Premise selection is a critical bottleneck in interactive theorem proving, particularly with large libraries. Existing methods, primarily relying on semantic embeddings, often fail to effectively leverage the rich structural information inherent in mathematical expressions. This paper proposes a novel framework for premise selection based on the structure of expression trees. The framework enhances premise selection ability by explicitly utilizing the structural information of Lean expressions and by means of the simplified tree representation obtained via common subexpression elimination. Our method employs a multi-stage filtering pipeline, incorporating structure-aware similarity measures including the Weisfeiler-Lehman kernel, tree edit distance, Constnode Jaccard similarity, and collapse-match similarity. An adaptive fusion strategy combines these metrics for refined ranking. To handle large-scale data efficiently, we incorporate cluster-based search space optimization and structural compatibility constraints. Comprehensive evaluation on a large theorem library extracted from Mathlib4 demonstrates that our method significantly outperforms existing premise retrieval tools across various metrics. Experimental analysis, including ablation studies and parameter sensitivity analysis, validates the contribution of individual components and highlights the efficacy of our structure-aware approach and multi-metric fusion.

Data augmentation using generative models has emerged as a powerful paradigm for enhancing performance in computer vision tasks. However, most existing augmentation approaches primarily focus on optimizing intrinsic data attributes - such as fidelity and diversity - to generate visually high-quality synthetic data, while often neglecting task-specific requirements. Yet, it is essential for data generators to account for the needs of downstream tasks, as training data requirements can vary significantly across different tasks and network architectures. To address these limitations, we propose UTILGEN, a novel utility-centric data augmentation framework that adaptively optimizes the data generation process to produce taskspecific, high-utility training data via downstream task feedback. Specifically, we first introduce a weight allocation network to evaluate the task-specific utility of each synthetic sample. Guided by these evaluations, UTILGEN iteratively refines the data generation process using a dual-level optimization strategy to maximize the synthetic data utility: (1) model-level optimization tailors the generative model to the downstream task, and (2) instance-level optimization adjusts generation policies - such as prompt embeddings and initial noise - at each generation round. Extensive experiments on eight benchmark datasets of varying complexity and granularity demonstrate that UTILGEN consistently achieves superior performance, with an average accuracy improvement of 3.87% over previous SOTA. Further analysis of data influence and distribution reveals that UTILGEN produces more impactful and task-relevant synthetic data, validating the effectiveness of the paradigm shift from visual characteristics-centric to task utility-centric data augmentation.

The development of reasoning capabilities represents a critical frontier in large language models (LLMs) research, where reinforcement learning (RL) and process reward models (PRMs) have emerged as predominant methodological frameworks. Contrary to conventional wisdom, empirical evidence from DeepSeek-R1 demonstrates that pure RL training focused on mathematical problem-solving can progressively enhance reasoning abilities without PRM integration, challenging the perceived necessity of process supervision. In this study, we conduct a systematic investigation of the relationship between RL training and PRM capabilities. Our findings demonstrate that problem-solving proficiency and process supervision capabilities represent complementary dimensions of reasoning that co-evolve synergistically during pure RL training. In particular, current PRMs underperform simple baselines like majority voting when applied to state-of-the-art models such as DeepSeek-R1 and QwQ-32B.

Uncertainty quantification (UQ) is essential for safe deployment of generative AI models such as large language models (LLMs), especially in high-stakes applications. Conformal prediction (CP) offers a principled uncertainty quantification framework, but classical methods focus on regression and classification, relying on geometric distances or softmax scores-tools that presuppose structured outputs. We depart from this paradigm by studying CP in a query-only setting, where prediction sets must be constructed solely from finite queries to a black-box generative model, introducing a new trade-off between coverage, test-time query budget, and informativeness. We introduce Conformal Prediction with Query Oracle (CPQ), a framework characterizing the optimal interplay between these objectives. Our finite-sample algorithm is built on two core principles: one governs the optimal query policy, and the other defines the optimal mapping from queried samples to prediction sets.

And based on this idea, we propose a powerful decomposition-based enhancement framework, namely DecompNet. Our method converts the time series decomposition into an implicit process, where it can give a time series model the decomposition-related knowledge during inference, even though this model does not actually decompose the input time series. Thus, our DecompNet can enable a model to inherit the performance promotion brought by time series decomposition but will not introduce any additional inference costs, successfully enhancing the model performance while enjoying better efficiency. Experimentally, our DecompNet exhibits promising enhancement capability and compelling framework generality. Especially, it can also enhance the performance of the latest and state-of-the-art models, greatly pushing the performance limit of time series forecasting. Through comprehensive comparisons, DecompNet also shows excellent performance and efficiency superiority, making the decomposition-based enhancement framework surpass the well-recognized normalization-based frameworks for the first time.

