Papers

This article evaluates an artificial intelligence (AI)-based Automatic Ground Collision Avoidance System (AGCAS) designed for advanced jet trainers to enhance operational effectiveness. In the continuously evolving field of aerospace engineering, the integration of AI is crucial for advancing operations with improved timing constraints and efficiency. Our study explores the design process of an AI-driven AGCAS, specifically tailored for advanced jet trainers, focusing on addressing the AGCAS problem within a limited observation space. The system utilizes line-of-sight queries on a terrain server to ensure precise and efficient collision avoidance. This approach aims to significantly improve the safety and operational capabilities of advanced jet trainers.

In the noisy intermediate-scale quantum (NISQ) regime, quantum devices contain hardware-specific noise sources which restrict device-invariant error mitigation strategies. We explore transfer learning approaches to apply noise models learned on one quantum device to a different device with the help of a small amount of data. We create a real-hardware dataset from two IBM quantum devices, ibm_fez (source) and ibm_marrakesh (target), comprising 170 noisy and ideal circuit output distributions, with device calibration features added. We train a residual neural network on the source device to map noisy to ideal outcomes. The zero-shot transfer test shows a KL divergence of 1.6706 (up from 0.3014), establishing device specificity. With K = 20 fine-tuning samples, KL drops to 1.1924 (28.6% improvement over zero-shot), recovering 34.9% of the gap between zero-shot and in-domain KL. Ablation studies reveal that the major cause of mismatches across devices is CX gate error, followed by readout error. The results show quantum noise can be learned and fine-tuned with minimal samples, and provide a plausible approach to cross-device quantum error mitigation.

There has been growing interest in studying the complexity of Rectified Linear Unit (ReLU) based activation networks. Recent work investigates the evolution of the number of piecewise-linear partitions (linear regions) that are formed during training. However, current research is limited to examining the complexity of models trained in a supervised way. Self-Supervised Learning (SSL) differs in that it directly optimises the representation space using a loss function to enhance the model's performance across multiple downstream tasks. This study investigates the local distribution of linear regions produced by SSL models. We demonstrate that the evolution of linear regions correlates with the representation quality by utilising SplineCam to extract two-dimensional polytopes near the data distribution. We track the number, area, eccentricity, and boundaries of regions throughout training. The study compares supervised, contrastive, and self-distillation methods over two standard benchmark datasets, MNIST and FashionMNIST. The analysis of the experimental results shows that self-supervised methods create substantially fewer regions to achieve comparable accuracy to supervised models. Contrastive methods rapidly expand regions over time, whereas self-distillation methods tend to consolidate by merging neighbouring regions. Lastly, we can detect representation collapse early within the geometric space of linear regions. Our analysis suggests that polytopal metrics can serve as reliable indicators of representation quality and model performance.

Safety-critical applications are required to perform as expected in normal operations. Image processing functions are often required to be insensitive to small geometric perturbations such as rotation, scaling, shearing or translation. This paper addresses the formal verification of neural networks against geometric perturbations on their image dataset. Our method Super-DeepG improves the reasoning used in linear relaxation techniques and Lipschitz optimization, and provides an implementation that leverages GPU hardware. By doing so, Super-DeepG achieves both precision and computational efficiency of robustness certification, to an extent that outperforms prior work. Super-DeepG is shared as an open-source tool on GitHub.

Cancer survival prediction from multi-omics data remains challenging because prognostic signals are high-dimensional, heterogeneous, and distributed across interacting genes and pathways. We propose PathMoG, a pathway-centric modular graph neural network for multi-omics survival prediction. PathMoG reorganizes genome-scale inputs into 354 KEGG-informed pathway modules, introduces a Hierarchical Omics Modulation module to condition gene-expression representations on mutation, copy number variation, pathway, and clinical context, and uses dual-level attention to capture both intra-pathway driver signals and inter-pathway clinical relevance. We evaluated PathMoG on 5,650 patients across 10 TCGA cancer types and observed consistent improvements over representative survival baselines. The framework further provides gene-level, pathway-level, and patient-level interpretability, supporting biologically grounded and clinically relevant risk stratification.

