The increasing reliance on cyber physical infrastructure in modern power systems has amplified the risk of targeted cyber attacks, necessitating robust and adaptive resilience strategies. This paper presents a mathematically rigorous game theoretic framework to evaluate and enhance microgrid resilience using a combination of quantitative resilience metrics Load Served Ratio LSR, Critical Load Resilience CLR, Topological Survivability Score TSS, and DER Resilience Score DRS. These are integrated into a unified payoff matrix using the Analytic Hierarchy Process AHP to assess attack defense interactions. The framework is formalized as a finite horizon Markov Decision Process MDP with formal convergence guarantees and computational complexity bounds. Three case studies are developed 1. static attacks analyzed via Nash equilibrium, 2. severe attacks incorporating high impact strategies, and 3. adaptive attacks using Stackelberg games, regret matching, softmax heuristics, and Multi Agent Q Learning. Rigorous theoretical analysis provides convergence proofs with explicit rates , PAC learning sample complexity bounds, and computational complexity analysis. The framework is tested on an enhanced IEEE 33bus distribution system with DERs and control switches, demonstrating the effectiveness of adaptive and strategic defenses in improving cyber physical resilience with statistically significant improvements of 18.7% 2.1% over static approaches.

Transformers have exhibited exceptional capabilities in sequence modeling tasks, leveraging self-attention and in-context learning. Critical to this success are induction heads, attention circuits that enable copying tokens based on their previous occurrences. In this work, we introduce a novel framework that showcases transformers' ability to dynamically handle causal structures. Existing works rely on Markov Chains to study the formation of induction heads, revealing how transformers capture causal dependencies and learn transition probabilities in-context. However, they rely on a fixed causal structure that fails to capture the complexity of natural languages, where the relationship between tokens dynamically changes with context. To this end, our framework varies the causal structure through interleaved Markov chains with different lags while keeping the transition probabilities fixed. This setting unveils the formation of Selective Induction Heads, a new circuit that endows transformers with the ability to select the correct causal structure in-context. We empirically demonstrate that transformers learn this mechanism to predict the next token by identifying the correct lag and copying the corresponding token from the past. We provide a detailed construction of a 3-layer transformer to implement the selective induction head, and a theoretical analysis proving that this mechanism asymptotically converges to the maximum likelihood solution. Our findings advance the understanding of how transformers select causal structures, providing new insights into their functioning and interpretability.

Deep neural networks have transformed machine learning, excelling in tasks such as image classification and natural language processing through hierarchical feature learning [1]. However, traditional neurons, which compute deterministic weighted sums followed by nonlinear activations (e.g., ReLU, sigmoid), struggle to model uncertainty or multimodal distributions prevalent in real-world data. This limitation has historically been addressed by probabilistic graphical models, such as Bayesian Networks [2] and Markov Random Fields [3], which offer robust frameworks for uncertainty quantification and complex dependency modeling [4]. These models provide a strong conceptual foundation, but often lack the deep hierarchical feature learning capabilities of modern neural networks. A key research focus has therefore been on bridging the gap between these two paradigms. This has led to approaches that incorporate the probabilistic principles of graphical models directly into deep learning architectures. For example, Bayesian Neural Networks (BNNs) embed uncertainty into network weights [5], while Probabilistic Circuits (PCs), including Sum-Product Networks (SPNs) [7, 8, 9], are deep probabilistic models that build on a formal probabilistic structure, fusing the representational power of graphical models with the hierarchical feature learning of neural networks. In contrast, this paper introduces a novel approach by embedding a univariate Gaussian Mixture Model (uGMM) directly into the network's computational units, enabling each neuron to represent 1

