To answer this, we introduce Tropical Attention, an attention mechanism grounded in tropical geometry that lifts the attention kernel into tropical projective space, where reasoning is piecewise-linear and 1-Lipschitz, thus preserving the polyhedral decision structure inherent to combinatorial reasoning. We prove that multi-head Tropical Attention (MHTA) stacks universally approximate tropical circuits and realize tropical transitive closure through composition, achieving polynomial resource bounds without invoking recurrent mechanisms. These guarantees explain why the induced polyhedral decision boundaries remain sharp and scale-invariant, rather than smoothed by Softmax. Empirically, we show that Tropical Attention delivers stronger out-of-distribution generalization in both length and value, with high robustness against perturbative noise, and substantially faster inference with fewer parameters compared to Softmax-based and recurrent attention baselines, respectively.

Scientific reasoning is an increasingly important capability of large language models, yet improving the robustness and efficiency of training such reasoning remains a key open challenge. We study this problem in instruction-based molecular optimization, where answer-only supervised fine-tuning (SFT) collapses multi-step reasoning and reinforcement learning with verifiable rewards (RLVR) suffers from sparse feedback. Reference-guided Policy Optimization (RePO) mitigates both by anchoring policy updates to dataset-provided references, but its effectiveness is tightly coupled to reference quality: weak or misaligned references impose a performance ceiling. To overcome this ceiling, we propose active reasoning, a paradigm in which the policy actively decides, on a per-instance basis, when to imitate a reference and when to reinforce its own discoveries, while continuously upgrading what it imitates. We instantiate this paradigm as Active Group Relative Policy Optimization (Active-GRPO), realized through two coupled mechanisms: active imitate-reinforce and active referencing. The former performs imitation learning when the reference still outperforms the policy's own candidates, and shifts to self-improvement via reinforcement learning once the policy has generated molecules that surpass the reference. The latter continuously upgrades the reference itself by replacing it with the best policy-generated candidate discovered so far, progressively raising the imitation target and ensuring that reference guidance remains informative--rather than restrictive--throughout training. Across TOMG-Bench MOLOPT, Active-GRPO improves average SR Sim from 0.0959 for GRPO and 0.1665 for RePO to 0.1773 under matched three-seed evaluation, with statistically significant gains on LogP, MR, and QED.

Adaptive gradient methods are among the standard tools for training machine learning models. Their appeal is that they reduce the need to tune a fixed learning rate by adjusting the effective stepsize using information observed along the optimization trajectory. AdaGrad, introduced by Duchi et al. [2011], is a prototypical example: it rescales the update by the square root of the cumulative sum of past squared subgradients, coordinate by coordinate. The method was originally proposed for nonsmooth Lipschitz-continuous composite convex optimization, achieving the optimal rate O(1/ n) in the objective gap. Later works considered the smooth setting and asked whether AdaGrad can adapt to the unknown smoothness level of the objective, while attaining the corresponding standard rate.

We consider sparse multivariate stochastic systems that evolve in continuous time according to a causal mechanism and present methodology to recover the system's time-infinitesimal transition mechanism from mere cross-sectional data. This observational paradigm is motivated by applications such as gene expression analysis, where destructive experimental techniques may only allow recording data once over a cell's lifetime. Precisely, we assume the system follows a time-homogeneous diffusion process that has reached an equilibrium distribution at observation time. Further, we assume the causal mechanism is fully described by the diffusion drift, is acyclic, and its causal structure graph is known. In this setting, we prove that the full causal mechanism, i.e., the drift function, can be non-parametrically identified under a weak non-explosion criterion. We derive a non-parametric kernel estimator for this challenging inverse problem and prove its consistency. Moreover, we propose a cross-validation scheme for hyperparameter tuning, illustrate the behavior of our estimator in simulations, and we discuss connections with irreversible generative diffusion models and low-frequency sampled data.

In 2020, the U.S. Census Bureau adopted differential privacy for the Decennial Census by injecting integer-valued Gaussian noise into published census tabulations. Exactly evaluating the privacy guarantees of these data releases would enable the Bureau to determine the absolute minimum noise required to satisfy a given privacy budget, preventing the injection of unnecessary excess noise and thereby substantially enhancing the statistical utility of the data for downstream applications such as federal funding allocation and political redistricting. In this paper, we introduce a computationally efficient and mathematically rigorous quadrature method to evaluate the exact privacy profile of practical, large-scale census releases under the composition of heterogeneous discrete Gaussian mechanisms. Mathematically, this problem reduces to evaluating the tail probabilities of high-dimensional convolutions of integer-valued random variables sampled from heterogeneous discrete Gaussian distributions under exceptionally stringent numerical error tolerances (e.g., $10^{-35}$). By recasting the exact privacy accounting as a numerical integration problem via the discrete Fourier transform, we explicitly exploit the exponential convergence of the trapezoidal rule for complex analytic, periodic characteristic functions. Furthermore, to overcome the computational bottleneck of evaluating highly oscillatory integrands in high dimensions, we develop a sieve algorithm that identifies and prunes negligible quadrature nodes, accelerating the computation by three orders of magnitude. Taken together, these numerical innovations enable the first exact, assumption-free privacy accounting for the 2020 Census Demographic and Housing Characteristics File, achieving a 1,824-fold speedup over prior methods while maintaining census-mandated error tolerances.