Generating high-fidelity synthetic tabular data remains a critical challenge for enhancing data availability in privacy-sensitive and low-resource domains. Recent approaches leverage LLMs by representing table rows as sequences, yet suffer from two fundamental limitations: (1) they model feature dependencies densely, introducing spurious correlations; and (2) they assume static relationships between features, ignoring how these dependencies vary with feature values. To overcome these limitations, we introduce SAGE (Sparse Adaptive Guidance), a novel LLM-based generation framework that enforces sparse and dynamic dependency guidance. SAGE discretizes features into value-aware pseudo-features and constructs a mutual information-based sparse dependency graph. This graph adaptively guides generation through explicit context selection or implicit logit correction, enabling LLMs to focus on truly relevant information during synthesis. Our extensive experiments across six datasets and multiple tasks reveal that SAGE not only improves data fidelity and downstream utility, boosting F1 scores by 10% compared to previous LLM-based methods, but also reduces policy violations by one point. These results highlight the importance of adaptive structure in tabular data generation and provide new insights into context-sensitive control of LLMs.

Discrete diffusion models admit many token orders, yet most systems rely on confidence-based decoding. Confidence is a strong and efficient heuristic, but it can be myopic because local certainty does not measure a position's effect on terminal quality. We introduce DPRM (Doob h-transform Process Reward Model), a plug-in token-ordering module induced by a terminal-reward-tilted trajectory distribution. The Doob transform converts terminal reward into a process reward for each candidate position. A compact estimator indexed by generation progress, confidence, and optional positional or task state makes this correction reusable across partial states. DPRM changes only token order; the host architecture, denoising objective, supervision, and token sampler remain fixed. We derive the exact reward-tilted law and bound its shortlist and online approximations. Across nine recent open-source hosts spanning language, multimodal generation, and scientific sequences, DPRM improves reasoning, numeric VQA, visual-codebook ordering, and preference-conditioned generation. Relative to confidence, it improves PUMA GSM8K by 14.6%, Omni-Diffusion CLIP-L/14 by 13.2%, Prism voted accuracy from 82.41% to 83.85%, and RealWorldQA numeric/count accuracy by 8.97 points. Matched scientific gains include 8.3% in DCM nonzero recovery, 15.0% in GenMol QED, and 53.3% in SDPO-DNA total utility. Entropy and random-order controls show that the gains do not come from favoring uncertain tokens indiscriminately. Token-order traces and shared-state interventions instead show how terminal reward can correct a locally confident but globally poor choice. Code is available at: https://github.com/DakeBU/DPRM-DLLM.

What do recurrent neural networks, polynomial ODEs, and discrete polynomial maps each bring to computation, and what do they lack? All three operate over the continuum–real-valued states evolved by real-valued dynamics–even when the target functions are discrete. We study them through primitive recursion. We prove that primitive recursion admits equivalent characterizations in all three frameworks: bounded iteration of a fixed recurrent ReLU network, robust computation by a fixed polynomial ODE, and iteration of a fixed polynomial map with an externally supplied step-size parameter. In each, the time bound is itself primitive recursive, composition emerges from the dynamics rather than as a closure rule, and inputs are raw integer vectors. Every primitive recursive function is first compiled into bounded iteration of a single threshold-affine normal form, then interpreted as a ReLU computation and as a polynomial ODE. The equivalences expose a structural asymmetry: no fixed polynomial map can round uniformly to the nearest integer or realize exact phase selection–operations polynomial ODEs perform robustly via continuous-time flow. Each formalism compensates for a limitation the others lack: the ReLU gate provides exact branching, continuous time provides autonomous rounding and control, and the step-size parameter recovers both at the cost of discretization precision. This opens dynamical characterizations of subrecursive hierarchies and complexity classes by restricting time bounds, polynomial degrees, or discretization resources within one framework. More broadly, these models do not compute by composing subroutines: they shape the trajectory of a dynamical system through clocks, phase selectors, and error correction built into the dynamics. This differs structurally from symbolic programming, and our theorem gives a precise framework to study the difference.

This article explores the progress made in the creation of a pilot activated recovery system (PARS) for advanced jet trainers that utilizes artificial intelligence (AI) in an effort to enhance operational efficiency. The PARS model employs an advanced reinforcement learning (RL) architecture, incorporating a cutting-edge soft-actor critic (SAC) model and hyper-parameter optimization methods. Negative-g punishments and other handcrafted features remarked upon by control engineers and domain experts regarding PARS are also taken into account by the system. When evaluated by them, the AI model's behavior is deemed more desirable than that of conventional control methods.

