We propose a geometric extension of restricted Boltzmann machines (RBMs) by allowing weights to take values in abstract groups such as \( \mathrm{GL}_n(\mathbb{R}) \), \( \mathrm{SU}(2) \), or even infinite-dimensional operator groups. This generalization enables the modeling of complex relational structures, including projective transformations, spinor dynamics, and functional symmetries, with direct applications to vision, language, and quantum learning. A central contribution of this work is the introduction of a \emph{contextuality index} based on group-valued holonomies computed along cycles in the RBM graph. This index quantifies the global inconsistency or "curvature" induced by local weights, generalizing classical notions of coherence, consistency, and geometric flatness. We establish links with sheaf-theoretic contextuality, gauge theory, and noncommutative geometry, and provide numerical and diagrammatic examples in both finite and infinite dimensions. This framework opens novel directions in AI, from curvature-aware learning architectures to topological regularization in uncertain or adversarial environments.

Recently, federated learning (FL) has emerged as a novel framework for distributed model training. In FL, the task publisher (TP) releases tasks, and local model owners (LMOs) use their local data to train models. Sometimes, FL suffers from the lack of training data, and thus workers are recruited for gathering data. To this end, this paper proposes an adaptive incentive mechanism from a service-oriented perspective, with the objective of maximizing the utilities of TP, LMOs and workers. Specifically, a Stackelberg game is theoretically established between the LMOs and TP, positioning TP as the leader and the LMOs as followers. An analytical Nash equilibrium solution is derived to maximize their utilities. The interaction between LMOs and workers is formulated by a multi-agent Markov decision process (MAMDP), with the optimal strategy identified via deep reinforcement learning (DRL). Additionally, an Adaptively Searching the Optimal Strategy Algorithm (ASOSA) is designed to stabilize the strategies of each participant and solve the coupling problems. Extensive numerical experiments are conducted to validate the efficacy of the proposed method.

B. Observation History The core contribution of this work is a novel history encoder for processing historical observations, which integrates two key operations: depthwise separable convolution and multi-head attention. The background of these operations is briefly reviewed below. Depthwise separable convolution [33] is a streamlined variant of standard convolution that reduces both parameter count and computational cost. It decomposes the operation into two steps: (1) a depthwise convolution, which applies a single filter to each input channel, and (2) a pointwise convolution, which uses a 1 1 convolution to linearly combine the outputs of the depthwise stage. This factorization enables efficient extraction of spatial and cross-channel features while maintaining strong representational capacity. It has been widely adopted in lightweight neural architectures such as MobileNet [34] and is particularly well suited to real-time and resource-constrained applications. Multi-head attention [9] is a fundamental component of Transformer architectures, enabling the model to capture diverse patterns across different representation subspaces.

We propose a method for inferring entropy production (EP) in high-dimensional stochastic systems, including many-body systems and non-Markovian systems with long memory. Standard techniques for estimating EP become intractable in such systems due to computational and statistical limitations. We infer trajectory-level EP and lower bounds on average EP by exploiting a nonequilibrium analogue of the Maximum Entropy principle, along with convex duality. Our approach uses only samples of trajectory observables, such as spatiotemporal correlations. It does not require reconstruction of high-dimensional probability distributions or rate matrices, nor impose any special assumptions such as discrete states or multipartite dynamics. In addition, it may be used to compute a hierarchical decomposition of EP, reflecting contributions from different interaction orders, and it has an intuitive physical interpretation as a "thermodynamic uncertainty relation." We demonstrate its numerical performance on a disordered nonequilibrium spin model with 1000 spins and a large neural spike-train dataset.

We introduce a comprehensive framework for the detection and demodulation of covert electromagnetic signals using solid-state spin sensors. Our approach, named RAPID, is a two-stage hybrid strategy that leverages nitrogen-vacancy (NV) centers to operate below the classical noise floor employing a robust adaptive policy via imitation and distillation. We first formulate the joint detection and estimation task as a unified stochastic optimal control problem, optimizing a composite Bayesian risk objective under realistic physical constraints. The RAPID algorithm solves this by first computing a robust, non-adaptive baseline protocol grounded in the quantum Fisher information matrix (QFIM), and then using this baseline to warm-start an online, adaptive policy learned via deep reinforcement learning (Soft Actor-Critic). This method dynamically optimizes control pulses, interrogation times, and measurement bases to maximize information gain while actively suppressing non-Markovian noise and decoherence. Numerical simulations demonstrate that the protocol achieves a significant sensitivity gain over static methods, maintains high estimation precision in correlated noise environments, and, when applied to sensor arrays, enables coherent quantum beamforming that achieves Heisenberg-like scaling in precision. This work establishes a theoretically rigorous and practically viable pathway for deploying quantum sensors in security-critical applications such as electronic warfare and covert surveillance.

We consider two-player zero-sum concurrent stochastic games (CSGs) played on graphs with reachability and safety objectives. These include degenerate classes such as Markov decision processes or turn-based stochastic games, which can be solved by linear or quadratic programming; however, in practice, value iteration (VI) outperforms the other approaches and is the most implemented method. Similarly, for CSGs, this practical performance makes VI an attractive alternative to the standard theoretical solution via the existential theory of reals. VI starts with an under-approximation of the sought values for each state and iteratively updates them, traditionally terminating once two consecutive approximations are $ε$-close. However, this stopping criterion lacks guarantees on the precision of the approximation, which is the goal of this work. We provide bounded (a.k.a. interval) VI for CSGs: it complements standard VI with a converging sequence of over-approximations and terminates once the over- and under-approximations are $ε$-close.

This paper presents a feature-based Partially Observable Markov Decision Process (POMDP) framework for quantum network routing, combining belief-state planning with Graph Neural Networks (GNNs) to address partial observability, decoherence, and scalability challenges in dynamic quantum systems. Our approach encodes complex quantum network dynamics, including entanglement degradation and time-varying channel noise, into a low-dimensional feature space, enabling efficient belief updates and scalable policy learning. The core of our framework is a hybrid GNN-POMDP architecture that processes graph-structured representations of entangled links to learn routing policies, coupled with a noise-adaptive mechanism that fuses POMDP belief updates with GNN outputs for robust decision making. We provide a theoretical analysis establishing guarantees for belief convergence, policy improvement, and robustness to noise. Experiments on simulated quantum networks with up to 100 nodes demonstrate significant improvements in routing fidelity and entanglement delivery rates compared to state-of-the-art baselines, particularly under high decoherence and nonstationary conditions.

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.

In time-critical settings such as assistive driving, assistants often rely on alerts or haptic signals to prompt rapid human attention, but these cues usually leave humans to interpret situations and decide responses independently, introducing potential delays or ambiguity in meaning. Language-based assistive systems can instead provide instructions backed by context, offering more informative guidance. However, current approaches (e.g., social assistive robots) largely prioritize content generation while overlooking critical timing factors such as verbal conveyance duration, human comprehension delays, and subsequent follow-through duration. These timing considerations are crucial in time-critical settings, where even minor delays can substantially affect outcomes. We aim to study this inherent trade-off between timeliness and informativeness by framing the challenge as a sequential decision-making problem using an augmented-state Markov Decision Process. We design a framework combining reinforcement learning and a generated offline taxonomy dataset, where we balance the trade-off while enabling a scalable taxonomy dataset generation pipeline. Empirical evaluation with synthetic humans shows our framework improves success rates by over 40% compared to methods that ignore time delays, while effectively balancing timeliness and informativeness. It also exposes an often-overlooked trade-off between these two factors, opening new directions for optimizing communication in time-critical human-AI assistance.