Sparse regression problems, where the goal is to identify a small set of relevant predictors, often require modeling not only main effects but also meaningful interactions through other variables. While the pliable lasso has emerged as a powerful frequentist tool for modeling such interactions under strong heredity constraints, it lacks a natural framework for uncertainty quantification and incorporation of prior knowledge. In this paper, we propose a Bayesian pliable lasso that extends this approach by placing sparsity-inducing priors, such as the horseshoe, on both main and interaction effects. The hierarchical prior structure enforces heredity constraints while adaptively shrinking irrelevant coefficients and allowing important effects to persist. We extend this framework to Generalized Linear Models (GLMs) and develop a tailored approach to handle missing responses. To facilitate posterior inference, we develop an efficient Gibbs sampling algorithm based on a reparameterization of the horseshoe prior. Our Bayesian framework yields sparse, interpretable interaction structures, and principled measures of uncertainty. Through simulations and real-data studies, we demonstrate its advantages over existing methods in recovering complex interaction patterns under both complete and incomplete data. Our method is implemented in the package \texttt{hspliable} available on Github.

The Organization for Economic Cooperation and Development (OECD) Working Party on Nuclear Criticality Safety (WPNCS) proposed a benchmark exercise to assess the performance of current nuclear data adjustment techniques applied to nonlinear applications and experiments with low correlation to applications. This work introduces Bayesian Inverse Uncertainty Quantification (IUQ) as a method for nuclear data adjustments in this benchmark, and compares IUQ to the more traditional methods of Generalized Linear Least Squares (GLLS) and Monte Carlo Bayes (MOCABA). Posterior predictions from IUQ showed agreement with GLLS and MOCABA for linear applications. When comparing GLLS, MOCABA, and IUQ posterior predictions to computed model responses using adjusted parameters, we observe that GLLS predictions fail to replicate computed response distributions for nonlinear applications, while MOCABA shows near agreement, and IUQ uses computed model responses directly. We also discuss observations on why experiments with low correlation to applications can be informative to nuclear data adjustments and identify some properties useful in selecting experiments for inclusion in nuclear data adjustment. Performance in this benchmark indicates potential for Bayesian IUQ in nuclear data adjustments.

Causal relationships play a fundamental role in understanding the world around us. The ability to identify and understand cause-effect relationships is critical to making informed decisions, predicting outcomes, and developing effective strategies. However, deciphering causal relationships from observational data is a difficult task, as correlations alone may not provide definitive evidence of causality. In recent years, the field of machine learning (ML) has emerged as a powerful tool, offering new opportunities for uncovering hidden causal mechanisms and better understanding complex systems. In this work, we address the issue of detecting the intrinsic causal links of a large class of complex systems in the framework of the response theory in physics. We develop some theoretical ideas put forward by [1], and technically we use state-of-the-art ML techniques to build up models from data. We consider both linear stochastic and non-linear systems. Finally, we compute the asymptotic efficiency of the linear response based causal predictor in a case of large scale Markov process network of linear interactions.

Explainability is a critical tool in helping stakeholders understand robots. In particular, the ability for robots to explain why they have made a particular decision or behaved in a certain way is useful in this regard. Behaviour trees are a popular framework for controlling the decision-making of robots and other software systems, and thus a natural question to ask is whether or not a system driven by a behaviour tree is capable of answering "why" questions. While explainability for behaviour trees has seen some prior attention, no existing methods are capable of generating causal, counterfactual explanations which detail the reasons for robot decisions and behaviour. Therefore, in this work, we introduce a novel approach which automatically generates counterfactual explanations in response to contrastive "why" questions. Our method achieves this by first automatically building a causal model from the structure of the behaviour tree as well as domain knowledge about the state and individual behaviour tree nodes. The resultant causal model is then queried and searched to find a set of diverse counterfactual explanations. We demonstrate that our approach is able to correctly explain the behaviour of a wide range of behaviour tree structures and states. By being able to answer a wide range of causal queries, our approach represents a step towards more transparent, understandable and ultimately trustworthy robotic systems.