The continuous advancement of autonomous driving (AD) introduces challenges across multiple disciplines to ensure safe and efficient driving. One such challenge is the generation of High-Definition (HD) maps, which must remain up to date and highly accurate for downstream automotive tasks. One promising approach is the use of crowdsourced data from a vehicle fleet, representing road topology and lane-level features. This work focuses on the generation of centerlines and lane dividers from crowdsourced vehicle trajectories. We adopt a Detection Transformer (DETR)-based approach, where a rasterized representation of vehicle trajectories is used as input to predict vectorized lane representations. Each lane consists of a centerline with an associated direction and corresponding lane dividers that are geometrically constrained by the centerline. Our method includes the extraction of local tiles, from which crowdsourced vehicle trajectories are aggregated. Each tile undergoes a transformation into a rasterized representation encoding both the presence and direction of each trajectory, enabling the prediction of vectorized directed lanes. Experiments are conducted on an internal dataset as well as on the public datasets nuScenes and nuPlan.

Controllable diffusion methods have substantially expanded the practical utility of diffusion models, but they are typically developed as isolated, backbone-specific systems with incompatible training pipelines, parameter formats, and runtime hooks. This fragmentation makes it difficult to reuse infrastructure across tasks, transfer capabilities across backbones, or compose multiple controls within a single generation pipeline. We present Diffusion Templates, a unified and open plugin framework that decouples base-model inference from controllable capability injection. The framework is organized around three components: Template models that map arbitrary task-specific inputs to an intermediate capability representation, a Template cache that functions as a standardized interface for capability injection, and a Template pipeline that loads, merges, and injects one or more Template caches into the base diffusion runtime. Because the interface is defined at the systems level rather than tied to a specific control architecture, heterogeneous capability carriers such as KV-Cache and LoRA can be supported under the same abstraction. Based on this design, we build a diverse model zoo spanning structural control, brightness adjustment, color adjustment, image editing, super-resolution, sharpness enhancement, aesthetic alignment, content reference, local inpainting, and age control. These case studies show that Diffusion Templates can unify a broad range of controllable generation tasks while preserving modularity, composability, and practical extensibility across rapidly evolving diffusion backbones. All resources will be open sourced, including code, models, and datasets.

Fast Adversarial Training (FAT) has attracted significant attention due to its efficiency in enhancing neural network robustness against adversarial attacks. However, FAT is prone to catastrophic overfitting (CO), wherein models overfit to the specific attack used during training and fail to generalize to others. While existing methods introduce diverse hypotheses and propose various strategies to mitigate CO, a systematic and intuitive explanation of CO remains absent. In this work, we innovatively interpret CO through the lens of backdoor. Through validations on pathway division, diverse feature predictions, and universal class distinguishable triggers in CO, we conceptualize CO as a weak trigger variant of unlearnable tasks, unifying CO, backdoor attacks, and unlearnable tasks under a common theoretical framework. Guided by this, we leverage several backdoor inspired strategies to mitigate CO: (i) Recalibrate CO affected model parameters using vanilla fine tuning, linear probing, or reinitialization-based techniques; (ii) Introduce a weight outlier suppression constraint to regulate abnormal deviations in model weights. Extensive experiments support our interpretation of CO and show the efficacy of the proposed mitigation strategies.

In pathology, the spatial distribution and proportions of tissue types are key indicators of disease progression, and are more readily available than fine-grained annotations. However, these assessments are rarely mapped to pixel-wise segmentation. The task is fundamentally underdetermined, as many spatially distinct segmentations can satisfy the same global proportions in the absence of pixel-wise constraints. To address this, we introduce Variational Segmentation from Label Proportions (VSLP), a two-stage framework that infers dense segmentations from global label proportions, without any pixel-level annotations. This framework first leverages a pre-trained transformer model with test-time augmentation to produce a pixel-wise confidence estimate. In the second stage, these estimates are fused by solving a variational optimization problem that incorporates a Wasserstein data fidelity term alongside a learned regularizer. Unlike end-to-end networks, our variational method can visualize the fidelity-regularization energy, resulting in more interpretable segmentation. We validate our approach on two public datasets, achieving superior performance over existing weakly supervised and unsupervised methods. For one of these datasets, proportions have been estimated by an experienced pathologist to provide a realistic benchmark to the community. Furthermore, the method scales to an in-house dataset with noisy pathologist labels, severely outperforming state-of-the-art methods, thereby demonstrating practical applicability. The code and data will be made publicly available upon acceptance at https://github.com/xiaoliangpi/VSLP.

