Expressive Power of Deep Networks on Manifolds: Simultaneous Approximation

A key challenge in scientific machine learning is solving partial differential equations (PDEs) on complex domains, where the curved geometry complicates the approximation of functions and their derivatives required by differential operators. This paper establishes the first simultaneous approximation theory for deep neural networks on manifolds. We prove that a constant-depth $\mathrm{ReLU}^{k-1}$ network with bounded weights--a property that plays a crucial role in controlling generalization error--can approximate any function in the Sobolev space $\mathcal{W}_p^{k}(\mathcal{M}^d)$ to an error of $\varepsilon$ in the $\mathcal{W}_p^{s}(\mathcal{M}^d)$ norm, for $k\geq 3$ and $s<k$, using $\mathcal{O}(\varepsilon^{-d/(k-s)})$ nonzero parameters, a rate that overcomes the curse of dimensionality by depending only on the intrinsic dimension $d$. These results readily extend to functions in H\"older-Zygmund spaces. We complement this result with a matching lower bound, proving our construction is nearly optimal by showing the required number of parameters matches up to a logarithmic factor. Our proof of the lower bound introduces novel estimates for the Vapnik-Chervonenkis dimension and pseudo-dimension of the network's high-order derivative classes. These complexity bounds provide a theoretical cornerstone for learning PDEs on manifolds involving derivatives. Our analysis reveals that the network architecture leverages a sparse structure to efficiently exploit the manifold's low-dimensional geometry.

ALLabel: Three-stage Active Learning for LLM-based Entity Recognition using Demonstration Retrieval

Many contemporary data-driven research efforts in the natural sciences, such as chemistry and materials science, require large-scale, high-performance entity recognition from scientific datasets. Large language models (LLMs) have increasingly been adopted to solve the entity recognition task, with the same trend being observed on all-spectrum NLP tasks. The prevailing entity recognition LLMs rely on fine-tuned technology, yet the fine-tuning process often incurs significant cost. To achieve a best performance-cost trade-off, we propose ALLabel, a three-stage framework designed to select the most informative and representative samples in preparing the demonstrations for LLM modeling. The annotated examples are used to construct a ground-truth retrieval corpus for LLM in-context learning. By sequentially employing three distinct active learning strategies, ALLabel consistently outperforms all baselines under the same annotation budget across three specialized domain datasets. Experimental results also demonstrate that selectively annotating only 5\%-10\% of the dataset with ALLabel can achieve performance comparable to the method annotating the entire dataset. Further analyses and ablation studies verify the effectiveness and generalizability of our proposal.

Optimal Convergence Rates of Deep Neural Network Classifiers

In this paper, we study the binary classification problem on $[0,1]^d$ under the Tsybakov noise condition (with exponent $s \in [0,\infty]$) and the compositional assumption. This assumption requires the conditional class probability function of the data distribution to be the composition of $q+1$ vector-valued multivariate functions, where each component function is either a maximum value function or a H\"{o}lder-$\beta$ smooth function that depends only on $d_*$ of its input variables. Notably, $d_*$ can be significantly smaller than the input dimension $d$. We prove that, under these conditions, the optimal convergence rate for the excess 0-1 risk of classifiers is $$ \left( \frac{1}{n} \right)^{\frac{\beta\cdot(1\wedge\beta)^q}{{\frac{d_*}{s+1}+(1+\frac{1}{s+1})\cdot\beta\cdot(1\wedge\beta)^q}}}\;\;\;, $$ which is independent of the input dimension $d$. Additionally, we demonstrate that ReLU deep neural networks (DNNs) trained with hinge loss can achieve this optimal convergence rate up to a logarithmic factor. This result provides theoretical justification for the excellent performance of ReLU DNNs in practical classification tasks, particularly in high-dimensional settings. The technique used to establish these results extends the oracle inequality presented in our previous work. The generalized approach is of independent interest.

