Papers

Gradient boosted decision trees require a stopping rule to avoid overfitting. The standard rule monitors a validation loss and stops if the loss fails to improve for a fixed patience period. However, the patience parameter has no interpretable scale and validation losses can be noisy or implicitly defined by a user-specified gradient. We propose ScoreStop, a gradient-based early-stopping rule that casts the stopping decision at each iteration as a test of the null hypothesis that the current predictor is the population risk minimizer. We use a functional score test, computed on validation data, with a statistic that is scale-invariant in the update direction, with a known asymptotic distribution under the null. Because our test uses gradients rather than loss values, the same construction applies to implicit losses such as LambdaRank, and data-dependent losses such as Cox regression via influence functions. In synthetic experiments and real-data benchmarks, we show that ScoreStop is competitive with loss-based methods.

Generalization remains a central bottleneck for vision-language-action (VLA) models: under distractors, appearance shifts, and semantically similar tasks, the policy must often infer local execution details from coarse instructions while also deciding which parts of the image matter for control. We present S2 (See Less, Specify More), a framework for improving VLA generalization by training the executor under a cleaner interface. Specify More preserves the original instruction as a stable high-level goal while relabeling each trajectory into refined trajectory- and subtask-level language that disambiguates the current execution mode. Unlike native attention, See Less imposes an explicit visual evidence budget, training the executor to act from task-sufficient evidence rather than unconstrained visual context, without any region or mask annotation. This interface lets the executor follow detailed guidance without relying on distracting visual patches or resolving avoidable ambiguity on its own, and it remains compatible with off-the-shelf VLM planners through in-context learning. Across our main evaluation settings, S2 improves overall generalization metrics by changing the executor's learning problem: coarse instructions induce avoidable supervision aliasing, goal-preserving local guidance outperforms instruction replacement in our main ablations, and explicit evidence budgeting reduces dependence on broad visual context beyond efficiency considerations. Across eight real-robot tasks on TX-G2 (an AgiBot G2-compatible variant) and HSR, S2 raises mean subtask success from 54.2% to 79.0% over pi0.5. Together, these results suggest that VLA generalization improves when the executor is trained to act from informative local guidance and task-sufficient visual evidence, rather than recovering both from weak supervision.

On-Policy distillation (OPD) in large language models is shifting from full-trace KL supervision toward more selective training paradigms. Recent OPD methods increasingly focus on selecting which trajectories to learn from, which tokens are most informative, and which supervision signals are most reliable. Motivated by this trend, we rethink optimization granularity of OPD and propose \fireicon\ FiRe-OPD (Filter, then Reweight), which jointly adjusts supervision signals at both trajectory and token levels. In details, FiRe-OPD first filters trajectories to remove low-quality rollout samples, and then applies soft reweighting within the retained trajectories to emphasize informative tokens. Compared with hard token selection, FiRe-OPD leverages a soft-weighting mechanism to effectively mitigate information loss and enhance optimization stability, thereby achieving finer-grained OPD optimization. We validate the effectiveness of FiRe-OPD across strong-to-weak, single-teacher, and multi-teacher settings, and demonstrate its superiority over recent token-level OPD methods ( (e.g., +6.25 on AIME 2024 in strong-to-weak, +18.81 on Miner in multi-teacher). Our code is available at https://github.com/YuYingLi0/FiRe-OPD.

Hybrid Transformers reduce the cost of long-context modeling by combining local attention with periodic full-attention layers. When global communication is already available, however, the best use of local computation remains unclear. We introduce Post-Boundary Bridge (PBB), which preserves causal attention within blocks and adds direct connections across their boundaries. Rather than extending the range of information relayed through successive local layers, PBB prioritizes within-block modeling and nearby exchange, leaving long-range communication to full-attention layers. Across dense and mixture-of-experts models with 205 million to 2.07 billion stored parameters, PBB hybrids retain near-Full perplexity and competitive performance on standard downstream benchmarks while improving controlled source retrieval. At 205 million parameters, PBB also matches a hybrid using sliding-window attention (SWA) in perplexity and achieves higher source-retrieval accuracy. The same boundary-aligned structure enables Flash-PBB, an exact decoding implementation that updates its key-value cache without relocating retained entries. Compared with Flash-SWA, Flash-PBB delivers 1.82x decode attention-core throughput with half the allocated local key-value cache. These results show that targeted boundary exchange can preserve model quality while enabling faster, more memory-efficient local attention between global layers.

