The Koopman operator provides a principled framework for analyzing nonlinear dynamical systems through linear operator theory. Recent advances in dynamic mode decomposition (DMD) have shown that trajectory data can be used to identify dominant modes of a system in a data-driven manner. Building on this idea, deep learning methods such as VAMPnet and DPNet have been proposed to learn the leading singular subspaces of the Koopman operator. However, these methods require backpropagation through potentially numerically unstable operations on empirical second moment matrices, such as singular value decomposition and matrix inversion, during objective computation, which can introduce biased gradient estimates and hinder scalability to large systems. In this work, we propose a scalable and conceptually simple method for learning the top-k singular functions of the Koopman operator for stochastic dynamical systems based on the idea of lowrank approximation. Our approach eliminates the need for unstable linear-algebraic operations and integrates easily into modern deep learning pipelines. Empirical results demonstrate that the learned singular subspaces are both reliable and effective for downstream tasks such as eigen-analysis and multi-step prediction.

Large language models (LLMs) have shown remarkable progress in mathematical problem-solving, but evaluation has largely focused on problems that have exact analytical solutions or involve formal proofs, often overlooking approximationbased problems ubiquitous in applied science and engineering. To fill this gap, we build on prior work and present HARDMath2, a dataset of 211 original problems covering the core topics in an introductory graduate applied math class, including boundary-layer analysis, WKB methods, asymptotic solutions of nonlinear partial differential equations, and the asymptotics of oscillatory integrals. This dataset was designed and verified by the students and instructors of a core graduate applied mathematics course at Harvard. We built the dataset through a novel collaborative environment that challenges students to write and refine difficult problems consistent with the class syllabus, peer-validate solutions, test different models, and automatically check LLM-generated solutions against their own answers and numerical ground truths. Evaluation results show that leading frontier models still struggle with many of the problems in the dataset, highlighting a gap in the mathematical reasoning skills of current LLMs. Importantly, students identified strategies to create increasingly difficult problems by interacting with the models and exploiting common failure modes. This back-and-forth with the models not only resulted in a richer and more challenging benchmark but also led to qualitative improvements in the students' understanding of the course material, which is increasingly important as we enter an age where state-of-the-art language models can solve many challenging problems across a wide domain of fields.

Multi-source transfer learning provides an effective solution to data scarcity in realworld supervised learning scenarios by leveraging multiple source tasks. In this field, existing works typically use all available samples from sources in training, which constrains their training efficiency and may lead to suboptimal results. To address this, we propose a theoretical framework that answers the question: what is the optimal quantity of source samples needed from each source task to jointly train the target model? Specifically, we introduce a generalization error measure based on K-L divergence, and minimize it based on high-dimensional statistical analysis to determine the optimal transfer quantity for each source task. Additionally, we develop an architecture-agnostic and data-efficient algorithm OTQMS to implement our theoretical results for target model training in multisource transfer learning. Experimental studies on diverse architectures and two real-world benchmark datasets show that our proposed algorithm significantly outperforms state-of-the-art approaches in both accuracy and data efficiency. The code is available at https://github.com/zqy0126/OTQMS.

More than 1,000 civilians in Sudan have been killed in drone strikes in the first five months of 2026, according to the United Nations. The death toll is due to a "sharp" increase in the use of drone warfare in the country's vicious civil war, UN High Commissioner for Human Rights (UNHCHR) Volker Turk said in a speech on Monday. On top of documenting more than 1,000 civilians being killed in the first five months of this year, the UN office also reported "rampant" levels of sexual violence, including rape. The war in the African nation started in April 2023 when a rivalry between Sudan's army chief, Abdel Fattah al-Burhan, and the commander of the paramilitary Rapid Support Forces, Mohamed Hamdan "Hemedti" Dagalo, exploded into war. The conflict, which had first started in the capital Khartoum, soon spread to several areas of the country.

Breast cancer remains a leading cause of death among women, with early de(NCCT) tection significantly scans of the impro chest, ving routinely prognosis.