Weak Physics Informed Neural Networks for Geometry Compatible Hyperbolic Conservation Laws on Manifolds

Physics-informed neural networks (PINNs), owing to their mesh-free nature, offer a powerful approach for solving high-dimensional partial differential equations (PDEs) in complex geometries, including irregular domains. This capability effectively circumvents the challenges of mesh generation that traditional numerical methods face in high-dimensional or geometrically intricate settings. While recent studies have extended PINNs to manifolds, the theoretical foundations remain scarce. Existing theoretical analyses of PINNs in Euclidean space often rely on smoothness assumptions for the solutions. However, recent empirical evidence indicates that PINNs may struggle to approximate solutions with low regularity, such as those arising from nonlinear hyperbolic equations. In this paper, we develop a framework for PINNs tailored to the efficient approximation of weak solutions, particularly nonlinear hyperbolic equations defined on manifolds. We introduce a novel weak PINN (wPINN) formulation on manifolds that leverages the well-posedness theory to approximate entropy solutions of geometry-compatible hyperbolic conservation laws on manifolds. Employing tools from approximation theory, we establish a convergence analysis of the algorithm, including an analysis of approximation errors for time-dependent entropy solutions. This analysis provides insight into the accumulation of approximation errors over long time horizons. Notably, the network complexity depends only on the intrinsic dimension, independent of the ambient space dimension. Our results match the minimax rate in the d-dimensional Euclidean space, demonstrating that PINNs can alleviate the curse of dimensionality in the context of low-dimensional manifolds. Finally, we validate the performance of the proposed wPINN framework through numerical experiments, confirming its ability to efficiently approximate entropy solutions on manifolds.

Semi-Supervised Multi-Label Feature Selection with Consistent Sparse Graph Learning

In practical domains, high-dimensional data are usually associated with diverse semantic labels, whereas traditional feature selection methods are designed for single-label data. Moreover, existing multi-label methods encounter two main challenges in semi-supervised scenarios: (1). Most semi-supervised methods fail to evaluate the label correlations without enough labeled samples, which are the critical information of multi-label feature selection, making label-specific features discarded. (2). The similarity graph structure directly derived from the original feature space is suboptimal for multi-label problems in existing graph-based methods, leading to unreliable soft labels and degraded feature selection performance. To overcome them, we propose a consistent sparse graph learning method for multi-label semi-supervised feature selection (SGMFS), which can enhance the feature selection performance by maintaining space consistency and learning label correlations in semi-supervised scenarios. Specifically, for Challenge (1), SGMFS learns a low-dimensional and independent label subspace from the projected features, which can compatibly cross multiple labels and effectively achieve the label correlations. For Challenge (2), instead of constructing a fixed similarity graph for semi-supervised learning, SGMFS thoroughly explores the intrinsic structure of the data by performing sparse reconstruction of samples in both the label space and the learned subspace simultaneously. In this way, the similarity graph can be adaptively learned to maintain the consistency between label space and the learned subspace, which can promote propagating proper soft labels for unlabeled samples, facilitating the ultimate feature selection. An effective solution with fast convergence is designed to optimize the objective function. Extensive experiments validate the superiority of SGMFS.

SHARP: Synthesizing High-quality Aligned Reasoning Problems for Large Reasoning Models Reinforcement Learning

Training large reasoning models (LRMs) with reinforcement learning in STEM domains is hindered by the scarcity of high-quality, diverse, and verifiable problem sets. Existing synthesis methods, such as Chain-of-Thought prompting, often generate oversimplified or uncheckable data, limiting model advancement on complex tasks. To address these challenges, we introduce SHARP, a unified approach to Synthesizing High-quality Aligned Reasoning Problems for LRMs reinforcement learning with verifiable rewards (RLVR). SHARP encompasses a strategic set of self-alignment principles -- targeting graduate and Olympiad-level difficulty, rigorous logical consistency, and unambiguous, verifiable answers -- and a structured three-phase framework (Alignment, Instantiation, Inference) that ensures thematic diversity and fine-grained control over problem generation. We implement SHARP by leveraging a state-of-the-art LRM to infer and verify challenging STEM questions, then employ a reinforcement learning loop to refine the model's reasoning through verifiable reward signals. Experiments on benchmarks such as GPQA demonstrate that SHARP-augmented training substantially outperforms existing methods, markedly improving complex reasoning accuracy and pushing LRM performance closer to expert-level proficiency. Our contributions include the SHARP strategy, framework design, end-to-end implementation, and experimental evaluation of its effectiveness in elevating LRM reasoning capabilities.