Multimodal systems often benefit from combining information across language, sound, and visual streams, but this benefit is not guaranteed. A modality that is useful for one input may become distracting for another, and local feature responses within the same modality can disagree with evidence from other sources. This work investigates how to adjust multimodal representations before they are merged by a downstream predictor. We develop a compact calibration module that compares each modality with the others at the summary level, extracts cues of cross-source support and conflict, and converts these cues into instance-wise and dimension-wise modulation signals. The calibration is applied to the original modality features rather than to already fused representations, enabling the model to suppress misleading components, preserve weak but useful evidence, and emphasize responses that are better supported by the current multimodal context. The module is designed as a plug-in component and can be attached to different fusion backbones without changing their prediction heads. Across five benchmarks covering sentiment understanding, action recognition, audio-visual event detection, and audio-visual emotion classification, the proposed pre-combination calibration strategy improves performance under both sequence-based and convolutional fusion settings. Additional analyses under modality removal, synthetic corruption, training dynamics, and feature-level visualization show that calibrating signals before fusion can reduce interference from unreliable modalities and produce more stable multimodal optimization.

Graphs have been used to enhance large language models (LLMs) for structured reasoning, mostly as external knowledge sources are provided to models at test time. In this paper, we take a different view: the value of graphs for LLMs lie not only in supplying information, but also in organizing reasoning. Inspired by how humans use graph-structured mind maps to organize branching and converging thoughts, we ask whether graphs can serve as an internal form of reasoning assistance. We study this question on multi-hop question answering tasks, where teacher-provided reasoning traces are rewritten as graph mind maps and used to guide a student model. Our experiments reveal a clear modality gap. When graph structures are flattened into text, their benefits become limited once direct answer hints are removed. Under this abstract guidance setting, both reasoning efficiency and answer quality degrade substantially. In contrast, visual graph guidance remains effective without direct answer clues, and its advantage persists after supervised fine-tuning and KL-based distillation. The above findings support the claim that graphs should be studied not only as external knowledge structures for LLMs, but also as visual scaffolds for organizing reasoning.

Machine learning predictors have become essential tools for guiding automated decision making. However, a major misalignment persists: predictive models are typically optimized in terms of standard statistical metrics in isolation from the algorithmic tasks they inform. We highlight this incongruity in the high-stakes domain of organ allocation by demonstrating that any algorithm relying on (even highly accurate) survival predictors optimized for standard metrics – such as the Concordance index (C-index) – can yield arbitrarily poor outcomes when used for allocation, failing to guarantee utility better than a uniform random selection. To bridge the gap between survival analysis and policy optimization, we introduce a decision-focused learning approach based on optimizing normalized discounted cumulative gain (NDCG), a mainstay metric in information retrieval. We establish the utility of NDCG in survival analysis by proving that it translates to guarantees on the performance of allocation. Empirically, we propose a bootstrapping approach to optimize the NDCG of existing survival models. Unlike prior work, we also address the challenge of right censorship when evaluating ranking. On historical heart transplant data from the US, our method dramatically boosts the NDCG of baseline models by 50-100%, which translates to tens of thousands of additional life years gained annually when deployed for transplant allocation. We anticipate that our framework will find broader applications in decision making with predictions.

Many recent multivariate time series anomaly detection (MTSAD) models incorporate cross-channel modeling, under the implicit assumption that the structure of anomalies may be spread across multiple channels. We evaluate this assumption on eight widely used public benchmarks by introducing a per-segment diagnostic framework that flags, for each labeled anomaly, whether at least one channel deviates individually from its normal history, whether the cross-channel correlation structure changes, or both. The framework shows that no cross-channel rupture occurs without an accompanying univariate deviation across a range of reasonable thresholds. A complementary metric also reveals that on six of the eight benchmarks, at least half of the labeled anomaly segments deviate univariately on 89% to 100% of their timesteps, reaching 100% on three of these datasets. To verify that our framework captures cross-channel structure when present, we construct synthetic data of phase-shifted sinusoidal channels with shared noise. Each anomalous segment is altered through one of two channel-wise corruptions that preserve the per-channel marginal distribution while breaking cross-channel structure, and our framework correctly characterizes these segments as cross-channel-only. On these data, channel-dependent (CD) models successfully exploit the cross-channel signal whereas channel-independent (CI) ones fail. The CI/CD comparison of a recent SOTA detector on real benchmarks further confirms that CD modeling brings no measurable gain. We conclude that current MTSAD benchmarks are unsuitable for validating cross-channel modeling capabilities, and we call for the development of more structurally diverse evaluation sets. The code for this study is publicly available.