How to integrate and verify spatial intelligence in foundation models remains an open challenge. Current practice often proxies Visual-Spatial Intelligence (VSI) with purely textual prompts and VQA-style scoring, which obscures geometry, invites linguistic shortcuts, and weakens attribution to genuinely spatial skills. We introduce Spatial Intelligence Grid (SIG): a structured, grid-based schema that explicitly encodes object layouts, inter-object relations, and physically grounded priors. As a complementary channel to text, SIG provides a faithful, compositional representation of scene structure for foundation-model reasoning. Building on SIG, we derive SIG-informed evaluation metrics that quantify a model's intrinsic VSI, which separates spatial capability from language priors.

The rapid adoption of generative AI and Large Language Models (LLMs) has spurred interest in synthetic data as a privacy-preserving alternative to sensitive real-world datasets. However, generating high-utility synthetic data often carries the risk of memorizing and regurgitating private information from the training corpus. In this work, we present a customizable empirical auditing framework designed to detect and explain such data disclosures. Our framework introduces a mechanism to distinguish between "true disclosures"-where the system directly reproduces a user's information-and "phantom disclosures''-where the system incidentally generates a user's data. By partitioning input data into training and holdout sets and applying rigorous statistical hypothesis testing, we determine if observed disclosures are consistent with strict privacy baselines, such as zero-learning or specific Differential Privacy (DP) bounds. Crucially, this approach requires no model access, no canary insertion, and no reference model training -only the synthetic output and a held-out control set. We demonstrate that this framework effectively functions as a membership inference attack, providing empirical lower bounds on privacy leakage that are tighter than prior data-based auditing methods. Our approach is model-agnostic, applies to any synthetic data generation mechanism, and requires orders of magnitude fewer computational resources than shadow-model or canary-based alternatives.

Molecular dynamics (MD) simulations generate trajectories in a high-dimensional configuration space whose analysis critically depends on molecular descriptors, typically handcrafted observables or learned kinetic embeddings. Designing descriptors that are both expressive and broadly applicable, however, remains challenging. We study persistent homology (PH) as a general-purpose representation for MD and introduce the masked Flood complex, a protein-tailored modification of a recently introduced simplicial complex construction that emphasizes inter-residue structure at low computational cost. Vectorized persistence diagrams then provide information-rich, geometry-aware summaries of protein conformations, which we evaluate on protein class prediction, frame-level observable regression, and Markov state model (MSM) estimation from learned low-dimensional coordinates in a single shared representation space. Results on the mdCATH dataset show that PH-based descriptors are competitive across tasks, with masked Flood PH yielding the most consistent overall performance. Further, when using topologically-informed MSMs as a drop-in replacement within the recent MarS-FM framework for generative modeling of protein conformations, we obtain consistently better ensemble statistics than MSMs based on physical observables. Finally, we explore the transferability of the generative model to qualitatively different, fast folding, proteins.

An artificial intelligence must have a model of its environment that is causal, supporting reasoning about interventions and counterfactuals, and also combinatorial, supporting generalization to unseen combinations of objects. In this work, we formally study when and how such a model can be learned. We develop relational structural causal models, extending structural causal models (Pearl 2009) to settings where objects and their relations vary. First, we show how answers to not only causal but also observational queries about unseen combinations of objects can not be identified without further assumptions. To enable such identification--including in the presence of unobserved confounding--we define relational causal graphs and derive symbolic identification criteria. Finally, we propose relational neural causal models, a provably correct approach that outperforms non-relational baselines on simulated traffic scenes with varying cars, signals, and pedestrians.

A significant gap exists between theory and practice in deep learning. Generalization and approximation error bounds are often derived for simplified models or are too loose to be informative. Many rely on the manifold hypothesis and on geometric regularity such as intrinsic dimension, curvature, and reach. Progress requires insight into data-manifold geometry and suitable benchmarks, yet existing options are polarized: analytic manifolds with known geometry but limited applicability, or real-world datasets where geometry is only coarsely estimable. We introduce a benchmarking framework for studying data geometry. We repurpose and extend dSprites and COIL-20 with additional transformation dimensions and dense, axis-aligned sampling, and pair them with finite-difference estimators that recover curvature, reach, and volume at near-ground-truth accuracy in a regime where general-purpose estimators are unreliable or difficult to deploy. The framework is intended as a controlled testbed, useful as a calibration environment for geometric estimators and a sandbox for probing theoretical assumptions. To illustrate its use, we present two application studies, namely assessing the scaling behavior of the bounds of Genovese et al. and Fefferman et al., and tracking the layer-wise geometry of a $β$-VAE, highlighting the behavior of current bounds and the value of controlled benchmarks for guiding and validating future theory. A reference implementation is available at https://github.com/koulakis/manifold-microscope.