Papers

In simulation-based inference (SBI), neural posterior estimators trained on a bounded range of sample sizes can become inaccurate on larger datasets because their parameterizations do not specify how posterior location, uncertainty and shape must change as observations accumulate. We propose center-estimated NPE (CE-NPE), which combines a statistical center with an explicit contraction rate and learns a local center correction and residual densities in posterior-scale coordinates. The center is a statistical estimate, such as a closed-form statistic or a simulator-based moment match, which does not require an evaluable likelihood. Our analysis explains how contraction amplifies center error and decomposes Gaussian posterior error into shape, mean and covariance contributions. On six regular benchmarks, CE-NPE gives the best reference-posterior agreement of all tested amortized estimators at every out-of-range size, up to five times the largest training size, without retraining; it also fits better within the training range on most of them. Matched controls and nonregular examples characterize the scope of these gains and identify the conditions on which they depend: an appropriate contraction rate, an accurate center, and a stable residual representation.

Matching one distribution to many targets can require repeated evaluations of every target. CAL-$\alpha$-MODM aggregates a nonlinear predictor into one barycenter, calibrates its residual derivatives, and checks steps against the original $f$-divergence objective. Under smoothness assumptions, calibration controls parameter-step model error without requiring targets to be close to the model. Exact-oracle stationarity and local convergence guarantees are established. At common matching, canonical geometry cancels the leading quadratic iteration error, giving cubic convergence with second-order calibration and sufficiently accurate inner solves. High-precision experiments support this prediction. Sensitivity-based initialization also accelerates preference continuation along regular stationary branches. On 300-target Jensen–Shannon problems, CAL-p2 achieves median paired speedups of $3.15\times$ over L-BFGS and $1.10\times$ over second-order Direct-TR, across 23/25 and 24/25 objective-comparable stationary pairs. The savings arise when cheaper predictor queries offset calibration overhead.

We obtain very high genus asymptotics for double Hurwitz numbers when both the size $n$ and the genus $g$ go to infinity, under the constraint that $g$ grows much faster than $n$, more precisely that $\frac{2g}{n}-\ln n\to\infty$. Our proof relies on adapting arguments about the mixing time of the transposition random walk introduced by Diaconis and Shahshahani.

Progressive depth growth builds large networks by starting with shallow models and inserting layers during training. Recent advances in large language models have renewed interest in this longstanding approach, demonstrating benefits for training efficiency and reasoning alongside changes in internal computations. However, how growing affects representation quality and computational patterns in self-supervised vision transformers remains unclear. In this work, we systematically investigate progressive depth growth in two popular self-supervised families, namely masked autoencoders (MAE) and DINOv3, and analyze how growing affects downstream performance, intermediate representations, and internal computations. Across global and dense tasks, we show that growing improves downstream performance while reducing training FLOPs by 13.9–24.8%, with ImageNet linear-probe accuracy gains of up to 3.36 percentage points. Growing also improves downstream performance at intermediate layers and reshapes internal computational patterns, introducing recurring alternations between global and localized attention in MAE and earlier local-to-global attention progression in self-distilled DINOv3. Together, these findings establish progressive depth growth as an effective and efficient pretraining strategy for improving downstream performance across global and dense visual tasks with less computation, and advance our understanding of how growing shapes visual computation.

Global reanalyses are too coarse to resolve the convective systems that produce most West African monsoon rainfall. In this work, we use U-Net based architectures to downscale daily ERA5 fields (0.25{\deg}) to IMERG V07 satellite precipitation (0.1{\deg}) over West Africa for the June to October seasons of 2000 to 2024, using geopotential, temperature, humidity and wind at three pressure levels together with elevation and land fraction. Adding ERA5 precipitation as input slightly improves performance for this particular challenging region. We find that all the deterministic networks investigated for this work are too dry and lose almost all observed variance below a few hundred kilometers, which motivated our use of a GAN-based alternative. We also investigated permutation importance for the U-Net that ranks 500 hPa specific humidity first, consistent with the sensitivity of deep convection to mid-tropospheric moisture.