In recent years, fueled by the rapid advancement of diffusion models, text-to-video (T2V) generation models have achieved remarkable progress, with notable examples including Pika, Luma, Kling, and Open-Sora. Although these models exhibit impressive generative capabilities, they also expose significant security risks due to their vulnerability to jailbreak attacks, where the models are manipulated to produce unsafe content such as pornography, violence, or discrimination. Existing works such as T2VSafetyBench provide preliminary benchmarks for safety evaluation, but lack systematic methods for thoroughly exploring model vulnerabilities. To address this gap, we are the first to formalize the T2V jailbreak attack as a discrete optimization problem and propose a joint objective-based optimization framework, called \emph{T2V-OptJail}. This framework consists of two key optimization goals: bypassing the built-in safety filtering mechanisms to increase the attack success rate, preserving semantic consistency between the adversarial prompt and the unsafe input prompt, as well as between the generated video and the unsafe input prompt, to enhance content controllability. In addition, we introduce an iterative optimization strategy guided by prompt variants, where multiple semantically equivalent candidates are generated in each round, and their scores are aggregated to robustly guide the search toward optimal adversarial prompts. We conduct large-scale experiments on several T2V models, covering both open-source models (\textit{e.g.}, Open-Sora) and real commercial closed-source models (\textit{e.g.}, Pika, Luma, Kling). The experimental results show that the proposed method improves 11.4\% and 10.0\% over the existing state-of-the-art method (SoTA) in terms of attack success rate assessed by GPT-4, attack success rate assessed by human accessors, respectively, verifying the significant advantages of the method in terms of attack effectiveness and content control. This study reveals the potential abuse risk of the semantic alignment mechanism in the current T2V model and provides a basis for the design of subsequent jailbreak defense methods.

Abstract--As black-box models become foundational to mod-solution that can be applied across a wide range of scientific ern research, ensuring their stability is paramount for the realiza-and industrial domains. The inherent diversity of inputs--ranging from structured Gaussian distributions to Notwithstanding its widespread application, the framework complex data with unknown structures--poses a significantexhibits certain shortcomings when dealing with complex challenge: how to stabilize black-box outputs while effectivelydatasets. First, standard resampling schemes often fail to leveraging available prior information. This paper introduces aaccount for the underlying data structures; as a result, the task-oriented randomization methodology that adaptively tailorsdrawn samples cannot reflect the true data distribution, thereby its strategy to the underlying generative mechanisms of the input data, specifically addressing unstructured complexities. Second, effective sampling requires prior comprehensive suite of stability guarantees is proposed. Beyondknowledge of the distribution, which is often unattainable establishing rigorous theoretical foundations for stability, thein practical environments.

Attention mechanisms have demonstrated remarkable empirical success in identifying relevant information from large collections of tokens, yet the theoretical principles underlying this behavior remain poorly understood. We study a stylized softmax-attention model in which a query vector is learned by stochastic gradient ascent from a collection of informative and nuisance tokens. Exploiting the symmetry of the model, we derive a population objective and characterize the limiting ordinary differential equation governing the learning dynamics. Using tools from stochastic approximation and dynamical systems theory, we establish a rigorous connection between the stochastic learning algorithm and its deterministic limit. Our main result shows that, under suitable high-dimensional scaling assumptions and standard step-size conditions, the learned query converges almost surely to the one-dimensional signal subspace spanned by the latent informative direction. Equivalently, the query asymptotically recovers the latent signal up to the intrinsic sign ambiguity. These results provide a rigorous theoretical foundation for understanding attention mechanisms as signal extraction procedures in high-dimensional noisy environments and offer a dynamical-systems perspective on how attention discovers relevant information in the presence of substantial noise.

This paper introduces the Real-DRL framework for safety-critical autonomous systems, enabling runtime learning of a deep reinforcement learning (DRL) agent to develop safe and high-performance action policies in real plants (i.e., real physical systems to be controlled), while prioritizing safety! The Real-DRL consists of three interactive components: a DRL-Student, a PHY-Teacher, and a Trigger. The DRL-Student is a DRL agent that innovates in the dual self-learning and teaching-to-learn paradigm and the real-time safety-informed batch sampling. On the other hand, PHY-Teacher is a physics-model-based design of action policies that focuses solely on safety-critical functions. PHY-Teacher is novel in its realtime patch for two key missions: i) fostering the teaching-to-learn paradigm for DRL-Student and ii) backing up the safety of real plants. The Trigger manages the interaction between the DRL-Student and the PHY-Teacher. Powered by the three interactive components, the Real-DRL can effectively address safety challenges that arise from the unknown unknowns and the Sim2Real gap. Additionally, Real-DRL notably features i) assured safety, ii) automatic hierarchy learning (i.e., safety-first learning and then high-performance learning), and iii) safety-informed batch sampling to address the learning experience imbalance caused by corner cases. Experiments with a real quadruped robot, a quadruped robot in NVIDIA Isaac Gym, and a cart-pole system, along with comparisons and ablation studies, demonstrate the Real-DRL's effectiveness and unique features.

Accurately modeling complex dynamic spatio-temporal systems requires capturing flow-mediated interdependencies and context-sensitive interaction dynamics. Existing methods, predominantly graph-based or attention-driven, rely on similaritydriven connectivity assumptions, neglecting asymmetric flow exchanges that govern system evolution. We propose Spatio-Temporal Flow, a physics-inspired paradigm that explicitly models dynamic node couplings through quantifiable flow transfers governed by conservation principles. Building on this, we design FlowNet, a novel architecture leveraging flow tokens as information carriers to simulate source-todestination transfers via Flow Allocation Modules, ensuring state redistribution aligns with conservation laws. FlowNet dynamically adjusts the interaction radius through an Adaptive Spatial Masking module, suppressing irrelevant noise while enabling context-aware propagation. A cascaded architecture enhances scalability and nonlinear representation capacity. Experiments demonstrate that FlowNet significantly outperforms existing state-of-the-art approaches on seven metrics in the modeling of three real-world systems, validating its efficiency and physical interpretability. We establish a principled methodology for modeling complex systems through spatio-temporal flow interactions.