For continuous systems modeled by dynamical equations such as ODEs and SDEs, Bellman's principle of optimality takes the form of the Hamilton-Jacobi-Bellman (HJB) equation, which provides the theoretical target of reinforcement learning (RL). Although recent advances in RL successfully leverage this formulation, the existing methods typically assume the underlying dynamics are known a priori because they need explicit access to the drift and diffusion coefficients to update the value function following the HJB equation. We address this inherent limitation of HJB-based RL; we propose a model-free approach still targeting the HJB equation and the corresponding temporal difference method. We prove exponential stability of the induced continuous-time dynamics, and we empirically demonstrate the resulting advantages over transition-kernel-based formulations. The proposed formulation paves the way toward bridging stochastic control and model-free reinforcement learning.

In multimodal large language models (MLLMs), the length of input visual tokens is often significantly greater than that of their textual counterparts, leading to a high inference cost. Many works aim to address this issue by removing redundant visual tokens. However, current approaches either rely on attention-based pruning, which retains numerous duplicate tokens, or use similarity-based pruning, overlooking the instruction relevance, consequently causing suboptimal performance. In this paper, we go beyond attention or similarity by proposing a novel visual token pruning method named CDPruner, which maximizes the conditional diversity of retained tokens. We first define the conditional similarity between visual tokens conditioned on the instruction, and then reformulate the token pruning problem with determinantal point process (DPP) to maximize the conditional diversity of the selected subset. The proposed CDPruner is training-free and model-agnostic, allowing easy application to various MLLMs. Extensive experiments across diverse MLLMs show that CDPruner establishes new state-of-the-art on various visionlanguage benchmarks. By maximizing conditional diversity through DPP, the selected subset better represents the input images while closely adhering to user instructions, thereby preserving strong performance even with high reduction ratios. When applied to LLaVA, CDPruner reduces FLOPs by 95% and CUDA latency by 78%, while maintaining 94% of the original accuracy.

We study deployment-efficient reward-free exploration with linear function approximation, where the goal is to explore a linear Markov Decision Process (MDP) without revealing the reward function, while minimizing the number of distinct policies implemented during learning. By "deployment efficient", we mean algorithms that require few policies deployed during exploration - crucial in real-world applications where such deployments are costly or disruptive. We design a novel reinforcement learning algorithm that achieves near-optimal deployment efficiency for linear MDPs in the reward-free setting, using at most H exploration policies during execution (where H is the horizon length), while maintaining sample complexity polynomial in feature dimension and horizon length. Unlike previous approaches with similar deployment efficiency guarantees, our algorithm's sample complexity is independent of the reachability or explorability coefficients of the underlying MDP, which can be arbitrarily small and lead to unbounded sample complexity in certain cases - directly addressing an open problem from prior work. Our technical contributions include a data-dependent method for truncating stateaction pairs in linear MDPs, efficient offline policy evaluation and optimization algorithms for these truncated MDPs, and a careful integration of these components to implement reward-free exploration with linear function approximation without sacrificing deployment efficiency.

Multimodal Large Language Models (MLLMs) are increasingly deployed in finetuning-as-a-service (FTaaS) settings, where user-submitted datasets adapt generalpurpose models to downstream tasks. This flexibility, however, introduces serious security risks, as malicious fine-tuning can implant backdoors into MLLMs with minimal effort. In this paper, we observe that backdoor triggers systematically disrupt cross-modal processing by causing abnormal attention concentration on non-semantic regions--a phenomenon we term attention collapse. Based on this insight, we propose Believe Your Eyes (BYE), a data filtering framework that leverages attention entropy patterns as self-supervised signals to identify and filter backdoor samples. BYE operates via a three-stage pipeline: (1) extracting attention maps using the fine-tuned model, (2) computing entropy scores and profiling sensitive layers via bimodal separation, and (3) performing unsupervised clustering to remove suspicious samples. Unlike prior defenses, BYE requires no clean supervision, auxiliary labels, or model modifications. Extensive experiments across various datasets, models, and diverse trigger types validate BYE's effectiveness: it achieves near-zero attack success rates while maintaining clean-task performance, offering a robust and generalizable solution against backdoor threats in MLLMs.

Graph neural networks have shown promising results in weather forecasting, which is critical for human activity such as agriculture planning and extreme weather preparation.