Calibration has been extensively studied for classifiers, but remains largely unexplored for crowdsourcing aggregation methods, which are increasingly used for making probabilistic decisions. To the best of our knowledge, we provide the first theoretical and empirical study of calibration in label aggregation. To our great surprise, we find that widely used aggregation methods can be severely miscalibrated, and in opposite directions: Majority Vote is underconfident while model-based aggregators are overconfident. We prove that the Bayesian aggregator is perfectly calibrated under its data-generating model. We also show that exact calibration of weighted majority votes requires highly restrictive conditions, which helps explain why Majority Vote is generally miscalibrated in realistic crowds. Empirically, we study calibration across $39$ human crowdsourcing datasets and LLM-based crowds. In both cases, we observe systematic miscalibration for aggregation models such as Majority Vote, Dawid–Skene, GLAD, and CrowdFM. Finally, we propose a post-hoc calibration procedure that aggregates the aggregators: for any aggregation function $\phi$, we introduce a transformation that first combines its predictive distribution with those of complementary aggregators, before applying temperature scaling. This simple and interpretable transformation makes temperature scaling more effective than when applied directly to $\phi$. It improves calibration, especially in the high-redundancy regime where there are many annotations per item, while generally preserving or even improving accuracy.

Large language models (LLMs) are increasingly used as judges to assign scalar scores that drive downstream decision-making, including tasks such as paper review and recommendation. However, LLM-as-a-Judge pipelines are vulnerable to score-hijacking prompt injection embedded in evaluated content, which corrupts scoring signals and distorts induced rankings. Existing defenses often enforce robustness by modifying the judge model itself, making safety dependent on a specific model's scale and training and requiring repeated effort when the judge is updated or replaced. We propose StableJudge, a lightweight defense framework that externalizes prompt-injection mitigation as a reusable pre-scoring capability. StableJudge centers on a compact 0.13B Sentry that localizes injected spans and performs targeted sanitization prior to evaluation, and is trained using adversarially evolving attacks to improve robustness beyond a fixed prompt set. To stabilize measurement, StableJudge derives deterministic, fine-grained scalar scores from the judge's output distribution. Across held-out scientific abstracts and out-of-distribution attack variants, StableJudge improves score recovery under attack while largely preserving general-purpose performance, demonstrating that robust LLM-based judging can be achieved by decoupling safety from judge scale via reusable defense capabilities. Code and data will be released.

Vision foundation models (VFMs) provide transferable visual representations, but their inference cost limits deployment in many settings. Fine-tuning specializes a VFM without shrinking it; distillation shrinks it, but through expensive teacher-guided training. Structured pruning offers a more direct route by extracting a compact subnetwork from the VFM. However, as compression gets aggressive, existing methods fail because they treat subnetwork selection and recovery as separate axes. Across layers, capacity must follow task sensitivity rather than spread uniformly. Within a layer, a removal cannot be judged by importance alone; its impact also depends on how well the lost contribution can be recovered from what remains. Selection and recovery must therefore be decided jointly. We introduce Carve VFM, which resolves this interdependence. Given a VFM, a calibration set, and a parameter budget, it (i) searches for task-specific capacity allocations across layers; (ii) guides the search via an adaptive recovery-aware fitness criterion; and (iii) reconstructs removed behavior through iterative closed-form low-rank corrections merged into the surviving weights. The result is a dense encoder produced in minutes on a single GPU without backbone retraining. Across twelve benchmarks and four models from DINOv3 and MetaCLIP-2, Carve VFM achieves the highest reported mean accuracy in seven of eight backbone-budget configurations among the evaluated dense structured-pruning methods. Further fine-tuning surpasses the strongest evaluated edge model (88.1% vs. 85.8%), and carved models cut compute and memory while delivering 1.3-2.0x higher GPU throughput than the uncompressed one. These results show that the compact model you need is inside the VFM; it only has to be carved out.

Symbolic regression (SR) aims to discover interpretable mathematical expressions from data and plays a key role in scientific discovery. However, existing methods face a common bottleneck: the enormous search space and severe combinatorial explosion make it difficult to achieve a favorable trade-off between accuracy and algorithmic efficiency. We argue that, in addition to designing increasingly complex solver-specific SR strategies, an equally promising direction is to equip them with a shareable auxiliary strategy that can provide reliable search guidance. To this end, we propose DASS, a solver-adaptable Dynamic Auxiliary Search Strategy that improves SR by constructing and refining a high-potential auxiliary search subspace. DASS represents the auxiliary search space as a functional subspace spanned by a set of high-potential basis function terms. It estimates term utility through multi-environment evaluation, filters unstable guidance, and updates term potential scores with a Gibbs posterior, thereby calibrating the subspace to provide more reliable guidance for downstream solvers. The high-quality solutions produced by the solver are used to update the set of basis terms. This forms a virtuous closed-loop optimization process: the subspace guides the downstream solver, and the solver feedback reshapes the subspace for subsequent exploration. DASS can be easily integrated with various types of SR solvers. Extensive experiments on LLM-SRBench show that DASS significantly improves the accuracy of original solvers with a manageable increase in runtime.