Latent state-space models are widely used to study partially observed dynamical systems, yet most formulations assume that process variability is independent of latent-state position. In many biological, behavioral, and physiological systems, however, variability may depend systematically on the underlying dynamical state, producing structured stochasticity that is not captured by constant-variance models. We introduce a state-coupled stochastic volatility framework in which latent process variance depends on displacement from a latent equilibrium. To estimate this relationship under partial observation, we develop a particle expectation-maximization procedure combining bootstrap particle filtering and backward trajectory smoothing. The model includes a coupling parameter, $\gamma$, that quantifies the strength of association between latent-state position and process variability. A large-scale simulation benchmark evaluated recovery and detection performance across varying coupling strengths, observation noise levels, trajectory lengths, and persistence regimes. The proposed framework consistently reduced recovery bias relative to an observed-state heteroskedastic proxy, with the largest improvements occurring under strong coupling. Recovery performance improved with increasing latent persistence, while detection performance remained competitive across a broad range of conditions and became increasingly advantageous as observation noise increased. Taken together, the results demonstrate that state-coupled volatility can be identified and estimated under partial observation when latent-state structure is explicitly modeled. The framework provides a practical methodological foundation for studying state-dependent variability and evaluating whether structured stochasticity contributes information about system dynamics beyond that contained in mean-state trajectories alone.

Recent advances in machine learning have produced probabilistic weather forecasting models comparable to state-of-the-art numerical weather predictors. But no model consistently dominates spatio-temporally, and relative performance is highly context-dependent. This motivates adaptive methods for combining multiple forecasts to obtain improvements and robustness. While combined forecasts have been proposed in the literature, these are achieved either through supervised learning or through prediction with expert advice methods. We introduce AdaWeather, an adaptive framework that combines many probabilistic forecasts using both machine learning as well as mixture of experts to arrive at a unified improved probabilistic forecast. While traditional expert methods develop the regret bounds with respect to the best single expert in hindsight, we extend the algorithm and analysis to show our method has logarithmic regret compared to the best static mixture of experts in hindsight. Empirically, we focus on forecasting temperature, and observe improvements over existing methods.

Machine learning has accelerated quantum chemistry but is hindered by the prohibitive cost of generating high fidelity training data. Multifidelity machine learning (MFML) mitigates this overhead by systematically combining abundant low fidelity data with sparse high fidelity data. In spite of its success, standard MFML schemes rely on pre-defined scaling factors to determine sparse data ratio across fidelities, often generating redundant multifidelity data resulting in a loss of efficiency. Here, we introduce an adaptive on-the-fly multifidelity framework for machine learning that autonomously determines training dataset composition. By dynamically querying training samples at each fidelity, the algorithm saturates model accuracy at lower fidelities before moving up to more expensive reference calculations. We benchmark the novel adaptive-MFML across diverse chemical properties including theoretical best estimate ab initio potential energy surfaces, golden standard coupled cluster energies, and the more chemically challenging excitation energies. In our numerical experiments we show that our adaptive algorithm reduces data generation costs by up to a factor of 20 compared to single fidelity methods and improves upon standard MFML by up to a factor of 3. The mitigation of data redundancy establishes a high-accuracy low-cost pathway for sustainable cost-aware machine learning in quantum chemistry.