This paper evaluates an advanced jet trainer's utilization of artificial intelligence (AI)-based aircraft aerobatic maneuvers with the intention of developing an AI-assisted pilot training module for specific aircraft maneuvers. A multitude of aircraft maneuvers have been simulated using reinforcement learning (RL) agents, which will serve as a training tool for future pilots.

In this work, we construct new non-Euclidean neural quantum states (NQS) based on hyperbolic Lorentz recurrent architectures (RNN/GRU). These constructions, together with the Poincare RNN NQS also newly constructed here, extend the class of previously introduced non-Eucllidean NQS which consists only of Poincare hyperbolic GRU. Using the Heisenberg J1J2 and J1J2J3 models consisting of 100 spins in the Variational Monte Carlo (VMC) setting, we show that the four hyperbolic RNN/GRU NQS variants are always able to furnish better representations of the ground state wavefunctions of the quantum systems than their respective Euclidean counterparts with the same architecture. In our experiments, among the four hyperbolic NQS, Lorentz RNN stands out in particular because despite having almost three times fewer parameters, it is capable of surpassing the more complex Poincare GRU and Lorentz GRU to emerge as the best overall hyperbolic NQS ansatz on many instances involving different J2 and (J2,J3) couplings. Given the findings from this work showing that the four newly constructed hyperbolic RNN/GRU NQS ansatze are able to outperform the well-established Euclidean RNN/GRU NQS in Heisenberg spin models, we establish the utility and efficiency of the hyperbolic Lorentz RNN/GRU NQS as well as the Poincare RNN/GRU NQS for future variational studies of quantum many-body systems, especially those exhibiting a hierarchical structure in the form of the different degrees of nearest-neighbor interactions.

Fast Adversarial Training (FAT) has proven effective in enhancing model robustness by encouraging networks to learn perturbation-invariant representations. However, FAT often suffers from catastrophic overfitting (CO), where the model overfits to the training attack and fails to generalize to unseen ones. Moreover, robustness oriented optimization typically leads to notable performance degradation on clean inputs, and such degradation becomes increasingly severe as the perturbation budget grows. In this work, we conduct a comprehensive analysis of how guidance strength affects model performance by modulating perturbation and supervision levels across distinct confidence groups. The findings reveal that low confidence samples are the primary contributors to CO and the robustness accuracy trade off. Building on this insight, we propose a Distribution-aware Dynamic Guidance (DDG) strategy that dynamically adjusts both the perturbation budget and supervision signal. Specifically, DDG scales the perturbation magnitude according to the sample confidence at the ground truth class, thereby guiding samples toward consistent decision boundaries while mitigating the influence of learning spurious correlations. Simultaneously, it dynamically adjusts the supervision signal based on the prediction state of each sample, preventing overemphasis on incorrect signals. To alleviate potential gradient instability arising from dynamic guidance, we further design a weighted regularization constraint. Extensive experiments on standard benchmarks demonstrate that DDG effectively alleviates both CO and the robustness accuracy trade off.

Training large-scale deep neural networks effectively and stably is essential for applying deep learning across various fields. However, conventional methods, which rely on training a single large network, often encounter challenges such as gradient vanishing, overfitting and unstable learning. To overcome these limitations, we introduce Self-Abstraction Learning (SAL), a hierarchical framework. In SAL, networks are arranged by structural complexity, where the simplest topmost network is trained first and its hidden and output layers serve as guidance for the successively more complex networks below. This top-down sequential guidance effectively mitigates optimization issues, enabling stable training of deep architectures. Various experiments across MLP, CNN, and RNN architectures demonstrate that SAL consistently outperforms conventional methods, ensuring robust generalization even in data-scarce and complex network regimes.