Chest X-ray (CXR) interpretation often requires integrating visual findings with task-specific evidence rather than relying on direct visual recognition alone. While tool-augmented medical agents can supply such evidence via external modules, they often lack explicit mechanisms for learning when auxiliary evidence is needed and for preventing inapplicable quantitative evidence from entering diagnostic reasoning. To address this, we propose X-Ragent, a post-training framework for tool-augmented reinforcement learning that decouples learned auxiliary-evidence seeking from validity-gated quantitative measurement acquisition for evidence-grounded CXR reasoning. We construct Rad-CoT, a 7.7K-sample corpus combining causally ordered tool-augmented trajectories with report-style diagnostic reasoning, and use it to initialize Qwen3-VL-8B via agentic supervised fine-tuning (SFT). We then optimize the policy with Group Relative Policy Optimization (GRPO) using an accuracy-gated reward that applies process-level terms for measurement consistency and auxiliary-tool cost only to correct trajectories, preventing plausible process behavior from compensating for an incorrect diagnosis. Across three CXR benchmarks, X-Ragent achieves the strongest results, including an 8.6-point improvement on the fully held-out ChestAgentBench. By seeking evidence selectively and admitting only valid measurements, X-Ragent supports more faithful and evidence-grounded CXR reasoning.

Hybrid analog–digital arrays reduce hardware cost in integrated sensing and communications (ISAC), but spatial compression and beam-dependent scan responses complicate multi-target estimation with few radio-frequency (RF) chains. To address these challenges, this paper proposes a burst beam-scanning multiple-input multiple-output orthogonal frequency-division multiplexing (MIMO-OFDM) framework that reuses downlink reference signals for joint range, radial-velocity, elevation, and azimuth estimation. Specifically, a structured global–local dual-tensor estimator separates targets by using scanning observations and refines their parameters from selected-beam RF-chain responses, with delay–Doppler signatures maintaining cross-stage target correspondence. The recovered subarray responses yield closed-form elevation and azimuth estimates through exact sum–difference relations, avoiding small-angle approximations and fine angular grid searches. To further improve Doppler accuracy, spatial-mask-aided weighted least squares exploits the longer inter-burst temporal aperture while resolving phase ambiguity using intra-burst estimates. The joint multi-target Cramér–Rao lower bound (CRLB) and corresponding three-dimensional (3D) positioning bound are also derived to characterize estimation performance. Simulation results show that the proposed scheme achieves near-CRLB estimation accuracy at high transmit powers and centimeter- to millimeter-level 3D positioning with only four RF chains, while maintaining low computational cost.

This study addresses the prediction of audio popularity on an online sharing platform by leveraging both metadata and acoustic feature representations. Two distinct categories of audio content are considered: music and sound effects. Popularity is formulated as a supervised classification task based on two target variables: the number of downloads and the average user rating. We conduct a comparative evaluation of multiple machine learning models, including Random Forest, XGBoost, and a Neural Network model. Experiments are performed across different feature sets, including metadata, platform-provided descriptors, and eGeMAPS acoustic features. Results indicate that metadata consistently provide the most informative representation for predicting popularity. Nevertheless, acoustic features alone achieve performance significantly above chance level, demonstrating that intrinsic properties of the audio signal contribute to popularity prediction.

We prove that the classical Bloch constant is strictly smaller than the Ahlfors–Grunsky value, disproving the famous Bloch constant conjecture. The Lean 4 formalization of the main results are also provided.