Neurosymbolic AI (NeSy) aims to integrate the statistical strengths of neural networks with the interpretability and structure of symbolic reasoning. However, current NeSy frameworks like DeepProbLog enforce a fixed flow where symbolic reasoning always follows neural processing. This restricts their ability to model complex dependencies, especially in irregular data structures such as graphs. In this work, we introduce DeepGraphLog, a novel NeSy framework that extends ProbLog with Graph Neural Predicates. DeepGraphLog enables multi-layer neural-symbolic reasoning, allowing neural and symbolic components to be layered in arbitrary order. In contrast to DeepProbLog, which cannot handle symbolic reasoning via neural methods, DeepGraphLog treats symbolic representations as graphs, enabling them to be processed by Graph Neural Networks (GNNs). We showcase the capabilities of DeepGraphLog on tasks in planning, knowledge graph completion with distant supervision, and GNN expressivity. Our results demonstrate that DeepGraphLog effectively captures complex relational dependencies, overcoming key limitations of existing NeSy systems. By broadening the applicability of neurosymbolic AI to graph-structured domains, DeepGraphLog offers a more expressive and flexible framework for neural-symbolic integration.

Class imbalance poses a significant challenge to supervised classification, particularly in critical domains like medical diagnostics and anomaly detection where minority class instances are rare. While numerous studies have explored rebalancing techniques to address this issue, less attention has been given to evaluating the performance of binary classifiers under imbalance when no such techniques are applied. Therefore, the goal of this study is to assess the performance of binary classifiers "as-is", without performing any explicit rebalancing. Specifically, we systematically evaluate the robustness of a diverse set of binary classifiers across both real-world and synthetic datasets, under progressively reduced minority class sizes, using one-shot and few-shot scenarios as baselines. Our approach also explores varying data complexities through synthetic decision boundary generation to simulate real-world conditions. In addition to standard classifiers, we include experiments using undersampling, oversampling strategies, and one-class classification (OCC) methods to examine their behavior under severe imbalance. The results confirm that classification becomes more difficult as data complexity increases and the minority class size decreases. While traditional classifiers deteriorate under extreme imbalance, advanced models like TabPFN and boosting-based ensembles retain relatively higher performance and better generalization compared to traditional classifiers. Visual interpretability and evaluation metrics further validate these findings. Our work offers valuable guidance on model selection for imbalanced learning, providing insights into classifier robustness without dependence on explicit rebalancing techniques.

Figure 1: We introduce OmniMap, a general online mapping framework integrating optics, geometry, and semantics. OmniMap incrementally maintains an open-vocabulary instance-level voxel representation and a 3DGS (3D Gaussian Splatting) representation, from which color and geometric meshes are derived. OmniMap supports multi-modal rendering (RGB / depth / normal / instance), and achieves state-of-the-art performance in rendering fidelity, mesh quality, and semantic understanding. This holistic framework enables versatile support for a wide range of downstream applications. Abstract--Robotic systems demand accurate and comprehensive 3D environment perception, requiring simultaneous capture of photo-realistic appearance (optical), precise layout shape (geometric), and open-vocabulary scene understanding (semantic). Existing methods typically achieve only partial fulfillment of these requirements while exhibiting optical blurring, geometric irregularities, and semantic ambiguities. T o address these challenges, we propose OmniMap. Overall, OmniMap represents the first online mapping framework that simultaneously captures optical, geometric, and semantic scene attributes while maintaining real-time performance and model compactness. This work is supported by the National Natural Science Foundation of China under Grant 92370203, 62473050, 62233002, Beijing Natural Science Foundation Undergraduate Research Program QY24180. Mengyin Fu is with the School of Automation, Beijing Institute of Technology, Beijing 100081, China, and the School of Automation, Nanjing University of Science and Technology, Nanjing 210018, China (e-mail: fumy@bit.edu.cn). The project page of OmniMap is available at https://omni-map.github.io/. At the implementation level, OmniMap identifies key challenges across different modalities and introduces several innovations: adaptive camera modeling for motion blur and exposure compensation, hybrid incremental representation with normal constraints, and probabilistic fusion for robust instance-level understanding. Extensive experiments show OmniMap's superior performance in rendering fidelity, geometric accuracy, and zero-shot semantic segmentation compared to state-of-the-art methods across diverse scenes. The framework's versatility is further evidenced through a variety of downstream applications, including multi-domain scene Q&A, interactive editing, perception-guided manipulation, and map-assisted navigation. The quality of a robot's 3D environmental representation, measured by its accuracy and dimensionality, fundamentally impacts the robot's task operational performance and execution capabilities.