Accurate precipitation nowcasting is vital for disaster mitigation, but deep learning methods face a key trade-off: regression models produce over-smoothed, spectrally decaying predictions that blur convective details and violate turbulence power laws; diffusion models generate realistic yet unanchored hallucinations lacking physical grounding. We propose Spectral-Decoupled Iterative Refinement (SDIR), a deterministic framework that reformulates nowcasting as progressive frequency-decoupled refinement. SDIR first extracts a stable low-frequency synoptic skeleton, then iteratively refines high-frequency textures under physical constraints, eliminating both blurring and hallucinations. It features a dual-path design: the Synoptic Frequency-Guided Former (SFG-Former) with Scale-Adaptive Transformers for global structure, and the Fourier Residual Refiner (FR-Refiner) with Scale-Conditioned Fourier Neural Operators for fine residuals. A Physically Consistent Power Spectral Density (PCPSD) loss with dynamic masking enforces a turbulence-consistent spectral distribution. Experiments on three benchmarks show SDIR significantly outperforms SOTA methods in spatial accuracy while achieving spectral fidelity competitive with diffusion-based methods, enabling reliable high-resolution operational nowcasting. Code link: https://github.com/RuntimeWarning/SDIR.

Multimodal data fusion involves integrating and analyzing information from multiple modalities to uncover latent correlations and complementary patterns, thereby enhancing data processing and decision-making. While existing methods for structured multimodal inputs are typically designed around specific tasks and assume fully observed modalities, real-world applications often suffer from uncertain or missing modality inputs due to various factors. Some traditional models overly emphasize local interactions within missing modalities, neglecting the global complementary cues embedded in multimodal representations. To overcome these limitations, we propose a Dynamic Multimodal Data Fusion model based on Contrastive Learning (CL-DMDF). CL-DMDF introduces a novel attention mechanism that operates across both feature and modality dimensions to compute reliable attention scores, effectively reflecting importance at each level. The CL-DMDF further incorporates an entity-centroid contrastive learning module that constructs centroid-based positive samples from entity features to enhance discriminative learning. Additionally, an adaptive fusion module is employed to improve the efficiency and accuracy of dynamic fusion strategies. Extensive experiments conducted on three datasets demonstrate the effectiveness of the CL-DMDF across diverse multimodal fusion tasks.

The standard generalization bounds assume that the training and deployment distributions are the same, or are static, and don't consider regime switching environments where the ratio of calm vs crisis states is different. This paper proposes a framework that generalizes regime-aware models by quantifying the extra risk due to regime composition mismatch, when distribution shifts are Markov-switching. We obtain an exact decomposition, separating regime mismatch from regime sensitivity; we extend the bound to beta-mixing data using the effective sample size corrected for the spectral gap; and we show a minimax lower bound for synthetic data and on 25 years of global equity indices. The proposed penalty is an ex post realized generalization gap, whereas the training-only estimator does not show significant correlation: the feature geometry of crises can be detected, but not the temporal arrival. Thus, the framework is not a forecast machine. Forecasting the composition of the future regime is an open question in the rare cases of regime change.

External regret certifies stability only against replacing one's behavior by a fixed alternative. In a quantum game, this misses a natural physical move: a player can apply a local completely positive trace-preserving (CPTP) map to the state it actually received or prepared. We introduce coherent swap regret as the regret benchmark against all such local CPTP deviations, and give an algorithm achieving $O(\sqrt{dT\log d})$ coherent swap regret via entropic mirror ascent on the CPTP Choi slice with a fixed-point play rule. The main result is a three-level deviation-class landscape. Replacement channels recover ordinary external regret at rate $\Theta(\sqrt{T\log d})$. Unital channels, including unitary deviations and mixtures of unitaries, have zero minimax regret. Deterministic measurement-and-preparation channels already force $\Omega(\sqrt{dT\log d})$ regret in the moderate-horizon regime, and this rate is also sufficient for all CPTP deviations. Thus the hardness comes from non-unital use of the recommendation register, not from quantum coherence alone. As an application, decentralized full-information learning in finite quantum games reaches an $\varepsilon$-approximate separable quantum correlated equilibrium after $T=O(\max_i d_i\log d_i/\varepsilon^2)$ rounds. We identify these equilibria with channel-proofness of mediated quantum recommendation protocols, give an SDP audit for local CPTP exploitability applicable to arbitrary finite-dimensional states, and include a probing-bandit extension with pseudo-regret $O(d^{4/3}T^{2/3}(\log d)^{1/3})$ under Haar-random pure-state probes.