Driven by the scientific value of Venusian atmospheric studies, numerous missions, such as Magellan and Venus Express, have relied on single-spacecraft radio occultation (RO) experiments. Despite their proven success, data throughput is inherently limited by the use of a single transmitter and its relative geometry with respect to Earth-based receivers. Therefore, the literature proposes leveraging a constellation of spacecraft to increase RO measurement opportunities and improve spatial and temporal coverage; however, current efforts impose strict simplifying assumptions that limit their applicability and responsiveness to complex mission narratives. Beyond initial constellation design, evolving mission objectives may require event-based constellation reconfiguration during operations. To address both design and operational needs, we propose two problems formulated as integer linear programs: the RO Constellation Configuration Design Problem and the RO Constellation Reconfiguration Problem. Further, we address the programs' solution-space explosion with a Benders decomposition approach. Numerical experiments show the applicability of our problems for constellation configuration design, tackle the full mission pipeline, transitioning from global atmospheric profiling to on-demand regional profiling above areas of volcanic interest, and show that the Benders decomposition approach efficiently obtains high-quality solutions to large-scale instances.

Static dipole polarizabilities of H$_2$, LiH, and helium confined in a single-mode optical cavity are computed using quantum-electrodynamical full configuration interaction (QED-FCI). Response properties are obtained as finite-field derivatives of the coupled electron-photon energy. The cavity-induced modification is found to arise from the competition of two contributions of opposite sign, the dipole self-energy and photon exchange, whose relative strength depends on both the cavity frequency and the number of photons involved in the exchange. The residual change suppresses the response of the molecular ground electronic states. For the metastable states of helium atom this balance changes qualitatively, and the differential response of the $2\,^3S_1 \to 2\,^1S_0$ clock transition, which governs its sensitivity to static electric fields and to thermal radiation, is strongly enhanced instead. Cavity confinement can therefore either reduce or increase the dc Stark and blackbody-radiation shifts of a frequency standard, depending on the electronic structure of the states involved. The reported data serve as a reference for cavity-modified response and underscore the need for an exact treatment of light-matter correlation in cavity-QED predictions.

We find recurring signatures of intermittency in molecular-gas velocities from cloud to galactic scales. In 31 selected galaxies from PHANGS-ALMA, velocity-centroid exponents through sixth order show nonlinear order dependence and nearly linear relations between consecutive exponents across galaxies. Positive estimated offsets from single-exponent scaling occur in all 52 galaxies with a valid power law fit and 99 of 100 clouds. A generalized She-Leveque hierarchy accommodates these signatures, supporting a hierarchical description of turbulent scaling in molecular gas across these environments.

In this paper we develop several aspects of the algebraic theory of Girard algebras, regarded as the equivalent algebraic semantics of multiplicative additive Linear Logic $\mathsf{MALL}$. The main structural results concern the bottom of the subvariety lattice of the variety of Girard algebras: varieties generated by simple Girard algebras generated by the constants correspond exactly to the atoms of the lattice. Moreover the lattice itself is atomic, i.e. every nontrivial subvariety contains an atom. These atoms can be represented uniformly by the maximal filters of the free algebra generated by the constants. Moreover every congruence of this free algebra is realized by a subvariety, and the resulting atom-support topology records the behavior of arbitrary joins: the atoms below a join of atoms are precisely the points in the closure of the corresponding subset of the spectrum. The concrete constructions show that the bottom of the lattice is already quite rich. Besides determining the integral and degenerate boundary cases and exhibiting finite modular and nonmodular atoms and chains of covers, we constructed infinite families of atoms whose topology reflects genuinely infinitary behavior; thus finite and infinite joins behave essentially differently at the bottom of the subvariety lattice. On the logical side, these results yield $2^{\aleph_0}$ Post-complete extensions of $\mathsf{MALL}$.

Ball's weak cylinder conjecture asserts that a set of $p^2$ points in $\mathrm{AG}(3,p)$ with at least $p$ non-determined directions is a union of $p$ parallel lines. We prove the conjecture for sets admitting a polynomial graph representation of total degree at most $D$, provided that $p>D^{9/2}$. More precisely, a non-cylindrical graph of degree $D\ge3$ satisfies $|\mathcal{N}_S|\le p+1-\lceil p/(D-1)\rceil$ under the same hypothesis. This bound is attained when $p\equiv1\pmod{D-1}$. The proof relates non-determined directions to permutation polynomials on parallel affine lines. A rigidity lemma for moving line restrictions, together with a reconstruction of their compositional components, bounds the number of contributing directions uniformly in $p$. Directions with affine restrictions are treated separately; all other contributions are bounded using geometric indecomposability and the Weil bound for plane curves.

