the true hypothesis and the likelihood models can change over time. This goal is achieved by exploiting two adaptation stages: i) a stochastic gradient descent update to learn and track the drifts in the decision model; ii) and an adaptive belief update to track the true hypothesis changing over time. These stages are controlled by two adaptation parameters that govern the evolution of the error probability for each agent. We show that all agents learn consistently for sufficiently small adaptation parameters, in the sense that they ultimately place all their belief mass on the true hypothesis. Index T erms Social learning, belief formation, decision making, distributed optimization, online leaerning, opinion diffusion over graphs. Marco Carpentiero and Vincenzo Matta are with the Department of Information and Electrical Engineering and Applied Mathematics (DIEM), University of Salerno, via Giovanni Paolo II, I-84084, Fisciano (SA), Italy, and Vincenzo Matta is also with the National Inter-University Consortium for Telecommunications (CNIT), Italy (e-mails: { mcarpentiero, vmatta }@unisa.it). Matta was partially supported by the European Union under the Italian National Recovery and Resilience Plan (NRRP) of NextGenerationEU, partnership on "Telecommunications of the Future" (PE00000001 - program "REST ART"). This work was produced while Virginia Bordignon was a post-doc with the Ecole Polytechnique F ed erale de Lausanne EPFL, School of Engineering, CH-1015 Lausanne, Switzerland (e-mail: virginia.bordignon@alumni.epfl.ch).

Unresolved questions about how autonomous vehicles (AVs) should meet the informational needs of riders hinder real-world adoption. Complicating our ability to satisfy rider needs is that different people, goals, and driving contexts have different criteria for what constitutes interaction success. Unfortunately, most human-AV research and design today treats all people and situations uniformly. It is crucial to understand how an AV should communicate to meet rider needs, and how communications should change when the human-AV complex system changes. I argue that understanding the relationships between different aspects of the human-AV system can help us build improved and adaptable AV communications. I support this argument using three empirical studies. First, I identify optimal communication strategies that enhance driving performance, confidence, and trust for learning in extreme driving environments. Findings highlight the need for task-sensitive, modality-appropriate communications tuned to learner cognitive limits and goals. Next, I highlight the consequences of deploying faulty communication systems and demonstrate the need for context-sensitive communications. Third, I use machine learning (ML) to illuminate personal factors predicting trust in AVs, emphasizing the importance of tailoring designs to individual traits and concerns. Together, this dissertation supports the necessity of transparent, adaptable, and personalized AV systems that cater to individual needs, goals, and contextual demands. By considering the complex system within which human-AV interactions occur, we can deliver valuable insights for designers, researchers, and policymakers. This dissertation also provides a concrete domain to study theories of human-machine joint action and situational awareness, and can be used to guide future human-AI interaction research. [shortened for arxiv]