Multimodal Large Language Models (MLLMs) achieve strong performance through instruction tuning, but real-world deployment requires them to continually acquire new vision-language capabilities, making Multimodal Continual Instruction Tuning (MCIT) essential. To reduce inter-task interference and promote collaboration, recent methods often employ sparse architectures like Mixture of LoRA Experts with image-text similarity routing. However, tasks with distinct response structures could share highly similar visual-linguistic semantics and thus be wrongly routed to the same expert; image-text similarity alone is insufficient for reliable task assignment. For example, an expert in a grounding task requiring coordinate prediction may be biased toward producing short textual answers after learning semantically similar VQA tasks. This format-blind task assignment integrates heterogeneous response types into shared parameters, inducing gradient interference and ineffective expert collaboration. To address this problem, we propose ProtoAda, a prototype-guided adaptive tuning framework. ProtoAda introduces format-aware task prototypes to align task assignment and routing with both task semantics and output structure, and further consolidates format-compatible updates in a geometry-aware manner to effectively reuse and progressively refine existing parameters. Extensive experiments on multiple benchmarks demonstrate that ProtoAda achieves superior performance, especially on tasks whose answer structures are easily corrupted by sequential tuning.

Heterogeneous Differential Privacy (HDP) in Federated Learning (FL) allows clients to select individual privacy budgets ($\varepsilon_i$) according to institutional policies and data sensitivity. In practice, many HDP-FL systems employ $\varepsilon$-aware server aggregation to improve model utility by re-weighting client updates according to their declared privacy budgets. However, gradient updates in FL retain structural patterns induced by non-independent and identically-distributed (non-IID) data, and these additional signals exposed by $\varepsilon$-aware aggregation create new opportunities for inference by an honest-but-curious server. In this work, we first show that a server equipped with gradient denoising and surrogate modeling can mount a Privacy Inference Attack that infers distributional attributes of clients and links updates from the same client across training rounds, measured via surrogate inference accuracy and linkage success, under realistic knowledge constraints. The Shuffle-Model has been widely studied as a defense against such inference risks by anonymizing update sources, but it is fundamentally incompatible with HDP-FL $\varepsilon$-aware aggregation. To address this challenge, we propose IntraShuffler, a middleware defense framework designed for HDP-FL systems. IntraShuffler introduces a privacy-aware shuffling mechanism that groups clients into privacy-compatible buckets and performs parameter-level shuffling within each bucket to disrupt persistent gradient structure while preserving $\varepsilon$-aware aggregation. Experiments across four different datasets show that IntraShuffler reduces gradient recoverability by over 60% and decreases surrogate inference accuracy from 0.78 to 0.33 while maintaining comparable model utility across multiple FL aggregation rules.

Autonomous robots that interact with people must make safe and efficient decisions under human-induced uncertainty, such as their preferences, goals, competency, and willingness to cooperate. Safety filters are a popular approach for ensuring safety in interactive robotics, since their modular design separates safety from performance, allowing robots to operate safely around people with minimal impact on task efficiency. While traditional safety filters typically operate only in the physical space, neglecting the robot's ability to learn and adapt online, the recently proposed belief-space safety filter (BeliefSF) reasons about robot safety in closed-loop with runtime inference that actively reduces the robot's uncertainty online, thereby reducing conservativeness in filtering. However, providing formal safety guarantees for robots deploying BeliefSF remains a significant challenge due to errors in runtime inference and neural approximation of safety filters required to handle the high dimensionality of belief spaces. In this paper, we propose an algorithmic approach to certify high-probability safety of BeliefSF using conformal prediction, while explicitly accounting for the reliability of the robot's runtime inference module. Our method leverages the structure of belief-space safety filtering by focusing verification on a region where inference is expected to be reliable. It preserves the simplicity and sample complexity of standard conformal prediction, yet can certify a substantially less conservative safety filter. Through a simulated human-vehicle interaction benchmark, we show that our approach verifies a significantly more permissive belief-space safety filter than a standard conformal prediction baseline.