Accurate forecasting of solar power output is essential for efficient integration of renewable energy into the grid. In this study, an attention-based deep learning model, inspired by transformer architecture, is used for short-term solar power forecasting. Our proposed model, "SolarTformer", is designed to predict solar power output from meteorological data. Unlike traditional models, SolarTformer leverages self-attention mechanisms to effectively capture temporal dependencies and spatial variability in solar irradiance. In addition, the proposed methodology includes feeding power station-specific metadata into the model, which helps to generalize between power stations located at different locations and with different panel configurations and in different seasons. Our experiments demonstrate that SolarTformer significantly outperforms previous models on the same data set. In particular, the model exhibits strong performance on both clear and cloudy days, indicating high robustness and generalizability. These findings highlight the potential of attention-based architectures in enhancing the accuracy of solar forecasting, contributing to a more reliable management of renewable energy.

Graph neural ordinary differential equations provide continuous-time propagation on graphs, but their long-time behavior is largely determined by the asymptotic structure of the mixing operator. We show that diffusion-style Graph ODEs with positive irreducible mixing face a monostability trap, where node features converge to a rank-one consensus subspace. This reveals a structural limitation of positive diffusion-style continuous graph propagation: within this consensus class, exact long-time avoidance requires changing the effective support; otherwise, smooth suppression should be interpreted as finite-horizon or metastable mitigation. We address this by introducing a hysteretic mechanism for topology evolution, where candidate edges carry latent bistable potentials in a double-well landscape. A node-feature-conditioned force field evolves these potentials, and a smooth sigmoid gate converts them into differentiable propagation weights. We prove global consensus under the stated mixing assumptions and block-wise consensus under idealised support separation. For consensus-form dynamics, we further establish a finite-horizon perturbation bound under small cross-block leakage, and empirically evaluate the resulting nonlinear HGODE on node- and graph-level benchmarks.

We present a framework using Relative Entropy Inverse Reinforcement Learning (RE-IRL) to recover investor reward functions from observed investment actions and market conditions. Unlike traditional IRL algorithms, RE-IRL is employed to account for environments where transition probabilities are unknown or inaccessible. To address the challenge of data sparsity, we utilize a $K$-nearest neighbor approach to estimate the observed behavior policy. Furthermore, we propose a statistical testing framework to evaluate the validity and robustness of the estimated results.

The deployment of intelligent reinforcement learning (RL) agents on resource-constrained edge devices remains a fundamental challenge due to the substantial memory, computational, and energy requirements of modern deep learning systems. While large language models (LLMs) have emerged as powerful architectures for decision-making agents, their multi-billion parameter scale confines them to cloud-based deployment, raising concerns about latency, privacy, and connectivity dependence. We introduce BitRL, a framework for building RL agents using 1-bit quantized language models that enables practical on-device learning and inference under severe resource constraints. Leveraging the BitNet b1.58 architecture with ternary weights (-1, 0, +1) and an optimized inference stack, BitRL achieves 10-16x memory reduction and 3-5x energy efficiency improvements over full-precision baselines while maintaining 85-98 percent of task performance across benchmarks. We provide theoretical analysis of quantization as structured parameter perturbation, derive convergence bounds for quantized policy gradients under frozen-backbone architectures, and identify the exploration-stability trade-off in extreme quantization. Our framework systematically integrates 1-bit quantized language models with reinforcement learning for edge deployment and demonstrates effectiveness on commodity hardware.

Diffusion models are a leading paradigm for data generation, but training-free editing typically re-runs the full denoising trajectory for every edit strength, making iterative refinement expensive. To address this issue, we instead edit near the data manifold, where small local updates can replace repeated re-synthesis. To enable this, we estimate a local manifold tangent space directly from perturbed samples and prove that this sample-based estimator closely approximates the true tangent. Building on this guarantee, we devise a Jacobian-free algorithm that constructs a tangent frame via small perturbations to the initial noise and alternates small tangent moves with diffusion-based projections. Updates within this frame follow principled on-manifold directions while suppressing off-manifold drift, enabling fine-grained edits without full re-diffusion or additional training. Edit strength is controlled by the number of steps for rapid, continuous adjustments that preserve fidelity and plug into existing samplers. Empirically, the resulting tangent directions yield smooth, semantic unsupervised traversals and effective CLIP-guided optimization, demonstrating practical interactive continuous editing.