Online Iterative Self-Alignment for Radiology Report Generation

Radiology Report Generation (RRG) is an important research topic for relieving radiologist' heavy workload. Existing RRG models mainly rely on supervised fine-tuning (SFT) based on different model architectures using data pairs of radiological images and corresponding radiologist-annotated reports. Recent research has shifted focus to post-training improvements, aligning RRG model outputs with human preferences using reinforcement learning (RL). However, the limited data coverage of high-quality annotated data poses risks of overfitting and generalization. This paper proposes a novel Online Iterative Self-Alignment (OISA) method for RRG that consists of four stages: self-generation of diverse data, self-evaluation for multi-objective preference data,self-alignment for multi-objective optimization and self-iteration for further improvement. Our approach allows for generating varied reports tailored to specific clinical objectives, enhancing the overall performance of the RRG model iteratively. Unlike existing methods, our frame-work significantly increases data quality and optimizes performance through iterative multi-objective optimization. Experimental results demonstrate that our method surpasses previous approaches, achieving state-of-the-art performance across multiple evaluation metrics.

Seed1.5-VL Technical Report

We present Seed1.5-VL, a vision-language foundation model designed to advance general-purpose multimodal understanding and reasoning. Seed1.5-VL is composed with a 532M-parameter vision encoder and a Mixture-of-Experts (MoE) LLM of 20B active parameters. Despite its relatively compact architecture, it delivers strong performance across a wide spectrum of public VLM benchmarks and internal evaluation suites, achieving the state-of-the-art performance on 38 out of 60 public benchmarks. Moreover, in agent-centric tasks such as GUI control and gameplay, Seed1.5-VL outperforms leading multimodal systems, including OpenAI CUA and Claude 3.7. Beyond visual and video understanding, it also demonstrates strong reasoning abilities, making it particularly effective for multimodal reasoning challenges such as visual puzzles. We believe these capabilities will empower broader applications across diverse tasks. In this report, we mainly provide a comprehensive review of our experiences in building Seed1.5-VL across model design, data construction, and training at various stages, hoping that this report can inspire further research. Seed1.5-VL is now accessible at https://www.volcengine.com/ (Volcano Engine Model ID: doubao-1-5-thinking-vision-pro-250428)

Learning Operators by Regularized Stochastic Gradient Descent with Operator-valued Kernels

This paper investigates regularized stochastic gradient descent (SGD) algorithms for estimating nonlinear operators from a Polish space to a separable Hilbert space. We assume that the regression operator lies in a vector-valued reproducing kernel Hilbert space induced by an operator-valued kernel. Two significant settings are considered: an online setting with polynomially decaying step sizes and regularization parameters, and a finite-horizon setting with constant step sizes and regularization parameters. We introduce regularity conditions on the structure and smoothness of the target operator and the input random variables. Under these conditions, we provide a dimension-free convergence analysis for the prediction and estimation errors, deriving both expectation and high-probability error bounds. Our analysis demonstrates that these convergence rates are nearly optimal. Furthermore, we present a new technique for deriving bounds with high probability for general SGD schemes, which also ensures almost-sure convergence. Finally, we discuss potential extensions to more general operator-valued kernels and the encoder-decoder framework.

RiskNet: Interaction-Aware Risk Forecasting for Autonomous Driving in Long-Tail Scenarios

Ensuring the safety of autonomous vehicles (AVs) in long-tail scenarios remains a critical challenge, particularly under high uncertainty and complex multi-agent interactions. To address this, we propose RiskNet, an interaction-aware risk forecasting framework, which integrates deterministic risk modeling with probabilistic behavior prediction for comprehensive risk assessment. At its core, RiskNet employs a field-theoretic model that captures interactions among ego vehicle, surrounding agents, and infrastructure via interaction fields and force. This model supports multidimensional risk evaluation across diverse scenarios (highways, intersections, and roundabouts), and shows robustness under high-risk and long-tail settings. To capture the behavioral uncertainty, we incorporate a graph neural network (GNN)-based trajectory prediction module, which learns multi-modal future motion distributions. Coupled with the deterministic risk field, it enables dynamic, probabilistic risk inference across time, enabling proactive safety assessment under uncertainty. Evaluations on the highD, inD, and rounD datasets, spanning lane changes, turns, and complex merges, demonstrate that our method significantly outperforms traditional approaches (e.g., TTC, THW, RSS, NC Field) in terms of accuracy, responsiveness, and directional sensitivity, while maintaining strong generalization across scenarios. This framework supports real-time, scenario-adaptive risk forecasting and demonstrates strong generalization across uncertain driving environments. It offers a unified foundation for safety-critical decision-making in long-tail scenarios.