Large language models now power robo-advisors and trading agents, yet whether they carry built-in biases toward specific assets is largely untested. We ask three questions: do LLMs systematically prefer certain financial instruments; can an internal representation with causal leverage over those preferences be identified; and does that representation affect downstream financial decisions? We develop a three-level audit protocol and apply it to Bitcoin. First, a behavioral audit of nine frontier LLMs shows that Bitcoin's ranking among money-like instruments is frame-dependent: models place it around rank 5 of 8 as "reliable money" but near the top under crisis and autonomous-agent frames, and an attribute-swap experiment shows that rankings track functional properties, not names. Second, we open a model's internals: a search across thousands of sparse-autoencoder features in Gemma 3 identifies a dominant Bitcoin-selective feature. Amplifying it shifts the model toward the asset and suppressing it shifts the model away, even when "Bitcoin" never appears in the prompt. Third, we test financial consequences: amplification raises Bitcoin's portfolio share by 5.2 percentage points while suppression lowers it by 4.6 pp, with amplification reallocating within crypto and suppression cutting total crypto exposure. We characterize this as bounded behavioral leverage (leverage meaning causal influence over outputs, not financial leverage): an identifiable internal feature can be perturbed to move financial choices, but only within measurable limits. The framework links internal representations to external recommendations, validated with random controls and mechanism boundaries. As LLMs become autonomous financial agents, this is a first step toward a behavioral layer for emerging know-your-agent (KYA) standards: knowing what an agent prefers, and how far that preference can be moved.

One-step text-to-image generators are attractive for deployment because they generate an image with a single forward pass, but preference finetuning them remains difficult: standard alignment methods often rely on policy likelihoods, denoising trajectories, differentiable reward gradients, or test-time optimization. We propose Drifting Preference Optimization (DrPO), an online preference-finetuning method for deterministic one-step generators. For each prompt, DrPO samples candidates from the current generator, ranks them with a target reward, and uses high- and low-scoring samples to synthesize a feature-space update direction. The update is a non-parametric dipole preference field plus a reference drift estimated from the frozen base generator, and is optimized through a detached feature-space regression target. The target reward is used only for ranking, so DrPO can train with large, black-box, or non-differentiable rewards while inference remains a single generator call. We evaluate DrPO on SD-Turbo and SDXL-Turbo with multiple target rewards and benchmarks, including HPSv3 and GenEval. DrPO improves alignment over reward-gradient-free one-step preference baselines and reduces HPSv3 training computation by $3.51\times$ under the matched effective-batch setting by removing reward-model backpropagation. Initial offline experiments suggest that sample-based gradient synthesis can also be used beyond online reward ranking.

Optimal transport (OT) provides a principled framework for mapping between probability distributions. Despite extensive progress, applying OT to large-scale data remains computationally demanding, and the resulting pointwise transport plans are often difficult to interpret. We introduce Optimal Mixture Transport (OMT), a scalable framework that shifts the transport paradigm from individual samples to mixtures of subpopulations, reformulating the transport problem as a strictly biconvex optimization with a unique global minimizer. We further establish theoretical guarantees on the stability of the OMT map, showing that bounded perturbations of the underlying distributions lead to bounded changes in the transport plan. By formulating subpopulations as exponential-family distributions, OMT decouples computational complexity from the sample size, scaling solely with the number of mixture components. We demonstrate the effectiveness and practicality of OMT on a wide range of synthetic benchmarks and real-world datasets, including image data and large-scale single-cell RNA sequencing measurements.

Inverse materials design is shifting materials discovery from forward prediction toward targeted proposal of candidates that satisfy objectives under physical constraints. Here, we review advances in generative crystal structure modeling, multimodal learning, and closed-loop design pipelines for crystalline solids. We survey how generators learn chemical-structural priors from databases to enable controllable sampling of periodic structures, comparing variational autoencoders, normalizing flows, autoregressive models, and diffusion models. Across these families, we examine where feasibility constraints and physical priors enter, from representations and training objectives to sampling-time guidance, screening, and relaxation. We also discuss multimodal learning combining crystal structures, thermodynamic and electronic information, microscopy, spectroscopy, processing context, and scientific text to construct materials representations. Inverse-design strategies integrating conditional generation with latent optimization, Bayesian optimization, reinforcement learning, and active learning are also examined. We highlight recurring failure modes, including surrogate exploitation, diversity collapse, distribution shift, and the stability-synthesizability gap, and outline evaluation based on validity, novelty, uniqueness, stability, and cost. To support credible claims, we define a nine-rung discovery-credibility ladder and propose a minimum reporting standard: declared matching tolerances and database snapshots; separate reporting of uniqueness, training-set memorization, and external rediscovery; novelty as a continuous distance distribution; energy-above-hull distributions with functional and hull version; relaxation-survival and dynamical stability rates; and validation cost per credible hit. Headline validity or S.U.N. rates without these disclosures should be treated as uninformative.