We study the vehicle routing problem (VRP) with stochastic demands, where each client has an independent demand with a known distribution, and its realized demand is observed only on the first visit. We consider a priori and restocking policies that fix the client order before observing demands and serve each client's entire demand before first visiting the next. Gupta, Nagarajan, and Ravi (Oper. Res., 2012) developed randomized a priori $(\alpha+1)$-approximation and $(\alpha+2)$-approximation algorithms for splittable and unsplittable deliveries, respectively, where $\alpha$ is the approximation ratio of a metric traveling salesman problem (TSP) algorithm. We improve these approximation ratios to $2$ and $3$ by using their replenishment rules with a TSP tour constructed via matroid parity. We also obtain deterministic a priori $(2+\varepsilon)$-approximation and $(3+\varepsilon)$-approximation algorithms by compressing the distributions of cumulative demands to select an initial stock in polynomial time. All guarantees are relative to the general adaptive optimum. We then establish matching lower bounds of $2$ and $3$ that hold even for the more general class of randomized restocking policies. Finally, we apply our matroid-parity construction to improve approximation ratios for several VRP variants with deterministic demands.

Tabular foundation models (TFMs) achieve state-of-the-art performance on IID benchmarks, but how do they respond to repeated measurements at the same inputs? Across six TFMs, synthetic and semi-synthetic experiments show that additional measurements can improve predictions at measured inputs while degrading them elsewhere. Harm is most frequent at low noise, and irrelevant features amplify it in several models; real repeated-measurement datasets show similar patterns. Statistically, the IID rule targeted by PFN pretraining treats each repeated input as fresh evidence about the input distribution. When the prior links this distribution to the response function, the resulting protocol mismatch can outweigh the benefit of additional measurements, even under exact inference. Computationally, interventions identify row-count dependence in TabPFN-v3's attention scaling as a contributor to harm. Resetting the count largely reverses distortion in a noiseless linear diagnostic and attenuates amplification by irrelevant features. Finally, we evaluate ways to retain the benefits of repeated measurements while limiting their harm. Averaging measurements usually outperforms supplying them separately at larger repetition counts, whereas count reset can suppress both harm and useful gains. Code, saved results and reproduction instructions are available at https://github.com/drizmiz/PFN-Repeated-Measurement-Supp

4Cast Lab is a free, open-source application for machine learning, econometrics and statistical analysis that runs on the user's own computer, with no installation, no programming and no data leaving the machine. It was developed at the E|M|E|F Lab, Department of Economics, Democritus University of Thrace, for researchers, analysts, instructors and students in any field. Its Forecasting mode trains eight learning algorithms (with three kernels for SVM/SVR), implemented from scratch, under leakage-safe validation for time-ordered data; it selects models by cross-validation, compares the selected model with every alternative and with naive benchmarks through formal tests of predictive accuracy, and documents each run in a methods report and a Python script that the user can export. Its Estimation mode covers the core of applied econometrics, from OLS with robust inference to ARIMA, ARDL, VAR/VECM and GARCH. This paper describes the design, the interface and the workflow of version 4.6.2, reports a validation of 1,234 numerical results against statsmodels, linearmodels, arch, scikit-learn and NIST certified values, illustrates the workflow with three examples, and discusses current limitations.

General Parallel Relativity is a dynamically equivalent formulation of Einstein gravity in which the parallel frame supplies explicit boundary reference data. In this formulation, finite-region energy-momentum and horizon entropy arise as distinct Hamiltonian surface charges. Diffeomorphisms along canonical frame directions define a quasi-local energy-momentum including a matter surface term, while normal-plane boosts at a smooth horizon give entropy. These covariant formulas are applied to the Reissner-Nordström black hole and to a dilatonic charged black hole. In both examples, the matter contribution and its Legendre transform complete the Misner-Sharp mass into the Hamiltonian energy appropriate to the chosen electromagnetic boundary datum. The resulting horizon charges satisfy the thermodynamic relation obtained independently from the Euclidean boundary action.