Short-range prediction of convective precipitation from weather radar observations is essential for severe weather warnings. However, deep learning models trained with pixel-wise error metrics tend to produce overly smooth forecasts that suppress intense echoes critical for hazard detection. This issue is exacerbated by insufficient multi-scale feature interaction and suboptimal fusion of heterogeneous geophysical inputs. We propose IMPA-Net (Integrated Multi-scale Predictive Attention Network), a deterministic 0-2 hour nowcasting framework that addresses these limitations through meteorologically-informed designs at the input, architecture, and loss function levels. A parameter-free Spatial Mixer reorganizes heterogeneous input channels at the mesoscale-$\gamma$ neighborhood (~2 km) via deterministic channel permutation, providing a structured cross-field prior. An integrated multi-scale predictive attention module serves as the spatiotemporal translator, capturing dynamics from mesoscale-$\beta$ to mesoscale-$\gamma$ scales. A Meteorologically-Aware Dynamic Loss employs three-level asymmetric weighting – adapting across training epochs, storm intensity, and forecast lead time – to counteract regression-to-the-mean. Evaluated against seven baselines on a multi-source radar dataset over eastern China, IMPA-Net raises the Heidke Skill Score at $\geq$45 dBZ from 0.049 (SimVP baseline) to 0.143 under matched settings. Relative to pySTEPS, it provides a better trade-off between severe-event detection and false-alarm control. Spectral analysis confirms preserved energy across mesoscale bands where competing methods show progressive smoothing. These improvements are shown within a single domain and convective regime; generalizability to other orographic and climatic regions remains to be tested.

Motivation: Multi-omics integration can improve cancer subtyping, but modality informativeness and noise vary across cancer types and patients. Most graph methods for multi-omics data learn modality contributions within the downstream classification objective, leaving predictive reliability for each patient implicit. As a result, uninformative modalities can weaken the fused representation, while unreliable omics can introduce noisy patient relationships into graph propagation. To address these two problems, we propose CMGL, which produces a separate reliability estimate before fusion and uses consensus patient neighborhoods for graph classification. Results: CMGL estimates modality confidence for each patient through evidential deep learning, fixes these values during fusion across omics, and performs classification on an independently specified consistency graph. On four MLOmics cancer-subtype tasks and the 32-class pan-cancer task, CMGL consistently improves over the strongest baseline, surpassing it by 4.03% in average accuracy on the four single-cancer tasks. Its representations recover the PAM50 intrinsic subtypes of breast invasive carcinoma (BRCA), and the model trained on BRCA transfers without fine tuning to kidney renal clear cell carcinoma (KIRC), stratifying patients into prognostically distinct groups.

Process Reward Models (PRMs) have achieved remarkable success in augmenting the reasoning capabilities of Large Language Models (LLMs) within static domains such as mathematics. However, their potential in dynamic data analysis tasks remains underexplored. In this work, we first present a empirical study revealing that general-domain PRMs struggle to supervise data analysis agents. Specifically, they fail to detect silent errors, logical flaws that yield incorrect results without triggering interpreter exceptions, and erroneously penalize exploratory actions, mistaking necessary trial-and-error exploration for grounding failures. To bridge this gap, we introduce DataPRM, a novel environment-aware generative process reward model that (1) can serve as an active verifier, autonomously interacting with the environment to probe intermediate execution states and uncover silent errors, and (2) employs a reflection-aware ternary reward strategy that distinguishes between correctable grounding errors and irrecoverable mistakes. We design a scalable pipeline to construct over 8K high-quality training instances for DataPRM via diversity-driven trajectory generation and knowledge-augmented step-level annotation. Experimental results demonstrate that DataPRM improves downstream policy LLMs by 7.21% on ScienceAgentBench and 11.28% on DABStep using Best-of-N inference. Notably, with only 4B parameters, DataPRM outperforms strong baselines, and exhibits robust generalizability across diverse Test-Time Scaling strategies. Furthermore, integrating DataPRM into Reinforcement Learning yields substantial gains over outcome-reward baselines, achieving 78.73% on DABench and 64.84% on TableBench, validating the effectiveness of process reward supervision. Code is available at https://github.com/zjunlp/DataMind.