This work studies neural architectures for classifying symmetric positive-definite matrices, focusing on congruence-like layers, in which the input matrix is multiplied on the left and right by a (possibly rectangular) weight matrix $W$ and its transpose. Such layers lie at the core of the celebrated SPDNet and have also been employed independently for dimensionality reduction on positive-definite data. We show that the (semi)-orthogonality constraint commonly imposed on $W$ limits the expressivity of these layers: for certain activation functions, the resulting architecture collapses to a one-hidden-layer equivalent. This lack of expressivity follows from a loss of spectral diversity in congruence-like layers for semi-orthogonal $W$ and is a direct consequence of Poincaré's separation theorem. We then examine the choice of the final classifier, comparing several Riemannian classifiers and discussing their compatibility with the feature maps produced by congruence-like layers.

Recent advances in large language models and agentic AI systems have enabled significant progress in mathematical discovery, from solving competition problems to tackling research-level conjectures. However, open problems in computational mathematics have received comparatively less attention: research in this area often requires not only proofs but also numerical experimentation, adversarial constructions, and algorithm design. In this paper, we introduce an agentic research system, Iteris, designed for open problems in computational mathematics. We apply Iteris to two open problems from a recent Simons Workshop collection (arXiv:2602.05394). In these case studies, Iteris generated numerical evidence, constructions, and proof drafts that led, after expert review and correction, to verified results. The first result is a phase diagram for the asymptotic comparison between conjugate gradient and randomized coordinate descent on power-law spectra; the second is a counterexample showing that QR factorization with column pivoting can fail to select well-conditioned submatrices even under low coherence. These case studies suggest that agentic AI systems can participate meaningfully in research workflows for open problems in computational mathematics, while human validation remains essential.

Classical finite-difference solvers remain reliable tools for partial differential equations, but their efficiency depends on where mesh resolution is placed. Uniform refinement can waste degrees of freedom when solution difficulty is localised near sharp gradients, fronts, oscillations, or constraint-sensitive regions. This paper studies a hybrid strategy in which a physics-informed neural network (PINN) is used not as the final solver, but as an off-grid residual probe for adaptive mesh refinement. The PINN residual is sampled over the domain, converted into cellwise indicators, and used to guide refinement before the final approximation is computed by a finite-difference solver. The method is evaluated on three benchmarks. The main full-solver validation uses the one-dimensional viscous Burgers equation with a nonuniform finite-difference solve on the adapted meshes. PINN-threshold refinement attains final relative $L^2$ error $0.021067$ with $60$ degrees of freedom, compared with $0.022617$ for uniform refinement with $192$ degrees of freedom. At matched mesh size, PINN-threshold reduces the error by about $67.5\%$. PINN-Dörfler refinement gives similar performance, with error $0.021264$ using $58$ degrees of freedom. A gradient indicator remains slightly more accurate, so the result supports usefulness rather than universal superiority. Manufactured 2D and 3D proxy tests, based on a nonlinear Schrödinger equation and an incompressible Navier–Stokes system, show that PINN residuals can organise structured refinement and improve over random refinement, although they do not consistently outperform gradient or uniform baselines. The results support PINN-guided AMR as a residual-indicator strategy for transferring physics-informed diagnostic information into finite-difference mesh adaptation while preserving the classical solver as the final approximation engine.

Molecular dynamics (MD) is a key tool for simulating the dynamical behavior of atomic systems. However, MD is inherently serial, which makes it difficult to increase single-system throughput with concurrent compute. To address this, we introduce Langevin Speculative Dynamics (LSD), a distributed and model-agnostic speculative sampler for accelerating MD without adding relative error. Inspired by speculative methods in language and diffusion modeling, LSD uses a draft model to propose fast simulation steps and verifies them in parallel with a slower target model, applying a transport map from the draft to the target distribution. We extend speculative sampling to second-order Langevin dynamics, derive the achievable speedup as a function of physical parameters, show that LSD generalizes across different systems and draft-target combinations with a 3-9x speedup, and confirm theoretically and empirically that LSD samples trajectories from its target model distribution.