Chain-of-Thought (CoT) enables transformers to solve complex reasoning tasks by generating intermediate reasoning steps. Despite strong empirical success, the mechanisms by which such reasoning abilities emerge during training remain poorly understood, particularly from a theoretical perspective. In this work, we study this question in a stylized yet fully analyzable setting: in-context $\mathrm{K}$-nearest-neighbor ($\mathrm{K}$-NN) prediction with a one-layer transformer. Earlier work has demonstrated that one-layer transformers can be trained to perform $1$-NN without CoT (Li et al., 2024a) by having the softmax attention attend to the nearest neighbor in context. However, this result on $1$-NN cannot be extended to $\mathrm{K}$-NN for $K>1$, as softmax attention is much less naturally suited to attending to the 2nd through $K$-th nearest neighbors. We first give a rigorous negative result showing that this limitation already appears on a well-separated class of binary classification tasks for odd $K>1$. We then give an explicit construction of a one-layer transformer, and show that it can solve $\mathrm{K}$-NN via CoT reasoning. We further prove that, under a stylized training setup, gradient descent can recover this construction, thereby showing that transformers can acquire the capability to solve $\mathrm{K}$-NN through training. In addition, we show that the resulting trained model applies to a broader test class than that assumed in the training analysis. Our results shed light on how CoT expands the ability of transformers to learn and solve tasks that are otherwise hard to solve.

We study the information-theoretic limits of inference in rectangular rank-one spiked matrix models with additive rotationally invariant noise, thereby going beyond the standard assumption of independent noise entries. Using the replica method from statistical physics, we derive a variational formula for the mutual information between the signal and the data, together with a system of saddle point equations predicting the minimum mean squared error in estimating the rank-one spike. We further show that, in the Bayes-optimal setting, these saddle point equations coincide with the fixed point equations of the state evolution of a recently proposed orthogonal approximate message passing algorithm with Bayes-optimal denoisers. This equivalence provides an algorithmic interpretation of the replica prediction and supports the Bayes-optimality of orthogonal approximate message passing whenever the relevant fixed point is unique. Numerical evaluations of the saddle point equations for several rotationally invariant noise ensembles and signal priors, together with finite-size simulations of the algorithm, confirm the asymptotic predictions. They illustrate both regimes with a unique informative fixed point, where orthogonal approximate message passing is predicted to attain the minimum mean squared error, and regimes with multiple fixed points, where a statistical-to-computational gap exists.

Existing deep-learning-based channel state information (CSI) feedback methods reduce feedback overhead by compressing high-dimensional CSI into low-dimensional representations and/or exploiting historical CSI to predict future CSI. However, these methods may become unreliable when user equipment (UE) suddenly changes its motion. To address this issue, we propose a prediction-aided coding (PAC) framework for generative CSI feedback, which adaptively encodes CSI based on predicted channel/CSI distributions and the observed CSI in real time. Each CSI frame is represented by multiple CSI tokens obtained without using historical CSI, making token generation robust to abrupt UE motion. In the first stage, the conditional distribution, instead of a deterministic value, of the next-frame CSI tokens is predicted based on historical tokens. In the second stage, once the next frame arrives, the predicted distribution is used for variable-length entropy coding of its CSI tokens. To further reduce feedback overhead and inference latency, we propose efficient designs for both stages. For the first stage, a customized causal Transformer with a new positional embedding strategy accurately predicts next-frame token probabilities. For the second stage, a parallel causal decoding strategy reduces the inference latency caused by sequential token decoding. Extensive experiments show that, compared with existing history-aware baselines, PAC reduces feedback overhead by approximately 72% to 84% while achieving higher reconstruction accuracy. Additionally, PAC maintains CSI reconstruction accuracy under random-walk trajectories, whereas existing history-aware baselines suffer sharp degradation in reconstruction accuracy.

Established interpolation-based methods efficiently compute visibility from one source to every grid cell using values computed closer to the source. Interpolation yields fractional values even on binary grids of free or fully blocking cells. Two-dimensional path-counting theory interprets these values as fractions of unobstructed grid paths, but three-dimensional methods use different predecessor structures. Interpreting these values consistently across dimensions and on nonbinary grids remains unresolved. We provide a unified mathematical account that gives every visibility value a precise meaning in any dimension. We establish when grid refinement yields eventual agreement with continuous visibility, implement parallel solvers, and compare four three-dimensional methods in accuracy and throughput. We define cell transmission: zero blocks visibility, one preserves it, and intermediate values attenuate it proportionally. We prove each computed value equals the uniform finite-path average of transmission products over finitely many source-to-cell grid paths. It is the fraction of clear paths on binary grids and the mean path transmission on nonbinary grids. For a source–target query, the fractions of paths passing through cells at each step form a hypergeometric distribution in 2D, which we call the hypergeometric bridge. On the last step into the target, these probabilities equal the interpolation weights. Higher-dimensional interpolation weights arise from products of hypergeometric bridges. For any fixed dimension, the full-field algorithm remains linear in grid size. Exact path enumeration reproduces every checked visibility value. NumPy, OpenMP, and CUDA agree numerically, and our OpenMP solver achieves the highest throughput in the four-method CPU comparison at every tested resolution.