A drifting model is a one-step generator trained by moving each sample along a field of kernel-weighted attraction toward data samples and repulsion between model samples; training halts once this field vanishes. The soundness of this scheme rests on two questions: whether a zero-field equilibrium guarantees agreement with the data distribution, and how the error is controlled when the field is small. We answer both questions. We introduce the companion-elliptic kernel class, which contains the Laplace kernel, and prove that a vanishing field identifies arbitrary Borel probability measures within this class and its blockwise product extension. The scalar companion-elliptic kernels are exactly the Gaussian and Matérn families, and deconvolution, the recovery of a measure from its kernel-smoothed density, is possible precisely when the zero set of the kernel's Fourier transform has empty interior. At approximate equilibria, a mass-escape counterexample shows that field information on a bounded region cannot control the global error. We therefore prove an explicit stability inequality bounding the error by the field residual on the observation region plus the convolution mass outside it. In this inequality the Gaussian kernel requires only field values but amplifies high-frequency errors exponentially, whereas the Matérn kernel requires one additional order of derivative information in exchange for polynomial amplification. Since this gap widens exponentially with frequency, the balance favors the Matérn family and hence the Laplace kernel.

Multi-Objective Alignment aims to align Large Language Models (LLMs) with diverse and often conflicting human values by optimizing multiple objectives simultaneously. Existing methods predominantly rely on static preference weight construction strategies. However, rigidly aligning to fixed targets discards valuable intermediate information, as training responses inherently embody valid preference trade-offs even when deviating from the target. To address this limitation, we propose Meal, i.e., MEta ALigner, a bi-level meta-learning framework enabling bidirectional optimization between preferences and policy responses, generating instructive dynamic preferences for steadier training. Specifically, we introduce a preference-weight-net as a meta-learner to generate adaptive preference weights based on input prompts and update the preference weights as learnable parameters, while the LLM policy acts as a base-learner optimizing response generation conditioned on these preferences with rejection sampling strategy. Extensive empirical results demonstrate that our method achieves superior performance on several multi-objective benchmarks, validating the effectiveness of the dynamic bidirectional preference-policy optimization framework.

This paper uses a minimum divergence framework to introduce a new way of calculating model weights that can be used to average probabilistic predictions from statistical and machine learning models. The method is general and can be applied regardless of whether the models under consideration are fit to data using frequentist, Bayesian, or some other fitting method. The proposed method is motivated in two different ways and is shown empirically to perform better than or on a par with standard model averaging methods, including model stacking and model averaging that relies on Akaike-style negative exponentiated model weighting, especially when the sample size is small. Our theoretical analysis explains why the method has a small-sample advantage.

Implicit neural representations (INRs) are increasingly being used as tools to map coordinates to signals, encompassing applications from neural fields to texture compression, shape representations, and beyond. Most INR methods are based on using high-dimensional projections of the initial coordinates through encoders such as grid or positional encoding. Nevertheless, positional encoding is often insufficient and grids, as we show in this paper, require high resolution for being able to learn. In this paper, we demonstrate that positional encoding can be used not only as a high-dimensional embedding but also decomposed as a series of meaningful points. We propose the Positional Encoding Projected Sampling, where we treat the projection of the original coordinate at each frequency as a point of interest. We describe the motion of each point with respect to the frequencies and show that it follows a unique pattern. Finally, we use the unique motion of each point as a basis decomposition for doing learned positional encoding using grids. We prove, using three competitive applications; image representation, texture compression, and signed distance function; that the proposed approach outperforms the current state of the art methods, and often requires 25% less parameters for equivalent reconstruction error or rendering.

The Universal Approximation Theorem (UAT) guarantees universal function approximation but does not explain how residual models distribute approximation across layers. We reframe residual networks as a layer-wise approximation process that builds an approximation trajectory from input to target, and prove the existence of progressive trajectories where error decreases monotonically with depth. It reveals that residual networks can implement structured, step-by-step refinement rather than end-to-end (E2E) black-box mapping. Building on this, we propose Layer-wise Progressive Approximation (LPA), a theoretically grounded training principle that explicitly aligns each layer with its residual target to realize such trajectories. LPA is architecture-agnostic: we observe progressive behavior in residual FNNs, ResNets, and Transformers across tasks including complex surface fitting, image classification, and NLP with LLMs for generation and classification. Crucially, this enables “train once, use $N$ models": a single network yields useful predictions at every depth, supporting efficient shallow inference without retraining. Our work unifies approximation theory with practical deep learning, providing a new lens on representation learning and a flexible framework for multi-depth deployment. The source code will be released unpon acceptance at https://(open\_upon\_acceptance).