Multimodal agents are increasingly expected to operate interfaces on behalf of users, raising a central deployment question: can they truly substitute for humans in workflows that services deliberately protect against automation? CAPTCHA verification makes this question concrete. It is not merely a visual puzzle, but a human-verification boundary placed before account creation, content access, form submission, and other protected actions. We introduce Humanity's Last Line of Verification (HLL), a controlled benchmark that uses interactive CAPTCHA verification to evaluate whether agents can cross this boundary through grounded, human-like interaction rather than recognition alone. HLL covers diverse CAPTCHA interactions and exposes agents to controlled realism stressors, including cluttered webpages, harder task variants, and trace-conditioned validation of the solving process. We evaluate eight frontier multimodal agents in a closed-loop GUI environment. The results show that current agents remain brittle at this human-substitution boundary: performance varies sharply across verification types, degrades under realistic interface conditions, and drops further when correct answers must be supported by valid action traces. By exposing gaps in localization, action calibration, state tracking, and process consistency, HLL provides a concrete testbed for measuring how close multimodal agents are to acting as human substitutes in protected real-world workflows. Our code is available at https://github.com/XinhaoS0101/HLL

Parameter-efficient fine-tuning (PEFT) is usually treated as a cheaper alternative to full fine-tuning. We study a broader role: small trainable adapters as persistent local state on top of strong shared foundation models. In this framing, the base model provides shared competence while adapters carry instance-specific behavior such as preferences, skills, tool habits, and memory-like updates. We organize the problem around three scaling axes: Scale Up, where stronger shared priors make small local updates more useful; Scale Down, where we study how small adapters can be while remaining reliable; and Scale Out, where many persistent adapted instances coexist. MinT provides one infrastructure example for managing adapter identity, revision, provenance, evaluation, and serving residency. Together, the results suggest that PEFT can be a compact substrate for persistent personal models rather than only a budget substitute for full fine-tuning.

The rapid development of LLMs has significantly advanced tabular question answering, but most systems cannot perform future-oriented numerical prediction. To address this gap, we introduce a novel task, Open-Domain Tabular Question Answering for Future Data Forecasting and Reasoning, and propose the first dataset to cover time-series forecasting and forecast-based reasoning scenarios using real estate data. This task poses challenges in retrieving precise historical data, overcoming the forecasting limitations of LLMs, and standardizing responses for diverse queries. To solve the above challenges, we propose TimeFore, an LLM agent-based framework that decomposes the problem into three collaborative roles: a Retriever autonomously generates SQL to fetch data, a Forecaster invokes external time-series models for higher accuracy, and an Analyzer synthesizes the results to construct a precise and consistent final answer. Extensive experiments demonstrate the effectiveness of our TimeFore.

Existing evaluations of neural operators and in-context operator learning rely primarily on prediction error, but accurate output prediction does not guarantee the correct local dynamical structure. A model may match solutions while exhibiting incorrect sensitivities, distorted frequency response, spurious mode coupling, or unstable tangent behavior. We introduce a Jacobian-based spectral audit for in-context operator learning. For a fixed prompt, we differentiate the network output with respect to the query function and view the resulting Jacobian as a learned tangent operator. Projecting it onto Fourier modes, we obtain a local spectral characterization of the inferred operator, including frequency-dependent gains, phase structure, and cross-mode coupling. The audit complements standard prediction metrics by testing whether the model reproduces local mechanisms of the underlying PDE operator rather than only outputs. Across benchmarks, the audit reveals distinct operator-level phenomena, including phase transport, viscosity-dependent damping, nonlinear mode coupling, and reaction–diffusion stability structure. It also detects failures partially hidden by prediction-error metrics, including high-frequency degradation, incorrect phase recovery, and prompt–operator inconsistencies. Corrupted or internally inconsistent prompts lead to degraded tangent-operator structure even when pointwise predictions remain partially accurate. Our results suggest that prediction accuracy and local operator fidelity are distinct properties of learned neural operators. Our framework also provides a diagnostic for stability, sensitivity, and operator consistency.