Proximal Policy Optimization (PPO) is widely used in reinforcement learning with verifiable rewards for language-model reasoning. While value evolution along trajectory characterizes relative token-level credit, yet standard critics often fail to track it under sparse terminal rewards. To model this evolution, we analyze the relation between successive prefix values and show that they admit a Bayesian decomposition with additive updates in log-odds space. We use this additive structure to construct Tracking-Value PPO (\method), a cumulative value parameterization that sums position-wise scalar outputs into value logits. These outputs are learned through terminal-outcome supervision on the reconstructed values. \method improves aggregate mathematical reasoning scores over PPO by 7.0 and 8.4 points on Qwen3-4B-Base and Qwen3-8B-Base, respectively, while improving out-of-distribution generalization. Gains extend to Qwen3-4B, Qwen2.5-Math-1.5B, Qwen3.5-9B and to countdown, and scientific coding tasks. Further analysis shows more accurate tracking of value evolution and more reliable advantage estimates. Our code is publicly available in https://github.com/LINs-lab/TPPO.

Interacting with unfamiliar objects requires understanding how their parts connect and move. We introduce ViewStruct, a framework for jointly predicting connectivity and articulation in objects and scenes. Given an RGB image with aligned part segmentation and depth, optionally supplemented by images or video, ViewStruct predicts rigid-body groupings, attachments, and joint parameters. Its view-grounded representation links predictions to observed regions and expresses joint geometry in a gravity-aligned, view-relative frame, without requiring category-specific canonical orientations. A pretrained vision–language model is adapted to predict structure in seconds, without requiring observed articulation motion or per-instance optimisation. Experiments demonstrate improved structural prediction over evaluated baselines and transfer to unseen categories, asset sources, and real-world human-interaction videos, despite task-specific training exclusively on synthetic data. Simulated manipulation experiments further demonstrate the utility of predicted joints for generating robot motion trajectories.

Adaptive optimization methods have received considerable attention in recent years due to their strong empirical performance. However, they generally require maintaining a preconditioning matrix, which incurs prohibitive computational overhead in large-scale applications. To address these limitations, we investigate the acceleration of preconditioned optimization methods through frequent directions (FD), a widely used matrix sketching technique. We first propose FTSL by integrating FD with adaptive subgradient methods under the primal-dual framework and adding back the cumulative information discarded by FD. FTSL obtains a tighter regret bound than existing state-of-the-art results. We then apply a doubling trick that further reduces the time complexity of FTSL, while maintaining the same regret guarantee. We also consider optimization over matrix-valued decision variables and exploit their structural properties to construct sketched preconditioners. We develop FTSL-Shampoo by adapting FD to Shampoo and give a novel analysis under the primal-dual framework. To further mitigate its computational complexity, we design one-sided variants of FTSL-Shampoo under both mirror descent and primal-dual frameworks. Our algorithms not only reduce the computational cost of FTSL-Shampoo, but also achieve tighter regret bounds. We extend our methods to offline stochastic convex optimization and exploit smoothness to achieve faster convergence rates. Finally, we introduce practical variants of our algorithms and empirically demonstrate their efficiency and effectiveness across a broad range of tasks.

We study the spontaneous motion and collective dynamics of self-phoretic particles on a ring. The particles act as localized sources of a chemical field, with particle-dependent source strengths, and respond to gradients of this field through particle-dependent phoretic mobilities. The chemical field undergoes diffusion and evaporation. For a single particle, we derive the exact threshold for spontaneous propulsion and establish the emergence of stable constant-velocity states near onset through a supercritical pitchfork bifurcation. We then consider an active source coupled to a non-emitting tracer and determine the conditions for slipping and stable phase-locked motion. For two mutually coupled sources, the stationary antipodal configuration can lose stability through either a pitchfork or a Hopf bifurcation, depending on the relative source strengths and phoretic mobilities. We determine the competing instability thresholds and compare them with direct numerical simulations of the coupled particle-field equations. Beyond onset, the resulting translating and oscillatory states provide qualitative counterparts to modes of motion observed experimentally for camphor disks in annular water channels. These results connect the onset of individual propulsion, tracer entrainment, and collective motion within a common analytically tractable framework.